{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b4 : \u03b1 \u2192 Type u_2\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nx\u271d\u00b9 x\u271d : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a\n\u22a2 (fun f a => f a (_ : a \u2208 univ)) x\u271d\u00b9 = (fun f a => f a (_ : a \u2208 univ)) x\u271d \u2192 x\u271d\u00b9 = x\u271d\n[PROOFSTEP]\nsimp (config := { contextual := true }) [Function.funext_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b4 : \u03b1 \u2192 Type u_2\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a\n\u22a2 f \u2208 piFinset t \u2194 \u2200 (a : \u03b1), f a \u2208 t a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b4 : \u03b1 \u2192 Type u_2\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a\n\u22a2 f \u2208 piFinset t \u2192 \u2200 (a : \u03b1), f a \u2208 t a\n[PROOFSTEP]\nsimp only [piFinset, mem_map, and_imp, forall_prop_of_true, exists_prop, mem_univ, exists_imp, mem_pi]\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b4 : \u03b1 \u2192 Type u_2\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a\n\u22a2 \u2200 (x : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a),\n (\u2200 (a : \u03b1), x a (_ : a \u2208 univ) \u2208 t a) \u2192\n \u2191{ toFun := fun f a => f a (_ : a \u2208 univ),\n inj' :=\n (_ :\n \u2200 (x x_1 : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a),\n ((fun a => x a (_ : a \u2208 univ)) = fun a => x_1 a (_ : a \u2208 univ)) \u2192 x = x_1) }\n x =\n f \u2192\n \u2200 (a : \u03b1), f a \u2208 t a\n[PROOFSTEP]\nrintro g hg hgf a\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b4 : \u03b1 \u2192 Type u_2\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a\ng : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a\nhg : \u2200 (a : \u03b1), g a (_ : a \u2208 univ) \u2208 t a\nhgf :\n \u2191{ toFun := fun f a => f a (_ : a \u2208 univ),\n inj' :=\n (_ :\n \u2200 (x x_1 : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a),\n ((fun a => x a (_ : a \u2208 univ)) = fun a => x_1 a (_ : a \u2208 univ)) \u2192 x = x_1) }\n g =\n f\na : \u03b1\n\u22a2 f a \u2208 t a\n[PROOFSTEP]\nrw [\u2190 hgf]\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b4 : \u03b1 \u2192 Type u_2\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a\ng : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a\nhg : \u2200 (a : \u03b1), g a (_ : a \u2208 univ) \u2208 t a\nhgf :\n \u2191{ toFun := fun f a => f a (_ : a \u2208 univ),\n inj' :=\n (_ :\n \u2200 (x x_1 : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a),\n ((fun a => x a (_ : a \u2208 univ)) = fun a => x_1 a (_ : a \u2208 univ)) \u2192 x = x_1) }\n g =\n f\na : \u03b1\n\u22a2 \u2191{ toFun := fun f a => f a (_ : a \u2208 univ),\n inj' :=\n (_ :\n \u2200 (x x_1 : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a),\n ((fun a => x a (_ : a \u2208 univ)) = fun a => x_1 a (_ : a \u2208 univ)) \u2192 x = x_1) }\n g a \u2208\n t a\n[PROOFSTEP]\nexact hg a\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b4 : \u03b1 \u2192 Type u_2\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a\n\u22a2 (\u2200 (a : \u03b1), f a \u2208 t a) \u2192 f \u2208 piFinset t\n[PROOFSTEP]\nsimp only [piFinset, mem_map, forall_prop_of_true, exists_prop, mem_univ, mem_pi]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b4 : \u03b1 \u2192 Type u_2\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a\n\u22a2 (\u2200 (a : \u03b1), f a \u2208 t a) \u2192\n \u2203 a,\n (\u2200 (a_1 : \u03b1), a a_1 (_ : a_1 \u2208 univ) \u2208 t a_1) \u2227\n \u2191{ toFun := fun f a => f a (_ : a \u2208 univ),\n inj' :=\n (_ :\n \u2200 (x x_1 : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b4 a),\n ((fun a => x a (_ : a \u2208 univ)) = fun a => x_1 a (_ : a \u2208 univ)) \u2192 x = x_1) }\n a =\n f\n[PROOFSTEP]\nexact fun hf => \u27e8fun a _ => f a, hf, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b4 : \u03b1 \u2192 Type u_2\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nx : (a : \u03b1) \u2192 \u03b4 a\n\u22a2 x \u2208 \u2191(piFinset t) \u2194 x \u2208 Set.pi Set.univ fun a => \u2191(t a)\n[PROOFSTEP]\nrw [Set.mem_univ_pi]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b4 : \u03b1 \u2192 Type u_2\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nx : (a : \u03b1) \u2192 \u03b4 a\n\u22a2 x \u2208 \u2191(piFinset t) \u2194 \u2200 (i : \u03b1), x i \u2208 \u2191(t i)\n[PROOFSTEP]\nexact Fintype.mem_piFinset\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\n\u03b4 : \u03b1 \u2192 Type u_2\ninst\u271d : Nonempty \u03b1\nx\u271d : (a : \u03b1) \u2192 \u03b4 a\n\u22a2 \u00acx\u271d \u2208 piFinset fun x => \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b4 : \u03b1 \u2192 Type u_2\nf x\u271d : (a : \u03b1) \u2192 \u03b4 a\n\u22a2 (x\u271d \u2208 piFinset fun i => {f i}) \u2194 x\u271d \u2208 {f}\n[PROOFSTEP]\nsimp only [Function.funext_iff, Fintype.mem_piFinset, mem_singleton]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_3\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : (a : \u03b1) \u2192 Fintype (\u03b2 a)\n\u22a2 \u2200 (x : (a : \u03b1) \u2192 \u03b2 a), x \u2208 Fintype.piFinset fun x => univ\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type ?u.21193\n\u03b2 : Type ?u.21196\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u03b2\n\u22a2 Fintype (\u03b1 \u21aa \u03b2)\n[PROOFSTEP]\nclassical. exact Fintype.ofEquiv _ (Equiv.subtypeInjectiveEquivEmbedding \u03b1 \u03b2)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type ?u.21193\n\u03b2 : Type ?u.21196\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u03b2\n\u22a2 Fintype (\u03b1 \u21aa \u03b2)\n[PROOFSTEP]\nexact Fintype.ofEquiv _ (Equiv.subtypeInjectiveEquivEmbedding \u03b1 \u03b2)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_3\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : (a : \u03b1) \u2192 Fintype (\u03b2 a)\n\u22a2 (pi univ fun a => univ) = univ\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_3\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : (a : \u03b1) \u2192 Fintype (\u03b2 a)\na\u271d : (a : \u03b1) \u2192 a \u2208 univ \u2192 \u03b2 a\n\u22a2 (a\u271d \u2208 pi univ fun a => univ) \u2194 a\u271d \u2208 univ\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Data.Fintype.Pi", "llama_tokens": 2694, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.48047867804790706, "lm_q2_score": 0.04146227034253319, "lm_q1q2_score": 0.019921736843045292}} {"text": "[GOAL]\nC : Type (u + 1)\ninst\u271d\u2075 : Category.{u_1, u + 1} C\ninst\u271d\u2074 : ConcreteCategory C\nD : Type (u + 1)\ninst\u271d\u00b3 : Category.{u_2, u + 1} D\ninst\u271d\u00b2 : ConcreteCategory D\ninst\u271d\u00b9 : HasForget\u2082 C D\ninst\u271d : ReflectsIsomorphisms (forget C)\nX Y : C\nf : X \u27f6 Y\ni : IsIso ((forget\u2082 C D).map f)\n\u22a2 IsIso f\n[PROOFSTEP]\nskip\n[GOAL]\nC : Type (u + 1)\ninst\u271d\u2075 : Category.{u_1, u + 1} C\ninst\u271d\u2074 : ConcreteCategory C\nD : Type (u + 1)\ninst\u271d\u00b3 : Category.{u_2, u + 1} D\ninst\u271d\u00b2 : ConcreteCategory D\ninst\u271d\u00b9 : HasForget\u2082 C D\ninst\u271d : ReflectsIsomorphisms (forget C)\nX Y : C\nf : X \u27f6 Y\ni : IsIso ((forget\u2082 C D).map f)\n\u22a2 IsIso f\n[PROOFSTEP]\nhaveI i' : IsIso ((forget D).map ((forget\u2082 C D).map f)) := Functor.map_isIso (forget D) _\n[GOAL]\nC : Type (u + 1)\ninst\u271d\u2075 : Category.{u_1, u + 1} C\ninst\u271d\u2074 : ConcreteCategory C\nD : Type (u + 1)\ninst\u271d\u00b3 : Category.{u_2, u + 1} D\ninst\u271d\u00b2 : ConcreteCategory D\ninst\u271d\u00b9 : HasForget\u2082 C D\ninst\u271d : ReflectsIsomorphisms (forget C)\nX Y : C\nf : X \u27f6 Y\ni : IsIso ((forget\u2082 C D).map f)\ni' : IsIso ((forget D).map ((forget\u2082 C D).map f))\n\u22a2 IsIso f\n[PROOFSTEP]\nhaveI : IsIso ((forget C).map f) := by\n have := @HasForget\u2082.forget_comp C D\n rw [\u2190 this]\n exact i'\n[GOAL]\nC : Type (u + 1)\ninst\u271d\u2075 : Category.{u_1, u + 1} C\ninst\u271d\u2074 : ConcreteCategory C\nD : Type (u + 1)\ninst\u271d\u00b3 : Category.{u_2, u + 1} D\ninst\u271d\u00b2 : ConcreteCategory D\ninst\u271d\u00b9 : HasForget\u2082 C D\ninst\u271d : ReflectsIsomorphisms (forget C)\nX Y : C\nf : X \u27f6 Y\ni : IsIso ((forget\u2082 C D).map f)\ni' : IsIso ((forget D).map ((forget\u2082 C D).map f))\n\u22a2 IsIso ((forget C).map f)\n[PROOFSTEP]\nhave := @HasForget\u2082.forget_comp C D\n[GOAL]\nC : Type (u + 1)\ninst\u271d\u2075 : Category.{u_1, u + 1} C\ninst\u271d\u2074 : ConcreteCategory C\nD : Type (u + 1)\ninst\u271d\u00b3 : Category.{u_2, u + 1} D\ninst\u271d\u00b2 : ConcreteCategory D\ninst\u271d\u00b9 : HasForget\u2082 C D\ninst\u271d : ReflectsIsomorphisms (forget C)\nX Y : C\nf : X \u27f6 Y\ni : IsIso ((forget\u2082 C D).map f)\ni' : IsIso ((forget D).map ((forget\u2082 C D).map f))\nthis :\n \u2200 [inst : Category.{?u.1872, u + 1} C] [inst_1 : ConcreteCategory C] [inst_2 : Category.{?u.1871, u + 1} D]\n [inst_3 : ConcreteCategory D] [self : HasForget\u2082 C D], HasForget\u2082.forget\u2082 \u22d9 forget D = forget C\n\u22a2 IsIso ((forget C).map f)\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\nC : Type (u + 1)\ninst\u271d\u2075 : Category.{u_1, u + 1} C\ninst\u271d\u2074 : ConcreteCategory C\nD : Type (u + 1)\ninst\u271d\u00b3 : Category.{u_2, u + 1} D\ninst\u271d\u00b2 : ConcreteCategory D\ninst\u271d\u00b9 : HasForget\u2082 C D\ninst\u271d : ReflectsIsomorphisms (forget C)\nX Y : C\nf : X \u27f6 Y\ni : IsIso ((forget\u2082 C D).map f)\ni' : IsIso ((forget D).map ((forget\u2082 C D).map f))\nthis :\n \u2200 [inst : Category.{u_1, u + 1} C] [inst_1 : ConcreteCategory C] [inst_2 : Category.{u_2, u + 1} D]\n [inst_3 : ConcreteCategory D] [self : HasForget\u2082 C D], HasForget\u2082.forget\u2082 \u22d9 forget D = forget C\n\u22a2 IsIso ((HasForget\u2082.forget\u2082 \u22d9 forget D).map f)\n[PROOFSTEP]\nexact i'\n[GOAL]\nC : Type (u + 1)\ninst\u271d\u2075 : Category.{u_1, u + 1} C\ninst\u271d\u2074 : ConcreteCategory C\nD : Type (u + 1)\ninst\u271d\u00b3 : Category.{u_2, u + 1} D\ninst\u271d\u00b2 : ConcreteCategory D\ninst\u271d\u00b9 : HasForget\u2082 C D\ninst\u271d : ReflectsIsomorphisms (forget C)\nX Y : C\nf : X \u27f6 Y\ni : IsIso ((forget\u2082 C D).map f)\ni' : IsIso ((forget D).map ((forget\u2082 C D).map f))\nthis : IsIso ((forget C).map f)\n\u22a2 IsIso f\n[PROOFSTEP]\napply isIso_of_reflects_iso f (forget C)\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.ConcreteCategory.ReflectsIso", "llama_tokens": 1589, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.48438008427698437, "lm_q2_score": 0.03358950698030741, "lm_q1q2_score": 0.016270088221943656}} {"text": "[GOAL]\n\u22a2 1 = 1\n[PROOFSTEP]\nsleep_heartbeats 1000\n[GOAL]\n\u22a2 1 = 1\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Util.SleepHeartbeats", "llama_tokens": 52, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO\n\n", "lm_q1_score": 0.35936414516010196, "lm_q2_score": 0.0446808737709169, "lm_q1q2_score": 0.016056704007691974}} {"text": "[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.4128, u_1} C\nX Y : PresheafedSpace C\n\u03b1 \u03b2 : Hom X Y\nw : \u03b1.base = \u03b2.base\n\u22a2 (Opens.map \u03b1.base).op = (Opens.map \u03b2.base).op\n[PROOFSTEP]\nrw [w]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : PresheafedSpace C\n\u03b1 \u03b2 : Hom X Y\nw : \u03b1.base = \u03b2.base\nh : \u03b1.c \u226b whiskerRight (eqToHom (_ : (Opens.map \u03b1.base).op = (Opens.map \u03b2.base).op)) X.presheaf = \u03b2.c\n\u22a2 \u03b1 = \u03b2\n[PROOFSTEP]\nrcases \u03b1 with \u27e8base, c\u27e9\n[GOAL]\ncase mk\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : PresheafedSpace C\n\u03b2 : Hom X Y\nbase : \u2191X \u27f6 \u2191Y\nc : Y.presheaf \u27f6 base _* X.presheaf\nw : { base := base, c := c }.base = \u03b2.base\nh :\n { base := base, c := c }.c \u226b\n whiskerRight (eqToHom (_ : (Opens.map { base := base, c := c }.base).op = (Opens.map \u03b2.base).op)) X.presheaf =\n \u03b2.c\n\u22a2 { base := base, c := c } = \u03b2\n[PROOFSTEP]\nrcases \u03b2 with \u27e8base', c'\u27e9\n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : PresheafedSpace C\nbase : \u2191X \u27f6 \u2191Y\nc : Y.presheaf \u27f6 base _* X.presheaf\nbase' : \u2191X \u27f6 \u2191Y\nc' : Y.presheaf \u27f6 base' _* X.presheaf\nw : { base := base, c := c }.base = { base := base', c := c' }.base\nh :\n { base := base, c := c }.c \u226b\n whiskerRight\n (eqToHom (_ : (Opens.map { base := base, c := c }.base).op = (Opens.map { base := base', c := c' }.base).op))\n X.presheaf =\n { base := base', c := c' }.c\n\u22a2 { base := base, c := c } = { base := base', c := c' }\n[PROOFSTEP]\ndsimp at w \n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : PresheafedSpace C\nbase : \u2191X \u27f6 \u2191Y\nc : Y.presheaf \u27f6 base _* X.presheaf\nbase' : \u2191X \u27f6 \u2191Y\nc' : Y.presheaf \u27f6 base' _* X.presheaf\nw : base = base'\nh :\n { base := base, c := c }.c \u226b\n whiskerRight\n (eqToHom (_ : (Opens.map { base := base, c := c }.base).op = (Opens.map { base := base', c := c' }.base).op))\n X.presheaf =\n { base := base', c := c' }.c\n\u22a2 { base := base, c := c } = { base := base', c := c' }\n[PROOFSTEP]\nsubst w\n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : PresheafedSpace C\nbase : \u2191X \u27f6 \u2191Y\nc c' : Y.presheaf \u27f6 base _* X.presheaf\nh :\n { base := base, c := c }.c \u226b\n whiskerRight\n (eqToHom (_ : (Opens.map { base := base, c := c }.base).op = (Opens.map { base := base, c := c' }.base).op))\n X.presheaf =\n { base := base, c := c' }.c\n\u22a2 { base := base, c := c } = { base := base, c := c' }\n[PROOFSTEP]\ndsimp at h \n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : PresheafedSpace C\nbase : \u2191X \u27f6 \u2191Y\nc c' : Y.presheaf \u27f6 base _* X.presheaf\nh : c \u226b whiskerRight (\ud835\udfd9 (Opens.map base).op) X.presheaf = c'\n\u22a2 { base := base, c := c } = { base := base, c := c' }\n[PROOFSTEP]\nerw [whiskerRight_id', comp_id] at h \n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : PresheafedSpace C\nbase : \u2191X \u27f6 \u2191Y\nc c' : Y.presheaf \u27f6 base _* X.presheaf\nh : c = c'\n\u22a2 { base := base, c := c } = { base := base, c := c' }\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : PresheafedSpace C\nbase : \u2191X \u27f6 \u2191Y\nc : Y.presheaf \u27f6 base _* X.presheaf\n\u22a2 { base := base, c := c } = { base := base, c := c }\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : PresheafedSpace C\n\u03b1 \u03b2 : Hom X Y\nw : \u03b1.base = \u03b2.base\nh : HEq \u03b1.c \u03b2.c\n\u22a2 \u03b1 = \u03b2\n[PROOFSTEP]\ncases \u03b1\n[GOAL]\ncase mk\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : PresheafedSpace C\n\u03b2 : Hom X Y\nbase\u271d : \u2191X \u27f6 \u2191Y\nc\u271d : Y.presheaf \u27f6 base\u271d _* X.presheaf\nw : { base := base\u271d, c := c\u271d }.base = \u03b2.base\nh : HEq { base := base\u271d, c := c\u271d }.c \u03b2.c\n\u22a2 { base := base\u271d, c := c\u271d } = \u03b2\n[PROOFSTEP]\ncases \u03b2\n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : PresheafedSpace C\nbase\u271d\u00b9 : \u2191X \u27f6 \u2191Y\nc\u271d\u00b9 : Y.presheaf \u27f6 base\u271d\u00b9 _* X.presheaf\nbase\u271d : \u2191X \u27f6 \u2191Y\nc\u271d : Y.presheaf \u27f6 base\u271d _* X.presheaf\nw : { base := base\u271d\u00b9, c := c\u271d\u00b9 }.base = { base := base\u271d, c := c\u271d }.base\nh : HEq { base := base\u271d\u00b9, c := c\u271d\u00b9 }.c { base := base\u271d, c := c\u271d }.c\n\u22a2 { base := base\u271d\u00b9, c := c\u271d\u00b9 } = { base := base\u271d, c := c\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.68893, u_1} C\nX\u271d Y\u271d : PresheafedSpace C\nx\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 \ud835\udfd9 X\u271d \u226b x\u271d = x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.68893, u_1} C\nX\u271d Y\u271d : PresheafedSpace C\nx\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 { base := \ud835\udfd9 \u2191X\u271d \u226b x\u271d.base, c := x\u271d.c \u226b (Presheaf.pushforward C x\u271d.base).map (\ud835\udfd9 X\u271d.presheaf) } = x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w.w\nC : Type u_1\ninst\u271d : Category.{?u.68893, u_1} C\nX\u271d Y\u271d : PresheafedSpace C\nx\u271d\u00b9 : X\u271d \u27f6 Y\u271d\nx\u271d : (forget TopCat).obj \u2191X\u271d\n\u22a2 \u2191{ base := \ud835\udfd9 \u2191X\u271d \u226b x\u271d\u00b9.base, c := x\u271d\u00b9.c \u226b (Presheaf.pushforward C x\u271d\u00b9.base).map (\ud835\udfd9 X\u271d.presheaf) }.base x\u271d =\n \u2191x\u271d\u00b9.base x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.w\nC : Type u_1\ninst\u271d : Category.{?u.68893, u_1} C\nX\u271d Y\u271d : PresheafedSpace C\nx\u271d\u00b9 : X\u271d \u27f6 Y\u271d\nx\u271d : (forget TopCat).obj \u2191X\u271d\n\u22a2 \u2191(\ud835\udfd9 \u2191X\u271d \u226b x\u271d\u00b9.base) x\u271d = \u2191x\u271d\u00b9.base x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.w\nC : Type u_1\ninst\u271d : Category.{?u.68893, u_1} C\nX\u271d Y\u271d : PresheafedSpace C\nx\u271d : X\u271d \u27f6 Y\u271d\nU\u271d : Opens \u2191\u2191Y\u271d\n\u22a2 NatTrans.app\n ({ base := \ud835\udfd9 \u2191X\u271d \u226b x\u271d.base, c := x\u271d.c \u226b (Presheaf.pushforward C x\u271d.base).map (\ud835\udfd9 X\u271d.presheaf) }.c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (Opens.map\n { base := \ud835\udfd9 \u2191X\u271d \u226b x\u271d.base,\n c := x\u271d.c \u226b (Presheaf.pushforward C x\u271d.base).map (\ud835\udfd9 X\u271d.presheaf) }.base).op =\n (Opens.map x\u271d.base).op))\n X\u271d.presheaf)\n (op U\u271d) =\n NatTrans.app x\u271d.c (op U\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.w\nC : Type u_1\ninst\u271d : Category.{?u.68893, u_1} C\nX\u271d Y\u271d : PresheafedSpace C\nx\u271d : X\u271d \u27f6 Y\u271d\nU\u271d : Opens \u2191\u2191Y\u271d\n\u22a2 NatTrans.app\n ((x\u271d.c \u226b (Presheaf.pushforward C x\u271d.base).map (\ud835\udfd9 X\u271d.presheaf)) \u226b\n whiskerRight (\ud835\udfd9 (Opens.map (\ud835\udfd9 \u2191X\u271d \u226b x\u271d.base)).op) X\u271d.presheaf)\n (op U\u271d) =\n NatTrans.app x\u271d.c (op U\u271d)\n[PROOFSTEP]\nsimp only [map_id, whiskerRight_id', assoc]\n[GOAL]\ncase h.w\nC : Type u_1\ninst\u271d : Category.{?u.68893, u_1} C\nX\u271d Y\u271d : PresheafedSpace C\nx\u271d : X\u271d \u27f6 Y\u271d\nU\u271d : Opens \u2191\u2191Y\u271d\n\u22a2 NatTrans.app\n (x\u271d.c \u226b \ud835\udfd9 ((Presheaf.pushforward C x\u271d.base).obj X\u271d.presheaf) \u226b \ud835\udfd9 ((Opens.map (\ud835\udfd9 \u2191X\u271d \u226b x\u271d.base)).op \u22d9 X\u271d.presheaf))\n (op U\u271d) =\n NatTrans.app x\u271d.c (op U\u271d)\n[PROOFSTEP]\nerw [comp_id, comp_id]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.77226, u_1} C\nX Y : PresheafedSpace C\n\u03b1 \u03b2 : X \u27f6 Y\nw : \u03b1.base = \u03b2.base\n\u22a2 (Opens.map \u03b1.base).op = (Opens.map \u03b2.base).op\n[PROOFSTEP]\nrw [w]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX : PresheafedSpace C\nU : (Opens \u2191\u2191X)\u1d52\u1d56\n\u22a2 NatTrans.app (\ud835\udfd9 X).c U = X.presheaf.map (\ud835\udfd9 U)\n[PROOFSTEP]\nrw [id_c, map_id]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX : PresheafedSpace C\nU : (Opens \u2191\u2191X)\u1d52\u1d56\n\u22a2 NatTrans.app (\ud835\udfd9 X.presheaf) U = \ud835\udfd9 (X.presheaf.obj U)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.143246, u_1} C\nX Y : PresheafedSpace C\n\u03b1 \u03b2 : X \u27f6 Y\nh : \u03b1 = \u03b2\nU : (Opens \u2191\u2191Y)\u1d52\u1d56\n\u22a2 (Opens.map \u03b2.base).op.obj U = (Opens.map \u03b1.base).op.obj U\n[PROOFSTEP]\nsubst h\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.143246, u_1} C\nX Y : PresheafedSpace C\n\u03b1 : X \u27f6 Y\nU : (Opens \u2191\u2191Y)\u1d52\u1d56\n\u22a2 (Opens.map \u03b1.base).op.obj U = (Opens.map \u03b1.base).op.obj U\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : PresheafedSpace C\n\u03b1 \u03b2 : X \u27f6 Y\nh : \u03b1 = \u03b2\nU : (Opens \u2191\u2191Y)\u1d52\u1d56\n\u22a2 NatTrans.app \u03b1.c U =\n NatTrans.app \u03b2.c U \u226b X.presheaf.map (eqToHom (_ : (Opens.map \u03b2.base).op.obj U = (Opens.map \u03b1.base).op.obj U))\n[PROOFSTEP]\nsubst h\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : PresheafedSpace C\n\u03b1 : X \u27f6 Y\nU : (Opens \u2191\u2191Y)\u1d52\u1d56\n\u22a2 NatTrans.app \u03b1.c U =\n NatTrans.app \u03b1.c U \u226b X.presheaf.map (eqToHom (_ : (Opens.map \u03b1.base).op.obj U = (Opens.map \u03b1.base).op.obj U))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.204326, u_1} C\nX Y : PresheafedSpace C\nH : \u2191X \u2245 \u2191Y\n\u03b1 : H.hom _* X.presheaf \u2245 Y.presheaf\n\u22a2 { base := H.hom, c := \u03b1.inv } \u226b { base := H.inv, c := Presheaf.toPushforwardOfIso H \u03b1.hom } = \ud835\udfd9 X\n[PROOFSTEP]\next\n[GOAL]\ncase w.w\nC : Type u_1\ninst\u271d : Category.{?u.204326, u_1} C\nX Y : PresheafedSpace C\nH : \u2191X \u2245 \u2191Y\n\u03b1 : H.hom _* X.presheaf \u2245 Y.presheaf\nx\u271d : (CategoryTheory.forget TopCat).obj \u2191X\n\u22a2 \u2191({ base := H.hom, c := \u03b1.inv } \u226b { base := H.inv, c := Presheaf.toPushforwardOfIso H \u03b1.hom }).base x\u271d =\n \u2191(\ud835\udfd9 X).base x\u271d\ncase h.w\nC : Type u_1\ninst\u271d : Category.{?u.204326, u_1} C\nX Y : PresheafedSpace C\nH : \u2191X \u2245 \u2191Y\n\u03b1 : H.hom _* X.presheaf \u2245 Y.presheaf\nU\u271d : Opens \u2191\u2191X\n\u22a2 NatTrans.app\n (({ base := H.hom, c := \u03b1.inv } \u226b { base := H.inv, c := Presheaf.toPushforwardOfIso H \u03b1.hom }).c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (Opens.map\n ({ base := H.hom, c := \u03b1.inv } \u226b\n { base := H.inv, c := Presheaf.toPushforwardOfIso H \u03b1.hom }).base).op =\n (Opens.map (\ud835\udfd9 X).base).op))\n X.presheaf)\n (op U\u271d) =\n NatTrans.app (\ud835\udfd9 X).c (op U\u271d)\n[PROOFSTEP]\nsimp only [comp_base, Iso.hom_inv_id, FunctorToTypes.map_id_apply, id_base]\n[GOAL]\ncase h.w\nC : Type u_1\ninst\u271d : Category.{?u.204326, u_1} C\nX Y : PresheafedSpace C\nH : \u2191X \u2245 \u2191Y\n\u03b1 : H.hom _* X.presheaf \u2245 Y.presheaf\nU\u271d : Opens \u2191\u2191X\n\u22a2 NatTrans.app\n (({ base := H.hom, c := \u03b1.inv } \u226b { base := H.inv, c := Presheaf.toPushforwardOfIso H \u03b1.hom }).c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (Opens.map\n ({ base := H.hom, c := \u03b1.inv } \u226b\n { base := H.inv, c := Presheaf.toPushforwardOfIso H \u03b1.hom }).base).op =\n (Opens.map (\ud835\udfd9 X).base).op))\n X.presheaf)\n (op U\u271d) =\n NatTrans.app (\ud835\udfd9 X).c (op U\u271d)\n[PROOFSTEP]\nrw [NatTrans.comp_app]\n[GOAL]\ncase h.w\nC : Type u_1\ninst\u271d : Category.{?u.204326, u_1} C\nX Y : PresheafedSpace C\nH : \u2191X \u2245 \u2191Y\n\u03b1 : H.hom _* X.presheaf \u2245 Y.presheaf\nU\u271d : Opens \u2191\u2191X\n\u22a2 NatTrans.app ({ base := H.hom, c := \u03b1.inv } \u226b { base := H.inv, c := Presheaf.toPushforwardOfIso H \u03b1.hom }).c (op U\u271d) \u226b\n NatTrans.app\n (whiskerRight\n (eqToHom\n (_ :\n (Opens.map\n ({ base := H.hom, c := \u03b1.inv } \u226b\n { base := H.inv, c := Presheaf.toPushforwardOfIso H \u03b1.hom }).base).op =\n (Opens.map (\ud835\udfd9 X).base).op))\n X.presheaf)\n (op U\u271d) =\n NatTrans.app (\ud835\udfd9 X).c (op U\u271d)\n[PROOFSTEP]\nsimp only [id_base, comp_obj, op_obj, comp_base, Presheaf.pushforwardObj_obj, Opens.map_comp_obj, comp_c_app, unop_op,\n Presheaf.toPushforwardOfIso_app, assoc, Iso.hom_inv_id_app, comp_id, whiskerRight_app, eqToHom_app, id_c_app, map_id,\n \u2190 Functor.map_comp, eqToHom_trans, eqToHom_refl]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.204326, u_1} C\nX Y : PresheafedSpace C\nH : \u2191X \u2245 \u2191Y\n\u03b1 : H.hom _* X.presheaf \u2245 Y.presheaf\n\u22a2 { base := H.inv, c := Presheaf.toPushforwardOfIso H \u03b1.hom } \u226b { base := H.hom, c := \u03b1.inv } = \ud835\udfd9 Y\n[PROOFSTEP]\next\n[GOAL]\ncase w.w\nC : Type u_1\ninst\u271d : Category.{?u.204326, u_1} C\nX Y : PresheafedSpace C\nH : \u2191X \u2245 \u2191Y\n\u03b1 : H.hom _* X.presheaf \u2245 Y.presheaf\nx\u271d : (CategoryTheory.forget TopCat).obj \u2191Y\n\u22a2 \u2191({ base := H.inv, c := Presheaf.toPushforwardOfIso H \u03b1.hom } \u226b { base := H.hom, c := \u03b1.inv }).base x\u271d =\n \u2191(\ud835\udfd9 Y).base x\u271d\ncase h.w\nC : Type u_1\ninst\u271d : Category.{?u.204326, u_1} C\nX Y : PresheafedSpace C\nH : \u2191X \u2245 \u2191Y\n\u03b1 : H.hom _* X.presheaf \u2245 Y.presheaf\nU\u271d : Opens \u2191\u2191Y\n\u22a2 NatTrans.app\n (({ base := H.inv, c := Presheaf.toPushforwardOfIso H \u03b1.hom } \u226b { base := H.hom, c := \u03b1.inv }).c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (Opens.map\n ({ base := H.inv, c := Presheaf.toPushforwardOfIso H \u03b1.hom } \u226b\n { base := H.hom, c := \u03b1.inv }).base).op =\n (Opens.map (\ud835\udfd9 Y).base).op))\n Y.presheaf)\n (op U\u271d) =\n NatTrans.app (\ud835\udfd9 Y).c (op U\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.w\nC : Type u_1\ninst\u271d : Category.{?u.204326, u_1} C\nX Y : PresheafedSpace C\nH : \u2191X \u2245 \u2191Y\n\u03b1 : H.hom _* X.presheaf \u2245 Y.presheaf\nx\u271d : (CategoryTheory.forget TopCat).obj \u2191Y\n\u22a2 \u2191(H.inv \u226b H.hom) x\u271d = \u2191(\ud835\udfd9 \u2191Y) x\u271d\ncase h.w\nC : Type u_1\ninst\u271d : Category.{?u.204326, u_1} C\nX Y : PresheafedSpace C\nH : \u2191X \u2245 \u2191Y\n\u03b1 : H.hom _* X.presheaf \u2245 Y.presheaf\nU\u271d : Opens \u2191\u2191Y\n\u22a2 NatTrans.app\n (({ base := H.inv, c := Presheaf.toPushforwardOfIso H \u03b1.hom } \u226b { base := H.hom, c := \u03b1.inv }).c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (Opens.map\n ({ base := H.inv, c := Presheaf.toPushforwardOfIso H \u03b1.hom } \u226b\n { base := H.hom, c := \u03b1.inv }).base).op =\n (Opens.map (\ud835\udfd9 Y).base).op))\n Y.presheaf)\n (op U\u271d) =\n NatTrans.app (\ud835\udfd9 Y).c (op U\u271d)\n[PROOFSTEP]\nrw [H.inv_hom_id]\n[GOAL]\ncase h.w\nC : Type u_1\ninst\u271d : Category.{?u.204326, u_1} C\nX Y : PresheafedSpace C\nH : \u2191X \u2245 \u2191Y\n\u03b1 : H.hom _* X.presheaf \u2245 Y.presheaf\nU\u271d : Opens \u2191\u2191Y\n\u22a2 NatTrans.app\n (({ base := H.inv, c := Presheaf.toPushforwardOfIso H \u03b1.hom } \u226b { base := H.hom, c := \u03b1.inv }).c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (Opens.map\n ({ base := H.inv, c := Presheaf.toPushforwardOfIso H \u03b1.hom } \u226b\n { base := H.hom, c := \u03b1.inv }).base).op =\n (Opens.map (\ud835\udfd9 Y).base).op))\n Y.presheaf)\n (op U\u271d) =\n NatTrans.app (\ud835\udfd9 Y).c (op U\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.w\nC : Type u_1\ninst\u271d : Category.{?u.204326, u_1} C\nX Y : PresheafedSpace C\nH : \u2191X \u2245 \u2191Y\n\u03b1 : H.hom _* X.presheaf \u2245 Y.presheaf\nU\u271d : Opens \u2191\u2191Y\n\u22a2 NatTrans.app\n (({ base := H.inv, c := Presheaf.toPushforwardOfIso H \u03b1.hom } \u226b { base := H.hom, c := \u03b1.inv }).c \u226b\n whiskerRight (eqToHom (_ : (Opens.map (H.inv \u226b H.hom)).op = (Opens.map (\ud835\udfd9 \u2191Y)).op)) Y.presheaf)\n (op U\u271d) =\n NatTrans.app (\ud835\udfd9 Y).c (op U\u271d)\n[PROOFSTEP]\nrw [NatTrans.comp_app]\n[GOAL]\ncase h.w\nC : Type u_1\ninst\u271d : Category.{?u.204326, u_1} C\nX Y : PresheafedSpace C\nH : \u2191X \u2245 \u2191Y\n\u03b1 : H.hom _* X.presheaf \u2245 Y.presheaf\nU\u271d : Opens \u2191\u2191Y\n\u22a2 NatTrans.app ({ base := H.inv, c := Presheaf.toPushforwardOfIso H \u03b1.hom } \u226b { base := H.hom, c := \u03b1.inv }).c (op U\u271d) \u226b\n NatTrans.app (whiskerRight (eqToHom (_ : (Opens.map (H.inv \u226b H.hom)).op = (Opens.map (\ud835\udfd9 \u2191Y)).op)) Y.presheaf)\n (op U\u271d) =\n NatTrans.app (\ud835\udfd9 Y).c (op U\u271d)\n[PROOFSTEP]\nsimp only [Presheaf.pushforwardObj_obj, op_obj, Opens.map_comp_obj, comp_obj, comp_c_app, unop_op,\n Presheaf.toPushforwardOfIso_app, whiskerRight_app, eqToHom_app, assoc, id_c_app, map_id]\n[GOAL]\ncase h.w\nC : Type u_1\ninst\u271d : Category.{?u.204326, u_1} C\nX Y : PresheafedSpace C\nH : \u2191X \u2245 \u2191Y\n\u03b1 : H.hom _* X.presheaf \u2245 Y.presheaf\nU\u271d : Opens \u2191\u2191Y\n\u22a2 NatTrans.app \u03b1.inv (op U\u271d) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n op ((Opens.map H.hom).obj U\u271d) =\n op\n {\n carrier :=\n \u2191H.hom \u207b\u00b9'\n \u2191(op\n { carrier := \u2191H.inv \u207b\u00b9' \u2191(op ((Opens.map H.hom).obj U\u271d)).unop,\n is_open' := (_ : IsOpen (\u2191H.inv \u207b\u00b9' \u2191(op ((Opens.map H.hom).obj U\u271d)).unop)) }).unop,\n is_open' :=\n (_ :\n IsOpen\n (\u2191H.hom \u207b\u00b9'\n \u2191(op\n { carrier := \u2191H.inv \u207b\u00b9' \u2191(op ((Opens.map H.hom).obj U\u271d)).unop,\n is_open' :=\n (_ : IsOpen (\u2191H.inv \u207b\u00b9' \u2191(op ((Opens.map H.hom).obj U\u271d)).unop)) }).unop)) })) \u226b\n NatTrans.app \u03b1.hom (op ((Opens.map H.inv).obj ((Opens.map H.hom).obj U\u271d))) \u226b\n Y.presheaf.map\n (eqToHom (_ : (Opens.map (H.inv \u226b H.hom)).op.obj (op U\u271d) = (Opens.map (\ud835\udfd9 \u2191Y)).op.obj (op U\u271d))) =\n \ud835\udfd9 (Y.presheaf.obj (op U\u271d))\n[PROOFSTEP]\nrw [\u2190 \u03b1.hom.naturality, Presheaf.pushforwardObj_map, eqToHom_map, eqToHom_map, eqToHom_map, eqToHom_trans_assoc,\n eqToHom_refl, id_comp]\n[GOAL]\ncase h.w\nC : Type u_1\ninst\u271d : Category.{?u.204326, u_1} C\nX Y : PresheafedSpace C\nH : \u2191X \u2245 \u2191Y\n\u03b1 : H.hom _* X.presheaf \u2245 Y.presheaf\nU\u271d : Opens \u2191\u2191Y\n\u22a2 NatTrans.app \u03b1.inv (op U\u271d) \u226b NatTrans.app \u03b1.hom (op ((Opens.map (\ud835\udfd9 \u2191Y)).obj U\u271d)) = \ud835\udfd9 (Y.presheaf.obj (op U\u271d))\n[PROOFSTEP]\napply Iso.inv_hom_id_app\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.217173, u_1} C\nX Y : PresheafedSpace C\nH : X \u2245 Y\n\u22a2 H.hom.c \u226b Presheaf.pushforwardToOfIso ((forget C).mapIso H).symm H.inv.c = \ud835\udfd9 Y.presheaf\n[PROOFSTEP]\next U\n[GOAL]\ncase w\nC : Type u_1\ninst\u271d : Category.{?u.217173, u_1} C\nX Y : PresheafedSpace C\nH : X \u2245 Y\nU : Opens \u2191\u2191Y\n\u22a2 NatTrans.app (H.hom.c \u226b Presheaf.pushforwardToOfIso ((forget C).mapIso H).symm H.inv.c) (op U) =\n NatTrans.app (\ud835\udfd9 Y.presheaf) (op U)\n[PROOFSTEP]\nrw [NatTrans.comp_app]\n[GOAL]\ncase w\nC : Type u_1\ninst\u271d : Category.{?u.217173, u_1} C\nX Y : PresheafedSpace C\nH : X \u2245 Y\nU : Opens \u2191\u2191Y\n\u22a2 NatTrans.app H.hom.c (op U) \u226b NatTrans.app (Presheaf.pushforwardToOfIso ((forget C).mapIso H).symm H.inv.c) (op U) =\n NatTrans.app (\ud835\udfd9 Y.presheaf) (op U)\n[PROOFSTEP]\nsimpa using congr_arg (fun f => f \u226b eqToHom _) (congr_app H.inv_hom_id (op U))\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.217173, u_1} C\nX Y : PresheafedSpace C\nH : X \u2245 Y\n\u22a2 Presheaf.pushforwardToOfIso ((forget C).mapIso H).symm H.inv.c \u226b H.hom.c = \ud835\udfd9 (H.hom.base _* X.presheaf)\n[PROOFSTEP]\next U\n[GOAL]\ncase w\nC : Type u_1\ninst\u271d : Category.{?u.217173, u_1} C\nX Y : PresheafedSpace C\nH : X \u2245 Y\nU : Opens \u2191\u2191Y\n\u22a2 NatTrans.app (Presheaf.pushforwardToOfIso ((forget C).mapIso H).symm H.inv.c \u226b H.hom.c) (op U) =\n NatTrans.app (\ud835\udfd9 (H.hom.base _* X.presheaf)) (op U)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w\nC : Type u_1\ninst\u271d : Category.{?u.217173, u_1} C\nX Y : PresheafedSpace C\nH : X \u2245 Y\nU : Opens \u2191\u2191Y\n\u22a2 NatTrans.app (Presheaf.pushforwardToOfIso ((forget C).mapIso H).symm H.inv.c \u226b H.hom.c) (op U) =\n NatTrans.app (\ud835\udfd9 (H.hom.base _* X.presheaf)) (op U)\n[PROOFSTEP]\nrw [NatTrans.comp_app, NatTrans.id_app]\n[GOAL]\ncase w\nC : Type u_1\ninst\u271d : Category.{?u.217173, u_1} C\nX Y : PresheafedSpace C\nH : X \u2245 Y\nU : Opens \u2191\u2191Y\n\u22a2 NatTrans.app (Presheaf.pushforwardToOfIso ((forget C).mapIso H).symm H.inv.c) (op U) \u226b NatTrans.app H.hom.c (op U) =\n \ud835\udfd9 ((H.hom.base _* X.presheaf).obj (op U))\n[PROOFSTEP]\nsimp only [Presheaf.pushforwardObj_obj, op_obj, Presheaf.pushforwardToOfIso_app, Iso.symm_inv, mapIso_hom, forget_map,\n Iso.symm_hom, mapIso_inv, unop_op, eqToHom_map, assoc]\n[GOAL]\ncase w\nC : Type u_1\ninst\u271d : Category.{?u.217173, u_1} C\nX Y : PresheafedSpace C\nH : X \u2245 Y\nU : Opens \u2191\u2191Y\n\u22a2 NatTrans.app H.inv.c (op ((Opens.map H.hom.base).obj U)) \u226b\n eqToHom\n (_ :\n Y.presheaf.obj (op ((Opens.map H.inv.base).obj ((Opens.map H.hom.base).obj U))) = Y.presheaf.obj (op U)) \u226b\n NatTrans.app H.hom.c (op U) =\n \ud835\udfd9 (X.presheaf.obj (op ((Opens.map H.hom.base).obj U)))\n[PROOFSTEP]\nhave eq\u2081 := congr_app H.hom_inv_id (op ((Opens.map H.hom.base).obj U))\n[GOAL]\ncase w\nC : Type u_1\ninst\u271d : Category.{?u.217173, u_1} C\nX Y : PresheafedSpace C\nH : X \u2245 Y\nU : Opens \u2191\u2191Y\neq\u2081 :\n NatTrans.app (H.hom \u226b H.inv).c (op ((Opens.map H.hom.base).obj U)) =\n NatTrans.app (\ud835\udfd9 X).c (op ((Opens.map H.hom.base).obj U)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (Opens.map (\ud835\udfd9 X).base).op.obj (op ((Opens.map H.hom.base).obj U)) =\n (Opens.map (H.hom \u226b H.inv).base).op.obj (op ((Opens.map H.hom.base).obj U))))\n\u22a2 NatTrans.app H.inv.c (op ((Opens.map H.hom.base).obj U)) \u226b\n eqToHom\n (_ :\n Y.presheaf.obj (op ((Opens.map H.inv.base).obj ((Opens.map H.hom.base).obj U))) = Y.presheaf.obj (op U)) \u226b\n NatTrans.app H.hom.c (op U) =\n \ud835\udfd9 (X.presheaf.obj (op ((Opens.map H.hom.base).obj U)))\n[PROOFSTEP]\nhave eq\u2082 := H.hom.c.naturality (eqToHom (congr_obj (congr_arg Opens.map ((forget C).congr_map H.inv_hom_id.symm)) U)).op\n[GOAL]\ncase w\nC : Type u_1\ninst\u271d : Category.{?u.217173, u_1} C\nX Y : PresheafedSpace C\nH : X \u2245 Y\nU : Opens \u2191\u2191Y\neq\u2081 :\n NatTrans.app (H.hom \u226b H.inv).c (op ((Opens.map H.hom.base).obj U)) =\n NatTrans.app (\ud835\udfd9 X).c (op ((Opens.map H.hom.base).obj U)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (Opens.map (\ud835\udfd9 X).base).op.obj (op ((Opens.map H.hom.base).obj U)) =\n (Opens.map (H.hom \u226b H.inv).base).op.obj (op ((Opens.map H.hom.base).obj U))))\neq\u2082 :\n Y.presheaf.map\n (eqToHom\n (_ : (Opens.map ((forget C).map (\ud835\udfd9 Y))).obj U = (Opens.map ((forget C).map (H.inv \u226b H.hom))).obj U)).op \u226b\n NatTrans.app H.hom.c (op ((Opens.map ((forget C).map (\ud835\udfd9 Y))).obj U)) =\n NatTrans.app H.hom.c (op ((Opens.map ((forget C).map (H.inv \u226b H.hom))).obj U)) \u226b\n (H.hom.base _* X.presheaf).map\n (eqToHom (_ : (Opens.map ((forget C).map (\ud835\udfd9 Y))).obj U = (Opens.map ((forget C).map (H.inv \u226b H.hom))).obj U)).op\n\u22a2 NatTrans.app H.inv.c (op ((Opens.map H.hom.base).obj U)) \u226b\n eqToHom\n (_ :\n Y.presheaf.obj (op ((Opens.map H.inv.base).obj ((Opens.map H.hom.base).obj U))) = Y.presheaf.obj (op U)) \u226b\n NatTrans.app H.hom.c (op U) =\n \ud835\udfd9 (X.presheaf.obj (op ((Opens.map H.hom.base).obj U)))\n[PROOFSTEP]\nrw [id_c, NatTrans.id_app, id_comp, eqToHom_map, comp_c_app] at eq\u2081 \n[GOAL]\ncase w\nC : Type u_1\ninst\u271d : Category.{?u.217173, u_1} C\nX Y : PresheafedSpace C\nH : X \u2245 Y\nU : Opens \u2191\u2191Y\neq\u2081 :\n NatTrans.app H.inv.c (op ((Opens.map H.hom.base).obj U)) \u226b\n NatTrans.app H.hom.c (op ((Opens.map H.inv.base).obj (op ((Opens.map H.hom.base).obj U)).unop)) =\n eqToHom\n (_ :\n X.presheaf.obj ((Opens.map (\ud835\udfd9 X).base).op.obj (op ((Opens.map H.hom.base).obj U))) =\n X.presheaf.obj ((Opens.map (H.hom \u226b H.inv).base).op.obj (op ((Opens.map H.hom.base).obj U))))\neq\u2082 :\n Y.presheaf.map\n (eqToHom\n (_ : (Opens.map ((forget C).map (\ud835\udfd9 Y))).obj U = (Opens.map ((forget C).map (H.inv \u226b H.hom))).obj U)).op \u226b\n NatTrans.app H.hom.c (op ((Opens.map ((forget C).map (\ud835\udfd9 Y))).obj U)) =\n NatTrans.app H.hom.c (op ((Opens.map ((forget C).map (H.inv \u226b H.hom))).obj U)) \u226b\n (H.hom.base _* X.presheaf).map\n (eqToHom (_ : (Opens.map ((forget C).map (\ud835\udfd9 Y))).obj U = (Opens.map ((forget C).map (H.inv \u226b H.hom))).obj U)).op\n\u22a2 NatTrans.app H.inv.c (op ((Opens.map H.hom.base).obj U)) \u226b\n eqToHom\n (_ :\n Y.presheaf.obj (op ((Opens.map H.inv.base).obj ((Opens.map H.hom.base).obj U))) = Y.presheaf.obj (op U)) \u226b\n NatTrans.app H.hom.c (op U) =\n \ud835\udfd9 (X.presheaf.obj (op ((Opens.map H.hom.base).obj U)))\n[PROOFSTEP]\nrw [eqToHom_op, eqToHom_map] at eq\u2082 \n[GOAL]\ncase w\nC : Type u_1\ninst\u271d : Category.{?u.217173, u_1} C\nX Y : PresheafedSpace C\nH : X \u2245 Y\nU : Opens \u2191\u2191Y\neq\u2081 :\n NatTrans.app H.inv.c (op ((Opens.map H.hom.base).obj U)) \u226b\n NatTrans.app H.hom.c (op ((Opens.map H.inv.base).obj (op ((Opens.map H.hom.base).obj U)).unop)) =\n eqToHom\n (_ :\n X.presheaf.obj ((Opens.map (\ud835\udfd9 X).base).op.obj (op ((Opens.map H.hom.base).obj U))) =\n X.presheaf.obj ((Opens.map (H.hom \u226b H.inv).base).op.obj (op ((Opens.map H.hom.base).obj U))))\neq\u2082 :\n eqToHom\n (_ :\n Y.presheaf.obj (op ((Opens.map ((forget C).map (H.inv \u226b H.hom))).obj U)) =\n Y.presheaf.obj (op ((Opens.map ((forget C).map (\ud835\udfd9 Y))).obj U))) \u226b\n NatTrans.app H.hom.c (op ((Opens.map ((forget C).map (\ud835\udfd9 Y))).obj U)) =\n NatTrans.app H.hom.c (op ((Opens.map ((forget C).map (H.inv \u226b H.hom))).obj U)) \u226b\n (H.hom.base _* X.presheaf).map\n (eqToHom\n (_ : op ((Opens.map ((forget C).map (H.inv \u226b H.hom))).obj U) = op ((Opens.map ((forget C).map (\ud835\udfd9 Y))).obj U)))\n\u22a2 NatTrans.app H.inv.c (op ((Opens.map H.hom.base).obj U)) \u226b\n eqToHom\n (_ :\n Y.presheaf.obj (op ((Opens.map H.inv.base).obj ((Opens.map H.hom.base).obj U))) = Y.presheaf.obj (op U)) \u226b\n NatTrans.app H.hom.c (op U) =\n \ud835\udfd9 (X.presheaf.obj (op ((Opens.map H.hom.base).obj U)))\n[PROOFSTEP]\nerw [eq\u2082, reassoc_of% eq\u2081]\n[GOAL]\ncase w\nC : Type u_1\ninst\u271d : Category.{?u.217173, u_1} C\nX Y : PresheafedSpace C\nH : X \u2245 Y\nU : Opens \u2191\u2191Y\neq\u2081 :\n NatTrans.app H.inv.c (op ((Opens.map H.hom.base).obj U)) \u226b\n NatTrans.app H.hom.c (op ((Opens.map H.inv.base).obj (op ((Opens.map H.hom.base).obj U)).unop)) =\n eqToHom\n (_ :\n X.presheaf.obj ((Opens.map (\ud835\udfd9 X).base).op.obj (op ((Opens.map H.hom.base).obj U))) =\n X.presheaf.obj ((Opens.map (H.hom \u226b H.inv).base).op.obj (op ((Opens.map H.hom.base).obj U))))\neq\u2082 :\n eqToHom\n (_ :\n Y.presheaf.obj (op ((Opens.map ((forget C).map (H.inv \u226b H.hom))).obj U)) =\n Y.presheaf.obj (op ((Opens.map ((forget C).map (\ud835\udfd9 Y))).obj U))) \u226b\n NatTrans.app H.hom.c (op ((Opens.map ((forget C).map (\ud835\udfd9 Y))).obj U)) =\n NatTrans.app H.hom.c (op ((Opens.map ((forget C).map (H.inv \u226b H.hom))).obj U)) \u226b\n (H.hom.base _* X.presheaf).map\n (eqToHom\n (_ : op ((Opens.map ((forget C).map (H.inv \u226b H.hom))).obj U) = op ((Opens.map ((forget C).map (\ud835\udfd9 Y))).obj U)))\n\u22a2 eqToHom\n (_ :\n X.presheaf.obj ((Opens.map (\ud835\udfd9 X).base).op.obj (op ((Opens.map H.hom.base).obj U))) =\n X.presheaf.obj ((Opens.map (H.hom \u226b H.inv).base).op.obj (op ((Opens.map H.hom.base).obj U)))) \u226b\n (H.hom.base _* X.presheaf).map\n (eqToHom\n (_ :\n op ((Opens.map ((forget C).map (H.inv \u226b H.hom))).obj U) = op ((Opens.map ((forget C).map (\ud835\udfd9 Y))).obj U))) =\n \ud835\udfd9 (X.presheaf.obj (op ((Opens.map H.hom.base).obj U)))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.242573, u_1} C\nU\u271d : TopCat\nX : PresheafedSpace C\nf\u271d : U\u271d \u27f6 \u2191X\nh : OpenEmbedding \u2191f\u271d\nU V : (Opens \u2191\u2191X)\u1d52\u1d56\nf : U \u27f6 V\n\u22a2 X.presheaf.map f \u226b\n (fun V => X.presheaf.map (NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191f\u271d)).counit V.unop).op) V =\n (fun V => X.presheaf.map (NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191f\u271d)).counit V.unop).op) U \u226b\n X.presheaf.map ((IsOpenMap.functor (_ : IsOpenMap \u2191f\u271d)).op.map ((Opens.map f\u271d).op.map f))\n[PROOFSTEP]\nrw [\u2190 map_comp, \u2190 map_comp]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.242573, u_1} C\nU\u271d : TopCat\nX : PresheafedSpace C\nf\u271d : U\u271d \u27f6 \u2191X\nh : OpenEmbedding \u2191f\u271d\nU V : (Opens \u2191\u2191X)\u1d52\u1d56\nf : U \u27f6 V\n\u22a2 X.presheaf.map (f \u226b (NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191f\u271d)).counit V.unop).op) =\n X.presheaf.map\n ((NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191f\u271d)).counit U.unop).op \u226b\n (IsOpenMap.functor (_ : IsOpenMap \u2191f\u271d)).op.map ((Opens.map f\u271d).op.map f))\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\n\u22a2 Mono (ofRestrict X hf)\n[PROOFSTEP]\nhaveI : Mono f := (TopCat.mono_iff_injective _).mpr hf.inj\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\nthis : Mono f\n\u22a2 Mono (ofRestrict X hf)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase right_cancellation\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\nthis : Mono f\n\u22a2 \u2200 {Z : PresheafedSpace C} (g h : Z \u27f6 restrict X hf), g \u226b ofRestrict X hf = h \u226b ofRestrict X hf \u2192 g = h\n[PROOFSTEP]\nintro Z g\u2081 g\u2082 eq\n[GOAL]\ncase right_cancellation\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\nthis : Mono f\nZ : PresheafedSpace C\ng\u2081 g\u2082 : Z \u27f6 restrict X hf\neq : g\u2081 \u226b ofRestrict X hf = g\u2082 \u226b ofRestrict X hf\n\u22a2 g\u2081 = g\u2082\n[PROOFSTEP]\next1\n[GOAL]\ncase right_cancellation.w\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\nthis : Mono f\nZ : PresheafedSpace C\ng\u2081 g\u2082 : Z \u27f6 restrict X hf\neq : g\u2081 \u226b ofRestrict X hf = g\u2082 \u226b ofRestrict X hf\n\u22a2 g\u2081.base = g\u2082.base\n[PROOFSTEP]\nhave := congr_arg PresheafedSpace.Hom.base eq\n[GOAL]\ncase right_cancellation.w\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\nthis\u271d : Mono f\nZ : PresheafedSpace C\ng\u2081 g\u2082 : Z \u27f6 restrict X hf\neq : g\u2081 \u226b ofRestrict X hf = g\u2082 \u226b ofRestrict X hf\nthis : (g\u2081 \u226b ofRestrict X hf).base = (g\u2082 \u226b ofRestrict X hf).base\n\u22a2 g\u2081.base = g\u2082.base\n[PROOFSTEP]\nsimp only [PresheafedSpace.comp_base, PresheafedSpace.ofRestrict_base] at this \n[GOAL]\ncase right_cancellation.w\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\nthis\u271d : Mono f\nZ : PresheafedSpace C\ng\u2081 g\u2082 : Z \u27f6 restrict X hf\neq : g\u2081 \u226b ofRestrict X hf = g\u2082 \u226b ofRestrict X hf\nthis : g\u2081.base \u226b f = g\u2082.base \u226b f\n\u22a2 g\u2081.base = g\u2082.base\n[PROOFSTEP]\nrw [cancel_mono] at this \n[GOAL]\ncase right_cancellation.w\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\nthis\u271d : Mono f\nZ : PresheafedSpace C\ng\u2081 g\u2082 : Z \u27f6 restrict X hf\neq : g\u2081 \u226b ofRestrict X hf = g\u2082 \u226b ofRestrict X hf\nthis : g\u2081.base = g\u2082.base\n\u22a2 g\u2081.base = g\u2082.base\n[PROOFSTEP]\nexact this\n[GOAL]\ncase right_cancellation.h\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\nthis : Mono f\nZ : PresheafedSpace C\ng\u2081 g\u2082 : Z \u27f6 restrict X hf\neq : g\u2081 \u226b ofRestrict X hf = g\u2082 \u226b ofRestrict X hf\n\u22a2 g\u2081.c \u226b whiskerRight (eqToHom (_ : (Opens.map g\u2081.base).op = (Opens.map g\u2082.base).op)) Z.presheaf = g\u2082.c\n[PROOFSTEP]\next V\n[GOAL]\ncase right_cancellation.h.w\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\nthis : Mono f\nZ : PresheafedSpace C\ng\u2081 g\u2082 : Z \u27f6 restrict X hf\neq : g\u2081 \u226b ofRestrict X hf = g\u2082 \u226b ofRestrict X hf\nV : Opens \u2191\u2191(restrict X hf)\n\u22a2 NatTrans.app (g\u2081.c \u226b whiskerRight (eqToHom (_ : (Opens.map g\u2081.base).op = (Opens.map g\u2082.base).op)) Z.presheaf) (op V) =\n NatTrans.app g\u2082.c (op V)\n[PROOFSTEP]\nhave hV : (Opens.map (X.ofRestrict hf).base).obj (hf.isOpenMap.functor.obj V) = V :=\n by\n ext1\n exact Set.preimage_image_eq _ hf.inj\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\nthis : Mono f\nZ : PresheafedSpace C\ng\u2081 g\u2082 : Z \u27f6 restrict X hf\neq : g\u2081 \u226b ofRestrict X hf = g\u2082 \u226b ofRestrict X hf\nV : Opens \u2191\u2191(restrict X hf)\n\u22a2 (Opens.map (ofRestrict X hf).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V) = V\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\nthis : Mono f\nZ : PresheafedSpace C\ng\u2081 g\u2082 : Z \u27f6 restrict X hf\neq : g\u2081 \u226b ofRestrict X hf = g\u2082 \u226b ofRestrict X hf\nV : Opens \u2191\u2191(restrict X hf)\n\u22a2 \u2191((Opens.map (ofRestrict X hf).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)) = \u2191V\n[PROOFSTEP]\nexact Set.preimage_image_eq _ hf.inj\n[GOAL]\ncase right_cancellation.h.w\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\nthis : Mono f\nZ : PresheafedSpace C\ng\u2081 g\u2082 : Z \u27f6 restrict X hf\neq : g\u2081 \u226b ofRestrict X hf = g\u2082 \u226b ofRestrict X hf\nV : Opens \u2191\u2191(restrict X hf)\nhV : (Opens.map (ofRestrict X hf).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V) = V\n\u22a2 NatTrans.app (g\u2081.c \u226b whiskerRight (eqToHom (_ : (Opens.map g\u2081.base).op = (Opens.map g\u2082.base).op)) Z.presheaf) (op V) =\n NatTrans.app g\u2082.c (op V)\n[PROOFSTEP]\nhaveI : IsIso (hf.isOpenMap.adjunction.counit.app (unop (op (hf.isOpenMap.functor.obj V)))) :=\n NatIso.isIso_app_of_isIso (whiskerLeft hf.isOpenMap.functor hf.isOpenMap.adjunction.counit) V\n[GOAL]\ncase right_cancellation.h.w\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\nthis\u271d : Mono f\nZ : PresheafedSpace C\ng\u2081 g\u2082 : Z \u27f6 restrict X hf\neq : g\u2081 \u226b ofRestrict X hf = g\u2082 \u226b ofRestrict X hf\nV : Opens \u2191\u2191(restrict X hf)\nhV : (Opens.map (ofRestrict X hf).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V) = V\nthis :\n IsIso\n (NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191f)).counit\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)\n\u22a2 NatTrans.app (g\u2081.c \u226b whiskerRight (eqToHom (_ : (Opens.map g\u2081.base).op = (Opens.map g\u2082.base).op)) Z.presheaf) (op V) =\n NatTrans.app g\u2082.c (op V)\n[PROOFSTEP]\nhave := PresheafedSpace.congr_app eq (op (hf.isOpenMap.functor.obj V))\n[GOAL]\ncase right_cancellation.h.w\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\nthis\u271d\u00b9 : Mono f\nZ : PresheafedSpace C\ng\u2081 g\u2082 : Z \u27f6 restrict X hf\neq : g\u2081 \u226b ofRestrict X hf = g\u2082 \u226b ofRestrict X hf\nV : Opens \u2191\u2191(restrict X hf)\nhV : (Opens.map (ofRestrict X hf).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V) = V\nthis\u271d :\n IsIso\n (NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191f)).counit\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)\nthis :\n NatTrans.app (g\u2081 \u226b ofRestrict X hf).c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)) =\n NatTrans.app (g\u2082 \u226b ofRestrict X hf).c (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)) \u226b\n Z.presheaf.map\n (eqToHom\n (_ :\n (Opens.map (g\u2082 \u226b ofRestrict X hf).base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)) =\n (Opens.map (g\u2081 \u226b ofRestrict X hf).base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V))))\n\u22a2 NatTrans.app (g\u2081.c \u226b whiskerRight (eqToHom (_ : (Opens.map g\u2081.base).op = (Opens.map g\u2082.base).op)) Z.presheaf) (op V) =\n NatTrans.app g\u2082.c (op V)\n[PROOFSTEP]\nsimp only [PresheafedSpace.comp_c_app, PresheafedSpace.ofRestrict_c_app, Category.assoc, cancel_epi] at this \n[GOAL]\ncase right_cancellation.h.w\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\nthis\u271d\u00b9 : Mono f\nZ : PresheafedSpace C\ng\u2081 g\u2082 : Z \u27f6 restrict X hf\neq : g\u2081 \u226b ofRestrict X hf = g\u2082 \u226b ofRestrict X hf\nV : Opens \u2191\u2191(restrict X hf)\nhV : (Opens.map (ofRestrict X hf).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V) = V\nthis\u271d :\n IsIso\n (NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191f)).counit\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)\nthis :\n NatTrans.app g\u2081.c\n (op ((Opens.map (ofRestrict X hf).base).obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)) =\n NatTrans.app g\u2082.c\n (op ((Opens.map (ofRestrict X hf).base).obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)) \u226b\n Z.presheaf.map\n (eqToHom\n (_ :\n (Opens.map (g\u2082 \u226b ofRestrict X hf).base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)) =\n (Opens.map (g\u2081 \u226b ofRestrict X hf).base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V))))\n\u22a2 NatTrans.app (g\u2081.c \u226b whiskerRight (eqToHom (_ : (Opens.map g\u2081.base).op = (Opens.map g\u2082.base).op)) Z.presheaf) (op V) =\n NatTrans.app g\u2082.c (op V)\n[PROOFSTEP]\nhave h : _ \u226b _ = _ \u226b _ \u226b _ := congr_arg (fun f => (X.restrict hf).presheaf.map (eqToHom hV).op \u226b f) this\n[GOAL]\ncase right_cancellation.h.w\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\nthis\u271d\u00b9 : Mono f\nZ : PresheafedSpace C\ng\u2081 g\u2082 : Z \u27f6 restrict X hf\neq : g\u2081 \u226b ofRestrict X hf = g\u2082 \u226b ofRestrict X hf\nV : Opens \u2191\u2191(restrict X hf)\nhV : (Opens.map (ofRestrict X hf).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V) = V\nthis\u271d :\n IsIso\n (NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191f)).counit\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)\nthis :\n NatTrans.app g\u2081.c\n (op ((Opens.map (ofRestrict X hf).base).obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)) =\n NatTrans.app g\u2082.c\n (op ((Opens.map (ofRestrict X hf).base).obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)) \u226b\n Z.presheaf.map\n (eqToHom\n (_ :\n (Opens.map (g\u2082 \u226b ofRestrict X hf).base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)) =\n (Opens.map (g\u2081 \u226b ofRestrict X hf).base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V))))\nh :\n (restrict X hf).presheaf.map (eqToHom hV).op \u226b\n NatTrans.app g\u2081.c\n (op ((Opens.map (ofRestrict X hf).base).obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)) =\n (restrict X hf).presheaf.map (eqToHom hV).op \u226b\n NatTrans.app g\u2082.c\n (op ((Opens.map (ofRestrict X hf).base).obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)) \u226b\n Z.presheaf.map\n (eqToHom\n (_ :\n (Opens.map (g\u2082 \u226b ofRestrict X hf).base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)) =\n (Opens.map (g\u2081 \u226b ofRestrict X hf).base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V))))\n\u22a2 NatTrans.app (g\u2081.c \u226b whiskerRight (eqToHom (_ : (Opens.map g\u2081.base).op = (Opens.map g\u2082.base).op)) Z.presheaf) (op V) =\n NatTrans.app g\u2082.c (op V)\n[PROOFSTEP]\nerw [g\u2081.c.naturality, g\u2082.c.naturality_assoc] at h \n[GOAL]\ncase right_cancellation.h.w\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\nthis\u271d\u00b9 : Mono f\nZ : PresheafedSpace C\ng\u2081 g\u2082 : Z \u27f6 restrict X hf\neq : g\u2081 \u226b ofRestrict X hf = g\u2082 \u226b ofRestrict X hf\nV : Opens \u2191\u2191(restrict X hf)\nhV : (Opens.map (ofRestrict X hf).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V) = V\nthis\u271d :\n IsIso\n (NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191f)).counit\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)\nthis :\n NatTrans.app g\u2081.c\n (op ((Opens.map (ofRestrict X hf).base).obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)) =\n NatTrans.app g\u2082.c\n (op ((Opens.map (ofRestrict X hf).base).obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)) \u226b\n Z.presheaf.map\n (eqToHom\n (_ :\n (Opens.map (g\u2082 \u226b ofRestrict X hf).base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)) =\n (Opens.map (g\u2081 \u226b ofRestrict X hf).base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V))))\nh :\n NatTrans.app g\u2081.c (op V) \u226b (g\u2081.base _* Z.presheaf).map (eqToHom hV).op =\n NatTrans.app g\u2082.c (op V) \u226b\n (g\u2082.base _* Z.presheaf).map (eqToHom hV).op \u226b\n Z.presheaf.map\n (eqToHom\n (_ :\n (Opens.map (g\u2082 \u226b ofRestrict X hf).base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)) =\n (Opens.map (g\u2081 \u226b ofRestrict X hf).base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V))))\n\u22a2 NatTrans.app (g\u2081.c \u226b whiskerRight (eqToHom (_ : (Opens.map g\u2081.base).op = (Opens.map g\u2082.base).op)) Z.presheaf) (op V) =\n NatTrans.app g\u2082.c (op V)\n[PROOFSTEP]\nsimp only [Presheaf.pushforwardObj_map, eqToHom_op, Category.assoc, eqToHom_map, eqToHom_trans] at h \n[GOAL]\ncase right_cancellation.h.w\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\nthis\u271d\u00b9 : Mono f\nZ : PresheafedSpace C\ng\u2081 g\u2082 : Z \u27f6 restrict X hf\neq : g\u2081 \u226b ofRestrict X hf = g\u2082 \u226b ofRestrict X hf\nV : Opens \u2191\u2191(restrict X hf)\nhV : (Opens.map (ofRestrict X hf).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V) = V\nthis\u271d :\n IsIso\n (NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191f)).counit\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)\nthis :\n NatTrans.app g\u2081.c\n (op ((Opens.map (ofRestrict X hf).base).obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)) =\n NatTrans.app g\u2082.c\n (op ((Opens.map (ofRestrict X hf).base).obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)) \u226b\n Z.presheaf.map\n (eqToHom\n (_ :\n (Opens.map (g\u2082 \u226b ofRestrict X hf).base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)) =\n (Opens.map (g\u2081 \u226b ofRestrict X hf).base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V))))\nh :\n NatTrans.app g\u2081.c (op V) \u226b\n eqToHom\n (_ :\n Z.presheaf.obj ((Opens.map g\u2081.base).op.obj (op V)) =\n Z.presheaf.obj\n ((Opens.map g\u2081.base).op.obj\n (op ((Opens.map (ofRestrict X hf).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V))))) =\n NatTrans.app g\u2082.c (op V) \u226b\n eqToHom\n (_ :\n (g\u2082.base _* Z.presheaf).obj (op V) =\n (g\u2081.base _* Z.presheaf).obj\n (op ((Opens.map (ofRestrict X hf).base).obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)))\n\u22a2 NatTrans.app (g\u2081.c \u226b whiskerRight (eqToHom (_ : (Opens.map g\u2081.base).op = (Opens.map g\u2082.base).op)) Z.presheaf) (op V) =\n NatTrans.app g\u2082.c (op V)\n[PROOFSTEP]\nrw [\u2190 IsIso.comp_inv_eq, inv_eqToHom, Category.assoc, eqToHom_trans] at h \n[GOAL]\ncase right_cancellation.h.w\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\nthis\u271d\u00b9 : Mono f\nZ : PresheafedSpace C\ng\u2081 g\u2082 : Z \u27f6 restrict X hf\neq : g\u2081 \u226b ofRestrict X hf = g\u2082 \u226b ofRestrict X hf\nV : Opens \u2191\u2191(restrict X hf)\nhV : (Opens.map (ofRestrict X hf).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V) = V\nthis\u271d :\n IsIso\n (NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191f)).counit\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)\nthis :\n NatTrans.app g\u2081.c\n (op ((Opens.map (ofRestrict X hf).base).obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)) =\n NatTrans.app g\u2082.c\n (op ((Opens.map (ofRestrict X hf).base).obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)) \u226b\n Z.presheaf.map\n (eqToHom\n (_ :\n (Opens.map (g\u2082 \u226b ofRestrict X hf).base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)) =\n (Opens.map (g\u2081 \u226b ofRestrict X hf).base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V))))\nh :\n NatTrans.app g\u2081.c (op V) \u226b eqToHom (_ : (g\u2081.base _* Z.presheaf).obj (op V) = (g\u2082.base _* Z.presheaf).obj (op V)) =\n NatTrans.app g\u2082.c (op V)\n\u22a2 NatTrans.app (g\u2081.c \u226b whiskerRight (eqToHom (_ : (Opens.map g\u2081.base).op = (Opens.map g\u2082.base).op)) Z.presheaf) (op V) =\n NatTrans.app g\u2082.c (op V)\n[PROOFSTEP]\nrw [NatTrans.comp_app]\n[GOAL]\ncase right_cancellation.h.w\nC : Type u_1\ninst\u271d : Category.{?u.247153, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U \u27f6 \u2191X\nhf : OpenEmbedding \u2191f\nthis\u271d\u00b9 : Mono f\nZ : PresheafedSpace C\ng\u2081 g\u2082 : Z \u27f6 restrict X hf\neq : g\u2081 \u226b ofRestrict X hf = g\u2082 \u226b ofRestrict X hf\nV : Opens \u2191\u2191(restrict X hf)\nhV : (Opens.map (ofRestrict X hf).base).obj ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V) = V\nthis\u271d :\n IsIso\n (NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191f)).counit\n (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)\nthis :\n NatTrans.app g\u2081.c\n (op ((Opens.map (ofRestrict X hf).base).obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)) =\n NatTrans.app g\u2082.c\n (op ((Opens.map (ofRestrict X hf).base).obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)).unop)) \u226b\n Z.presheaf.map\n (eqToHom\n (_ :\n (Opens.map (g\u2082 \u226b ofRestrict X hf).base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V)) =\n (Opens.map (g\u2081 \u226b ofRestrict X hf).base).op.obj (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f)).obj V))))\nh :\n NatTrans.app g\u2081.c (op V) \u226b eqToHom (_ : (g\u2081.base _* Z.presheaf).obj (op V) = (g\u2082.base _* Z.presheaf).obj (op V)) =\n NatTrans.app g\u2082.c (op V)\n\u22a2 NatTrans.app g\u2081.c (op V) \u226b\n NatTrans.app (whiskerRight (eqToHom (_ : (Opens.map g\u2081.base).op = (Opens.map g\u2082.base).op)) Z.presheaf) (op V) =\n NatTrans.app g\u2082.c (op V)\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX : PresheafedSpace C\n\u22a2 (restrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4))).presheaf = (Opens.inclusionTopIso \u2191X).inv _* X.presheaf\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX : PresheafedSpace C\n\u22a2 (IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion \u22a4))).op \u22d9 X.presheaf =\n (Opens.inclusionTopIso \u2191X).inv _* X.presheaf\n[PROOFSTEP]\nrw [Opens.inclusion_top_functor X.carrier]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX : PresheafedSpace C\n\u22a2 (Opens.map (Opens.inclusionTopIso \u2191X).inv).op \u22d9 X.presheaf = (Opens.inclusionTopIso \u2191X).inv _* X.presheaf\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.265841, u_1} C\nX : PresheafedSpace C\n\u22a2 X.presheaf =\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4))).base _*\n (restrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4))).presheaf\n[PROOFSTEP]\nrw [restrict_top_presheaf, \u2190 Presheaf.Pushforward.comp_eq]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.265841, u_1} C\nX : PresheafedSpace C\n\u22a2 X.presheaf =\n ((Opens.inclusionTopIso \u2191X).inv \u226b (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4))).base) _* X.presheaf\n[PROOFSTEP]\nerw [Iso.inv_hom_id]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.265841, u_1} C\nX : PresheafedSpace C\n\u22a2 X.presheaf = \ud835\udfd9 \u2191X _* X.presheaf\n[PROOFSTEP]\nrw [Presheaf.Pushforward.id_eq]\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX : PresheafedSpace C\n\u22a2 (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4))).c =\n eqToHom\n (_ :\n X.presheaf =\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4))).base _*\n (restrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4))).presheaf)\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX : PresheafedSpace C\nU\u271d : Opens \u2191\u2191X\n\u22a2 NatTrans.app (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4))).c (op U\u271d) =\n NatTrans.app\n (eqToHom\n (_ :\n X.presheaf =\n (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4))).base _*\n (restrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4))).presheaf))\n (op U\u271d)\n[PROOFSTEP]\ndsimp [ofRestrict]\n[GOAL]\ncase w\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX : PresheafedSpace C\nU\u271d : Opens \u2191\u2191X\n\u22a2 X.presheaf.map (NatTrans.app (IsOpenMap.adjunction (_ : IsOpenMap \u2191(Opens.inclusion \u22a4))).counit U\u271d).op =\n NatTrans.app\n (eqToHom\n (_ :\n X.presheaf = Opens.inclusion \u22a4 _* ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion \u22a4))).op \u22d9 X.presheaf)))\n (op U\u271d)\n[PROOFSTEP]\nerw [eqToHom_map, eqToHom_app]\n[GOAL]\ncase w.p\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX : PresheafedSpace C\nU\u271d : Opens \u2191\u2191X\n\u22a2 op U\u271d = op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion \u22a4))).obj ((Opens.map (Opens.inclusion \u22a4)).obj U\u271d))\ncase w.p\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX : PresheafedSpace C\nU\u271d : Opens \u2191\u2191X\n\u22a2 op U\u271d = op ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion \u22a4))).obj ((Opens.map (Opens.inclusion \u22a4)).obj U\u271d))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.273541, u_1} C\nX : PresheafedSpace C\n\u22a2 ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4)) \u226b toRestrictTop X =\n \ud835\udfd9 (restrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4)))\n[PROOFSTEP]\next\n[GOAL]\ncase w.w\nC : Type u_1\ninst\u271d : Category.{?u.273541, u_1} C\nX : PresheafedSpace C\nx\u271d : (CategoryTheory.forget TopCat).obj \u2191(restrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4)))\n\u22a2 \u2191(ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4)) \u226b toRestrictTop X).base x\u271d =\n \u2191(\ud835\udfd9 (restrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4)))).base x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.w\nC : Type u_1\ninst\u271d : Category.{?u.273541, u_1} C\nX : PresheafedSpace C\nU\u271d : Opens \u2191\u2191(restrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4)))\n\u22a2 NatTrans.app\n ((ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4)) \u226b toRestrictTop X).c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (Opens.map (ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4)) \u226b toRestrictTop X).base).op =\n (Opens.map (\ud835\udfd9 (restrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4)))).base).op))\n (restrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4))).presheaf)\n (op U\u271d) =\n NatTrans.app (\ud835\udfd9 (restrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4)))).c (op U\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.w\nC : Type u_1\ninst\u271d : Category.{?u.273541, u_1} C\nX : PresheafedSpace C\nU\u271d : Opens \u2191\u2191(restrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4)))\n\u22a2 NatTrans.app\n ((ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4)) \u226b toRestrictTop X).c \u226b\n whiskerRight (\ud835\udfd9 (Opens.map (Opens.inclusion \u22a4 \u226b (Opens.inclusionTopIso \u2191X).inv)).op)\n ((IsOpenMap.functor (_ : IsOpenMap \u2191(Opens.inclusion \u22a4))).op \u22d9 X.presheaf))\n (op U\u271d) =\n NatTrans.app (\ud835\udfd9 (restrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4)))).c (op U\u271d)\n[PROOFSTEP]\nerw [comp_c, toRestrictTop_c, whiskerRight_id', comp_id, ofRestrict_top_c, eqToHom_map, eqToHom_trans, eqToHom_refl]\n[GOAL]\ncase h.w\nC : Type u_1\ninst\u271d : Category.{?u.273541, u_1} C\nX : PresheafedSpace C\nU\u271d : Opens \u2191\u2191(restrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4)))\n\u22a2 NatTrans.app (\ud835\udfd9 (restrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4))).presheaf) (op U\u271d) =\n NatTrans.app (\ud835\udfd9 (restrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4)))).c (op U\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\ninst\u271d : Category.{?u.273541, u_1} C\nX : PresheafedSpace C\n\u22a2 toRestrictTop X \u226b ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4)) = \ud835\udfd9 X\n[PROOFSTEP]\next\n[GOAL]\ncase w.w\nC : Type u_1\ninst\u271d : Category.{?u.273541, u_1} C\nX : PresheafedSpace C\nx\u271d : (CategoryTheory.forget TopCat).obj \u2191X\n\u22a2 \u2191(toRestrictTop X \u226b ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4))).base x\u271d = \u2191(\ud835\udfd9 X).base x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.w\nC : Type u_1\ninst\u271d : Category.{?u.273541, u_1} C\nX : PresheafedSpace C\nU\u271d : Opens \u2191\u2191X\n\u22a2 NatTrans.app\n ((toRestrictTop X \u226b ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4))).c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (Opens.map (toRestrictTop X \u226b ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4))).base).op =\n (Opens.map (\ud835\udfd9 X).base).op))\n X.presheaf)\n (op U\u271d) =\n NatTrans.app (\ud835\udfd9 X).c (op U\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.w\nC : Type u_1\ninst\u271d : Category.{?u.273541, u_1} C\nX : PresheafedSpace C\nU\u271d : Opens \u2191\u2191X\n\u22a2 NatTrans.app\n ((toRestrictTop X \u226b ofRestrict X (_ : OpenEmbedding \u2191(Opens.inclusion \u22a4))).c \u226b\n whiskerRight (\ud835\udfd9 (Opens.map ((Opens.inclusionTopIso \u2191X).inv \u226b Opens.inclusion \u22a4)).op) X.presheaf)\n (op U\u271d) =\n NatTrans.app (\ud835\udfd9 X).c (op U\u271d)\n[PROOFSTEP]\nerw [comp_c, ofRestrict_top_c, toRestrictTop_c, eqToHom_map, whiskerRight_id', comp_id, eqToHom_trans, eqToHom_refl]\n[GOAL]\ncase h.w\nC : Type u_1\ninst\u271d : Category.{?u.273541, u_1} C\nX : PresheafedSpace C\nU\u271d : Opens \u2191\u2191X\n\u22a2 NatTrans.app (\ud835\udfd9 X.presheaf) (op U\u271d) = NatTrans.app (\ud835\udfd9 X).c (op U\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.293007, u_1} C\nD : Type u_2\ninst\u271d : Category.{?u.293014, u_2} D\nF : C \u2964 D\nX : PresheafedSpace C\n\u22a2 { obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n (\ud835\udfd9 X) =\n \ud835\udfd9\n ({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.obj\n X)\n[PROOFSTEP]\next U\n[GOAL]\ncase w.w\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.293007, u_1} C\nD : Type u_2\ninst\u271d : Category.{?u.293014, u_2} D\nF : C \u2964 D\nX : PresheafedSpace C\nU :\n (forget TopCat).obj\n \u2191({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.obj\n X)\n\u22a2 \u2191({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n (\ud835\udfd9 X)).base\n U =\n \u2191(\ud835\udfd9\n ({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.obj\n X)).base\n U\ncase h.w\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.293007, u_1} C\nD : Type u_2\ninst\u271d : Category.{?u.293014, u_2} D\nF : C \u2964 D\nX : PresheafedSpace C\nU :\n Opens\n \u2191\u2191({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.obj\n X)\n\u22a2 NatTrans.app\n (({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n (\ud835\udfd9 X)).c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (Opens.map\n ({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n (\ud835\udfd9 X)).base).op =\n (Opens.map\n (\ud835\udfd9\n ({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.obj\n X)).base).op))\n ({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.obj\n X).presheaf)\n (op U) =\n NatTrans.app\n (\ud835\udfd9\n ({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.obj\n X)).c\n (op U)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.w\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.293007, u_1} C\nD : Type u_2\ninst\u271d : Category.{?u.293014, u_2} D\nF : C \u2964 D\nX : PresheafedSpace C\nU :\n Opens\n \u2191\u2191({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.obj\n X)\n\u22a2 NatTrans.app\n (({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n (\ud835\udfd9 X)).c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (Opens.map\n ({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n (\ud835\udfd9 X)).base).op =\n (Opens.map\n (\ud835\udfd9\n ({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.obj\n X)).base).op))\n ({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.obj\n X).presheaf)\n (op U) =\n NatTrans.app\n (\ud835\udfd9\n ({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.obj\n X)).c\n (op U)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.293007, u_1} C\nD : Type u_2\ninst\u271d : Category.{?u.293014, u_2} D\nF : C \u2964 D\nX\u271d Y\u271d Z\u271d : PresheafedSpace C\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n (f \u226b g) =\n { obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n f \u226b\n { obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n g\n[PROOFSTEP]\next U\n[GOAL]\ncase w.w\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.293007, u_1} C\nD : Type u_2\ninst\u271d : Category.{?u.293014, u_2} D\nF : C \u2964 D\nX\u271d Y\u271d Z\u271d : PresheafedSpace C\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\nU :\n (forget TopCat).obj\n \u2191({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.obj\n X\u271d)\n\u22a2 \u2191({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n (f \u226b g)).base\n U =\n \u2191({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n f \u226b\n { obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n g).base\n U\ncase h.w\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.293007, u_1} C\nD : Type u_2\ninst\u271d : Category.{?u.293014, u_2} D\nF : C \u2964 D\nX\u271d Y\u271d Z\u271d : PresheafedSpace C\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\nU :\n Opens\n \u2191\u2191({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.obj\n Z\u271d)\n\u22a2 NatTrans.app\n (({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n (f \u226b g)).c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (Opens.map\n ({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n (f \u226b g)).base).op =\n (Opens.map\n ({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n f \u226b\n { obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n g).base).op))\n ({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.obj\n X\u271d).presheaf)\n (op U) =\n NatTrans.app\n ({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n f \u226b\n { obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n g).c\n (op U)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.w\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.293007, u_1} C\nD : Type u_2\ninst\u271d : Category.{?u.293014, u_2} D\nF : C \u2964 D\nX\u271d Y\u271d Z\u271d : PresheafedSpace C\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\nU :\n Opens\n \u2191\u2191({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.obj\n Z\u271d)\n\u22a2 NatTrans.app\n (({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n (f \u226b g)).c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (Opens.map\n ({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n (f \u226b g)).base).op =\n (Opens.map\n ({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n f \u226b\n { obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n g).base).op))\n ({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.obj\n X\u271d).presheaf)\n (op U) =\n NatTrans.app\n ({ obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n f \u226b\n { obj := fun X => { carrier := \u2191X, presheaf := X.presheaf \u22d9 F },\n map := fun {X Y} f => { base := f.base, c := whiskerRight f.c F } }.map\n g).c\n (op U)\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.PresheafedSpace", "llama_tokens": 28360, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.48047867804790706, "lm_q2_score": 0.03258974605797164, "lm_q1q2_score": 0.015658678103851204}} {"text": "[GOAL]\nR : Type u\ninst\u271d : Ring R\nM : FGModuleCat R\n\u22a2 AddCommGroup \u2191M\n[PROOFSTEP]\nchange AddCommGroup M.obj\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nM : FGModuleCat R\n\u22a2 AddCommGroup \u2191M.obj\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nM : FGModuleCat R\n\u22a2 Module R \u2191M\n[PROOFSTEP]\nchange Module R M.obj\n[GOAL]\nR : Type u\ninst\u271d : Ring R\nM : FGModuleCat R\n\u22a2 Module R \u2191M.obj\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u\ninst\u271d : Ring R\n\u22a2 LargeCategory (FGModuleCat R)\n[PROOFSTEP]\ndsimp [FGModuleCat]\n[GOAL]\nR : Type u\ninst\u271d : Ring R\n\u22a2 LargeCategory (FullSubcategory fun V => Module.Finite R \u2191V)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u\ninst\u271d : Ring R\n\u22a2 ConcreteCategory (FGModuleCat R)\n[PROOFSTEP]\ndsimp [FGModuleCat]\n[GOAL]\nR : Type u\ninst\u271d : Ring R\n\u22a2 ConcreteCategory (FullSubcategory fun V => Module.Finite R \u2191V)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u\ninst\u271d : Ring R\n\u22a2 Preadditive (FGModuleCat R)\n[PROOFSTEP]\ndsimp [FGModuleCat]\n[GOAL]\nR : Type u\ninst\u271d : Ring R\n\u22a2 Preadditive (FullSubcategory fun V => Module.Finite R \u2191V)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : Ring R\nV : Type u\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : Module.Finite R V\n\u22a2 Module.Finite R \u2191(ModuleCat.of R V)\n[PROOFSTEP]\nchange Module.Finite R V\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : Ring R\nV : Type u\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : Module.Finite R V\n\u22a2 Module.Finite R V\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u\ninst\u271d : Ring R\n\u22a2 HasForget\u2082 (FGModuleCat R) (ModuleCat R)\n[PROOFSTEP]\ndsimp [FGModuleCat]\n[GOAL]\nR : Type u\ninst\u271d : Ring R\n\u22a2 HasForget\u2082 (FullSubcategory fun V => Module.Finite R \u2191V) (ModuleCat R)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u\ninst\u271d\u2076 : Ring R\nV W : Type u\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : Module.Finite R V\ninst\u271d\u00b2 : AddCommGroup W\ninst\u271d\u00b9 : Module R W\ninst\u271d : Module.Finite R W\ne : V \u2243\u2097[R] W\n\u22a2 \u2191e \u226b \u2191(LinearEquiv.symm e) = \ud835\udfd9 (of R V)\n[PROOFSTEP]\next x\n[GOAL]\ncase w\nR : Type u\ninst\u271d\u2076 : Ring R\nV W : Type u\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : Module.Finite R V\ninst\u271d\u00b2 : AddCommGroup W\ninst\u271d\u00b9 : Module R W\ninst\u271d : Module.Finite R W\ne : V \u2243\u2097[R] W\nx : (forget (FGModuleCat R)).obj (of R V)\n\u22a2 \u2191(\u2191e \u226b \u2191(LinearEquiv.symm e)) x = \u2191(\ud835\udfd9 (of R V)) x\n[PROOFSTEP]\nexact e.left_inv x\n[GOAL]\nR : Type u\ninst\u271d\u2076 : Ring R\nV W : Type u\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : Module.Finite R V\ninst\u271d\u00b2 : AddCommGroup W\ninst\u271d\u00b9 : Module R W\ninst\u271d : Module.Finite R W\ne : V \u2243\u2097[R] W\n\u22a2 \u2191(LinearEquiv.symm e) \u226b \u2191e = \ud835\udfd9 (of R W)\n[PROOFSTEP]\next x\n[GOAL]\ncase w\nR : Type u\ninst\u271d\u2076 : Ring R\nV W : Type u\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : Module.Finite R V\ninst\u271d\u00b2 : AddCommGroup W\ninst\u271d\u00b9 : Module R W\ninst\u271d : Module.Finite R W\ne : V \u2243\u2097[R] W\nx : (forget (FGModuleCat R)).obj (of R W)\n\u22a2 \u2191(\u2191(LinearEquiv.symm e) \u226b \u2191e) x = \u2191(\ud835\udfd9 (of R W)) x\n[PROOFSTEP]\nexact e.right_inv x\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u22a2 Linear R (FGModuleCat R)\n[PROOFSTEP]\ndsimp [FGModuleCat]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u22a2 Linear R (FullSubcategory fun V => Module.Finite R \u2191V)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u22a2 MonoidalCategory (FGModuleCat R)\n[PROOFSTEP]\ndsimp [FGModuleCat]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u22a2 MonoidalCategory (FullSubcategory fun V => Module.Finite R \u2191V)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u22a2 SymmetricCategory (FGModuleCat R)\n[PROOFSTEP]\ndsimp [FGModuleCat]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u22a2 SymmetricCategory (FullSubcategory fun V => Module.Finite R \u2191V)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u22a2 MonoidalPreadditive (FGModuleCat R)\n[PROOFSTEP]\ndsimp [FGModuleCat]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u22a2 MonoidalPreadditive (FullSubcategory fun V => Module.Finite R \u2191V)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u22a2 MonoidalLinear R (FGModuleCat R)\n[PROOFSTEP]\ndsimp [FGModuleCat]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u22a2 MonoidalLinear R (FullSubcategory fun V => Module.Finite R \u2191V)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u22a2 Faithful (forget\u2082Monoidal R).toLaxMonoidalFunctor.toFunctor\n[PROOFSTEP]\ndsimp [forget\u2082Monoidal]\n -- Porting note: was `infer_instance`\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u22a2 Faithful (fullSubcategoryInclusion fun V => Module.Finite R \u2191V)\n[PROOFSTEP]\nexact FullSubcategory.faithful _\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u22a2 Functor.Additive (forget\u2082Monoidal R).toLaxMonoidalFunctor.toFunctor\n[PROOFSTEP]\ndsimp [forget\u2082Monoidal]\n -- Porting note: was `infer_instance`\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u22a2 Functor.Additive (fullSubcategoryInclusion fun V => Module.Finite R \u2191V)\n[PROOFSTEP]\nexact Functor.fullSubcategoryInclusion_additive _\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u22a2 Functor.Linear R (forget\u2082Monoidal R).toLaxMonoidalFunctor.toFunctor\n[PROOFSTEP]\ndsimp [forget\u2082Monoidal]\n -- Porting note: was `infer_instance`\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\n\u22a2 Functor.Linear R (fullSubcategoryInclusion fun V => Module.Finite R \u2191V)\n[PROOFSTEP]\nexact Functor.fullSubcategoryInclusionLinear _ _\n[GOAL]\nK : Type u\ninst\u271d : Field K\nV W : FGModuleCat K\n\u22a2 Module.Finite K (\u2191V \u2192\u2097[K] \u2191W)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nK : Type u\ninst\u271d : Field K\n\u22a2 MonoidalClosed (FGModuleCat K)\n[PROOFSTEP]\ndsimp [FGModuleCat]\n -- Porting note: was `infer_instance`\n[GOAL]\nK : Type u\ninst\u271d : Field K\n\u22a2 MonoidalClosed (FullSubcategory fun V => Module.Finite K \u2191V)\n[PROOFSTEP]\nexact MonoidalCategory.fullMonoidalClosedSubcategory _\n[GOAL]\nK : Type u\ninst\u271d : Field K\nV W : FGModuleCat K\n\u22a2 (\ud835\udfd9 (FGModuleCatDual K V) \u2297 FGModuleCatCoevaluation K V) \u226b\n (\u03b1_ (FGModuleCatDual K V) V (FGModuleCatDual K V)).inv \u226b (FGModuleCatEvaluation K V \u2297 \ud835\udfd9 (FGModuleCatDual K V)) =\n (\u03c1_ (FGModuleCatDual K V)).hom \u226b (\u03bb_ (FGModuleCatDual K V)).inv\n[PROOFSTEP]\napply contractLeft_assoc_coevaluation K V\n[GOAL]\nK : Type u\ninst\u271d : Field K\nV W : FGModuleCat K\n\u22a2 (FGModuleCatCoevaluation K V \u2297 \ud835\udfd9 V) \u226b (\u03b1_ V (FGModuleCatDual K V) V).hom \u226b (\ud835\udfd9 V \u2297 FGModuleCatEvaluation K V) =\n (\u03bb_ V).hom \u226b (\u03c1_ V).inv\n[PROOFSTEP]\napply contractLeft_assoc_coevaluation' K V\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.FGModuleCat.Basic", "llama_tokens": 2822, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.49609382947091946, "lm_q2_score": 0.030675799827571404, "lm_q1q2_score": 0.015218075008543268}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasStrictInitialObjects C\nI : C\nhI : IsInitial I\nA : C\nf g : A \u27f6 I\n\u22a2 f = g\n[PROOFSTEP]\nhaveI := hI.isIso_to f\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasStrictInitialObjects C\nI : C\nhI : IsInitial I\nA : C\nf g : A \u27f6 I\nthis : IsIso f\n\u22a2 f = g\n[PROOFSTEP]\nhaveI := hI.isIso_to g\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasStrictInitialObjects C\nI : C\nhI : IsInitial I\nA : C\nf g : A \u27f6 I\nthis\u271d : IsIso f\nthis : IsIso g\n\u22a2 f = g\n[PROOFSTEP]\nexact eq_of_inv_eq_inv (hI.hom_ext (inv f) (inv g))\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasStrictInitialObjects C\nI X : C\ninst\u271d : HasBinaryProduct X I\nhI : IsInitial I\n\u22a2 X \u2a2f I \u2245 I\n[PROOFSTEP]\nhave := hI.isIso_to (prod.snd : X \u2a2f I \u27f6 I)\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasStrictInitialObjects C\nI X : C\ninst\u271d : HasBinaryProduct X I\nhI : IsInitial I\nthis : IsIso prod.snd\n\u22a2 X \u2a2f I \u2245 I\n[PROOFSTEP]\nexact asIso prod.snd\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasStrictInitialObjects C\nI X : C\ninst\u271d : HasBinaryProduct I X\nhI : IsInitial I\n\u22a2 I \u2a2f X \u2245 I\n[PROOFSTEP]\nhave := hI.isIso_to (prod.fst : I \u2a2f X \u27f6 I)\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : HasStrictInitialObjects C\nI X : C\ninst\u271d : HasBinaryProduct I X\nhI : IsInitial I\nthis : IsIso prod.fst\n\u22a2 I \u2a2f X \u2245 I\n[PROOFSTEP]\nexact asIso prod.fst\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasInitial C\nh : \u2200 (A : C) (f : A \u27f6 \u22a5_ C), IsIso f\nI A : C\nf : A \u27f6 I\nhI : IsInitial I\nthis : IsIso (f \u226b IsInitial.to hI (\u22a5_ C))\n\u22a2 f \u226b IsInitial.to hI (\u22a5_ C) \u226b inv (f \u226b IsInitial.to hI (\u22a5_ C)) = \ud835\udfd9 A\n[PROOFSTEP]\nrw [\u2190 assoc, IsIso.hom_inv_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasStrictTerminalObjects C\nI : C\nhI : IsTerminal I\nA : C\nf g : I \u27f6 A\n\u22a2 f = g\n[PROOFSTEP]\nhaveI := hI.isIso_from f\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasStrictTerminalObjects C\nI : C\nhI : IsTerminal I\nA : C\nf g : I \u27f6 A\nthis : IsIso f\n\u22a2 f = g\n[PROOFSTEP]\nhaveI := hI.isIso_from g\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasStrictTerminalObjects C\nI : C\nhI : IsTerminal I\nA : C\nf g : I \u27f6 A\nthis\u271d : IsIso f\nthis : IsIso g\n\u22a2 f = g\n[PROOFSTEP]\nexact eq_of_inv_eq_inv (hI.hom_ext (inv f) (inv g))\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\n\u22a2 IsIso (limit.\u03c0 F i)\n[PROOFSTEP]\nclassical\nrefine' \u27e8\u27e8limit.lift _ \u27e8_, \u27e8_, _\u27e9\u27e9, _, _\u27e9\u27e9\n\u00b7 exact fun j => dite (j = i) (fun h => eqToHom (by cases h; rfl)) fun h => (H _ h).from _\n\u00b7 intro j k f\n split_ifs with h h_1 h_1\n \u00b7 cases h\n cases h_1\n obtain rfl : f = \ud835\udfd9 _ := Subsingleton.elim _ _\n simp\n \u00b7 cases h\n erw [Category.comp_id]\n haveI : IsIso (F.map f) := (H _ h_1).isIso_from _\n rw [\u2190 IsIso.comp_inv_eq]\n apply (H _ h_1).hom_ext\n \u00b7 cases h_1\n apply (H _ h).hom_ext\n \u00b7 apply (H _ h).hom_ext\n\u00b7 ext\n rw [assoc, limit.lift_\u03c0]\n dsimp only\n split_ifs with h\n \u00b7 cases h\n rw [id_comp, eqToHom_refl]\n exact comp_id _\n \u00b7 apply (H _ h).hom_ext\n\u00b7 rw [limit.lift_\u03c0]\n simp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\n\u22a2 IsIso (limit.\u03c0 F i)\n[PROOFSTEP]\nrefine' \u27e8\u27e8limit.lift _ \u27e8_, \u27e8_, _\u27e9\u27e9, _, _\u27e9\u27e9\n[GOAL]\ncase refine'_1\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\n\u22a2 (X : J) \u2192 ((Functor.const J).obj (F.toPrefunctor.1 i)).obj X \u27f6 F.obj X\n[PROOFSTEP]\nexact fun j => dite (j = i) (fun h => eqToHom (by cases h; rfl)) fun h => (H _ h).from _\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\nj : J\nh : j = i\n\u22a2 ((Functor.const J).obj (F.toPrefunctor.1 i)).obj j = F.obj j\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\n\u22a2 ((Functor.const J).obj (F.toPrefunctor.1 i)).obj i = F.obj i\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\n\u22a2 \u2200 \u2983X Y : J\u2984 (f : X \u27f6 Y),\n (((Functor.const J).obj (F.toPrefunctor.1 i)).map f \u226b\n if h : Y = i then eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj Y = F.obj Y)\n else IsTerminal.from (H Y h) (((Functor.const J).obj (F.toPrefunctor.1 i)).obj Y)) =\n (if h : X = i then eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj X = F.obj X)\n else IsTerminal.from (H X h) (((Functor.const J).obj (F.toPrefunctor.1 i)).obj X)) \u226b\n F.map f\n[PROOFSTEP]\nintro j k f\n[GOAL]\ncase refine'_2\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\nj k : J\nf : j \u27f6 k\n\u22a2 (((Functor.const J).obj (F.toPrefunctor.1 i)).map f \u226b\n if h : k = i then eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj k = F.obj k)\n else IsTerminal.from (H k h) (((Functor.const J).obj (F.toPrefunctor.1 i)).obj k)) =\n (if h : j = i then eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj j = F.obj j)\n else IsTerminal.from (H j h) (((Functor.const J).obj (F.toPrefunctor.1 i)).obj j)) \u226b\n F.map f\n[PROOFSTEP]\nsplit_ifs with h h_1 h_1\n[GOAL]\ncase pos\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\nj k : J\nf : j \u27f6 k\nh : k = i\nh_1 : j = i\n\u22a2 ((Functor.const J).obj (F.toPrefunctor.1 i)).map f \u226b\n eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj k = F.obj k) =\n eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj j = F.obj j) \u226b F.map f\n[PROOFSTEP]\ncases h\n[GOAL]\ncase pos.refl\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\nj : J\nh_1 : j = i\nf : j \u27f6 i\n\u22a2 ((Functor.const J).obj (F.toPrefunctor.1 i)).map f \u226b\n eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj i = F.obj i) =\n eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj j = F.obj j) \u226b F.map f\n[PROOFSTEP]\ncases h_1\n[GOAL]\ncase pos.refl.refl\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\nf : i \u27f6 i\n\u22a2 ((Functor.const J).obj (F.toPrefunctor.1 i)).map f \u226b\n eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj i = F.obj i) =\n eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj i = F.obj i) \u226b F.map f\n[PROOFSTEP]\nobtain rfl : f = \ud835\udfd9 _ := Subsingleton.elim _ _\n[GOAL]\ncase pos.refl.refl\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\n\u22a2 ((Functor.const J).obj (F.toPrefunctor.1 i)).map (\ud835\udfd9 i) \u226b\n eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj i = F.obj i) =\n eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj i = F.obj i) \u226b F.map (\ud835\udfd9 i)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\nj k : J\nf : j \u27f6 k\nh : k = i\nh_1 : \u00acj = i\n\u22a2 ((Functor.const J).obj (F.toPrefunctor.1 i)).map f \u226b\n eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj k = F.obj k) =\n IsTerminal.from (H j h_1) (((Functor.const J).obj (F.toPrefunctor.1 i)).obj j) \u226b F.map f\n[PROOFSTEP]\ncases h\n[GOAL]\ncase neg.refl\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\nj : J\nh_1 : \u00acj = i\nf : j \u27f6 i\n\u22a2 ((Functor.const J).obj (F.toPrefunctor.1 i)).map f \u226b\n eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj i = F.obj i) =\n IsTerminal.from (H j h_1) (((Functor.const J).obj (F.toPrefunctor.1 i)).obj j) \u226b F.map f\n[PROOFSTEP]\nerw [Category.comp_id]\n[GOAL]\ncase neg.refl\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\nj : J\nh_1 : \u00acj = i\nf : j \u27f6 i\n\u22a2 ((Functor.const J).obj (F.toPrefunctor.1 i)).map f =\n IsTerminal.from (H j h_1) (((Functor.const J).obj (F.toPrefunctor.1 i)).obj j) \u226b F.map f\n[PROOFSTEP]\nhaveI : IsIso (F.map f) := (H _ h_1).isIso_from _\n[GOAL]\ncase neg.refl\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\nj : J\nh_1 : \u00acj = i\nf : j \u27f6 i\nthis : IsIso (F.map f)\n\u22a2 ((Functor.const J).obj (F.toPrefunctor.1 i)).map f =\n IsTerminal.from (H j h_1) (((Functor.const J).obj (F.toPrefunctor.1 i)).obj j) \u226b F.map f\n[PROOFSTEP]\nrw [\u2190 IsIso.comp_inv_eq]\n[GOAL]\ncase neg.refl\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\nj : J\nh_1 : \u00acj = i\nf : j \u27f6 i\nthis : IsIso (F.map f)\n\u22a2 ((Functor.const J).obj (F.toPrefunctor.1 i)).map f \u226b inv (F.map f) =\n IsTerminal.from (H j h_1) (((Functor.const J).obj (F.toPrefunctor.1 i)).obj j)\n[PROOFSTEP]\napply (H _ h_1).hom_ext\n[GOAL]\ncase pos\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\nj k : J\nf : j \u27f6 k\nh : \u00ack = i\nh_1 : j = i\n\u22a2 ((Functor.const J).obj (F.toPrefunctor.1 i)).map f \u226b\n IsTerminal.from (H k h) (((Functor.const J).obj (F.toPrefunctor.1 i)).obj k) =\n eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj j = F.obj j) \u226b F.map f\n[PROOFSTEP]\ncases h_1\n[GOAL]\ncase pos.refl\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\nk : J\nh : \u00ack = i\nf : i \u27f6 k\n\u22a2 ((Functor.const J).obj (F.toPrefunctor.1 i)).map f \u226b\n IsTerminal.from (H k h) (((Functor.const J).obj (F.toPrefunctor.1 i)).obj k) =\n eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj i = F.obj i) \u226b F.map f\n[PROOFSTEP]\napply (H _ h).hom_ext\n[GOAL]\ncase neg\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\nj k : J\nf : j \u27f6 k\nh : \u00ack = i\nh_1 : \u00acj = i\n\u22a2 ((Functor.const J).obj (F.toPrefunctor.1 i)).map f \u226b\n IsTerminal.from (H k h) (((Functor.const J).obj (F.toPrefunctor.1 i)).obj k) =\n IsTerminal.from (H j h_1) (((Functor.const J).obj (F.toPrefunctor.1 i)).obj j) \u226b F.map f\n[PROOFSTEP]\napply (H _ h).hom_ext\n[GOAL]\ncase refine'_3\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\n\u22a2 limit.\u03c0 F i \u226b\n limit.lift F\n { pt := F.toPrefunctor.1 i,\n \u03c0 :=\n NatTrans.mk fun j =>\n if h : j = i then eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj j = F.obj j)\n else IsTerminal.from (H j h) (((Functor.const J).obj (F.toPrefunctor.1 i)).obj j) } =\n \ud835\udfd9 (limit F)\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_3.w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\nj\u271d : J\n\u22a2 (limit.\u03c0 F i \u226b\n limit.lift F\n { pt := F.toPrefunctor.1 i,\n \u03c0 :=\n NatTrans.mk fun j =>\n if h : j = i then eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj j = F.obj j)\n else IsTerminal.from (H j h) (((Functor.const J).obj (F.toPrefunctor.1 i)).obj j) }) \u226b\n limit.\u03c0 F j\u271d =\n \ud835\udfd9 (limit F) \u226b limit.\u03c0 F j\u271d\n[PROOFSTEP]\nrw [assoc, limit.lift_\u03c0]\n[GOAL]\ncase refine'_3.w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\nj\u271d : J\n\u22a2 limit.\u03c0 F i \u226b\n NatTrans.app\n { pt := F.toPrefunctor.1 i,\n \u03c0 :=\n NatTrans.mk fun j =>\n if h : j = i then eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj j = F.obj j)\n else IsTerminal.from (H j h) (((Functor.const J).obj (F.toPrefunctor.1 i)).obj j) }.\u03c0\n j\u271d =\n \ud835\udfd9 (limit F) \u226b limit.\u03c0 F j\u271d\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase refine'_3.w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\nj\u271d : J\n\u22a2 (limit.\u03c0 F i \u226b\n if h : j\u271d = i then eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj j\u271d = F.obj j\u271d)\n else IsTerminal.from (H j\u271d h) (((Functor.const J).obj (F.toPrefunctor.1 i)).obj j\u271d)) =\n \ud835\udfd9 (limit F) \u226b limit.\u03c0 F j\u271d\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\nj\u271d : J\nh : j\u271d = i\n\u22a2 limit.\u03c0 F i \u226b eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj j\u271d = F.obj j\u271d) =\n \ud835\udfd9 (limit F) \u226b limit.\u03c0 F j\u271d\n[PROOFSTEP]\ncases h\n[GOAL]\ncase pos.refl\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\n\u22a2 limit.\u03c0 F i \u226b eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj i = F.obj i) = \ud835\udfd9 (limit F) \u226b limit.\u03c0 F i\n[PROOFSTEP]\nrw [id_comp, eqToHom_refl]\n[GOAL]\ncase pos.refl\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\n\u22a2 limit.\u03c0 F i \u226b \ud835\udfd9 (((Functor.const J).obj (F.toPrefunctor.1 i)).obj i) = limit.\u03c0 F i\n[PROOFSTEP]\nexact comp_id _\n[GOAL]\ncase neg\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\nj\u271d : J\nh : \u00acj\u271d = i\n\u22a2 limit.\u03c0 F i \u226b IsTerminal.from (H j\u271d h) (((Functor.const J).obj (F.toPrefunctor.1 i)).obj j\u271d) =\n \ud835\udfd9 (limit F) \u226b limit.\u03c0 F j\u271d\n[PROOFSTEP]\napply (H _ h).hom_ext\n[GOAL]\ncase refine'_4\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\n\u22a2 limit.lift F\n { pt := F.toPrefunctor.1 i,\n \u03c0 :=\n NatTrans.mk fun j =>\n if h : j = i then eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj j = F.obj j)\n else IsTerminal.from (H j h) (((Functor.const J).obj (F.toPrefunctor.1 i)).obj j) } \u226b\n limit.\u03c0 F i =\n \ud835\udfd9 (F.obj i)\n[PROOFSTEP]\nrw [limit.lift_\u03c0]\n[GOAL]\ncase refine'_4\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasStrictTerminalObjects C\nI : C\nJ : Type v\ninst\u271d\u00b2 : SmallCategory J\nF : J \u2964 C\ninst\u271d\u00b9 : HasLimit F\ni : J\nH : (j : J) \u2192 j \u2260 i \u2192 IsTerminal (F.obj j)\ninst\u271d : Subsingleton (i \u27f6 i)\n\u22a2 NatTrans.app\n { pt := F.toPrefunctor.1 i,\n \u03c0 :=\n NatTrans.mk fun j =>\n if h : j = i then eqToHom (_ : ((Functor.const J).obj (F.toPrefunctor.1 i)).obj j = F.obj j)\n else IsTerminal.from (H j h) (((Functor.const J).obj (F.toPrefunctor.1 i)).obj j) }.\u03c0\n i =\n \ud835\udfd9 (F.obj i)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nI : C\nh : \u2200 (A : C) (f : I \u27f6 A), IsIso f\nI' A : C\nf : I' \u27f6 A\nhI' : IsTerminal I'\nthis : IsIso (IsTerminal.from hI' I \u226b f)\n\u22a2 (inv (IsTerminal.from hI' I \u226b f) \u226b IsTerminal.from hI' I) \u226b f = \ud835\udfd9 A\n[PROOFSTEP]\nrw [assoc, IsIso.inv_hom_id]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Shapes.StrictInitial", "llama_tokens": 8784, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.45713673161914675, "lm_q2_score": 0.030675801156820354, "lm_q1q2_score": 0.014023035480627697}} {"text": "[GOAL]\n\u03b1\u271d \u03b2\u271d : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\n\u03b1 \u03b2 : Type ?u.384\n\u22a2 Functor.mapConst = Functor.map \u2218 const \u03b2\n[PROOFSTEP]\nsimp only [Functor.mapConst, Functor.map]\n[GOAL]\n\u03b1\u271d \u03b2\u271d : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\n\u03b1 \u03b2 \u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Finset \u03b1\nt : Finset \u03b2\n\u22a2 image\u2082 f s t = Seq.seq (f <$> s) fun x => t\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1\u271d \u03b2\u271d : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\n\u03b1 \u03b2 \u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Finset \u03b1\nt : Finset \u03b2\na\u271d : \u03b3\n\u22a2 a\u271d \u2208 image\u2082 f s t \u2194 a\u271d \u2208 Seq.seq (f <$> s) fun x => t\n[PROOFSTEP]\nsimp [mem_sup]\n[GOAL]\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\n\u22a2 (SeqLeft.seqLeft s fun x => t) = Seq.seq (const \u03b2\u271d <$> s) fun x => t\n[PROOFSTEP]\nrw [seq_def, fmap_def, seqLeft_def]\n[GOAL]\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\n\u22a2 (if t = \u2205 then \u2205 else s) = sup (image (const \u03b2\u271d) s) fun f => image f t\n[PROOFSTEP]\nobtain rfl | ht := t.eq_empty_or_nonempty\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\n\u22a2 (if \u2205 = \u2205 then \u2205 else s) = sup (image (const \u03b2\u271d) s) fun f => image f \u2205\n[PROOFSTEP]\nsimp_rw [image_empty, if_true]\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\n\u22a2 \u2205 = sup (image (const \u03b2\u271d) s) fun f => \u2205\n[PROOFSTEP]\nexact (sup_bot _).symm\n[GOAL]\ncase inr\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\nht : Finset.Nonempty t\n\u22a2 (if t = \u2205 then \u2205 else s) = sup (image (const \u03b2\u271d) s) fun f => image f t\n[PROOFSTEP]\next a\n[GOAL]\ncase inr.a\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\nht : Finset.Nonempty t\na : \u03b1\u271d\n\u22a2 (a \u2208 if t = \u2205 then \u2205 else s) \u2194 a \u2208 sup (image (const \u03b2\u271d) s) fun f => image f t\n[PROOFSTEP]\nrw [if_neg ht.ne_empty, mem_sup]\n[GOAL]\ncase inr.a\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\nht : Finset.Nonempty t\na : \u03b1\u271d\n\u22a2 a \u2208 s \u2194 \u2203 v, v \u2208 image (const \u03b2\u271d) s \u2227 a \u2208 image v t\n[PROOFSTEP]\nrefine' \u27e8fun ha => \u27e8const _ a, mem_image_of_mem _ ha, mem_image_const_self.2 ht\u27e9, _\u27e9\n[GOAL]\ncase inr.a\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\nht : Finset.Nonempty t\na : \u03b1\u271d\n\u22a2 (\u2203 v, v \u2208 image (const \u03b2\u271d) s \u2227 a \u2208 image v t) \u2192 a \u2208 s\n[PROOFSTEP]\nrintro \u27e8f, hf, ha\u27e9\n[GOAL]\ncase inr.a.intro.intro\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\nht : Finset.Nonempty t\na : \u03b1\u271d\nf : \u03b2\u271d \u2192 \u03b1\u271d\nhf : f \u2208 image (const \u03b2\u271d) s\nha : a \u2208 image f t\n\u22a2 a \u2208 s\n[PROOFSTEP]\nrw [mem_image] at hf ha \n[GOAL]\ncase inr.a.intro.intro\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\nht : Finset.Nonempty t\na : \u03b1\u271d\nf : \u03b2\u271d \u2192 \u03b1\u271d\nhf : \u2203 a, a \u2208 s \u2227 const \u03b2\u271d a = f\nha : \u2203 a_1, a_1 \u2208 t \u2227 f a_1 = a\n\u22a2 a \u2208 s\n[PROOFSTEP]\nobtain \u27e8b, hb, rfl\u27e9 := hf\n[GOAL]\ncase inr.a.intro.intro.intro.intro\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\nht : Finset.Nonempty t\na b : \u03b1\u271d\nhb : b \u2208 s\nha : \u2203 a_1, a_1 \u2208 t \u2227 const \u03b2\u271d b a_1 = a\n\u22a2 a \u2208 s\n[PROOFSTEP]\nobtain \u27e8_, _, rfl\u27e9 := ha\n[GOAL]\ncase inr.a.intro.intro.intro.intro.intro.intro\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\nht : Finset.Nonempty t\nb : \u03b1\u271d\nhb : b \u2208 s\nw\u271d : \u03b2\u271d\nleft\u271d : w\u271d \u2208 t\n\u22a2 const \u03b2\u271d b w\u271d \u2208 s\n[PROOFSTEP]\nexact hb\n[GOAL]\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\n\u22a2 (SeqRight.seqRight s fun x => t) = Seq.seq (const \u03b1\u271d id <$> s) fun x => t\n[PROOFSTEP]\nrw [seq_def, fmap_def, seqRight_def]\n[GOAL]\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\n\u22a2 (if s = \u2205 then \u2205 else t) = sup (image (const \u03b1\u271d id) s) fun f => image f t\n[PROOFSTEP]\nobtain rfl | hs := s.eq_empty_or_nonempty\n[GOAL]\ncase inl\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\nt : Finset \u03b2\u271d\n\u22a2 (if \u2205 = \u2205 then \u2205 else t) = sup (image (const \u03b1\u271d id) \u2205) fun f => image f t\n[PROOFSTEP]\nrw [if_pos rfl, image_empty, sup_empty, bot_eq_empty]\n[GOAL]\ncase inr\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\nhs : Finset.Nonempty s\n\u22a2 (if s = \u2205 then \u2205 else t) = sup (image (const \u03b1\u271d id) s) fun f => image f t\n[PROOFSTEP]\next a\n[GOAL]\ncase inr.a\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\nhs : Finset.Nonempty s\na : \u03b2\u271d\n\u22a2 (a \u2208 if s = \u2205 then \u2205 else t) \u2194 a \u2208 sup (image (const \u03b1\u271d id) s) fun f => image f t\n[PROOFSTEP]\nrw [if_neg hs.ne_empty, mem_sup]\n[GOAL]\ncase inr.a\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\nhs : Finset.Nonempty s\na : \u03b2\u271d\n\u22a2 a \u2208 t \u2194 \u2203 v, v \u2208 image (const \u03b1\u271d id) s \u2227 a \u2208 image v t\n[PROOFSTEP]\nrefine' \u27e8fun ha => \u27e8id, mem_image_const_self.2 hs, by rwa [image_id]\u27e9, _\u27e9\n[GOAL]\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\nhs : Finset.Nonempty s\na : \u03b2\u271d\nha : a \u2208 t\n\u22a2 a \u2208 image id t\n[PROOFSTEP]\nrwa [image_id]\n[GOAL]\ncase inr.a\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\nhs : Finset.Nonempty s\na : \u03b2\u271d\n\u22a2 (\u2203 v, v \u2208 image (const \u03b1\u271d id) s \u2227 a \u2208 image v t) \u2192 a \u2208 t\n[PROOFSTEP]\nrintro \u27e8f, hf, ha\u27e9\n[GOAL]\ncase inr.a.intro.intro\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\nhs : Finset.Nonempty s\na : \u03b2\u271d\nf : \u03b2\u271d \u2192 \u03b2\u271d\nhf : f \u2208 image (const \u03b1\u271d id) s\nha : a \u2208 image f t\n\u22a2 a \u2208 t\n[PROOFSTEP]\nrw [mem_image] at hf ha \n[GOAL]\ncase inr.a.intro.intro\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\nhs : Finset.Nonempty s\na : \u03b2\u271d\nf : \u03b2\u271d \u2192 \u03b2\u271d\nhf : \u2203 a, a \u2208 s \u2227 const \u03b1\u271d id a = f\nha : \u2203 a_1, a_1 \u2208 t \u2227 f a_1 = a\n\u22a2 a \u2208 t\n[PROOFSTEP]\nobtain \u27e8b, hb, rfl\u27e9 := ha\n[GOAL]\ncase inr.a.intro.intro.intro.intro\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\nhs : Finset.Nonempty s\nf : \u03b2\u271d \u2192 \u03b2\u271d\nhf : \u2203 a, a \u2208 s \u2227 const \u03b1\u271d id a = f\nb : \u03b2\u271d\nhb : b \u2208 t\n\u22a2 f b \u2208 t\n[PROOFSTEP]\nobtain \u27e8_, _, rfl\u27e9 := hf\n[GOAL]\ncase inr.a.intro.intro.intro.intro.intro.intro\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\nhs : Finset.Nonempty s\nb : \u03b2\u271d\nhb : b \u2208 t\nw\u271d : \u03b1\u271d\nleft\u271d : w\u271d \u2208 s\n\u22a2 const \u03b1\u271d id w\u271d b \u2208 t\n[PROOFSTEP]\nexact hb\n[GOAL]\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d : Type ?u.13031\nf : \u03b1\u271d \u2192 \u03b2\u271d\ns : Finset \u03b1\u271d\n\u22a2 (Seq.seq (pure f) fun x => s) = f <$> s\n[PROOFSTEP]\nsimp only [pure_def, seq_def, sup_singleton, fmap_def]\n[GOAL]\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset (\u03b1\u271d \u2192 \u03b2\u271d)\nu : Finset (\u03b2\u271d \u2192 \u03b3\u271d)\n\u22a2 (Seq.seq u fun x => Seq.seq t fun x => s) = Seq.seq (Seq.seq (comp <$> u) fun x => t) fun x => s\n[PROOFSTEP]\next a\n[GOAL]\ncase a\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset (\u03b1\u271d \u2192 \u03b2\u271d)\nu : Finset (\u03b2\u271d \u2192 \u03b3\u271d)\na : \u03b3\u271d\n\u22a2 (a \u2208 Seq.seq u fun x => Seq.seq t fun x => s) \u2194 a \u2208 Seq.seq (Seq.seq (comp <$> u) fun x => t) fun x => s\n[PROOFSTEP]\nsimp_rw [seq_def, fmap_def]\n[GOAL]\ncase a\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset (\u03b1\u271d \u2192 \u03b2\u271d)\nu : Finset (\u03b2\u271d \u2192 \u03b3\u271d)\na : \u03b3\u271d\n\u22a2 (a \u2208 sup u fun f => image f (sup t fun f => image f s)) \u2194\n a \u2208 sup (sup (image comp u) fun f => image f t) fun f => image f s\n[PROOFSTEP]\nsimp only [exists_prop, mem_sup, mem_image]\n[GOAL]\ncase a\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset (\u03b1\u271d \u2192 \u03b2\u271d)\nu : Finset (\u03b2\u271d \u2192 \u03b3\u271d)\na : \u03b3\u271d\n\u22a2 (\u2203 v, v \u2208 u \u2227 \u2203 a_1, (\u2203 v, v \u2208 t \u2227 \u2203 a, a \u2208 s \u2227 v a = a_1) \u2227 v a_1 = a) \u2194\n \u2203 v, (\u2203 v_1, (\u2203 a, a \u2208 u \u2227 comp a = v_1) \u2227 \u2203 a, a \u2208 t \u2227 v_1 a = v) \u2227 \u2203 a_1, a_1 \u2208 s \u2227 v a_1 = a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.mp\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset (\u03b1\u271d \u2192 \u03b2\u271d)\nu : Finset (\u03b2\u271d \u2192 \u03b3\u271d)\na : \u03b3\u271d\n\u22a2 (\u2203 v, v \u2208 u \u2227 \u2203 a_1, (\u2203 v, v \u2208 t \u2227 \u2203 a, a \u2208 s \u2227 v a = a_1) \u2227 v a_1 = a) \u2192\n \u2203 v, (\u2203 v_1, (\u2203 a, a \u2208 u \u2227 comp a = v_1) \u2227 \u2203 a, a \u2208 t \u2227 v_1 a = v) \u2227 \u2203 a_2, a_2 \u2208 s \u2227 v a_2 = a\n[PROOFSTEP]\nrintro \u27e8g, hg, b, \u27e8f, hf, a, ha, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase a.mp.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset (\u03b1\u271d \u2192 \u03b2\u271d)\nu : Finset (\u03b2\u271d \u2192 \u03b3\u271d)\ng : \u03b2\u271d \u2192 \u03b3\u271d\nhg : g \u2208 u\nf : \u03b1\u271d \u2192 \u03b2\u271d\nhf : f \u2208 t\na : \u03b1\u271d\nha : a \u2208 s\n\u22a2 \u2203 v, (\u2203 v_1, (\u2203 a, a \u2208 u \u2227 comp a = v_1) \u2227 \u2203 a, a \u2208 t \u2227 v_1 a = v) \u2227 \u2203 a_1, a_1 \u2208 s \u2227 v a_1 = g (f a)\n[PROOFSTEP]\nexact \u27e8g \u2218 f, \u27e8comp g, \u27e8g, hg, rfl\u27e9, f, hf, rfl\u27e9, a, ha, rfl\u27e9\n[GOAL]\ncase a.mpr\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset (\u03b1\u271d \u2192 \u03b2\u271d)\nu : Finset (\u03b2\u271d \u2192 \u03b3\u271d)\na : \u03b3\u271d\n\u22a2 (\u2203 v, (\u2203 v_1, (\u2203 a, a \u2208 u \u2227 comp a = v_1) \u2227 \u2203 a, a \u2208 t \u2227 v_1 a = v) \u2227 \u2203 a_1, a_1 \u2208 s \u2227 v a_1 = a) \u2192\n \u2203 v, v \u2208 u \u2227 \u2203 a_2, (\u2203 v, v \u2208 t \u2227 \u2203 a, a \u2208 s \u2227 v a = a_2) \u2227 v a_2 = a\n[PROOFSTEP]\nrintro \u27e8c, \u27e8_, \u27e8g, hg, rfl\u27e9, f, hf, rfl\u27e9, a, ha, rfl\u27e9\n[GOAL]\ncase a.mpr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulFunctor Finset := lawfulFunctor\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.13031\ns : Finset \u03b1\u271d\nt : Finset (\u03b1\u271d \u2192 \u03b2\u271d)\nu : Finset (\u03b2\u271d \u2192 \u03b3\u271d)\ng : \u03b2\u271d \u2192 \u03b3\u271d\nhg : g \u2208 u\nf : \u03b1\u271d \u2192 \u03b2\u271d\nhf : f \u2208 t\na : \u03b1\u271d\nha : a \u2208 s\n\u22a2 \u2203 v, v \u2208 u \u2227 \u2203 a_1, (\u2203 v, v \u2208 t \u2227 \u2203 a, a \u2208 s \u2227 v a = a_1) \u2227 v a_1 = (g \u2218 f) a\n[PROOFSTEP]\nexact \u27e8g, hg, f a, \u27e8f, hf, a, ha, rfl\u27e9, rfl\u27e9\n[GOAL]\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulApplicative Finset := lawfulApplicative\n\u03b1\u271d \u03b2\u271d : Type ?u.21428\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\n\u22a2 (Seq.seq (Prod.mk <$> s) fun x => t) = Seq.seq ((fun b a => (a, b)) <$> t) fun x => s\n[PROOFSTEP]\nsimp_rw [seq_def, fmap_def, sup_image, sup_eq_biUnion]\n[GOAL]\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulApplicative Finset := lawfulApplicative\n\u03b1\u271d \u03b2\u271d : Type ?u.21428\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\n\u22a2 Finset.biUnion s ((fun f => image f t) \u2218 Prod.mk) = Finset.biUnion t ((fun f => image f s) \u2218 fun b a => (a, b))\n[PROOFSTEP]\nchange (s.biUnion fun a => t.image fun b => (a, b)) = t.biUnion fun b => s.image fun a => (a, b)\n[GOAL]\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulApplicative Finset := lawfulApplicative\n\u03b1\u271d \u03b2\u271d : Type ?u.21428\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\n\u22a2 (Finset.biUnion s fun a => image (fun b => (a, b)) t) = Finset.biUnion t fun b => image (fun a => (a, b)) s\n[PROOFSTEP]\ntrans s \u00d7\u02e2 t <;> [rw [product_eq_biUnion]; rw [product_eq_biUnion_right]]\n[GOAL]\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulApplicative Finset := lawfulApplicative\n\u03b1\u271d \u03b2\u271d : Type ?u.21428\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\n\u22a2 (Finset.biUnion s fun a => image (fun b => (a, b)) t) = Finset.biUnion t fun b => image (fun a => (a, b)) s\n[PROOFSTEP]\ntrans s \u00d7\u02e2 t\n[GOAL]\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulApplicative Finset := lawfulApplicative\n\u03b1\u271d \u03b2\u271d : Type ?u.23242\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\n\u22a2 (Finset.biUnion s fun a => image (fun b => (a, b)) t) = s \u00d7\u02e2 t\n[PROOFSTEP]\nrw [product_eq_biUnion]\n[GOAL]\n\u03b1 \u03b2 : Type u\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulApplicative Finset := lawfulApplicative\n\u03b1\u271d \u03b2\u271d : Type ?u.23242\ns : Finset \u03b1\u271d\nt : Finset \u03b2\u271d\n\u22a2 s \u00d7\u02e2 t = Finset.biUnion t fun b => image (fun a => (a, b)) s\n[PROOFSTEP]\nrw [product_eq_biUnion_right]\n[GOAL]\ninst\u271d : (P : Prop) \u2192 Decidable P\nsrc\u271d : LawfulApplicative Finset := lawfulApplicative\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.24298\ns : Finset \u03b1\u271d\nf : \u03b1\u271d \u2192 Finset \u03b2\u271d\ng : \u03b2\u271d \u2192 Finset \u03b3\u271d\n\u22a2 s >>= f >>= g = s >>= fun x => f x >>= g\n[PROOFSTEP]\nsimp only [bind, \u2190 sup_biUnion, sup_eq_biUnion, biUnion_biUnion]\n[GOAL]\n\u03b1 \u03b2 \u03b3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : CommApplicative F\ninst\u271d\u00b9 : CommApplicative G\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\n\u22a2 traverse pure s = s\n[PROOFSTEP]\nrw [traverse, Multiset.id_traverse]\n[GOAL]\n\u03b1 \u03b2 \u03b3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u2074 : Applicative F\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : CommApplicative F\ninst\u271d\u00b9 : CommApplicative G\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\n\u22a2 Multiset.toFinset <$> s.val = s\n[PROOFSTEP]\nexact s.val_toFinset\n[GOAL]\n\u03b1 \u03b2 \u03b3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : CommApplicative F\ninst\u271d : CommApplicative G\ng : \u03b1 \u2192 G \u03b2\nh : \u03b2 \u2192 \u03b3\ns : Finset \u03b1\n\u22a2 Functor.map h <$> traverse g s = traverse (Functor.map h \u2218 g) s\n[PROOFSTEP]\nunfold traverse\n[GOAL]\n\u03b1 \u03b2 \u03b3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : CommApplicative F\ninst\u271d : CommApplicative G\ng : \u03b1 \u2192 G \u03b2\nh : \u03b2 \u2192 \u03b3\ns : Finset \u03b1\n\u22a2 Functor.map h <$> Multiset.toFinset <$> Multiset.traverse g s.val =\n Multiset.toFinset <$> Multiset.traverse (Functor.map h \u2218 g) s.val\n[PROOFSTEP]\nsimp only [map_comp_coe, functor_norm]\n[GOAL]\n\u03b1 \u03b2 \u03b3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : CommApplicative F\ninst\u271d : CommApplicative G\ng : \u03b1 \u2192 G \u03b2\nh : \u03b2 \u2192 \u03b3\ns : Finset \u03b1\n\u22a2 (Multiset.toFinset \u2218 Functor.map h) <$> Multiset.traverse g s.val =\n Multiset.toFinset <$> Multiset.traverse (Functor.map h \u2218 g) s.val\n[PROOFSTEP]\nrw [LawfulFunctor.comp_map, Multiset.map_traverse]\n", "meta": {"mathlib_filename": "Mathlib.Data.Finset.Functor", "llama_tokens": 8140, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.42632159254749036, "lm_q2_score": 0.032589744766025344, "lm_q1q2_score": 0.013893711889368164}} {"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : IsNoetherianRing \u03b1\n\u22a2 WellFounded DvdNotUnit\n[PROOFSTEP]\nconvert InvImage.wf (fun a => Ideal.span ({ a } : Set \u03b1)) (wellFounded_submodule_gt _ _)\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : IsNoetherianRing \u03b1\n\u22a2 DvdNotUnit = InvImage (fun x x_1 => x > x_1) fun a => Ideal.span {a}\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_2.h.h.a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : IsDomain \u03b1\ninst\u271d : IsNoetherianRing \u03b1\nx\u271d\u00b9 x\u271d : \u03b1\n\u22a2 DvdNotUnit x\u271d\u00b9 x\u271d \u2194 InvImage (fun x x_1 => x > x_1) (fun a => Ideal.span {a}) x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nexact Ideal.span_singleton_lt_span_singleton.symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommMonoidWithZero \u03b1\nh : WfDvdMonoid (Associates \u03b1)\n\u22a2 WellFounded DvdNotUnit\n[PROOFSTEP]\nrefine' (Surjective.wellFounded_iff mk_surjective _).2 wellFounded_dvdNotUnit\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommMonoidWithZero \u03b1\nh : WfDvdMonoid (Associates \u03b1)\n\u22a2 \u2200 {a b : \u03b1}, DvdNotUnit a b \u2194 DvdNotUnit (Associates.mk a) (Associates.mk b)\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommMonoidWithZero \u03b1\nh : WfDvdMonoid (Associates \u03b1)\na\u271d b\u271d : \u03b1\n\u22a2 DvdNotUnit a\u271d b\u271d \u2194 DvdNotUnit (Associates.mk a\u271d) (Associates.mk b\u271d)\n[PROOFSTEP]\nrw [mk_dvdNotUnit_mk_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero \u03b1\ninst\u271d : WfDvdMonoid \u03b1\n\u22a2 WellFounded DvdNotUnit\n[PROOFSTEP]\nrefine' (Surjective.wellFounded_iff mk_surjective _).1 wellFounded_dvdNotUnit\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero \u03b1\ninst\u271d : WfDvdMonoid \u03b1\n\u22a2 \u2200 {a b : \u03b1}, DvdNotUnit a b \u2194 DvdNotUnit (Associates.mk a) (Associates.mk b)\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero \u03b1\ninst\u271d : WfDvdMonoid \u03b1\na\u271d b\u271d : \u03b1\n\u22a2 DvdNotUnit a\u271d b\u271d \u2194 DvdNotUnit (Associates.mk a\u271d) (Associates.mk b\u271d)\n[PROOFSTEP]\nrw [mk_dvdNotUnit_mk_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero \u03b1\ninst\u271d : WfDvdMonoid \u03b1\na\u271d a i : \u03b1\nha0 : a \u2260 0\nhi : Irreducible i\nih : a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nx\u271d : i * a \u2260 0\ns : Multiset \u03b1\nhs : (\u2200 (b : \u03b1), b \u2208 s \u2192 Irreducible b) \u2227 Multiset.prod s ~\u1d64 a\n\u22a2 Multiset.prod (i ::\u2098 s) ~\u1d64 i * a\n[PROOFSTEP]\nrw [s.prod_cons i]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero \u03b1\ninst\u271d : WfDvdMonoid \u03b1\na\u271d a i : \u03b1\nha0 : a \u2260 0\nhi : Irreducible i\nih : a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nx\u271d : i * a \u2260 0\ns : Multiset \u03b1\nhs : (\u2200 (b : \u03b1), b \u2208 s \u2192 Irreducible b) \u2227 Multiset.prod s ~\u1d64 a\n\u22a2 i * Multiset.prod s ~\u1d64 i * a\n[PROOFSTEP]\nexact hs.2.mul_left i\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero \u03b1\ninst\u271d : WfDvdMonoid \u03b1\na : \u03b1\nhn0 : a \u2260 0\nhnu : \u00acIsUnit a\n\u22a2 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f = a \u2227 f \u2260 \u2205\n[PROOFSTEP]\nobtain \u27e8f, hi, u, rfl\u27e9 := exists_factors a hn0\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero \u03b1\ninst\u271d : WfDvdMonoid \u03b1\nf : Multiset \u03b1\nhi : \u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b\nu : \u03b1\u02e3\nhn0 : Multiset.prod f * \u2191u \u2260 0\nhnu : \u00acIsUnit (Multiset.prod f * \u2191u)\n\u22a2 \u2203 f_1, (\u2200 (b : \u03b1), b \u2208 f_1 \u2192 Irreducible b) \u2227 Multiset.prod f_1 = Multiset.prod f * \u2191u \u2227 f_1 \u2260 \u2205\n[PROOFSTEP]\nobtain \u27e8b, h\u27e9 := Multiset.exists_mem_of_ne_zero fun h : f = 0 => hnu <| by simp [h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero \u03b1\ninst\u271d : WfDvdMonoid \u03b1\nf : Multiset \u03b1\nhi : \u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b\nu : \u03b1\u02e3\nhn0 : Multiset.prod f * \u2191u \u2260 0\nhnu : \u00acIsUnit (Multiset.prod f * \u2191u)\nh : f = 0\n\u22a2 IsUnit (Multiset.prod f * \u2191u)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero \u03b1\ninst\u271d : WfDvdMonoid \u03b1\nf : Multiset \u03b1\nhi : \u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b\nu : \u03b1\u02e3\nhn0 : Multiset.prod f * \u2191u \u2260 0\nhnu : \u00acIsUnit (Multiset.prod f * \u2191u)\nb : \u03b1\nh : b \u2208 f\n\u22a2 \u2203 f_1, (\u2200 (b : \u03b1), b \u2208 f_1 \u2192 Irreducible b) \u2227 Multiset.prod f_1 = Multiset.prod f * \u2191u \u2227 f_1 \u2260 \u2205\n[PROOFSTEP]\nclassical\nrefine' \u27e8(f.erase b).cons (b * u), fun a ha => _, _, Multiset.cons_ne_zero\u27e9\n\u00b7 obtain rfl | ha := Multiset.mem_cons.1 ha\n exacts [Associated.irreducible \u27e8u, rfl\u27e9 (hi b h), hi a (Multiset.mem_of_mem_erase ha)]\n\u00b7 rw [Multiset.prod_cons, mul_comm b, mul_assoc, Multiset.prod_erase h, mul_comm]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero \u03b1\ninst\u271d : WfDvdMonoid \u03b1\nf : Multiset \u03b1\nhi : \u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b\nu : \u03b1\u02e3\nhn0 : Multiset.prod f * \u2191u \u2260 0\nhnu : \u00acIsUnit (Multiset.prod f * \u2191u)\nb : \u03b1\nh : b \u2208 f\n\u22a2 \u2203 f_1, (\u2200 (b : \u03b1), b \u2208 f_1 \u2192 Irreducible b) \u2227 Multiset.prod f_1 = Multiset.prod f * \u2191u \u2227 f_1 \u2260 \u2205\n[PROOFSTEP]\nrefine' \u27e8(f.erase b).cons (b * u), fun a ha => _, _, Multiset.cons_ne_zero\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero \u03b1\ninst\u271d : WfDvdMonoid \u03b1\nf : Multiset \u03b1\nhi : \u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b\nu : \u03b1\u02e3\nhn0 : Multiset.prod f * \u2191u \u2260 0\nhnu : \u00acIsUnit (Multiset.prod f * \u2191u)\nb : \u03b1\nh : b \u2208 f\na : \u03b1\nha : a \u2208 b * \u2191u ::\u2098 Multiset.erase f b\n\u22a2 Irreducible a\n[PROOFSTEP]\nobtain rfl | ha := Multiset.mem_cons.1 ha\n[GOAL]\ncase intro.intro.intro.intro.refine'_1.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero \u03b1\ninst\u271d : WfDvdMonoid \u03b1\nf : Multiset \u03b1\nhi : \u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b\nu : \u03b1\u02e3\nhn0 : Multiset.prod f * \u2191u \u2260 0\nhnu : \u00acIsUnit (Multiset.prod f * \u2191u)\nb : \u03b1\nh : b \u2208 f\nha : b * \u2191u \u2208 b * \u2191u ::\u2098 Multiset.erase f b\n\u22a2 Irreducible (b * \u2191u)\ncase intro.intro.intro.intro.refine'_1.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero \u03b1\ninst\u271d : WfDvdMonoid \u03b1\nf : Multiset \u03b1\nhi : \u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b\nu : \u03b1\u02e3\nhn0 : Multiset.prod f * \u2191u \u2260 0\nhnu : \u00acIsUnit (Multiset.prod f * \u2191u)\nb : \u03b1\nh : b \u2208 f\na : \u03b1\nha\u271d : a \u2208 b * \u2191u ::\u2098 Multiset.erase f b\nha : a \u2208 Multiset.erase f b\n\u22a2 Irreducible a\n[PROOFSTEP]\nexacts [Associated.irreducible \u27e8u, rfl\u27e9 (hi b h), hi a (Multiset.mem_of_mem_erase ha)]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero \u03b1\ninst\u271d : WfDvdMonoid \u03b1\nf : Multiset \u03b1\nhi : \u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b\nu : \u03b1\u02e3\nhn0 : Multiset.prod f * \u2191u \u2260 0\nhnu : \u00acIsUnit (Multiset.prod f * \u2191u)\nb : \u03b1\nh : b \u2208 f\n\u22a2 Multiset.prod (b * \u2191u ::\u2098 Multiset.erase f b) = Multiset.prod f * \u2191u\n[PROOFSTEP]\nrw [Multiset.prod_cons, mul_comm b, mul_assoc, Multiset.prod_erase h, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\nh : WellFounded fun x x_1 => x < x_1\n\u22a2 WellFounded DvdNotUnit\n[PROOFSTEP]\nconvert h\n[GOAL]\ncase h.e'_2.h.h.h.e\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\nh : WellFounded fun x x_1 => x < x_1\nx\u271d\u00b9 x\u271d : Associates \u03b1\n\u22a2 DvdNotUnit = LT.lt\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_2.h.h.h.e.h.h.a\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\nh : WellFounded fun x x_1 => x < x_1\nx\u271d\u00b3 x\u271d\u00b2 x\u271d\u00b9 x\u271d : Associates \u03b1\n\u22a2 DvdNotUnit x\u271d\u00b9 x\u271d \u2194 x\u271d\u00b9 < x\u271d\n[PROOFSTEP]\nexact Associates.dvdNotUnit_iff_lt\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\n\u22a2 WfDvdMonoid \u03b1 \u2192 WellFounded fun x x_1 => x < x_1\n[PROOFSTEP]\napply WfDvdMonoid.wellFounded_associates\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nsrc\u271d : WfDvdMonoid (Associates \u03b1) := WfDvdMonoid.wfDvdMonoid_associates\n\u22a2 \u2200 {a : Associates \u03b1}, Irreducible a \u2194 Prime a\n[PROOFSTEP]\nrw [\u2190 Associates.irreducible_iff_prime_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nsrc\u271d : WfDvdMonoid (Associates \u03b1) := WfDvdMonoid.wfDvdMonoid_associates\n\u22a2 \u2200 (a : \u03b1), Irreducible a \u2194 Prime a\n[PROOFSTEP]\napply UniqueFactorizationMonoid.irreducible_iff_prime\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\n\u22a2 a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\n[PROOFSTEP]\nsimp_rw [\u2190 UniqueFactorizationMonoid.irreducible_iff_prime]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\n\u22a2 a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\n[PROOFSTEP]\napply WfDvdMonoid.exists_factors a\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nP : \u03b1 \u2192 Prop\na : \u03b1\nh\u2081 : P 0\nh\u2082 : \u2200 (x : \u03b1), IsUnit x \u2192 P x\nh\u2083 : \u2200 (a p : \u03b1), a \u2260 0 \u2192 Prime p \u2192 P a \u2192 P (p * a)\n\u22a2 P a\n[PROOFSTEP]\nsimp_rw [\u2190 UniqueFactorizationMonoid.irreducible_iff_prime] at h\u2083 \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nP : \u03b1 \u2192 Prop\na : \u03b1\nh\u2081 : P 0\nh\u2082 : \u2200 (x : \u03b1), IsUnit x \u2192 P x\nh\u2083 : \u2200 (a p : \u03b1), a \u2260 0 \u2192 Irreducible p \u2192 P a \u2192 P (p * a)\n\u22a2 P a\n[PROOFSTEP]\nexact WfDvdMonoid.induction_on_irreducible a h\u2081 h\u2082 h\u2083\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\n\u22a2 \u2200 {f g : Multiset \u03b1},\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Prime x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Prime x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\n[PROOFSTEP]\nclassical\nintro f\ninduction' f using Multiset.induction_on with p f ih\n\u00b7 intros g _ hg h\n exact\n Multiset.rel_zero_left.2 <|\n Multiset.eq_zero_of_forall_not_mem fun x hx =>\n have : IsUnit g.prod := by simpa [associated_one_iff_isUnit] using h.symm\n (hg x hx).not_unit <| isUnit_iff_dvd_one.2 <| (Multiset.dvd_prod hx).trans (isUnit_iff_dvd_one.1 this)\n\u00b7 intros g hf hg hfg\n let \u27e8b, hbg, hb\u27e9 :=\n (exists_associated_mem_of_dvd_prod (hf p (by simp)) fun q hq => hg _ hq) <|\n hfg.dvd_iff_dvd_right.1 (show p \u2223 (p ::\u2098 f).prod by simp)\n haveI := Classical.decEq \u03b1\n rw [\u2190 Multiset.cons_erase hbg]\n exact\n Multiset.Rel.cons hb\n (ih (fun q hq => hf _ (by simp [hq])) (fun {q} (hq : q \u2208 g.erase b) => hg q (Multiset.mem_of_mem_erase hq))\n (Associated.of_mul_left (by rwa [\u2190 Multiset.prod_cons, \u2190 Multiset.prod_cons, Multiset.cons_erase hbg]) hb\n (hf p (by simp)).ne_zero))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\n\u22a2 \u2200 {f g : Multiset \u03b1},\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Prime x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Prime x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\n[PROOFSTEP]\nintro f\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\nf : Multiset \u03b1\n\u22a2 \u2200 {g : Multiset \u03b1},\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Prime x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Prime x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\n[PROOFSTEP]\ninduction' f using Multiset.induction_on with p f ih\n[GOAL]\ncase empty\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\n\u22a2 \u2200 {g : Multiset \u03b1},\n (\u2200 (x : \u03b1), x \u2208 0 \u2192 Prime x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Prime x) \u2192 Multiset.prod 0 ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated 0 g\n[PROOFSTEP]\nintros g _ hg h\n[GOAL]\ncase empty\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\ng : Multiset \u03b1\na\u271d : \u2200 (x : \u03b1), x \u2208 0 \u2192 Prime x\nhg : \u2200 (x : \u03b1), x \u2208 g \u2192 Prime x\nh : Multiset.prod 0 ~\u1d64 Multiset.prod g\n\u22a2 Multiset.Rel Associated 0 g\n[PROOFSTEP]\nexact\n Multiset.rel_zero_left.2 <|\n Multiset.eq_zero_of_forall_not_mem fun x hx =>\n have : IsUnit g.prod := by simpa [associated_one_iff_isUnit] using h.symm\n (hg x hx).not_unit <| isUnit_iff_dvd_one.2 <| (Multiset.dvd_prod hx).trans (isUnit_iff_dvd_one.1 this)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\ng : Multiset \u03b1\na\u271d : \u2200 (x : \u03b1), x \u2208 0 \u2192 Prime x\nhg : \u2200 (x : \u03b1), x \u2208 g \u2192 Prime x\nh : Multiset.prod 0 ~\u1d64 Multiset.prod g\nx : \u03b1\nhx : x \u2208 g\n\u22a2 IsUnit (Multiset.prod g)\n[PROOFSTEP]\nsimpa [associated_one_iff_isUnit] using h.symm\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\np : \u03b1\nf : Multiset \u03b1\nih :\n \u2200 {g : Multiset \u03b1},\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Prime x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Prime x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\n\u22a2 \u2200 {g : Multiset \u03b1},\n (\u2200 (x : \u03b1), x \u2208 p ::\u2098 f \u2192 Prime x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Prime x) \u2192 Multiset.prod (p ::\u2098 f) ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated (p ::\u2098 f) g\n[PROOFSTEP]\nintros g hf hg hfg\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\np : \u03b1\nf : Multiset \u03b1\nih :\n \u2200 {g : Multiset \u03b1},\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Prime x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Prime x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\ng : Multiset \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 p ::\u2098 f \u2192 Prime x\nhg : \u2200 (x : \u03b1), x \u2208 g \u2192 Prime x\nhfg : Multiset.prod (p ::\u2098 f) ~\u1d64 Multiset.prod g\n\u22a2 Multiset.Rel Associated (p ::\u2098 f) g\n[PROOFSTEP]\nlet \u27e8b, hbg, hb\u27e9 :=\n (exists_associated_mem_of_dvd_prod (hf p (by simp)) fun q hq => hg _ hq) <|\n hfg.dvd_iff_dvd_right.1 (show p \u2223 (p ::\u2098 f).prod by simp)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\np : \u03b1\nf : Multiset \u03b1\nih :\n \u2200 {g : Multiset \u03b1},\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Prime x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Prime x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\ng : Multiset \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 p ::\u2098 f \u2192 Prime x\nhg : \u2200 (x : \u03b1), x \u2208 g \u2192 Prime x\nhfg : Multiset.prod (p ::\u2098 f) ~\u1d64 Multiset.prod g\n\u22a2 p \u2208 p ::\u2098 f\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\np : \u03b1\nf : Multiset \u03b1\nih :\n \u2200 {g : Multiset \u03b1},\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Prime x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Prime x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\ng : Multiset \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 p ::\u2098 f \u2192 Prime x\nhg : \u2200 (x : \u03b1), x \u2208 g \u2192 Prime x\nhfg : Multiset.prod (p ::\u2098 f) ~\u1d64 Multiset.prod g\n\u22a2 p \u2223 Multiset.prod (p ::\u2098 f)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\np : \u03b1\nf : Multiset \u03b1\nih :\n \u2200 {g : Multiset \u03b1},\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Prime x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Prime x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\ng : Multiset \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 p ::\u2098 f \u2192 Prime x\nhg : \u2200 (x : \u03b1), x \u2208 g \u2192 Prime x\nhfg : Multiset.prod (p ::\u2098 f) ~\u1d64 Multiset.prod g\nb : \u03b1\nhbg : b \u2208 g\nhb : p ~\u1d64 b\n\u22a2 Multiset.Rel Associated (p ::\u2098 f) g\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b1\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\np : \u03b1\nf : Multiset \u03b1\nih :\n \u2200 {g : Multiset \u03b1},\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Prime x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Prime x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\ng : Multiset \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 p ::\u2098 f \u2192 Prime x\nhg : \u2200 (x : \u03b1), x \u2208 g \u2192 Prime x\nhfg : Multiset.prod (p ::\u2098 f) ~\u1d64 Multiset.prod g\nb : \u03b1\nhbg : b \u2208 g\nhb : p ~\u1d64 b\nthis : DecidableEq \u03b1\n\u22a2 Multiset.Rel Associated (p ::\u2098 f) g\n[PROOFSTEP]\nrw [\u2190 Multiset.cons_erase hbg]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\np : \u03b1\nf : Multiset \u03b1\nih :\n \u2200 {g : Multiset \u03b1},\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Prime x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Prime x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\ng : Multiset \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 p ::\u2098 f \u2192 Prime x\nhg : \u2200 (x : \u03b1), x \u2208 g \u2192 Prime x\nhfg : Multiset.prod (p ::\u2098 f) ~\u1d64 Multiset.prod g\nb : \u03b1\nhbg : b \u2208 g\nhb : p ~\u1d64 b\nthis : DecidableEq \u03b1\n\u22a2 Multiset.Rel Associated (p ::\u2098 f) (b ::\u2098 Multiset.erase g b)\n[PROOFSTEP]\nexact\n Multiset.Rel.cons hb\n (ih (fun q hq => hf _ (by simp [hq])) (fun {q} (hq : q \u2208 g.erase b) => hg q (Multiset.mem_of_mem_erase hq))\n (Associated.of_mul_left (by rwa [\u2190 Multiset.prod_cons, \u2190 Multiset.prod_cons, Multiset.cons_erase hbg]) hb\n (hf p (by simp)).ne_zero))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\np : \u03b1\nf : Multiset \u03b1\nih :\n \u2200 {g : Multiset \u03b1},\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Prime x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Prime x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\ng : Multiset \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 p ::\u2098 f \u2192 Prime x\nhg : \u2200 (x : \u03b1), x \u2208 g \u2192 Prime x\nhfg : Multiset.prod (p ::\u2098 f) ~\u1d64 Multiset.prod g\nb : \u03b1\nhbg : b \u2208 g\nhb : p ~\u1d64 b\nthis : DecidableEq \u03b1\nq : \u03b1\nhq : q \u2208 f\n\u22a2 q \u2208 p ::\u2098 f\n[PROOFSTEP]\nsimp [hq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\np : \u03b1\nf : Multiset \u03b1\nih :\n \u2200 {g : Multiset \u03b1},\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Prime x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Prime x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\ng : Multiset \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 p ::\u2098 f \u2192 Prime x\nhg : \u2200 (x : \u03b1), x \u2208 g \u2192 Prime x\nhfg : Multiset.prod (p ::\u2098 f) ~\u1d64 Multiset.prod g\nb : \u03b1\nhbg : b \u2208 g\nhb : p ~\u1d64 b\nthis : DecidableEq \u03b1\n\u22a2 p * Multiset.prod f ~\u1d64 b * Multiset.prod (Multiset.erase g b)\n[PROOFSTEP]\nrwa [\u2190 Multiset.prod_cons, \u2190 Multiset.prod_cons, Multiset.cons_erase hbg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\np : \u03b1\nf : Multiset \u03b1\nih :\n \u2200 {g : Multiset \u03b1},\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Prime x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Prime x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\ng : Multiset \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 p ::\u2098 f \u2192 Prime x\nhg : \u2200 (x : \u03b1), x \u2208 g \u2192 Prime x\nhfg : Multiset.prod (p ::\u2098 f) ~\u1d64 Multiset.prod g\nb : \u03b1\nhbg : b \u2208 g\nhb : p ~\u1d64 b\nthis : DecidableEq \u03b1\n\u22a2 p \u2208 p ::\u2098 f\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : \u03b1\nf : Multiset \u03b1\nha : Irreducible a\npfa : (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\n\u22a2 \u2203 p, a ~\u1d64 p \u2227 f = {p}\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : \u03b1\nf : Multiset \u03b1\nha : Irreducible a\npfa : (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\nthis : DecidableEq \u03b1\n\u22a2 \u2203 p, a ~\u1d64 p \u2227 f = {p}\n[PROOFSTEP]\nrefine\n @Multiset.induction_on _ (fun g => (g.prod ~\u1d64 a) \u2192 (\u2200 b \u2208 g, Prime b) \u2192 \u2203 p, a ~\u1d64 p \u2227 g = { p }) f ?_ ?_ pfa.2 pfa.1\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : \u03b1\nf : Multiset \u03b1\nha : Irreducible a\npfa : (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\nthis : DecidableEq \u03b1\n\u22a2 (fun g => Multiset.prod g ~\u1d64 a \u2192 (\u2200 (b : \u03b1), b \u2208 g \u2192 Prime b) \u2192 \u2203 p, a ~\u1d64 p \u2227 g = {p}) 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : \u03b1\nf : Multiset \u03b1\nha : Irreducible a\npfa : (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\nthis : DecidableEq \u03b1\nh : Multiset.prod 0 ~\u1d64 a\n\u22a2 (\u2200 (b : \u03b1), b \u2208 0 \u2192 Prime b) \u2192 \u2203 p, a ~\u1d64 p \u2227 0 = {p}\n[PROOFSTEP]\nexact (ha.not_unit (associated_one_iff_isUnit.1 (Associated.symm h))).elim\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : \u03b1\nf : Multiset \u03b1\nha : Irreducible a\npfa : (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\nthis : DecidableEq \u03b1\n\u22a2 \u2200 \u2983a_1 : \u03b1\u2984 {s : Multiset \u03b1},\n (fun g => Multiset.prod g ~\u1d64 a \u2192 (\u2200 (b : \u03b1), b \u2208 g \u2192 Prime b) \u2192 \u2203 p, a ~\u1d64 p \u2227 g = {p}) s \u2192\n (fun g => Multiset.prod g ~\u1d64 a \u2192 (\u2200 (b : \u03b1), b \u2208 g \u2192 Prime b) \u2192 \u2203 p, a ~\u1d64 p \u2227 g = {p}) (a_1 ::\u2098 s)\n[PROOFSTEP]\nrintro p s _ \u27e8u, hu\u27e9 hs\n[GOAL]\ncase refine_2.intro\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : \u03b1\nf : Multiset \u03b1\nha : Irreducible a\npfa : (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\nthis : DecidableEq \u03b1\np : \u03b1\ns : Multiset \u03b1\na\u271d : Multiset.prod s ~\u1d64 a \u2192 (\u2200 (b : \u03b1), b \u2208 s \u2192 Prime b) \u2192 \u2203 p, a ~\u1d64 p \u2227 s = {p}\nu : \u03b1\u02e3\nhu : Multiset.prod (p ::\u2098 s) * \u2191u = a\nhs : \u2200 (b : \u03b1), b \u2208 p ::\u2098 s \u2192 Prime b\n\u22a2 \u2203 p_1, a ~\u1d64 p_1 \u2227 p ::\u2098 s = {p_1}\n[PROOFSTEP]\nuse p\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : \u03b1\nf : Multiset \u03b1\nha : Irreducible a\npfa : (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\nthis : DecidableEq \u03b1\np : \u03b1\ns : Multiset \u03b1\na\u271d : Multiset.prod s ~\u1d64 a \u2192 (\u2200 (b : \u03b1), b \u2208 s \u2192 Prime b) \u2192 \u2203 p, a ~\u1d64 p \u2227 s = {p}\nu : \u03b1\u02e3\nhu : Multiset.prod (p ::\u2098 s) * \u2191u = a\nhs : \u2200 (b : \u03b1), b \u2208 p ::\u2098 s \u2192 Prime b\n\u22a2 a ~\u1d64 p \u2227 p ::\u2098 s = {p}\n[PROOFSTEP]\nhave hs0 : s = 0 := by\n by_contra hs0\n obtain \u27e8q, hq\u27e9 := Multiset.exists_mem_of_ne_zero hs0\n apply (hs q (by simp [hq])).2.1\n refine' (ha.isUnit_or_isUnit (_ : _ = p * \u2191u * (s.erase q).prod * _)).resolve_left _\n \u00b7 rw [mul_right_comm _ _ q, mul_assoc, \u2190 Multiset.prod_cons, Multiset.cons_erase hq, \u2190 hu, mul_comm, mul_comm p _,\n mul_assoc]\n simp\n apply mt isUnit_of_mul_isUnit_left (mt isUnit_of_mul_isUnit_left _)\n apply (hs p (Multiset.mem_cons_self _ _)).2.1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : \u03b1\nf : Multiset \u03b1\nha : Irreducible a\npfa : (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\nthis : DecidableEq \u03b1\np : \u03b1\ns : Multiset \u03b1\na\u271d : Multiset.prod s ~\u1d64 a \u2192 (\u2200 (b : \u03b1), b \u2208 s \u2192 Prime b) \u2192 \u2203 p, a ~\u1d64 p \u2227 s = {p}\nu : \u03b1\u02e3\nhu : Multiset.prod (p ::\u2098 s) * \u2191u = a\nhs : \u2200 (b : \u03b1), b \u2208 p ::\u2098 s \u2192 Prime b\n\u22a2 s = 0\n[PROOFSTEP]\nby_contra hs0\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : \u03b1\nf : Multiset \u03b1\nha : Irreducible a\npfa : (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\nthis : DecidableEq \u03b1\np : \u03b1\ns : Multiset \u03b1\na\u271d : Multiset.prod s ~\u1d64 a \u2192 (\u2200 (b : \u03b1), b \u2208 s \u2192 Prime b) \u2192 \u2203 p, a ~\u1d64 p \u2227 s = {p}\nu : \u03b1\u02e3\nhu : Multiset.prod (p ::\u2098 s) * \u2191u = a\nhs : \u2200 (b : \u03b1), b \u2208 p ::\u2098 s \u2192 Prime b\nhs0 : \u00acs = 0\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8q, hq\u27e9 := Multiset.exists_mem_of_ne_zero hs0\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : \u03b1\nf : Multiset \u03b1\nha : Irreducible a\npfa : (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\nthis : DecidableEq \u03b1\np : \u03b1\ns : Multiset \u03b1\na\u271d : Multiset.prod s ~\u1d64 a \u2192 (\u2200 (b : \u03b1), b \u2208 s \u2192 Prime b) \u2192 \u2203 p, a ~\u1d64 p \u2227 s = {p}\nu : \u03b1\u02e3\nhu : Multiset.prod (p ::\u2098 s) * \u2191u = a\nhs : \u2200 (b : \u03b1), b \u2208 p ::\u2098 s \u2192 Prime b\nhs0 : \u00acs = 0\nq : \u03b1\nhq : q \u2208 s\n\u22a2 False\n[PROOFSTEP]\napply (hs q (by simp [hq])).2.1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : \u03b1\nf : Multiset \u03b1\nha : Irreducible a\npfa : (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\nthis : DecidableEq \u03b1\np : \u03b1\ns : Multiset \u03b1\na\u271d : Multiset.prod s ~\u1d64 a \u2192 (\u2200 (b : \u03b1), b \u2208 s \u2192 Prime b) \u2192 \u2203 p, a ~\u1d64 p \u2227 s = {p}\nu : \u03b1\u02e3\nhu : Multiset.prod (p ::\u2098 s) * \u2191u = a\nhs : \u2200 (b : \u03b1), b \u2208 p ::\u2098 s \u2192 Prime b\nhs0 : \u00acs = 0\nq : \u03b1\nhq : q \u2208 s\n\u22a2 q \u2208 p ::\u2098 s\n[PROOFSTEP]\nsimp [hq]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : \u03b1\nf : Multiset \u03b1\nha : Irreducible a\npfa : (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\nthis : DecidableEq \u03b1\np : \u03b1\ns : Multiset \u03b1\na\u271d : Multiset.prod s ~\u1d64 a \u2192 (\u2200 (b : \u03b1), b \u2208 s \u2192 Prime b) \u2192 \u2203 p, a ~\u1d64 p \u2227 s = {p}\nu : \u03b1\u02e3\nhu : Multiset.prod (p ::\u2098 s) * \u2191u = a\nhs : \u2200 (b : \u03b1), b \u2208 p ::\u2098 s \u2192 Prime b\nhs0 : \u00acs = 0\nq : \u03b1\nhq : q \u2208 s\n\u22a2 IsUnit q\n[PROOFSTEP]\nrefine' (ha.isUnit_or_isUnit (_ : _ = p * \u2191u * (s.erase q).prod * _)).resolve_left _\n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : \u03b1\nf : Multiset \u03b1\nha : Irreducible a\npfa : (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\nthis : DecidableEq \u03b1\np : \u03b1\ns : Multiset \u03b1\na\u271d : Multiset.prod s ~\u1d64 a \u2192 (\u2200 (b : \u03b1), b \u2208 s \u2192 Prime b) \u2192 \u2203 p, a ~\u1d64 p \u2227 s = {p}\nu : \u03b1\u02e3\nhu : Multiset.prod (p ::\u2098 s) * \u2191u = a\nhs : \u2200 (b : \u03b1), b \u2208 p ::\u2098 s \u2192 Prime b\nhs0 : \u00acs = 0\nq : \u03b1\nhq : q \u2208 s\n\u22a2 a = p * \u2191u * Multiset.prod (Multiset.erase s q) * q\n[PROOFSTEP]\nrw [mul_right_comm _ _ q, mul_assoc, \u2190 Multiset.prod_cons, Multiset.cons_erase hq, \u2190 hu, mul_comm, mul_comm p _,\n mul_assoc]\n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : \u03b1\nf : Multiset \u03b1\nha : Irreducible a\npfa : (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\nthis : DecidableEq \u03b1\np : \u03b1\ns : Multiset \u03b1\na\u271d : Multiset.prod s ~\u1d64 a \u2192 (\u2200 (b : \u03b1), b \u2208 s \u2192 Prime b) \u2192 \u2203 p, a ~\u1d64 p \u2227 s = {p}\nu : \u03b1\u02e3\nhu : Multiset.prod (p ::\u2098 s) * \u2191u = a\nhs : \u2200 (b : \u03b1), b \u2208 p ::\u2098 s \u2192 Prime b\nhs0 : \u00acs = 0\nq : \u03b1\nhq : q \u2208 s\n\u22a2 \u2191u * Multiset.prod (p ::\u2098 s) = \u2191u * (p * Multiset.prod s)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : \u03b1\nf : Multiset \u03b1\nha : Irreducible a\npfa : (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\nthis : DecidableEq \u03b1\np : \u03b1\ns : Multiset \u03b1\na\u271d : Multiset.prod s ~\u1d64 a \u2192 (\u2200 (b : \u03b1), b \u2208 s \u2192 Prime b) \u2192 \u2203 p, a ~\u1d64 p \u2227 s = {p}\nu : \u03b1\u02e3\nhu : Multiset.prod (p ::\u2098 s) * \u2191u = a\nhs : \u2200 (b : \u03b1), b \u2208 p ::\u2098 s \u2192 Prime b\nhs0 : \u00acs = 0\nq : \u03b1\nhq : q \u2208 s\n\u22a2 \u00acIsUnit (p * \u2191u * Multiset.prod (Multiset.erase s q))\n[PROOFSTEP]\napply mt isUnit_of_mul_isUnit_left (mt isUnit_of_mul_isUnit_left _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : \u03b1\nf : Multiset \u03b1\nha : Irreducible a\npfa : (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\nthis : DecidableEq \u03b1\np : \u03b1\ns : Multiset \u03b1\na\u271d : Multiset.prod s ~\u1d64 a \u2192 (\u2200 (b : \u03b1), b \u2208 s \u2192 Prime b) \u2192 \u2203 p, a ~\u1d64 p \u2227 s = {p}\nu : \u03b1\u02e3\nhu : Multiset.prod (p ::\u2098 s) * \u2191u = a\nhs : \u2200 (b : \u03b1), b \u2208 p ::\u2098 s \u2192 Prime b\nhs0 : \u00acs = 0\nq : \u03b1\nhq : q \u2208 s\n\u22a2 \u00acIsUnit p\n[PROOFSTEP]\napply (hs p (Multiset.mem_cons_self _ _)).2.1\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : \u03b1\nf : Multiset \u03b1\nha : Irreducible a\npfa : (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\nthis : DecidableEq \u03b1\np : \u03b1\ns : Multiset \u03b1\na\u271d : Multiset.prod s ~\u1d64 a \u2192 (\u2200 (b : \u03b1), b \u2208 s \u2192 Prime b) \u2192 \u2203 p, a ~\u1d64 p \u2227 s = {p}\nu : \u03b1\u02e3\nhu : Multiset.prod (p ::\u2098 s) * \u2191u = a\nhs : \u2200 (b : \u03b1), b \u2208 p ::\u2098 s \u2192 Prime b\nhs0 : s = 0\n\u22a2 a ~\u1d64 p \u2227 p ::\u2098 s = {p}\n[PROOFSTEP]\nsimp only [mul_one, Multiset.prod_cons, Multiset.prod_zero, hs0] at *\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : \u03b1\nf : Multiset \u03b1\nha : Irreducible a\npfa : (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\nthis : DecidableEq \u03b1\np : \u03b1\ns : Multiset \u03b1\nu : \u03b1\u02e3\na\u271d : 1 ~\u1d64 a \u2192 (\u2200 (b : \u03b1), b \u2208 0 \u2192 Prime b) \u2192 \u2203 p, a ~\u1d64 p \u2227 0 = {p}\nhu : p * \u2191u = a\nhs : \u2200 (b : \u03b1), b \u2208 p ::\u2098 0 \u2192 Prime b\nhs0 : True\n\u22a2 a ~\u1d64 p \u2227 p ::\u2098 0 = {p}\n[PROOFSTEP]\nexact \u27e8Associated.symm \u27e8u, hu\u27e9, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\n\u22a2 WellFounded DvdNotUnit\n[PROOFSTEP]\nclassical\nrefine'\n RelHomClass.wellFounded (RelHom.mk _ _ : (DvdNotUnit : \u03b1 \u2192 \u03b1 \u2192 Prop) \u2192r ((\u00b7 < \u00b7) : \u2115\u221e \u2192 \u2115\u221e \u2192 Prop))\n (WithTop.wellFounded_lt Nat.lt_wfRel.wf)\n\u00b7 intro a\n by_cases h : a = 0\n \u00b7 exact \u22a4\n exact \u2191(Multiset.card (Classical.choose (pf a h)))\nrintro a b \u27e8ane0, \u27e8c, hc, b_eq\u27e9\u27e9\nrw [dif_neg ane0]\nby_cases h : b = 0\n\u00b7 simp [h, lt_top_iff_ne_top]\n\u00b7 rw [dif_neg h]\n erw [WithTop.coe_lt_coe]\n have cne0 : c \u2260 0 := by\n refine' mt (fun con => _) h\n rw [b_eq, con, mul_zero]\n calc\n Multiset.card (Classical.choose (pf a ane0)) < _ + Multiset.card (Classical.choose (pf c cne0)) :=\n lt_add_of_pos_right _ (Multiset.card_pos.mpr fun con => hc (associated_one_iff_isUnit.mp ?_))\n _ = Multiset.card (Classical.choose (pf a ane0) + Classical.choose (pf c cne0)) := (Multiset.card_add _ _).symm\n _ = Multiset.card (Classical.choose (pf b h)) :=\n Multiset.card_eq_card_of_rel (prime_factors_unique ?_ (Classical.choose_spec (pf _ h)).1 ?_)\n \u00b7 convert (Classical.choose_spec (pf c cne0)).2.symm\n rw [con, Multiset.prod_zero]\n \u00b7 intro x hadd\n rw [Multiset.mem_add] at hadd \n cases' hadd with h h <;> apply (Classical.choose_spec (pf _ _)).1 _ h <;> assumption\n \u00b7 rw [Multiset.prod_add]\n trans a * c\n \u00b7 apply Associated.mul_mul <;> apply (Classical.choose_spec (pf _ _)).2 <;> assumption\n \u00b7 rw [\u2190 b_eq]\n apply (Classical.choose_spec (pf _ _)).2.symm; assumption\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\n\u22a2 WellFounded DvdNotUnit\n[PROOFSTEP]\nrefine'\n RelHomClass.wellFounded (RelHom.mk _ _ : (DvdNotUnit : \u03b1 \u2192 \u03b1 \u2192 Prop) \u2192r ((\u00b7 < \u00b7) : \u2115\u221e \u2192 \u2115\u221e \u2192 Prop))\n (WithTop.wellFounded_lt Nat.lt_wfRel.wf)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\n\u22a2 \u03b1 \u2192 \u2115\u221e\n[PROOFSTEP]\nintro a\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na : \u03b1\n\u22a2 \u2115\u221e\n[PROOFSTEP]\nby_cases h : a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na : \u03b1\nh : a = 0\n\u22a2 \u2115\u221e\n[PROOFSTEP]\nexact \u22a4\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na : \u03b1\nh : \u00aca = 0\n\u22a2 \u2115\u221e\n[PROOFSTEP]\nexact \u2191(Multiset.card (Classical.choose (pf a h)))\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\n\u22a2 \u2200 {a b : \u03b1},\n DvdNotUnit a b \u2192\n (if h : a = 0 then \u22a4\n else \u2191(\u2191Multiset.card (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a)))) <\n if h : b = 0 then \u22a4\n else \u2191(\u2191Multiset.card (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 b)))\n[PROOFSTEP]\nrintro a b \u27e8ane0, \u27e8c, hc, b_eq\u27e9\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\n\u22a2 (if h : a = 0 then \u22a4\n else \u2191(\u2191Multiset.card (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a)))) <\n if h : b = 0 then \u22a4\n else \u2191(\u2191Multiset.card (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 b)))\n[PROOFSTEP]\nrw [dif_neg ane0]\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\n\u22a2 \u2191(\u2191Multiset.card (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a))) <\n if h : b = 0 then \u22a4\n else \u2191(\u2191Multiset.card (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 b)))\n[PROOFSTEP]\nby_cases h : b = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : b = 0\n\u22a2 \u2191(\u2191Multiset.card (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a))) <\n if h : b = 0 then \u22a4\n else \u2191(\u2191Multiset.card (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 b)))\n[PROOFSTEP]\nsimp [h, lt_top_iff_ne_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\n\u22a2 \u2191(\u2191Multiset.card (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a))) <\n if h : b = 0 then \u22a4\n else \u2191(\u2191Multiset.card (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 b)))\n[PROOFSTEP]\nrw [dif_neg h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\n\u22a2 \u2191(\u2191Multiset.card (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a))) <\n \u2191(\u2191Multiset.card (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 b)))\n[PROOFSTEP]\nerw [WithTop.coe_lt_coe]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\n\u22a2 \u2191(\u2191Multiset.card (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a))) <\n \u2191(\u2191Multiset.card (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 b)))\n[PROOFSTEP]\nhave cne0 : c \u2260 0 := by\n refine' mt (fun con => _) h\n rw [b_eq, con, mul_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\n\u22a2 c \u2260 0\n[PROOFSTEP]\nrefine' mt (fun con => _) h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\ncon : c = 0\n\u22a2 b = 0\n[PROOFSTEP]\nrw [b_eq, con, mul_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\ncne0 : c \u2260 0\n\u22a2 \u2191(\u2191Multiset.card (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a))) <\n \u2191(\u2191Multiset.card (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 b)))\n[PROOFSTEP]\ncalc\n Multiset.card (Classical.choose (pf a ane0)) < _ + Multiset.card (Classical.choose (pf c cne0)) :=\n lt_add_of_pos_right _ (Multiset.card_pos.mpr fun con => hc (associated_one_iff_isUnit.mp ?_))\n _ = Multiset.card (Classical.choose (pf a ane0) + Classical.choose (pf c cne0)) := (Multiset.card_add _ _).symm\n _ = Multiset.card (Classical.choose (pf b h)) :=\n Multiset.card_eq_card_of_rel (prime_factors_unique ?_ (Classical.choose_spec (pf _ h)).1 ?_)\n[GOAL]\ncase neg.calc_1\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\ncne0 : c \u2260 0\ncon : Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 c) = 0\n\u22a2 c ~\u1d64 1\n[PROOFSTEP]\nconvert (Classical.choose_spec (pf c cne0)).2.symm\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\ncne0 : c \u2260 0\ncon : Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 c) = 0\n\u22a2 1 = Multiset.prod (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 c))\n[PROOFSTEP]\nrw [con, Multiset.prod_zero]\n[GOAL]\ncase neg.calc_2\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\ncne0 : c \u2260 0\n\u22a2 \u2200 (x : \u03b1),\n x \u2208\n Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a) +\n Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 c) \u2192\n Prime x\n[PROOFSTEP]\nintro x hadd\n[GOAL]\ncase neg.calc_2\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\ncne0 : c \u2260 0\nx : \u03b1\nhadd :\n x \u2208\n Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a) +\n Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 c)\n\u22a2 Prime x\n[PROOFSTEP]\nrw [Multiset.mem_add] at hadd \n[GOAL]\ncase neg.calc_2\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\ncne0 : c \u2260 0\nx : \u03b1\nhadd :\n x \u2208 Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a) \u2228\n x \u2208 Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 c)\n\u22a2 Prime x\n[PROOFSTEP]\ncases' hadd with h h\n[GOAL]\ncase neg.calc_2.inl\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh\u271d : \u00acb = 0\ncne0 : c \u2260 0\nx : \u03b1\nh : x \u2208 Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a)\n\u22a2 Prime x\n[PROOFSTEP]\napply (Classical.choose_spec (pf _ _)).1 _ h\n[GOAL]\ncase neg.calc_2.inr\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh\u271d : \u00acb = 0\ncne0 : c \u2260 0\nx : \u03b1\nh : x \u2208 Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 c)\n\u22a2 Prime x\n[PROOFSTEP]\napply (Classical.choose_spec (pf _ _)).1 _ h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh\u271d : \u00acb = 0\ncne0 : c \u2260 0\nx : \u03b1\nh : x \u2208 Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a)\n\u22a2 a \u2260 0\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh\u271d : \u00acb = 0\ncne0 : c \u2260 0\nx : \u03b1\nh : x \u2208 Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 c)\n\u22a2 c \u2260 0\n[PROOFSTEP]\nassumption\n[GOAL]\ncase neg.calc_3\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\ncne0 : c \u2260 0\n\u22a2 Multiset.prod\n (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a) +\n Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 c)) ~\u1d64\n Multiset.prod (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 b))\n[PROOFSTEP]\nrw [Multiset.prod_add]\n[GOAL]\ncase neg.calc_3\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\ncne0 : c \u2260 0\n\u22a2 Multiset.prod (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a)) *\n Multiset.prod (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 c)) ~\u1d64\n Multiset.prod (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 b))\n[PROOFSTEP]\ntrans a * c\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\ncne0 : c \u2260 0\n\u22a2 Multiset.prod (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a)) *\n Multiset.prod (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 c)) ~\u1d64\n a * c\n[PROOFSTEP]\napply Associated.mul_mul\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\ncne0 : c \u2260 0\n\u22a2 Multiset.prod (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a)) ~\u1d64 a\n[PROOFSTEP]\napply (Classical.choose_spec (pf _ _)).2\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\ncne0 : c \u2260 0\n\u22a2 Multiset.prod (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 c)) ~\u1d64 c\n[PROOFSTEP]\napply (Classical.choose_spec (pf _ _)).2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\ncne0 : c \u2260 0\n\u22a2 a \u2260 0\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\ncne0 : c \u2260 0\n\u22a2 c \u2260 0\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\ncne0 : c \u2260 0\n\u22a2 a * c ~\u1d64 Multiset.prod (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 b))\n[PROOFSTEP]\nrw [\u2190 b_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\ncne0 : c \u2260 0\n\u22a2 b ~\u1d64 Multiset.prod (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 b))\n[PROOFSTEP]\napply (Classical.choose_spec (pf _ _)).2.symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na b : \u03b1\nane0 : a \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nb_eq : b = a * c\nh : \u00acb = 0\ncne0 : c \u2260 0\n\u22a2 b \u2260 0\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\np : \u03b1\n\u22a2 Irreducible p \u2194 Prime p\n[PROOFSTEP]\nby_cases hp0 : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\np : \u03b1\nhp0 : p = 0\n\u22a2 Irreducible p \u2194 Prime p\n[PROOFSTEP]\nsimp [hp0]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\np : \u03b1\nhp0 : \u00acp = 0\n\u22a2 Irreducible p \u2194 Prime p\n[PROOFSTEP]\nrefine' \u27e8fun h => _, Prime.irreducible\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\np : \u03b1\nhp0 : \u00acp = 0\nh : Irreducible p\n\u22a2 Prime p\n[PROOFSTEP]\nobtain \u27e8f, hf\u27e9 := pf p hp0\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\np : \u03b1\nhp0 : \u00acp = 0\nh : Irreducible p\nf : Multiset \u03b1\nhf : (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 p\n\u22a2 Prime p\n[PROOFSTEP]\nobtain \u27e8q, hq, rfl\u27e9 := prime_factors_irreducible h hf\n[GOAL]\ncase neg.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\np : \u03b1\nhp0 : \u00acp = 0\nh : Irreducible p\nq : \u03b1\nhq : p ~\u1d64 q\nhf : (\u2200 (b : \u03b1), b \u2208 {q} \u2192 Prime b) \u2227 Multiset.prod {q} ~\u1d64 p\n\u22a2 Prime p\n[PROOFSTEP]\nrw [hq.prime_iff]\n[GOAL]\ncase neg.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\npf : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\np : \u03b1\nhp0 : \u00acp = 0\nh : Irreducible p\nq : \u03b1\nhq : p ~\u1d64 q\nhf : (\u2200 (b : \u03b1), b \u2208 {q} \u2192 Prime b) \u2227 Multiset.prod {q} ~\u1d64 p\n\u22a2 Prime q\n[PROOFSTEP]\nexact hf.1 q (Multiset.mem_singleton_self _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : CancelCommMonoidWithZero \u03b2\ne : \u03b1 \u2243* \u03b2\nh\u03b1 : UniqueFactorizationMonoid \u03b1\n\u22a2 UniqueFactorizationMonoid \u03b2\n[PROOFSTEP]\nrw [UniqueFactorizationMonoid.iff_exists_prime_factors] at h\u03b1 \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : CancelCommMonoidWithZero \u03b2\ne : \u03b1 \u2243* \u03b2\nh\u03b1 : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\n\u22a2 \u2200 (a : \u03b2), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b2), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\n[PROOFSTEP]\nintro a ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : CancelCommMonoidWithZero \u03b2\ne : \u03b1 \u2243* \u03b2\nh\u03b1 : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na : \u03b2\nha : a \u2260 0\n\u22a2 \u2203 f, (\u2200 (b : \u03b2), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\n[PROOFSTEP]\nobtain \u27e8w, hp, u, h\u27e9 :=\n h\u03b1 (e.symm a) fun h =>\n ha <| by\n convert \u2190 map_zero e\n simp [\u2190 h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : CancelCommMonoidWithZero \u03b2\ne : \u03b1 \u2243* \u03b2\nh\u03b1 : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na : \u03b2\nha : a \u2260 0\nh : \u2191(symm e) a = 0\n\u22a2 a = 0\n[PROOFSTEP]\nconvert \u2190 map_zero e\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : CancelCommMonoidWithZero \u03b2\ne : \u03b1 \u2243* \u03b2\nh\u03b1 : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na : \u03b2\nha : a \u2260 0\nh : \u2191(symm e) a = 0\n\u22a2 \u2191e 0 = a\n[PROOFSTEP]\nsimp [\u2190 h]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : CancelCommMonoidWithZero \u03b2\ne : \u03b1 \u2243* \u03b2\nh\u03b1 : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na : \u03b2\nha : a \u2260 0\nw : Multiset \u03b1\nhp : \u2200 (b : \u03b1), b \u2208 w \u2192 Prime b\nu : \u03b1\u02e3\nh : Multiset.prod w * \u2191u = \u2191(symm e) a\n\u22a2 \u2203 f, (\u2200 (b : \u03b2), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\n[PROOFSTEP]\nexact\n \u27e8w.map e, fun b hb =>\n let \u27e8c, hc, he\u27e9 := Multiset.mem_map.1 hb\n he \u25b8 e.prime_iff.1 (hp c hc),\n Units.map e.toMonoidHom u, by\n erw [Multiset.prod_hom, \u2190 e.map_mul, h]\n simp\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : CancelCommMonoidWithZero \u03b2\ne : \u03b1 \u2243* \u03b2\nh\u03b1 : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na : \u03b2\nha : a \u2260 0\nw : Multiset \u03b1\nhp : \u2200 (b : \u03b1), b \u2208 w \u2192 Prime b\nu : \u03b1\u02e3\nh : Multiset.prod w * \u2191u = \u2191(symm e) a\n\u22a2 Multiset.prod (Multiset.map (\u2191e) w) * \u2191(\u2191(Units.map (toMonoidHom e)) u) = a\n[PROOFSTEP]\nerw [Multiset.prod_hom, \u2190 e.map_mul, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : CancelCommMonoidWithZero \u03b2\ne : \u03b1 \u2243* \u03b2\nh\u03b1 : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\na : \u03b2\nha : a \u2260 0\nw : Multiset \u03b1\nhp : \u2200 (b : \u03b1), b \u2208 w \u2192 Prime b\nu : \u03b1\u02e3\nh : Multiset.prod w * \u2191u = \u2191(symm e) a\n\u22a2 \u2191e (\u2191(symm e) a) = a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : a * b = 0\nha0 : a = 0\n\u22a2 p \u2223 a \u2228 p \u2223 b\n[PROOFSTEP]\nsimp [ha0]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : a * b = 0\nhb0 : b = 0\n\u22a2 p \u2223 a \u2228 p \u2223 b\n[PROOFSTEP]\nsimp [hb0]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\n\u22a2 p \u2223 a \u2228 p \u2223 b\n[PROOFSTEP]\nhave hx0 : x \u2260 0 := fun hx0 => by simp_all\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x = 0\n\u22a2 False\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\n\u22a2 p \u2223 a \u2228 p \u2223 b\n[PROOFSTEP]\nhave ha0 : a \u2260 0 := left_ne_zero_of_mul hab0\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\n\u22a2 p \u2223 a \u2228 p \u2223 b\n[PROOFSTEP]\nhave hb0 : b \u2260 0 := right_ne_zero_of_mul hab0\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\n\u22a2 p \u2223 a \u2228 p \u2223 b\n[PROOFSTEP]\ncases' eif x hx0 with fx hfx\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\n\u22a2 p \u2223 a \u2228 p \u2223 b\n[PROOFSTEP]\ncases' eif a ha0 with fa hfa\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\n\u22a2 p \u2223 a \u2228 p \u2223 b\n[PROOFSTEP]\ncases' eif b hb0 with fb hfb\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\n\u22a2 p \u2223 a \u2228 p \u2223 b\n[PROOFSTEP]\nhave h : Multiset.Rel Associated (p ::\u2098 fx) (fa + fb) :=\n by\n apply uif\n \u00b7 exact fun i hi => (Multiset.mem_cons.1 hi).elim (fun hip => hip.symm \u25b8 hpi) (hfx.1 _)\n \u00b7 exact fun i hi => (Multiset.mem_add.1 hi).elim (hfa.1 _) (hfb.1 _)\n calc\n Multiset.prod (p ::\u2098 fx) ~\u1d64 a * b := by rw [hx, Multiset.prod_cons]; exact hfx.2.mul_left _\n _ ~\u1d64 fa.prod * fb.prod := (hfa.2.symm.mul_mul hfb.2.symm)\n _ = _ := by rw [Multiset.prod_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\n\u22a2 Multiset.Rel Associated (p ::\u2098 fx) (fa + fb)\n[PROOFSTEP]\napply uif\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\n\u22a2 \u2200 (x : \u03b1), x \u2208 p ::\u2098 fx \u2192 Irreducible x\n[PROOFSTEP]\nexact fun i hi => (Multiset.mem_cons.1 hi).elim (fun hip => hip.symm \u25b8 hpi) (hfx.1 _)\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\n\u22a2 \u2200 (x : \u03b1), x \u2208 fa + fb \u2192 Irreducible x\n[PROOFSTEP]\nexact fun i hi => (Multiset.mem_add.1 hi).elim (hfa.1 _) (hfb.1 _)\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\n\u22a2 Multiset.prod (p ::\u2098 fx) ~\u1d64 Multiset.prod (fa + fb)\n[PROOFSTEP]\ncalc\n Multiset.prod (p ::\u2098 fx) ~\u1d64 a * b := by rw [hx, Multiset.prod_cons]; exact hfx.2.mul_left _\n _ ~\u1d64 fa.prod * fb.prod := (hfa.2.symm.mul_mul hfb.2.symm)\n _ = _ := by rw [Multiset.prod_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\n\u22a2 Multiset.prod (p ::\u2098 fx) ~\u1d64 a * b\n[PROOFSTEP]\nrw [hx, Multiset.prod_cons]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\n\u22a2 p * Multiset.prod fx ~\u1d64 p * x\n[PROOFSTEP]\nexact hfx.2.mul_left _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\n\u22a2 Multiset.prod fa * Multiset.prod fb = Multiset.prod (fa + fb)\n[PROOFSTEP]\nrw [Multiset.prod_add]\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\nh : Multiset.Rel Associated (p ::\u2098 fx) (fa + fb)\n\u22a2 p \u2223 a \u2228 p \u2223 b\n[PROOFSTEP]\nexact\n let \u27e8q, hqf, hq\u27e9 := Multiset.exists_mem_of_rel_of_mem h (Multiset.mem_cons_self p _)\n (Multiset.mem_add.1 hqf).elim\n (fun hqa => Or.inl <| hq.dvd_iff_dvd_left.2 <| hfa.2.dvd_iff_dvd_right.1 (Multiset.dvd_prod hqa)) fun hqb =>\n Or.inr <| hq.dvd_iff_dvd_left.2 <| hfb.2.dvd_iff_dvd_right.1 (Multiset.dvd_prod hqb)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\n\u22a2 \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a\n[PROOFSTEP]\nconvert eif using 7\n[GOAL]\ncase h.h'.h.e'_2.h.h.e'_1.h.h'.a\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\na\u271d\u00b3 : \u03b1\na\u271d\u00b2 : a\u271d\u00b3 \u2260 0\nx\u271d : Multiset \u03b1\na\u271d\u00b9 : \u03b1\na\u271d : a\u271d\u00b9 \u2208 x\u271d\n\u22a2 Prime a\u271d\u00b9 \u2194 Irreducible a\u271d\u00b9\n[PROOFSTEP]\nsimp_rw [irreducible_iff_prime_of_exists_unique_irreducible_factors eif uif]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nane0 : a \u2260 0\n\u22a2 Multiset.prod (factors a) ~\u1d64 a\n[PROOFSTEP]\nrw [factors, dif_neg ane0]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nane0 : a \u2260 0\n\u22a2 Multiset.prod (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a)) ~\u1d64 a\n[PROOFSTEP]\nexact (Classical.choose_spec (exists_prime_factors a ane0)).2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\np a : \u03b1\nh : p \u2208 factors a\n\u22a2 a \u2260 0\n[PROOFSTEP]\nintro ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\np a : \u03b1\nh : p \u2208 factors a\nha : a = 0\n\u22a2 False\n[PROOFSTEP]\nrw [factors, dif_pos ha] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\np a : \u03b1\nh : p \u2208 0\nha : a = 0\n\u22a2 False\n[PROOFSTEP]\nexact Multiset.not_mem_zero _ h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na x : \u03b1\nhx : x \u2208 factors a\n\u22a2 Prime x\n[PROOFSTEP]\nhave ane0 := ne_zero_of_mem_factors hx\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na x : \u03b1\nhx : x \u2208 factors a\nane0 : a \u2260 0\n\u22a2 Prime x\n[PROOFSTEP]\nrw [factors, dif_neg ane0] at hx \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na x : \u03b1\nane0 : a \u2260 0\nhx : x \u2208 Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a)\n\u22a2 Prime x\n[PROOFSTEP]\nexact (Classical.choose_spec (UniqueFactorizationMonoid.exists_prime_factors a ane0)).1 x hx\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\n\u22a2 factors 0 = 0\n[PROOFSTEP]\nsimp [factors]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\n\u22a2 factors 1 = 0\n[PROOFSTEP]\nnontriviality \u03b1 using factors\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\n\u271d : Nontrivial \u03b1\n\u22a2 factors 1 = 0\n[PROOFSTEP]\nrw [\u2190 Multiset.rel_zero_right]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\n\u271d : Nontrivial \u03b1\n\u22a2 Multiset.Rel ?m.413739 (factors 1) 0\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\n\u271d : Nontrivial \u03b1\n\u22a2 Type ?u.413736\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\n\u271d : Nontrivial \u03b1\n\u22a2 \u03b1 \u2192 ?m.413738 \u2192 Prop\n[PROOFSTEP]\nrefine' factors_unique irreducible_of_factor (fun x hx => (Multiset.not_mem_zero x hx).elim) _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\n\u271d : Nontrivial \u03b1\n\u22a2 Multiset.prod (factors 1) ~\u1d64 Multiset.prod 0\n[PROOFSTEP]\nrw [Multiset.prod_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\n\u271d : Nontrivial \u03b1\n\u22a2 Multiset.prod (factors 1) ~\u1d64 1\n[PROOFSTEP]\nexact factors_prod one_ne_zero\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\nx\u271d : p \u2223 a\nb : \u03b1\nhb : a = p * b\nhb0 : b = 0\n\u22a2 False\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\nx\u271d : p \u2223 a\nb : \u03b1\nhb : a = p * b\nhb0 : b \u2260 0\n\u22a2 p * b ~\u1d64 Multiset.prod (p ::\u2098 factors b)\n[PROOFSTEP]\nrw [Multiset.prod_cons]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\nx\u271d : p \u2223 a\nb : \u03b1\nhb : a = p * b\nhb0 : b \u2260 0\n\u22a2 p * b ~\u1d64 p * Multiset.prod (factors b)\n[PROOFSTEP]\nexact (factors_prod hb0).symm.mul_left _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\nx\u271d : p \u2223 a\nb : \u03b1\nhb : a = p * b\nhb0 : b \u2260 0\nthis : Multiset.Rel Associated (p ::\u2098 factors b) (factors a)\n\u22a2 p \u2208 p ::\u2098 factors b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx : x \u2260 0\nh : \u00acIsUnit x\n\u22a2 \u2203 p, p \u2208 factors x\n[PROOFSTEP]\nobtain \u27e8p', hp', hp'x\u27e9 := WfDvdMonoid.exists_irreducible_factor h hx\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx : x \u2260 0\nh : \u00acIsUnit x\np' : \u03b1\nhp' : Irreducible p'\nhp'x : p' \u2223 x\n\u22a2 \u2203 p, p \u2208 factors x\n[PROOFSTEP]\nobtain \u27e8p, hp, _\u27e9 := exists_mem_factors_of_dvd hx hp' hp'x\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx : x \u2260 0\nh : \u00acIsUnit x\np' : \u03b1\nhp' : Irreducible p'\nhp'x : p' \u2223 x\np : \u03b1\nhp : p \u2208 factors x\nright\u271d : p' ~\u1d64 p\n\u22a2 \u2203 p, p \u2208 factors x\n[PROOFSTEP]\nexact \u27e8p, hp\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 Multiset.Rel Associated (factors (x * y)) (factors x + factors y)\n[PROOFSTEP]\nrefine'\n factors_unique irreducible_of_factor\n (fun a ha => (Multiset.mem_add.mp ha).by_cases (irreducible_of_factor _) (irreducible_of_factor _))\n ((factors_prod (mul_ne_zero hx hy)).trans _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 x * y ~\u1d64 Multiset.prod (factors x + factors y)\n[PROOFSTEP]\nrw [Multiset.prod_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 x * y ~\u1d64 Multiset.prod (factors x) * Multiset.prod (factors y)\n[PROOFSTEP]\nexact (Associated.mul_mul (factors_prod hx) (factors_prod hy)).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nn : \u2115\n\u22a2 Multiset.Rel Associated (factors (x ^ n)) (n \u2022 factors x)\n[PROOFSTEP]\nmatch n with\n| 0 => rw [zero_smul, pow_zero, factors_one, Multiset.rel_zero_right]\n| n + 1 =>\n \u00b7 by_cases h0 : x = 0\n \u00b7 simp [h0, zero_pow n.succ_pos, smul_zero]\n \u00b7 rw [pow_succ, succ_nsmul]\n refine' Multiset.Rel.trans _ (factors_mul h0 (pow_ne_zero n h0)) _\n refine' Multiset.Rel.add _ <| factors_pow n\n exact Multiset.rel_refl_of_refl_on fun y _ => Associated.refl _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nn : \u2115\n\u22a2 Multiset.Rel Associated (factors (x ^ 0)) (0 \u2022 factors x)\n[PROOFSTEP]\nrw [zero_smul, pow_zero, factors_one, Multiset.rel_zero_right]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nn\u271d n : \u2115\n\u22a2 Multiset.Rel Associated (factors (x ^ (n + 1))) ((n + 1) \u2022 factors x)\n[PROOFSTEP]\nby_cases h0 : x = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nn\u271d n : \u2115\nh0 : x = 0\n\u22a2 Multiset.Rel Associated (factors (x ^ (n + 1))) ((n + 1) \u2022 factors x)\n[PROOFSTEP]\nsimp [h0, zero_pow n.succ_pos, smul_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nn\u271d n : \u2115\nh0 : \u00acx = 0\n\u22a2 Multiset.Rel Associated (factors (x ^ (n + 1))) ((n + 1) \u2022 factors x)\n[PROOFSTEP]\nrw [pow_succ, succ_nsmul]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nn\u271d n : \u2115\nh0 : \u00acx = 0\n\u22a2 Multiset.Rel Associated (factors (x * x ^ n)) (factors x + n \u2022 factors x)\n[PROOFSTEP]\nrefine' Multiset.Rel.trans _ (factors_mul h0 (pow_ne_zero n h0)) _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nn\u271d n : \u2115\nh0 : \u00acx = 0\n\u22a2 Multiset.Rel Associated (factors x + factors (x ^ n)) (factors x + n \u2022 factors x)\n[PROOFSTEP]\nrefine' Multiset.Rel.add _ <| factors_pow n\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nn\u271d n : \u2115\nh0 : \u00acx = 0\n\u22a2 Multiset.Rel Associated (factors x) (factors x)\n[PROOFSTEP]\nexact Multiset.rel_refl_of_refl_on fun y _ => Associated.refl _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx : x \u2260 0\n\u22a2 0 < factors x \u2194 \u00acIsUnit x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx : x \u2260 0\n\u22a2 0 < factors x \u2192 \u00acIsUnit x\n[PROOFSTEP]\nintro h hx\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx\u271d : x \u2260 0\nh : 0 < factors x\nhx : IsUnit x\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8p, hp\u27e9 := Multiset.exists_mem_of_ne_zero h.ne'\n[GOAL]\ncase mp.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx\u271d : x \u2260 0\nh : 0 < factors x\nhx : IsUnit x\np : \u03b1\nhp : p \u2208 factors x\n\u22a2 False\n[PROOFSTEP]\nexact (prime_of_factor _ hp).not_unit (isUnit_of_dvd_unit (dvd_of_mem_factors hp) hx)\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx : x \u2260 0\n\u22a2 \u00acIsUnit x \u2192 0 < factors x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx : x \u2260 0\nh : \u00acIsUnit x\n\u22a2 0 < factors x\n[PROOFSTEP]\nobtain \u27e8p, hp\u27e9 := exists_mem_factors hx h\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx : x \u2260 0\nh : \u00acIsUnit x\np : \u03b1\nhp : p \u2208 factors x\n\u22a2 0 < factors x\n[PROOFSTEP]\nexact bot_lt_iff_ne_bot.mpr (mt Multiset.eq_zero_iff_forall_not_mem.mp (not_forall.mpr \u27e8p, not_not.mpr hp\u27e9))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u2076 : DecidableEq \u03b1\ninst\u271d\u2075 : NormalizationMonoid \u03b1\ninst\u271d\u2074 : UniqueFactorizationMonoid \u03b1\nM : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero M\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Unique M\u02e3\nx : M\n\u22a2 factors x = normalizedFactors x\n[PROOFSTEP]\nunfold normalizedFactors\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u2076 : DecidableEq \u03b1\ninst\u271d\u2075 : NormalizationMonoid \u03b1\ninst\u271d\u2074 : UniqueFactorizationMonoid \u03b1\nM : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero M\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Unique M\u02e3\nx : M\n\u22a2 factors x = Multiset.map (\u2191normalize) (factors x)\n[PROOFSTEP]\nconvert (Multiset.map_id (factors x)).symm\n[GOAL]\ncase h.e'_3.a.h.e\n\u03b1 : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u2076 : DecidableEq \u03b1\ninst\u271d\u2075 : NormalizationMonoid \u03b1\ninst\u271d\u2074 : UniqueFactorizationMonoid \u03b1\nM : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero M\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Unique M\u02e3\nx x\u271d : M\na\u271d : x\u271d \u2208 factors x\n\u22a2 \u2191normalize = id\n[PROOFSTEP]\next p\n[GOAL]\ncase h.e'_3.a.h.e.h\n\u03b1 : Type u_1\ninst\u271d\u2077 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u2076 : DecidableEq \u03b1\ninst\u271d\u2075 : NormalizationMonoid \u03b1\ninst\u271d\u2074 : UniqueFactorizationMonoid \u03b1\nM : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero M\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Unique M\u02e3\nx x\u271d : M\na\u271d : x\u271d \u2208 factors x\np : M\n\u22a2 \u2191normalize p = id p\n[PROOFSTEP]\nexact normalize_eq p\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nane0 : a \u2260 0\n\u22a2 Multiset.prod (normalizedFactors a) ~\u1d64 a\n[PROOFSTEP]\nrw [normalizedFactors, factors, dif_neg ane0]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nane0 : a \u2260 0\n\u22a2 Multiset.prod\n (Multiset.map (\u2191normalize) (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a))) ~\u1d64\n a\n[PROOFSTEP]\nrefine' Associated.trans _ (Classical.choose_spec (exists_prime_factors a ane0)).2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nane0 : a \u2260 0\n\u22a2 Multiset.prod\n (Multiset.map (\u2191normalize) (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a))) ~\u1d64\n Multiset.prod (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a))\n[PROOFSTEP]\nrw [\u2190 Associates.mk_eq_mk_iff_associated, \u2190 Associates.prod_mk, \u2190 Associates.prod_mk, Multiset.map_map]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nane0 : a \u2260 0\n\u22a2 Multiset.prod\n (Multiset.map (Associates.mk \u2218 \u2191normalize)\n (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a))) =\n Multiset.prod\n (Multiset.map Associates.mk (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a)))\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_a.e_f\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nane0 : a \u2260 0\n\u22a2 Associates.mk \u2218 \u2191normalize = Associates.mk\n[PROOFSTEP]\next\n[GOAL]\ncase e_a.e_f.h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nane0 : a \u2260 0\nx\u271d : \u03b1\n\u22a2 (Associates.mk \u2218 \u2191normalize) x\u271d = Associates.mk x\u271d\n[PROOFSTEP]\nrw [Function.comp_apply, Associates.mk_normalize]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\n\u22a2 \u2200 (x : \u03b1), x \u2208 normalizedFactors a \u2192 Prime x\n[PROOFSTEP]\nrw [normalizedFactors, factors]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\n\u22a2 \u2200 (x : \u03b1),\n x \u2208\n Multiset.map (\u2191normalize)\n (if h : a = 0 then 0 else Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a)) \u2192\n Prime x\n[PROOFSTEP]\nsplit_ifs with ane0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nane0 : a = 0\n\u22a2 \u2200 (x : \u03b1), x \u2208 Multiset.map (fun x => \u2191normalize x) 0 \u2192 Prime x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nane0 : \u00aca = 0\n\u22a2 \u2200 (x : \u03b1),\n x \u2208\n Multiset.map (fun x => \u2191normalize x)\n (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a)) \u2192\n Prime x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nane0 : \u00aca = 0\nx : \u03b1\nhx :\n x \u2208\n Multiset.map (fun x => \u2191normalize x)\n (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a))\n\u22a2 Prime x\n[PROOFSTEP]\nrcases Multiset.mem_map.1 hx with \u27e8y, \u27e8hy, rfl\u27e9\u27e9\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nane0 : \u00aca = 0\ny : \u03b1\nhy : y \u2208 Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a)\nhx :\n \u2191normalize y \u2208\n Multiset.map (fun x => \u2191normalize x)\n (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a))\n\u22a2 Prime (\u2191normalize y)\n[PROOFSTEP]\nrw [(normalize_associated _).prime_iff]\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nane0 : \u00aca = 0\ny : \u03b1\nhy : y \u2208 Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a)\nhx :\n \u2191normalize y \u2208\n Multiset.map (fun x => \u2191normalize x)\n (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a))\n\u22a2 Prime y\n[PROOFSTEP]\nexact (Classical.choose_spec (UniqueFactorizationMonoid.exists_prime_factors a ane0)).1 y hy\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\n\u22a2 \u2200 (x : \u03b1), x \u2208 normalizedFactors a \u2192 \u2191normalize x = x\n[PROOFSTEP]\nrw [normalizedFactors, factors]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\n\u22a2 \u2200 (x : \u03b1),\n x \u2208\n Multiset.map (\u2191normalize)\n (if h : a = 0 then 0 else Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a)) \u2192\n \u2191normalize x = x\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nh : a = 0\n\u22a2 \u2200 (x : \u03b1), x \u2208 Multiset.map (fun x => \u2191normalize x) 0 \u2192 \u2191normalize x = x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nh : \u00aca = 0\n\u22a2 \u2200 (x : \u03b1),\n x \u2208\n Multiset.map (fun x => \u2191normalize x)\n (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a)) \u2192\n \u2191normalize x = x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nh : \u00aca = 0\nx : \u03b1\nhx :\n x \u2208\n Multiset.map (fun x => \u2191normalize x)\n (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a))\n\u22a2 \u2191normalize x = x\n[PROOFSTEP]\nobtain \u27e8y, _, rfl\u27e9 := Multiset.mem_map.1 hx\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nh : \u00aca = 0\ny : \u03b1\nleft\u271d : y \u2208 Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a)\nhx :\n \u2191normalize y \u2208\n Multiset.map (fun x => \u2191normalize x)\n (Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 a))\n\u22a2 \u2191normalize (\u2191normalize y) = \u2191normalize y\n[PROOFSTEP]\napply normalize_idem\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nha : Irreducible a\n\u22a2 normalizedFactors a = {\u2191normalize a}\n[PROOFSTEP]\nobtain \u27e8p, a_assoc, hp\u27e9 := prime_factors_irreducible ha \u27e8prime_of_normalized_factor, normalizedFactors_prod ha.ne_zero\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nha : Irreducible a\np : \u03b1\na_assoc : a ~\u1d64 p\nhp : normalizedFactors a = {p}\n\u22a2 normalizedFactors a = {\u2191normalize a}\n[PROOFSTEP]\nhave p_mem : p \u2208 normalizedFactors a := by\n rw [hp]\n exact Multiset.mem_singleton_self _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nha : Irreducible a\np : \u03b1\na_assoc : a ~\u1d64 p\nhp : normalizedFactors a = {p}\n\u22a2 p \u2208 normalizedFactors a\n[PROOFSTEP]\nrw [hp]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nha : Irreducible a\np : \u03b1\na_assoc : a ~\u1d64 p\nhp : normalizedFactors a = {p}\n\u22a2 p \u2208 {p}\n[PROOFSTEP]\nexact Multiset.mem_singleton_self _\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nha : Irreducible a\np : \u03b1\na_assoc : a ~\u1d64 p\nhp : normalizedFactors a = {p}\np_mem : p \u2208 normalizedFactors a\n\u22a2 normalizedFactors a = {\u2191normalize a}\n[PROOFSTEP]\nconvert hp\n[GOAL]\ncase h.e'_3.h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\nha : Irreducible a\np : \u03b1\na_assoc : a ~\u1d64 p\nhp : normalizedFactors a = {p}\np_mem : p \u2208 normalizedFactors a\n\u22a2 \u2191normalize a = p\n[PROOFSTEP]\nrwa [\u2190 normalize_normalized_factor p p_mem, normalize_eq_normalize_iff, dvd_dvd_iff_associated]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na : \u03b1\n\u22a2 \u2200 (p : \u03b1), p \u2208 normalizedFactors a \u2192 \u2200 (q : \u03b1), q \u2208 normalizedFactors a \u2192 p \u2223 q \u2192 p = q\n[PROOFSTEP]\nintro p hp q hq hdvd\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na p : \u03b1\nhp : p \u2208 normalizedFactors a\nq : \u03b1\nhq : q \u2208 normalizedFactors a\nhdvd : p \u2223 q\n\u22a2 p = q\n[PROOFSTEP]\nconvert\n normalize_eq_normalize hdvd\n ((prime_of_normalized_factor _ hp).irreducible.dvd_symm (prime_of_normalized_factor _ hq).irreducible hdvd)\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na p : \u03b1\nhp : p \u2208 normalizedFactors a\nq : \u03b1\nhq : q \u2208 normalizedFactors a\nhdvd : p \u2223 q\n\u22a2 p = \u2191normalize p\n[PROOFSTEP]\napply (normalize_normalized_factor _ \u2039_\u203a).symm\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na p : \u03b1\nhp : p \u2208 normalizedFactors a\nq : \u03b1\nhq : q \u2208 normalizedFactors a\nhdvd : p \u2223 q\n\u22a2 q = \u2191normalize q\n[PROOFSTEP]\napply (normalize_normalized_factor _ \u2039_\u203a).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\nx\u271d : p \u2223 a\nb : \u03b1\nhb : a = p * b\nhb0 : b = 0\n\u22a2 False\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\nx\u271d : p \u2223 a\nb : \u03b1\nhb : a = p * b\nhb0 : b \u2260 0\n\u22a2 p * b ~\u1d64 Multiset.prod (p ::\u2098 normalizedFactors b)\n[PROOFSTEP]\nrw [Multiset.prod_cons]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\nx\u271d : p \u2223 a\nb : \u03b1\nhb : a = p * b\nhb0 : b \u2260 0\n\u22a2 p * b ~\u1d64 p * Multiset.prod (normalizedFactors b)\n[PROOFSTEP]\nexact (normalizedFactors_prod hb0).symm.mul_left _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\nx\u271d : p \u2223 a\nb : \u03b1\nhb : a = p * b\nhb0 : b \u2260 0\nthis : Multiset.Rel Associated (p ::\u2098 normalizedFactors b) (normalizedFactors a)\n\u22a2 p \u2208 p ::\u2098 normalizedFactors b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx : x \u2260 0\nh : \u00acIsUnit x\n\u22a2 \u2203 p, p \u2208 normalizedFactors x\n[PROOFSTEP]\nobtain \u27e8p', hp', hp'x\u27e9 := WfDvdMonoid.exists_irreducible_factor h hx\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx : x \u2260 0\nh : \u00acIsUnit x\np' : \u03b1\nhp' : Irreducible p'\nhp'x : p' \u2223 x\n\u22a2 \u2203 p, p \u2208 normalizedFactors x\n[PROOFSTEP]\nobtain \u27e8p, hp, _\u27e9 := exists_mem_normalizedFactors_of_dvd hx hp' hp'x\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx : x \u2260 0\nh : \u00acIsUnit x\np' : \u03b1\nhp' : Irreducible p'\nhp'x : p' \u2223 x\np : \u03b1\nhp : p \u2208 normalizedFactors x\nright\u271d : p' ~\u1d64 p\n\u22a2 \u2203 p, p \u2208 normalizedFactors x\n[PROOFSTEP]\nexact \u27e8p, hp\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\n\u22a2 normalizedFactors 0 = 0\n[PROOFSTEP]\nsimp [normalizedFactors, factors]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\n\u22a2 normalizedFactors 1 = 0\n[PROOFSTEP]\ncases' subsingleton_or_nontrivial \u03b1 with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Subsingleton \u03b1\n\u22a2 normalizedFactors 1 = 0\n[PROOFSTEP]\ndsimp [normalizedFactors, factors]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Subsingleton \u03b1\n\u22a2 Multiset.map (\u2191normalize)\n (if h : 1 = 0 then 0 else Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 1)) =\n 0\n[PROOFSTEP]\nsimp [Subsingleton.elim (1 : \u03b1) 0]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\n\u22a2 normalizedFactors 1 = 0\n[PROOFSTEP]\nrw [\u2190 Multiset.rel_zero_right]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\n\u22a2 Multiset.Rel ?m.629311 (normalizedFactors 1) 0\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\n\u22a2 Type ?u.629308\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\n\u22a2 \u03b1 \u2192 ?m.629310 \u2192 Prop\n[PROOFSTEP]\napply factors_unique irreducible_of_normalized_factor\n[GOAL]\ncase inr.hg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\n\u22a2 \u2200 (x : \u03b1), x \u2208 0 \u2192 Irreducible x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase inr.hg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\nx : \u03b1\nhx : x \u2208 0\n\u22a2 Irreducible x\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase inr.hg.h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\nx : \u03b1\nhx : x \u2208 0\n\u22a2 False\n[PROOFSTEP]\napply Multiset.not_mem_zero x hx\n[GOAL]\ncase inr.h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\n\u22a2 Multiset.prod (normalizedFactors 1) ~\u1d64 Multiset.prod 0\n[PROOFSTEP]\napply normalizedFactors_prod one_ne_zero\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 normalizedFactors (x * y) = normalizedFactors x + normalizedFactors y\n[PROOFSTEP]\nhave h : (normalize : \u03b1 \u2192 \u03b1) = Associates.out \u2218 Associates.mk :=\n by\n ext\n rw [Function.comp_apply, Associates.out_mk]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 \u2191normalize = Associates.out \u2218 Associates.mk\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nx\u271d : \u03b1\n\u22a2 \u2191normalize x\u271d = (Associates.out \u2218 Associates.mk) x\u271d\n[PROOFSTEP]\nrw [Function.comp_apply, Associates.out_mk]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 normalizedFactors (x * y) = normalizedFactors x + normalizedFactors y\n[PROOFSTEP]\nrw [\u2190 Multiset.map_id' (normalizedFactors (x * y)), \u2190 Multiset.map_id' (normalizedFactors x), \u2190\n Multiset.map_id' (normalizedFactors y), \u2190 Multiset.map_congr rfl normalize_normalized_factor, \u2190\n Multiset.map_congr rfl normalize_normalized_factor, \u2190 Multiset.map_congr rfl normalize_normalized_factor, \u2190\n Multiset.map_add, h, \u2190 Multiset.map_map Associates.out, eq_comm, \u2190 Multiset.map_map Associates.out]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 Multiset.map Associates.out (Multiset.map Associates.mk (normalizedFactors x + normalizedFactors y)) =\n Multiset.map Associates.out (Multiset.map Associates.mk (normalizedFactors (x * y)))\n[PROOFSTEP]\nrefine' congr rfl _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 Multiset.map Associates.mk (normalizedFactors x + normalizedFactors y) =\n Multiset.map Associates.mk (normalizedFactors (x * y))\n[PROOFSTEP]\napply Multiset.map_mk_eq_map_mk_of_rel\n[GOAL]\ncase hst\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 Multiset.Rel Setoid.r (normalizedFactors x + normalizedFactors y) (normalizedFactors (x * y))\n[PROOFSTEP]\napply factors_unique\n[GOAL]\ncase hst.hf\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 \u2200 (x_1 : \u03b1), x_1 \u2208 normalizedFactors x + normalizedFactors y \u2192 Irreducible x_1\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase hst.hf\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx\u271d y : \u03b1\nhx\u271d : x\u271d \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\nx : \u03b1\nhx : x \u2208 normalizedFactors x\u271d + normalizedFactors y\n\u22a2 Irreducible x\n[PROOFSTEP]\nrcases Multiset.mem_add.1 hx with (hx | hx)\n[GOAL]\ncase hst.hf.inl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx\u271d y : \u03b1\nhx\u271d\u00b9 : x\u271d \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\nx : \u03b1\nhx\u271d : x \u2208 normalizedFactors x\u271d + normalizedFactors y\nhx : x \u2208 normalizedFactors x\u271d\n\u22a2 Irreducible x\n[PROOFSTEP]\nexact irreducible_of_normalized_factor x hx\n[GOAL]\ncase hst.hf.inr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx\u271d y : \u03b1\nhx\u271d\u00b9 : x\u271d \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\nx : \u03b1\nhx\u271d : x \u2208 normalizedFactors x\u271d + normalizedFactors y\nhx : x \u2208 normalizedFactors y\n\u22a2 Irreducible x\n[PROOFSTEP]\nexact irreducible_of_normalized_factor x hx\n[GOAL]\ncase hst.hg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 \u2200 (x_1 : \u03b1), x_1 \u2208 normalizedFactors (x * y) \u2192 Irreducible x_1\n[PROOFSTEP]\nexact irreducible_of_normalized_factor\n[GOAL]\ncase hst.h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 Multiset.prod (normalizedFactors x + normalizedFactors y) ~\u1d64 Multiset.prod (normalizedFactors (x * y))\n[PROOFSTEP]\nrw [Multiset.prod_add]\n[GOAL]\ncase hst.h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 Multiset.prod (normalizedFactors x) * Multiset.prod (normalizedFactors y) ~\u1d64 Multiset.prod (normalizedFactors (x * y))\n[PROOFSTEP]\nexact\n ((normalizedFactors_prod hx).mul_mul (normalizedFactors_prod hy)).trans\n (normalizedFactors_prod (mul_ne_zero hx hy)).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nn : \u2115\n\u22a2 normalizedFactors (x ^ n) = n \u2022 normalizedFactors x\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\n\u22a2 normalizedFactors (x ^ Nat.zero) = Nat.zero \u2022 normalizedFactors x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nn : \u2115\nih : normalizedFactors (x ^ n) = n \u2022 normalizedFactors x\n\u22a2 normalizedFactors (x ^ Nat.succ n) = Nat.succ n \u2022 normalizedFactors x\n[PROOFSTEP]\nby_cases h0 : x = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nn : \u2115\nih : normalizedFactors (x ^ n) = n \u2022 normalizedFactors x\nh0 : x = 0\n\u22a2 normalizedFactors (x ^ Nat.succ n) = Nat.succ n \u2022 normalizedFactors x\n[PROOFSTEP]\nsimp [h0, zero_pow n.succ_pos, smul_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nn : \u2115\nih : normalizedFactors (x ^ n) = n \u2022 normalizedFactors x\nh0 : \u00acx = 0\n\u22a2 normalizedFactors (x ^ Nat.succ n) = Nat.succ n \u2022 normalizedFactors x\n[PROOFSTEP]\nrw [pow_succ, succ_nsmul, normalizedFactors_mul h0 (pow_ne_zero _ h0), ih]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\np : \u03b1\nhp : Irreducible p\nk : \u2115\n\u22a2 normalizedFactors (p ^ k) = Multiset.replicate k (\u2191normalize p)\n[PROOFSTEP]\nrw [UniqueFactorizationMonoid.normalizedFactors_pow, normalizedFactors_irreducible hp, Multiset.nsmul_singleton]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\ns : Multiset \u03b1\nhs : \u2200 (a : \u03b1), a \u2208 s \u2192 Irreducible a\n\u22a2 normalizedFactors (Multiset.prod s) = Multiset.map (\u2191normalize) s\n[PROOFSTEP]\ninduction' s using Multiset.induction with a s ih\n[GOAL]\ncase empty\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\ns : Multiset \u03b1\nhs\u271d : \u2200 (a : \u03b1), a \u2208 s \u2192 Irreducible a\nhs : \u2200 (a : \u03b1), a \u2208 0 \u2192 Irreducible a\n\u22a2 normalizedFactors (Multiset.prod 0) = Multiset.map (\u2191normalize) 0\n[PROOFSTEP]\nrw [Multiset.prod_zero, normalizedFactors_one, Multiset.map_zero]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\ns\u271d : Multiset \u03b1\nhs\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Irreducible a\na : \u03b1\ns : Multiset \u03b1\nih : (\u2200 (a : \u03b1), a \u2208 s \u2192 Irreducible a) \u2192 normalizedFactors (Multiset.prod s) = Multiset.map (\u2191normalize) s\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 Irreducible a_1\n\u22a2 normalizedFactors (Multiset.prod (a ::\u2098 s)) = Multiset.map (\u2191normalize) (a ::\u2098 s)\n[PROOFSTEP]\nhave ia := hs a (Multiset.mem_cons_self a _)\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\ns\u271d : Multiset \u03b1\nhs\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Irreducible a\na : \u03b1\ns : Multiset \u03b1\nih : (\u2200 (a : \u03b1), a \u2208 s \u2192 Irreducible a) \u2192 normalizedFactors (Multiset.prod s) = Multiset.map (\u2191normalize) s\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 Irreducible a_1\nia : Irreducible a\n\u22a2 normalizedFactors (Multiset.prod (a ::\u2098 s)) = Multiset.map (\u2191normalize) (a ::\u2098 s)\n[PROOFSTEP]\nhave ib := fun b h => hs b (Multiset.mem_cons_of_mem h)\n[GOAL]\ncase cons\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\ns\u271d : Multiset \u03b1\nhs\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Irreducible a\na : \u03b1\ns : Multiset \u03b1\nih : (\u2200 (a : \u03b1), a \u2208 s \u2192 Irreducible a) \u2192 normalizedFactors (Multiset.prod s) = Multiset.map (\u2191normalize) s\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 Irreducible a_1\nia : Irreducible a\nib : \u2200 (b : \u03b1), b \u2208 s \u2192 Irreducible b\n\u22a2 normalizedFactors (Multiset.prod (a ::\u2098 s)) = Multiset.map (\u2191normalize) (a ::\u2098 s)\n[PROOFSTEP]\nobtain rfl | \u27e8b, hb\u27e9 := s.empty_or_exists_mem\n[GOAL]\ncase cons.inl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\ns : Multiset \u03b1\nhs\u271d : \u2200 (a : \u03b1), a \u2208 s \u2192 Irreducible a\na : \u03b1\nia : Irreducible a\nih : (\u2200 (a : \u03b1), a \u2208 0 \u2192 Irreducible a) \u2192 normalizedFactors (Multiset.prod 0) = Multiset.map (\u2191normalize) 0\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 0 \u2192 Irreducible a_1\nib : \u2200 (b : \u03b1), b \u2208 0 \u2192 Irreducible b\n\u22a2 normalizedFactors (Multiset.prod (a ::\u2098 0)) = Multiset.map (\u2191normalize) (a ::\u2098 0)\n[PROOFSTEP]\nrw [Multiset.cons_zero, Multiset.prod_singleton, Multiset.map_singleton, normalizedFactors_irreducible ia]\n[GOAL]\ncase cons.inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\ns\u271d : Multiset \u03b1\nhs\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Irreducible a\na : \u03b1\ns : Multiset \u03b1\nih : (\u2200 (a : \u03b1), a \u2208 s \u2192 Irreducible a) \u2192 normalizedFactors (Multiset.prod s) = Multiset.map (\u2191normalize) s\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 Irreducible a_1\nia : Irreducible a\nib : \u2200 (b : \u03b1), b \u2208 s \u2192 Irreducible b\nb : \u03b1\nhb : b \u2208 s\n\u22a2 normalizedFactors (Multiset.prod (a ::\u2098 s)) = Multiset.map (\u2191normalize) (a ::\u2098 s)\n[PROOFSTEP]\nhaveI := nontrivial_of_ne b 0 (ib b hb).ne_zero\n[GOAL]\ncase cons.inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\ns\u271d : Multiset \u03b1\nhs\u271d : \u2200 (a : \u03b1), a \u2208 s\u271d \u2192 Irreducible a\na : \u03b1\ns : Multiset \u03b1\nih : (\u2200 (a : \u03b1), a \u2208 s \u2192 Irreducible a) \u2192 normalizedFactors (Multiset.prod s) = Multiset.map (\u2191normalize) s\nhs : \u2200 (a_1 : \u03b1), a_1 \u2208 a ::\u2098 s \u2192 Irreducible a_1\nia : Irreducible a\nib : \u2200 (b : \u03b1), b \u2208 s \u2192 Irreducible b\nb : \u03b1\nhb : b \u2208 s\nthis : Nontrivial \u03b1\n\u22a2 normalizedFactors (Multiset.prod (a ::\u2098 s)) = Multiset.map (\u2191normalize) (a ::\u2098 s)\n[PROOFSTEP]\nrw [Multiset.prod_cons, Multiset.map_cons,\n normalizedFactors_mul ia.ne_zero (Multiset.prod_ne_zero fun h => (ib 0 h).ne_zero rfl),\n normalizedFactors_irreducible ia, ih ib, Multiset.singleton_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 x \u2223 y \u2194 normalizedFactors x \u2264 normalizedFactors y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 x \u2223 y \u2192 normalizedFactors x \u2264 normalizedFactors y\n[PROOFSTEP]\nrintro \u27e8c, rfl\u27e9\n[GOAL]\ncase mp.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx : x \u2260 0\nc : \u03b1\nhy : x * c \u2260 0\n\u22a2 normalizedFactors x \u2264 normalizedFactors (x * c)\n[PROOFSTEP]\nsimp [hx, right_ne_zero_of_mul hy]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 normalizedFactors x \u2264 normalizedFactors y \u2192 x \u2223 y\n[PROOFSTEP]\nrw [\u2190 (normalizedFactors_prod hx).dvd_iff_dvd_left, \u2190 (normalizedFactors_prod hy).dvd_iff_dvd_right]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 normalizedFactors x \u2264 normalizedFactors y \u2192 Multiset.prod (normalizedFactors x) \u2223 Multiset.prod (normalizedFactors y)\n[PROOFSTEP]\napply Multiset.prod_dvd_prod_of_le\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 x ~\u1d64 y \u2194 normalizedFactors x = normalizedFactors y\n[PROOFSTEP]\nrefine'\n \u27e8fun h => _, fun h => (normalizedFactors_prod hx).symm.trans (_root_.trans (by rw [h]) (normalizedFactors_prod hy))\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : normalizedFactors x = normalizedFactors y\n\u22a2 Multiset.prod (normalizedFactors x) ~\u1d64 Multiset.prod (normalizedFactors y)\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : x ~\u1d64 y\n\u22a2 normalizedFactors x = normalizedFactors y\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : x ~\u1d64 y\n\u22a2 normalizedFactors x \u2264 normalizedFactors y\n[PROOFSTEP]\nrw [\u2190 dvd_iff_normalizedFactors_le_normalizedFactors]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : x ~\u1d64 y\n\u22a2 normalizedFactors y \u2264 normalizedFactors x\n[PROOFSTEP]\nrw [\u2190 dvd_iff_normalizedFactors_le_normalizedFactors]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : x ~\u1d64 y\n\u22a2 x \u2223 y\ncase a.hx\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : x ~\u1d64 y\n\u22a2 x \u2260 0\ncase a.hy\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : x ~\u1d64 y\n\u22a2 y \u2260 0\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : x ~\u1d64 y\n\u22a2 y \u2223 x\ncase a.hx\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : x ~\u1d64 y\n\u22a2 y \u2260 0\ncase a.hy\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : x ~\u1d64 y\n\u22a2 x \u2260 0\n[PROOFSTEP]\nall_goals simp [*, h.dvd, h.symm.dvd]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : x ~\u1d64 y\n\u22a2 x \u2223 y\n[PROOFSTEP]\nsimp [*, h.dvd, h.symm.dvd]\n[GOAL]\ncase a.hx\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : x ~\u1d64 y\n\u22a2 x \u2260 0\n[PROOFSTEP]\nsimp [*, h.dvd, h.symm.dvd]\n[GOAL]\ncase a.hy\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : x ~\u1d64 y\n\u22a2 y \u2260 0\n[PROOFSTEP]\nsimp [*, h.dvd, h.symm.dvd]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : x ~\u1d64 y\n\u22a2 y \u2223 x\n[PROOFSTEP]\nsimp [*, h.dvd, h.symm.dvd]\n[GOAL]\ncase a.hx\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : x ~\u1d64 y\n\u22a2 y \u2260 0\n[PROOFSTEP]\nsimp [*, h.dvd, h.symm.dvd]\n[GOAL]\ncase a.hy\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : x ~\u1d64 y\n\u22a2 x \u2260 0\n[PROOFSTEP]\nsimp [*, h.dvd, h.symm.dvd]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\np : \u03b1\nhp : Irreducible p\nk : \u2115\n\u22a2 normalizedFactors (p ^ k) = Multiset.replicate k (\u2191normalize p)\n[PROOFSTEP]\nrw [normalizedFactors_pow, normalizedFactors_irreducible hp, Multiset.nsmul_singleton]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na p : \u03b1\nH : p \u2208 normalizedFactors a\n\u22a2 p \u2223 a\n[PROOFSTEP]\nby_cases hcases : a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na p : \u03b1\nH : p \u2208 normalizedFactors a\nhcases : a = 0\n\u22a2 p \u2223 a\n[PROOFSTEP]\nrw [hcases]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na p : \u03b1\nH : p \u2208 normalizedFactors a\nhcases : a = 0\n\u22a2 p \u2223 0\n[PROOFSTEP]\nexact dvd_zero p\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na p : \u03b1\nH : p \u2208 normalizedFactors a\nhcases : \u00aca = 0\n\u22a2 p \u2223 a\n[PROOFSTEP]\nexact dvd_trans (Multiset.dvd_prod H) (Associated.dvd (normalizedFactors_prod hcases))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\np r : \u03b1\nh : \u2200 {m : \u03b1}, m \u2208 normalizedFactors r \u2192 m = p\nhr : r \u2260 0\n\u22a2 \u2203 i, p ^ i ~\u1d64 r\n[PROOFSTEP]\nuse Multiset.card.toFun (normalizedFactors r)\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\np r : \u03b1\nh : \u2200 {m : \u03b1}, m \u2208 normalizedFactors r \u2192 m = p\nhr : r \u2260 0\n\u22a2 p ^ ZeroHom.toFun (\u2191Multiset.card) (normalizedFactors r) ~\u1d64 r\n[PROOFSTEP]\nhave := UniqueFactorizationMonoid.normalizedFactors_prod hr\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\np r : \u03b1\nh : \u2200 {m : \u03b1}, m \u2208 normalizedFactors r \u2192 m = p\nhr : r \u2260 0\nthis : Multiset.prod (normalizedFactors r) ~\u1d64 r\n\u22a2 p ^ ZeroHom.toFun (\u2191Multiset.card) (normalizedFactors r) ~\u1d64 r\n[PROOFSTEP]\nrwa [Multiset.eq_replicate_of_mem fun b => h, Multiset.prod_replicate] at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : NormalizationMonoid \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : Unique \u03b1\u02e3\nm : Multiset \u03b1\nh : \u2200 (p : \u03b1), p \u2208 m \u2192 Prime p\n\u22a2 normalizedFactors (Multiset.prod m) = m\n[PROOFSTEP]\nsimpa only [\u2190 Multiset.rel_eq, \u2190 associated_eq_eq] using\n prime_factors_unique prime_of_normalized_factor h (normalizedFactors_prod (m.prod_ne_zero_of_prime h))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na b c : \u03b1\nha : a \u2208 normalizedFactors c\nhb : b \u2208 normalizedFactors c\nh : a ~\u1d64 b\n\u22a2 a = b\n[PROOFSTEP]\nrw [\u2190 normalize_normalized_factor a ha, \u2190 normalize_normalized_factor b hb, normalize_eq_normalize_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na b c : \u03b1\nha : a \u2208 normalizedFactors c\nhb : b \u2208 normalizedFactors c\nh : a ~\u1d64 b\n\u22a2 a \u2223 b \u2227 b \u2223 a\n[PROOFSTEP]\nexact Associated.dvd_dvd h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx : x \u2260 0\n\u22a2 0 < normalizedFactors x \u2194 \u00acIsUnit x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx : x \u2260 0\n\u22a2 0 < normalizedFactors x \u2192 \u00acIsUnit x\n[PROOFSTEP]\nintro h hx\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx\u271d : x \u2260 0\nh : 0 < normalizedFactors x\nhx : IsUnit x\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8p, hp\u27e9 := Multiset.exists_mem_of_ne_zero h.ne'\n[GOAL]\ncase mp.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx\u271d : x \u2260 0\nh : 0 < normalizedFactors x\nhx : IsUnit x\np : \u03b1\nhp : p \u2208 normalizedFactors x\n\u22a2 False\n[PROOFSTEP]\nexact (prime_of_normalized_factor _ hp).not_unit (isUnit_of_dvd_unit (dvd_of_mem_normalizedFactors hp) hx)\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx : x \u2260 0\n\u22a2 \u00acIsUnit x \u2192 0 < normalizedFactors x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx : x \u2260 0\nh : \u00acIsUnit x\n\u22a2 0 < normalizedFactors x\n[PROOFSTEP]\nobtain \u27e8p, hp\u27e9 := exists_mem_normalizedFactors hx h\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx : x \u2260 0\nh : \u00acIsUnit x\np : \u03b1\nhp : p \u2208 normalizedFactors x\n\u22a2 0 < normalizedFactors x\n[PROOFSTEP]\nexact bot_lt_iff_ne_bot.mpr (mt Multiset.eq_zero_iff_forall_not_mem.mp (not_forall.mpr \u27e8p, not_not.mpr hp\u27e9))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 DvdNotUnit x y \u2194 normalizedFactors x < normalizedFactors y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 DvdNotUnit x y \u2192 normalizedFactors x < normalizedFactors y\n[PROOFSTEP]\nrintro \u27e8_, c, hc, rfl\u27e9\n[GOAL]\ncase mp.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : \u03b1\nhx left\u271d : x \u2260 0\nc : \u03b1\nhc : \u00acIsUnit c\nhy : x * c \u2260 0\n\u22a2 normalizedFactors x < normalizedFactors (x * c)\n[PROOFSTEP]\nsimp only [hx, right_ne_zero_of_mul hy, normalizedFactors_mul, Ne.def, not_false_iff, lt_add_iff_pos_right,\n normalizedFactors_pos, hc]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 normalizedFactors x < normalizedFactors y \u2192 DvdNotUnit x y\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : normalizedFactors x < normalizedFactors y\n\u22a2 DvdNotUnit x y\n[PROOFSTEP]\nexact\n dvdNotUnit_of_dvd_of_not_dvd ((dvd_iff_normalizedFactors_le_normalizedFactors hx hy).mpr h.le)\n (mt (dvd_iff_normalizedFactors_le_normalizedFactors hy hx).mp h.not_le)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\n\u22a2 (fun a =>\n if a = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)))\n 1 =\n 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : Associates \u03b1\n\u22a2 OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =\n 1) }\n y\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : Associates \u03b1\nhx : x = 0\n\u22a2 OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =\n 1) }\n y\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : Associates \u03b1\nhx : \u00acx = 0\n\u22a2 OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =\n 1) }\n y\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : Associates \u03b1\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =\n 1) }\n y\n[PROOFSTEP]\nsimp [hy]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : Associates \u03b1\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =\n 1) }\n y\n[PROOFSTEP]\nsimp [hx, hy]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\n\u22a2 Function.RightInverse\n (\u2191{\n toOneHom :=\n {\n toFun := fun a =>\n if a = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else\n prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : Associates \u03b1),\n OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else\n prod\n (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else\n prod\n (map (Classical.choose (_ : Function.HasRightInverse Associates.mk))\n (normalizedFactors 1))) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else\n prod\n (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else\n prod\n (map (Classical.choose (_ : Function.HasRightInverse Associates.mk))\n (normalizedFactors 1))) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else\n prod\n (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else\n prod\n (map (Classical.choose (_ : Function.HasRightInverse Associates.mk))\n (normalizedFactors 1))) =\n 1) }\n y) })\n Associates.mk\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : Associates \u03b1\n\u22a2 Associates.mk\n (\u2191{\n toOneHom :=\n {\n toFun := fun a =>\n if a = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else\n prod\n (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors 1))) =\n 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : Associates \u03b1),\n OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else\n prod\n (map (Classical.choose (_ : Function.HasRightInverse Associates.mk))\n (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else\n prod\n (map (Classical.choose (_ : Function.HasRightInverse Associates.mk))\n (normalizedFactors 1))) =\n 1) }\n (x * y) =\n OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else\n prod\n (map (Classical.choose (_ : Function.HasRightInverse Associates.mk))\n (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else\n prod\n (map (Classical.choose (_ : Function.HasRightInverse Associates.mk))\n (normalizedFactors 1))) =\n 1) }\n x *\n OneHom.toFun\n {\n toFun := fun a =>\n if a = 0 then 0\n else\n prod\n (map (Classical.choose (_ : Function.HasRightInverse Associates.mk))\n (normalizedFactors a)),\n map_one' :=\n (_ :\n (if 1 = 0 then 0\n else\n prod\n (map (Classical.choose (_ : Function.HasRightInverse Associates.mk))\n (normalizedFactors 1))) =\n 1) }\n y) }\n x) =\n x\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : Associates \u03b1\n\u22a2 Associates.mk\n (if x = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors x))) =\n x\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : Associates \u03b1\nhx : x = 0\n\u22a2 Associates.mk\n (if x = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors x))) =\n x\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : Associates \u03b1\nhx : \u00acx = 0\n\u22a2 Associates.mk\n (if x = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors x))) =\n x\n[PROOFSTEP]\nhave h : Associates.mkMonoidHom \u2218 Classical.choose mk_surjective.hasRightInverse = (id : Associates \u03b1 \u2192 Associates \u03b1) :=\n by\n ext x\n rw [Function.comp_apply, mkMonoidHom_apply, Classical.choose_spec mk_surjective.hasRightInverse x]\n rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : Associates \u03b1\nhx : \u00acx = 0\n\u22a2 \u2191Associates.mkMonoidHom \u2218 Classical.choose (_ : Function.HasRightInverse Associates.mk) = id\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx\u271d : Associates \u03b1\nhx : \u00acx\u271d = 0\nx : Associates \u03b1\n\u22a2 (\u2191Associates.mkMonoidHom \u2218 Classical.choose (_ : Function.HasRightInverse Associates.mk)) x = id x\n[PROOFSTEP]\nrw [Function.comp_apply, mkMonoidHom_apply, Classical.choose_spec mk_surjective.hasRightInverse x]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx\u271d : Associates \u03b1\nhx : \u00acx\u271d = 0\nx : Associates \u03b1\n\u22a2 x = id x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : Associates \u03b1\nhx : \u00acx = 0\nh : \u2191Associates.mkMonoidHom \u2218 Classical.choose (_ : Function.HasRightInverse Associates.mk) = id\n\u22a2 Associates.mk\n (if x = 0 then 0\n else prod (map (Classical.choose (_ : Function.HasRightInverse Associates.mk)) (normalizedFactors x))) =\n x\n[PROOFSTEP]\nrw [if_neg hx, \u2190 mkMonoidHom_apply, MonoidHom.map_multiset_prod, map_map, h, map_id, \u2190 associated_iff_eq]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : Nontrivial \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx : Associates \u03b1\nhx : \u00acx = 0\nh : \u2191Associates.mkMonoidHom \u2218 Classical.choose (_ : Function.HasRightInverse Associates.mk) = id\n\u22a2 prod (normalizedFactors x) ~\u1d64 x\n[PROOFSTEP]\napply normalizedFactors_prod hx\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na b : R\nha : a \u2260 0\nh : \u2200 {d : R}, d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nd : R\n\u22a2 0 \u2223 a \u2192 0 \u2223 b \u2192 IsUnit 0\n[PROOFSTEP]\nsimp only [zero_dvd_iff]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na b : R\nha : a \u2260 0\nh : \u2200 {d : R}, d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nd : R\n\u22a2 a = 0 \u2192 b = 0 \u2192 IsUnit 0\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na b : R\nha : a \u2260 0\nh : \u2200 {d : R}, d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nd : R\na\u271d\u00b9 : a = 0\na\u271d : b = 0\n\u22a2 IsUnit 0\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na b c : R\nha : a \u2260 0\n\u22a2 (\u2200 {d : R}, d \u2223 a \u2192 d \u2223 c \u2192 \u00acPrime d) \u2192 a \u2223 b * c \u2192 a \u2223 b\n[PROOFSTEP]\nrefine' induction_on_prime c _ _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na b c : R\nha : a \u2260 0\n\u22a2 (\u2200 {d : R}, d \u2223 a \u2192 d \u2223 0 \u2192 \u00acPrime d) \u2192 a \u2223 b * 0 \u2192 a \u2223 b\n[PROOFSTEP]\nintro no_factors\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na b c : R\nha : a \u2260 0\nno_factors : \u2200 {d : R}, d \u2223 a \u2192 d \u2223 0 \u2192 \u00acPrime d\n\u22a2 a \u2223 b * 0 \u2192 a \u2223 b\n[PROOFSTEP]\nsimp only [dvd_zero, mul_zero, forall_prop_of_true]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na b c : R\nha : a \u2260 0\nno_factors : \u2200 {d : R}, d \u2223 a \u2192 d \u2223 0 \u2192 \u00acPrime d\n\u22a2 a \u2223 b\n[PROOFSTEP]\nhaveI := Classical.propDecidable\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na b c : R\nha : a \u2260 0\nno_factors : \u2200 {d : R}, d \u2223 a \u2192 d \u2223 0 \u2192 \u00acPrime d\nthis : (a : Prop) \u2192 Decidable a\n\u22a2 a \u2223 b\n[PROOFSTEP]\nexact isUnit_iff_forall_dvd.mp (no_factors_of_no_prime_factors ha (@no_factors) (dvd_refl a) (dvd_zero a)) _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na b c : R\nha : a \u2260 0\n\u22a2 \u2200 (x : R), IsUnit x \u2192 (\u2200 {d : R}, d \u2223 a \u2192 d \u2223 x \u2192 \u00acPrime d) \u2192 a \u2223 b * x \u2192 a \u2223 b\n[PROOFSTEP]\nrintro _ \u27e8x, rfl\u27e9 _ a_dvd_bx\n[GOAL]\ncase refine'_2.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na b c : R\nha : a \u2260 0\nx : R\u02e3\na\u271d : \u2200 {d : R}, d \u2223 a \u2192 d \u2223 \u2191x \u2192 \u00acPrime d\na_dvd_bx : a \u2223 b * \u2191x\n\u22a2 a \u2223 b\n[PROOFSTEP]\napply Units.dvd_mul_right.mp a_dvd_bx\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na b c : R\nha : a \u2260 0\n\u22a2 \u2200 (a_1 p : R),\n a_1 \u2260 0 \u2192\n Prime p \u2192\n ((\u2200 {d : R}, d \u2223 a \u2192 d \u2223 a_1 \u2192 \u00acPrime d) \u2192 a \u2223 b * a_1 \u2192 a \u2223 b) \u2192\n (\u2200 {d : R}, d \u2223 a \u2192 d \u2223 p * a_1 \u2192 \u00acPrime d) \u2192 a \u2223 b * (p * a_1) \u2192 a \u2223 b\n[PROOFSTEP]\nintro c p _ hp ih no_factors a_dvd_bpc\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na b c\u271d : R\nha : a \u2260 0\nc p : R\na\u271d : c \u2260 0\nhp : Prime p\nih : (\u2200 {d : R}, d \u2223 a \u2192 d \u2223 c \u2192 \u00acPrime d) \u2192 a \u2223 b * c \u2192 a \u2223 b\nno_factors : \u2200 {d : R}, d \u2223 a \u2192 d \u2223 p * c \u2192 \u00acPrime d\na_dvd_bpc : a \u2223 b * (p * c)\n\u22a2 a \u2223 b\n[PROOFSTEP]\napply ih fun {q} dvd_a dvd_c hq => no_factors dvd_a (dvd_c.mul_left _) hq\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na b c\u271d : R\nha : a \u2260 0\nc p : R\na\u271d : c \u2260 0\nhp : Prime p\nih : (\u2200 {d : R}, d \u2223 a \u2192 d \u2223 c \u2192 \u00acPrime d) \u2192 a \u2223 b * c \u2192 a \u2223 b\nno_factors : \u2200 {d : R}, d \u2223 a \u2192 d \u2223 p * c \u2192 \u00acPrime d\na_dvd_bpc : a \u2223 b * (p * c)\n\u22a2 a \u2223 b * c\n[PROOFSTEP]\nrw [mul_left_comm] at a_dvd_bpc \n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na b c\u271d : R\nha : a \u2260 0\nc p : R\na\u271d : c \u2260 0\nhp : Prime p\nih : (\u2200 {d : R}, d \u2223 a \u2192 d \u2223 c \u2192 \u00acPrime d) \u2192 a \u2223 b * c \u2192 a \u2223 b\nno_factors : \u2200 {d : R}, d \u2223 a \u2192 d \u2223 p * c \u2192 \u00acPrime d\na_dvd_bpc : a \u2223 p * (b * c)\n\u22a2 a \u2223 b * c\n[PROOFSTEP]\nrefine' Or.resolve_left (hp.left_dvd_or_dvd_right_of_dvd_mul a_dvd_bpc) fun h => _\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na b c\u271d : R\nha : a \u2260 0\nc p : R\na\u271d : c \u2260 0\nhp : Prime p\nih : (\u2200 {d : R}, d \u2223 a \u2192 d \u2223 c \u2192 \u00acPrime d) \u2192 a \u2223 b * c \u2192 a \u2223 b\nno_factors : \u2200 {d : R}, d \u2223 a \u2192 d \u2223 p * c \u2192 \u00acPrime d\na_dvd_bpc : a \u2223 p * (b * c)\nh : p \u2223 a\n\u22a2 False\n[PROOFSTEP]\nexact no_factors h (dvd_mul_right p c) hp\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na b c : R\nha : a \u2260 0\nno_factors : \u2200 {d : R}, d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\n\u22a2 a \u2223 b * c \u2192 a \u2223 c\n[PROOFSTEP]\nsimpa [mul_comm b c] using dvd_of_dvd_mul_left_of_no_prime_factors ha @no_factors\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\n\u22a2 \u2200 (a : R), a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b\n[PROOFSTEP]\nhaveI := Classical.propDecidable\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\n\u22a2 \u2200 (a : R), a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b\n[PROOFSTEP]\nintro a\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na : R\n\u22a2 a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b\n[PROOFSTEP]\nrefine' induction_on_prime a _ _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na : R\n\u22a2 0 \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = 0 \u2227 c' * b' = b\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na : R\nx\u271d : 0 \u2260 0\nb\u271d : R\n\u22a2 \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = 0 \u2227 c' * b' = b\u271d\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na : R\n\u22a2 \u2200 (x : R),\n IsUnit x \u2192 x \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = x \u2227 c' * b' = b\n[PROOFSTEP]\nintro a a_unit _ b\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na\u271d a : R\na_unit : IsUnit a\nx\u271d : a \u2260 0\nb : R\n\u22a2 \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b\n[PROOFSTEP]\nuse a, b, 1\n[GOAL]\ncase h\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na\u271d a : R\na_unit : IsUnit a\nx\u271d : a \u2260 0\nb : R\n\u22a2 (\u2200 {d : R}, d \u2223 a \u2192 d \u2223 b \u2192 IsUnit d) \u2227 1 * a = a \u2227 1 * b = b\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na\u271d a : R\na_unit : IsUnit a\nx\u271d : a \u2260 0\nb : R\n\u22a2 \u2200 {d : R}, d \u2223 a \u2192 d \u2223 b \u2192 IsUnit d\n[PROOFSTEP]\nintro p p_dvd_a _\n[GOAL]\ncase h.left\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na\u271d\u00b9 a : R\na_unit : IsUnit a\nx\u271d : a \u2260 0\nb p : R\np_dvd_a : p \u2223 a\na\u271d : p \u2223 b\n\u22a2 IsUnit p\n[PROOFSTEP]\nexact isUnit_of_dvd_unit p_dvd_a a_unit\n[GOAL]\ncase h.right\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na\u271d a : R\na_unit : IsUnit a\nx\u271d : a \u2260 0\nb : R\n\u22a2 1 * a = a \u2227 1 * b = b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na : R\n\u22a2 \u2200 (a p : R),\n a \u2260 0 \u2192\n Prime p \u2192\n (a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b) \u2192\n p * a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = p * a \u2227 c' * b' = b\n[PROOFSTEP]\nintro a p a_ne_zero p_prime ih_a pa_ne_zero b\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na\u271d a p : R\na_ne_zero : a \u2260 0\np_prime : Prime p\nih_a : a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b\npa_ne_zero : p * a \u2260 0\nb : R\n\u22a2 \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = p * a \u2227 c' * b' = b\n[PROOFSTEP]\nby_cases p \u2223 b\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na\u271d a p : R\na_ne_zero : a \u2260 0\np_prime : Prime p\nih_a : a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b\npa_ne_zero : p * a \u2260 0\nb : R\n\u22a2 \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = p * a \u2227 c' * b' = b\n[PROOFSTEP]\nby_cases p \u2223 b\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na\u271d a p : R\na_ne_zero : a \u2260 0\np_prime : Prime p\nih_a : a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b\npa_ne_zero : p * a \u2260 0\nb : R\nh : p \u2223 b\n\u22a2 \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = p * a \u2227 c' * b' = b\n[PROOFSTEP]\nrcases h with \u27e8b, rfl\u27e9\n[GOAL]\ncase pos.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na\u271d a p : R\na_ne_zero : a \u2260 0\np_prime : Prime p\nih_a : a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b\npa_ne_zero : p * a \u2260 0\nb : R\n\u22a2 \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = p * a \u2227 c' * b' = p * b\n[PROOFSTEP]\nobtain \u27e8a', b', c', no_factor, ha', hb'\u27e9 := ih_a a_ne_zero b\n[GOAL]\ncase pos.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na\u271d a p : R\na_ne_zero : a \u2260 0\np_prime : Prime p\nih_a : a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b\npa_ne_zero : p * a \u2260 0\nb a' b' c' : R\nno_factor : \u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d\nha' : c' * a' = a\nhb' : c' * b' = b\n\u22a2 \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = p * a \u2227 c' * b' = p * b\n[PROOFSTEP]\nrefine' \u27e8a', b', p * c', @no_factor, _, _\u27e9\n[GOAL]\ncase pos.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na\u271d a p : R\na_ne_zero : a \u2260 0\np_prime : Prime p\nih_a : a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b\npa_ne_zero : p * a \u2260 0\nb a' b' c' : R\nno_factor : \u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d\nha' : c' * a' = a\nhb' : c' * b' = b\n\u22a2 p * c' * a' = p * a\n[PROOFSTEP]\nrw [mul_assoc, ha']\n[GOAL]\ncase pos.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na\u271d a p : R\na_ne_zero : a \u2260 0\np_prime : Prime p\nih_a : a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b\npa_ne_zero : p * a \u2260 0\nb a' b' c' : R\nno_factor : \u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d\nha' : c' * a' = a\nhb' : c' * b' = b\n\u22a2 p * c' * b' = p * b\n[PROOFSTEP]\nrw [mul_assoc, hb']\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na\u271d a p : R\na_ne_zero : a \u2260 0\np_prime : Prime p\nih_a : a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b\npa_ne_zero : p * a \u2260 0\nb : R\nh : \u00acp \u2223 b\n\u22a2 \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = p * a \u2227 c' * b' = b\n[PROOFSTEP]\nobtain \u27e8a', b', c', coprime, rfl, rfl\u27e9 := ih_a a_ne_zero b\n[GOAL]\ncase neg.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na p : R\np_prime : Prime p\na' b' c' : R\ncoprime : \u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d\na_ne_zero : c' * a' \u2260 0\nih_a :\n c' * a' \u2260 0 \u2192\n \u2200 (b : R), \u2203 a'_1 b' c'_1, (\u2200 {d : R}, d \u2223 a'_1 \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c'_1 * a'_1 = c' * a' \u2227 c'_1 * b' = b\npa_ne_zero : p * (c' * a') \u2260 0\nh : \u00acp \u2223 c' * b'\n\u22a2 \u2203 a'_1 b'_1 c'_1, (\u2200 {d : R}, d \u2223 a'_1 \u2192 d \u2223 b'_1 \u2192 IsUnit d) \u2227 c'_1 * a'_1 = p * (c' * a') \u2227 c'_1 * b'_1 = c' * b'\n[PROOFSTEP]\nrefine' \u27e8p * a', b', c', _, mul_left_comm _ _ _, rfl\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na p : R\np_prime : Prime p\na' b' c' : R\ncoprime : \u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d\na_ne_zero : c' * a' \u2260 0\nih_a :\n c' * a' \u2260 0 \u2192\n \u2200 (b : R), \u2203 a'_1 b' c'_1, (\u2200 {d : R}, d \u2223 a'_1 \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c'_1 * a'_1 = c' * a' \u2227 c'_1 * b' = b\npa_ne_zero : p * (c' * a') \u2260 0\nh : \u00acp \u2223 c' * b'\n\u22a2 \u2200 {d : R}, d \u2223 p * a' \u2192 d \u2223 b' \u2192 IsUnit d\n[PROOFSTEP]\nintro q q_dvd_pa' q_dvd_b'\n[GOAL]\ncase neg.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na p : R\np_prime : Prime p\na' b' c' : R\ncoprime : \u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d\na_ne_zero : c' * a' \u2260 0\nih_a :\n c' * a' \u2260 0 \u2192\n \u2200 (b : R), \u2203 a'_1 b' c'_1, (\u2200 {d : R}, d \u2223 a'_1 \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c'_1 * a'_1 = c' * a' \u2227 c'_1 * b' = b\npa_ne_zero : p * (c' * a') \u2260 0\nh : \u00acp \u2223 c' * b'\nq : R\nq_dvd_pa' : q \u2223 p * a'\nq_dvd_b' : q \u2223 b'\n\u22a2 IsUnit q\n[PROOFSTEP]\ncases' p_prime.left_dvd_or_dvd_right_of_dvd_mul q_dvd_pa' with p_dvd_q q_dvd_a'\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.inl\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na p : R\np_prime : Prime p\na' b' c' : R\ncoprime : \u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d\na_ne_zero : c' * a' \u2260 0\nih_a :\n c' * a' \u2260 0 \u2192\n \u2200 (b : R), \u2203 a'_1 b' c'_1, (\u2200 {d : R}, d \u2223 a'_1 \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c'_1 * a'_1 = c' * a' \u2227 c'_1 * b' = b\npa_ne_zero : p * (c' * a') \u2260 0\nh : \u00acp \u2223 c' * b'\nq : R\nq_dvd_pa' : q \u2223 p * a'\nq_dvd_b' : q \u2223 b'\np_dvd_q : p \u2223 q\n\u22a2 IsUnit q\n[PROOFSTEP]\nhave : p \u2223 c' * b' := dvd_mul_of_dvd_right (p_dvd_q.trans q_dvd_b') _\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.inl\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis\u271d : (a : Prop) \u2192 Decidable a\na p : R\np_prime : Prime p\na' b' c' : R\ncoprime : \u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d\na_ne_zero : c' * a' \u2260 0\nih_a :\n c' * a' \u2260 0 \u2192\n \u2200 (b : R), \u2203 a'_1 b' c'_1, (\u2200 {d : R}, d \u2223 a'_1 \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c'_1 * a'_1 = c' * a' \u2227 c'_1 * b' = b\npa_ne_zero : p * (c' * a') \u2260 0\nh : \u00acp \u2223 c' * b'\nq : R\nq_dvd_pa' : q \u2223 p * a'\nq_dvd_b' : q \u2223 b'\np_dvd_q : p \u2223 q\nthis : p \u2223 c' * b'\n\u22a2 IsUnit q\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\nthis : (a : Prop) \u2192 Decidable a\na p : R\np_prime : Prime p\na' b' c' : R\ncoprime : \u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d\na_ne_zero : c' * a' \u2260 0\nih_a :\n c' * a' \u2260 0 \u2192\n \u2200 (b : R), \u2203 a'_1 b' c'_1, (\u2200 {d : R}, d \u2223 a'_1 \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c'_1 * a'_1 = c' * a' \u2227 c'_1 * b' = b\npa_ne_zero : p * (c' * a') \u2260 0\nh : \u00acp \u2223 c' * b'\nq : R\nq_dvd_pa' : q \u2223 p * a'\nq_dvd_b' : q \u2223 b'\nq_dvd_a' : q \u2223 a'\n\u22a2 IsUnit q\n[PROOFSTEP]\nexact coprime q_dvd_a' q_dvd_b'\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na : R\nha0 : a \u2260 0\nha1 : \u00acIsUnit a\n\u22a2 Function.Injective ((fun x x_1 => x ^ x_1) a)\n[PROOFSTEP]\nletI := Classical.decEq R\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na : R\nha0 : a \u2260 0\nha1 : \u00acIsUnit a\nthis : DecidableEq R := Classical.decEq R\n\u22a2 Function.Injective ((fun x x_1 => x ^ x_1) a)\n[PROOFSTEP]\nintro i j hij\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na : R\nha0 : a \u2260 0\nha1 : \u00acIsUnit a\nthis : DecidableEq R := Classical.decEq R\ni j : \u2115\nhij : (fun x x_1 => x ^ x_1) a i = (fun x x_1 => x ^ x_1) a j\n\u22a2 i = j\n[PROOFSTEP]\nletI : Nontrivial R := \u27e8\u27e8a, 0, ha0\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na : R\nha0 : a \u2260 0\nha1 : \u00acIsUnit a\nthis\u271d : DecidableEq R := Classical.decEq R\ni j : \u2115\nhij : (fun x x_1 => x ^ x_1) a i = (fun x x_1 => x ^ x_1) a j\nthis : Nontrivial R := { exists_pair_ne := Exists.intro a (Exists.intro 0 ha0) }\n\u22a2 i = j\n[PROOFSTEP]\nletI : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na : R\nha0 : a \u2260 0\nha1 : \u00acIsUnit a\nthis\u271d\u00b9 : DecidableEq R := Classical.decEq R\ni j : \u2115\nhij : (fun x x_1 => x ^ x_1) a i = (fun x x_1 => x ^ x_1) a j\nthis\u271d : Nontrivial R := { exists_pair_ne := Exists.intro a (Exists.intro 0 ha0) }\nthis : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 i = j\n[PROOFSTEP]\nobtain \u27e8p', hp', dvd'\u27e9 := WfDvdMonoid.exists_irreducible_factor ha1 ha0\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na : R\nha0 : a \u2260 0\nha1 : \u00acIsUnit a\nthis\u271d\u00b9 : DecidableEq R := Classical.decEq R\ni j : \u2115\nhij : (fun x x_1 => x ^ x_1) a i = (fun x x_1 => x ^ x_1) a j\nthis\u271d : Nontrivial R := { exists_pair_ne := Exists.intro a (Exists.intro 0 ha0) }\nthis : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid\np' : R\nhp' : Irreducible p'\ndvd' : p' \u2223 a\n\u22a2 i = j\n[PROOFSTEP]\nobtain \u27e8p, mem, _\u27e9 := exists_mem_normalizedFactors_of_dvd ha0 hp' dvd'\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na : R\nha0 : a \u2260 0\nha1 : \u00acIsUnit a\nthis\u271d\u00b9 : DecidableEq R := Classical.decEq R\ni j : \u2115\nhij : (fun x x_1 => x ^ x_1) a i = (fun x x_1 => x ^ x_1) a j\nthis\u271d : Nontrivial R := { exists_pair_ne := Exists.intro a (Exists.intro 0 ha0) }\nthis : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid\np' : R\nhp' : Irreducible p'\ndvd' : p' \u2223 a\np : R\nmem : p \u2208 normalizedFactors a\nright\u271d : p' ~\u1d64 p\n\u22a2 i = j\n[PROOFSTEP]\nhave := congr_arg (fun x => Multiset.count p (normalizedFactors x)) hij\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na : R\nha0 : a \u2260 0\nha1 : \u00acIsUnit a\nthis\u271d\u00b2 : DecidableEq R := Classical.decEq R\ni j : \u2115\nhij : (fun x x_1 => x ^ x_1) a i = (fun x x_1 => x ^ x_1) a j\nthis\u271d\u00b9 : Nontrivial R := { exists_pair_ne := Exists.intro a (Exists.intro 0 ha0) }\nthis\u271d : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid\np' : R\nhp' : Irreducible p'\ndvd' : p' \u2223 a\np : R\nmem : p \u2208 normalizedFactors a\nright\u271d : p' ~\u1d64 p\nthis :\n (fun x => Multiset.count p (normalizedFactors x)) ((fun x x_1 => x ^ x_1) a i) =\n (fun x => Multiset.count p (normalizedFactors x)) ((fun x x_1 => x ^ x_1) a j)\n\u22a2 i = j\n[PROOFSTEP]\nsimp only [normalizedFactors_pow, Multiset.count_nsmul] at this \n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na : R\nha0 : a \u2260 0\nha1 : \u00acIsUnit a\nthis\u271d\u00b2 : DecidableEq R := Classical.decEq R\ni j : \u2115\nhij : (fun x x_1 => x ^ x_1) a i = (fun x x_1 => x ^ x_1) a j\nthis\u271d\u00b9 : Nontrivial R := { exists_pair_ne := Exists.intro a (Exists.intro 0 ha0) }\nthis\u271d : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid\np' : R\nhp' : Irreducible p'\ndvd' : p' \u2223 a\np : R\nmem : p \u2208 normalizedFactors a\nright\u271d : p' ~\u1d64 p\nthis : i * Multiset.count p (normalizedFactors a) = j * Multiset.count p (normalizedFactors a)\n\u22a2 i = j\n[PROOFSTEP]\nexact mul_right_cancel\u2080 (Multiset.count_ne_zero.mpr mem) this\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\na b : R\nn : \u2115\nha : Irreducible a\nhb : b \u2260 0\n\u22a2 \u2191n \u2264 multiplicity a b \u2194 replicate n (\u2191normalize a) \u2264 normalizedFactors b\n[PROOFSTEP]\nrw [\u2190 pow_dvd_iff_le_multiplicity]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\na b : R\nn : \u2115\nha : Irreducible a\nhb : b \u2260 0\n\u22a2 a ^ n \u2223 b \u2194 replicate n (\u2191normalize a) \u2264 normalizedFactors b\n[PROOFSTEP]\nrevert b\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\na : R\nn : \u2115\nha : Irreducible a\n\u22a2 \u2200 {b : R}, b \u2260 0 \u2192 (a ^ n \u2223 b \u2194 replicate n (\u2191normalize a) \u2264 normalizedFactors b)\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\na : R\nha : Irreducible a\n\u22a2 \u2200 {b : R}, b \u2260 0 \u2192 (a ^ Nat.zero \u2223 b \u2194 replicate Nat.zero (\u2191normalize a) \u2264 normalizedFactors b)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\na : R\nha : Irreducible a\nn : \u2115\nih : \u2200 {b : R}, b \u2260 0 \u2192 (a ^ n \u2223 b \u2194 replicate n (\u2191normalize a) \u2264 normalizedFactors b)\n\u22a2 \u2200 {b : R}, b \u2260 0 \u2192 (a ^ Nat.succ n \u2223 b \u2194 replicate (Nat.succ n) (\u2191normalize a) \u2264 normalizedFactors b)\n[PROOFSTEP]\nintro b hb\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\na : R\nha : Irreducible a\nn : \u2115\nih : \u2200 {b : R}, b \u2260 0 \u2192 (a ^ n \u2223 b \u2194 replicate n (\u2191normalize a) \u2264 normalizedFactors b)\nb : R\nhb : b \u2260 0\n\u22a2 a ^ Nat.succ n \u2223 b \u2194 replicate (Nat.succ n) (\u2191normalize a) \u2264 normalizedFactors b\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase succ.mp\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\na : R\nha : Irreducible a\nn : \u2115\nih : \u2200 {b : R}, b \u2260 0 \u2192 (a ^ n \u2223 b \u2194 replicate n (\u2191normalize a) \u2264 normalizedFactors b)\nb : R\nhb : b \u2260 0\n\u22a2 a ^ Nat.succ n \u2223 b \u2192 replicate (Nat.succ n) (\u2191normalize a) \u2264 normalizedFactors b\n[PROOFSTEP]\nrintro \u27e8c, rfl\u27e9\n[GOAL]\ncase succ.mp.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\na : R\nha : Irreducible a\nn : \u2115\nih : \u2200 {b : R}, b \u2260 0 \u2192 (a ^ n \u2223 b \u2194 replicate n (\u2191normalize a) \u2264 normalizedFactors b)\nc : R\nhb : a ^ Nat.succ n * c \u2260 0\n\u22a2 replicate (Nat.succ n) (\u2191normalize a) \u2264 normalizedFactors (a ^ Nat.succ n * c)\n[PROOFSTEP]\nrw [Ne.def, pow_succ, mul_assoc, mul_eq_zero, not_or] at hb \n[GOAL]\ncase succ.mp.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\na : R\nha : Irreducible a\nn : \u2115\nih : \u2200 {b : R}, b \u2260 0 \u2192 (a ^ n \u2223 b \u2194 replicate n (\u2191normalize a) \u2264 normalizedFactors b)\nc : R\nhb : \u00aca = 0 \u2227 \u00aca ^ n * c = 0\n\u22a2 replicate (Nat.succ n) (\u2191normalize a) \u2264 normalizedFactors (a ^ Nat.succ n * c)\n[PROOFSTEP]\nrw [pow_succ, mul_assoc, normalizedFactors_mul hb.1 hb.2, replicate_succ, normalizedFactors_irreducible ha,\n singleton_add, cons_le_cons_iff, \u2190 ih hb.2]\n[GOAL]\ncase succ.mp.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\na : R\nha : Irreducible a\nn : \u2115\nih : \u2200 {b : R}, b \u2260 0 \u2192 (a ^ n \u2223 b \u2194 replicate n (\u2191normalize a) \u2264 normalizedFactors b)\nc : R\nhb : \u00aca = 0 \u2227 \u00aca ^ n * c = 0\n\u22a2 a ^ n \u2223 a ^ n * c\n[PROOFSTEP]\napply Dvd.intro _ rfl\n[GOAL]\ncase succ.mpr\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\na : R\nha : Irreducible a\nn : \u2115\nih : \u2200 {b : R}, b \u2260 0 \u2192 (a ^ n \u2223 b \u2194 replicate n (\u2191normalize a) \u2264 normalizedFactors b)\nb : R\nhb : b \u2260 0\n\u22a2 replicate (Nat.succ n) (\u2191normalize a) \u2264 normalizedFactors b \u2192 a ^ Nat.succ n \u2223 b\n[PROOFSTEP]\nrw [Multiset.le_iff_exists_add]\n[GOAL]\ncase succ.mpr\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\na : R\nha : Irreducible a\nn : \u2115\nih : \u2200 {b : R}, b \u2260 0 \u2192 (a ^ n \u2223 b \u2194 replicate n (\u2191normalize a) \u2264 normalizedFactors b)\nb : R\nhb : b \u2260 0\n\u22a2 (\u2203 u, normalizedFactors b = replicate (Nat.succ n) (\u2191normalize a) + u) \u2192 a ^ Nat.succ n \u2223 b\n[PROOFSTEP]\nrintro \u27e8u, hu\u27e9\n[GOAL]\ncase succ.mpr.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\na : R\nha : Irreducible a\nn : \u2115\nih : \u2200 {b : R}, b \u2260 0 \u2192 (a ^ n \u2223 b \u2194 replicate n (\u2191normalize a) \u2264 normalizedFactors b)\nb : R\nhb : b \u2260 0\nu : Multiset ((fun x => R) a)\nhu : normalizedFactors b = replicate (Nat.succ n) (\u2191normalize a) + u\n\u22a2 a ^ Nat.succ n \u2223 b\n[PROOFSTEP]\nrw [\u2190 (normalizedFactors_prod hb).dvd_iff_dvd_right, hu, prod_add, prod_replicate]\n[GOAL]\ncase succ.mpr.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\na : R\nha : Irreducible a\nn : \u2115\nih : \u2200 {b : R}, b \u2260 0 \u2192 (a ^ n \u2223 b \u2194 replicate n (\u2191normalize a) \u2264 normalizedFactors b)\nb : R\nhb : b \u2260 0\nu : Multiset ((fun x => R) a)\nhu : normalizedFactors b = replicate (Nat.succ n) (\u2191normalize a) + u\n\u22a2 a ^ Nat.succ n \u2223 \u2191normalize a ^ Nat.succ n * prod u\n[PROOFSTEP]\nexact (Associated.pow_pow <| associated_normalize a).dvd.trans (Dvd.intro u.prod rfl)\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\na b : R\nha : Irreducible a\nhb : b \u2260 0\n\u22a2 multiplicity a b = \u2191(count (\u2191normalize a) (normalizedFactors b))\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\na b : R\nha : Irreducible a\nhb : b \u2260 0\n\u22a2 multiplicity a b \u2264 \u2191(count (\u2191normalize a) (normalizedFactors b))\n[PROOFSTEP]\napply PartENat.le_of_lt_add_one\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\na b : R\nha : Irreducible a\nhb : b \u2260 0\n\u22a2 multiplicity a b < \u2191(count (\u2191normalize a) (normalizedFactors b)) + 1\n[PROOFSTEP]\nrw [\u2190 Nat.cast_one, \u2190 Nat.cast_add, lt_iff_not_ge, ge_iff_le, le_multiplicity_iff_replicate_le_normalizedFactors ha hb,\n \u2190 le_count_iff_replicate_le]\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\na b : R\nha : Irreducible a\nhb : b \u2260 0\n\u22a2 \u00account (\u2191normalize a) (normalizedFactors b) + 1 \u2264 count (\u2191normalize a) (normalizedFactors b)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\na b : R\nha : Irreducible a\nhb : b \u2260 0\n\u22a2 \u2191(count (\u2191normalize a) (normalizedFactors b)) \u2264 multiplicity a b\n[PROOFSTEP]\nrw [le_multiplicity_iff_replicate_le_normalizedFactors ha hb, \u2190 le_count_iff_replicate_le]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\np x : R\nhp : Irreducible p\nhnorm : \u2191normalize p = p\nn : \u2115\nhle : p ^ n \u2223 x\nhlt : \u00acp ^ (n + 1) \u2223 x\n\u22a2 count p (normalizedFactors x) = n\n[PROOFSTEP]\nletI : DecidableRel ((\u00b7 \u2223 \u00b7) : R \u2192 R \u2192 Prop) := fun _ _ => Classical.propDecidable _\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\np x : R\nhp : Irreducible p\nhnorm : \u2191normalize p = p\nn : \u2115\nhle : p ^ n \u2223 x\nhlt : \u00acp ^ (n + 1) \u2223 x\nthis : DecidableRel fun x x_1 => x \u2223 x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x \u2223 x_2) x x_1)\n\u22a2 count p (normalizedFactors x) = n\n[PROOFSTEP]\nby_cases hx0 : x = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\np x : R\nhp : Irreducible p\nhnorm : \u2191normalize p = p\nn : \u2115\nhle : p ^ n \u2223 x\nhlt : \u00acp ^ (n + 1) \u2223 x\nthis : DecidableRel fun x x_1 => x \u2223 x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x \u2223 x_2) x x_1)\nhx0 : x = 0\n\u22a2 count p (normalizedFactors x) = n\n[PROOFSTEP]\nsimp [hx0] at hlt \n[GOAL]\ncase neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\np x : R\nhp : Irreducible p\nhnorm : \u2191normalize p = p\nn : \u2115\nhle : p ^ n \u2223 x\nhlt : \u00acp ^ (n + 1) \u2223 x\nthis : DecidableRel fun x x_1 => x \u2223 x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x \u2223 x_2) x x_1)\nhx0 : \u00acx = 0\n\u22a2 count p (normalizedFactors x) = n\n[PROOFSTEP]\nrw [\u2190 PartENat.natCast_inj]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\np x : R\nhp : Irreducible p\nhnorm : \u2191normalize p = p\nn : \u2115\nhle : p ^ n \u2223 x\nhlt : \u00acp ^ (n + 1) \u2223 x\nthis : DecidableRel fun x x_1 => x \u2223 x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x \u2223 x_2) x x_1)\nhx0 : \u00acx = 0\n\u22a2 \u2191(count p (normalizedFactors x)) = \u2191n\n[PROOFSTEP]\nconvert (multiplicity_eq_count_normalizedFactors hp hx0).symm\n[GOAL]\ncase h.e'_2.h.e'_3.h.e'_3\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\np x : R\nhp : Irreducible p\nhnorm : \u2191normalize p = p\nn : \u2115\nhle : p ^ n \u2223 x\nhlt : \u00acp ^ (n + 1) \u2223 x\nthis : DecidableRel fun x x_1 => x \u2223 x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x \u2223 x_2) x x_1)\nhx0 : \u00acx = 0\n\u22a2 p = \u2191normalize p\n[PROOFSTEP]\nexact hnorm.symm\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\np x : R\nhp : Irreducible p\nhnorm : \u2191normalize p = p\nn : \u2115\nhle : p ^ n \u2223 x\nhlt : \u00acp ^ (n + 1) \u2223 x\nthis : DecidableRel fun x x_1 => x \u2223 x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x \u2223 x_2) x x_1)\nhx0 : \u00acx = 0\n\u22a2 \u2191n = multiplicity p x\n[PROOFSTEP]\nexact (multiplicity.eq_coe_iff.mpr \u27e8hle, hlt\u27e9).symm\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\np x : R\nhp : p = 0 \u2228 Irreducible p\nhnorm : \u2191normalize p = p\nn : \u2115\nhle : p ^ n \u2223 x\nhlt : \u00acp ^ (n + 1) \u2223 x\n\u22a2 count p (normalizedFactors x) = n\n[PROOFSTEP]\nrcases hp with (rfl | hp)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\nx : R\nn : \u2115\nhnorm : \u2191normalize 0 = 0\nhle : 0 ^ n \u2223 x\nhlt : \u00ac0 ^ (n + 1) \u2223 x\n\u22a2 count 0 (normalizedFactors x) = n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase inl.zero\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\nx : R\nhnorm : \u2191normalize 0 = 0\nhle : 0 ^ Nat.zero \u2223 x\nhlt : \u00ac0 ^ (Nat.zero + 1) \u2223 x\n\u22a2 count 0 (normalizedFactors x) = Nat.zero\n[PROOFSTEP]\nexact count_eq_zero.2 (zero_not_mem_normalizedFactors _)\n[GOAL]\ncase inl.succ\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\nx : R\nhnorm : \u2191normalize 0 = 0\nn\u271d : \u2115\nhle : 0 ^ Nat.succ n\u271d \u2223 x\nhlt : \u00ac0 ^ (Nat.succ n\u271d + 1) \u2223 x\n\u22a2 count 0 (normalizedFactors x) = Nat.succ n\u271d\n[PROOFSTEP]\nrw [zero_pow (Nat.succ_pos _)] at hle hlt \n[GOAL]\ncase inl.succ\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\nx : R\nhnorm : \u2191normalize 0 = 0\nn\u271d : \u2115\nhle : 0 \u2223 x\nhlt : \u00ac0 \u2223 x\n\u22a2 count 0 (normalizedFactors x) = Nat.succ n\u271d\n[PROOFSTEP]\nexact absurd hle hlt\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\ninst\u271d : DecidableEq R\np x : R\nhnorm : \u2191normalize p = p\nn : \u2115\nhle : p ^ n \u2223 x\nhlt : \u00acp ^ (n + 1) \u2223 x\nhp : Irreducible p\n\u22a2 count p (normalizedFactors x) = n\n[PROOFSTEP]\nexact count_normalizedFactors_eq hp hnorm hle hlt\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero R\ninst\u271d\u00b2 : UniqueFactorizationMonoid R\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\na\u2080 x : R\nh : a\u2080 \u2260 0\nhx : Irreducible x\n\u22a2 \u2203 n a, \u00acx \u2223 a \u2227 a\u2080 = x ^ n * a\n[PROOFSTEP]\nclassical\nlet n := (normalizedFactors a\u2080).count (normalize x)\nobtain \u27e8a, ha1, ha2\u27e9 :=\n @exists_eq_pow_mul_and_not_dvd R _ _ x a\u2080\n (ne_top_iff_finite.mp (PartENat.ne_top_iff.mpr \u27e8n, multiplicity_eq_count_normalizedFactors hx h\u27e9))\nsimp_rw [\u2190 (multiplicity_eq_count_normalizedFactors hx h).symm] at ha1 \nuse n, a, ha2, ha1\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero R\ninst\u271d\u00b2 : UniqueFactorizationMonoid R\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\na\u2080 x : R\nh : a\u2080 \u2260 0\nhx : Irreducible x\n\u22a2 \u2203 n a, \u00acx \u2223 a \u2227 a\u2080 = x ^ n * a\n[PROOFSTEP]\nlet n := (normalizedFactors a\u2080).count (normalize x)\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero R\ninst\u271d\u00b2 : UniqueFactorizationMonoid R\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\na\u2080 x : R\nh : a\u2080 \u2260 0\nhx : Irreducible x\nn : \u2115 := count (\u2191normalize x) (normalizedFactors a\u2080)\n\u22a2 \u2203 n a, \u00acx \u2223 a \u2227 a\u2080 = x ^ n * a\n[PROOFSTEP]\nobtain \u27e8a, ha1, ha2\u27e9 :=\n @exists_eq_pow_mul_and_not_dvd R _ _ x a\u2080\n (ne_top_iff_finite.mp (PartENat.ne_top_iff.mpr \u27e8n, multiplicity_eq_count_normalizedFactors hx h\u27e9))\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero R\ninst\u271d\u00b2 : UniqueFactorizationMonoid R\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\na\u2080 x : R\nh : a\u2080 \u2260 0\nhx : Irreducible x\nn : \u2115 := count (\u2191normalize x) (normalizedFactors a\u2080)\na : R\nha1 : a\u2080 = x ^ Part.get (multiplicity x a\u2080) (_ : multiplicity.Finite x a\u2080) * a\nha2 : \u00acx \u2223 a\n\u22a2 \u2203 n a, \u00acx \u2223 a \u2227 a\u2080 = x ^ n * a\n[PROOFSTEP]\nsimp_rw [\u2190 (multiplicity_eq_count_normalizedFactors hx h).symm] at ha1 \n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero R\ninst\u271d\u00b2 : UniqueFactorizationMonoid R\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NormalizationMonoid R\ndec_dvd : DecidableRel Dvd.dvd\na\u2080 x : R\nh : a\u2080 \u2260 0\nhx : Irreducible x\nn : \u2115 := count (\u2191normalize x) (normalizedFactors a\u2080)\na : R\nha2 : \u00acx \u2223 a\nha1 :\n a\u2080 =\n x ^\n Part.get \u2191(count (\u2191normalize x) (normalizedFactors a\u2080))\n (_ : (\u2191(count (\u2191normalize x) (normalizedFactors a\u2080))).Dom) *\n a\n\u22a2 \u2203 n a, \u00acx \u2223 a \u2227 a\u2080 = x ^ n * a\n[PROOFSTEP]\nuse n, a, ha2, ha1\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero R\ninst\u271d\u2074 : UniqueFactorizationMonoid R\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b2\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\ni : \u03b1 \u2192 \u2115\np : \u03b1\nhps : \u00acp \u2208 s\nis_prime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 Prime q\nis_coprime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 \u2200 (q' : \u03b1), q' \u2208 insert p s \u2192 q \u2223 q' \u2192 q = q'\n\u22a2 \u2200 (q : \u03b1), q \u2223 p ^ i p \u2192 q \u2223 \u220f p' in s, p' ^ i p' \u2192 IsUnit q\n[PROOFSTEP]\nhave hp := is_prime _ (Finset.mem_insert_self _ _)\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero R\ninst\u271d\u2074 : UniqueFactorizationMonoid R\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b2\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\ni : \u03b1 \u2192 \u2115\np : \u03b1\nhps : \u00acp \u2208 s\nis_prime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 Prime q\nis_coprime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 \u2200 (q' : \u03b1), q' \u2208 insert p s \u2192 q \u2223 q' \u2192 q = q'\nhp : Prime p\n\u22a2 \u2200 (q : \u03b1), q \u2223 p ^ i p \u2192 q \u2223 \u220f p' in s, p' ^ i p' \u2192 IsUnit q\n[PROOFSTEP]\nrefine' fun _ => no_factors_of_no_prime_factors (pow_ne_zero _ hp.ne_zero) _\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero R\ninst\u271d\u2074 : UniqueFactorizationMonoid R\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b2\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\ni : \u03b1 \u2192 \u2115\np : \u03b1\nhps : \u00acp \u2208 s\nis_prime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 Prime q\nis_coprime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 \u2200 (q' : \u03b1), q' \u2208 insert p s \u2192 q \u2223 q' \u2192 q = q'\nhp : Prime p\nx\u271d : \u03b1\n\u22a2 \u2200 {d : \u03b1}, d \u2223 p ^ i p \u2192 d \u2223 \u220f p' in s, p' ^ i p' \u2192 \u00acPrime d\n[PROOFSTEP]\nintro d hdp hdprod hd\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero R\ninst\u271d\u2074 : UniqueFactorizationMonoid R\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b2\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\ni : \u03b1 \u2192 \u2115\np : \u03b1\nhps : \u00acp \u2208 s\nis_prime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 Prime q\nis_coprime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 \u2200 (q' : \u03b1), q' \u2208 insert p s \u2192 q \u2223 q' \u2192 q = q'\nhp : Prime p\nx\u271d d : \u03b1\nhdp : d \u2223 p ^ i p\nhdprod : d \u2223 \u220f p' in s, p' ^ i p'\nhd : Prime d\n\u22a2 False\n[PROOFSTEP]\napply hps\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero R\ninst\u271d\u2074 : UniqueFactorizationMonoid R\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b2\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\ni : \u03b1 \u2192 \u2115\np : \u03b1\nhps : \u00acp \u2208 s\nis_prime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 Prime q\nis_coprime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 \u2200 (q' : \u03b1), q' \u2208 insert p s \u2192 q \u2223 q' \u2192 q = q'\nhp : Prime p\nx\u271d d : \u03b1\nhdp : d \u2223 p ^ i p\nhdprod : d \u2223 \u220f p' in s, p' ^ i p'\nhd : Prime d\n\u22a2 p \u2208 s\n[PROOFSTEP]\nreplace hdp := hd.dvd_of_dvd_pow hdp\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero R\ninst\u271d\u2074 : UniqueFactorizationMonoid R\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b2\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\ni : \u03b1 \u2192 \u2115\np : \u03b1\nhps : \u00acp \u2208 s\nis_prime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 Prime q\nis_coprime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 \u2200 (q' : \u03b1), q' \u2208 insert p s \u2192 q \u2223 q' \u2192 q = q'\nhp : Prime p\nx\u271d d : \u03b1\nhdprod : d \u2223 \u220f p' in s, p' ^ i p'\nhd : Prime d\nhdp : d \u2223 p\n\u22a2 p \u2208 s\n[PROOFSTEP]\nobtain \u27e8q, q_mem', hdq\u27e9 := hd.exists_mem_multiset_dvd hdprod\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero R\ninst\u271d\u2074 : UniqueFactorizationMonoid R\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b2\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\ni : \u03b1 \u2192 \u2115\np : \u03b1\nhps : \u00acp \u2208 s\nis_prime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 Prime q\nis_coprime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 \u2200 (q' : \u03b1), q' \u2208 insert p s \u2192 q \u2223 q' \u2192 q = q'\nhp : Prime p\nx\u271d d : \u03b1\nhdprod : d \u2223 \u220f p' in s, p' ^ i p'\nhd : Prime d\nhdp : d \u2223 p\nq : \u03b1\nq_mem' : q \u2208 Multiset.map (fun p' => p' ^ i p') s.val\nhdq : d \u2223 q\n\u22a2 p \u2208 s\n[PROOFSTEP]\nobtain \u27e8q, q_mem, rfl\u27e9 := Multiset.mem_map.mp q_mem'\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero R\ninst\u271d\u2074 : UniqueFactorizationMonoid R\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b2\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\ni : \u03b1 \u2192 \u2115\np : \u03b1\nhps : \u00acp \u2208 s\nis_prime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 Prime q\nis_coprime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 \u2200 (q' : \u03b1), q' \u2208 insert p s \u2192 q \u2223 q' \u2192 q = q'\nhp : Prime p\nx\u271d d : \u03b1\nhdprod : d \u2223 \u220f p' in s, p' ^ i p'\nhd : Prime d\nhdp : d \u2223 p\nq : \u03b1\nq_mem : q \u2208 s.val\nq_mem' : q ^ i q \u2208 Multiset.map (fun p' => p' ^ i p') s.val\nhdq : d \u2223 q ^ i q\n\u22a2 p \u2208 s\n[PROOFSTEP]\nreplace hdq := hd.dvd_of_dvd_pow hdq\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero R\ninst\u271d\u2074 : UniqueFactorizationMonoid R\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b2\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\ni : \u03b1 \u2192 \u2115\np : \u03b1\nhps : \u00acp \u2208 s\nis_prime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 Prime q\nis_coprime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 \u2200 (q' : \u03b1), q' \u2208 insert p s \u2192 q \u2223 q' \u2192 q = q'\nhp : Prime p\nx\u271d d : \u03b1\nhdprod : d \u2223 \u220f p' in s, p' ^ i p'\nhd : Prime d\nhdp : d \u2223 p\nq : \u03b1\nq_mem : q \u2208 s.val\nq_mem' : q ^ i q \u2208 Multiset.map (fun p' => p' ^ i p') s.val\nhdq : d \u2223 q\n\u22a2 p \u2208 s\n[PROOFSTEP]\nhave : p \u2223 q := dvd_trans (hd.irreducible.dvd_symm hp.irreducible hdp) hdq\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero R\ninst\u271d\u2074 : UniqueFactorizationMonoid R\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b2\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\ni : \u03b1 \u2192 \u2115\np : \u03b1\nhps : \u00acp \u2208 s\nis_prime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 Prime q\nis_coprime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 \u2200 (q' : \u03b1), q' \u2208 insert p s \u2192 q \u2223 q' \u2192 q = q'\nhp : Prime p\nx\u271d d : \u03b1\nhdprod : d \u2223 \u220f p' in s, p' ^ i p'\nhd : Prime d\nhdp : d \u2223 p\nq : \u03b1\nq_mem : q \u2208 s.val\nq_mem' : q ^ i q \u2208 Multiset.map (fun p' => p' ^ i p') s.val\nhdq : d \u2223 q\nthis : p \u2223 q\n\u22a2 p \u2208 s\n[PROOFSTEP]\nconvert q_mem\n[GOAL]\ncase a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : CancelCommMonoidWithZero R\ninst\u271d\u2074 : UniqueFactorizationMonoid R\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b2\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\ni : \u03b1 \u2192 \u2115\np : \u03b1\nhps : \u00acp \u2208 s\nis_prime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 Prime q\nis_coprime : \u2200 (q : \u03b1), q \u2208 insert p s \u2192 \u2200 (q' : \u03b1), q' \u2208 insert p s \u2192 q \u2223 q' \u2192 q = q'\nhp : Prime p\nx\u271d d : \u03b1\nhdprod : d \u2223 \u220f p' in s, p' ^ i p'\nhd : Prime d\nhdp : d \u2223 p\nq : \u03b1\nq_mem : q \u2208 s.val\nq_mem' : q ^ i q \u2208 Multiset.map (fun p' => p' ^ i p') s.val\nhdq : d \u2223 q\nthis : p \u2223 q\n\u22a2 p \u2208 s \u2194 q \u2208 s.val\n[PROOFSTEP]\nrw [Finset.mem_val, is_coprime _ (Finset.mem_insert_self p s) _ (Finset.mem_insert_of_mem q_mem) this]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nP : \u03b1 \u2192 Prop\ns : Finset \u03b1\ni : \u03b1 \u2192 \u2115\nis_prime : \u2200 (p : \u03b1), p \u2208 s \u2192 Prime p\nis_coprime : \u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q\nh1 : \u2200 {x : \u03b1}, IsUnit x \u2192 P x\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 P (p ^ i)\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 P x \u2192 P y \u2192 P (x * y)\n\u22a2 P (\u220f p in s, p ^ i p)\n[PROOFSTEP]\nletI := Classical.decEq \u03b1\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nP : \u03b1 \u2192 Prop\ns : Finset \u03b1\ni : \u03b1 \u2192 \u2115\nis_prime : \u2200 (p : \u03b1), p \u2208 s \u2192 Prime p\nis_coprime : \u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q\nh1 : \u2200 {x : \u03b1}, IsUnit x \u2192 P x\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 P (p ^ i)\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 P x \u2192 P y \u2192 P (x * y)\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\n\u22a2 P (\u220f p in s, p ^ i p)\n[PROOFSTEP]\ninduction' s using Finset.induction_on with p f' hpf' ih\n[GOAL]\ncase empty\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nP : \u03b1 \u2192 Prop\ns : Finset \u03b1\ni : \u03b1 \u2192 \u2115\nis_prime\u271d : \u2200 (p : \u03b1), p \u2208 s \u2192 Prime p\nis_coprime\u271d : \u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q\nh1 : \u2200 {x : \u03b1}, IsUnit x \u2192 P x\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 P (p ^ i)\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 P x \u2192 P y \u2192 P (x * y)\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nis_prime : \u2200 (p : \u03b1), p \u2208 \u2205 \u2192 Prime p\nis_coprime : \u2200 (p : \u03b1), p \u2208 \u2205 \u2192 \u2200 (q : \u03b1), q \u2208 \u2205 \u2192 p \u2223 q \u2192 p = q\n\u22a2 P (\u220f p in \u2205, p ^ i p)\n[PROOFSTEP]\nsimpa using h1 isUnit_one\n[GOAL]\ncase insert\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nP : \u03b1 \u2192 Prop\ns : Finset \u03b1\ni : \u03b1 \u2192 \u2115\nis_prime\u271d : \u2200 (p : \u03b1), p \u2208 s \u2192 Prime p\nis_coprime\u271d : \u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q\nh1 : \u2200 {x : \u03b1}, IsUnit x \u2192 P x\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 P (p ^ i)\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 P x \u2192 P y \u2192 P (x * y)\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\np : \u03b1\nf' : Finset \u03b1\nhpf' : \u00acp \u2208 f'\nih : (\u2200 (p : \u03b1), p \u2208 f' \u2192 Prime p) \u2192 (\u2200 (p : \u03b1), p \u2208 f' \u2192 \u2200 (q : \u03b1), q \u2208 f' \u2192 p \u2223 q \u2192 p = q) \u2192 P (\u220f p in f', p ^ i p)\nis_prime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p f' \u2192 Prime p_1\nis_coprime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p f' \u2192 \u2200 (q : \u03b1), q \u2208 insert p f' \u2192 p_1 \u2223 q \u2192 p_1 = q\n\u22a2 P (\u220f p in insert p f', p ^ i p)\n[PROOFSTEP]\nrw [Finset.prod_insert hpf']\n[GOAL]\ncase insert\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nP : \u03b1 \u2192 Prop\ns : Finset \u03b1\ni : \u03b1 \u2192 \u2115\nis_prime\u271d : \u2200 (p : \u03b1), p \u2208 s \u2192 Prime p\nis_coprime\u271d : \u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q\nh1 : \u2200 {x : \u03b1}, IsUnit x \u2192 P x\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 P (p ^ i)\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 P x \u2192 P y \u2192 P (x * y)\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\np : \u03b1\nf' : Finset \u03b1\nhpf' : \u00acp \u2208 f'\nih : (\u2200 (p : \u03b1), p \u2208 f' \u2192 Prime p) \u2192 (\u2200 (p : \u03b1), p \u2208 f' \u2192 \u2200 (q : \u03b1), q \u2208 f' \u2192 p \u2223 q \u2192 p = q) \u2192 P (\u220f p in f', p ^ i p)\nis_prime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p f' \u2192 Prime p_1\nis_coprime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p f' \u2192 \u2200 (q : \u03b1), q \u2208 insert p f' \u2192 p_1 \u2223 q \u2192 p_1 = q\n\u22a2 P (p ^ i p * \u220f x in f', x ^ i x)\n[PROOFSTEP]\nexact\n hcp (prime_pow_coprime_prod_of_coprime_insert i p hpf' is_prime is_coprime)\n (hpr (i p) (is_prime _ (Finset.mem_insert_self _ _)))\n (ih (fun q hq => is_prime _ (Finset.mem_insert_of_mem hq)) fun q hq q' hq' =>\n is_coprime _ (Finset.mem_insert_of_mem hq) _ (Finset.mem_insert_of_mem hq'))\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nP : \u03b1 \u2192 Prop\na : \u03b1\nh0 : P 0\nh1 : \u2200 {x : \u03b1}, IsUnit x \u2192 P x\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 P (p ^ i)\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 P x \u2192 P y \u2192 P (x * y)\n\u22a2 P a\n[PROOFSTEP]\nletI := Classical.decEq \u03b1\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nP : \u03b1 \u2192 Prop\na : \u03b1\nh0 : P 0\nh1 : \u2200 {x : \u03b1}, IsUnit x \u2192 P x\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 P (p ^ i)\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 P x \u2192 P y \u2192 P (x * y)\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\n\u22a2 P a\n[PROOFSTEP]\nhave P_of_associated : \u2200 {x y}, Associated x y \u2192 P x \u2192 P y :=\n by\n rintro x y \u27e8u, rfl\u27e9 hx\n exact hcp (fun p _ hpx => isUnit_of_dvd_unit hpx u.isUnit) hx (h1 u.isUnit)\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nP : \u03b1 \u2192 Prop\na : \u03b1\nh0 : P 0\nh1 : \u2200 {x : \u03b1}, IsUnit x \u2192 P x\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 P (p ^ i)\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 P x \u2192 P y \u2192 P (x * y)\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\n\u22a2 \u2200 {x y : \u03b1}, x ~\u1d64 y \u2192 P x \u2192 P y\n[PROOFSTEP]\nrintro x y \u27e8u, rfl\u27e9 hx\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nP : \u03b1 \u2192 Prop\na : \u03b1\nh0 : P 0\nh1 : \u2200 {x : \u03b1}, IsUnit x \u2192 P x\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 P (p ^ i)\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 P x \u2192 P y \u2192 P (x * y)\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nx : \u03b1\nu : \u03b1\u02e3\nhx : P x\n\u22a2 P (x * \u2191u)\n[PROOFSTEP]\nexact hcp (fun p _ hpx => isUnit_of_dvd_unit hpx u.isUnit) hx (h1 u.isUnit)\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nP : \u03b1 \u2192 Prop\na : \u03b1\nh0 : P 0\nh1 : \u2200 {x : \u03b1}, IsUnit x \u2192 P x\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 P (p ^ i)\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 P x \u2192 P y \u2192 P (x * y)\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nP_of_associated : \u2200 {x y : \u03b1}, x ~\u1d64 y \u2192 P x \u2192 P y\n\u22a2 P a\n[PROOFSTEP]\nby_cases ha0 : a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nP : \u03b1 \u2192 Prop\na : \u03b1\nh0 : P 0\nh1 : \u2200 {x : \u03b1}, IsUnit x \u2192 P x\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 P (p ^ i)\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 P x \u2192 P y \u2192 P (x * y)\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nP_of_associated : \u2200 {x y : \u03b1}, x ~\u1d64 y \u2192 P x \u2192 P y\nha0 : a = 0\n\u22a2 P a\n[PROOFSTEP]\nrwa [ha0]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nP : \u03b1 \u2192 Prop\na : \u03b1\nh0 : P 0\nh1 : \u2200 {x : \u03b1}, IsUnit x \u2192 P x\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 P (p ^ i)\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 P x \u2192 P y \u2192 P (x * y)\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nP_of_associated : \u2200 {x y : \u03b1}, x ~\u1d64 y \u2192 P x \u2192 P y\nha0 : \u00aca = 0\n\u22a2 P a\n[PROOFSTEP]\nhaveI : Nontrivial \u03b1 := \u27e8\u27e8_, _, ha0\u27e9\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nP : \u03b1 \u2192 Prop\na : \u03b1\nh0 : P 0\nh1 : \u2200 {x : \u03b1}, IsUnit x \u2192 P x\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 P (p ^ i)\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 P x \u2192 P y \u2192 P (x * y)\nthis\u271d : DecidableEq \u03b1 := Classical.decEq \u03b1\nP_of_associated : \u2200 {x y : \u03b1}, x ~\u1d64 y \u2192 P x \u2192 P y\nha0 : \u00aca = 0\nthis : Nontrivial \u03b1\n\u22a2 P a\n[PROOFSTEP]\nletI : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nP : \u03b1 \u2192 Prop\na : \u03b1\nh0 : P 0\nh1 : \u2200 {x : \u03b1}, IsUnit x \u2192 P x\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 P (p ^ i)\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 P x \u2192 P y \u2192 P (x * y)\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nP_of_associated : \u2200 {x y : \u03b1}, x ~\u1d64 y \u2192 P x \u2192 P y\nha0 : \u00aca = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 P a\n[PROOFSTEP]\nrefine' P_of_associated (normalizedFactors_prod ha0) _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nP : \u03b1 \u2192 Prop\na : \u03b1\nh0 : P 0\nh1 : \u2200 {x : \u03b1}, IsUnit x \u2192 P x\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 P (p ^ i)\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 P x \u2192 P y \u2192 P (x * y)\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nP_of_associated : \u2200 {x y : \u03b1}, x ~\u1d64 y \u2192 P x \u2192 P y\nha0 : \u00aca = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 P (Multiset.prod (normalizedFactors a))\n[PROOFSTEP]\nrw [\u2190 (normalizedFactors a).map_id, Finset.prod_multiset_map_count]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nP : \u03b1 \u2192 Prop\na : \u03b1\nh0 : P 0\nh1 : \u2200 {x : \u03b1}, IsUnit x \u2192 P x\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 P (p ^ i)\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 P x \u2192 P y \u2192 P (x * y)\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nP_of_associated : \u2200 {x y : \u03b1}, x ~\u1d64 y \u2192 P x \u2192 P y\nha0 : \u00aca = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 P (\u220f m in Multiset.toFinset (normalizedFactors a), id m ^ Multiset.count m (normalizedFactors a))\n[PROOFSTEP]\nrefine' induction_on_prime_power _ _ _ _ @h1 @hpr @hcp\n[GOAL]\ncase neg.refine'_1\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nP : \u03b1 \u2192 Prop\na : \u03b1\nh0 : P 0\nh1 : \u2200 {x : \u03b1}, IsUnit x \u2192 P x\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 P (p ^ i)\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 P x \u2192 P y \u2192 P (x * y)\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nP_of_associated : \u2200 {x y : \u03b1}, x ~\u1d64 y \u2192 P x \u2192 P y\nha0 : \u00aca = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 \u2200 (p : \u03b1), p \u2208 Multiset.toFinset (normalizedFactors a) \u2192 Prime p\n[PROOFSTEP]\nsimp only [Multiset.mem_toFinset]\n[GOAL]\ncase neg.refine'_2\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nP : \u03b1 \u2192 Prop\na : \u03b1\nh0 : P 0\nh1 : \u2200 {x : \u03b1}, IsUnit x \u2192 P x\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 P (p ^ i)\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 P x \u2192 P y \u2192 P (x * y)\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nP_of_associated : \u2200 {x y : \u03b1}, x ~\u1d64 y \u2192 P x \u2192 P y\nha0 : \u00aca = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 \u2200 (p : \u03b1),\n p \u2208 Multiset.toFinset (normalizedFactors a) \u2192 \u2200 (q : \u03b1), q \u2208 Multiset.toFinset (normalizedFactors a) \u2192 p \u2223 q \u2192 p = q\n[PROOFSTEP]\nsimp only [Multiset.mem_toFinset]\n[GOAL]\ncase neg.refine'_1\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nP : \u03b1 \u2192 Prop\na : \u03b1\nh0 : P 0\nh1 : \u2200 {x : \u03b1}, IsUnit x \u2192 P x\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 P (p ^ i)\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 P x \u2192 P y \u2192 P (x * y)\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nP_of_associated : \u2200 {x y : \u03b1}, x ~\u1d64 y \u2192 P x \u2192 P y\nha0 : \u00aca = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 \u2200 (p : \u03b1), p \u2208 normalizedFactors a \u2192 Prime p\n[PROOFSTEP]\napply prime_of_normalized_factor\n[GOAL]\ncase neg.refine'_2\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nP : \u03b1 \u2192 Prop\na : \u03b1\nh0 : P 0\nh1 : \u2200 {x : \u03b1}, IsUnit x \u2192 P x\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 P (p ^ i)\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 P x \u2192 P y \u2192 P (x * y)\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nP_of_associated : \u2200 {x y : \u03b1}, x ~\u1d64 y \u2192 P x \u2192 P y\nha0 : \u00aca = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 \u2200 (p : \u03b1), p \u2208 normalizedFactors a \u2192 \u2200 (q : \u03b1), q \u2208 normalizedFactors a \u2192 p \u2223 q \u2192 p = q\n[PROOFSTEP]\napply normalizedFactors_eq_of_dvd\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\ni j : \u03b1 \u2192 \u2115\nis_prime : \u2200 (p : \u03b1), p \u2208 s \u2192 Prime p\nis_coprime : \u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\n\u22a2 f (\u220f p in s, p ^ (i p + j p)) = f (\u220f p in s, p ^ i p) * f (\u220f p in s, p ^ j p)\n[PROOFSTEP]\nletI := Classical.decEq \u03b1\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\ni j : \u03b1 \u2192 \u2115\nis_prime : \u2200 (p : \u03b1), p \u2208 s \u2192 Prime p\nis_coprime : \u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\n\u22a2 f (\u220f p in s, p ^ (i p + j p)) = f (\u220f p in s, p ^ i p) * f (\u220f p in s, p ^ j p)\n[PROOFSTEP]\ninduction' s using Finset.induction_on with p s hps ih\n[GOAL]\ncase empty\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\ni j : \u03b1 \u2192 \u2115\nis_prime\u271d : \u2200 (p : \u03b1), p \u2208 s \u2192 Prime p\nis_coprime\u271d : \u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nis_prime : \u2200 (p : \u03b1), p \u2208 \u2205 \u2192 Prime p\nis_coprime : \u2200 (p : \u03b1), p \u2208 \u2205 \u2192 \u2200 (q : \u03b1), q \u2208 \u2205 \u2192 p \u2223 q \u2192 p = q\n\u22a2 f (\u220f p in \u2205, p ^ (i p + j p)) = f (\u220f p in \u2205, p ^ i p) * f (\u220f p in \u2205, p ^ j p)\n[PROOFSTEP]\nsimpa using h1 isUnit_one\n[GOAL]\ncase insert\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Finset \u03b1\ni j : \u03b1 \u2192 \u2115\nis_prime\u271d : \u2200 (p : \u03b1), p \u2208 s\u271d \u2192 Prime p\nis_coprime\u271d : \u2200 (p : \u03b1), p \u2208 s\u271d \u2192 \u2200 (q : \u03b1), q \u2208 s\u271d \u2192 p \u2223 q \u2192 p = q\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\np : \u03b1\ns : Finset \u03b1\nhps : \u00acp \u2208 s\nih :\n (\u2200 (p : \u03b1), p \u2208 s \u2192 Prime p) \u2192\n (\u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q) \u2192\n f (\u220f p in s, p ^ (i p + j p)) = f (\u220f p in s, p ^ i p) * f (\u220f p in s, p ^ j p)\nis_prime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p s \u2192 Prime p_1\nis_coprime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p s \u2192 \u2200 (q : \u03b1), q \u2208 insert p s \u2192 p_1 \u2223 q \u2192 p_1 = q\n\u22a2 f (\u220f p in insert p s, p ^ (i p + j p)) = f (\u220f p in insert p s, p ^ i p) * f (\u220f p in insert p s, p ^ j p)\n[PROOFSTEP]\nhave hpr_p := is_prime _ (Finset.mem_insert_self _ _)\n[GOAL]\ncase insert\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Finset \u03b1\ni j : \u03b1 \u2192 \u2115\nis_prime\u271d : \u2200 (p : \u03b1), p \u2208 s\u271d \u2192 Prime p\nis_coprime\u271d : \u2200 (p : \u03b1), p \u2208 s\u271d \u2192 \u2200 (q : \u03b1), q \u2208 s\u271d \u2192 p \u2223 q \u2192 p = q\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\np : \u03b1\ns : Finset \u03b1\nhps : \u00acp \u2208 s\nih :\n (\u2200 (p : \u03b1), p \u2208 s \u2192 Prime p) \u2192\n (\u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q) \u2192\n f (\u220f p in s, p ^ (i p + j p)) = f (\u220f p in s, p ^ i p) * f (\u220f p in s, p ^ j p)\nis_prime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p s \u2192 Prime p_1\nis_coprime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p s \u2192 \u2200 (q : \u03b1), q \u2208 insert p s \u2192 p_1 \u2223 q \u2192 p_1 = q\nhpr_p : Prime p\n\u22a2 f (\u220f p in insert p s, p ^ (i p + j p)) = f (\u220f p in insert p s, p ^ i p) * f (\u220f p in insert p s, p ^ j p)\n[PROOFSTEP]\nhave hpr_s : \u2200 p \u2208 s, Prime p := fun p hp => is_prime _ (Finset.mem_insert_of_mem hp)\n[GOAL]\ncase insert\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Finset \u03b1\ni j : \u03b1 \u2192 \u2115\nis_prime\u271d : \u2200 (p : \u03b1), p \u2208 s\u271d \u2192 Prime p\nis_coprime\u271d : \u2200 (p : \u03b1), p \u2208 s\u271d \u2192 \u2200 (q : \u03b1), q \u2208 s\u271d \u2192 p \u2223 q \u2192 p = q\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\np : \u03b1\ns : Finset \u03b1\nhps : \u00acp \u2208 s\nih :\n (\u2200 (p : \u03b1), p \u2208 s \u2192 Prime p) \u2192\n (\u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q) \u2192\n f (\u220f p in s, p ^ (i p + j p)) = f (\u220f p in s, p ^ i p) * f (\u220f p in s, p ^ j p)\nis_prime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p s \u2192 Prime p_1\nis_coprime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p s \u2192 \u2200 (q : \u03b1), q \u2208 insert p s \u2192 p_1 \u2223 q \u2192 p_1 = q\nhpr_p : Prime p\nhpr_s : \u2200 (p : \u03b1), p \u2208 s \u2192 Prime p\n\u22a2 f (\u220f p in insert p s, p ^ (i p + j p)) = f (\u220f p in insert p s, p ^ i p) * f (\u220f p in insert p s, p ^ j p)\n[PROOFSTEP]\nhave hcp_p := fun i => prime_pow_coprime_prod_of_coprime_insert i p hps is_prime is_coprime\n[GOAL]\ncase insert\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Finset \u03b1\ni j : \u03b1 \u2192 \u2115\nis_prime\u271d : \u2200 (p : \u03b1), p \u2208 s\u271d \u2192 Prime p\nis_coprime\u271d : \u2200 (p : \u03b1), p \u2208 s\u271d \u2192 \u2200 (q : \u03b1), q \u2208 s\u271d \u2192 p \u2223 q \u2192 p = q\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\np : \u03b1\ns : Finset \u03b1\nhps : \u00acp \u2208 s\nih :\n (\u2200 (p : \u03b1), p \u2208 s \u2192 Prime p) \u2192\n (\u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q) \u2192\n f (\u220f p in s, p ^ (i p + j p)) = f (\u220f p in s, p ^ i p) * f (\u220f p in s, p ^ j p)\nis_prime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p s \u2192 Prime p_1\nis_coprime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p s \u2192 \u2200 (q : \u03b1), q \u2208 insert p s \u2192 p_1 \u2223 q \u2192 p_1 = q\nhpr_p : Prime p\nhpr_s : \u2200 (p : \u03b1), p \u2208 s \u2192 Prime p\nhcp_p : \u2200 (i : \u03b1 \u2192 \u2115) (q : \u03b1), q \u2223 p ^ i p \u2192 q \u2223 \u220f p' in s, p' ^ i p' \u2192 IsUnit q\n\u22a2 f (\u220f p in insert p s, p ^ (i p + j p)) = f (\u220f p in insert p s, p ^ i p) * f (\u220f p in insert p s, p ^ j p)\n[PROOFSTEP]\nhave hcp_s : \u2200 (p) (_ : p \u2208 s) (q) (_ : q \u2208 s), p \u2223 q \u2192 p = q := fun p hp q hq =>\n is_coprime p (Finset.mem_insert_of_mem hp) q (Finset.mem_insert_of_mem hq)\n[GOAL]\ncase insert\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Finset \u03b1\ni j : \u03b1 \u2192 \u2115\nis_prime\u271d : \u2200 (p : \u03b1), p \u2208 s\u271d \u2192 Prime p\nis_coprime\u271d : \u2200 (p : \u03b1), p \u2208 s\u271d \u2192 \u2200 (q : \u03b1), q \u2208 s\u271d \u2192 p \u2223 q \u2192 p = q\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\np : \u03b1\ns : Finset \u03b1\nhps : \u00acp \u2208 s\nih :\n (\u2200 (p : \u03b1), p \u2208 s \u2192 Prime p) \u2192\n (\u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q) \u2192\n f (\u220f p in s, p ^ (i p + j p)) = f (\u220f p in s, p ^ i p) * f (\u220f p in s, p ^ j p)\nis_prime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p s \u2192 Prime p_1\nis_coprime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p s \u2192 \u2200 (q : \u03b1), q \u2208 insert p s \u2192 p_1 \u2223 q \u2192 p_1 = q\nhpr_p : Prime p\nhpr_s : \u2200 (p : \u03b1), p \u2208 s \u2192 Prime p\nhcp_p : \u2200 (i : \u03b1 \u2192 \u2115) (q : \u03b1), q \u2223 p ^ i p \u2192 q \u2223 \u220f p' in s, p' ^ i p' \u2192 IsUnit q\nhcp_s : \u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q\n\u22a2 f (\u220f p in insert p s, p ^ (i p + j p)) = f (\u220f p in insert p s, p ^ i p) * f (\u220f p in insert p s, p ^ j p)\n[PROOFSTEP]\nrw [Finset.prod_insert hps, Finset.prod_insert hps, Finset.prod_insert hps, hcp (hcp_p _), hpr _ hpr_p, hcp (hcp_p _),\n hpr _ hpr_p, hcp (hcp_p (fun p => i p + j p)), hpr _ hpr_p, ih hpr_s hcp_s, pow_add, mul_assoc,\n mul_left_comm (f p ^ j p), mul_assoc]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\n\u22a2 f (a * b) = f a * f b\n[PROOFSTEP]\nletI := Classical.decEq \u03b1\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\n\u22a2 f (a * b) = f a * f b\n[PROOFSTEP]\nby_cases ha0 : a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : a = 0\n\u22a2 f (a * b) = f a * f b\n[PROOFSTEP]\nrw [ha0, zero_mul, h0, zero_mul]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\n\u22a2 f (a * b) = f a * f b\n[PROOFSTEP]\nby_cases hb0 : b = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : b = 0\n\u22a2 f (a * b) = f a * f b\n[PROOFSTEP]\nrw [hb0, mul_zero, h0, mul_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\n\u22a2 f (a * b) = f a * f b\n[PROOFSTEP]\nby_cases hf1 : f 1 = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : f 1 = 0\n\u22a2 f (a * b) = f a * f b\n[PROOFSTEP]\ncalc\n f (a * b) = f (a * b * 1) := by rw [mul_one]\n _ = 0 := by simp only [h1 isUnit_one, hf1, mul_zero]\n _ = f a * f (b * 1) := by simp only [h1 isUnit_one, hf1, mul_zero]\n _ = f a * f b := by rw [mul_one]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : f 1 = 0\n\u22a2 f (a * b) = f (a * b * 1)\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : f 1 = 0\n\u22a2 f (a * b * 1) = 0\n[PROOFSTEP]\nsimp only [h1 isUnit_one, hf1, mul_zero]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : f 1 = 0\n\u22a2 0 = f a * f (b * 1)\n[PROOFSTEP]\nsimp only [h1 isUnit_one, hf1, mul_zero]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : f 1 = 0\n\u22a2 f a * f (b * 1) = f a * f b\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\n\u22a2 f (a * b) = f a * f b\n[PROOFSTEP]\nhaveI : Nontrivial \u03b1 := \u27e8\u27e8_, _, ha0\u27e9\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis : Nontrivial \u03b1\n\u22a2 f (a * b) = f a * f b\n[PROOFSTEP]\nletI : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 f (a * b) = f a * f b\n[PROOFSTEP]\nsuffices\n f\n (\u220f p in (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n p ^ ((normalizedFactors a).count p + (normalizedFactors b).count p)) =\n f (\u220f p in (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset, p ^ (normalizedFactors a).count p) *\n f (\u220f p : \u03b1 in (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset, p ^ (normalizedFactors b).count p)\n by\n obtain \u27e8ua, a_eq\u27e9 := normalizedFactors_prod ha0\n obtain \u27e8ub, b_eq\u27e9 := normalizedFactors_prod hb0\n rw [\u2190 a_eq, \u2190 b_eq,\n mul_right_comm (Multiset.prod (normalizedFactors a)) ua (Multiset.prod (normalizedFactors b) * ub), h1 ua.isUnit,\n h1 ub.isUnit, h1 ua.isUnit, \u2190 mul_assoc, h1 ub.isUnit, mul_right_comm _ (f ua), \u2190 mul_assoc]\n congr\n rw [\u2190 (normalizedFactors a).map_id, \u2190 (normalizedFactors b).map_id, Finset.prod_multiset_map_count,\n Finset.prod_multiset_map_count, Finset.prod_subset (Finset.subset_union_left _ (normalizedFactors b).toFinset),\n Finset.prod_subset (Finset.subset_union_right _ (normalizedFactors b).toFinset), \u2190 Finset.prod_mul_distrib]\n simp_rw [id.def, \u2190 pow_add, this]\n all_goals simp only [Multiset.mem_toFinset]\n \u00b7 intro p _ hpb\n simp [hpb]\n \u00b7 intro p _ hpa\n simp [hpa]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\n\u22a2 f (a * b) = f a * f b\n[PROOFSTEP]\nobtain \u27e8ua, a_eq\u27e9 := normalizedFactors_prod ha0\n[GOAL]\ncase intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : Multiset.prod (normalizedFactors a) * \u2191ua = a\n\u22a2 f (a * b) = f a * f b\n[PROOFSTEP]\nobtain \u27e8ub, b_eq\u27e9 := normalizedFactors_prod hb0\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : Multiset.prod (normalizedFactors a) * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : Multiset.prod (normalizedFactors b) * \u2191ub = b\n\u22a2 f (a * b) = f a * f b\n[PROOFSTEP]\nrw [\u2190 a_eq, \u2190 b_eq, mul_right_comm (Multiset.prod (normalizedFactors a)) ua (Multiset.prod (normalizedFactors b) * ub),\n h1 ua.isUnit, h1 ub.isUnit, h1 ua.isUnit, \u2190 mul_assoc, h1 ub.isUnit, mul_right_comm _ (f ua), \u2190 mul_assoc]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : Multiset.prod (normalizedFactors a) * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : Multiset.prod (normalizedFactors b) * \u2191ub = b\n\u22a2 f (Multiset.prod (normalizedFactors a) * Multiset.prod (normalizedFactors b)) * f \u2191ub * f \u2191ua =\n f (Multiset.prod (normalizedFactors a)) * f (Multiset.prod (normalizedFactors b)) * f \u2191ub * f \u2191ua\n[PROOFSTEP]\ncongr\n[GOAL]\ncase intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : Multiset.prod (normalizedFactors a) * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : Multiset.prod (normalizedFactors b) * \u2191ub = b\n\u22a2 f (Multiset.prod (normalizedFactors a) * Multiset.prod (normalizedFactors b)) =\n f (Multiset.prod (normalizedFactors a)) * f (Multiset.prod (normalizedFactors b))\n[PROOFSTEP]\nrw [\u2190 (normalizedFactors a).map_id, \u2190 (normalizedFactors b).map_id, Finset.prod_multiset_map_count,\n Finset.prod_multiset_map_count, Finset.prod_subset (Finset.subset_union_left _ (normalizedFactors b).toFinset),\n Finset.prod_subset (Finset.subset_union_right _ (normalizedFactors b).toFinset), \u2190 Finset.prod_mul_distrib]\n[GOAL]\ncase intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : Multiset.prod (normalizedFactors a) * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : Multiset.prod (normalizedFactors b) * \u2191ub = b\n\u22a2 f\n (\u220f x in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n id x ^ Multiset.count x (normalizedFactors a) * id x ^ Multiset.count x (normalizedFactors b)) =\n f\n (\u220f x in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n id x ^ Multiset.count x (normalizedFactors a)) *\n f\n (\u220f x in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n id x ^ Multiset.count x (normalizedFactors b))\ncase intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : Multiset.prod (normalizedFactors a) * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : Multiset.prod (normalizedFactors b) * \u2191ub = b\n\u22a2 \u2200 (x : \u03b1),\n x \u2208 Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b) \u2192\n \u00acx \u2208 Multiset.toFinset (normalizedFactors b) \u2192 id x ^ Multiset.count x (normalizedFactors b) = 1\ncase intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : Multiset.prod (normalizedFactors a) * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : Multiset.prod (normalizedFactors b) * \u2191ub = b\n\u22a2 \u2200 (x : \u03b1),\n x \u2208 Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b) \u2192\n \u00acx \u2208 Multiset.toFinset (normalizedFactors a) \u2192 id x ^ Multiset.count x (normalizedFactors a) = 1\n[PROOFSTEP]\nsimp_rw [id.def, \u2190 pow_add, this]\n[GOAL]\ncase intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : Multiset.prod (normalizedFactors a) * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : Multiset.prod (normalizedFactors b) * \u2191ub = b\n\u22a2 \u2200 (x : \u03b1),\n x \u2208 Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b) \u2192\n \u00acx \u2208 Multiset.toFinset (normalizedFactors b) \u2192 id x ^ Multiset.count x (normalizedFactors b) = 1\ncase intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : Multiset.prod (normalizedFactors a) * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : Multiset.prod (normalizedFactors b) * \u2191ub = b\n\u22a2 \u2200 (x : \u03b1),\n x \u2208 Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b) \u2192\n \u00acx \u2208 Multiset.toFinset (normalizedFactors a) \u2192 id x ^ Multiset.count x (normalizedFactors a) = 1\n[PROOFSTEP]\nall_goals simp only [Multiset.mem_toFinset]\n[GOAL]\ncase intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : Multiset.prod (normalizedFactors a) * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : Multiset.prod (normalizedFactors b) * \u2191ub = b\n\u22a2 \u2200 (x : \u03b1),\n x \u2208 Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b) \u2192\n \u00acx \u2208 Multiset.toFinset (normalizedFactors b) \u2192 id x ^ Multiset.count x (normalizedFactors b) = 1\n[PROOFSTEP]\nsimp only [Multiset.mem_toFinset]\n[GOAL]\ncase intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : Multiset.prod (normalizedFactors a) * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : Multiset.prod (normalizedFactors b) * \u2191ub = b\n\u22a2 \u2200 (x : \u03b1),\n x \u2208 Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b) \u2192\n \u00acx \u2208 Multiset.toFinset (normalizedFactors a) \u2192 id x ^ Multiset.count x (normalizedFactors a) = 1\n[PROOFSTEP]\nsimp only [Multiset.mem_toFinset]\n[GOAL]\ncase intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : Multiset.prod (normalizedFactors a) * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : Multiset.prod (normalizedFactors b) * \u2191ub = b\n\u22a2 \u2200 (x : \u03b1),\n x \u2208 Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b) \u2192\n \u00acx \u2208 normalizedFactors b \u2192 id x ^ Multiset.count x (normalizedFactors b) = 1\n[PROOFSTEP]\nintro p _ hpb\n[GOAL]\ncase intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : Multiset.prod (normalizedFactors a) * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : Multiset.prod (normalizedFactors b) * \u2191ub = b\np : \u03b1\na\u271d : p \u2208 Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b)\nhpb : \u00acp \u2208 normalizedFactors b\n\u22a2 id p ^ Multiset.count p (normalizedFactors b) = 1\n[PROOFSTEP]\nsimp [hpb]\n[GOAL]\ncase intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : Multiset.prod (normalizedFactors a) * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : Multiset.prod (normalizedFactors b) * \u2191ub = b\n\u22a2 \u2200 (x : \u03b1),\n x \u2208 Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b) \u2192\n \u00acx \u2208 normalizedFactors a \u2192 id x ^ Multiset.count x (normalizedFactors a) = 1\n[PROOFSTEP]\nintro p _ hpa\n[GOAL]\ncase intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : Multiset.prod (normalizedFactors a) * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : Multiset.prod (normalizedFactors b) * \u2191ub = b\np : \u03b1\na\u271d : p \u2208 Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b)\nhpa : \u00acp \u2208 normalizedFactors a\n\u22a2 id p ^ Multiset.count p (normalizedFactors a) = 1\n[PROOFSTEP]\nsimp [hpa]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors a)) *\n f\n (\u220f p in Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b),\n p ^ Multiset.count p (normalizedFactors b))\n[PROOFSTEP]\nrefine' multiplicative_prime_power _ _ _ _ _ @h1 @hpr @hcp\n[GOAL]\ncase neg.refine'_1\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 \u2200 (p : \u03b1), p \u2208 Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b) \u2192 Prime p\ncase neg.refine'_2\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 \u2200 (p : \u03b1),\n p \u2208 Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b) \u2192\n \u2200 (q : \u03b1), q \u2208 Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b) \u2192 p \u2223 q \u2192 p = q\n[PROOFSTEP]\nall_goals simp only [Multiset.mem_toFinset, Finset.mem_union]\n[GOAL]\ncase neg.refine'_1\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 \u2200 (p : \u03b1), p \u2208 Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b) \u2192 Prime p\n[PROOFSTEP]\nsimp only [Multiset.mem_toFinset, Finset.mem_union]\n[GOAL]\ncase neg.refine'_2\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 \u2200 (p : \u03b1),\n p \u2208 Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b) \u2192\n \u2200 (q : \u03b1), q \u2208 Multiset.toFinset (normalizedFactors a) \u222a Multiset.toFinset (normalizedFactors b) \u2192 p \u2223 q \u2192 p = q\n[PROOFSTEP]\nsimp only [Multiset.mem_toFinset, Finset.mem_union]\n[GOAL]\ncase neg.refine'_1\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 \u2200 (p : \u03b1), p \u2208 normalizedFactors a \u2228 p \u2208 normalizedFactors b \u2192 Prime p\n[PROOFSTEP]\nrintro p (hpa | hpb)\n[GOAL]\ncase neg.refine'_1.inl\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\np : \u03b1\nhpa : p \u2208 normalizedFactors a\n\u22a2 Prime p\n[PROOFSTEP]\napply prime_of_normalized_factor\n[GOAL]\ncase neg.refine'_1.inr\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\np : \u03b1\nhpb : p \u2208 normalizedFactors b\n\u22a2 Prime p\n[PROOFSTEP]\napply prime_of_normalized_factor\n[GOAL]\ncase neg.refine'_1.inl.a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\np : \u03b1\nhpa : p \u2208 normalizedFactors a\n\u22a2 p \u2208 normalizedFactors ?neg.refine'_1.inl.a\u271d\n[PROOFSTEP]\nassumption\n[GOAL]\ncase neg.refine'_1.inr.a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\np : \u03b1\nhpb : p \u2208 normalizedFactors b\n\u22a2 p \u2208 normalizedFactors ?neg.refine'_1.inr.a\u271d\n[PROOFSTEP]\nassumption\n[GOAL]\ncase neg.refine'_2\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 \u2200 (p : \u03b1),\n p \u2208 normalizedFactors a \u2228 p \u2208 normalizedFactors b \u2192\n \u2200 (q : \u03b1), q \u2208 normalizedFactors a \u2228 q \u2208 normalizedFactors b \u2192 p \u2223 q \u2192 p = q\n[PROOFSTEP]\nrintro p (hp | hp) q (hq | hq) hdvd\n[GOAL]\ncase neg.refine'_2.inl.inl\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\np : \u03b1\nhp : p \u2208 normalizedFactors a\nq : \u03b1\nhq : q \u2208 normalizedFactors a\nhdvd : p \u2223 q\n\u22a2 p = q\n[PROOFSTEP]\nrw [\u2190 normalize_normalized_factor _ hp, \u2190 normalize_normalized_factor _ hq]\n[GOAL]\ncase neg.refine'_2.inl.inr\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\np : \u03b1\nhp : p \u2208 normalizedFactors a\nq : \u03b1\nhq : q \u2208 normalizedFactors b\nhdvd : p \u2223 q\n\u22a2 p = q\n[PROOFSTEP]\nrw [\u2190 normalize_normalized_factor _ hp, \u2190 normalize_normalized_factor _ hq]\n[GOAL]\ncase neg.refine'_2.inr.inl\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\np : \u03b1\nhp : p \u2208 normalizedFactors b\nq : \u03b1\nhq : q \u2208 normalizedFactors a\nhdvd : p \u2223 q\n\u22a2 p = q\n[PROOFSTEP]\nrw [\u2190 normalize_normalized_factor _ hp, \u2190 normalize_normalized_factor _ hq]\n[GOAL]\ncase neg.refine'_2.inr.inr\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\np : \u03b1\nhp : p \u2208 normalizedFactors b\nq : \u03b1\nhq : q \u2208 normalizedFactors b\nhdvd : p \u2223 q\n\u22a2 p = q\n[PROOFSTEP]\nrw [\u2190 normalize_normalized_factor _ hp, \u2190 normalize_normalized_factor _ hq]\n[GOAL]\ncase neg.refine'_2.inl.inl\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\np : \u03b1\nhp : p \u2208 normalizedFactors a\nq : \u03b1\nhq : q \u2208 normalizedFactors a\nhdvd : p \u2223 q\n\u22a2 \u2191normalize p = \u2191normalize q\n[PROOFSTEP]\nexact\n normalize_eq_normalize hdvd\n ((prime_of_normalized_factor _ hp).irreducible.dvd_symm (prime_of_normalized_factor _ hq).irreducible hdvd)\n[GOAL]\ncase neg.refine'_2.inl.inr\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\np : \u03b1\nhp : p \u2208 normalizedFactors a\nq : \u03b1\nhq : q \u2208 normalizedFactors b\nhdvd : p \u2223 q\n\u22a2 \u2191normalize p = \u2191normalize q\n[PROOFSTEP]\nexact\n normalize_eq_normalize hdvd\n ((prime_of_normalized_factor _ hp).irreducible.dvd_symm (prime_of_normalized_factor _ hq).irreducible hdvd)\n[GOAL]\ncase neg.refine'_2.inr.inl\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\np : \u03b1\nhp : p \u2208 normalizedFactors b\nq : \u03b1\nhq : q \u2208 normalizedFactors a\nhdvd : p \u2223 q\n\u22a2 \u2191normalize p = \u2191normalize q\n[PROOFSTEP]\nexact\n normalize_eq_normalize hdvd\n ((prime_of_normalized_factor _ hp).irreducible.dvd_symm (prime_of_normalized_factor _ hq).irreducible hdvd)\n[GOAL]\ncase neg.refine'_2.inr.inr\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\np : \u03b1\nhp : p \u2208 normalizedFactors b\nq : \u03b1\nhq : q \u2208 normalizedFactors b\nhdvd : p \u2223 q\n\u22a2 \u2191normalize p = \u2191normalize q\n[PROOFSTEP]\nexact\n normalize_eq_normalize hdvd\n ((prime_of_normalized_factor _ hp).irreducible.dvd_symm (prime_of_normalized_factor _ hq).irreducible hdvd)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na b : Multiset { a // Irreducible a }\n\u22a2 \u2191(a + b) = \u2191a + \u2191b\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq (Associates \u03b1)\nb : FactorSet \u03b1\n\u22a2 \u22a4 \u2294 b + \u22a4 \u2293 b = \u22a4 + b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq (Associates \u03b1)\na : FactorSet \u03b1\n\u22a2 a \u2294 \u22a4 + a \u2293 \u22a4 = a + \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq (Associates \u03b1)\na b : Multiset { a // Irreducible a }\n\u22a2 \u2191a \u2294 \u2191b + \u2191a \u2293 \u2191b = \u2191a + \u2191b\n[PROOFSTEP]\nrw [\u2190 WithTop.coe_sup, \u2190 WithTop.coe_inf, \u2190 WithTop.coe_add, \u2190 WithTop.coe_add, WithTop.coe_eq_coe]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : DecidableEq (Associates \u03b1)\na b : Multiset { a // Irreducible a }\n\u22a2 a \u2294 b + a \u2293 b = a + b\n[PROOFSTEP]\nexact Multiset.union_add_inter _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\nb : FactorSet \u03b1\n\u22a2 FactorSet.prod (\u22a4 + b) = FactorSet.prod \u22a4 * FactorSet.prod b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : FactorSet \u03b1\n\u22a2 FactorSet.prod (a + \u22a4) = FactorSet.prod a * FactorSet.prod \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na b : Multiset { a // Irreducible a }\n\u22a2 FactorSet.prod (\u2191a + \u2191b) = FactorSet.prod \u2191a * FactorSet.prod \u2191b\n[PROOFSTEP]\nrw [\u2190 FactorSet.coe_add, prod_coe, prod_coe, prod_coe, Multiset.map_add, Multiset.prod_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\nb : FactorSet \u03b1\nh : none \u2264 b\n\u22a2 FactorSet.prod none \u2264 FactorSet.prod b\n[PROOFSTEP]\nhave : b = \u22a4 := top_unique h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\nb : FactorSet \u03b1\nh : none \u2264 b\nthis : b = \u22a4\n\u22a2 FactorSet.prod none \u2264 FactorSet.prod b\n[PROOFSTEP]\nrw [this, prod_top]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\nb : FactorSet \u03b1\nh : none \u2264 b\nthis : b = \u22a4\n\u22a2 FactorSet.prod none \u2264 0\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : FactorSet \u03b1\nx\u271d : a \u2264 none\n\u22a2 FactorSet.prod a \u2264 FactorSet.prod \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\na : FactorSet \u03b1\nx\u271d : a \u2264 none\n\u22a2 FactorSet.prod a \u2264 0\n[PROOFSTEP]\nexact le_top\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\np : FactorSet \u03b1\n\u22a2 prod p = 0 \u2194 p = \u22a4\n[PROOFSTEP]\ninduction p using WithTop.recTopCoe\n[GOAL]\ncase top\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\n\u22a2 prod \u22a4 = 0 \u2194 \u22a4 = \u22a4\n[PROOFSTEP]\nsimp only [iff_self_iff, eq_self_iff_true, Associates.prod_top]\n[GOAL]\ncase coe\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\na\u271d : Multiset { a // Irreducible a }\n\u22a2 prod \u2191a\u271d = 0 \u2194 \u2191a\u271d = \u22a4\n[PROOFSTEP]\nrw [prod_coe, Multiset.prod_eq_zero_iff, Multiset.mem_map, eq_false WithTop.coe_ne_top, iff_false_iff, not_exists]\n[GOAL]\ncase coe\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Nontrivial \u03b1\na\u271d : Multiset { a // Irreducible a }\n\u22a2 \u2200 (x : { a // Irreducible a }), \u00ac(x \u2208 a\u271d \u2227 \u2191x = 0)\n[PROOFSTEP]\nexact fun a => not_and_of_not_right _ a.prop.ne_zero\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : DecidableEq (Associates \u03b1)\np : Associates \u03b1\nhp : Irreducible p\ns : Multiset { a // Irreducible a }\n\u22a2 count p (some s) = Multiset.count { val := p, property := hp } s\n[PROOFSTEP]\ndsimp only [count]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : DecidableEq (Associates \u03b1)\np : Associates \u03b1\nhp : Irreducible p\ns : Multiset { a // Irreducible a }\n\u22a2 dite (Irreducible p) (fun hp => bcount { val := p, property := hp }) (fun hp => 0) (some s) =\n Multiset.count { val := p, property := hp } s\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : DecidableEq (Associates \u03b1)\np : Associates \u03b1\nhp : Irreducible p\ns : Multiset { a // Irreducible a }\n\u22a2 bcount { val := p, property := hp } (some s) = Multiset.count { val := p, property := hp } s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : DecidableEq (Associates \u03b1)\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 count p 0 = 0\n[PROOFSTEP]\ndsimp only [count]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : DecidableEq (Associates \u03b1)\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 dite (Irreducible p) (fun hp => bcount { val := p, property := hp }) (fun hp => 0) 0 = 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : DecidableEq (Associates \u03b1)\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 bcount { val := p, property := hp } 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 p \u2208 \u22a4\n[PROOFSTEP]\ndsimp only [Membership.mem]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 FactorSetMem p \u22a4\n[PROOFSTEP]\ndsimp only [FactorSetMem]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 if hp : Irreducible p then BfactorSetMem { val := p, property := hp } \u22a4 else False\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 BfactorSetMem { val := p, property := hp } \u22a4\n[PROOFSTEP]\nexact trivial\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\np : Associates \u03b1\nhp : Irreducible p\nl : Multiset { a // Irreducible a }\n\u22a2 p \u2208 \u2191l \u2194 { val := p, property := hp } \u2208 l\n[PROOFSTEP]\ndsimp only [Membership.mem]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\np : Associates \u03b1\nhp : Irreducible p\nl : Multiset { a // Irreducible a }\n\u22a2 FactorSetMem p \u2191l \u2194 Mem { val := p, property := hp } l\n[PROOFSTEP]\ndsimp only [FactorSetMem]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\np : Associates \u03b1\nhp : Irreducible p\nl : Multiset { a // Irreducible a }\n\u22a2 (if hp : Irreducible p then BfactorSetMem { val := p, property := hp } \u2191l else False) \u2194\n Mem { val := p, property := hp } l\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\np : Associates \u03b1\nhp : Irreducible p\nl : Multiset { a // Irreducible a }\n\u22a2 BfactorSetMem { val := p, property := hp } \u2191l \u2194 Mem { val := p, property := hp } l\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\np : Associates \u03b1\nhp : \u00acIrreducible p\ns : FactorSet \u03b1\nh : if hp : Irreducible p then BfactorSetMem { val := p, property := hp } s else False\n\u22a2 False\n[PROOFSTEP]\nrwa [dif_neg hp] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\np q : Multiset (Associates \u03b1)\n\u22a2 (\u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a) \u2192\n (\u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a) \u2192 prod p = prod q \u2192 p = q\n[PROOFSTEP]\napply Multiset.induction_on_multiset_quot p\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\np q : Multiset (Associates \u03b1)\n\u22a2 \u2200 (s : Multiset \u03b1),\n (\u2200 (a : Associates \u03b1), a \u2208 map (Quot.mk Setoid.r) s \u2192 Irreducible a) \u2192\n (\u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a) \u2192\n prod (map (Quot.mk Setoid.r) s) = prod q \u2192 map (Quot.mk Setoid.r) s = q\n[PROOFSTEP]\napply Multiset.induction_on_multiset_quot q\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\np q : Multiset (Associates \u03b1)\n\u22a2 \u2200 (s s_1 : Multiset \u03b1),\n (\u2200 (a : Associates \u03b1), a \u2208 map (Quot.mk Setoid.r) s_1 \u2192 Irreducible a) \u2192\n (\u2200 (a : Associates \u03b1), a \u2208 map (Quot.mk Setoid.r) s \u2192 Irreducible a) \u2192\n prod (map (Quot.mk Setoid.r) s_1) = prod (map (Quot.mk Setoid.r) s) \u2192\n map (Quot.mk Setoid.r) s_1 = map (Quot.mk Setoid.r) s\n[PROOFSTEP]\nintro s t hs ht eq\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\np q : Multiset (Associates \u03b1)\ns t : Multiset \u03b1\nhs : \u2200 (a : Associates \u03b1), a \u2208 map (Quot.mk Setoid.r) t \u2192 Irreducible a\nht : \u2200 (a : Associates \u03b1), a \u2208 map (Quot.mk Setoid.r) s \u2192 Irreducible a\neq : prod (map (Quot.mk Setoid.r) t) = prod (map (Quot.mk Setoid.r) s)\n\u22a2 map (Quot.mk Setoid.r) t = map (Quot.mk Setoid.r) s\n[PROOFSTEP]\nrefine' Multiset.map_mk_eq_map_mk_of_rel (UniqueFactorizationMonoid.factors_unique _ _ _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\np q : Multiset (Associates \u03b1)\ns t : Multiset \u03b1\nhs : \u2200 (a : Associates \u03b1), a \u2208 map (Quot.mk Setoid.r) t \u2192 Irreducible a\nht : \u2200 (a : Associates \u03b1), a \u2208 map (Quot.mk Setoid.r) s \u2192 Irreducible a\neq : prod (map (Quot.mk Setoid.r) t) = prod (map (Quot.mk Setoid.r) s)\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 Irreducible x\n[PROOFSTEP]\nexact fun a ha => (irreducible_mk _).1 <| hs _ <| Multiset.mem_map_of_mem _ ha\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\np q : Multiset (Associates \u03b1)\ns t : Multiset \u03b1\nhs : \u2200 (a : Associates \u03b1), a \u2208 map (Quot.mk Setoid.r) t \u2192 Irreducible a\nht : \u2200 (a : Associates \u03b1), a \u2208 map (Quot.mk Setoid.r) s \u2192 Irreducible a\neq : prod (map (Quot.mk Setoid.r) t) = prod (map (Quot.mk Setoid.r) s)\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 Irreducible x\n[PROOFSTEP]\nexact fun a ha => (irreducible_mk _).1 <| ht _ <| Multiset.mem_map_of_mem _ ha\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\np q : Multiset (Associates \u03b1)\ns t : Multiset \u03b1\nhs : \u2200 (a : Associates \u03b1), a \u2208 map (Quot.mk Setoid.r) t \u2192 Irreducible a\nht : \u2200 (a : Associates \u03b1), a \u2208 map (Quot.mk Setoid.r) s \u2192 Irreducible a\neq : prod (map (Quot.mk Setoid.r) t) = prod (map (Quot.mk Setoid.r) s)\n\u22a2 prod t ~\u1d64 prod s\n[PROOFSTEP]\nhave eq' : (Quot.mk Setoid.r : \u03b1 \u2192 Associates \u03b1) = Associates.mk := funext quot_mk_eq_mk\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\np q : Multiset (Associates \u03b1)\ns t : Multiset \u03b1\nhs : \u2200 (a : Associates \u03b1), a \u2208 map (Quot.mk Setoid.r) t \u2192 Irreducible a\nht : \u2200 (a : Associates \u03b1), a \u2208 map (Quot.mk Setoid.r) s \u2192 Irreducible a\neq : prod (map (Quot.mk Setoid.r) t) = prod (map (Quot.mk Setoid.r) s)\neq' : Quot.mk Setoid.r = Associates.mk\n\u22a2 prod t ~\u1d64 prod s\n[PROOFSTEP]\nrwa [eq', prod_mk, prod_mk, mk_eq_mk_iff_associated] at eq \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\np q : FactorSet \u03b1\nh : prod p = prod q\n\u22a2 p = q\n[PROOFSTEP]\ninduction p using WithTop.recTopCoe\n[GOAL]\ncase top\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\nq : FactorSet \u03b1\nh : prod \u22a4 = prod q\n\u22a2 \u22a4 = q\n[PROOFSTEP]\ninduction q using WithTop.recTopCoe\n[GOAL]\ncase coe\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\nq : FactorSet \u03b1\na\u271d : Multiset { a // Irreducible a }\nh : prod \u2191a\u271d = prod q\n\u22a2 \u2191a\u271d = q\n[PROOFSTEP]\ninduction q using WithTop.recTopCoe\n[GOAL]\ncase top.top\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\nh : prod \u22a4 = prod \u22a4\n\u22a2 \u22a4 = \u22a4\n[PROOFSTEP]\nrfl\n[GOAL]\ncase top.coe\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\na\u271d : Multiset { a // Irreducible a }\nh : prod \u22a4 = prod \u2191a\u271d\n\u22a2 \u22a4 = \u2191a\u271d\n[PROOFSTEP]\nrw [eq_comm, \u2190 FactorSet.prod_eq_zero_iff, \u2190 h, Associates.prod_top]\n[GOAL]\ncase coe.top\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\na\u271d : Multiset { a // Irreducible a }\nh : prod \u2191a\u271d = prod \u22a4\n\u22a2 \u2191a\u271d = \u22a4\n[PROOFSTEP]\nrw [\u2190 FactorSet.prod_eq_zero_iff, h, Associates.prod_top]\n[GOAL]\ncase coe.coe\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\na\u271d\u00b9 a\u271d : Multiset { a // Irreducible a }\nh : prod \u2191a\u271d\u00b9 = prod \u2191a\u271d\n\u22a2 \u2191a\u271d\u00b9 = \u2191a\u271d\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase coe.coe.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\na\u271d\u00b9 a\u271d : Multiset { a // Irreducible a }\nh : prod \u2191a\u271d\u00b9 = prod \u2191a\u271d\n\u22a2 a\u271d\u00b9 = a\u271d\n[PROOFSTEP]\nrw [\u2190 Multiset.map_eq_map Subtype.coe_injective]\n[GOAL]\ncase coe.coe.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\na\u271d\u00b9 a\u271d : Multiset { a // Irreducible a }\nh : prod \u2191a\u271d\u00b9 = prod \u2191a\u271d\n\u22a2 map (fun a => \u2191a) a\u271d\u00b9 = map (fun a => \u2191a) a\u271d\n[PROOFSTEP]\napply unique' _ _ h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\na\u271d\u00b9 a\u271d : Multiset { a // Irreducible a }\nh : prod \u2191a\u271d\u00b9 = prod \u2191a\u271d\n\u22a2 \u2200 (a : Associates \u03b1), a \u2208 map Subtype.val a\u271d\u00b9 \u2192 Irreducible a\n[PROOFSTEP]\nintro a ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\na\u271d\u00b9 a\u271d : Multiset { a // Irreducible a }\nh : prod \u2191a\u271d\u00b9 = prod \u2191a\u271d\na : Associates \u03b1\nha : a \u2208 map Subtype.val a\u271d\u00b9\n\u22a2 Irreducible a\n[PROOFSTEP]\nobtain \u27e8\u27e8a', irred\u27e9, -, rfl\u27e9 := Multiset.mem_map.mp ha\n[GOAL]\ncase intro.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\na\u271d\u00b9 a\u271d : Multiset { a // Irreducible a }\nh : prod \u2191a\u271d\u00b9 = prod \u2191a\u271d\na' : Associates \u03b1\nirred : Irreducible a'\nha : \u2191{ val := a', property := irred } \u2208 map Subtype.val a\u271d\u00b9\n\u22a2 Irreducible \u2191{ val := a', property := irred }\n[PROOFSTEP]\nrwa [Subtype.coe_mk]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\na\u271d\u00b9 a\u271d : Multiset { a // Irreducible a }\nh : prod \u2191a\u271d\u00b9 = prod \u2191a\u271d\n\u22a2 \u2200 (a : Associates \u03b1), a \u2208 map Subtype.val a\u271d \u2192 Irreducible a\n[PROOFSTEP]\nintro a ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\na\u271d\u00b9 a\u271d : Multiset { a // Irreducible a }\nh : prod \u2191a\u271d\u00b9 = prod \u2191a\u271d\na : Associates \u03b1\nha : a \u2208 map Subtype.val a\u271d\n\u22a2 Irreducible a\n[PROOFSTEP]\nobtain \u27e8\u27e8a', irred\u27e9, -, rfl\u27e9 := Multiset.mem_map.mp ha\n[GOAL]\ncase intro.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\na\u271d\u00b9 a\u271d : Multiset { a // Irreducible a }\nh : prod \u2191a\u271d\u00b9 = prod \u2191a\u271d\na' : Associates \u03b1\nirred : Irreducible a'\nha : \u2191{ val := a', property := irred } \u2208 map Subtype.val a\u271d\n\u22a2 Irreducible \u2191{ val := a', property := irred }\n[PROOFSTEP]\nrwa [Subtype.coe_mk]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\np q : Multiset (Associates \u03b1)\nhp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a\nhq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a\n\u22a2 prod p \u2264 prod q \u2192 p \u2264 q\n[PROOFSTEP]\nclassical\nrintro \u27e8c, eqc\u27e9\nrefine' Multiset.le_iff_exists_add.2 \u27e8factors c, unique' hq (fun x hx => _) _\u27e9\n\u00b7 obtain h | h := Multiset.mem_add.1 hx\n \u00b7 exact hp x h\n \u00b7 exact irreducible_of_factor _ h\n\u00b7 rw [eqc, Multiset.prod_add]\n congr\n refine' associated_iff_eq.mp (factors_prod fun hc => _).symm\n refine' not_irreducible_zero (hq _ _)\n rw [\u2190 prod_eq_zero_iff, eqc, hc, mul_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\np q : Multiset (Associates \u03b1)\nhp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a\nhq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a\n\u22a2 prod p \u2264 prod q \u2192 p \u2264 q\n[PROOFSTEP]\nrintro \u27e8c, eqc\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\np q : Multiset (Associates \u03b1)\nhp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a\nhq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a\nc : Associates \u03b1\neqc : prod q = prod p * c\n\u22a2 p \u2264 q\n[PROOFSTEP]\nrefine' Multiset.le_iff_exists_add.2 \u27e8factors c, unique' hq (fun x hx => _) _\u27e9\n[GOAL]\ncase intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\np q : Multiset (Associates \u03b1)\nhp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a\nhq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a\nc : Associates \u03b1\neqc : prod q = prod p * c\nx : Associates \u03b1\nhx : x \u2208 p + factors c\n\u22a2 Irreducible x\n[PROOFSTEP]\nobtain h | h := Multiset.mem_add.1 hx\n[GOAL]\ncase intro.refine'_1.inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\np q : Multiset (Associates \u03b1)\nhp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a\nhq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a\nc : Associates \u03b1\neqc : prod q = prod p * c\nx : Associates \u03b1\nhx : x \u2208 p + factors c\nh : x \u2208 p\n\u22a2 Irreducible x\n[PROOFSTEP]\nexact hp x h\n[GOAL]\ncase intro.refine'_1.inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\np q : Multiset (Associates \u03b1)\nhp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a\nhq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a\nc : Associates \u03b1\neqc : prod q = prod p * c\nx : Associates \u03b1\nhx : x \u2208 p + factors c\nh : x \u2208 factors c\n\u22a2 Irreducible x\n[PROOFSTEP]\nexact irreducible_of_factor _ h\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\np q : Multiset (Associates \u03b1)\nhp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a\nhq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a\nc : Associates \u03b1\neqc : prod q = prod p * c\n\u22a2 prod q = prod (p + factors c)\n[PROOFSTEP]\nrw [eqc, Multiset.prod_add]\n[GOAL]\ncase intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\np q : Multiset (Associates \u03b1)\nhp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a\nhq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a\nc : Associates \u03b1\neqc : prod q = prod p * c\n\u22a2 prod p * c = prod p * prod (factors c)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase intro.refine'_2.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\np q : Multiset (Associates \u03b1)\nhp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a\nhq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a\nc : Associates \u03b1\neqc : prod q = prod p * c\n\u22a2 c = prod (factors c)\n[PROOFSTEP]\nrefine' associated_iff_eq.mp (factors_prod fun hc => _).symm\n[GOAL]\ncase intro.refine'_2.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\np q : Multiset (Associates \u03b1)\nhp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a\nhq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a\nc : Associates \u03b1\neqc : prod q = prod p * c\nhc : c = 0\n\u22a2 False\n[PROOFSTEP]\nrefine' not_irreducible_zero (hq _ _)\n[GOAL]\ncase intro.refine'_2.e_a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ninst\u271d : Nontrivial \u03b1\np q : Multiset (Associates \u03b1)\nhp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a\nhq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a\nc : Associates \u03b1\neqc : prod q = prod p * c\nhc : c = 0\n\u22a2 0 \u2208 q\n[PROOFSTEP]\nrw [\u2190 prod_eq_zero_iff, eqc, hc, mul_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : \u03b1\n\u22a2 map Subtype.val (factors' a) = map Associates.mk (factors a)\n[PROOFSTEP]\nsimp [factors', Multiset.map_pmap, Multiset.pmap_eq_map]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nh : a ~\u1d64 b\n\u22a2 factors' a = factors' b\n[PROOFSTEP]\nobtain rfl | hb := eq_or_ne b 0\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : \u03b1\nh : a ~\u1d64 0\n\u22a2 factors' a = factors' 0\n[PROOFSTEP]\nrw [associated_zero_iff_eq_zero] at h \n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : \u03b1\nh : a = 0\n\u22a2 factors' a = factors' 0\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nh : a ~\u1d64 b\nhb : b \u2260 0\n\u22a2 factors' a = factors' b\n[PROOFSTEP]\nhave ha : a \u2260 0 := by\n contrapose! hb with ha\n rw [\u2190 associated_zero_iff_eq_zero, \u2190 ha]\n exact h.symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nh : a ~\u1d64 b\nhb : b \u2260 0\n\u22a2 a \u2260 0\n[PROOFSTEP]\ncontrapose! hb with ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nh : a ~\u1d64 b\nha : a = 0\n\u22a2 b = 0\n[PROOFSTEP]\nrw [\u2190 associated_zero_iff_eq_zero, \u2190 ha]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nh : a ~\u1d64 b\nha : a = 0\n\u22a2 b ~\u1d64 a\n[PROOFSTEP]\nexact h.symm\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nh : a ~\u1d64 b\nhb : b \u2260 0\nha : a \u2260 0\n\u22a2 factors' a = factors' b\n[PROOFSTEP]\nrw [\u2190 Multiset.map_eq_map Subtype.coe_injective, map_subtype_coe_factors', map_subtype_coe_factors', \u2190\n rel_associated_iff_map_eq_map]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nh : a ~\u1d64 b\nhb : b \u2260 0\nha : a \u2260 0\n\u22a2 Rel Associated (factors a) (factors b)\n[PROOFSTEP]\nexact\n factors_unique irreducible_of_factor irreducible_of_factor\n ((factors_prod ha).trans <| h.trans <| (factors_prod hb).symm)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : Associates \u03b1\n\u22a2 FactorSet \u03b1\n[PROOFSTEP]\nrefine' if h : a = 0 then \u22a4 else Quotient.hrecOn a (fun x _ => some <| factors' x) _ h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : Associates \u03b1\nh : \u00aca = 0\n\u22a2 \u2200 (a b : \u03b1), a \u2248 b \u2192 HEq (fun x => some (factors' a)) fun x => some (factors' b)\n[PROOFSTEP]\nintro a b hab\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na\u271d : Associates \u03b1\nh : \u00aca\u271d = 0\na b : \u03b1\nhab : a \u2248 b\n\u22a2 HEq (fun x => some (factors' a)) fun x => some (factors' b)\n[PROOFSTEP]\napply Function.hfunext\n[GOAL]\ncase h\u03b1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na\u271d : Associates \u03b1\nh : \u00aca\u271d = 0\na b : \u03b1\nhab : a \u2248 b\n\u22a2 (\u00acQuotient.mk (Associated.setoid \u03b1) a = 0) = \u00acQuotient.mk (Associated.setoid \u03b1) b = 0\n[PROOFSTEP]\nhave : a ~\u1d64 0 \u2194 b ~\u1d64 0 := Iff.intro (fun ha0 => hab.symm.trans ha0) fun hb0 => hab.trans hb0\n[GOAL]\ncase h\u03b1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na\u271d : Associates \u03b1\nh : \u00aca\u271d = 0\na b : \u03b1\nhab : a \u2248 b\nthis : a ~\u1d64 0 \u2194 b ~\u1d64 0\n\u22a2 (\u00acQuotient.mk (Associated.setoid \u03b1) a = 0) = \u00acQuotient.mk (Associated.setoid \u03b1) b = 0\n[PROOFSTEP]\nsimp only [associated_zero_iff_eq_zero] at this \n[GOAL]\ncase h\u03b1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na\u271d : Associates \u03b1\nh : \u00aca\u271d = 0\na b : \u03b1\nhab : a \u2248 b\nthis : a = 0 \u2194 b = 0\n\u22a2 (\u00acQuotient.mk (Associated.setoid \u03b1) a = 0) = \u00acQuotient.mk (Associated.setoid \u03b1) b = 0\n[PROOFSTEP]\nsimp only [quotient_mk_eq_mk, this, mk_eq_zero]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na\u271d : Associates \u03b1\nh : \u00aca\u271d = 0\na b : \u03b1\nhab : a \u2248 b\n\u22a2 \u2200 (a_1 : \u00acQuotient.mk (Associated.setoid \u03b1) a = 0) (a' : \u00acQuotient.mk (Associated.setoid \u03b1) b = 0),\n HEq a_1 a' \u2192 HEq (some (factors' a)) (some (factors' b))\n[PROOFSTEP]\nexact fun ha hb _ => heq_of_eq <| congr_arg some <| factors'_cong hab\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : \u03b1\nh : a \u2260 0\n\u22a2 factors (Associates.mk a) = \u2191(factors' a)\n[PROOFSTEP]\nclassical\napply dif_neg\napply mt mk_eq_zero.1 h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : \u03b1\nh : a \u2260 0\n\u22a2 factors (Associates.mk a) = \u2191(factors' a)\n[PROOFSTEP]\napply dif_neg\n[GOAL]\ncase hnc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : \u03b1\nh : a \u2260 0\n\u22a2 \u00acAssociates.mk a = 0\n[PROOFSTEP]\napply mt mk_eq_zero.1 h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : Associates \u03b1\nb : \u03b1\nthis : Associates.mk b = 0\n\u22a2 FactorSet.prod (factors (Quotient.mk (Associated.setoid \u03b1) b)) = Quotient.mk (Associated.setoid \u03b1) b\n[PROOFSTEP]\nsimp [quotient_mk_eq_mk, this]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : Associates \u03b1\nb : \u03b1\nx\u271d : Associates.mk b \u2260 0\n\u22a2 FactorSet.prod (factors (Quotient.mk (Associated.setoid \u03b1) b)) = Quotient.mk (Associated.setoid \u03b1) b\n[PROOFSTEP]\nhave : b \u2260 0 := by simp_all\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : Associates \u03b1\nb : \u03b1\nx\u271d : Associates.mk b \u2260 0\n\u22a2 b \u2260 0\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : Associates \u03b1\nb : \u03b1\nx\u271d : Associates.mk b \u2260 0\nthis : b \u2260 0\n\u22a2 FactorSet.prod (factors (Quotient.mk (Associated.setoid \u03b1) b)) = Quotient.mk (Associated.setoid \u03b1) b\n[PROOFSTEP]\nsimp [this, quotient_mk_eq_mk, prod_mk, mk_eq_mk_iff_associated.2 (UniqueFactorizationMonoid.factors_prod this)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Subsingleton \u03b1\na : Associates \u03b1\n\u22a2 factors a = none\n[PROOFSTEP]\nconvert @factors_0 _ _ _ _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : Associates \u03b1\n\u22a2 factors a = none \u2194 a = 0\n[PROOFSTEP]\nnontriviality \u03b1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : Associates \u03b1\n\u271d : Nontrivial \u03b1\n\u22a2 factors a = none \u2194 a = 0\n[PROOFSTEP]\nexact \u27e8fun h => by rwa [\u2190 factors_prod a, FactorSet.prod_eq_zero_iff], fun h => h.symm \u25b8 factors_0\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : Associates \u03b1\n\u271d : Nontrivial \u03b1\nh : factors a = none\n\u22a2 a = 0\n[PROOFSTEP]\nrwa [\u2190 factors_prod a, FactorSet.prod_eq_zero_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : Associates \u03b1\n\u22a2 (\u2203 s, factors a = some s) \u2194 a \u2260 0\n[PROOFSTEP]\nrw [\u2190 Option.isSome_iff_exists, \u2190 Option.ne_none_iff_isSome, Ne.def, Ne.def, factors_eq_none_iff_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nh : factors a = factors b\n\u22a2 a = b\n[PROOFSTEP]\nhave : a.factors.prod = b.factors.prod := by rw [h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nh : factors a = factors b\n\u22a2 FactorSet.prod (factors a) = FactorSet.prod (factors b)\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nh : factors a = factors b\nthis : FactorSet.prod (factors a) = FactorSet.prod (factors b)\n\u22a2 a = b\n[PROOFSTEP]\nrwa [factors_prod, factors_prod] at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b : FactorSet \u03b1\nh : FactorSet.prod a = FactorSet.prod b\n\u22a2 a = b\n[PROOFSTEP]\nclassical\nhave : a.prod.factors = b.prod.factors := by rw [h]\nrwa [prod_factors, prod_factors] at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b : FactorSet \u03b1\nh : FactorSet.prod a = FactorSet.prod b\n\u22a2 a = b\n[PROOFSTEP]\nhave : a.prod.factors = b.prod.factors := by rw [h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b : FactorSet \u03b1\nh : FactorSet.prod a = FactorSet.prod b\n\u22a2 factors (FactorSet.prod a) = factors (FactorSet.prod b)\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b : FactorSet \u03b1\nh : FactorSet.prod a = FactorSet.prod b\nthis : factors (FactorSet.prod a) = factors (FactorSet.prod b)\n\u22a2 a = b\n[PROOFSTEP]\nrwa [prod_factors, prod_factors] at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nh : \u2200 (p : Associates \u03b1), Irreducible p \u2192 count p (factors a) = count p (factors b)\n\u22a2 factors a = factors b\n[PROOFSTEP]\nobtain \u27e8sa, h_sa\u27e9 := factors_eq_some_iff_ne_zero.mpr ha\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nh : \u2200 (p : Associates \u03b1), Irreducible p \u2192 count p (factors a) = count p (factors b)\nsa : Multiset { p // Irreducible p }\nh_sa : factors a = some sa\n\u22a2 factors a = factors b\n[PROOFSTEP]\nobtain \u27e8sb, h_sb\u27e9 := factors_eq_some_iff_ne_zero.mpr hb\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nh : \u2200 (p : Associates \u03b1), Irreducible p \u2192 count p (factors a) = count p (factors b)\nsa : Multiset { p // Irreducible p }\nh_sa : factors a = some sa\nsb : Multiset { p // Irreducible p }\nh_sb : factors b = some sb\n\u22a2 factors a = factors b\n[PROOFSTEP]\nrw [h_sa, h_sb] at h \u22a2\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nsa : Multiset { p // Irreducible p }\nh_sa : factors a = some sa\nsb : Multiset { p // Irreducible p }\nh : \u2200 (p : Associates \u03b1), Irreducible p \u2192 count p (some sa) = count p (some sb)\nh_sb : factors b = some sb\n\u22a2 some sa = some sb\n[PROOFSTEP]\nrw [Option.some_inj]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nsa : Multiset { p // Irreducible p }\nh_sa : factors a = some sa\nsb : Multiset { p // Irreducible p }\nh : \u2200 (p : Associates \u03b1), Irreducible p \u2192 count p (some sa) = count p (some sb)\nh_sb : factors b = some sb\n\u22a2 sa = sb\n[PROOFSTEP]\nhave h_count : \u2200 (p : Associates \u03b1) (hp : Irreducible p), sa.count \u27e8p, hp\u27e9 = sb.count \u27e8p, hp\u27e9 :=\n by\n intro p hp\n rw [\u2190 count_some, \u2190 count_some, h p hp]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nsa : Multiset { p // Irreducible p }\nh_sa : factors a = some sa\nsb : Multiset { p // Irreducible p }\nh : \u2200 (p : Associates \u03b1), Irreducible p \u2192 count p (some sa) = count p (some sb)\nh_sb : factors b = some sb\n\u22a2 \u2200 (p : Associates \u03b1) (hp : Irreducible p),\n Multiset.count { val := p, property := hp } sa = Multiset.count { val := p, property := hp } sb\n[PROOFSTEP]\nintro p hp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nsa : Multiset { p // Irreducible p }\nh_sa : factors a = some sa\nsb : Multiset { p // Irreducible p }\nh : \u2200 (p : Associates \u03b1), Irreducible p \u2192 count p (some sa) = count p (some sb)\nh_sb : factors b = some sb\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 Multiset.count { val := p, property := hp } sa = Multiset.count { val := p, property := hp } sb\n[PROOFSTEP]\nrw [\u2190 count_some, \u2190 count_some, h p hp]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nsa : Multiset { p // Irreducible p }\nh_sa : factors a = some sa\nsb : Multiset { p // Irreducible p }\nh : \u2200 (p : Associates \u03b1), Irreducible p \u2192 count p (some sa) = count p (some sb)\nh_sb : factors b = some sb\nh_count :\n \u2200 (p : Associates \u03b1) (hp : Irreducible p),\n Multiset.count { val := p, property := hp } sa = Multiset.count { val := p, property := hp } sb\n\u22a2 sa = sb\n[PROOFSTEP]\napply Multiset.toFinsupp.injective\n[GOAL]\ncase intro.intro.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nsa : Multiset { p // Irreducible p }\nh_sa : factors a = some sa\nsb : Multiset { p // Irreducible p }\nh : \u2200 (p : Associates \u03b1), Irreducible p \u2192 count p (some sa) = count p (some sb)\nh_sb : factors b = some sb\nh_count :\n \u2200 (p : Associates \u03b1) (hp : Irreducible p),\n Multiset.count { val := p, property := hp } sa = Multiset.count { val := p, property := hp } sb\n\u22a2 \u2191toFinsupp sa = \u2191toFinsupp sb\n[PROOFSTEP]\next \u27e8p, hp\u27e9\n[GOAL]\ncase intro.intro.a.h.mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nsa : Multiset { p // Irreducible p }\nh_sa : factors a = some sa\nsb : Multiset { p // Irreducible p }\nh : \u2200 (p : Associates \u03b1), Irreducible p \u2192 count p (some sa) = count p (some sb)\nh_sb : factors b = some sb\nh_count :\n \u2200 (p : Associates \u03b1) (hp : Irreducible p),\n Multiset.count { val := p, property := hp } sa = Multiset.count { val := p, property := hp } sb\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 \u2191(\u2191toFinsupp sa) { val := p, property := hp } = \u2191(\u2191toFinsupp sb) { val := p, property := hp }\n[PROOFSTEP]\nrw [Multiset.toFinsupp_apply, Multiset.toFinsupp_apply, h_count p hp]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b p : Associates \u03b1\nhb : b \u2260 0\nhp : Irreducible p\nh : factors a \u2264 factors b\n\u22a2 count p (factors a) \u2264 count p (factors b)\n[PROOFSTEP]\nby_cases ha : a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b p : Associates \u03b1\nhb : b \u2260 0\nhp : Irreducible p\nh : factors a \u2264 factors b\nha : a = 0\n\u22a2 count p (factors a) \u2264 count p (factors b)\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b p : Associates \u03b1\nhb : b \u2260 0\nhp : Irreducible p\nh : factors a \u2264 factors b\nha : \u00aca = 0\n\u22a2 count p (factors a) \u2264 count p (factors b)\n[PROOFSTEP]\nobtain \u27e8sa, h_sa\u27e9 := factors_eq_some_iff_ne_zero.mpr ha\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b p : Associates \u03b1\nhb : b \u2260 0\nhp : Irreducible p\nh : factors a \u2264 factors b\nha : \u00aca = 0\nsa : Multiset { p // Irreducible p }\nh_sa : factors a = some sa\n\u22a2 count p (factors a) \u2264 count p (factors b)\n[PROOFSTEP]\nobtain \u27e8sb, h_sb\u27e9 := factors_eq_some_iff_ne_zero.mpr hb\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b p : Associates \u03b1\nhb : b \u2260 0\nhp : Irreducible p\nh : factors a \u2264 factors b\nha : \u00aca = 0\nsa : Multiset { p // Irreducible p }\nh_sa : factors a = some sa\nsb : Multiset { p // Irreducible p }\nh_sb : factors b = some sb\n\u22a2 count p (factors a) \u2264 count p (factors b)\n[PROOFSTEP]\nrw [h_sa, h_sb] at h \u22a2\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b p : Associates \u03b1\nhb : b \u2260 0\nhp : Irreducible p\nha : \u00aca = 0\nsa : Multiset { p // Irreducible p }\nh_sa : factors a = some sa\nsb : Multiset { p // Irreducible p }\nh : some sa \u2264 some sb\nh_sb : factors b = some sb\n\u22a2 count p (some sa) \u2264 count p (some sb)\n[PROOFSTEP]\nrw [count_some hp, count_some hp]\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b p : Associates \u03b1\nhb : b \u2260 0\nhp : Irreducible p\nha : \u00aca = 0\nsa : Multiset { p // Irreducible p }\nh_sa : factors a = some sa\nsb : Multiset { p // Irreducible p }\nh : some sa \u2264 some sb\nh_sb : factors b = some sb\n\u22a2 Multiset.count { val := p, property := hp } sa \u2264 Multiset.count { val := p, property := hp } sb\n[PROOFSTEP]\nrw [WithTop.some_le_some] at h \n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b p : Associates \u03b1\nhb : b \u2260 0\nhp : Irreducible p\nha : \u00aca = 0\nsa : Multiset { p // Irreducible p }\nh_sa : factors a = some sa\nsb : Multiset { p // Irreducible p }\nh : sa \u2264 sb\nh_sb : factors b = some sb\n\u22a2 Multiset.count { val := p, property := hp } sa \u2264 Multiset.count { val := p, property := hp } sb\n[PROOFSTEP]\nexact Multiset.count_le_of_le _ h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\n\u22a2 factors (a * b) = factors a + factors b\n[PROOFSTEP]\ncases subsingleton_or_nontrivial \u03b1\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nh\u271d : Subsingleton \u03b1\n\u22a2 factors (a * b) = factors a + factors b\n[PROOFSTEP]\nsimp [Subsingleton.elim a 0]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nh\u271d : Nontrivial \u03b1\n\u22a2 factors (a * b) = factors a + factors b\n[PROOFSTEP]\nrefine' eq_of_prod_eq_prod (eq_of_factors_eq_factors _)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nh\u271d : Nontrivial \u03b1\n\u22a2 factors (FactorSet.prod (factors (a * b))) = factors (FactorSet.prod (factors a + factors b))\n[PROOFSTEP]\nrw [prod_add, factors_prod, factors_prod, factors_prod]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ns t : Associates \u03b1\nd : Associates \u03b1\neq : t = s * d\n\u22a2 factors s \u2264 factors t\n[PROOFSTEP]\nrw [eq, factors_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ns t : Associates \u03b1\nd : Associates \u03b1\neq : t = s * d\n\u22a2 factors s \u2264 factors s + factors d\n[PROOFSTEP]\nexact le_add_of_nonneg_right bot_le\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nh : factors a \u2264 factors b\n\u22a2 a \u2264 b\n[PROOFSTEP]\nhave : a.factors.prod \u2264 b.factors.prod := prod_mono h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nh : factors a \u2264 factors b\nthis : FactorSet.prod (factors a) \u2264 FactorSet.prod (factors b)\n\u22a2 a \u2264 b\n[PROOFSTEP]\nrwa [factors_prod, factors_prod] at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b : FactorSet \u03b1\n\u22a2 FactorSet.prod a \u2264 FactorSet.prod b \u2194 a \u2264 b\n[PROOFSTEP]\nclassical exact\n Iff.intro\n (fun h => by\n have : a.prod.factors \u2264 b.prod.factors := factors_mono h\n rwa [prod_factors, prod_factors] at this )\n prod_mono\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b : FactorSet \u03b1\n\u22a2 FactorSet.prod a \u2264 FactorSet.prod b \u2194 a \u2264 b\n[PROOFSTEP]\nexact\n Iff.intro\n (fun h => by\n have : a.prod.factors \u2264 b.prod.factors := factors_mono h\n rwa [prod_factors, prod_factors] at this )\n prod_mono\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b : FactorSet \u03b1\nh : FactorSet.prod a \u2264 FactorSet.prod b\n\u22a2 a \u2264 b\n[PROOFSTEP]\nhave : a.prod.factors \u2264 b.prod.factors := factors_mono h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b : FactorSet \u03b1\nh : FactorSet.prod a \u2264 FactorSet.prod b\nthis : factors (FactorSet.prod a) \u2264 factors (FactorSet.prod b)\n\u22a2 a \u2264 b\n[PROOFSTEP]\nrwa [prod_factors, prod_factors] at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\n\u22a2 FactorSet.prod (factors a \u2294 factors b) * FactorSet.prod (factors a \u2293 factors b) = a * b\n[PROOFSTEP]\nnontriviality \u03b1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\n\u271d : Nontrivial \u03b1\n\u22a2 FactorSet.prod (factors a \u2294 factors b) * FactorSet.prod (factors a \u2293 factors b) = a * b\n[PROOFSTEP]\nrefine' eq_of_factors_eq_factors _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\n\u271d : Nontrivial \u03b1\n\u22a2 factors (FactorSet.prod (factors a \u2294 factors b) * FactorSet.prod (factors a \u2293 factors b)) = factors (a * b)\n[PROOFSTEP]\nrw [\u2190 prod_add, prod_factors, factors_mul, FactorSet.sup_add_inf_eq_add]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\nhm : p \u2208 factors a\n\u22a2 p \u2223 a\n[PROOFSTEP]\nby_cases ha0 : a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\nhm : p \u2208 factors a\nha0 : a = 0\n\u22a2 p \u2223 a\n[PROOFSTEP]\nrw [ha0]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\nhm : p \u2208 factors a\nha0 : a = 0\n\u22a2 p \u2223 0\n[PROOFSTEP]\nexact dvd_zero p\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\nhm : p \u2208 factors a\nha0 : \u00aca = 0\n\u22a2 p \u2223 a\n[PROOFSTEP]\nobtain \u27e8a0, nza, ha'\u27e9 := exists_non_zero_rep ha0\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\nhm : p \u2208 factors a\nha0 : \u00aca = 0\na0 : \u03b1\nnza : a0 \u2260 0\nha' : Associates.mk a0 = a\n\u22a2 p \u2223 a\n[PROOFSTEP]\nrw [\u2190 Associates.factors_prod a]\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\nhm : p \u2208 factors a\nha0 : \u00aca = 0\na0 : \u03b1\nnza : a0 \u2260 0\nha' : Associates.mk a0 = a\n\u22a2 p \u2223 FactorSet.prod (factors a)\n[PROOFSTEP]\nrw [\u2190 ha', factors_mk a0 nza] at hm \u22a2\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\nha0 : \u00aca = 0\na0 : \u03b1\nhm : p \u2208 \u2191(factors' a0)\nnza : a0 \u2260 0\nha' : Associates.mk a0 = a\n\u22a2 p \u2223 FactorSet.prod \u2191(factors' a0)\n[PROOFSTEP]\nrw [prod_coe]\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\nha0 : \u00aca = 0\na0 : \u03b1\nhm : p \u2208 \u2191(factors' a0)\nnza : a0 \u2260 0\nha' : Associates.mk a0 = a\n\u22a2 p \u2223 prod (map Subtype.val (factors' a0))\n[PROOFSTEP]\napply Multiset.dvd_prod\n[GOAL]\ncase neg.intro.intro.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\nha0 : \u00aca = 0\na0 : \u03b1\nhm : p \u2208 \u2191(factors' a0)\nnza : a0 \u2260 0\nha' : Associates.mk a0 = a\n\u22a2 p \u2208 map Subtype.val (factors' a0)\n[PROOFSTEP]\napply Multiset.mem_map.mpr\n[GOAL]\ncase neg.intro.intro.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\nha0 : \u00aca = 0\na0 : \u03b1\nhm : p \u2208 \u2191(factors' a0)\nnza : a0 \u2260 0\nha' : Associates.mk a0 = a\n\u22a2 \u2203 a, a \u2208 factors' a0 \u2227 \u2191a = p\n[PROOFSTEP]\nexact \u27e8\u27e8p, hp\u27e9, mem_factorSet_some.mp hm, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : \u03b1\np : Associates \u03b1\nhp : Irreducible p\nhz : a \u2260 0\nh_mem : { val := p, property := hp } \u2208 factors' a\n\u22a2 p \u2223 Associates.mk a\n[PROOFSTEP]\nhaveI := Classical.decEq (Associates \u03b1)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : \u03b1\np : Associates \u03b1\nhp : Irreducible p\nhz : a \u2260 0\nh_mem : { val := p, property := hp } \u2208 factors' a\nthis : DecidableEq (Associates \u03b1)\n\u22a2 p \u2223 Associates.mk a\n[PROOFSTEP]\napply dvd_of_mem_factors (hp := hp)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : \u03b1\np : Associates \u03b1\nhp : Irreducible p\nhz : a \u2260 0\nh_mem : { val := p, property := hp } \u2208 factors' a\nthis : DecidableEq (Associates \u03b1)\n\u22a2 p \u2208 factors (Associates.mk a)\n[PROOFSTEP]\nrw [factors_mk _ hz]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : \u03b1\np : Associates \u03b1\nhp : Irreducible p\nhz : a \u2260 0\nh_mem : { val := p, property := hp } \u2208 factors' a\nthis : DecidableEq (Associates \u03b1)\n\u22a2 p \u2208 \u2191(factors' a)\n[PROOFSTEP]\napply mem_factorSet_some.2 h_mem\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\nhd : p \u2223 a\n\u22a2 { val := Associates.mk p, property := (_ : Irreducible (Associates.mk p)) } \u2208 factors' a\n[PROOFSTEP]\nobtain \u27e8q, hq, hpq\u27e9 := exists_mem_factors_of_dvd ha0 hp hd\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\nhd : p \u2223 a\nq : \u03b1\nhq : q \u2208 UniqueFactorizationMonoid.factors a\nhpq : p ~\u1d64 q\n\u22a2 { val := Associates.mk p, property := (_ : Irreducible (Associates.mk p)) } \u2208 factors' a\n[PROOFSTEP]\napply Multiset.mem_pmap.mpr\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\nhd : p \u2223 a\nq : \u03b1\nhq : q \u2208 UniqueFactorizationMonoid.factors a\nhpq : p ~\u1d64 q\n\u22a2 \u2203 a_1 h,\n { val := Associates.mk a_1, property := (_ : Irreducible (Associates.mk a_1)) } =\n { val := Associates.mk p, property := (_ : Irreducible (Associates.mk p)) }\n[PROOFSTEP]\nuse q\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\nhd : p \u2223 a\nq : \u03b1\nhq : q \u2208 UniqueFactorizationMonoid.factors a\nhpq : p ~\u1d64 q\n\u22a2 \u2203 h,\n { val := Associates.mk q, property := (_ : Irreducible (Associates.mk q)) } =\n { val := Associates.mk p, property := (_ : Irreducible (Associates.mk p)) }\n[PROOFSTEP]\nuse hq\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\nhd : p \u2223 a\nq : \u03b1\nhq : q \u2208 UniqueFactorizationMonoid.factors a\nhpq : p ~\u1d64 q\n\u22a2 { val := Associates.mk q, property := (_ : Irreducible (Associates.mk q)) } =\n { val := Associates.mk p, property := (_ : Irreducible (Associates.mk p)) }\n[PROOFSTEP]\nexact Subtype.eq (Eq.symm (mk_eq_mk_iff_associated.mpr hpq))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\n\u22a2 { val := Associates.mk p, property := (_ : Irreducible (Associates.mk p)) } \u2208 factors' a \u2194 p \u2223 a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\n\u22a2 { val := Associates.mk p, property := (_ : Irreducible (Associates.mk p)) } \u2208 factors' a \u2192 p \u2223 a\n[PROOFSTEP]\nrw [\u2190 mk_dvd_mk]\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\n\u22a2 { val := Associates.mk p, property := (_ : Irreducible (Associates.mk p)) } \u2208 factors' a \u2192\n Associates.mk p \u2223 Associates.mk a\n[PROOFSTEP]\napply dvd_of_mem_factors'\n[GOAL]\ncase mp.hz\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\n\u22a2 a \u2260 0\n[PROOFSTEP]\napply ha0\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\n\u22a2 p \u2223 a \u2192 { val := Associates.mk p, property := (_ : Irreducible (Associates.mk p)) } \u2208 factors' a\n[PROOFSTEP]\napply mem_factors'_of_dvd ha0 hp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\nhd : p \u2223 a\n\u22a2 Associates.mk p \u2208 factors (Associates.mk a)\n[PROOFSTEP]\nrw [factors_mk _ ha0]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\nhd : p \u2223 a\n\u22a2 Associates.mk p \u2208 \u2191(factors' a)\n[PROOFSTEP]\nexact mem_factorSet_some.mpr (mem_factors'_of_dvd ha0 hp hd)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\n\u22a2 Associates.mk p \u2208 factors (Associates.mk a) \u2194 p \u2223 a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\n\u22a2 Associates.mk p \u2208 factors (Associates.mk a) \u2192 p \u2223 a\n[PROOFSTEP]\nrw [\u2190 mk_dvd_mk]\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\n\u22a2 Associates.mk p \u2208 factors (Associates.mk a) \u2192 Associates.mk p \u2223 Associates.mk a\n[PROOFSTEP]\napply dvd_of_mem_factors\n[GOAL]\ncase mp.hp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\n\u22a2 Irreducible (Associates.mk p)\n[PROOFSTEP]\nexact (irreducible_mk p).mpr hp\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\n\u22a2 p \u2223 a \u2192 Associates.mk p \u2208 factors (Associates.mk a)\n[PROOFSTEP]\napply mem_factors_of_dvd ha0 hp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nh : Associates.mk a \u2293 Associates.mk b \u2260 1\n\u22a2 \u2203 p, Prime p \u2227 p \u2223 a \u2227 p \u2223 b\n[PROOFSTEP]\nhave hz : factors (Associates.mk a) \u2293 factors (Associates.mk b) \u2260 0 :=\n by\n contrapose! h with hf\n change (factors (Associates.mk a) \u2293 factors (Associates.mk b)).prod = 1\n rw [hf]\n exact Multiset.prod_zero\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nh : Associates.mk a \u2293 Associates.mk b \u2260 1\n\u22a2 factors (Associates.mk a) \u2293 factors (Associates.mk b) \u2260 0\n[PROOFSTEP]\ncontrapose! h with hf\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nhf : factors (Associates.mk a) \u2293 factors (Associates.mk b) = 0\n\u22a2 Associates.mk a \u2293 Associates.mk b = 1\n[PROOFSTEP]\nchange (factors (Associates.mk a) \u2293 factors (Associates.mk b)).prod = 1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nhf : factors (Associates.mk a) \u2293 factors (Associates.mk b) = 0\n\u22a2 FactorSet.prod (factors (Associates.mk a) \u2293 factors (Associates.mk b)) = 1\n[PROOFSTEP]\nrw [hf]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nhf : factors (Associates.mk a) \u2293 factors (Associates.mk b) = 0\n\u22a2 FactorSet.prod 0 = 1\n[PROOFSTEP]\nexact Multiset.prod_zero\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nh : Associates.mk a \u2293 Associates.mk b \u2260 1\nhz : factors (Associates.mk a) \u2293 factors (Associates.mk b) \u2260 0\n\u22a2 \u2203 p, Prime p \u2227 p \u2223 a \u2227 p \u2223 b\n[PROOFSTEP]\nrw [factors_mk a ha, factors_mk b hb, \u2190 WithTop.coe_inf] at hz \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nh : Associates.mk a \u2293 Associates.mk b \u2260 1\nhz : \u2191(factors' a \u2293 factors' b) \u2260 0\n\u22a2 \u2203 p, Prime p \u2227 p \u2223 a \u2227 p \u2223 b\n[PROOFSTEP]\nobtain \u27e8\u27e8p0, p0_irr\u27e9, p0_mem\u27e9 := Multiset.exists_mem_of_ne_zero ((mt WithTop.coe_eq_coe.mpr) hz)\n[GOAL]\ncase intro.mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nh : Associates.mk a \u2293 Associates.mk b \u2260 1\nhz : \u2191(factors' a \u2293 factors' b) \u2260 0\np0 : Associates \u03b1\np0_irr : Irreducible p0\np0_mem : { val := p0, property := p0_irr } \u2208 factors' a \u2293 factors' b\n\u22a2 \u2203 p, Prime p \u2227 p \u2223 a \u2227 p \u2223 b\n[PROOFSTEP]\nrw [Multiset.inf_eq_inter] at p0_mem \n[GOAL]\ncase intro.mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nh : Associates.mk a \u2293 Associates.mk b \u2260 1\nhz : \u2191(factors' a \u2293 factors' b) \u2260 0\np0 : Associates \u03b1\np0_irr : Irreducible p0\np0_mem : { val := p0, property := p0_irr } \u2208 factors' a \u2229 factors' b\n\u22a2 \u2203 p, Prime p \u2227 p \u2223 a \u2227 p \u2223 b\n[PROOFSTEP]\nobtain \u27e8p, rfl\u27e9 : \u2203 p, Associates.mk p = p0 := Quot.exists_rep p0\n[GOAL]\ncase intro.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nh : Associates.mk a \u2293 Associates.mk b \u2260 1\nhz : \u2191(factors' a \u2293 factors' b) \u2260 0\np : \u03b1\np0_irr : Irreducible (Associates.mk p)\np0_mem : { val := Associates.mk p, property := p0_irr } \u2208 factors' a \u2229 factors' b\n\u22a2 \u2203 p, Prime p \u2227 p \u2223 a \u2227 p \u2223 b\n[PROOFSTEP]\nrefine' \u27e8p, _, _, _\u27e9\n[GOAL]\ncase intro.mk.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nh : Associates.mk a \u2293 Associates.mk b \u2260 1\nhz : \u2191(factors' a \u2293 factors' b) \u2260 0\np : \u03b1\np0_irr : Irreducible (Associates.mk p)\np0_mem : { val := Associates.mk p, property := p0_irr } \u2208 factors' a \u2229 factors' b\n\u22a2 Prime p\n[PROOFSTEP]\nrw [\u2190 irreducible_iff_prime, \u2190 irreducible_mk]\n[GOAL]\ncase intro.mk.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nh : Associates.mk a \u2293 Associates.mk b \u2260 1\nhz : \u2191(factors' a \u2293 factors' b) \u2260 0\np : \u03b1\np0_irr : Irreducible (Associates.mk p)\np0_mem : { val := Associates.mk p, property := p0_irr } \u2208 factors' a \u2229 factors' b\n\u22a2 Irreducible (Associates.mk p)\n[PROOFSTEP]\nexact p0_irr\n[GOAL]\ncase intro.mk.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nh : Associates.mk a \u2293 Associates.mk b \u2260 1\nhz : \u2191(factors' a \u2293 factors' b) \u2260 0\np : \u03b1\np0_irr : Irreducible (Associates.mk p)\np0_mem : { val := Associates.mk p, property := p0_irr } \u2208 factors' a \u2229 factors' b\n\u22a2 p \u2223 a\n[PROOFSTEP]\napply dvd_of_mk_le_mk\n[GOAL]\ncase intro.mk.intro.refine'_2.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nh : Associates.mk a \u2293 Associates.mk b \u2260 1\nhz : \u2191(factors' a \u2293 factors' b) \u2260 0\np : \u03b1\np0_irr : Irreducible (Associates.mk p)\np0_mem : { val := Associates.mk p, property := p0_irr } \u2208 factors' a \u2229 factors' b\n\u22a2 Associates.mk p \u2264 Associates.mk a\n[PROOFSTEP]\napply dvd_of_mem_factors' (Multiset.mem_inter.mp p0_mem).left\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nh : Associates.mk a \u2293 Associates.mk b \u2260 1\nhz : \u2191(factors' a \u2293 factors' b) \u2260 0\np : \u03b1\np0_irr : Irreducible (Associates.mk p)\np0_mem : { val := Associates.mk p, property := p0_irr } \u2208 factors' a \u2229 factors' b\n\u22a2 a \u2260 0\n[PROOFSTEP]\napply ha\n[GOAL]\ncase intro.mk.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nh : Associates.mk a \u2293 Associates.mk b \u2260 1\nhz : \u2191(factors' a \u2293 factors' b) \u2260 0\np : \u03b1\np0_irr : Irreducible (Associates.mk p)\np0_mem : { val := Associates.mk p, property := p0_irr } \u2208 factors' a \u2229 factors' b\n\u22a2 p \u2223 b\n[PROOFSTEP]\napply dvd_of_mk_le_mk\n[GOAL]\ncase intro.mk.intro.refine'_3.a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nh : Associates.mk a \u2293 Associates.mk b \u2260 1\nhz : \u2191(factors' a \u2293 factors' b) \u2260 0\np : \u03b1\np0_irr : Irreducible (Associates.mk p)\np0_mem : { val := Associates.mk p, property := p0_irr } \u2208 factors' a \u2229 factors' b\n\u22a2 Associates.mk p \u2264 Associates.mk b\n[PROOFSTEP]\napply dvd_of_mem_factors' (Multiset.mem_inter.mp p0_mem).right\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nh : Associates.mk a \u2293 Associates.mk b \u2260 1\nhz : \u2191(factors' a \u2293 factors' b) \u2260 0\np : \u03b1\np0_irr : Irreducible (Associates.mk p)\np0_mem : { val := Associates.mk p, property := p0_irr } \u2208 factors' a \u2229 factors' b\n\u22a2 b \u2260 0\n[PROOFSTEP]\napply hb\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha0 : a \u2260 0\nhb0 : b \u2260 0\n\u22a2 Associates.mk a \u2293 Associates.mk b = 1 \u2194 \u2200 {d : \u03b1}, d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha0 : a \u2260 0\nhb0 : b \u2260 0\n\u22a2 Associates.mk a \u2293 Associates.mk b = 1 \u2192 \u2200 {d : \u03b1}, d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\n[PROOFSTEP]\nintro hg p ha hb hp\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha0 : a \u2260 0\nhb0 : b \u2260 0\nhg : Associates.mk a \u2293 Associates.mk b = 1\np : \u03b1\nha : p \u2223 a\nhb : p \u2223 b\nhp : Prime p\n\u22a2 False\n[PROOFSTEP]\nrefine' ((Associates.prime_mk _).mpr hp).not_unit (isUnit_of_dvd_one _)\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha0 : a \u2260 0\nhb0 : b \u2260 0\nhg : Associates.mk a \u2293 Associates.mk b = 1\np : \u03b1\nha : p \u2223 a\nhb : p \u2223 b\nhp : Prime p\n\u22a2 Associates.mk p \u2223 1\n[PROOFSTEP]\nrw [\u2190 hg]\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha0 : a \u2260 0\nhb0 : b \u2260 0\nhg : Associates.mk a \u2293 Associates.mk b = 1\np : \u03b1\nha : p \u2223 a\nhb : p \u2223 b\nhp : Prime p\n\u22a2 Associates.mk p \u2223 Associates.mk a \u2293 Associates.mk b\n[PROOFSTEP]\nexact le_inf (mk_le_mk_of_dvd ha) (mk_le_mk_of_dvd hb)\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha0 : a \u2260 0\nhb0 : b \u2260 0\n\u22a2 (\u2200 {d : \u03b1}, d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d) \u2192 Associates.mk a \u2293 Associates.mk b = 1\n[PROOFSTEP]\ncontrapose\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha0 : a \u2260 0\nhb0 : b \u2260 0\n\u22a2 \u00acAssociates.mk a \u2293 Associates.mk b = 1 \u2192 \u00ac\u2200 {d : \u03b1}, d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\n[PROOFSTEP]\nintro hg hc\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha0 : a \u2260 0\nhb0 : b \u2260 0\nhg : \u00acAssociates.mk a \u2293 Associates.mk b = 1\nhc : \u2200 {d : \u03b1}, d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8p, hp, hpa, hpb\u27e9 := exists_prime_dvd_of_not_inf_one ha0 hb0 hg\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : \u03b1\nha0 : a \u2260 0\nhb0 : b \u2260 0\nhg : \u00acAssociates.mk a \u2293 Associates.mk b = 1\nhc : \u2200 {d : \u03b1}, d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\np : \u03b1\nhp : Prime p\nhpa : p \u2223 a\nhpb : p \u2223 b\n\u22a2 False\n[PROOFSTEP]\nexact hc hpa hpb hp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 FactorSet.prod (factors p) = FactorSet.prod (some {{ val := p, property := hp }})\n[PROOFSTEP]\nrw [factors_prod, FactorSet.prod]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 p =\n match some {{ val := p, property := hp }} with\n | none => 0\n | some s => prod (map Subtype.val s)\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 p = prod {p}\n[PROOFSTEP]\nrw [prod_singleton]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\np : Associates \u03b1\nhp : Irreducible p\nk : \u2115\n\u22a2 FactorSet.prod (factors (p ^ k)) = FactorSet.prod (some (replicate k { val := p, property := hp }))\n[PROOFSTEP]\nrw [Associates.factors_prod, FactorSet.prod]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\np : Associates \u03b1\nhp : Irreducible p\nk : \u2115\n\u22a2 p ^ k =\n match some (replicate k { val := p, property := hp }) with\n | none => 0\n | some s => prod (map Subtype.val s)\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\np : Associates \u03b1\nhp : Irreducible p\nk : \u2115\n\u22a2 p ^ k = prod (map Subtype.val (replicate k { val := p, property := hp }))\n[PROOFSTEP]\nrw [Multiset.map_replicate, Multiset.prod_replicate, Subtype.coe_mk]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\nm p : Associates \u03b1\nh\u2081 : m \u2260 0\nh\u2082 : Irreducible p\nk : \u2115\n\u22a2 p ^ k \u2264 m \u2194 k \u2264 count p (factors m)\n[PROOFSTEP]\nobtain \u27e8a, nz, rfl\u27e9 := Associates.exists_non_zero_rep h\u2081\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\np : Associates \u03b1\nh\u2082 : Irreducible p\nk : \u2115\na : \u03b1\nnz : a \u2260 0\nh\u2081 : Associates.mk a \u2260 0\n\u22a2 p ^ k \u2264 Associates.mk a \u2194 k \u2264 count p (factors (Associates.mk a))\n[PROOFSTEP]\nrw [factors_mk _ nz, \u2190 WithTop.some_eq_coe, count_some, Multiset.le_count_iff_replicate_le, \u2190 factors_le,\n factors_prime_pow h\u2082, factors_mk _ nz]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\np : Associates \u03b1\nh\u2082 : Irreducible p\nk : \u2115\na : \u03b1\nnz : a \u2260 0\nh\u2081 : Associates.mk a \u2260 0\n\u22a2 some (replicate k { val := p, property := h\u2082 }) \u2264 \u2191(factors' a) \u2194\n replicate k { val := p, property := ?intro.intro.hp } \u2264 factors' a\ncase intro.intro.hp\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\np : Associates \u03b1\nh\u2082 : Irreducible p\nk : \u2115\na : \u03b1\nnz : a \u2260 0\nh\u2081 : Associates.mk a \u2260 0\n\u22a2 Irreducible p\ncase intro.intro.hp\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\np : Associates \u03b1\nh\u2082 : Irreducible p\nk : \u2115\na : \u03b1\nnz : a \u2260 0\nh\u2081 : Associates.mk a \u2260 0\n\u22a2 Irreducible p\ncase intro.intro.hp\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\np : Associates \u03b1\nh\u2082 : Irreducible p\nk : \u2115\na : \u03b1\nnz : a \u2260 0\nh\u2081 : Associates.mk a \u2260 0\n\u22a2 Irreducible p\ncase intro.intro.hp\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\np : Associates \u03b1\nh\u2082 : Irreducible p\nk : \u2115\na : \u03b1\nnz : a \u2260 0\nh\u2081 : Associates.mk a \u2260 0\n\u22a2 Irreducible p\ncase intro.intro.hp\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\np : Associates \u03b1\nh\u2082 : Irreducible p\nk : \u2115\na : \u03b1\nnz : a \u2260 0\nh\u2081 : Associates.mk a \u2260 0\n\u22a2 Irreducible p\n[PROOFSTEP]\nexact WithTop.coe_le_coe\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\nm p : Associates \u03b1\nh0 : m \u2260 0\nhp : Irreducible p\n\u22a2 count p (factors m) \u2260 0 \u2192 p \u2264 m\n[PROOFSTEP]\nnontriviality \u03b1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\nm p : Associates \u03b1\nh0 : m \u2260 0\nhp : Irreducible p\n\u271d : Nontrivial \u03b1\n\u22a2 count p (factors m) \u2260 0 \u2192 p \u2264 m\n[PROOFSTEP]\nrw [\u2190 pos_iff_ne_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\nm p : Associates \u03b1\nh0 : m \u2260 0\nhp : Irreducible p\n\u271d : Nontrivial \u03b1\n\u22a2 0 < count p (factors m) \u2192 p \u2264 m\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\nm p : Associates \u03b1\nh0 : m \u2260 0\nhp : Irreducible p\n\u271d : Nontrivial \u03b1\nh : 0 < count p (factors m)\n\u22a2 p \u2264 m\n[PROOFSTEP]\nrw [\u2190 pow_one p]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\nm p : Associates \u03b1\nh0 : m \u2260 0\nhp : Irreducible p\n\u271d : Nontrivial \u03b1\nh : 0 < count p (factors m)\n\u22a2 p ^ 1 \u2264 m\n[PROOFSTEP]\napply (prime_pow_dvd_iff_le h0 hp).2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\nm p : Associates \u03b1\nh0 : m \u2260 0\nhp : Irreducible p\n\u271d : Nontrivial \u03b1\nh : 0 < count p (factors m)\n\u22a2 1 \u2264 count p (factors m)\n[PROOFSTEP]\nsimpa only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\n\u22a2 count (Associates.mk p) (factors (Associates.mk a)) \u2260 0 \u2194 p \u2223 a\n[PROOFSTEP]\nnontriviality \u03b1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\ninst\u271d : Nontrivial \u03b1\n\u22a2 count (Associates.mk p) (factors (Associates.mk a)) \u2260 0 \u2194 p \u2223 a\n[PROOFSTEP]\nrw [\u2190 Associates.mk_le_mk_iff_dvd_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\ninst\u271d : Nontrivial \u03b1\n\u22a2 count (Associates.mk p) (factors (Associates.mk a)) \u2260 0 \u2194 Associates.mk p \u2264 Associates.mk a\n[PROOFSTEP]\nrefine'\n \u27e8fun h => Associates.le_of_count_ne_zero (Associates.mk_ne_zero.mpr ha0) ((Associates.irreducible_mk p).mpr hp) h,\n fun h => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\ninst\u271d : Nontrivial \u03b1\nh : Associates.mk p \u2264 Associates.mk a\n\u22a2 count (Associates.mk p) (factors (Associates.mk a)) \u2260 0\n[PROOFSTEP]\nrw [\u2190 pow_one (Associates.mk p),\n Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero.mpr ha0) ((Associates.irreducible_mk p).mpr hp)] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\ninst\u271d : Nontrivial \u03b1\nh : 1 \u2264 count (Associates.mk p) (factors (Associates.mk a))\n\u22a2 count (Associates.mk p) (factors (Associates.mk a)) \u2260 0\n[PROOFSTEP]\nexact (zero_lt_one.trans_le h).ne'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 count p (factors p) = 1\n[PROOFSTEP]\nsimp [factors_self hp, Associates.count_some hp]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\np q : Associates \u03b1\nhp : Irreducible p\nhq : Irreducible q\nh : p \u2260 q\nh' : count p (factors q) \u2260 0\n\u22a2 p \u2223 q\n[PROOFSTEP]\nnontriviality \u03b1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\np q : Associates \u03b1\nhp : Irreducible p\nhq : Irreducible q\nh : p \u2260 q\nh' : count p (factors q) \u2260 0\n\u271d : Nontrivial \u03b1\n\u22a2 p \u2223 q\n[PROOFSTEP]\nexact le_of_count_ne_zero hq.ne_zero hp h'\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : Associates \u03b1\nha : a \u2260 0\nb : Associates \u03b1\nhb : b \u2260 0\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 count p (factors (a * b)) = count p (factors a) + count p (factors b)\n[PROOFSTEP]\nobtain \u27e8a0, nza, ha'\u27e9 := exists_non_zero_rep ha\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : Associates \u03b1\nha : a \u2260 0\nb : Associates \u03b1\nhb : b \u2260 0\np : Associates \u03b1\nhp : Irreducible p\na0 : \u03b1\nnza : a0 \u2260 0\nha' : Associates.mk a0 = a\n\u22a2 count p (factors (a * b)) = count p (factors a) + count p (factors b)\n[PROOFSTEP]\nobtain \u27e8b0, nzb, hb'\u27e9 := exists_non_zero_rep hb\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : Associates \u03b1\nha : a \u2260 0\nb : Associates \u03b1\nhb : b \u2260 0\np : Associates \u03b1\nhp : Irreducible p\na0 : \u03b1\nnza : a0 \u2260 0\nha' : Associates.mk a0 = a\nb0 : \u03b1\nnzb : b0 \u2260 0\nhb' : Associates.mk b0 = b\n\u22a2 count p (factors (a * b)) = count p (factors a) + count p (factors b)\n[PROOFSTEP]\nrw [factors_mul, \u2190 ha', \u2190 hb', factors_mk a0 nza, factors_mk b0 nzb, \u2190 FactorSet.coe_add, \u2190 WithTop.some_eq_coe, \u2190\n WithTop.some_eq_coe, \u2190 WithTop.some_eq_coe, count_some hp, Multiset.count_add, count_some hp, count_some hp]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : Associates \u03b1\nha : a \u2260 0\nb : Associates \u03b1\nhb : b \u2260 0\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 count p (factors a) = 0 \u2228 count p (factors b) = 0\n[PROOFSTEP]\nrw [or_iff_not_imp_left, \u2190 Ne.def]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : Associates \u03b1\nha : a \u2260 0\nb : Associates \u03b1\nhb : b \u2260 0\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 count p (factors a) \u2260 0 \u2192 count p (factors b) = 0\n[PROOFSTEP]\nintro hca\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : Associates \u03b1\nha : a \u2260 0\nb : Associates \u03b1\nhb : b \u2260 0\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\np : Associates \u03b1\nhp : Irreducible p\nhca : count p (factors a) \u2260 0\n\u22a2 count p (factors b) = 0\n[PROOFSTEP]\ncontrapose! hab with hcb\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na : Associates \u03b1\nha : a \u2260 0\nb : Associates \u03b1\nhb : b \u2260 0\np : Associates \u03b1\nhp : Irreducible p\nhca : count p (factors a) \u2260 0\nhcb : count p (factors b) \u2260 0\n\u22a2 \u2203 d, d \u2223 a \u2227 d \u2223 b \u2227 Prime d\n[PROOFSTEP]\nexact \u27e8p, le_of_count_ne_zero ha hp hca, le_of_count_ne_zero hb hp hcb, irreducible_iff_prime.mp hp\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nhb : b \u2260 0\np : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\n\u22a2 count p (factors a) = 0 \u2228 count p (factors a) = count p (factors (a * b))\n[PROOFSTEP]\nby_cases ha : a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nhb : b \u2260 0\np : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nha : a = 0\n\u22a2 count p (factors a) = 0 \u2228 count p (factors a) = count p (factors (a * b))\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nhb : b \u2260 0\np : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nha : \u00aca = 0\n\u22a2 count p (factors a) = 0 \u2228 count p (factors a) = count p (factors (a * b))\n[PROOFSTEP]\ncases' count_of_coprime ha hb hab hp with hz hb0\n[GOAL]\ncase neg.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nhb : b \u2260 0\np : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nha : \u00aca = 0\nhz : count p (factors a) = 0\n\u22a2 count p (factors a) = 0 \u2228 count p (factors a) = count p (factors (a * b))\n[PROOFSTEP]\ntauto\n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nhb : b \u2260 0\np : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nha : \u00aca = 0\nhb0 : count p (factors b) = 0\n\u22a2 count p (factors a) = 0 \u2228 count p (factors a) = count p (factors (a * b))\n[PROOFSTEP]\napply Or.intro_right\n[GOAL]\ncase neg.inr.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nhb : b \u2260 0\np : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nha : \u00aca = 0\nhb0 : count p (factors b) = 0\n\u22a2 count p (factors a) = count p (factors (a * b))\n[PROOFSTEP]\nrw [count_mul ha hb hp, hb0, add_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b p : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\n\u22a2 count p (factors (a * b)) = count p (factors a) \u2228 count p (factors (a * b)) = count p (factors b)\n[PROOFSTEP]\nby_cases ha : a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b p : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nha : a = 0\n\u22a2 count p (factors (a * b)) = count p (factors a) \u2228 count p (factors (a * b)) = count p (factors b)\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b p : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nha : \u00aca = 0\n\u22a2 count p (factors (a * b)) = count p (factors a) \u2228 count p (factors (a * b)) = count p (factors b)\n[PROOFSTEP]\nby_cases hb : b = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b p : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nha : \u00aca = 0\nhb : b = 0\n\u22a2 count p (factors (a * b)) = count p (factors a) \u2228 count p (factors (a * b)) = count p (factors b)\n[PROOFSTEP]\nsimp [hb]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b p : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nha : \u00aca = 0\nhb : \u00acb = 0\n\u22a2 count p (factors (a * b)) = count p (factors a) \u2228 count p (factors (a * b)) = count p (factors b)\n[PROOFSTEP]\nrw [count_mul ha hb hp]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b p : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nha : \u00aca = 0\nhb : \u00acb = 0\n\u22a2 count p (factors a) + count p (factors b) = count p (factors a) \u2228\n count p (factors a) + count p (factors b) = count p (factors b)\n[PROOFSTEP]\ncases' count_of_coprime ha hb hab hp with ha0 hb0\n[GOAL]\ncase neg.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b p : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nha : \u00aca = 0\nhb : \u00acb = 0\nha0 : count p (factors a) = 0\n\u22a2 count p (factors a) + count p (factors b) = count p (factors a) \u2228\n count p (factors a) + count p (factors b) = count p (factors b)\n[PROOFSTEP]\napply Or.intro_right\n[GOAL]\ncase neg.inl.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b p : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nha : \u00aca = 0\nhb : \u00acb = 0\nha0 : count p (factors a) = 0\n\u22a2 count p (factors a) + count p (factors b) = count p (factors b)\n[PROOFSTEP]\nrw [ha0, zero_add]\n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b p : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nha : \u00aca = 0\nhb : \u00acb = 0\nhb0 : count p (factors b) = 0\n\u22a2 count p (factors a) + count p (factors b) = count p (factors a) \u2228\n count p (factors a) + count p (factors b) = count p (factors b)\n[PROOFSTEP]\napply Or.intro_left\n[GOAL]\ncase neg.inr.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b p : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nha : \u00aca = 0\nhb : \u00acb = 0\nhb0 : count p (factors b) = 0\n\u22a2 count p (factors a) + count p (factors b) = count p (factors a)\n[PROOFSTEP]\nrw [hb0, add_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nhb : b \u2260 0\np : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nhabk : k \u2223 count p (factors (a * b))\n\u22a2 k \u2223 count p (factors a)\n[PROOFSTEP]\nby_cases ha : a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nhb : b \u2260 0\np : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nhabk : k \u2223 count p (factors (a * b))\nha : a = 0\n\u22a2 k \u2223 count p (factors a)\n[PROOFSTEP]\nsimpa [*] using habk\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nhb : b \u2260 0\np : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nhabk : k \u2223 count p (factors (a * b))\nha : \u00aca = 0\n\u22a2 k \u2223 count p (factors a)\n[PROOFSTEP]\ncases' count_of_coprime ha hb hab hp with hz h\n[GOAL]\ncase neg.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nhb : b \u2260 0\np : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nhabk : k \u2223 count p (factors (a * b))\nha : \u00aca = 0\nhz : count p (factors a) = 0\n\u22a2 k \u2223 count p (factors a)\n[PROOFSTEP]\nrw [hz]\n[GOAL]\ncase neg.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nhb : b \u2260 0\np : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nhabk : k \u2223 count p (factors (a * b))\nha : \u00aca = 0\nhz : count p (factors a) = 0\n\u22a2 k \u2223 0\n[PROOFSTEP]\nexact dvd_zero k\n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nhb : b \u2260 0\np : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nhabk : k \u2223 count p (factors (a * b))\nha : \u00aca = 0\nh : count p (factors b) = 0\n\u22a2 k \u2223 count p (factors a)\n[PROOFSTEP]\nrw [count_mul ha hb hp, h] at habk \n[GOAL]\ncase neg.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na b : Associates \u03b1\nhb : b \u2260 0\np : Associates \u03b1\nhp : Irreducible p\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nhabk : k \u2223 count p (factors a) + 0\nha : \u00aca = 0\nh : count p (factors b) = 0\n\u22a2 k \u2223 count p (factors a)\n[PROOFSTEP]\nexact habk\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\n\u22a2 factors 1 = 0\n[PROOFSTEP]\napply eq_of_prod_eq_prod\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\n\u22a2 FactorSet.prod (factors 1) = FactorSet.prod 0\n[PROOFSTEP]\nrw [Associates.factors_prod]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\n\u22a2 1 = FactorSet.prod 0\n[PROOFSTEP]\nexact Multiset.prod_zero\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na : Associates \u03b1\nk : \u2115\n\u22a2 factors (a ^ k) = k \u2022 factors a\n[PROOFSTEP]\ninduction' k with n h\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na : Associates \u03b1\n\u22a2 factors (a ^ Nat.zero) = Nat.zero \u2022 factors a\n[PROOFSTEP]\nrw [zero_nsmul, pow_zero]\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na : Associates \u03b1\n\u22a2 factors 1 = 0\n[PROOFSTEP]\nexact factors_one\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na : Associates \u03b1\nn : \u2115\nh : factors (a ^ n) = n \u2022 factors a\n\u22a2 factors (a ^ Nat.succ n) = Nat.succ n \u2022 factors a\n[PROOFSTEP]\nrw [pow_succ, succ_nsmul, factors_mul, h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na : Associates \u03b1\nha : a \u2260 0\np : Associates \u03b1\nhp : Irreducible p\nk : \u2115\n\u22a2 count p (factors (a ^ k)) = k * count p (factors a)\n[PROOFSTEP]\ninduction' k with n h\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na : Associates \u03b1\nha : a \u2260 0\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 count p (factors (a ^ Nat.zero)) = Nat.zero * count p (factors a)\n[PROOFSTEP]\nrw [pow_zero, factors_one, Nat.zero_eq, zero_mul, count_zero hp]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na : Associates \u03b1\nha : a \u2260 0\np : Associates \u03b1\nhp : Irreducible p\nn : \u2115\nh : count p (factors (a ^ n)) = n * count p (factors a)\n\u22a2 count p (factors (a ^ Nat.succ n)) = Nat.succ n * count p (factors a)\n[PROOFSTEP]\nrw [pow_succ, count_mul ha (pow_ne_zero _ ha) hp, h, Nat.succ_eq_add_one]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na : Associates \u03b1\nha : a \u2260 0\np : Associates \u03b1\nhp : Irreducible p\nn : \u2115\nh : count p (factors (a ^ n)) = n * count p (factors a)\n\u22a2 count p (factors a) + n * count p (factors a) = (n + 1) * count p (factors a)\n[PROOFSTEP]\nring\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na : Associates \u03b1\nha : a \u2260 0\np : Associates \u03b1\nhp : Irreducible p\nk : \u2115\n\u22a2 k \u2223 count p (factors (a ^ k))\n[PROOFSTEP]\nrw [count_pow ha hp]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na : Associates \u03b1\nha : a \u2260 0\np : Associates \u03b1\nhp : Irreducible p\nk : \u2115\n\u22a2 k \u2223 k * count p (factors a)\n[PROOFSTEP]\napply dvd_mul_right\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na : Associates \u03b1\nha : a \u2260 0\nk : \u2115\nhk : \u2200 (p : Associates \u03b1), Irreducible p \u2192 k \u2223 count p (factors a)\n\u22a2 \u2203 b, a = b ^ k\n[PROOFSTEP]\nobtain \u27e8a0, hz, rfl\u27e9 := exists_non_zero_rep ha\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\nk : \u2115\na0 : \u03b1\nhz : a0 \u2260 0\nha : Associates.mk a0 \u2260 0\nhk : \u2200 (p : Associates \u03b1), Irreducible p \u2192 k \u2223 count p (factors (Associates.mk a0))\n\u22a2 \u2203 b, Associates.mk a0 = b ^ k\n[PROOFSTEP]\nrw [factors_mk a0 hz] at hk \n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\nk : \u2115\na0 : \u03b1\nhz : a0 \u2260 0\nha : Associates.mk a0 \u2260 0\nhk : \u2200 (p : Associates \u03b1), Irreducible p \u2192 k \u2223 count p \u2191(factors' a0)\n\u22a2 \u2203 b, Associates.mk a0 = b ^ k\n[PROOFSTEP]\nhave hk' : \u2200 p, p \u2208 factors' a0 \u2192 k \u2223 (factors' a0).count p :=\n by\n rintro p -\n have pp : p = \u27e8p.val, p.2\u27e9 := by simp only [Subtype.coe_eta]\n rw [pp, \u2190 count_some p.2]\n exact hk p.val p.2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\nk : \u2115\na0 : \u03b1\nhz : a0 \u2260 0\nha : Associates.mk a0 \u2260 0\nhk : \u2200 (p : Associates \u03b1), Irreducible p \u2192 k \u2223 count p \u2191(factors' a0)\n\u22a2 \u2200 (p : { a // Irreducible a }), p \u2208 factors' a0 \u2192 k \u2223 Multiset.count p (factors' a0)\n[PROOFSTEP]\nrintro p -\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\nk : \u2115\na0 : \u03b1\nhz : a0 \u2260 0\nha : Associates.mk a0 \u2260 0\nhk : \u2200 (p : Associates \u03b1), Irreducible p \u2192 k \u2223 count p \u2191(factors' a0)\np : { a // Irreducible a }\n\u22a2 k \u2223 Multiset.count p (factors' a0)\n[PROOFSTEP]\nhave pp : p = \u27e8p.val, p.2\u27e9 := by simp only [Subtype.coe_eta]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\nk : \u2115\na0 : \u03b1\nhz : a0 \u2260 0\nha : Associates.mk a0 \u2260 0\nhk : \u2200 (p : Associates \u03b1), Irreducible p \u2192 k \u2223 count p \u2191(factors' a0)\np : { a // Irreducible a }\n\u22a2 p = { val := \u2191p, property := (_ : Irreducible \u2191p) }\n[PROOFSTEP]\nsimp only [Subtype.coe_eta]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\nk : \u2115\na0 : \u03b1\nhz : a0 \u2260 0\nha : Associates.mk a0 \u2260 0\nhk : \u2200 (p : Associates \u03b1), Irreducible p \u2192 k \u2223 count p \u2191(factors' a0)\np : { a // Irreducible a }\npp : p = { val := \u2191p, property := (_ : Irreducible \u2191p) }\n\u22a2 k \u2223 Multiset.count p (factors' a0)\n[PROOFSTEP]\nrw [pp, \u2190 count_some p.2]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\nk : \u2115\na0 : \u03b1\nhz : a0 \u2260 0\nha : Associates.mk a0 \u2260 0\nhk : \u2200 (p : Associates \u03b1), Irreducible p \u2192 k \u2223 count p \u2191(factors' a0)\np : { a // Irreducible a }\npp : p = { val := \u2191p, property := (_ : Irreducible \u2191p) }\n\u22a2 k \u2223 count (\u2191p) (some (factors' a0))\n[PROOFSTEP]\nexact hk p.val p.2\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\nk : \u2115\na0 : \u03b1\nhz : a0 \u2260 0\nha : Associates.mk a0 \u2260 0\nhk : \u2200 (p : Associates \u03b1), Irreducible p \u2192 k \u2223 count p \u2191(factors' a0)\nhk' : \u2200 (p : { a // Irreducible a }), p \u2208 factors' a0 \u2192 k \u2223 Multiset.count p (factors' a0)\n\u22a2 \u2203 b, Associates.mk a0 = b ^ k\n[PROOFSTEP]\nobtain \u27e8u, hu\u27e9 := Multiset.exists_smul_of_dvd_count _ hk'\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\nk : \u2115\na0 : \u03b1\nhz : a0 \u2260 0\nha : Associates.mk a0 \u2260 0\nhk : \u2200 (p : Associates \u03b1), Irreducible p \u2192 k \u2223 count p \u2191(factors' a0)\nhk' : \u2200 (p : { a // Irreducible a }), p \u2208 factors' a0 \u2192 k \u2223 Multiset.count p (factors' a0)\nu : Multiset { a // Irreducible a }\nhu : factors' a0 = k \u2022 u\n\u22a2 \u2203 b, Associates.mk a0 = b ^ k\n[PROOFSTEP]\nuse FactorSet.prod (u : FactorSet \u03b1)\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\nk : \u2115\na0 : \u03b1\nhz : a0 \u2260 0\nha : Associates.mk a0 \u2260 0\nhk : \u2200 (p : Associates \u03b1), Irreducible p \u2192 k \u2223 count p \u2191(factors' a0)\nhk' : \u2200 (p : { a // Irreducible a }), p \u2208 factors' a0 \u2192 k \u2223 Multiset.count p (factors' a0)\nu : Multiset { a // Irreducible a }\nhu : factors' a0 = k \u2022 u\n\u22a2 Associates.mk a0 = FactorSet.prod \u2191u ^ k\n[PROOFSTEP]\napply eq_of_factors_eq_factors\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\nk : \u2115\na0 : \u03b1\nhz : a0 \u2260 0\nha : Associates.mk a0 \u2260 0\nhk : \u2200 (p : Associates \u03b1), Irreducible p \u2192 k \u2223 count p \u2191(factors' a0)\nhk' : \u2200 (p : { a // Irreducible a }), p \u2208 factors' a0 \u2192 k \u2223 Multiset.count p (factors' a0)\nu : Multiset { a // Irreducible a }\nhu : factors' a0 = k \u2022 u\n\u22a2 factors (Associates.mk a0) = factors (FactorSet.prod \u2191u ^ k)\n[PROOFSTEP]\nrw [pow_factors, prod_factors, factors_mk a0 hz, \u2190 WithTop.some_eq_coe, hu]\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\nk : \u2115\na0 : \u03b1\nhz : a0 \u2260 0\nha : Associates.mk a0 \u2260 0\nhk : \u2200 (p : Associates \u03b1), Irreducible p \u2192 k \u2223 count p \u2191(factors' a0)\nhk' : \u2200 (p : { a // Irreducible a }), p \u2208 factors' a0 \u2192 k \u2223 Multiset.count p (factors' a0)\nu : Multiset { a // Irreducible a }\nhu : factors' a0 = k \u2022 u\n\u22a2 some (k \u2022 u) = k \u2022 \u2191u\n[PROOFSTEP]\nexact WithBot.coe_nsmul u k\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\np a : Associates \u03b1\nhp : Irreducible p\nn : \u2115\nh : a \u2223 p ^ n\n\u22a2 a = p ^ count p (factors a)\n[PROOFSTEP]\nnontriviality \u03b1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\np a : Associates \u03b1\nhp : Irreducible p\nn : \u2115\nh : a \u2223 p ^ n\n\u271d : Nontrivial \u03b1\n\u22a2 a = p ^ count p (factors a)\n[PROOFSTEP]\nhave hph := pow_ne_zero n hp.ne_zero\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\np a : Associates \u03b1\nhp : Irreducible p\nn : \u2115\nh : a \u2223 p ^ n\n\u271d : Nontrivial \u03b1\nhph : p ^ n \u2260 0\n\u22a2 a = p ^ count p (factors a)\n[PROOFSTEP]\nhave ha := ne_zero_of_dvd_ne_zero hph h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\np a : Associates \u03b1\nhp : Irreducible p\nn : \u2115\nh : a \u2223 p ^ n\n\u271d : Nontrivial \u03b1\nhph : p ^ n \u2260 0\nha : a \u2260 0\n\u22a2 a = p ^ count p (factors a)\n[PROOFSTEP]\napply eq_of_eq_counts ha (pow_ne_zero _ hp.ne_zero)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\np a : Associates \u03b1\nhp : Irreducible p\nn : \u2115\nh : a \u2223 p ^ n\n\u271d : Nontrivial \u03b1\nhph : p ^ n \u2260 0\nha : a \u2260 0\n\u22a2 \u2200 (p_1 : Associates \u03b1), Irreducible p_1 \u2192 count p_1 (factors a) = count p_1 (factors (p ^ count p (factors a)))\n[PROOFSTEP]\nhave eq_zero_of_ne : \u2200 q : Associates \u03b1, Irreducible q \u2192 q \u2260 p \u2192 _ = 0 := fun q hq h' =>\n Nat.eq_zero_of_le_zero <| by\n convert count_le_count_of_le hph hq h\n symm\n rw [count_pow hp.ne_zero hq, count_eq_zero_of_ne hq hp h', mul_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\np a : Associates \u03b1\nhp : Irreducible p\nn : \u2115\nh : a \u2223 p ^ n\n\u271d : Nontrivial \u03b1\nhph : p ^ n \u2260 0\nha : a \u2260 0\nq : Associates \u03b1\nhq : Irreducible q\nh' : q \u2260 p\n\u22a2 ?m.3109694 q \u2264 0\n[PROOFSTEP]\nconvert count_le_count_of_le hph hq h\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\np a : Associates \u03b1\nhp : Irreducible p\nn : \u2115\nh : a \u2223 p ^ n\n\u271d : Nontrivial \u03b1\nhph : p ^ n \u2260 0\nha : a \u2260 0\nq : Associates \u03b1\nhq : Irreducible q\nh' : q \u2260 p\n\u22a2 0 = count q (factors (p ^ n))\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\np a : Associates \u03b1\nhp : Irreducible p\nn : \u2115\nh : a \u2223 p ^ n\n\u271d : Nontrivial \u03b1\nhph : p ^ n \u2260 0\nha : a \u2260 0\nq : Associates \u03b1\nhq : Irreducible q\nh' : q \u2260 p\n\u22a2 count q (factors (p ^ n)) = 0\n[PROOFSTEP]\nrw [count_pow hp.ne_zero hq, count_eq_zero_of_ne hq hp h', mul_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\np a : Associates \u03b1\nhp : Irreducible p\nn : \u2115\nh : a \u2223 p ^ n\n\u271d : Nontrivial \u03b1\nhph : p ^ n \u2260 0\nha : a \u2260 0\neq_zero_of_ne : \u2200 (q : Associates \u03b1), Irreducible q \u2192 q \u2260 p \u2192 count q (factors a) = 0\n\u22a2 \u2200 (p_1 : Associates \u03b1), Irreducible p_1 \u2192 count p_1 (factors a) = count p_1 (factors (p ^ count p (factors a)))\n[PROOFSTEP]\nintro q hq\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\np a : Associates \u03b1\nhp : Irreducible p\nn : \u2115\nh : a \u2223 p ^ n\n\u271d : Nontrivial \u03b1\nhph : p ^ n \u2260 0\nha : a \u2260 0\neq_zero_of_ne : \u2200 (q : Associates \u03b1), Irreducible q \u2192 q \u2260 p \u2192 count q (factors a) = 0\nq : Associates \u03b1\nhq : Irreducible q\n\u22a2 count q (factors a) = count q (factors (p ^ count p (factors a)))\n[PROOFSTEP]\nrw [count_pow hp.ne_zero hq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\np a : Associates \u03b1\nhp : Irreducible p\nn : \u2115\nh : a \u2223 p ^ n\n\u271d : Nontrivial \u03b1\nhph : p ^ n \u2260 0\nha : a \u2260 0\neq_zero_of_ne : \u2200 (q : Associates \u03b1), Irreducible q \u2192 q \u2260 p \u2192 count q (factors a) = 0\nq : Associates \u03b1\nhq : Irreducible q\n\u22a2 count q (factors a) = count p (factors a) * count q (factors p)\n[PROOFSTEP]\nby_cases h : q = p\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\np a : Associates \u03b1\nhp : Irreducible p\nn : \u2115\nh\u271d : a \u2223 p ^ n\n\u271d : Nontrivial \u03b1\nhph : p ^ n \u2260 0\nha : a \u2260 0\neq_zero_of_ne : \u2200 (q : Associates \u03b1), Irreducible q \u2192 q \u2260 p \u2192 count q (factors a) = 0\nq : Associates \u03b1\nhq : Irreducible q\nh : q = p\n\u22a2 count q (factors a) = count p (factors a) * count q (factors p)\n[PROOFSTEP]\nrw [h, count_self hp, mul_one]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\np a : Associates \u03b1\nhp : Irreducible p\nn : \u2115\nh\u271d : a \u2223 p ^ n\n\u271d : Nontrivial \u03b1\nhph : p ^ n \u2260 0\nha : a \u2260 0\neq_zero_of_ne : \u2200 (q : Associates \u03b1), Irreducible q \u2192 q \u2260 p \u2192 count q (factors a) = 0\nq : Associates \u03b1\nhq : Irreducible q\nh : \u00acq = p\n\u22a2 count q (factors a) = count p (factors a) * count q (factors p)\n[PROOFSTEP]\nrw [count_eq_zero_of_ne hq hp h, mul_zero, eq_zero_of_ne q hq h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\ninst\u271d : (n : \u2115) \u2192 Decidable (a \u2223 p ^ n)\nn : \u2115\nh : a \u2223 p ^ n\n\u22a2 Nat.find (_ : \u2203 n, (fun n => a \u2223 p ^ n) n) = count p (factors a)\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\ninst\u271d : (n : \u2115) \u2192 Decidable (a \u2223 p ^ n)\nn : \u2115\nh : a \u2223 p ^ n\n\u22a2 Nat.find (_ : \u2203 n, (fun n => a \u2223 p ^ n) n) \u2264 count p (factors a)\n[PROOFSTEP]\nrefine' Nat.find_le \u27e81, _\u27e9\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\ninst\u271d : (n : \u2115) \u2192 Decidable (a \u2223 p ^ n)\nn : \u2115\nh : a \u2223 p ^ n\n\u22a2 p ^ count p (factors a) = a * 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\ninst\u271d : (n : \u2115) \u2192 Decidable (a \u2223 p ^ n)\nn : \u2115\nh : a \u2223 p ^ n\n\u22a2 p ^ count p (factors a) = a\n[PROOFSTEP]\nsymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\ninst\u271d : (n : \u2115) \u2192 Decidable (a \u2223 p ^ n)\nn : \u2115\nh : a \u2223 p ^ n\n\u22a2 a = p ^ count p (factors a)\n[PROOFSTEP]\nexact eq_pow_count_factors_of_dvd_pow hp h\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\ninst\u271d : (n : \u2115) \u2192 Decidable (a \u2223 p ^ n)\nn : \u2115\nh : a \u2223 p ^ n\n\u22a2 count p (factors a) \u2264 Nat.find (_ : \u2203 n, (fun n => a \u2223 p ^ n) n)\n[PROOFSTEP]\nhave hph := pow_ne_zero (@Nat.find (fun n => a \u2223 p ^ n) _ \u27e8n, h\u27e9) hp.ne_zero\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\ninst\u271d : (n : \u2115) \u2192 Decidable (a \u2223 p ^ n)\nn : \u2115\nh : a \u2223 p ^ n\nhph : p ^ Nat.find (_ : \u2203 n, (fun n => a \u2223 p ^ n) n) \u2260 0\n\u22a2 count p (factors a) \u2264 Nat.find (_ : \u2203 n, (fun n => a \u2223 p ^ n) n)\n[PROOFSTEP]\ncases' subsingleton_or_nontrivial \u03b1 with h\u03b1 h\u03b1\n[GOAL]\ncase a.inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\ninst\u271d : (n : \u2115) \u2192 Decidable (a \u2223 p ^ n)\nn : \u2115\nh : a \u2223 p ^ n\nhph : p ^ Nat.find (_ : \u2203 n, (fun n => a \u2223 p ^ n) n) \u2260 0\nh\u03b1 : Subsingleton \u03b1\n\u22a2 count p (factors a) \u2264 Nat.find (_ : \u2203 n, (fun n => a \u2223 p ^ n) n)\n[PROOFSTEP]\nsimp at hph \n[GOAL]\ncase a.inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\ninst\u271d : (n : \u2115) \u2192 Decidable (a \u2223 p ^ n)\nn : \u2115\nh : a \u2223 p ^ n\nhph : p ^ Nat.find (_ : \u2203 n, (fun n => a \u2223 p ^ n) n) \u2260 0\nh\u03b1 : Nontrivial \u03b1\n\u22a2 count p (factors a) \u2264 Nat.find (_ : \u2203 n, (fun n => a \u2223 p ^ n) n)\n[PROOFSTEP]\nconvert count_le_count_of_le hph hp (@Nat.find_spec (fun n => a \u2223 p ^ n) _ \u27e8n, h\u27e9)\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\ninst\u271d : (n : \u2115) \u2192 Decidable (a \u2223 p ^ n)\nn : \u2115\nh : a \u2223 p ^ n\nhph : p ^ Nat.find (_ : \u2203 n, (fun n => a \u2223 p ^ n) n) \u2260 0\nh\u03b1 : Nontrivial \u03b1\n\u22a2 Nat.find (_ : \u2203 n, (fun n => a \u2223 p ^ n) n) = count p (factors (p ^ Nat.find (_ : \u2203 n, (fun n => a \u2223 p ^ n) n)))\n[PROOFSTEP]\nrw [count_pow hp.ne_zero hp, count_self hp, mul_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b c : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nh : a * b = c ^ k\n\u22a2 \u2203 d, a = d ^ k\n[PROOFSTEP]\nclassical\nby_cases hk0 : k = 0\n\u00b7 use 1\n rw [hk0, pow_zero] at h \u22a2\n apply (mul_eq_one_iff.1 h).1\n\u00b7 refine' is_pow_of_dvd_count ha _\n intro p hp\n apply dvd_count_of_dvd_count_mul hb hp hab\n rw [h]\n apply dvd_count_pow _ hp\n rintro rfl\n rw [zero_pow' _ hk0] at h \n cases mul_eq_zero.mp h <;> contradiction\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b c : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nh : a * b = c ^ k\n\u22a2 \u2203 d, a = d ^ k\n[PROOFSTEP]\nby_cases hk0 : k = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b c : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nh : a * b = c ^ k\nhk0 : k = 0\n\u22a2 \u2203 d, a = d ^ k\n[PROOFSTEP]\nuse 1\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b c : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nh : a * b = c ^ k\nhk0 : k = 0\n\u22a2 a = 1 ^ k\n[PROOFSTEP]\nrw [hk0, pow_zero] at h \u22a2\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b c : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nh : a * b = 1\nhk0 : k = 0\n\u22a2 a = 1\n[PROOFSTEP]\napply (mul_eq_one_iff.1 h).1\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b c : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nh : a * b = c ^ k\nhk0 : \u00ack = 0\n\u22a2 \u2203 d, a = d ^ k\n[PROOFSTEP]\nrefine' is_pow_of_dvd_count ha _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b c : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nh : a * b = c ^ k\nhk0 : \u00ack = 0\n\u22a2 \u2200 (p : Associates \u03b1), Irreducible p \u2192 k \u2223 count p (factors a)\n[PROOFSTEP]\nintro p hp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b c : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nh : a * b = c ^ k\nhk0 : \u00ack = 0\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 k \u2223 count p (factors a)\n[PROOFSTEP]\napply dvd_count_of_dvd_count_mul hb hp hab\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b c : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nh : a * b = c ^ k\nhk0 : \u00ack = 0\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 k \u2223 count p (factors (a * b))\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b c : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nh : a * b = c ^ k\nhk0 : \u00ack = 0\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 k \u2223 count p (factors (c ^ k))\n[PROOFSTEP]\napply dvd_count_pow _ hp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b c : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nh : a * b = c ^ k\nhk0 : \u00ack = 0\np : Associates \u03b1\nhp : Irreducible p\n\u22a2 c \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nhk0 : \u00ack = 0\np : Associates \u03b1\nhp : Irreducible p\nh : a * b = 0 ^ k\n\u22a2 False\n[PROOFSTEP]\nrw [zero_pow' _ hk0] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nhk0 : \u00ack = 0\np : Associates \u03b1\nhp : Irreducible p\nh : a * b = 0\n\u22a2 False\n[PROOFSTEP]\ncases mul_eq_zero.mp h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nhk0 : \u00ack = 0\np : Associates \u03b1\nhp : Irreducible p\nh : a * b = 0\nh\u271d : a = 0\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\ninst\u271d : Nontrivial \u03b1\na b : Associates \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nhab : \u2200 (d : Associates \u03b1), d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\nk : \u2115\nhk0 : \u00ack = 0\np : Associates \u03b1\nhp : Irreducible p\nh : a * b = 0\nh\u271d : b = 0\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\ninst\u271d : (n : \u2115) \u2192 Decidable (a \u2223 p ^ n)\nn : \u2115\nh : a \u2223 p ^ n\n\u22a2 a = p ^ Nat.find (_ : \u2203 n, (fun n => a \u2223 p ^ n) n)\n[PROOFSTEP]\nclassical rw [count_factors_eq_find_of_dvd_pow hp, \u2190 eq_pow_count_factors_of_dvd_pow hp h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\ninst\u271d : (n : \u2115) \u2192 Decidable (a \u2223 p ^ n)\nn : \u2115\nh : a \u2223 p ^ n\n\u22a2 a = p ^ Nat.find (_ : \u2203 n, (fun n => a \u2223 p ^ n) n)\n[PROOFSTEP]\nrw [count_factors_eq_find_of_dvd_pow hp, \u2190 eq_pow_count_factors_of_dvd_pow hp h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\ninst\u271d : (n : \u2115) \u2192 Decidable (a \u2223 p ^ n)\nn : \u2115\nh : a \u2223 p ^ n\n\u22a2 a \u2223 p ^ ?m.3699339\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : Associates \u03b1\nhp : Irreducible p\ninst\u271d : (n : \u2115) \u2192 Decidable (a \u2223 p ^ n)\nn : \u2115\nh : a \u2223 p ^ n\n\u22a2 \u2115\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d : CommMonoid \u03b1\na : Associates \u03b1\n\u22a2 Associates.mk (Quot.out a) = a\n[PROOFSTEP]\nrw [\u2190 quot_mk_eq_mk, Quot.out_eq]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\n\u22a2 (fun a b => Quot.out (Associates.mk a \u2293 Associates.mk b)) a b \u2223 a\n[PROOFSTEP]\nrw [\u2190 mk_dvd_mk, (Associates.mk a \u2293 Associates.mk b).quot_out, congr_fun\u2082 dvd_eq_le]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\n\u22a2 Associates.mk a \u2293 Associates.mk b \u2264 Associates.mk a\n[PROOFSTEP]\nexact inf_le_left\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\n\u22a2 (fun a b => Quot.out (Associates.mk a \u2293 Associates.mk b)) a b \u2223 b\n[PROOFSTEP]\nrw [\u2190 mk_dvd_mk, (Associates.mk a \u2293 Associates.mk b).quot_out, congr_fun\u2082 dvd_eq_le]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\n\u22a2 Associates.mk a \u2293 Associates.mk b \u2264 Associates.mk b\n[PROOFSTEP]\nexact inf_le_right\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nhac : a \u2223 c\nhab : a \u2223 b\n\u22a2 a \u2223 (fun a b => Quot.out (Associates.mk a \u2293 Associates.mk b)) c b\n[PROOFSTEP]\nrw [\u2190 mk_dvd_mk, (Associates.mk c \u2293 Associates.mk b).quot_out, congr_fun\u2082 dvd_eq_le, le_inf_iff, mk_le_mk_iff_dvd_iff,\n mk_le_mk_iff_dvd_iff]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\nhac : a \u2223 c\nhab : a \u2223 b\n\u22a2 a \u2223 c \u2227 a \u2223 b\n[PROOFSTEP]\nexact \u27e8hac, hab\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\n\u22a2 (fun a b => Quot.out (Associates.mk a \u2293 Associates.mk b)) a b *\n (fun a b => Quot.out (Associates.mk a \u2294 Associates.mk b)) a b ~\u1d64\n a * b\n[PROOFSTEP]\nrw [\u2190 mk_eq_mk_iff_associated, \u2190 Associates.mk_mul_mk, \u2190 associated_iff_eq, Associates.quot_out, Associates.quot_out,\n mul_comm, sup_mul_inf, Associates.mk_mul_mk]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\n\u22a2 (fun a b => Quot.out (Associates.mk a \u2294 Associates.mk b)) 0 a = 0\n[PROOFSTEP]\nhave : Associates.mk (0 : \u03b1) = \u22a4 := rfl\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nthis : Associates.mk 0 = \u22a4\n\u22a2 (fun a b => Quot.out (Associates.mk a \u2294 Associates.mk b)) 0 a = 0\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nthis : Associates.mk 0 = \u22a4\n\u22a2 Quot.out (Associates.mk 0 \u2294 Associates.mk a) = 0\n[PROOFSTEP]\nrw [this, top_sup_eq, \u2190 this, \u2190 associated_zero_iff_eq_zero, \u2190 mk_eq_mk_iff_associated, \u2190 associated_iff_eq,\n Associates.quot_out]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\n\u22a2 (fun a b => Quot.out (Associates.mk a \u2294 Associates.mk b)) a 0 = 0\n[PROOFSTEP]\nhave : Associates.mk (0 : \u03b1) = \u22a4 := rfl\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nthis : Associates.mk 0 = \u22a4\n\u22a2 (fun a b => Quot.out (Associates.mk a \u2294 Associates.mk b)) a 0 = 0\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nthis : Associates.mk 0 = \u22a4\n\u22a2 Quot.out (Associates.mk a \u2294 Associates.mk 0) = 0\n[PROOFSTEP]\nrw [this, sup_top_eq, \u2190 this, \u2190 associated_zero_iff_eq_zero, \u2190 mk_eq_mk_iff_associated, \u2190 associated_iff_eq,\n Associates.quot_out]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b3 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b2 : NormalizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\nsrc\u271d : NormalizationMonoid \u03b1 := inst\u271d\u00b2\na b c : \u03b1\nhac : a \u2223 c\nhab : a \u2223 b\n\u22a2 a \u2223 Associates.out (Associates.mk c \u2293 Associates.mk b)\n[PROOFSTEP]\nrw [dvd_out_iff, le_inf_iff, mk_le_mk_iff_dvd_iff, mk_le_mk_iff_dvd_iff]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b3 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b2 : NormalizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\nsrc\u271d : NormalizationMonoid \u03b1 := inst\u271d\u00b2\na b c : \u03b1\nhac : a \u2223 c\nhab : a \u2223 b\n\u22a2 a \u2223 c \u2227 a \u2223 b\n[PROOFSTEP]\nexact \u27e8hac, hab\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b3 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b2 : NormalizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\nsrc\u271d : NormalizationMonoid \u03b1 := inst\u271d\u00b2\na b : \u03b1\n\u22a2 (fun a b => Associates.out (Associates.mk a \u2293 Associates.mk b)) a b *\n (fun a b => Associates.out (Associates.mk a \u2294 Associates.mk b)) a b ~\u1d64\n a * b\n[PROOFSTEP]\nrw [\u2190 out_mul, mul_comm, sup_mul_inf, mk_mul_mk, out_mk]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b3 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b2 : NormalizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\nsrc\u271d : NormalizationMonoid \u03b1 := inst\u271d\u00b2\na b : \u03b1\n\u22a2 \u2191normalize (a * b) ~\u1d64 a * b\n[PROOFSTEP]\nexact normalize_associated (a * b)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b3 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b2 : NormalizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\nsrc\u271d : NormalizationMonoid \u03b1 := inst\u271d\u00b2\na : \u03b1\n\u22a2 Associates.out (\u22a4 \u2294 Associates.mk a) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b3 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b2 : NormalizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\nsrc\u271d : NormalizationMonoid \u03b1 := inst\u271d\u00b2\na : \u03b1\n\u22a2 Associates.out (Associates.mk a \u2294 \u22a4) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b3 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b2 : NormalizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\nsrc\u271d : NormalizationMonoid \u03b1 := inst\u271d\u00b2\na b : \u03b1\n\u22a2 \u2191normalize (gcd a b) = gcd a b\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b3 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b2 : NormalizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\nsrc\u271d : NormalizationMonoid \u03b1 := inst\u271d\u00b2\na b : \u03b1\n\u22a2 \u2191normalize (gcd a b) = gcd a b\n[PROOFSTEP]\napply normalize_out _\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b3 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b2 : NormalizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\nsrc\u271d : NormalizationMonoid \u03b1 := inst\u271d\u00b2\na b : \u03b1\n\u22a2 \u2191normalize (lcm a b) = lcm a b\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b3 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b2 : NormalizationMonoid \u03b1\ninst\u271d\u00b9 : DecidableEq (Associates \u03b1)\ninst\u271d : DecidableEq \u03b1\nsrc\u271d : NormalizationMonoid \u03b1 := inst\u271d\u00b2\na b : \u03b1\n\u22a2 \u2191normalize (lcm a b) = lcm a b\n[PROOFSTEP]\napply normalize_out _\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Fintype M\u02e3\ny : M\nhy : y \u2260 0\n\u22a2 Fintype { x // x \u2223 y }\n[PROOFSTEP]\nhaveI : Nontrivial M := \u27e8\u27e8y, 0, hy\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Fintype M\u02e3\ny : M\nhy : y \u2260 0\nthis : Nontrivial M\n\u22a2 Fintype { x // x \u2223 y }\n[PROOFSTEP]\nhaveI : NormalizationMonoid M := UniqueFactorizationMonoid.normalizationMonoid\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Fintype M\u02e3\ny : M\nhy : y \u2260 0\nthis\u271d : Nontrivial M\nthis : NormalizationMonoid M\n\u22a2 Fintype { x // x \u2223 y }\n[PROOFSTEP]\nhaveI := Classical.decEq M\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Fintype M\u02e3\ny : M\nhy : y \u2260 0\nthis\u271d\u00b9 : Nontrivial M\nthis\u271d : NormalizationMonoid M\nthis : DecidableEq M\n\u22a2 Fintype { x // x \u2223 y }\n[PROOFSTEP]\nhaveI :=\n Classical.decEq\n (Associates M)\n -- We'll show `\u03bb (u : M\u02e3) (f \u2286 factors y) \u2192 u * \u03a0 f` is injective\n -- and has image exactly the divisors of `y`.\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Fintype M\u02e3\ny : M\nhy : y \u2260 0\nthis\u271d\u00b2 : Nontrivial M\nthis\u271d\u00b9 : NormalizationMonoid M\nthis\u271d : DecidableEq M\nthis : DecidableEq (Associates M)\n\u22a2 Fintype { x // x \u2223 y }\n[PROOFSTEP]\nrefine'\n Fintype.ofFinset\n (((normalizedFactors y).powerset.toFinset \u00d7\u02e2 (Finset.univ : Finset M\u02e3)).image fun s => (s.snd : M) * s.fst.prod)\n fun x => _\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Fintype M\u02e3\ny : M\nhy : y \u2260 0\nthis\u271d\u00b2 : Nontrivial M\nthis\u271d\u00b9 : NormalizationMonoid M\nthis\u271d : DecidableEq M\nthis : DecidableEq (Associates M)\nx : M\n\u22a2 x \u2208\n Finset.image (fun s => \u2191s.snd * Multiset.prod s.fst)\n (Multiset.toFinset (Multiset.powerset (normalizedFactors y)) \u00d7\u02e2 Finset.univ) \u2194\n x \u2208 fun x => \u2203 c, y = x * c\n[PROOFSTEP]\nsimp only [exists_prop, Finset.mem_image, Finset.mem_product, Finset.mem_univ, and_true_iff, Multiset.mem_toFinset,\n Multiset.mem_powerset, exists_eq_right, Multiset.mem_map]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Fintype M\u02e3\ny : M\nhy : y \u2260 0\nthis\u271d\u00b2 : Nontrivial M\nthis\u271d\u00b9 : NormalizationMonoid M\nthis\u271d : DecidableEq M\nthis : DecidableEq (Associates M)\nx : M\n\u22a2 (\u2203 a, a.fst \u2264 normalizedFactors y \u2227 \u2191a.snd * Multiset.prod a.fst = x) \u2194 x \u2208 fun x => \u2203 c, y = x * c\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Fintype M\u02e3\ny : M\nhy : y \u2260 0\nthis\u271d\u00b2 : Nontrivial M\nthis\u271d\u00b9 : NormalizationMonoid M\nthis\u271d : DecidableEq M\nthis : DecidableEq (Associates M)\nx : M\n\u22a2 (\u2203 a, a.fst \u2264 normalizedFactors y \u2227 \u2191a.snd * Multiset.prod a.fst = x) \u2192 x \u2208 fun x => \u2203 c, y = x * c\n[PROOFSTEP]\nrintro \u27e8s, hs, rfl\u27e9\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Fintype M\u02e3\ny : M\nhy : y \u2260 0\nthis\u271d\u00b2 : Nontrivial M\nthis\u271d\u00b9 : NormalizationMonoid M\nthis\u271d : DecidableEq M\nthis : DecidableEq (Associates M)\ns : Multiset M \u00d7 M\u02e3\nhs : s.fst \u2264 normalizedFactors y\n\u22a2 \u2191s.snd * Multiset.prod s.fst \u2208 fun x => \u2203 c, y = x * c\n[PROOFSTEP]\nshow (s.snd : M) * s.fst.prod \u2223 y\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Fintype M\u02e3\ny : M\nhy : y \u2260 0\nthis\u271d\u00b2 : Nontrivial M\nthis\u271d\u00b9 : NormalizationMonoid M\nthis\u271d : DecidableEq M\nthis : DecidableEq (Associates M)\ns : Multiset M \u00d7 M\u02e3\nhs : s.fst \u2264 normalizedFactors y\n\u22a2 \u2191s.snd * Multiset.prod s.fst \u2223 y\n[PROOFSTEP]\nrw [(unit_associated_one.mul_right s.fst.prod).dvd_iff_dvd_left, one_mul, \u2190\n (normalizedFactors_prod hy).dvd_iff_dvd_right]\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Fintype M\u02e3\ny : M\nhy : y \u2260 0\nthis\u271d\u00b2 : Nontrivial M\nthis\u271d\u00b9 : NormalizationMonoid M\nthis\u271d : DecidableEq M\nthis : DecidableEq (Associates M)\ns : Multiset M \u00d7 M\u02e3\nhs : s.fst \u2264 normalizedFactors y\n\u22a2 Multiset.prod s.fst \u2223 Multiset.prod (normalizedFactors y)\n[PROOFSTEP]\nexact Multiset.prod_dvd_prod_of_le hs\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Fintype M\u02e3\ny : M\nhy : y \u2260 0\nthis\u271d\u00b2 : Nontrivial M\nthis\u271d\u00b9 : NormalizationMonoid M\nthis\u271d : DecidableEq M\nthis : DecidableEq (Associates M)\nx : M\n\u22a2 (x \u2208 fun x => \u2203 c, y = x * c) \u2192 \u2203 a, a.fst \u2264 normalizedFactors y \u2227 \u2191a.snd * Multiset.prod a.fst = x\n[PROOFSTEP]\nrintro (h : x \u2223 y)\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Fintype M\u02e3\ny : M\nhy : y \u2260 0\nthis\u271d\u00b2 : Nontrivial M\nthis\u271d\u00b9 : NormalizationMonoid M\nthis\u271d : DecidableEq M\nthis : DecidableEq (Associates M)\nx : M\nh : x \u2223 y\n\u22a2 \u2203 a, a.fst \u2264 normalizedFactors y \u2227 \u2191a.snd * Multiset.prod a.fst = x\n[PROOFSTEP]\nhave hx : x \u2260 0 := by\n refine' mt (fun hx => _) hy\n rwa [hx, zero_dvd_iff] at h \n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Fintype M\u02e3\ny : M\nhy : y \u2260 0\nthis\u271d\u00b2 : Nontrivial M\nthis\u271d\u00b9 : NormalizationMonoid M\nthis\u271d : DecidableEq M\nthis : DecidableEq (Associates M)\nx : M\nh : x \u2223 y\n\u22a2 x \u2260 0\n[PROOFSTEP]\nrefine' mt (fun hx => _) hy\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Fintype M\u02e3\ny : M\nhy : y \u2260 0\nthis\u271d\u00b2 : Nontrivial M\nthis\u271d\u00b9 : NormalizationMonoid M\nthis\u271d : DecidableEq M\nthis : DecidableEq (Associates M)\nx : M\nh : x \u2223 y\nhx : x = 0\n\u22a2 y = 0\n[PROOFSTEP]\nrwa [hx, zero_dvd_iff] at h \n[GOAL]\ncase mpr\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Fintype M\u02e3\ny : M\nhy : y \u2260 0\nthis\u271d\u00b2 : Nontrivial M\nthis\u271d\u00b9 : NormalizationMonoid M\nthis\u271d : DecidableEq M\nthis : DecidableEq (Associates M)\nx : M\nh : x \u2223 y\nhx : x \u2260 0\n\u22a2 \u2203 a, a.fst \u2264 normalizedFactors y \u2227 \u2191a.snd * Multiset.prod a.fst = x\n[PROOFSTEP]\nobtain \u27e8u, hu\u27e9 := normalizedFactors_prod hx\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Fintype M\u02e3\ny : M\nhy : y \u2260 0\nthis\u271d\u00b2 : Nontrivial M\nthis\u271d\u00b9 : NormalizationMonoid M\nthis\u271d : DecidableEq M\nthis : DecidableEq (Associates M)\nx : M\nh : x \u2223 y\nhx : x \u2260 0\nu : M\u02e3\nhu : Multiset.prod (normalizedFactors x) * \u2191u = x\n\u22a2 \u2203 a, a.fst \u2264 normalizedFactors y \u2227 \u2191a.snd * Multiset.prod a.fst = x\n[PROOFSTEP]\nrefine' \u27e8\u27e8normalizedFactors x, u\u27e9, _, (mul_comm _ _).trans hu\u27e9\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CancelCommMonoidWithZero M\ninst\u271d\u00b9 : UniqueFactorizationMonoid M\ninst\u271d : Fintype M\u02e3\ny : M\nhy : y \u2260 0\nthis\u271d\u00b2 : Nontrivial M\nthis\u271d\u00b9 : NormalizationMonoid M\nthis\u271d : DecidableEq M\nthis : DecidableEq (Associates M)\nx : M\nh : x \u2223 y\nhx : x \u2260 0\nu : M\u02e3\nhu : Multiset.prod (normalizedFactors x) * \u2191u = x\n\u22a2 (normalizedFactors x, u).fst \u2264 normalizedFactors y\n[PROOFSTEP]\nexact (dvd_iff_normalizedFactors_le_normalizedFactors hx hy).mp h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nn p : \u03b1\n\u22a2 \u2191(factorization n) p = Multiset.count p (normalizedFactors n)\n[PROOFSTEP]\nsimp [factorization]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 factorization 0 = 0\n[PROOFSTEP]\nsimp [factorization]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 factorization 1 = 0\n[PROOFSTEP]\nsimp [factorization]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nn : \u03b1\n\u22a2 (factorization n).support = Multiset.toFinset (normalizedFactors n)\n[PROOFSTEP]\nsimp [factorization, Multiset.toFinsupp_support]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 factorization (a * b) = factorization a + factorization b\n[PROOFSTEP]\nsimp [factorization, normalizedFactors_mul ha hb]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nn : \u2115\n\u22a2 factorization (x ^ n) = n \u2022 factorization x\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nn : \u2115\na\u271d : \u03b1\n\u22a2 \u2191(factorization (x ^ n)) a\u271d = \u2191(n \u2022 factorization x) a\u271d\n[PROOFSTEP]\nsimp [factorization]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nh : factorization a = factorization b\n\u22a2 a ~\u1d64 b\n[PROOFSTEP]\nsimp_rw [factorization, AddEquiv.apply_eq_iff_eq] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : UniqueFactorizationMonoid \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\nh : normalizedFactors a = normalizedFactors b\n\u22a2 a ~\u1d64 b\n[PROOFSTEP]\nrwa [associated_iff_normalizedFactors_eq_normalizedFactors ha hb]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.UniqueFactorizationDomain", "llama_tokens": 185858, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.5, "lm_q2_score": 0.02716923320955869, "lm_q1q2_score": 0.013584616604779345}} {"text": "[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\n\u22a2 \u2191(closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S))) * R = \u22a4\n[PROOFSTEP]\nlet f : G \u2192 R := fun g => toFun hR g\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nf : G \u2192 \u2191R := fun g => toFun hR g\n\u22a2 \u2191(closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S))) * R = \u22a4\n[PROOFSTEP]\nlet U : Set G := (R * S).image fun g => g * (f g : G)\u207b\u00b9\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nf : G \u2192 \u2191R := fun g => toFun hR g\nU : Set G := (fun g => g * (\u2191(f g))\u207b\u00b9) '' (R * S)\n\u22a2 \u2191(closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S))) * R = \u22a4\n[PROOFSTEP]\nchange (closure U : Set G) * R = \u22a4\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nf : G \u2192 \u2191R := fun g => toFun hR g\nU : Set G := (fun g => g * (\u2191(f g))\u207b\u00b9) '' (R * S)\n\u22a2 \u2191(closure U) * R = \u22a4\n[PROOFSTEP]\nrefine' top_le_iff.mp fun g _ => _\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nf : G \u2192 \u2191R := fun g => toFun hR g\nU : Set G := (fun g => g * (\u2191(f g))\u207b\u00b9) '' (R * S)\ng : G\nx\u271d : g \u2208 \u22a4\n\u22a2 g \u2208 \u2191(closure U) * R\n[PROOFSTEP]\napply closure_induction_right (eq_top_iff.mp hS (mem_top g))\n[GOAL]\ncase H1\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nf : G \u2192 \u2191R := fun g => toFun hR g\nU : Set G := (fun g => g * (\u2191(f g))\u207b\u00b9) '' (R * S)\ng : G\nx\u271d : g \u2208 \u22a4\n\u22a2 1 \u2208 \u2191(closure U) * R\n[PROOFSTEP]\nexact \u27e81, 1, (closure U).one_mem, hR1, one_mul 1\u27e9\n[GOAL]\ncase Hmul\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nf : G \u2192 \u2191R := fun g => toFun hR g\nU : Set G := (fun g => g * (\u2191(f g))\u207b\u00b9) '' (R * S)\ng : G\nx\u271d : g \u2208 \u22a4\n\u22a2 \u2200 (x y : G), y \u2208 S \u2192 x \u2208 \u2191(closure U) * R \u2192 x * y \u2208 \u2191(closure U) * R\n[PROOFSTEP]\nrintro - s hs \u27e8u, r, hu, hr, rfl\u27e9\n[GOAL]\ncase Hmul.intro.intro.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nf : G \u2192 \u2191R := fun g => toFun hR g\nU : Set G := (fun g => g * (\u2191(f g))\u207b\u00b9) '' (R * S)\ng : G\nx\u271d : g \u2208 \u22a4\ns : G\nhs : s \u2208 S\nu r : G\nhu : u \u2208 \u2191(closure U)\nhr : r \u2208 R\n\u22a2 (fun x x_1 => x * x_1) u r * s \u2208 \u2191(closure U) * R\n[PROOFSTEP]\nrw [show u * r * s = u * (r * s * (f (r * s) : G)\u207b\u00b9) * f (r * s) by group]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nf : G \u2192 \u2191R := fun g => toFun hR g\nU : Set G := (fun g => g * (\u2191(f g))\u207b\u00b9) '' (R * S)\ng : G\nx\u271d : g \u2208 \u22a4\ns : G\nhs : s \u2208 S\nu r : G\nhu : u \u2208 \u2191(closure U)\nhr : r \u2208 R\n\u22a2 u * r * s = u * (r * s * (\u2191(f (r * s)))\u207b\u00b9) * \u2191(f (r * s))\n[PROOFSTEP]\ngroup\n[GOAL]\ncase Hmul.intro.intro.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nf : G \u2192 \u2191R := fun g => toFun hR g\nU : Set G := (fun g => g * (\u2191(f g))\u207b\u00b9) '' (R * S)\ng : G\nx\u271d : g \u2208 \u22a4\ns : G\nhs : s \u2208 S\nu r : G\nhu : u \u2208 \u2191(closure U)\nhr : r \u2208 R\n\u22a2 u * (r * s * (\u2191(f (r * s)))\u207b\u00b9) * \u2191(f (r * s)) \u2208 \u2191(closure U) * R\n[PROOFSTEP]\nrefine' Set.mul_mem_mul ((closure U).mul_mem hu _) (f (r * s)).coe_prop\n[GOAL]\ncase Hmul.intro.intro.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nf : G \u2192 \u2191R := fun g => toFun hR g\nU : Set G := (fun g => g * (\u2191(f g))\u207b\u00b9) '' (R * S)\ng : G\nx\u271d : g \u2208 \u22a4\ns : G\nhs : s \u2208 S\nu r : G\nhu : u \u2208 \u2191(closure U)\nhr : r \u2208 R\n\u22a2 r * s * (\u2191(f (r * s)))\u207b\u00b9 \u2208 closure U\n[PROOFSTEP]\nexact subset_closure \u27e8r * s, Set.mul_mem_mul hr hs, rfl\u27e9\n[GOAL]\ncase Hinv\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nf : G \u2192 \u2191R := fun g => toFun hR g\nU : Set G := (fun g => g * (\u2191(f g))\u207b\u00b9) '' (R * S)\ng : G\nx\u271d : g \u2208 \u22a4\n\u22a2 \u2200 (x y : G), y \u2208 S \u2192 x \u2208 \u2191(closure U) * R \u2192 x * y\u207b\u00b9 \u2208 \u2191(closure U) * R\n[PROOFSTEP]\nrintro - s hs \u27e8u, r, hu, hr, rfl\u27e9\n[GOAL]\ncase Hinv.intro.intro.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nf : G \u2192 \u2191R := fun g => toFun hR g\nU : Set G := (fun g => g * (\u2191(f g))\u207b\u00b9) '' (R * S)\ng : G\nx\u271d : g \u2208 \u22a4\ns : G\nhs : s \u2208 S\nu r : G\nhu : u \u2208 \u2191(closure U)\nhr : r \u2208 R\n\u22a2 (fun x x_1 => x * x_1) u r * s\u207b\u00b9 \u2208 \u2191(closure U) * R\n[PROOFSTEP]\nrw [show u * r * s\u207b\u00b9 = u * (f (r * s\u207b\u00b9) * s * r\u207b\u00b9)\u207b\u00b9 * f (r * s\u207b\u00b9) by group]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nf : G \u2192 \u2191R := fun g => toFun hR g\nU : Set G := (fun g => g * (\u2191(f g))\u207b\u00b9) '' (R * S)\ng : G\nx\u271d : g \u2208 \u22a4\ns : G\nhs : s \u2208 S\nu r : G\nhu : u \u2208 \u2191(closure U)\nhr : r \u2208 R\n\u22a2 u * r * s\u207b\u00b9 = u * (\u2191(f (r * s\u207b\u00b9)) * s * r\u207b\u00b9)\u207b\u00b9 * \u2191(f (r * s\u207b\u00b9))\n[PROOFSTEP]\ngroup\n[GOAL]\ncase Hinv.intro.intro.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nf : G \u2192 \u2191R := fun g => toFun hR g\nU : Set G := (fun g => g * (\u2191(f g))\u207b\u00b9) '' (R * S)\ng : G\nx\u271d : g \u2208 \u22a4\ns : G\nhs : s \u2208 S\nu r : G\nhu : u \u2208 \u2191(closure U)\nhr : r \u2208 R\n\u22a2 u * (\u2191(f (r * s\u207b\u00b9)) * s * r\u207b\u00b9)\u207b\u00b9 * \u2191(f (r * s\u207b\u00b9)) \u2208 \u2191(closure U) * R\n[PROOFSTEP]\nrefine' Set.mul_mem_mul ((closure U).mul_mem hu ((closure U).inv_mem _)) (f (r * s\u207b\u00b9)).2\n[GOAL]\ncase Hinv.intro.intro.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nf : G \u2192 \u2191R := fun g => toFun hR g\nU : Set G := (fun g => g * (\u2191(f g))\u207b\u00b9) '' (R * S)\ng : G\nx\u271d : g \u2208 \u22a4\ns : G\nhs : s \u2208 S\nu r : G\nhu : u \u2208 \u2191(closure U)\nhr : r \u2208 R\n\u22a2 \u2191(f (r * s\u207b\u00b9)) * s * r\u207b\u00b9 \u2208 closure U\n[PROOFSTEP]\nrefine' subset_closure \u27e8f (r * s\u207b\u00b9) * s, Set.mul_mem_mul (f (r * s\u207b\u00b9)).2 hs, _\u27e9\n[GOAL]\ncase Hinv.intro.intro.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nf : G \u2192 \u2191R := fun g => toFun hR g\nU : Set G := (fun g => g * (\u2191(f g))\u207b\u00b9) '' (R * S)\ng : G\nx\u271d : g \u2208 \u22a4\ns : G\nhs : s \u2208 S\nu r : G\nhu : u \u2208 \u2191(closure U)\nhr : r \u2208 R\n\u22a2 (fun g => g * (\u2191(f g))\u207b\u00b9) (\u2191(f (r * s\u207b\u00b9)) * s) = \u2191(f (r * s\u207b\u00b9)) * s * r\u207b\u00b9\n[PROOFSTEP]\nrw [mul_right_inj, inv_inj, \u2190 Subtype.coe_mk r hr, \u2190 Subtype.ext_iff, Subtype.coe_mk]\n[GOAL]\ncase Hinv.intro.intro.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nf : G \u2192 \u2191R := fun g => toFun hR g\nU : Set G := (fun g => g * (\u2191(f g))\u207b\u00b9) '' (R * S)\ng : G\nx\u271d : g \u2208 \u22a4\ns : G\nhs : s \u2208 S\nu r : G\nhu : u \u2208 \u2191(closure U)\nhr : r \u2208 R\n\u22a2 f (\u2191(toFun hR (\u2191{ val := r, property := hr } * s\u207b\u00b9)) * s) = { val := r, property := hr }\n[PROOFSTEP]\napply\n (mem_rightTransversals_iff_existsUnique_mul_inv_mem.mp hR (f (r * s\u207b\u00b9) * s)).unique\n (mul_inv_toFun_mem hR (f (r * s\u207b\u00b9) * s))\n[GOAL]\ncase Hinv.intro.intro.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nf : G \u2192 \u2191R := fun g => toFun hR g\nU : Set G := (fun g => g * (\u2191(f g))\u207b\u00b9) '' (R * S)\ng : G\nx\u271d : g \u2208 \u22a4\ns : G\nhs : s \u2208 S\nu r : G\nhu : u \u2208 \u2191(closure U)\nhr : r \u2208 R\n\u22a2 \u2191(f (r * s\u207b\u00b9)) * s * (\u2191{ val := r, property := hr })\u207b\u00b9 \u2208 \u2191H\n[PROOFSTEP]\nrw [mul_assoc, \u2190 inv_inv s, \u2190 mul_inv_rev, inv_inv]\n[GOAL]\ncase Hinv.intro.intro.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nf : G \u2192 \u2191R := fun g => toFun hR g\nU : Set G := (fun g => g * (\u2191(f g))\u207b\u00b9) '' (R * S)\ng : G\nx\u271d : g \u2208 \u22a4\ns : G\nhs : s \u2208 S\nu r : G\nhu : u \u2208 \u2191(closure U)\nhr : r \u2208 R\n\u22a2 \u2191(f (r * s\u207b\u00b9)) * (\u2191{ val := r, property := hr } * s\u207b\u00b9)\u207b\u00b9 \u2208 \u2191H\n[PROOFSTEP]\nexact toFun_mul_inv_mem hR (r * s\u207b\u00b9)\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\n\u22a2 closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S)) = H\n[PROOFSTEP]\nhave hU : closure ((R * S).image fun g => g * (toFun hR g : G)\u207b\u00b9) \u2264 H :=\n by\n rw [closure_le]\n rintro - \u27e8g, -, rfl\u27e9\n exact mul_inv_toFun_mem hR g\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\n\u22a2 closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S)) \u2264 H\n[PROOFSTEP]\nrw [closure_le]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\n\u22a2 (fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S) \u2286 \u2191H\n[PROOFSTEP]\nrintro - \u27e8g, -, rfl\u27e9\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\ng : G\n\u22a2 (fun g => g * (\u2191(toFun hR g))\u207b\u00b9) g \u2208 \u2191H\n[PROOFSTEP]\nexact mul_inv_toFun_mem hR g\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nhU : closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S)) \u2264 H\n\u22a2 closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S)) = H\n[PROOFSTEP]\nrefine' le_antisymm hU fun h hh => _\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nhU : closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S)) \u2264 H\nh : G\nhh : h \u2208 H\n\u22a2 h \u2208 closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S))\n[PROOFSTEP]\nobtain \u27e8g, r, hg, hr, rfl\u27e9 := show h \u2208 _ from eq_top_iff.mp (closure_mul_image_mul_eq_top hR hR1 hS) (mem_top h)\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nhU : closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S)) \u2264 H\ng r : G\nhg : g \u2208 \u2191(closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S)))\nhr : r \u2208 R\nhh : (fun x x_1 => x * x_1) g r \u2208 H\n\u22a2 (fun x x_1 => x * x_1) g r \u2208 closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S))\n[PROOFSTEP]\nsuffices (\u27e8r, hr\u27e9 : R) = (\u27e81, hR1\u27e9 : R) by simpa only [show r = 1 from Subtype.ext_iff.mp this, mul_one]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nhU : closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S)) \u2264 H\ng r : G\nhg : g \u2208 \u2191(closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S)))\nhr : r \u2208 R\nhh : (fun x x_1 => x * x_1) g r \u2208 H\nthis : { val := r, property := hr } = { val := 1, property := hR1 }\n\u22a2 (fun x x_1 => x * x_1) g r \u2208 closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S))\n[PROOFSTEP]\nsimpa only [show r = 1 from Subtype.ext_iff.mp this, mul_one]\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nhU : closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S)) \u2264 H\ng r : G\nhg : g \u2208 \u2191(closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S)))\nhr : r \u2208 R\nhh : (fun x x_1 => x * x_1) g r \u2208 H\n\u22a2 { val := r, property := hr } = { val := 1, property := hR1 }\n[PROOFSTEP]\napply (mem_rightTransversals_iff_existsUnique_mul_inv_mem.mp hR r).unique\n[GOAL]\ncase intro.intro.intro.intro.py\u2081\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nhU : closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S)) \u2264 H\ng r : G\nhg : g \u2208 \u2191(closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S)))\nhr : r \u2208 R\nhh : (fun x x_1 => x * x_1) g r \u2208 H\n\u22a2 r * (\u2191{ val := r, property := hr })\u207b\u00b9 \u2208 \u2191H\n[PROOFSTEP]\nrw [Subtype.coe_mk, mul_inv_self]\n[GOAL]\ncase intro.intro.intro.intro.py\u2081\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nhU : closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S)) \u2264 H\ng r : G\nhg : g \u2208 \u2191(closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S)))\nhr : r \u2208 R\nhh : (fun x x_1 => x * x_1) g r \u2208 H\n\u22a2 1 \u2208 \u2191H\n[PROOFSTEP]\nexact H.one_mem\n[GOAL]\ncase intro.intro.intro.intro.py\u2082\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nhU : closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S)) \u2264 H\ng r : G\nhg : g \u2208 \u2191(closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S)))\nhr : r \u2208 R\nhh : (fun x x_1 => x * x_1) g r \u2208 H\n\u22a2 r * (\u2191{ val := 1, property := hR1 })\u207b\u00b9 \u2208 \u2191H\n[PROOFSTEP]\nrw [Subtype.coe_mk, inv_one, mul_one]\n[GOAL]\ncase intro.intro.intro.intro.py\u2082\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\nhU : closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S)) \u2264 H\ng r : G\nhg : g \u2208 \u2191(closure ((fun g => g * (\u2191(toFun hR g))\u207b\u00b9) '' (R * S)))\nhr : r \u2208 R\nhh : (fun x x_1 => x * x_1) g r \u2208 H\n\u22a2 r \u2208 \u2191H\n[PROOFSTEP]\nexact (H.mul_mem_cancel_left (hU hg)).mp hh\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\n\u22a2 closure ((fun g => { val := g * (\u2191(toFun hR g))\u207b\u00b9, property := (_ : g * (\u2191(toFun hR g))\u207b\u00b9 \u2208 H) }) '' (R * S)) = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff, \u2190 map_subtype_le_map_subtype, MonoidHom.map_closure, Set.image_image]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\n\u22a2 map (Subgroup.subtype H) \u22a4 \u2264\n closure\n ((fun x => \u2191(Subgroup.subtype H) { val := x * (\u2191(toFun hR x))\u207b\u00b9, property := (_ : x * (\u2191(toFun hR x))\u207b\u00b9 \u2208 H) }) ''\n (R * S))\n[PROOFSTEP]\nexact (map_subtype_le \u22a4).trans (ge_of_eq (closure_mul_image_eq hR hR1 hS))\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR\u271d S\u271d : Set G\ninst\u271d : DecidableEq G\nR S : Finset G\nhR : \u2191R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure \u2191S = \u22a4\n\u22a2 closure\n \u2191(Finset.image (fun g => { val := g * (\u2191(toFun hR g))\u207b\u00b9, property := (_ : g * (\u2191(toFun hR g))\u207b\u00b9 \u2208 H) }) (R * S)) =\n \u22a4\n[PROOFSTEP]\nrw [Finset.coe_image, Finset.coe_mul]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR\u271d S\u271d : Set G\ninst\u271d : DecidableEq G\nR S : Finset G\nhR : \u2191R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure \u2191S = \u22a4\n\u22a2 closure ((fun g => { val := g * (\u2191(toFun hR g))\u207b\u00b9, property := (_ : g * (\u2191(toFun hR g))\u207b\u00b9 \u2208 H) }) '' (\u2191R * \u2191S)) = \u22a4\n[PROOFSTEP]\nexact closure_mul_image_eq_top hR hR1 hS\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S\u271d : Set G\ninst\u271d : FiniteIndex H\nS : Finset G\nhS : closure \u2191S = \u22a4\n\u22a2 \u2203 T, Finset.card T \u2264 index H * Finset.card S \u2227 closure \u2191T = \u22a4\n[PROOFSTEP]\nletI := H.fintypeQuotientOfFiniteIndex\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S\u271d : Set G\ninst\u271d : FiniteIndex H\nS : Finset G\nhS : closure \u2191S = \u22a4\nthis : Fintype (G \u29f8 H) := fintypeQuotientOfFiniteIndex H\n\u22a2 \u2203 T, Finset.card T \u2264 index H * Finset.card S \u2227 closure \u2191T = \u22a4\n[PROOFSTEP]\nhaveI : DecidableEq G := Classical.decEq G\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S\u271d : Set G\ninst\u271d : FiniteIndex H\nS : Finset G\nhS : closure \u2191S = \u22a4\nthis\u271d : Fintype (G \u29f8 H) := fintypeQuotientOfFiniteIndex H\nthis : DecidableEq G\n\u22a2 \u2203 T, Finset.card T \u2264 index H * Finset.card S \u2227 closure \u2191T = \u22a4\n[PROOFSTEP]\nobtain \u27e8R\u2080, hR : R\u2080 \u2208 rightTransversals (H : Set G), hR1\u27e9 := exists_right_transversal (1 : G)\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S\u271d : Set G\ninst\u271d : FiniteIndex H\nS : Finset G\nhS : closure \u2191S = \u22a4\nthis\u271d : Fintype (G \u29f8 H) := fintypeQuotientOfFiniteIndex H\nthis : DecidableEq G\nR\u2080 : Set G\nhR : R\u2080 \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\u2080\n\u22a2 \u2203 T, Finset.card T \u2264 index H * Finset.card S \u2227 closure \u2191T = \u22a4\n[PROOFSTEP]\nhaveI : Fintype R\u2080 := Fintype.ofEquiv _ (toEquiv hR)\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S\u271d : Set G\ninst\u271d : FiniteIndex H\nS : Finset G\nhS : closure \u2191S = \u22a4\nthis\u271d\u00b9 : Fintype (G \u29f8 H) := fintypeQuotientOfFiniteIndex H\nthis\u271d : DecidableEq G\nR\u2080 : Set G\nhR : R\u2080 \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\u2080\nthis : Fintype \u2191R\u2080\n\u22a2 \u2203 T, Finset.card T \u2264 index H * Finset.card S \u2227 closure \u2191T = \u22a4\n[PROOFSTEP]\nlet R : Finset G := Set.toFinset R\u2080\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR\u271d S\u271d : Set G\ninst\u271d : FiniteIndex H\nS : Finset G\nhS : closure \u2191S = \u22a4\nthis\u271d\u00b9 : Fintype (G \u29f8 H) := fintypeQuotientOfFiniteIndex H\nthis\u271d : DecidableEq G\nR\u2080 : Set G\nhR : R\u2080 \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\u2080\nthis : Fintype \u2191R\u2080\nR : Finset G := Set.toFinset R\u2080\n\u22a2 \u2203 T, Finset.card T \u2264 index H * Finset.card S \u2227 closure \u2191T = \u22a4\n[PROOFSTEP]\nreplace hR : (R : Set G) \u2208 rightTransversals (H : Set G) := by rwa [Set.coe_toFinset]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR\u271d S\u271d : Set G\ninst\u271d : FiniteIndex H\nS : Finset G\nhS : closure \u2191S = \u22a4\nthis\u271d\u00b9 : Fintype (G \u29f8 H) := fintypeQuotientOfFiniteIndex H\nthis\u271d : DecidableEq G\nR\u2080 : Set G\nhR : R\u2080 \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\u2080\nthis : Fintype \u2191R\u2080\nR : Finset G := Set.toFinset R\u2080\n\u22a2 \u2191R \u2208 rightTransversals \u2191H\n[PROOFSTEP]\nrwa [Set.coe_toFinset]\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR\u271d S\u271d : Set G\ninst\u271d : FiniteIndex H\nS : Finset G\nhS : closure \u2191S = \u22a4\nthis\u271d\u00b9 : Fintype (G \u29f8 H) := fintypeQuotientOfFiniteIndex H\nthis\u271d : DecidableEq G\nR\u2080 : Set G\nhR1 : 1 \u2208 R\u2080\nthis : Fintype \u2191R\u2080\nR : Finset G := Set.toFinset R\u2080\nhR : \u2191R \u2208 rightTransversals \u2191H\n\u22a2 \u2203 T, Finset.card T \u2264 index H * Finset.card S \u2227 closure \u2191T = \u22a4\n[PROOFSTEP]\nreplace hR1 : (1 : G) \u2208 R := by rwa [Set.mem_toFinset]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR\u271d S\u271d : Set G\ninst\u271d : FiniteIndex H\nS : Finset G\nhS : closure \u2191S = \u22a4\nthis\u271d\u00b9 : Fintype (G \u29f8 H) := fintypeQuotientOfFiniteIndex H\nthis\u271d : DecidableEq G\nR\u2080 : Set G\nhR1 : 1 \u2208 R\u2080\nthis : Fintype \u2191R\u2080\nR : Finset G := Set.toFinset R\u2080\nhR : \u2191R \u2208 rightTransversals \u2191H\n\u22a2 1 \u2208 R\n[PROOFSTEP]\nrwa [Set.mem_toFinset]\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR\u271d S\u271d : Set G\ninst\u271d : FiniteIndex H\nS : Finset G\nhS : closure \u2191S = \u22a4\nthis\u271d\u00b9 : Fintype (G \u29f8 H) := fintypeQuotientOfFiniteIndex H\nthis\u271d : DecidableEq G\nR\u2080 : Set G\nthis : Fintype \u2191R\u2080\nR : Finset G := Set.toFinset R\u2080\nhR : \u2191R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\n\u22a2 \u2203 T, Finset.card T \u2264 index H * Finset.card S \u2227 closure \u2191T = \u22a4\n[PROOFSTEP]\nrefine' \u27e8_, _, closure_mul_image_eq_top' hR hR1 hS\u27e9\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR\u271d S\u271d : Set G\ninst\u271d : FiniteIndex H\nS : Finset G\nhS : closure \u2191S = \u22a4\nthis\u271d\u00b9 : Fintype (G \u29f8 H) := fintypeQuotientOfFiniteIndex H\nthis\u271d : DecidableEq G\nR\u2080 : Set G\nthis : Fintype \u2191R\u2080\nR : Finset G := Set.toFinset R\u2080\nhR : \u2191R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\n\u22a2 Finset.card\n (Finset.image (fun g => { val := g * (\u2191(toFun hR g))\u207b\u00b9, property := (_ : g * (\u2191(toFun hR g))\u207b\u00b9 \u2208 H) }) (R * S)) \u2264\n index H * Finset.card S\n[PROOFSTEP]\ncalc\n _ \u2264 (R * S).card := Finset.card_image_le\n _ \u2264 (R \u00d7\u02e2 S).card := Finset.card_image_le\n _ = R.card * S.card := (R.card_product S)\n _ = H.index * S.card := congr_arg (\u00b7 * S.card) ?_\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR\u271d S\u271d : Set G\ninst\u271d : FiniteIndex H\nS : Finset G\nhS : closure \u2191S = \u22a4\nthis\u271d\u00b9 : Fintype (G \u29f8 H) := fintypeQuotientOfFiniteIndex H\nthis\u271d : DecidableEq G\nR\u2080 : Set G\nthis : Fintype \u2191R\u2080\nR : Finset G := Set.toFinset R\u2080\nhR : \u2191R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\n\u22a2 Finset.card R = index H\n[PROOFSTEP]\ncalc\n R.card = Fintype.card R := (Fintype.card_coe R).symm\n _ = _ := (Fintype.card_congr (toEquiv hR)).symm\n _ = Fintype.card (G \u29f8 H) := (QuotientGroup.card_quotient_rightRel H)\n _ = H.index := H.index_eq_card.symm\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\nhG : Group.FG G\ninst\u271d : FiniteIndex H\n\u22a2 Group.FG { x // x \u2208 H }\n[PROOFSTEP]\nobtain \u27e8S, hS\u27e9 := hG.1\n[GOAL]\ncase intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S\u271d : Set G\nhG : Group.FG G\ninst\u271d : FiniteIndex H\nS : Finset G\nhS : closure \u2191S = \u22a4\n\u22a2 Group.FG { x // x \u2208 H }\n[PROOFSTEP]\nobtain \u27e8T, -, hT\u27e9 := exists_finset_card_le_mul H hS\n[GOAL]\ncase intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S\u271d : Set G\nhG : Group.FG G\ninst\u271d : FiniteIndex H\nS : Finset G\nhS : closure \u2191S = \u22a4\nT : Finset { x // x \u2208 H }\nhT : closure \u2191T = \u22a4\n\u22a2 Group.FG { x // x \u2208 H }\n[PROOFSTEP]\nexact \u27e8\u27e8T, hT\u27e9\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\nhG : Group.FG G\ninst\u271d : FiniteIndex H\n\u22a2 Group.rank { x // x \u2208 H } \u2264 index H * Group.rank G\n[PROOFSTEP]\nhaveI := H.fg_of_index_ne_zero\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\nhG : Group.FG G\ninst\u271d : FiniteIndex H\nthis : Group.FG { x // x \u2208 H }\n\u22a2 Group.rank { x // x \u2208 H } \u2264 index H * Group.rank G\n[PROOFSTEP]\nobtain \u27e8S, hS\u2080, hS\u27e9 := Group.rank_spec G\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S\u271d : Set G\nhG : Group.FG G\ninst\u271d : FiniteIndex H\nthis : Group.FG { x // x \u2208 H }\nS : Finset G\nhS\u2080 : Finset.card S = Group.rank G\nhS : closure \u2191S = \u22a4\n\u22a2 Group.rank { x // x \u2208 H } \u2264 index H * Group.rank G\n[PROOFSTEP]\nobtain \u27e8T, hT\u2080, hT\u27e9 := exists_finset_card_le_mul H hS\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S\u271d : Set G\nhG : Group.FG G\ninst\u271d : FiniteIndex H\nthis : Group.FG { x // x \u2208 H }\nS : Finset G\nhS\u2080 : Finset.card S = Group.rank G\nhS : closure \u2191S = \u22a4\nT : Finset { x // x \u2208 H }\nhT\u2080 : Finset.card T \u2264 index H * Finset.card S\nhT : closure \u2191T = \u22a4\n\u22a2 Group.rank { x // x \u2208 H } \u2264 index H * Group.rank G\n[PROOFSTEP]\ncalc\n Group.rank H \u2264 T.card := Group.rank_le H hT\n _ \u2264 H.index * S.card := hT\u2080\n _ = H.index * Group.rank G := congr_arg ((\u00b7 * \u00b7) H.index) hS\u2080\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\n\u22a2 Nat.card { x // x \u2208 _root_.commutator G } \u2223 index (center G) ^ (index (center G) * Nat.card \u2191(commutatorSet G) + 1)\n[PROOFSTEP]\nby_cases hG : (center G).index = 0\n[GOAL]\ncase pos\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nhG : index (center G) = 0\n\u22a2 Nat.card { x // x \u2208 _root_.commutator G } \u2223 index (center G) ^ (index (center G) * Nat.card \u2191(commutatorSet G) + 1)\n[PROOFSTEP]\nsimp_rw [hG, zero_mul, zero_add, pow_one, dvd_zero]\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nhG : \u00acindex (center G) = 0\n\u22a2 Nat.card { x // x \u2208 _root_.commutator G } \u2223 index (center G) ^ (index (center G) * Nat.card \u2191(commutatorSet G) + 1)\n[PROOFSTEP]\nhaveI : FiniteIndex (center G) :=\n \u27e8hG\u27e9\n -- Rewrite as `|Z(G) \u2229 G'| * [G' : Z(G) \u2229 G'] \u2223 [G : Z(G)] ^ ([G : Z(G)] * n) * [G : Z(G)]`\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nhG : \u00acindex (center G) = 0\nthis : FiniteIndex (center G)\n\u22a2 Nat.card { x // x \u2208 _root_.commutator G } \u2223 index (center G) ^ (index (center G) * Nat.card \u2191(commutatorSet G) + 1)\n[PROOFSTEP]\nrw [\u2190 ((center G).subgroupOf (_root_.commutator G)).card_mul_index, pow_succ']\n -- We have `h1 : [G' : Z(G) \u2229 G'] \u2223 [G : Z(G)]`\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nhG : \u00acindex (center G) = 0\nthis : FiniteIndex (center G)\n\u22a2 Nat.card { x // x \u2208 subgroupOf (center G) (_root_.commutator G) } *\n index (subgroupOf (center G) (_root_.commutator G)) \u2223\n index (center G) ^ (index (center G) * Nat.card \u2191(commutatorSet G)) * index (center G)\n[PROOFSTEP]\nhave h1 :=\n relindex_dvd_index_of_normal (center G)\n (_root_.commutator G)\n -- So we can reduce to proving `|Z(G) \u2229 G'| \u2223 [G : Z(G)] ^ ([G : Z(G)] * n)`\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nhG : \u00acindex (center G) = 0\nthis : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) \u2223 index (center G)\n\u22a2 Nat.card { x // x \u2208 subgroupOf (center G) (_root_.commutator G) } *\n index (subgroupOf (center G) (_root_.commutator G)) \u2223\n index (center G) ^ (index (center G) * Nat.card \u2191(commutatorSet G)) * index (center G)\n[PROOFSTEP]\nrefine' mul_dvd_mul _ h1\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nhG : \u00acindex (center G) = 0\nthis : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) \u2223 index (center G)\n\u22a2 Nat.card { x // x \u2208 subgroupOf (center G) (_root_.commutator G) } \u2223\n index (center G) ^ (index (center G) * Nat.card \u2191(commutatorSet G))\n[PROOFSTEP]\nhaveI : FiniteIndex ((center G).subgroupOf (_root_.commutator G)) :=\n \u27e8ne_zero_of_dvd_ne_zero hG h1\u27e9\n -- We have `h2 : rank (Z(G) \u2229 G') \u2264 [G' : Z(G) \u2229 G'] * rank G'` by Schreier's lemma\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nhG : \u00acindex (center G) = 0\nthis\u271d : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) \u2223 index (center G)\nthis : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\n\u22a2 Nat.card { x // x \u2208 subgroupOf (center G) (_root_.commutator G) } \u2223\n index (center G) ^ (index (center G) * Nat.card \u2191(commutatorSet G))\n[PROOFSTEP]\nhave h2 :=\n rank_le_index_mul_rank\n ((center G).subgroupOf (_root_.commutator G))\n -- We have `h3 : [G' : Z(G) \u2229 G'] * rank G' \u2264 [G : Z(G)] * n` by `h1` and `rank G' \u2264 n`\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nhG : \u00acindex (center G) = 0\nthis\u271d : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) \u2223 index (center G)\nthis : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\nh2 :\n Group.rank { x // x \u2208 subgroupOf (center G) (_root_.commutator G) } \u2264\n index (subgroupOf (center G) (_root_.commutator G)) * Group.rank { x // x \u2208 _root_.commutator G }\n\u22a2 Nat.card { x // x \u2208 subgroupOf (center G) (_root_.commutator G) } \u2223\n index (center G) ^ (index (center G) * Nat.card \u2191(commutatorSet G))\n[PROOFSTEP]\nhave h3 :=\n Nat.mul_le_mul (Nat.le_of_dvd (Nat.pos_of_ne_zero hG) h1)\n (rank_commutator_le_card G)\n -- So we can reduce to proving `|Z(G) \u2229 G'| \u2223 [G : Z(G)] ^ rank (Z(G) \u2229 G')`\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nhG : \u00acindex (center G) = 0\nthis\u271d : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) \u2223 index (center G)\nthis : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\nh2 :\n Group.rank { x // x \u2208 subgroupOf (center G) (_root_.commutator G) } \u2264\n index (subgroupOf (center G) (_root_.commutator G)) * Group.rank { x // x \u2208 _root_.commutator G }\nh3 :\n relindex (center G) (_root_.commutator G) * Group.rank { x // x \u2208 _root_.commutator G } \u2264\n index (center G) * Nat.card \u2191(commutatorSet G)\n\u22a2 Nat.card { x // x \u2208 subgroupOf (center G) (_root_.commutator G) } \u2223\n index (center G) ^ (index (center G) * Nat.card \u2191(commutatorSet G))\n[PROOFSTEP]\nrefine'\n dvd_trans _\n (pow_dvd_pow (center G).index (h2.trans h3))\n -- `Z(G) \u2229 G'` is abelian, so it enough to prove that `g ^ [G : Z(G)] = 1` for `g \u2208 Z(G) \u2229 G'`\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nhG : \u00acindex (center G) = 0\nthis\u271d : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) \u2223 index (center G)\nthis : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\nh2 :\n Group.rank { x // x \u2208 subgroupOf (center G) (_root_.commutator G) } \u2264\n index (subgroupOf (center G) (_root_.commutator G)) * Group.rank { x // x \u2208 _root_.commutator G }\nh3 :\n relindex (center G) (_root_.commutator G) * Group.rank { x // x \u2208 _root_.commutator G } \u2264\n index (center G) * Nat.card \u2191(commutatorSet G)\n\u22a2 Nat.card { x // x \u2208 subgroupOf (center G) (_root_.commutator G) } \u2223\n index (center G) ^ Group.rank { x // x \u2208 subgroupOf (center G) (_root_.commutator G) }\n[PROOFSTEP]\napply card_dvd_exponent_pow_rank'\n[GOAL]\ncase neg.hG\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nhG : \u00acindex (center G) = 0\nthis\u271d : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) \u2223 index (center G)\nthis : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\nh2 :\n Group.rank { x // x \u2208 subgroupOf (center G) (_root_.commutator G) } \u2264\n index (subgroupOf (center G) (_root_.commutator G)) * Group.rank { x // x \u2208 _root_.commutator G }\nh3 :\n relindex (center G) (_root_.commutator G) * Group.rank { x // x \u2208 _root_.commutator G } \u2264\n index (center G) * Nat.card \u2191(commutatorSet G)\n\u22a2 \u2200 (g : { x // x \u2208 subgroupOf (center G) (_root_.commutator G) }), g ^ index (center G) = 1\n[PROOFSTEP]\nintro g\n[GOAL]\ncase neg.hG\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nhG : \u00acindex (center G) = 0\nthis\u271d : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) \u2223 index (center G)\nthis : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\nh2 :\n Group.rank { x // x \u2208 subgroupOf (center G) (_root_.commutator G) } \u2264\n index (subgroupOf (center G) (_root_.commutator G)) * Group.rank { x // x \u2208 _root_.commutator G }\nh3 :\n relindex (center G) (_root_.commutator G) * Group.rank { x // x \u2208 _root_.commutator G } \u2264\n index (center G) * Nat.card \u2191(commutatorSet G)\ng : { x // x \u2208 subgroupOf (center G) (_root_.commutator G) }\n\u22a2 g ^ index (center G) = 1\n[PROOFSTEP]\nhave :=\n Abelianization.commutator_subset_ker (MonoidHom.transferCenterPow G)\n g.1.2\n -- `transfer g` is defeq to `g ^ [G : Z(G)]`, so we are done\n[GOAL]\ncase neg.hG\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nhG : \u00acindex (center G) = 0\nthis\u271d\u00b9 : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) \u2223 index (center G)\nthis\u271d : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\nh2 :\n Group.rank { x // x \u2208 subgroupOf (center G) (_root_.commutator G) } \u2264\n index (subgroupOf (center G) (_root_.commutator G)) * Group.rank { x // x \u2208 _root_.commutator G }\nh3 :\n relindex (center G) (_root_.commutator G) * Group.rank { x // x \u2208 _root_.commutator G } \u2264\n index (center G) * Nat.card \u2191(commutatorSet G)\ng : { x // x \u2208 subgroupOf (center G) (_root_.commutator G) }\nthis : \u2191\u2191g \u2208 MonoidHom.ker (MonoidHom.transferCenterPow G)\n\u22a2 g ^ index (center G) = 1\n[PROOFSTEP]\nsimpa only [MonoidHom.mem_ker, Subtype.ext_iff] using this\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\n\u22a2 Nat.card { x // x \u2208 _root_.commutator G } \u2264 cardCommutatorBound (Nat.card \u2191(commutatorSet G))\n[PROOFSTEP]\nhave h1 := index_center_le_pow (closureCommutatorRepresentatives G)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nh1 :\n index (center { x // x \u2208 closureCommutatorRepresentatives G }) \u2264\n Nat.card \u2191(commutatorSet { x // x \u2208 closureCommutatorRepresentatives G }) ^\n Group.rank { x // x \u2208 closureCommutatorRepresentatives G }\n\u22a2 Nat.card { x // x \u2208 _root_.commutator G } \u2264 cardCommutatorBound (Nat.card \u2191(commutatorSet G))\n[PROOFSTEP]\nhave h2 := card_commutator_dvd_index_center_pow (closureCommutatorRepresentatives G)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nh1 :\n index (center { x // x \u2208 closureCommutatorRepresentatives G }) \u2264\n Nat.card \u2191(commutatorSet { x // x \u2208 closureCommutatorRepresentatives G }) ^\n Group.rank { x // x \u2208 closureCommutatorRepresentatives G }\nh2 :\n Nat.card { x // x \u2208 _root_.commutator { x // x \u2208 closureCommutatorRepresentatives G } } \u2223\n index (center { x // x \u2208 closureCommutatorRepresentatives G }) ^\n (index (center { x // x \u2208 closureCommutatorRepresentatives G }) *\n Nat.card \u2191(commutatorSet { x // x \u2208 closureCommutatorRepresentatives G }) +\n 1)\n\u22a2 Nat.card { x // x \u2208 _root_.commutator G } \u2264 cardCommutatorBound (Nat.card \u2191(commutatorSet G))\n[PROOFSTEP]\nrw [card_commutatorSet_closureCommutatorRepresentatives] at h1 h2 \n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nh1 :\n index (center { x // x \u2208 closureCommutatorRepresentatives G }) \u2264\n Nat.card \u2191(commutatorSet G) ^ Group.rank { x // x \u2208 closureCommutatorRepresentatives G }\nh2 :\n Nat.card { x // x \u2208 _root_.commutator { x // x \u2208 closureCommutatorRepresentatives G } } \u2223\n index (center { x // x \u2208 closureCommutatorRepresentatives G }) ^\n (index (center { x // x \u2208 closureCommutatorRepresentatives G }) * Nat.card \u2191(commutatorSet G) + 1)\n\u22a2 Nat.card { x // x \u2208 _root_.commutator G } \u2264 cardCommutatorBound (Nat.card \u2191(commutatorSet G))\n[PROOFSTEP]\nrw [card_commutator_closureCommutatorRepresentatives] at h2 \n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nh1 :\n index (center { x // x \u2208 closureCommutatorRepresentatives G }) \u2264\n Nat.card \u2191(commutatorSet G) ^ Group.rank { x // x \u2208 closureCommutatorRepresentatives G }\nh2 :\n Nat.card { x // x \u2208 _root_.commutator G } \u2223\n index (center { x // x \u2208 closureCommutatorRepresentatives G }) ^\n (index (center { x // x \u2208 closureCommutatorRepresentatives G }) * Nat.card \u2191(commutatorSet G) + 1)\n\u22a2 Nat.card { x // x \u2208 _root_.commutator G } \u2264 cardCommutatorBound (Nat.card \u2191(commutatorSet G))\n[PROOFSTEP]\nreplace h1 := h1.trans (Nat.pow_le_pow_of_le_right Finite.card_pos (rank_closureCommutatorRepresentatives_le G))\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nh2 :\n Nat.card { x // x \u2208 _root_.commutator G } \u2223\n index (center { x // x \u2208 closureCommutatorRepresentatives G }) ^\n (index (center { x // x \u2208 closureCommutatorRepresentatives G }) * Nat.card \u2191(commutatorSet G) + 1)\nh1 :\n index (center { x // x \u2208 closureCommutatorRepresentatives G }) \u2264\n Nat.card \u2191(commutatorSet G) ^ (2 * Nat.card \u2191(commutatorSet G))\n\u22a2 Nat.card { x // x \u2208 _root_.commutator G } \u2264 cardCommutatorBound (Nat.card \u2191(commutatorSet G))\n[PROOFSTEP]\nreplace h2 := h2.trans (pow_dvd_pow _ (add_le_add_right (mul_le_mul_right' h1 _) 1))\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nh1 :\n index (center { x // x \u2208 closureCommutatorRepresentatives G }) \u2264\n Nat.card \u2191(commutatorSet G) ^ (2 * Nat.card \u2191(commutatorSet G))\nh2 :\n Nat.card { x // x \u2208 _root_.commutator G } \u2223\n index (center { x // x \u2208 closureCommutatorRepresentatives G }) ^\n (Nat.card \u2191(commutatorSet G) ^ (2 * Nat.card \u2191(commutatorSet G)) * Nat.card \u2191(commutatorSet G) + 1)\n\u22a2 Nat.card { x // x \u2208 _root_.commutator G } \u2264 cardCommutatorBound (Nat.card \u2191(commutatorSet G))\n[PROOFSTEP]\nrw [\u2190 pow_succ'] at h2 \n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nh1 :\n index (center { x // x \u2208 closureCommutatorRepresentatives G }) \u2264\n Nat.card \u2191(commutatorSet G) ^ (2 * Nat.card \u2191(commutatorSet G))\nh2 :\n Nat.card { x // x \u2208 _root_.commutator G } \u2223\n index (center { x // x \u2208 closureCommutatorRepresentatives G }) ^\n (Nat.card \u2191(commutatorSet G) ^ (2 * Nat.card \u2191(commutatorSet G) + 1) + 1)\n\u22a2 Nat.card { x // x \u2208 _root_.commutator G } \u2264 cardCommutatorBound (Nat.card \u2191(commutatorSet G))\n[PROOFSTEP]\nrefine' (Nat.le_of_dvd _ h2).trans (Nat.pow_le_pow_of_le_left h1 _)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nh1 :\n index (center { x // x \u2208 closureCommutatorRepresentatives G }) \u2264\n Nat.card \u2191(commutatorSet G) ^ (2 * Nat.card \u2191(commutatorSet G))\nh2 :\n Nat.card { x // x \u2208 _root_.commutator G } \u2223\n index (center { x // x \u2208 closureCommutatorRepresentatives G }) ^\n (Nat.card \u2191(commutatorSet G) ^ (2 * Nat.card \u2191(commutatorSet G) + 1) + 1)\n\u22a2 0 <\n index (center { x // x \u2208 closureCommutatorRepresentatives G }) ^\n (Nat.card \u2191(commutatorSet G) ^ (2 * Nat.card \u2191(commutatorSet G) + 1) + 1)\n[PROOFSTEP]\nexact pow_pos (Nat.pos_of_ne_zero FiniteIndex.finiteIndex) _\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\n\u22a2 Finite { x // x \u2208 _root_.commutator G }\n[PROOFSTEP]\nhave h2 := card_commutator_dvd_index_center_pow (closureCommutatorRepresentatives G)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nh2 :\n Nat.card { x // x \u2208 _root_.commutator { x // x \u2208 closureCommutatorRepresentatives G } } \u2223\n index (center { x // x \u2208 closureCommutatorRepresentatives G }) ^\n (index (center { x // x \u2208 closureCommutatorRepresentatives G }) *\n Nat.card \u2191(commutatorSet { x // x \u2208 closureCommutatorRepresentatives G }) +\n 1)\n\u22a2 Finite { x // x \u2208 _root_.commutator G }\n[PROOFSTEP]\nrefine' Nat.finite_of_card_ne_zero fun h => _\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nh2 :\n Nat.card { x // x \u2208 _root_.commutator { x // x \u2208 closureCommutatorRepresentatives G } } \u2223\n index (center { x // x \u2208 closureCommutatorRepresentatives G }) ^\n (index (center { x // x \u2208 closureCommutatorRepresentatives G }) *\n Nat.card \u2191(commutatorSet { x // x \u2208 closureCommutatorRepresentatives G }) +\n 1)\nh : Nat.card { x // x \u2208 _root_.commutator G } = 0\n\u22a2 False\n[PROOFSTEP]\nrw [card_commutator_closureCommutatorRepresentatives, h, zero_dvd_iff] at h2 \n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\nR S : Set G\ninst\u271d : Finite \u2191(commutatorSet G)\nh2 :\n index (center { x // x \u2208 closureCommutatorRepresentatives G }) ^\n (index (center { x // x \u2208 closureCommutatorRepresentatives G }) *\n Nat.card \u2191(commutatorSet { x // x \u2208 closureCommutatorRepresentatives G }) +\n 1) =\n 0\nh : Nat.card { x // x \u2208 _root_.commutator G } = 0\n\u22a2 False\n[PROOFSTEP]\nexact FiniteIndex.finiteIndex (pow_eq_zero h2)\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Schreier", "llama_tokens": 17761, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.4960938294709195, "lm_q2_score": 0.026759284737740304, "lm_q1q2_score": 0.013275116039448318}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\n\u22a2 Injective toFun\n[PROOFSTEP]\nrintro \u27e8s, f, hf\u27e9 \u27e8t, g, hg\u27e9 (rfl : f = g)\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\ns : Finset \u03b1\nf : \u03b1 \u2192 M\nhf : \u2200 (a : \u03b1), a \u2208 s \u2194 f a \u2260 0\nt : Finset \u03b1\nhg : \u2200 (a : \u03b1), a \u2208 t \u2194 f a \u2260 0\n\u22a2 { support := s, toFun := f, mem_support_toFun := hf } = { support := t, toFun := f, mem_support_toFun := hg }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_support\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\ns : Finset \u03b1\nf : \u03b1 \u2192 M\nhf : \u2200 (a : \u03b1), a \u2208 s \u2194 f a \u2260 0\nt : Finset \u03b1\nhg : \u2200 (a : \u03b1), a \u2208 t \u2194 f a \u2260 0\n\u22a2 s = t\n[PROOFSTEP]\next a\n[GOAL]\ncase mk.mk.e_support.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\ns : Finset \u03b1\nf : \u03b1 \u2192 M\nhf : \u2200 (a : \u03b1), a \u2208 s \u2194 f a \u2260 0\nt : Finset \u03b1\nhg : \u2200 (a : \u03b1), a \u2208 t \u2194 f a \u2260 0\na : \u03b1\n\u22a2 a \u2208 s \u2194 a \u2208 t\n[PROOFSTEP]\nexact (hf _).trans (hg _).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\n\u22a2 \u2191f = 0 \u2194 f = 0\n[PROOFSTEP]\nrw [\u2190 coe_zero, FunLike.coe_fn_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf g : \u03b1 \u2192\u2080 M\nx\u271d : f.support = g.support \u2227 \u2200 (x : \u03b1), x \u2208 f.support \u2192 \u2191f x = \u2191g x\nh\u2081 : f.support = g.support\nh\u2082 : \u2200 (x : \u03b1), x \u2208 f.support \u2192 \u2191f x = \u2191g x\na : \u03b1\n\u22a2 \u2191f a = \u2191g a\n[PROOFSTEP]\nclassical exact\n if h : a \u2208 f.support then h\u2082 a h\n else by\n have hf : f a = 0 := not_mem_support_iff.1 h\n have hg : g a = 0 := by rwa [h\u2081, not_mem_support_iff] at h \n rw [hf, hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf g : \u03b1 \u2192\u2080 M\nx\u271d : f.support = g.support \u2227 \u2200 (x : \u03b1), x \u2208 f.support \u2192 \u2191f x = \u2191g x\nh\u2081 : f.support = g.support\nh\u2082 : \u2200 (x : \u03b1), x \u2208 f.support \u2192 \u2191f x = \u2191g x\na : \u03b1\n\u22a2 \u2191f a = \u2191g a\n[PROOFSTEP]\nexact\n if h : a \u2208 f.support then h\u2082 a h\n else by\n have hf : f a = 0 := not_mem_support_iff.1 h\n have hg : g a = 0 := by rwa [h\u2081, not_mem_support_iff] at h \n rw [hf, hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf g : \u03b1 \u2192\u2080 M\nx\u271d : f.support = g.support \u2227 \u2200 (x : \u03b1), x \u2208 f.support \u2192 \u2191f x = \u2191g x\nh\u2081 : f.support = g.support\nh\u2082 : \u2200 (x : \u03b1), x \u2208 f.support \u2192 \u2191f x = \u2191g x\na : \u03b1\nh : \u00aca \u2208 f.support\n\u22a2 \u2191f a = \u2191g a\n[PROOFSTEP]\nhave hf : f a = 0 := not_mem_support_iff.1 h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf g : \u03b1 \u2192\u2080 M\nx\u271d : f.support = g.support \u2227 \u2200 (x : \u03b1), x \u2208 f.support \u2192 \u2191f x = \u2191g x\nh\u2081 : f.support = g.support\nh\u2082 : \u2200 (x : \u03b1), x \u2208 f.support \u2192 \u2191f x = \u2191g x\na : \u03b1\nh : \u00aca \u2208 f.support\nhf : \u2191f a = 0\n\u22a2 \u2191f a = \u2191g a\n[PROOFSTEP]\nhave hg : g a = 0 := by rwa [h\u2081, not_mem_support_iff] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf g : \u03b1 \u2192\u2080 M\nx\u271d : f.support = g.support \u2227 \u2200 (x : \u03b1), x \u2208 f.support \u2192 \u2191f x = \u2191g x\nh\u2081 : f.support = g.support\nh\u2082 : \u2200 (x : \u03b1), x \u2208 f.support \u2192 \u2191f x = \u2191g x\na : \u03b1\nh : \u00aca \u2208 f.support\nhf : \u2191f a = 0\n\u22a2 \u2191g a = 0\n[PROOFSTEP]\nrwa [h\u2081, not_mem_support_iff] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf g : \u03b1 \u2192\u2080 M\nx\u271d : f.support = g.support \u2227 \u2200 (x : \u03b1), x \u2208 f.support \u2192 \u2191f x = \u2191g x\nh\u2081 : f.support = g.support\nh\u2082 : \u2200 (x : \u03b1), x \u2208 f.support \u2192 \u2191f x = \u2191g x\na : \u03b1\nh : \u00aca \u2208 f.support\nhf : \u2191f a = 0\nhg : \u2191g a = 0\n\u22a2 \u2191f a = \u2191g a\n[PROOFSTEP]\nrw [hf, hg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\n\u22a2 f.support = \u2205 \u2194 f = 0\n[PROOFSTEP]\nexact_mod_cast @Function.support_eq_empty_iff _ _ _ f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\n\u22a2 Finset.Nonempty f.support \u2194 f \u2260 0\n[PROOFSTEP]\nsimp only [Finsupp.support_eq_empty, Finset.nonempty_iff_ne_empty, Ne.def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\n\u22a2 f \u2260 0 \u2194 \u2203 a, \u2191f a \u2260 0\n[PROOFSTEP]\nsimp [\u2190 Finsupp.support_eq_empty, Finset.eq_empty_iff_forall_not_mem]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\n\u22a2 card f.support = 0 \u2194 f = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\ns : Set \u03b1\nf : \u03b1 \u2192\u2080 M\n\u22a2 \u2191f.support \u2286 s \u2194 \u2200 (a : \u03b1), \u00aca \u2208 s \u2192 \u2191f a = 0\n[PROOFSTEP]\nsimp only [Set.subset_def, mem_coe, mem_support_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\ns : Set \u03b1\nf : \u03b1 \u2192\u2080 M\n\u22a2 (\u2200 (x : \u03b1), \u2191f x \u2260 0 \u2192 x \u2208 s) \u2194 \u2200 (a : \u03b1), \u00aca \u2208 s \u2192 \u2191f a = 0\n[PROOFSTEP]\nexact forall_congr' fun a => not_imp_comm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Unique \u03b1\nf g : \u03b1 \u2192\u2080 M\nh : \u2191f default = \u2191g default\na : \u03b1\n\u22a2 \u2191f a = \u2191g a\n[PROOFSTEP]\nrwa [Unique.eq_default a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a'\u271d : \u03b1\nb\u271d : M\na : \u03b1\nb : M\na' : \u03b1\n\u22a2 (a' \u2208 if b = 0 then \u2205 else {a}) \u2194 Pi.single a b a' \u2260 0\n[PROOFSTEP]\nclassical\nobtain rfl | hb := eq_or_ne b 0\n\u00b7 simp [Pi.single, update]\nrw [if_neg hb, mem_singleton]\nobtain rfl | ha := eq_or_ne a' a\n\u00b7 simp [hb, Pi.single, update]\nsimp [Pi.single_eq_of_ne' ha.symm, ha]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a'\u271d : \u03b1\nb\u271d : M\na : \u03b1\nb : M\na' : \u03b1\n\u22a2 (a' \u2208 if b = 0 then \u2205 else {a}) \u2194 Pi.single a b a' \u2260 0\n[PROOFSTEP]\nobtain rfl | hb := eq_or_ne b 0\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a'\u271d : \u03b1\nb : M\na a' : \u03b1\n\u22a2 (a' \u2208 if 0 = 0 then \u2205 else {a}) \u2194 Pi.single a 0 a' \u2260 0\n[PROOFSTEP]\nsimp [Pi.single, update]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a'\u271d : \u03b1\nb\u271d : M\na : \u03b1\nb : M\na' : \u03b1\nhb : b \u2260 0\n\u22a2 (a' \u2208 if b = 0 then \u2205 else {a}) \u2194 Pi.single a b a' \u2260 0\n[PROOFSTEP]\nrw [if_neg hb, mem_singleton]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a'\u271d : \u03b1\nb\u271d : M\na : \u03b1\nb : M\na' : \u03b1\nhb : b \u2260 0\n\u22a2 a' = a \u2194 Pi.single a b a' \u2260 0\n[PROOFSTEP]\nobtain rfl | ha := eq_or_ne a' a\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a'\u271d : \u03b1\nb\u271d b : M\na' : \u03b1\nhb : b \u2260 0\n\u22a2 a' = a' \u2194 Pi.single a' b a' \u2260 0\n[PROOFSTEP]\nsimp [hb, Pi.single, update]\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a'\u271d : \u03b1\nb\u271d : M\na : \u03b1\nb : M\na' : \u03b1\nhb : b \u2260 0\nha : a' \u2260 a\n\u22a2 a' = a \u2194 Pi.single a b a' \u2260 0\n[PROOFSTEP]\nsimp [Pi.single_eq_of_ne' ha.symm, ha]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\na a' : \u03b1\nb : M\ninst\u271d : Decidable (a = a')\n\u22a2 \u2191(single a b) a' = if a = a' then b else 0\n[PROOFSTEP]\nclassical\nsimp_rw [@eq_comm _ a a']\nconvert Pi.single_apply a b a'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\na a' : \u03b1\nb : M\ninst\u271d : Decidable (a = a')\n\u22a2 \u2191(single a b) a' = if a = a' then b else 0\n[PROOFSTEP]\nsimp_rw [@eq_comm _ a a']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\na a' : \u03b1\nb : M\ninst\u271d : Decidable (a = a')\n\u22a2 \u2191(single a b) a' = if a' = a then b else 0\n[PROOFSTEP]\nconvert Pi.single_apply a b a'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nx z : \u03b1\ny : M\n\u22a2 \u2191(single (f x) y) (f z) = \u2191(single x y) z\n[PROOFSTEP]\nclassical simp only [single_apply, hf.eq_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nx z : \u03b1\ny : M\n\u22a2 \u2191(single (f x) y) (f z) = \u2191(single x y) z\n[PROOFSTEP]\nsimp only [single_apply, hf.eq_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\n\u22a2 \u2191(single a b) = Set.indicator {a} fun x => b\n[PROOFSTEP]\nclassical\next\nsimp [single_apply, Set.indicator, @eq_comm _ a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\n\u22a2 \u2191(single a b) = Set.indicator {a} fun x => b\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\nx\u271d : \u03b1\n\u22a2 \u2191(single a b) x\u271d = Set.indicator {a} (fun x => b) x\u271d\n[PROOFSTEP]\nsimp [single_apply, Set.indicator, @eq_comm _ a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\n\u22a2 \u2191(single a b) a = b\n[PROOFSTEP]\nclassical exact Pi.single_eq_same (f := \u03bb _ => M) a b\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\n\u22a2 \u2191(single a b) a = b\n[PROOFSTEP]\nexact Pi.single_eq_same (f := \u03bb _ => M) a b\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\nh : a \u2260 a'\n\u22a2 \u2191(single a b) a' = 0\n[PROOFSTEP]\nclassical exact Pi.single_eq_of_ne' h _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\nh : a \u2260 a'\n\u22a2 \u2191(single a b) a' = 0\n[PROOFSTEP]\nexact Pi.single_eq_of_ne' h _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\na\u271d a' : \u03b1\nb\u271d : M\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : M\n\u22a2 \u2191(single a b) = update 0 a b\n[PROOFSTEP]\nclassical rw [single_eq_set_indicator, \u2190 Set.piecewise_eq_indicator, Set.piecewise_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\na\u271d a' : \u03b1\nb\u271d : M\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : M\n\u22a2 \u2191(single a b) = update 0 a b\n[PROOFSTEP]\nrw [single_eq_set_indicator, \u2190 Set.piecewise_eq_indicator, Set.piecewise_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a' : \u03b1\nb : M\na : \u03b1\n\u22a2 (fun f => \u2191f) (single a 0) = (fun f => \u2191f) 0\n[PROOFSTEP]\nclassical simpa only [single_eq_update, coe_zero] using Function.update_eq_self a (0 : \u03b1 \u2192 M)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a' : \u03b1\nb : M\na : \u03b1\n\u22a2 (fun f => \u2191f) (single a 0) = (fun f => \u2191f) 0\n[PROOFSTEP]\nsimpa only [single_eq_update, coe_zero] using Function.update_eq_self a (0 : \u03b1 \u2192 M)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a'\u271d : \u03b1\nb\u271d : M\na a' : \u03b1\nb : M\n\u22a2 single a (\u2191(single a' b) a) = \u2191(single a' (single a' b)) a\n[PROOFSTEP]\nclassical\nrw [single_apply, single_apply]\next\nsplit_ifs with h\n\u00b7 rw [h]\n\u00b7 rw [zero_apply, single_apply, ite_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a'\u271d : \u03b1\nb\u271d : M\na a' : \u03b1\nb : M\n\u22a2 single a (\u2191(single a' b) a) = \u2191(single a' (single a' b)) a\n[PROOFSTEP]\nrw [single_apply, single_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a'\u271d : \u03b1\nb\u271d : M\na a' : \u03b1\nb : M\n\u22a2 single a (if a' = a then b else 0) = if a' = a then single a' b else 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d\u00b9 a'\u271d : \u03b1\nb\u271d : M\na a' : \u03b1\nb : M\na\u271d : \u03b1\n\u22a2 \u2191(single a (if a' = a then b else 0)) a\u271d = \u2191(if a' = a then single a' b else 0) a\u271d\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d\u00b9 a'\u271d : \u03b1\nb\u271d : M\na a' : \u03b1\nb : M\na\u271d : \u03b1\nh : a' = a\n\u22a2 \u2191(single a b) a\u271d = \u2191(single a' b) a\u271d\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d\u00b9 a'\u271d : \u03b1\nb\u271d : M\na a' : \u03b1\nb : M\na\u271d : \u03b1\nh : \u00aca' = a\n\u22a2 \u2191(single a 0) a\u271d = \u21910 a\u271d\n[PROOFSTEP]\nrw [zero_apply, single_apply, ite_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\n\u22a2 (single a b).support \u2286 {a}\n[PROOFSTEP]\nclassical show ite _ _ _ \u2286 _; split_ifs <;> [exact empty_subset _; exact Subset.refl _]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\n\u22a2 (single a b).support \u2286 {a}\n[PROOFSTEP]\nshow ite _ _ _ \u2286 _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\n\u22a2 (if b = 0 then \u2205 else {a}) \u2286 {a}\n[PROOFSTEP]\nsplit_ifs <;> [exact empty_subset _; exact Subset.refl _]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\n\u22a2 (if b = 0 then \u2205 else {a}) \u2286 {a}\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\nh\u271d : b = 0\n\u22a2 \u2205 \u2286 {a}\n[PROOFSTEP]\nexact empty_subset _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\nh\u271d : \u00acb = 0\n\u22a2 {a} \u2286 {a}\n[PROOFSTEP]\nexact Subset.refl _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\nx : \u03b1\n\u22a2 \u2191(single a b) x \u2208 {0, b}\n[PROOFSTEP]\nrcases em (a = x) with (rfl | hx) <;> [simp; simp [single_eq_of_ne hx]]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\nx : \u03b1\n\u22a2 \u2191(single a b) x \u2208 {0, b}\n[PROOFSTEP]\nrcases em (a = x) with (rfl | hx)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\n\u22a2 \u2191(single a b) a \u2208 {0, b}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\nx : \u03b1\nhx : \u00aca = x\n\u22a2 \u2191(single a b) x \u2208 {0, b}\n[PROOFSTEP]\nsimp [single_eq_of_ne hx]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a' : \u03b1\nb : M\na : \u03b1\nb\u2081 b\u2082 : M\neq : single a b\u2081 = single a b\u2082\n\u22a2 b\u2081 = b\u2082\n[PROOFSTEP]\nhave : (single a b\u2081 : \u03b1 \u2192\u2080 M) a = (single a b\u2082 : \u03b1 \u2192\u2080 M) a := by rw [eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a' : \u03b1\nb : M\na : \u03b1\nb\u2081 b\u2082 : M\neq : single a b\u2081 = single a b\u2082\n\u22a2 \u2191(single a b\u2081) a = \u2191(single a b\u2082) a\n[PROOFSTEP]\nrw [eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a' : \u03b1\nb : M\na : \u03b1\nb\u2081 b\u2082 : M\neq : single a b\u2081 = single a b\u2082\nthis : \u2191(single a b\u2081) a = \u2191(single a b\u2082) a\n\u22a2 b\u2081 = b\u2082\n[PROOFSTEP]\nrwa [single_eq_same, single_eq_same] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a' : \u03b1\nb\u271d : M\na x : \u03b1\nb : M\n\u22a2 \u2191(single a b) x = 0 \u2194 x = a \u2192 b = 0\n[PROOFSTEP]\nsimp [single_eq_set_indicator]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a' : \u03b1\nb\u271d : M\na x : \u03b1\nb : M\n\u22a2 \u2191(single a b) x \u2260 0 \u2194 x = a \u2227 b \u2260 0\n[PROOFSTEP]\nsimp [single_apply_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a'\u271d : \u03b1\nb\u271d : M\na a' : \u03b1\nb : M\n\u22a2 a \u2208 (single a' b).support \u2194 a = a' \u2227 b \u2260 0\n[PROOFSTEP]\nsimp [single_apply_eq_zero, not_or]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a' : \u03b1\nb\u271d : M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\n\u22a2 f = single a b \u2194 f.support \u2286 {a} \u2227 \u2191f a = b\n[PROOFSTEP]\nrefine' \u27e8fun h => h.symm \u25b8 \u27e8support_single_subset, single_eq_same\u27e9, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a' : \u03b1\nb\u271d : M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\n\u22a2 f.support \u2286 {a} \u2227 \u2191f a = b \u2192 f = single a b\n[PROOFSTEP]\nrintro \u27e8h, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a' : \u03b1\nb : M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nh : f.support \u2286 {a}\n\u22a2 f = single a (\u2191f a)\n[PROOFSTEP]\next x\n[GOAL]\ncase intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a' : \u03b1\nb : M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nh : f.support \u2286 {a}\nx : \u03b1\n\u22a2 \u2191f x = \u2191(single a (\u2191f a)) x\n[PROOFSTEP]\nby_cases hx : a = x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a' : \u03b1\nb : M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nh : f.support \u2286 {a}\nx : \u03b1\nhx : a = x\n\u22a2 \u2191f x = \u2191(single a (\u2191f a)) x\n[PROOFSTEP]\nsimp only [hx, single_eq_same, single_eq_of_ne, Ne.def, not_false_iff]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a' : \u03b1\nb : M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nh : f.support \u2286 {a}\nx : \u03b1\nhx : \u00aca = x\n\u22a2 \u2191f x = \u2191(single a (\u2191f a)) x\n[PROOFSTEP]\nsimp only [hx, single_eq_same, single_eq_of_ne, Ne.def, not_false_iff]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a' : \u03b1\nb : M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nh : f.support \u2286 {a}\nx : \u03b1\nhx : \u00aca = x\n\u22a2 \u2191f x = 0\n[PROOFSTEP]\nexact not_mem_support_iff.1 (mt (fun hx => (mem_singleton.1 (h hx)).symm) hx)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : M\n\u22a2 single a\u2081 b\u2081 = single a\u2082 b\u2082 \u2194 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082 \u2228 b\u2081 = 0 \u2227 b\u2082 = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : M\n\u22a2 single a\u2081 b\u2081 = single a\u2082 b\u2082 \u2192 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082 \u2228 b\u2081 = 0 \u2227 b\u2082 = 0\n[PROOFSTEP]\nintro eq\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : M\neq : single a\u2081 b\u2081 = single a\u2082 b\u2082\n\u22a2 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082 \u2228 b\u2081 = 0 \u2227 b\u2082 = 0\n[PROOFSTEP]\nby_cases h : a\u2081 = a\u2082\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : M\neq : single a\u2081 b\u2081 = single a\u2082 b\u2082\nh : a\u2081 = a\u2082\n\u22a2 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082 \u2228 b\u2081 = 0 \u2227 b\u2082 = 0\n[PROOFSTEP]\nrefine' Or.inl \u27e8h, _\u27e9\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : M\neq : single a\u2081 b\u2081 = single a\u2082 b\u2082\nh : a\u2081 = a\u2082\n\u22a2 b\u2081 = b\u2082\n[PROOFSTEP]\nrwa [h, (single_injective a\u2082).eq_iff] at eq \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : M\neq : single a\u2081 b\u2081 = single a\u2082 b\u2082\nh : \u00aca\u2081 = a\u2082\n\u22a2 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082 \u2228 b\u2081 = 0 \u2227 b\u2082 = 0\n[PROOFSTEP]\nrw [FunLike.ext_iff] at eq \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : M\neq\u271d : single a\u2081 b\u2081 = single a\u2082 b\u2082\neq : \u2200 (x : \u03b1), \u2191(single a\u2081 b\u2081) x = \u2191(single a\u2082 b\u2082) x\nh : \u00aca\u2081 = a\u2082\n\u22a2 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082 \u2228 b\u2081 = 0 \u2227 b\u2082 = 0\n[PROOFSTEP]\nhave h\u2081 := eq a\u2081\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : M\neq\u271d : single a\u2081 b\u2081 = single a\u2082 b\u2082\neq : \u2200 (x : \u03b1), \u2191(single a\u2081 b\u2081) x = \u2191(single a\u2082 b\u2082) x\nh : \u00aca\u2081 = a\u2082\nh\u2081 : \u2191(single a\u2081 b\u2081) a\u2081 = \u2191(single a\u2082 b\u2082) a\u2081\n\u22a2 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082 \u2228 b\u2081 = 0 \u2227 b\u2082 = 0\n[PROOFSTEP]\nhave h\u2082 := eq a\u2082\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : M\neq\u271d : single a\u2081 b\u2081 = single a\u2082 b\u2082\neq : \u2200 (x : \u03b1), \u2191(single a\u2081 b\u2081) x = \u2191(single a\u2082 b\u2082) x\nh : \u00aca\u2081 = a\u2082\nh\u2081 : \u2191(single a\u2081 b\u2081) a\u2081 = \u2191(single a\u2082 b\u2082) a\u2081\nh\u2082 : \u2191(single a\u2081 b\u2081) a\u2082 = \u2191(single a\u2082 b\u2082) a\u2082\n\u22a2 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082 \u2228 b\u2081 = 0 \u2227 b\u2082 = 0\n[PROOFSTEP]\nsimp only [single_eq_same, single_eq_of_ne h, single_eq_of_ne (Ne.symm h)] at h\u2081 h\u2082 \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : M\neq\u271d : single a\u2081 b\u2081 = single a\u2082 b\u2082\neq : \u2200 (x : \u03b1), \u2191(single a\u2081 b\u2081) x = \u2191(single a\u2082 b\u2082) x\nh : \u00aca\u2081 = a\u2082\nh\u2081 : b\u2081 = 0\nh\u2082 : 0 = b\u2082\n\u22a2 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082 \u2228 b\u2081 = 0 \u2227 b\u2082 = 0\n[PROOFSTEP]\nexact Or.inr \u27e8h\u2081, h\u2082.symm\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : M\n\u22a2 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082 \u2228 b\u2081 = 0 \u2227 b\u2082 = 0 \u2192 single a\u2081 b\u2081 = single a\u2082 b\u2082\n[PROOFSTEP]\nrintro (\u27e8rfl, rfl\u27e9 | \u27e8rfl, rfl\u27e9)\n[GOAL]\ncase mpr.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\na\u2081 : \u03b1\nb\u2081 : M\n\u22a2 single a\u2081 b\u2081 = single a\u2081 b\u2081\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mpr.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\na\u2081 a\u2082 : \u03b1\n\u22a2 single a\u2081 0 = single a\u2082 0\n[PROOFSTEP]\nrw [single_zero, single_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\ni : \u03b1\nh : b \u2260 0\n\u22a2 (single i b).support \u2260 \u22a5\n[PROOFSTEP]\nsimpa only [support_single_ne_zero _ h] using singleton_ne_empty _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb b' : M\nhb : b \u2260 0\nhb' : b' \u2260 0\ni j : \u03b1\n\u22a2 Disjoint (single i b).support (single j b').support \u2194 i \u2260 j\n[PROOFSTEP]\nrw [support_single_ne_zero _ hb, support_single_ne_zero _ hb', disjoint_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\n\u22a2 single a b = 0 \u2194 b = 0\n[PROOFSTEP]\nsimp [FunLike.ext_iff, single_eq_set_indicator]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb\u271d : M\na\u2081 a\u2082 : \u03b1\nb : M\n\u22a2 \u2191(single a\u2081 b) a\u2082 = \u2191(single a\u2082 b) a\u2081\n[PROOFSTEP]\nclassical simp only [single_apply, eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb\u271d : M\na\u2081 a\u2082 : \u03b1\nb : M\n\u22a2 \u2191(single a\u2081 b) a\u2082 = \u2191(single a\u2082 b) a\u2081\n[PROOFSTEP]\nsimp only [single_apply, eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\na a' : \u03b1\nb : M\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : Nontrivial M\n\u22a2 Nontrivial (\u03b1 \u2192\u2080 M)\n[PROOFSTEP]\ninhabit \u03b1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\na a' : \u03b1\nb : M\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : Nontrivial M\ninhabited_h : Inhabited \u03b1\n\u22a2 Nontrivial (\u03b1 \u2192\u2080 M)\n[PROOFSTEP]\nrcases exists_ne (0 : M) with \u27e8x, hx\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\na a' : \u03b1\nb : M\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : Nontrivial M\ninhabited_h : Inhabited \u03b1\nx : M\nhx : x \u2260 0\n\u22a2 Nontrivial (\u03b1 \u2192\u2080 M)\n[PROOFSTEP]\nexact nontrivial_of_ne (single default x) 0 (mt single_eq_zero.1 hx)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\na a' : \u03b1\nb : M\ninst\u271d : Unique \u03b1\nb' : M\n\u22a2 single a b = single a' b' \u2194 b = b'\n[PROOFSTEP]\nrw [unique_ext_iff, Unique.eq_default a, Unique.eq_default a', single_eq_same, single_eq_same]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\nf : \u03b1 \u2192\u2080 M\n\u22a2 card f.support = 1 \u2194 \u2203 a, \u2191f a \u2260 0 \u2227 f = single a (\u2191f a)\n[PROOFSTEP]\nsimp only [card_eq_one, support_eq_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\nf : \u03b1 \u2192\u2080 M\n\u22a2 card f.support = 1 \u2194 \u2203 a b x, f = single a b\n[PROOFSTEP]\nsimp only [card_eq_one, support_eq_singleton']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a' : \u03b1\nb\u271d : M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nx\u271d : \u2203 b, f = single a b\nb : M\nhb : f = single a b\n\u22a2 f.support \u2286 {a}\n[PROOFSTEP]\nrw [hb, support_subset_singleton, single_eq_same]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\na a' : \u03b1\nb : M\ninst\u271d : Nonempty \u03b1\nf : \u03b1 \u2192\u2080 M\n\u22a2 card f.support \u2264 1 \u2194 \u2203 a, f = single a (\u2191f a)\n[PROOFSTEP]\nsimp only [card_le_one_iff_subset_singleton, support_subset_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\na a' : \u03b1\nb : M\ninst\u271d : Nonempty \u03b1\nf : \u03b1 \u2192\u2080 M\n\u22a2 card f.support \u2264 1 \u2194 \u2203 a b, f = single a b\n[PROOFSTEP]\nsimp only [card_le_one_iff_subset_singleton, support_subset_singleton']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\na a' : \u03b1\nb : M\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Finite \u03b1\nx : \u03b1\nm : M\n\u22a2 \u2191equivFunOnFinite (single x m) = Pi.single x m\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\na a' : \u03b1\nb : M\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Finite \u03b1\nx : \u03b1\nm : M\nx\u271d : \u03b1\n\u22a2 \u2191equivFunOnFinite (single x m) x\u271d = Pi.single x m x\u271d\n[PROOFSTEP]\nsimp [Finsupp.single_eq_pi_single, equivFunOnFinite]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\na a' : \u03b1\nb : M\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Finite \u03b1\nx : \u03b1\nm : M\n\u22a2 \u2191equivFunOnFinite.symm (Pi.single x m) = single x m\n[PROOFSTEP]\nrw [\u2190 equivFunOnFinite_single, Equiv.symm_apply_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\n\u22a2 Finset \u03b1\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\nthis : DecidableEq \u03b1\n\u22a2 Finset \u03b1\n[PROOFSTEP]\nhaveI := Classical.decEq M\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\nthis\u271d : DecidableEq \u03b1\nthis : DecidableEq M\n\u22a2 Finset \u03b1\n[PROOFSTEP]\nexact if b = 0 then f.support.erase a else insert a f.support\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni\u271d : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\n\u22a2 (i \u2208 if b = 0 then erase f.support a else insert a f.support) \u2194 Function.update (\u2191f) a b i \u2260 0\n[PROOFSTEP]\nclassical\nsimp [Function.update, Ne.def]\nsplit_ifs with hb ha ha <;> try simp only [*, not_false_iff, iff_true, not_true, iff_false]\n\u00b7 rw [Finset.mem_erase]\n simp\n\u00b7 rw [Finset.mem_erase]\n simp [ha]\n\u00b7 rw [Finset.mem_insert]\n simp [ha]\n\u00b7 rw [Finset.mem_insert]\n simp [ha]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni\u271d : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\n\u22a2 (i \u2208 if b = 0 then erase f.support a else insert a f.support) \u2194 Function.update (\u2191f) a b i \u2260 0\n[PROOFSTEP]\nsimp [Function.update, Ne.def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni\u271d : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\n\u22a2 (i \u2208 if b = 0 then erase f.support a else insert a f.support) \u2194 \u00ac(if i = a then b else \u2191f i) = 0\n[PROOFSTEP]\nsplit_ifs with hb ha ha\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni\u271d : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\nhb : b = 0\nha : i = a\n\u22a2 i \u2208 erase f.support a \u2194 \u00acb = 0\n[PROOFSTEP]\ntry simp only [*, not_false_iff, iff_true, not_true, iff_false]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni\u271d : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\nhb : b = 0\nha : i = a\n\u22a2 i \u2208 erase f.support a \u2194 \u00acb = 0\n[PROOFSTEP]\nsimp only [*, not_false_iff, iff_true, not_true, iff_false]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni\u271d : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\nhb : b = 0\nha : \u00aci = a\n\u22a2 i \u2208 erase f.support a \u2194 \u00ac\u2191f i = 0\n[PROOFSTEP]\ntry simp only [*, not_false_iff, iff_true, not_true, iff_false]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni\u271d : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\nhb : b = 0\nha : \u00aci = a\n\u22a2 i \u2208 erase f.support a \u2194 \u00ac\u2191f i = 0\n[PROOFSTEP]\nsimp only [*, not_false_iff, iff_true, not_true, iff_false]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni\u271d : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\nhb : \u00acb = 0\nha : i = a\n\u22a2 i \u2208 insert a f.support \u2194 \u00acb = 0\n[PROOFSTEP]\ntry simp only [*, not_false_iff, iff_true, not_true, iff_false]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni\u271d : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\nhb : \u00acb = 0\nha : i = a\n\u22a2 i \u2208 insert a f.support \u2194 \u00acb = 0\n[PROOFSTEP]\nsimp only [*, not_false_iff, iff_true, not_true, iff_false]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni\u271d : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\nhb : \u00acb = 0\nha : \u00aci = a\n\u22a2 i \u2208 insert a f.support \u2194 \u00ac\u2191f i = 0\n[PROOFSTEP]\ntry simp only [*, not_false_iff, iff_true, not_true, iff_false]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni\u271d : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\nhb : \u00acb = 0\nha : \u00aci = a\n\u22a2 i \u2208 insert a f.support \u2194 \u00ac\u2191f i = 0\n[PROOFSTEP]\nsimp only [*, not_false_iff, iff_true, not_true, iff_false]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni\u271d : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\nhb : b = 0\nha : i = a\n\u22a2 \u00aca \u2208 erase f.support a\n[PROOFSTEP]\nrw [Finset.mem_erase]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni\u271d : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\nhb : b = 0\nha : i = a\n\u22a2 \u00ac(a \u2260 a \u2227 a \u2208 f.support)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni\u271d : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\nhb : b = 0\nha : \u00aci = a\n\u22a2 i \u2208 erase f.support a \u2194 \u00ac\u2191f i = 0\n[PROOFSTEP]\nrw [Finset.mem_erase]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni\u271d : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\nhb : b = 0\nha : \u00aci = a\n\u22a2 i \u2260 a \u2227 i \u2208 f.support \u2194 \u00ac\u2191f i = 0\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni\u271d : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\nhb : \u00acb = 0\nha : i = a\n\u22a2 a \u2208 insert a f.support\n[PROOFSTEP]\nrw [Finset.mem_insert]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni\u271d : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\nhb : \u00acb = 0\nha : i = a\n\u22a2 a = a \u2228 a \u2208 f.support\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni\u271d : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\nhb : \u00acb = 0\nha : \u00aci = a\n\u22a2 i \u2208 insert a f.support \u2194 \u00ac\u2191f i = 0\n[PROOFSTEP]\nrw [Finset.mem_insert]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf\u271d : \u03b1 \u2192\u2080 M\na\u271d : \u03b1\nb\u271d : M\ni\u271d : \u03b1\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\nhb : \u00acb = 0\nha : \u00aci = a\n\u22a2 i = a \u2228 i \u2208 f.support \u2194 \u00ac\u2191f i = 0\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 \u2191(update f a b) = Function.update (\u2191f) a b\n[PROOFSTEP]\ndelta update Function.update\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 \u2191{ support := if b = 0 then erase f.support a else insert a f.support,\n toFun := fun a_1 => if h : a_1 = a then (_ : a = a_1) \u25b8 b else \u2191f a_1,\n mem_support_toFun :=\n (_ :\n \u2200 (i : \u03b1),\n (i \u2208 if b = 0 then erase f.support a else insert a f.support) \u2194 Function.update (\u2191f) a b i \u2260 0) } =\n fun a_1 => if h : a_1 = a then (_ : a = a_1) \u25b8 b else \u2191f a_1\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\ninst\u271d : DecidableEq \u03b1\nx\u271d : \u03b1\n\u22a2 \u2191{ support := if b = 0 then erase f.support a else insert a f.support,\n toFun := fun a_1 => if h : a_1 = a then (_ : a = a_1) \u25b8 b else \u2191f a_1,\n mem_support_toFun :=\n (_ :\n \u2200 (i : \u03b1),\n (i \u2208 if b = 0 then erase f.support a else insert a f.support) \u2194 Function.update (\u2191f) a b i \u2260 0) }\n x\u271d =\n if h : x\u271d = a then (_ : a = x\u271d) \u25b8 b else \u2191f x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\ninst\u271d : DecidableEq \u03b1\nx\u271d : \u03b1\n\u22a2 (if h : x\u271d = a then (_ : a = x\u271d) \u25b8 b else \u2191f x\u271d) = if h : x\u271d = a then (_ : a = x\u271d) \u25b8 b else \u2191f x\u271d\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\ninst\u271d : DecidableEq \u03b1\nx\u271d : \u03b1\nh\u271d : x\u271d = a\n\u22a2 (_ : a = x\u271d) \u25b8 b = (_ : a = x\u271d) \u25b8 b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\ninst\u271d : DecidableEq \u03b1\nx\u271d : \u03b1\nh\u271d : \u00acx\u271d = a\n\u22a2 \u2191f x\u271d = \u2191f x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\n\u22a2 update f a (\u2191f a) = f\n[PROOFSTEP]\nclassical\next\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\n\u22a2 update f a (\u2191f a) = f\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni a\u271d : \u03b1\n\u22a2 \u2191(update f a (\u2191f a)) a\u271d = \u2191f a\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\n\u22a2 update 0 a b = single a b\n[PROOFSTEP]\nclassical\next\nrw [single_eq_update]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\n\u22a2 update 0 a b = single a b\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni a\u271d : \u03b1\n\u22a2 \u2191(update 0 a b) a\u271d = \u2191(single a b) a\u271d\n[PROOFSTEP]\nrw [single_eq_update]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni a\u271d : \u03b1\n\u22a2 \u2191(update 0 a b) a\u271d = Function.update 0 a b a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq M\n\u22a2 (update f a b).support = if b = 0 then erase f.support a else insert a f.support\n[PROOFSTEP]\nclassical dsimp [update]; congr <;> apply Subsingleton.elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq M\n\u22a2 (update f a b).support = if b = 0 then erase f.support a else insert a f.support\n[PROOFSTEP]\ndsimp [update]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq M\n\u22a2 (if b = 0 then erase f.support a else insert a f.support) = if b = 0 then erase f.support a else insert a f.support\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_t.h.e_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq M\n\u22a2 (fun a b => Classical.decEq \u03b1 a b) = fun a b => inst\u271d\u00b9 a b\n[PROOFSTEP]\napply Subsingleton.elim\n[GOAL]\ncase e_e.h.e_3.h.h.e_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq M\n\u22a2 (fun a b => Classical.decEq \u03b1 a b) = fun a b => inst\u271d\u00b9 a b\n[PROOFSTEP]\napply Subsingleton.elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 (update f a 0).support = erase f.support a\n[PROOFSTEP]\nclassical\nsimp only [update, ite_true, mem_support_iff, ne_eq, not_not]\ncongr; apply Subsingleton.elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 (update f a 0).support = erase f.support a\n[PROOFSTEP]\nsimp only [update, ite_true, mem_support_iff, ne_eq, not_not]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 erase f.support a = erase f.support a\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 (fun a b => Classical.decEq \u03b1 a b) = fun a b => inst\u271d a b\n[PROOFSTEP]\napply Subsingleton.elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : b \u2260 0\n\u22a2 (update f a b).support = insert a f.support\n[PROOFSTEP]\nclassical\nsimp only [update, h, ite_false, mem_support_iff, ne_eq]\ncongr; apply Subsingleton.elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : b \u2260 0\n\u22a2 (update f a b).support = insert a f.support\n[PROOFSTEP]\nsimp only [update, h, ite_false, mem_support_iff, ne_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : b \u2260 0\n\u22a2 insert a f.support = insert a f.support\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e_3.h.h.e_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\ni : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : b \u2260 0\n\u22a2 (fun a b => Classical.decEq \u03b1 a b) = fun a b => inst\u271d a b\n[PROOFSTEP]\napply Subsingleton.elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nf : \u03b1 \u2192\u2080 M\na' : \u03b1\n\u22a2 a' \u2208 Finset.erase f.support a \u2194 (fun a' => if a' = a then 0 else \u2191f a') a' \u2260 0\n[PROOFSTEP]\nclassical\nrw [mem_erase, mem_support_iff]; dsimp\nsplit_ifs with h\nexact \u27e8fun H _ => H.1 h, fun H => (H rfl).elim\u27e9\nexact and_iff_right h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nf : \u03b1 \u2192\u2080 M\na' : \u03b1\n\u22a2 a' \u2208 Finset.erase f.support a \u2194 (fun a' => if a' = a then 0 else \u2191f a') a' \u2260 0\n[PROOFSTEP]\nrw [mem_erase, mem_support_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nf : \u03b1 \u2192\u2080 M\na' : \u03b1\n\u22a2 a' \u2260 a \u2227 \u2191f a' \u2260 0 \u2194 (fun a' => if a' = a then 0 else \u2191f a') a' \u2260 0\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nf : \u03b1 \u2192\u2080 M\na' : \u03b1\n\u22a2 \u00aca' = a \u2227 \u00ac\u2191f a' = 0 \u2194 \u00ac(if a' = a then 0 else \u2191f a') = 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nf : \u03b1 \u2192\u2080 M\na' : \u03b1\nh : a' = a\n\u22a2 \u00aca' = a \u2227 \u00ac\u2191f a' = 0 \u2194 \u00ac0 = 0\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nf : \u03b1 \u2192\u2080 M\na' : \u03b1\nh : \u00aca' = a\n\u22a2 \u00aca' = a \u2227 \u00ac\u2191f a' = 0 \u2194 \u00ac\u2191f a' = 0\n[PROOFSTEP]\nexact \u27e8fun H _ => H.1 h, fun H => (H rfl).elim\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nf : \u03b1 \u2192\u2080 M\na' : \u03b1\nh : \u00aca' = a\n\u22a2 \u00aca' = a \u2227 \u00ac\u2191f a' = 0 \u2194 \u00ac\u2191f a' = 0\n[PROOFSTEP]\nexact and_iff_right h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nf : \u03b1 \u2192\u2080 M\n\u22a2 (erase a f).support = Finset.erase f.support a\n[PROOFSTEP]\nclassical\ndsimp [erase]\ncongr; apply Subsingleton.elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nf : \u03b1 \u2192\u2080 M\n\u22a2 (erase a f).support = Finset.erase f.support a\n[PROOFSTEP]\ndsimp [erase]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nf : \u03b1 \u2192\u2080 M\n\u22a2 Finset.erase f.support a = Finset.erase f.support a\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nf : \u03b1 \u2192\u2080 M\n\u22a2 (fun a b => Classical.decEq \u03b1 a b) = fun a b => inst\u271d a b\n[PROOFSTEP]\napply Subsingleton.elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nf : \u03b1 \u2192\u2080 M\n\u22a2 \u2191(erase a f) a = 0\n[PROOFSTEP]\nclassical simp only [erase, coe_mk, ite_true]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nf : \u03b1 \u2192\u2080 M\n\u22a2 \u2191(erase a f) a = 0\n[PROOFSTEP]\nsimp only [erase, coe_mk, ite_true]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nf : \u03b1 \u2192\u2080 M\nh : a' \u2260 a\n\u22a2 \u2191(erase a f) a' = \u2191f a'\n[PROOFSTEP]\nclassical simp only [erase, coe_mk, h, ite_false]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nf : \u03b1 \u2192\u2080 M\nh : a' \u2260 a\n\u22a2 \u2191(erase a f) a' = \u2191f a'\n[PROOFSTEP]\nsimp only [erase, coe_mk, h, ite_false]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nb : M\n\u22a2 erase a (single a b) = 0\n[PROOFSTEP]\next s\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nb : M\ns : \u03b1\n\u22a2 \u2191(erase a (single a b)) s = \u21910 s\n[PROOFSTEP]\nby_cases hs : s = a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nb : M\ns : \u03b1\nhs : s = a\n\u22a2 \u2191(erase a (single a b)) s = \u21910 s\n[PROOFSTEP]\nrw [hs, erase_same]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nb : M\ns : \u03b1\nhs : s = a\n\u22a2 0 = \u21910 a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nb : M\ns : \u03b1\nhs : \u00acs = a\n\u22a2 \u2191(erase a (single a b)) s = \u21910 s\n[PROOFSTEP]\nrw [erase_ne hs]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\nb : M\ns : \u03b1\nhs : \u00acs = a\n\u22a2 \u2191(single a b) s = \u21910 s\n[PROOFSTEP]\nexact single_eq_of_ne (Ne.symm hs)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\nh : a \u2260 a'\n\u22a2 erase a (single a' b) = single a' b\n[PROOFSTEP]\next s\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\nh : a \u2260 a'\ns : \u03b1\n\u22a2 \u2191(erase a (single a' b)) s = \u2191(single a' b) s\n[PROOFSTEP]\nby_cases hs : s = a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\nh : a \u2260 a'\ns : \u03b1\nhs : s = a\n\u22a2 \u2191(erase a (single a' b)) s = \u2191(single a' b) s\n[PROOFSTEP]\nrw [hs, erase_same, single_eq_of_ne h.symm]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\nh : a \u2260 a'\ns : \u03b1\nhs : \u00acs = a\n\u22a2 \u2191(erase a (single a' b)) s = \u2191(single a' b) s\n[PROOFSTEP]\nrw [erase_ne hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nhaf : \u00aca \u2208 f.support\n\u22a2 erase a f = f\n[PROOFSTEP]\next b\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nhaf : \u00aca \u2208 f.support\nb : \u03b1\n\u22a2 \u2191(erase a f) b = \u2191f b\n[PROOFSTEP]\nby_cases hab : b = a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nhaf : \u00aca \u2208 f.support\nb : \u03b1\nhab : b = a\n\u22a2 \u2191(erase a f) b = \u2191f b\n[PROOFSTEP]\nrwa [hab, erase_same, eq_comm, \u2190 not_mem_support_iff]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nhaf : \u00aca \u2208 f.support\nb : \u03b1\nhab : \u00acb = a\n\u22a2 \u2191(erase a f) b = \u2191f b\n[PROOFSTEP]\nrw [erase_ne hab]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\n\u22a2 erase a 0 = 0\n[PROOFSTEP]\nclassical rw [\u2190 support_eq_empty, support_erase, support_zero, erase_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na : \u03b1\n\u22a2 erase a 0 = 0\n[PROOFSTEP]\nrw [\u2190 support_eq_empty, support_erase, support_zero, erase_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\ns : Finset \u03b1\nf : \u03b1 \u2192 M\nhf : \u2200 (a : \u03b1), f a \u2260 0 \u2192 a \u2208 s\n\u22a2 \u2200 (a : \u03b1), a \u2208 filter (fun x => f x \u2260 0) s \u2194 f a \u2260 0\n[PROOFSTEP]\nclassical simpa\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\ns : Finset \u03b1\nf : \u03b1 \u2192 M\nhf : \u2200 (a : \u03b1), f a \u2260 0 \u2192 a \u2208 s\n\u22a2 \u2200 (a : \u03b1), a \u2208 filter (fun x => f x \u2260 0) s \u2194 f a \u2260 0\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\ns : Finset \u03b1\nf : \u03b1 \u2192 M\nhf : \u2200 (a : \u03b1), f a \u2260 0 \u2192 a \u2208 s\n\u22a2 (onFinset s f hf).support \u2286 s\n[PROOFSTEP]\nclassical convert filter_subset (f \u00b7 \u2260 0) s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\ns : Finset \u03b1\nf : \u03b1 \u2192 M\nhf : \u2200 (a : \u03b1), f a \u2260 0 \u2192 a \u2208 s\n\u22a2 (onFinset s f hf).support \u2286 s\n[PROOFSTEP]\nconvert filter_subset (f \u00b7 \u2260 0) s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\ns : Finset \u03b1\nf : \u03b1 \u2192 M\nhf : \u2200 (a : \u03b1), f a \u2260 0 \u2192 a \u2208 s\na : \u03b1\n\u22a2 a \u2208 (onFinset s f hf).support \u2194 f a \u2260 0\n[PROOFSTEP]\nrw [Finsupp.mem_support_iff, Finsupp.onFinset_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq M\ns : Finset \u03b1\nf : \u03b1 \u2192 M\nhf : \u2200 (a : \u03b1), f a \u2260 0 \u2192 a \u2208 s\n\u22a2 (onFinset s f hf).support = filter (fun a => f a \u2260 0) s\n[PROOFSTEP]\ndsimp [onFinset]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : DecidableEq M\ns : Finset \u03b1\nf : \u03b1 \u2192 M\nhf : \u2200 (a : \u03b1), f a \u2260 0 \u2192 a \u2208 s\n\u22a2 filter (fun x => \u00acf x = 0) s = filter (fun a => \u00acf a = 0) s\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2192 N\nhf : f 0 = 0\ng : \u03b1 \u2192\u2080 M\na : \u03b1\n\u22a2 (f \u2218 \u2191g) a \u2260 0 \u2192 a \u2208 g.support\n[PROOFSTEP]\nrw [mem_support_iff, not_imp_not]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2192 N\nhf : f 0 = 0\ng : \u03b1 \u2192\u2080 M\na : \u03b1\n\u22a2 \u2191g a = 0 \u2192 (f \u2218 \u2191g) a = 0\n[PROOFSTEP]\nexact fun H => (congr_arg f H).trans hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2192 N\nhf : f 0 = 0\na : \u03b1\n\u22a2 \u2191(mapRange f hf 0) a = \u21910 a\n[PROOFSTEP]\nsimp only [hf, zero_apply, mapRange_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2192 N\nhf : f 0 = 0\na : \u03b1\nb : M\na' : \u03b1\n\u22a2 \u2191(mapRange f hf (single a b)) a' = \u2191(single a (f b)) a'\n[PROOFSTEP]\nclassical simpa only [single_eq_pi_single] using Pi.apply_single _ (fun _ => hf) a _ a'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2192 N\nhf : f 0 = 0\na : \u03b1\nb : M\na' : \u03b1\n\u22a2 \u2191(mapRange f hf (single a b)) a' = \u2191(single a (f b)) a'\n[PROOFSTEP]\nsimpa only [single_eq_pi_single] using Pi.apply_single _ (fun _ => hf) a _ a'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\ne : M \u2192 N\nhe0 : e 0 = 0\nf : \u03b9 \u2192\u2080 M\nhe : Injective e\n\u22a2 (mapRange e he0 f).support = f.support\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\ne : M \u2192 N\nhe0 : e 0 = 0\nf : \u03b9 \u2192\u2080 M\nhe : Injective e\na\u271d : \u03b9\n\u22a2 a\u271d \u2208 (mapRange e he0 f).support \u2194 a\u271d \u2208 f.support\n[PROOFSTEP]\nsimp only [Finsupp.mem_support_iff, Ne.def, Finsupp.mapRange_apply]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\ne : M \u2192 N\nhe0 : e 0 = 0\nf : \u03b9 \u2192\u2080 M\nhe : Injective e\na\u271d : \u03b9\n\u22a2 \u00ace (\u2191f a\u271d) = 0 \u2194 \u00ac\u2191f a\u271d = 0\n[PROOFSTEP]\nexact he.ne_iff' he0\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na\u2082 : \u03b2\nthis : DecidableEq \u03b2\nh : a\u2082 \u2208 map f v.support\n\u22a2 \u2203! a, a \u2208 v.support \u2227 (fun a\u2081 => \u2191f a\u2081 = a\u2082) a\n[PROOFSTEP]\nrcases Finset.mem_map.1 h with \u27e8a, ha, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\nthis : DecidableEq \u03b2\na : \u03b1\nha : a \u2208 v.support\nh : \u2191f a \u2208 map f v.support\n\u22a2 \u2203! a_1, a_1 \u2208 v.support \u2227 (fun a\u2081 => \u2191f a\u2081 = \u2191f a) a_1\n[PROOFSTEP]\nexact ExistsUnique.intro a \u27e8ha, rfl\u27e9 fun b \u27e8_, hb\u27e9 => f.injective hb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na\u2082 : \u03b2\n\u22a2 a\u2082 \u2208 map f v.support \u2194\n (fun a\u2082 =>\n if h : a\u2082 \u2208 map f v.support then\n \u2191v (choose (fun a\u2081 => \u2191f a\u2081 = a\u2082) v.support (_ : \u2203! a, a \u2208 v.support \u2227 (fun a\u2081 => \u2191f a\u2081 = a\u2082) a))\n else 0)\n a\u2082 \u2260\n 0\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na\u2082 : \u03b2\n\u22a2 a\u2082 \u2208 map f v.support \u2194\n \u00ac(if h : a\u2082 \u2208 map f v.support then\n \u2191v (choose (fun a\u2081 => \u2191f a\u2081 = a\u2082) v.support (_ : \u2203! a, a \u2208 v.support \u2227 \u2191f a = a\u2082))\n else 0) =\n 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na\u2082 : \u03b2\nh : a\u2082 \u2208 map f v.support\n\u22a2 a\u2082 \u2208 map f v.support \u2194 \u00ac\u2191v (choose (fun a\u2081 => \u2191f a\u2081 = a\u2082) v.support (_ : \u2203! a, a \u2208 v.support \u2227 \u2191f a = a\u2082)) = 0\n[PROOFSTEP]\nsimp only [h, true_iff_iff, Ne.def]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na\u2082 : \u03b2\nh : a\u2082 \u2208 map f v.support\n\u22a2 \u00ac\u2191v (choose (fun a\u2081 => \u2191f a\u2081 = a\u2082) v.support (_ : \u2203! a, a \u2208 v.support \u2227 \u2191f a = a\u2082)) = 0\n[PROOFSTEP]\nrw [\u2190 not_mem_support_iff, not_not]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na\u2082 : \u03b2\nh : a\u2082 \u2208 map f v.support\n\u22a2 choose (fun a\u2081 => \u2191f a\u2081 = a\u2082) v.support (_ : \u2203! a, a \u2208 v.support \u2227 \u2191f a = a\u2082) \u2208 v.support\n[PROOFSTEP]\nclassical apply Finset.choose_mem\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na\u2082 : \u03b2\nh : a\u2082 \u2208 map f v.support\n\u22a2 choose (fun a\u2081 => \u2191f a\u2081 = a\u2082) v.support (_ : \u2203! a, a \u2208 v.support \u2227 \u2191f a = a\u2082) \u2208 v.support\n[PROOFSTEP]\napply Finset.choose_mem\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na\u2082 : \u03b2\nh : \u00aca\u2082 \u2208 map f v.support\n\u22a2 a\u2082 \u2208 map f v.support \u2194 \u00ac0 = 0\n[PROOFSTEP]\nsimp only [h, Ne.def, ne_self_iff_false]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na : \u03b1\n\u22a2 \u2191(embDomain f v) (\u2191f a) = \u2191v a\n[PROOFSTEP]\nclassical\nchange dite _ _ _ = _\nsplit_ifs with h <;> rw [Finset.mem_map' f] at h \n\u00b7 refine' congr_arg (v : \u03b1 \u2192 M) (f.inj' _)\n exact Finset.choose_property (fun a\u2081 => f a\u2081 = f a) _ _\n\u00b7 exact (not_mem_support_iff.1 h).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na : \u03b1\n\u22a2 \u2191(embDomain f v) (\u2191f a) = \u2191v a\n[PROOFSTEP]\nchange dite _ _ _ = _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na : \u03b1\n\u22a2 (if h : \u2191f a \u2208 map f v.support then\n (fun h =>\n \u2191v (choose (fun a\u2081 => \u2191f a\u2081 = \u2191f a) v.support (_ : \u2203! a_1, a_1 \u2208 v.support \u2227 (fun a\u2081 => \u2191f a\u2081 = \u2191f a) a_1)))\n h\n else (fun h => 0) h) =\n \u2191v a\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na : \u03b1\nh : \u2191f a \u2208 map f v.support\n\u22a2 (fun h => \u2191v (choose (fun a\u2081 => \u2191f a\u2081 = \u2191f a) v.support (_ : \u2203! a_1, a_1 \u2208 v.support \u2227 (fun a\u2081 => \u2191f a\u2081 = \u2191f a) a_1)))\n h =\n \u2191v a\n[PROOFSTEP]\nrw [Finset.mem_map' f] at h \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na : \u03b1\nh : \u00ac\u2191f a \u2208 map f v.support\n\u22a2 (fun h => 0) h = \u2191v a\n[PROOFSTEP]\nrw [Finset.mem_map' f] at h \n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na : \u03b1\nh\u271d : \u2191f a \u2208 map f v.support\nh : a \u2208 v.support\n\u22a2 (fun h => \u2191v (choose (fun a\u2081 => \u2191f a\u2081 = \u2191f a) v.support (_ : \u2203! a_1, a_1 \u2208 v.support \u2227 (fun a\u2081 => \u2191f a\u2081 = \u2191f a) a_1)))\n h\u271d =\n \u2191v a\n[PROOFSTEP]\nrefine' congr_arg (v : \u03b1 \u2192 M) (f.inj' _)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na : \u03b1\nh\u271d : \u2191f a \u2208 map f v.support\nh : a \u2208 v.support\n\u22a2 Embedding.toFun f\n (choose (fun a\u2081 => \u2191f a\u2081 = \u2191f a) v.support (_ : \u2203! a_1, a_1 \u2208 v.support \u2227 (fun a\u2081 => \u2191f a\u2081 = \u2191f a) a_1)) =\n Embedding.toFun f a\n[PROOFSTEP]\nexact Finset.choose_property (fun a\u2081 => f a\u2081 = f a) _ _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na : \u03b1\nh\u271d : \u00ac\u2191f a \u2208 map f v.support\nh : \u00aca \u2208 v.support\n\u22a2 (fun h => 0) h\u271d = \u2191v a\n[PROOFSTEP]\nexact (not_mem_support_iff.1 h).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na : \u03b2\nh : \u00aca \u2208 Set.range \u2191f\n\u22a2 \u2191(embDomain f v) a = 0\n[PROOFSTEP]\nclassical\nrefine' dif_neg (mt (fun h => _) h)\nrcases Finset.mem_map.1 h with \u27e8a, _h, rfl\u27e9\nexact Set.mem_range_self a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na : \u03b2\nh : \u00aca \u2208 Set.range \u2191f\n\u22a2 \u2191(embDomain f v) a = 0\n[PROOFSTEP]\nrefine' dif_neg (mt (fun h => _) h)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na : \u03b2\nh\u271d : \u00aca \u2208 Set.range \u2191f\nh : a \u2208 map f v.support\n\u22a2 a \u2208 Set.range \u2191f\n[PROOFSTEP]\nrcases Finset.mem_map.1 h with \u27e8a, _h, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nv : \u03b1 \u2192\u2080 M\na : \u03b1\n_h : a \u2208 v.support\nh\u271d : \u00ac\u2191f a \u2208 Set.range \u2191f\nh : \u2191f a \u2208 map f v.support\n\u22a2 \u2191f a \u2208 Set.range \u2191f\n[PROOFSTEP]\nexact Set.mem_range_self a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\nl\u2081 l\u2082 : \u03b1 \u2192\u2080 M\nh : embDomain f l\u2081 = embDomain f l\u2082\na : \u03b1\n\u22a2 \u2191l\u2081 a = \u2191l\u2082 a\n[PROOFSTEP]\nsimpa only [embDomain_apply] using FunLike.ext_iff.1 h (f a)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\ng : M \u2192 N\np : \u03b1 \u2192\u2080 M\nhg : g 0 = 0\n\u22a2 embDomain f (mapRange g hg p) = mapRange g hg (embDomain f p)\n[PROOFSTEP]\next a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\ng : M \u2192 N\np : \u03b1 \u2192\u2080 M\nhg : g 0 = 0\na : \u03b2\n\u22a2 \u2191(embDomain f (mapRange g hg p)) a = \u2191(mapRange g hg (embDomain f p)) a\n[PROOFSTEP]\nby_cases h : a \u2208 Set.range f\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\ng : M \u2192 N\np : \u03b1 \u2192\u2080 M\nhg : g 0 = 0\na : \u03b2\nh : a \u2208 Set.range \u2191f\n\u22a2 \u2191(embDomain f (mapRange g hg p)) a = \u2191(mapRange g hg (embDomain f p)) a\n[PROOFSTEP]\nrcases h with \u27e8a', rfl\u27e9\n[GOAL]\ncase pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\ng : M \u2192 N\np : \u03b1 \u2192\u2080 M\nhg : g 0 = 0\na' : \u03b1\n\u22a2 \u2191(embDomain f (mapRange g hg p)) (\u2191f a') = \u2191(mapRange g hg (embDomain f p)) (\u2191f a')\n[PROOFSTEP]\nrw [mapRange_apply, embDomain_apply, embDomain_apply, mapRange_apply]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\ng : M \u2192 N\np : \u03b1 \u2192\u2080 M\nhg : g 0 = 0\na : \u03b2\nh : \u00aca \u2208 Set.range \u2191f\n\u22a2 \u2191(embDomain f (mapRange g hg p)) a = \u2191(mapRange g hg (embDomain f p)) a\n[PROOFSTEP]\nrw [mapRange_apply, embDomain_notin_range, embDomain_notin_range, \u2190 hg]\n[GOAL]\ncase neg.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\ng : M \u2192 N\np : \u03b1 \u2192\u2080 M\nhg : g 0 = 0\na : \u03b2\nh : \u00aca \u2208 Set.range \u2191f\n\u22a2 \u00aca \u2208 Set.range \u2191f\n[PROOFSTEP]\nassumption\n[GOAL]\ncase neg.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\ng : M \u2192 N\np : \u03b1 \u2192\u2080 M\nhg : g 0 = 0\na : \u03b2\nh : \u00aca \u2208 Set.range \u2191f\n\u22a2 \u00aca \u2208 Set.range \u2191f\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nl : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\na : \u03b2\nb : M\nhb : b \u2260 0\nh : embDomain f l = single a b\n\u22a2 \u2203 x, l = single x b \u2227 \u2191f x = a\n[PROOFSTEP]\nclassical\nhave h_map_support : Finset.map f l.support = { a } := by rw [\u2190 support_embDomain, h, support_single_ne_zero _ hb]\nhave ha : a \u2208 Finset.map f l.support := by simp only [h_map_support, Finset.mem_singleton]\nrcases Finset.mem_map.1 ha with \u27e8c, _hc\u2081, hc\u2082\u27e9\nuse c\nconstructor\n\u00b7 ext d\n rw [\u2190 embDomain_apply f l, h]\n by_cases h_cases : c = d\n \u00b7 simp only [Eq.symm h_cases, hc\u2082, single_eq_same]\n \u00b7 rw [single_apply, single_apply, if_neg, if_neg h_cases]\n by_contra hfd\n exact h_cases (f.injective (hc\u2082.trans hfd))\n\u00b7 exact hc\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nl : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\na : \u03b2\nb : M\nhb : b \u2260 0\nh : embDomain f l = single a b\n\u22a2 \u2203 x, l = single x b \u2227 \u2191f x = a\n[PROOFSTEP]\nhave h_map_support : Finset.map f l.support = { a } := by rw [\u2190 support_embDomain, h, support_single_ne_zero _ hb]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nl : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\na : \u03b2\nb : M\nhb : b \u2260 0\nh : embDomain f l = single a b\n\u22a2 map f l.support = {a}\n[PROOFSTEP]\nrw [\u2190 support_embDomain, h, support_single_ne_zero _ hb]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nl : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\na : \u03b2\nb : M\nhb : b \u2260 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\n\u22a2 \u2203 x, l = single x b \u2227 \u2191f x = a\n[PROOFSTEP]\nhave ha : a \u2208 Finset.map f l.support := by simp only [h_map_support, Finset.mem_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nl : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\na : \u03b2\nb : M\nhb : b \u2260 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\n\u22a2 a \u2208 map f l.support\n[PROOFSTEP]\nsimp only [h_map_support, Finset.mem_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nl : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\na : \u03b2\nb : M\nhb : b \u2260 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a \u2208 map f l.support\n\u22a2 \u2203 x, l = single x b \u2227 \u2191f x = a\n[PROOFSTEP]\nrcases Finset.mem_map.1 ha with \u27e8c, _hc\u2081, hc\u2082\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nl : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\na : \u03b2\nb : M\nhb : b \u2260 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a \u2208 map f l.support\nc : \u03b1\n_hc\u2081 : c \u2208 l.support\nhc\u2082 : \u2191f c = a\n\u22a2 \u2203 x, l = single x b \u2227 \u2191f x = a\n[PROOFSTEP]\nuse c\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nl : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\na : \u03b2\nb : M\nhb : b \u2260 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a \u2208 map f l.support\nc : \u03b1\n_hc\u2081 : c \u2208 l.support\nhc\u2082 : \u2191f c = a\n\u22a2 l = single c b \u2227 \u2191f c = a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nl : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\na : \u03b2\nb : M\nhb : b \u2260 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a \u2208 map f l.support\nc : \u03b1\n_hc\u2081 : c \u2208 l.support\nhc\u2082 : \u2191f c = a\n\u22a2 l = single c b\n[PROOFSTEP]\next d\n[GOAL]\ncase h.left.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nl : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\na : \u03b2\nb : M\nhb : b \u2260 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a \u2208 map f l.support\nc : \u03b1\n_hc\u2081 : c \u2208 l.support\nhc\u2082 : \u2191f c = a\nd : \u03b1\n\u22a2 \u2191l d = \u2191(single c b) d\n[PROOFSTEP]\nrw [\u2190 embDomain_apply f l, h]\n[GOAL]\ncase h.left.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nl : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\na : \u03b2\nb : M\nhb : b \u2260 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a \u2208 map f l.support\nc : \u03b1\n_hc\u2081 : c \u2208 l.support\nhc\u2082 : \u2191f c = a\nd : \u03b1\n\u22a2 \u2191(single a b) (\u2191f d) = \u2191(single c b) d\n[PROOFSTEP]\nby_cases h_cases : c = d\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nl : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\na : \u03b2\nb : M\nhb : b \u2260 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a \u2208 map f l.support\nc : \u03b1\n_hc\u2081 : c \u2208 l.support\nhc\u2082 : \u2191f c = a\nd : \u03b1\nh_cases : c = d\n\u22a2 \u2191(single a b) (\u2191f d) = \u2191(single c b) d\n[PROOFSTEP]\nsimp only [Eq.symm h_cases, hc\u2082, single_eq_same]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nl : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\na : \u03b2\nb : M\nhb : b \u2260 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a \u2208 map f l.support\nc : \u03b1\n_hc\u2081 : c \u2208 l.support\nhc\u2082 : \u2191f c = a\nd : \u03b1\nh_cases : \u00acc = d\n\u22a2 \u2191(single a b) (\u2191f d) = \u2191(single c b) d\n[PROOFSTEP]\nrw [single_apply, single_apply, if_neg, if_neg h_cases]\n[GOAL]\ncase neg.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nl : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\na : \u03b2\nb : M\nhb : b \u2260 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a \u2208 map f l.support\nc : \u03b1\n_hc\u2081 : c \u2208 l.support\nhc\u2082 : \u2191f c = a\nd : \u03b1\nh_cases : \u00acc = d\n\u22a2 \u00aca = \u2191f d\n[PROOFSTEP]\nby_contra hfd\n[GOAL]\ncase neg.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nl : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\na : \u03b2\nb : M\nhb : b \u2260 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a \u2208 map f l.support\nc : \u03b1\n_hc\u2081 : c \u2208 l.support\nhc\u2082 : \u2191f c = a\nd : \u03b1\nh_cases : \u00acc = d\nhfd : a = \u2191f d\n\u22a2 False\n[PROOFSTEP]\nexact h_cases (f.injective (hc\u2082.trans hfd))\n[GOAL]\ncase h.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nl : \u03b1 \u2192\u2080 M\nf : \u03b1 \u21aa \u03b2\na : \u03b2\nb : M\nhb : b \u2260 0\nh : embDomain f l = single a b\nh_map_support : map f l.support = {a}\nha : a \u2208 map f l.support\nc : \u03b1\n_hc\u2081 : c \u2208 l.support\nhc\u2082 : \u2191f c = a\n\u22a2 \u2191f c = a\n[PROOFSTEP]\nexact hc\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nm : M\n\u22a2 embDomain f (single a m) = single (\u2191f a) m\n[PROOFSTEP]\nclassical\next b\nby_cases h : b \u2208 Set.range f\n\u00b7 rcases h with \u27e8a', rfl\u27e9\n simp [single_apply]\n\u00b7 simp only [embDomain_notin_range, h, single_apply, not_false_iff]\n rw [if_neg]\n rintro rfl\n simp at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nm : M\n\u22a2 embDomain f (single a m) = single (\u2191f a) m\n[PROOFSTEP]\next b\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nm : M\nb : \u03b2\n\u22a2 \u2191(embDomain f (single a m)) b = \u2191(single (\u2191f a) m) b\n[PROOFSTEP]\nby_cases h : b \u2208 Set.range f\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nm : M\nb : \u03b2\nh : b \u2208 Set.range \u2191f\n\u22a2 \u2191(embDomain f (single a m)) b = \u2191(single (\u2191f a) m) b\n[PROOFSTEP]\nrcases h with \u27e8a', rfl\u27e9\n[GOAL]\ncase pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nm : M\na' : \u03b1\n\u22a2 \u2191(embDomain f (single a m)) (\u2191f a') = \u2191(single (\u2191f a) m) (\u2191f a')\n[PROOFSTEP]\nsimp [single_apply]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nm : M\nb : \u03b2\nh : \u00acb \u2208 Set.range \u2191f\n\u22a2 \u2191(embDomain f (single a m)) b = \u2191(single (\u2191f a) m) b\n[PROOFSTEP]\nsimp only [embDomain_notin_range, h, single_apply, not_false_iff]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nm : M\nb : \u03b2\nh : \u00acb \u2208 Set.range \u2191f\n\u22a2 0 = if \u2191f a = b then m else 0\n[PROOFSTEP]\nrw [if_neg]\n[GOAL]\ncase neg.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nm : M\nb : \u03b2\nh : \u00acb \u2208 Set.range \u2191f\n\u22a2 \u00ac\u2191f a = b\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase neg.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : Zero M\ninst\u271d : Zero N\nf : \u03b1 \u21aa \u03b2\na : \u03b1\nm : M\nh : \u00ac\u2191f a \u2208 Set.range \u2191f\n\u22a2 False\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH\u271d : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2192 N \u2192 P\nhf : f 0 0 = 0\ng\u2081 : \u03b1 \u2192\u2080 M\ng\u2082 : \u03b1 \u2192\u2080 N\na : \u03b1\nH : f (\u2191g\u2081 a) (\u2191g\u2082 a) \u2260 0\n\u22a2 a \u2208 g\u2081.support \u222a g\u2082.support\n[PROOFSTEP]\nclassical\nrw [mem_union, mem_support_iff, mem_support_iff, \u2190 not_and_or]\nrintro \u27e8h\u2081, h\u2082\u27e9; rw [h\u2081, h\u2082] at H ; exact H hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH\u271d : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2192 N \u2192 P\nhf : f 0 0 = 0\ng\u2081 : \u03b1 \u2192\u2080 M\ng\u2082 : \u03b1 \u2192\u2080 N\na : \u03b1\nH : f (\u2191g\u2081 a) (\u2191g\u2082 a) \u2260 0\n\u22a2 a \u2208 g\u2081.support \u222a g\u2082.support\n[PROOFSTEP]\nrw [mem_union, mem_support_iff, mem_support_iff, \u2190 not_and_or]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH\u271d : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2192 N \u2192 P\nhf : f 0 0 = 0\ng\u2081 : \u03b1 \u2192\u2080 M\ng\u2082 : \u03b1 \u2192\u2080 N\na : \u03b1\nH : f (\u2191g\u2081 a) (\u2191g\u2082 a) \u2260 0\n\u22a2 \u00ac(\u2191g\u2081 a = 0 \u2227 \u2191g\u2082 a = 0)\n[PROOFSTEP]\nrintro \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH\u271d : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2192 N \u2192 P\nhf : f 0 0 = 0\ng\u2081 : \u03b1 \u2192\u2080 M\ng\u2082 : \u03b1 \u2192\u2080 N\na : \u03b1\nH : f (\u2191g\u2081 a) (\u2191g\u2082 a) \u2260 0\nh\u2081 : \u2191g\u2081 a = 0\nh\u2082 : \u2191g\u2082 a = 0\n\u22a2 False\n[PROOFSTEP]\nrw [h\u2081, h\u2082] at H \n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH\u271d : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nf : M \u2192 N \u2192 P\nhf : f 0 0 = 0\ng\u2081 : \u03b1 \u2192\u2080 M\ng\u2082 : \u03b1 \u2192\u2080 N\na : \u03b1\nH : f 0 0 \u2260 0\nh\u2081 : \u2191g\u2081 a = 0\nh\u2082 : \u2191g\u2082 a = 0\n\u22a2 False\n[PROOFSTEP]\nexact H hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nD : DecidableEq \u03b1\nf : M \u2192 N \u2192 P\nhf : f 0 0 = 0\ng\u2081 : \u03b1 \u2192\u2080 M\ng\u2082 : \u03b1 \u2192\u2080 N\n\u22a2 (zipWith f hf g\u2081 g\u2082).support \u2286 g\u2081.support \u222a g\u2082.support\n[PROOFSTEP]\nrw [Subsingleton.elim D]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero N\ninst\u271d : Zero P\nD : DecidableEq \u03b1\nf : M \u2192 N \u2192 P\nhf : f 0 0 = 0\ng\u2081 : \u03b1 \u2192\u2080 M\ng\u2082 : \u03b1 \u2192\u2080 N\n\u22a2 (zipWith f hf g\u2081 g\u2082).support \u2286 g\u2081.support \u222a g\u2082.support\n[PROOFSTEP]\nexact support_onFinset_subset\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddZeroClass M\ninst\u271d : DecidableEq \u03b1\ng\u2081 g\u2082 : \u03b1 \u2192\u2080 M\nh : Disjoint g\u2081.support g\u2082.support\na : \u03b1\nha\u271d : a \u2208 g\u2081.support \u222a g\u2082.support\nha : a \u2208 g\u2081.support\n\u22a2 a \u2208 (g\u2081 + g\u2082).support\n[PROOFSTEP]\nhave : a \u2209 g\u2082.support := disjoint_left.1 h ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddZeroClass M\ninst\u271d : DecidableEq \u03b1\ng\u2081 g\u2082 : \u03b1 \u2192\u2080 M\nh : Disjoint g\u2081.support g\u2082.support\na : \u03b1\nha\u271d : a \u2208 g\u2081.support \u222a g\u2082.support\nha : a \u2208 g\u2081.support\nthis : \u00aca \u2208 g\u2082.support\n\u22a2 a \u2208 (g\u2081 + g\u2082).support\n[PROOFSTEP]\nsimp only [mem_support_iff, not_not] at *\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddZeroClass M\ninst\u271d : DecidableEq \u03b1\ng\u2081 g\u2082 : \u03b1 \u2192\u2080 M\nh : Disjoint g\u2081.support g\u2082.support\na : \u03b1\nha\u271d : a \u2208 g\u2081.support \u222a g\u2082.support\nha : \u2191g\u2081 a \u2260 0\nthis : \u2191g\u2082 a = 0\n\u22a2 \u2191(g\u2081 + g\u2082) a \u2260 0\n[PROOFSTEP]\nsimpa only [add_apply, this, add_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddZeroClass M\ninst\u271d : DecidableEq \u03b1\ng\u2081 g\u2082 : \u03b1 \u2192\u2080 M\nh : Disjoint g\u2081.support g\u2082.support\na : \u03b1\nha\u271d : a \u2208 g\u2081.support \u222a g\u2082.support\nha : a \u2208 g\u2082.support\n\u22a2 a \u2208 (g\u2081 + g\u2082).support\n[PROOFSTEP]\nhave : a \u2209 g\u2081.support := disjoint_right.1 h ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddZeroClass M\ninst\u271d : DecidableEq \u03b1\ng\u2081 g\u2082 : \u03b1 \u2192\u2080 M\nh : Disjoint g\u2081.support g\u2082.support\na : \u03b1\nha\u271d : a \u2208 g\u2081.support \u222a g\u2082.support\nha : a \u2208 g\u2082.support\nthis : \u00aca \u2208 g\u2081.support\n\u22a2 a \u2208 (g\u2081 + g\u2082).support\n[PROOFSTEP]\nsimp only [mem_support_iff, not_not] at *\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddZeroClass M\ninst\u271d : DecidableEq \u03b1\ng\u2081 g\u2082 : \u03b1 \u2192\u2080 M\nh : Disjoint g\u2081.support g\u2082.support\na : \u03b1\nha\u271d : a \u2208 g\u2081.support \u222a g\u2082.support\nha : \u2191g\u2082 a \u2260 0\nthis : \u2191g\u2081 a = 0\n\u22a2 \u2191(g\u2081 + g\u2082) a \u2260 0\n[PROOFSTEP]\nsimpa only [add_apply, this, zero_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\na : \u03b1\nb\u2081 b\u2082 : M\na' : \u03b1\n\u22a2 \u2191(single a (b\u2081 + b\u2082)) a' = \u2191(single a b\u2081 + single a b\u2082) a'\n[PROOFSTEP]\nby_cases h : a = a'\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\na : \u03b1\nb\u2081 b\u2082 : M\na' : \u03b1\nh : a = a'\n\u22a2 \u2191(single a (b\u2081 + b\u2082)) a' = \u2191(single a b\u2081 + single a b\u2082) a'\n[PROOFSTEP]\nrw [h, add_apply, single_eq_same, single_eq_same, single_eq_same]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\na : \u03b1\nb\u2081 b\u2082 : M\na' : \u03b1\nh : \u00aca = a'\n\u22a2 \u2191(single a (b\u2081 + b\u2082)) a' = \u2191(single a b\u2081 + single a b\u2082) a'\n[PROOFSTEP]\nrw [add_apply, single_eq_of_ne h, single_eq_of_ne h, single_eq_of_ne h, zero_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\n\u22a2 update f a b = single a b + erase a f\n[PROOFSTEP]\nclassical\next j\nrcases eq_or_ne a j with (rfl | h)\n\u00b7 simp\n\u00b7 simp [Function.update_noteq h.symm, single_apply, h, erase_ne, h.symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\n\u22a2 update f a b = single a b + erase a f\n[PROOFSTEP]\next j\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\nj : \u03b1\n\u22a2 \u2191(update f a b) j = \u2191(single a b + erase a f) j\n[PROOFSTEP]\nrcases eq_or_ne a j with (rfl | h)\n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\n\u22a2 \u2191(update f a b) a = \u2191(single a b + erase a f) a\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\nj : \u03b1\nh : a \u2260 j\n\u22a2 \u2191(update f a b) j = \u2191(single a b + erase a f) j\n[PROOFSTEP]\nsimp [Function.update_noteq h.symm, single_apply, h, erase_ne, h.symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\n\u22a2 update f a b = erase a f + single a b\n[PROOFSTEP]\nclassical\next j\nrcases eq_or_ne a j with (rfl | h)\n\u00b7 simp\n\u00b7 simp [Function.update_noteq h.symm, single_apply, h, erase_ne, h.symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\n\u22a2 update f a b = erase a f + single a b\n[PROOFSTEP]\next j\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\nj : \u03b1\n\u22a2 \u2191(update f a b) j = \u2191(erase a f + single a b) j\n[PROOFSTEP]\nrcases eq_or_ne a j with (rfl | h)\n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\n\u22a2 \u2191(update f a b) a = \u2191(erase a f + single a b) a\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\nf : \u03b1 \u2192\u2080 M\na : \u03b1\nb : M\nj : \u03b1\nh : a \u2260 j\n\u22a2 \u2191(update f a b) j = \u2191(erase a f + single a b) j\n[PROOFSTEP]\nsimp [Function.update_noteq h.symm, single_apply, h, erase_ne, h.symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\na : \u03b1\nf : \u03b1 \u2192\u2080 M\n\u22a2 single a (\u2191f a) + erase a f = f\n[PROOFSTEP]\nrw [\u2190 update_eq_single_add_erase, update_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\na : \u03b1\nf : \u03b1 \u2192\u2080 M\n\u22a2 erase a f + single a (\u2191f a) = f\n[PROOFSTEP]\nrw [\u2190 update_eq_erase_add_single, update_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\na : \u03b1\nf f' : \u03b1 \u2192\u2080 M\n\u22a2 erase a (f + f') = erase a f + erase a f'\n[PROOFSTEP]\next s\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\na : \u03b1\nf f' : \u03b1 \u2192\u2080 M\ns : \u03b1\n\u22a2 \u2191(erase a (f + f')) s = \u2191(erase a f + erase a f') s\n[PROOFSTEP]\nby_cases hs : s = a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\na : \u03b1\nf f' : \u03b1 \u2192\u2080 M\ns : \u03b1\nhs : s = a\n\u22a2 \u2191(erase a (f + f')) s = \u2191(erase a f + erase a f') s\n[PROOFSTEP]\nrw [hs, add_apply, erase_same, erase_same, erase_same, add_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\na : \u03b1\nf f' : \u03b1 \u2192\u2080 M\ns : \u03b1\nhs : \u00acs = a\n\u22a2 \u2191(erase a (f + f')) s = \u2191(erase a f + erase a f') s\n[PROOFSTEP]\nrw [add_apply, erase_ne hs, erase_ne hs, erase_ne hs, add_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (single a b + f)\ns : Finset \u03b1\nf : \u03b1 \u2192\u2080 M\nhf : f.support = \u2205\n\u22a2 p f\n[PROOFSTEP]\nrwa [support_eq_empty.1 hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (single a b + f)\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f\nf : \u03b1 \u2192\u2080 M\nhf : f.support = cons a s has\n\u22a2 p f\n[PROOFSTEP]\nsuffices p (single a (f a) + f.erase a) by rwa [single_add_erase] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (single a b + f)\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f\nf : \u03b1 \u2192\u2080 M\nhf : f.support = cons a s has\nthis : p (single a (\u2191f a) + erase a f)\n\u22a2 p f\n[PROOFSTEP]\nrwa [single_add_erase] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (single a b + f)\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f\nf : \u03b1 \u2192\u2080 M\nhf : f.support = cons a s has\n\u22a2 p (single a (\u2191f a) + erase a f)\n[PROOFSTEP]\nclassical\napply ha\n\u00b7 rw [support_erase, mem_erase]\n exact fun H => H.1 rfl\n\u00b7 rw [\u2190 mem_support_iff, hf]\n exact mem_cons_self _ _\n\u00b7 apply ih _ _\n rw [support_erase, hf, Finset.erase_cons]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (single a b + f)\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f\nf : \u03b1 \u2192\u2080 M\nhf : f.support = cons a s has\n\u22a2 p (single a (\u2191f a) + erase a f)\n[PROOFSTEP]\napply ha\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (single a b + f)\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f\nf : \u03b1 \u2192\u2080 M\nhf : f.support = cons a s has\n\u22a2 \u00aca \u2208 (erase a f).support\n[PROOFSTEP]\nrw [support_erase, mem_erase]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (single a b + f)\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f\nf : \u03b1 \u2192\u2080 M\nhf : f.support = cons a s has\n\u22a2 \u00ac(a \u2260 a \u2227 a \u2208 f.support)\n[PROOFSTEP]\nexact fun H => H.1 rfl\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (single a b + f)\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f\nf : \u03b1 \u2192\u2080 M\nhf : f.support = cons a s has\n\u22a2 \u2191f a \u2260 0\n[PROOFSTEP]\nrw [\u2190 mem_support_iff, hf]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (single a b + f)\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f\nf : \u03b1 \u2192\u2080 M\nhf : f.support = cons a s has\n\u22a2 a \u2208 cons a s has\n[PROOFSTEP]\nexact mem_cons_self _ _\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (single a b + f)\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f\nf : \u03b1 \u2192\u2080 M\nhf : f.support = cons a s has\n\u22a2 p (erase a f)\n[PROOFSTEP]\napply ih _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (single a b + f)\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f\nf : \u03b1 \u2192\u2080 M\nhf : f.support = cons a s has\n\u22a2 (erase a f).support = s\n[PROOFSTEP]\nrw [support_erase, hf, Finset.erase_cons]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (f + single a b)\ns : Finset \u03b1\nf : \u03b1 \u2192\u2080 M\nhf : f.support = \u2205\n\u22a2 p f\n[PROOFSTEP]\nrwa [support_eq_empty.1 hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (f + single a b)\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f\nf : \u03b1 \u2192\u2080 M\nhf : f.support = cons a s has\n\u22a2 p f\n[PROOFSTEP]\nsuffices p (f.erase a + single a (f a)) by rwa [erase_add_single] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (f + single a b)\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f\nf : \u03b1 \u2192\u2080 M\nhf : f.support = cons a s has\nthis : p (erase a f + single a (\u2191f a))\n\u22a2 p f\n[PROOFSTEP]\nrwa [erase_add_single] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (f + single a b)\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f\nf : \u03b1 \u2192\u2080 M\nhf : f.support = cons a s has\n\u22a2 p (erase a f + single a (\u2191f a))\n[PROOFSTEP]\nclassical\napply ha\n\u00b7 rw [support_erase, mem_erase]\n exact fun H => H.1 rfl\n\u00b7 rw [\u2190 mem_support_iff, hf]\n exact mem_cons_self _ _\n\u00b7 apply ih _ _\n rw [support_erase, hf, Finset.erase_cons]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (f + single a b)\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f\nf : \u03b1 \u2192\u2080 M\nhf : f.support = cons a s has\n\u22a2 p (erase a f + single a (\u2191f a))\n[PROOFSTEP]\napply ha\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (f + single a b)\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f\nf : \u03b1 \u2192\u2080 M\nhf : f.support = cons a s has\n\u22a2 \u00aca \u2208 (erase a f).support\n[PROOFSTEP]\nrw [support_erase, mem_erase]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (f + single a b)\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f\nf : \u03b1 \u2192\u2080 M\nhf : f.support = cons a s has\n\u22a2 \u00ac(a \u2260 a \u2227 a \u2208 f.support)\n[PROOFSTEP]\nexact fun H => H.1 rfl\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (f + single a b)\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f\nf : \u03b1 \u2192\u2080 M\nhf : f.support = cons a s has\n\u22a2 \u2191f a \u2260 0\n[PROOFSTEP]\nrw [\u2190 mem_support_iff, hf]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (f + single a b)\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f\nf : \u03b1 \u2192\u2080 M\nhf : f.support = cons a s has\n\u22a2 a \u2208 cons a s has\n[PROOFSTEP]\nexact mem_cons_self _ _\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (f + single a b)\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f\nf : \u03b1 \u2192\u2080 M\nhf : f.support = cons a s has\n\u22a2 p (erase a f)\n[PROOFSTEP]\napply ih _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\np : (\u03b1 \u2192\u2080 M) \u2192 Prop\nf\u271d : \u03b1 \u2192\u2080 M\nh0 : p 0\nha : \u2200 (a : \u03b1) (b : M) (f : \u03b1 \u2192\u2080 M), \u00aca \u2208 f.support \u2192 b \u2260 0 \u2192 p f \u2192 p (f + single a b)\ns\u271d : Finset \u03b1\na : \u03b1\ns : Finset \u03b1\nhas : \u00aca \u2208 s\nih : \u2200 (f : \u03b1 \u2192\u2080 M), f.support = s \u2192 p f\nf : \u03b1 \u2192\u2080 M\nhf : f.support = cons a s has\n\u22a2 (erase a f).support = s\n[PROOFSTEP]\nrw [support_erase, hf, Finset.erase_cons]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH\u271d : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddZeroClass M\ninst\u271d : AddZeroClass N\nf g : (\u03b1 \u2192\u2080 M) \u2192+ N\nH : \u2200 (x : \u03b1) (y : M), \u2191f (single x y) = \u2191g (single x y)\n\u22a2 f = g\n[PROOFSTEP]\nrefine' AddMonoidHom.eq_of_eqOn_denseM add_closure_setOf_eq_single _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH\u271d : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddZeroClass M\ninst\u271d : AddZeroClass N\nf g : (\u03b1 \u2192\u2080 M) \u2192+ N\nH : \u2200 (x : \u03b1) (y : M), \u2191f (single x y) = \u2191g (single x y)\n\u22a2 Set.EqOn \u2191f \u2191g {f | \u2203 a b, f = single a b}\n[PROOFSTEP]\nrintro _ \u27e8x, y, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH\u271d : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddZeroClass M\ninst\u271d : AddZeroClass N\nf g : (\u03b1 \u2192\u2080 M) \u2192+ N\nH : \u2200 (x : \u03b1) (y : M), \u2191f (single x y) = \u2191g (single x y)\nx : \u03b1\ny : M\n\u22a2 \u2191f (single x y) = \u2191g (single x y)\n[PROOFSTEP]\napply H\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH\u271d : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddZeroClass M\ninst\u271d : MulOneClass N\nf g : Multiplicative (\u03b1 \u2192\u2080 M) \u2192* N\nH : \u2200 (x : \u03b1) (y : M), \u2191f (\u2191Multiplicative.ofAdd (single x y)) = \u2191g (\u2191Multiplicative.ofAdd (single x y))\n\u22a2 f = g\n[PROOFSTEP]\nhave := @addHom_ext \u03b1 M (Additive N) _ _ (MonoidHom.toAdditive'' f) (MonoidHom.toAdditive'' g) H\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH\u271d : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddZeroClass M\ninst\u271d : MulOneClass N\nf g : Multiplicative (\u03b1 \u2192\u2080 M) \u2192* N\nH : \u2200 (x : \u03b1) (y : M), \u2191f (\u2191Multiplicative.ofAdd (single x y)) = \u2191g (\u2191Multiplicative.ofAdd (single x y))\nthis : \u2191MonoidHom.toAdditive'' f = \u2191MonoidHom.toAdditive'' g\n\u22a2 f = g\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH\u271d : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddZeroClass M\ninst\u271d : MulOneClass N\nf g : Multiplicative (\u03b1 \u2192\u2080 M) \u2192* N\nH : \u2200 (x : \u03b1) (y : M), \u2191f (\u2191Multiplicative.ofAdd (single x y)) = \u2191g (\u2191Multiplicative.ofAdd (single x y))\nthis : \u2191MonoidHom.toAdditive'' f = \u2191MonoidHom.toAdditive'' g\nx\u271d : Multiplicative (\u03b1 \u2192\u2080 M)\n\u22a2 \u2191f x\u271d = \u2191g x\u271d\n[PROOFSTEP]\nrw [FunLike.ext_iff] at this \n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH\u271d : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddZeroClass M\ninst\u271d : MulOneClass N\nf g : Multiplicative (\u03b1 \u2192\u2080 M) \u2192* N\nH : \u2200 (x : \u03b1) (y : M), \u2191f (\u2191Multiplicative.ofAdd (single x y)) = \u2191g (\u2191Multiplicative.ofAdd (single x y))\nthis\u271d : \u2191MonoidHom.toAdditive'' f = \u2191MonoidHom.toAdditive'' g\nthis : \u2200 (x : \u03b1 \u2192\u2080 M), \u2191(\u2191MonoidHom.toAdditive'' f) x = \u2191(\u2191MonoidHom.toAdditive'' g) x\nx\u271d : Multiplicative (\u03b1 \u2192\u2080 M)\n\u22a2 \u2191f x\u271d = \u2191g x\u271d\n[PROOFSTEP]\napply this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : AddZeroClass M\ninst\u271d : AddZeroClass N\nf : M \u2192 N\nhf : f 0 = 0\nhf' : \u2200 (x y : M), f (x + y) = f x + f y\nv\u2081 v\u2082 : \u03b1 \u2192\u2080 M\nx\u271d : \u03b1\n\u22a2 \u2191(mapRange f hf (v\u2081 + v\u2082)) x\u271d = \u2191(mapRange f hf v\u2081 + mapRange f hf v\u2082) x\u271d\n[PROOFSTEP]\nsimp only [hf', add_apply, mapRange_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\nf : \u03b1 \u21aa \u03b2\n\u22a2 (fun v => embDomain f v) 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\nf : \u03b1 \u21aa \u03b2\nv w : \u03b1 \u2192\u2080 M\n\u22a2 ZeroHom.toFun { toFun := fun v => embDomain f v, map_zero' := (_ : 0 = 0) } (v + w) =\n ZeroHom.toFun { toFun := fun v => embDomain f v, map_zero' := (_ : 0 = 0) } v +\n ZeroHom.toFun { toFun := fun v => embDomain f v, map_zero' := (_ : 0 = 0) } w\n[PROOFSTEP]\next b\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\nf : \u03b1 \u21aa \u03b2\nv w : \u03b1 \u2192\u2080 M\nb : \u03b2\n\u22a2 \u2191(ZeroHom.toFun { toFun := fun v => embDomain f v, map_zero' := (_ : 0 = 0) } (v + w)) b =\n \u2191(ZeroHom.toFun { toFun := fun v => embDomain f v, map_zero' := (_ : 0 = 0) } v +\n ZeroHom.toFun { toFun := fun v => embDomain f v, map_zero' := (_ : 0 = 0) } w)\n b\n[PROOFSTEP]\nby_cases h : b \u2208 Set.range f\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\nf : \u03b1 \u21aa \u03b2\nv w : \u03b1 \u2192\u2080 M\nb : \u03b2\nh : b \u2208 Set.range \u2191f\n\u22a2 \u2191(ZeroHom.toFun { toFun := fun v => embDomain f v, map_zero' := (_ : 0 = 0) } (v + w)) b =\n \u2191(ZeroHom.toFun { toFun := fun v => embDomain f v, map_zero' := (_ : 0 = 0) } v +\n ZeroHom.toFun { toFun := fun v => embDomain f v, map_zero' := (_ : 0 = 0) } w)\n b\n[PROOFSTEP]\nrcases h with \u27e8a, rfl\u27e9\n[GOAL]\ncase pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\nf : \u03b1 \u21aa \u03b2\nv w : \u03b1 \u2192\u2080 M\na : \u03b1\n\u22a2 \u2191(ZeroHom.toFun { toFun := fun v => embDomain f v, map_zero' := (_ : 0 = 0) } (v + w)) (\u2191f a) =\n \u2191(ZeroHom.toFun { toFun := fun v => embDomain f v, map_zero' := (_ : 0 = 0) } v +\n ZeroHom.toFun { toFun := fun v => embDomain f v, map_zero' := (_ : 0 = 0) } w)\n (\u2191f a)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddZeroClass M\nf : \u03b1 \u21aa \u03b2\nv w : \u03b1 \u2192\u2080 M\nb : \u03b2\nh : \u00acb \u2208 Set.range \u2191f\n\u22a2 \u2191(ZeroHom.toFun { toFun := fun v => embDomain f v, map_zero' := (_ : 0 = 0) } (v + w)) b =\n \u2191(ZeroHom.toFun { toFun := fun v => embDomain f v, map_zero' := (_ : 0 = 0) } v +\n ZeroHom.toFun { toFun := fun v => embDomain f v, map_zero' := (_ : 0 = 0) } w)\n b\n[PROOFSTEP]\nsimp only [Set.mem_range, not_exists, coe_add, Pi.add_apply, embDomain_notin_range _ _ _ h, add_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : NegZeroClass G\ninst\u271d : NegZeroClass H\nf : G \u2192 H\nhf : f 0 = 0\nhf' : \u2200 (x : G), f (-x) = -f x\nv : \u03b1 \u2192\u2080 G\nx\u271d : \u03b1\n\u22a2 \u2191(mapRange f hf (-v)) x\u271d = \u2191(-mapRange f hf v) x\u271d\n[PROOFSTEP]\nsimp only [hf', neg_apply, mapRange_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : SubNegZeroMonoid G\ninst\u271d : SubNegZeroMonoid H\nf : G \u2192 H\nhf : f 0 = 0\nhf' : \u2200 (x y : G), f (x - y) = f x - f y\nv\u2081 v\u2082 : \u03b1 \u2192\u2080 G\nx\u271d : \u03b1\n\u22a2 \u2191(mapRange f hf (v\u2081 - v\u2082)) x\u271d = \u2191(mapRange f hf v\u2081 - mapRange f hf v\u2082) x\u271d\n[PROOFSTEP]\nsimp only [hf', sub_apply, mapRange_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nk l m n : \u03b1\nu v : M\nhu : u \u2260 0\nhv : v \u2260 0\n\u22a2 single k u + single l v = single m u + single n v \u2194 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u + v = 0 \u2227 k = l \u2227 m = n\n[PROOFSTEP]\nclassical\nsimp_rw [FunLike.ext_iff, coe_add, single_eq_pi_single, \u2190 funext_iff]\nexact Pi.single_add_single_eq_single_add_single hu hv\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nk l m n : \u03b1\nu v : M\nhu : u \u2260 0\nhv : v \u2260 0\n\u22a2 single k u + single l v = single m u + single n v \u2194 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u + v = 0 \u2227 k = l \u2227 m = n\n[PROOFSTEP]\nsimp_rw [FunLike.ext_iff, coe_add, single_eq_pi_single, \u2190 funext_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddCommMonoid M\nk l m n : \u03b1\nu v : M\nhu : u \u2260 0\nhv : v \u2260 0\n\u22a2 ((fun a => (Pi.single k u + Pi.single l v) a) = fun a => (Pi.single m u + Pi.single n v) a) \u2194\n k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u + v = 0 \u2227 k = l \u2227 m = n\n[PROOFSTEP]\nexact Pi.single_add_single_eq_single_add_single hu hv\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : AddGroup G\nf g : \u03b1 \u2192\u2080 G\n\u22a2 (f - g).support \u2286 f.support \u222a g.support\n[PROOFSTEP]\nrw [sub_eq_add_neg, \u2190 support_neg g]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : AddGroup G\nf g : \u03b1 \u2192\u2080 G\n\u22a2 (f + -g).support \u2286 f.support \u222a (-g).support\n[PROOFSTEP]\nexact support_add\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddGroup G\nf : \u03b1 \u2192\u2080 G\na : \u03b1\n\u22a2 erase a f = f - single a (\u2191f a)\n[PROOFSTEP]\next a'\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddGroup G\nf : \u03b1 \u2192\u2080 G\na a' : \u03b1\n\u22a2 \u2191(erase a f) a' = \u2191(f - single a (\u2191f a)) a'\n[PROOFSTEP]\nrcases eq_or_ne a a' with (rfl | h)\n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddGroup G\nf : \u03b1 \u2192\u2080 G\na : \u03b1\n\u22a2 \u2191(erase a f) a = \u2191(f - single a (\u2191f a)) a\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddGroup G\nf : \u03b1 \u2192\u2080 G\na a' : \u03b1\nh : a \u2260 a'\n\u22a2 \u2191(erase a f) a' = \u2191(f - single a (\u2191f a)) a'\n[PROOFSTEP]\nsimp [erase_ne h.symm, single_eq_of_ne h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : AddGroup G\nf : \u03b1 \u2192\u2080 G\na : \u03b1\nb : G\n\u22a2 update f a b = f - single a (\u2191f a) + single a b\n[PROOFSTEP]\nrw [update_eq_erase_add_single, erase_eq_sub_single]\n", "meta": {"mathlib_filename": "Mathlib.Data.Finsupp.Defs", "llama_tokens": 66341, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.44552952031526044, "lm_q2_score": 0.0297600957745801, "lm_q1q2_score": 0.01325900119498488}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX : C\nf g : MonoOver X\n\u22a2 \u2200 (a b : f \u27f6 g), a = b\n[PROOFSTEP]\nintro h\u2081 h\u2082\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX : C\nf g : MonoOver X\nh\u2081 h\u2082 : f \u27f6 g\n\u22a2 h\u2081 = h\u2082\n[PROOFSTEP]\napply Over.OverMorphism.ext\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX : C\nf g : MonoOver X\nh\u2081 h\u2082 : f \u27f6 g\n\u22a2 h\u2081.left = h\u2082.left\n[PROOFSTEP]\nerw [\u2190 cancel_mono g.arrow, Over.w h\u2081, Over.w h\u2082]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nf g : MonoOver X\nh : f.obj.left \u2245 g.obj.left\nw : autoParam (h.hom \u226b arrow g = arrow f) _auto\u271d\n\u22a2 h.inv \u226b arrow f = arrow g\n[PROOFSTEP]\nrw [h.inv_comp_eq, w]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasPullbacks C\nf : X \u27f6 Y\ng : MonoOver Y\n\u22a2 Mono ((Over.pullback f).obj ((forget Y).obj g)).hom\n[PROOFSTEP]\nhaveI : Mono ((forget Y).obj g).hom := (inferInstance : Mono g.arrow)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasPullbacks C\nf : X \u27f6 Y\ng : MonoOver Y\nthis : Mono ((forget Y).obj g).hom\n\u22a2 Mono ((Over.pullback f).obj ((forget Y).obj g)).hom\n[PROOFSTEP]\napply pullback.snd_of_mono\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nf : X \u27f6 Y\ninst\u271d : Mono f\ng : MonoOver X\n\u22a2 Mono ((Over.map f).obj ((forget X).obj g)).hom\n[PROOFSTEP]\napply mono_comp g.arrow f\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nf : X \u27f6 Y\ninst\u271d : Mono f\ng h : MonoOver X\ne : (map f).obj g \u27f6 (map f).obj h\n\u22a2 g \u27f6 h\n[PROOFSTEP]\nrefine' homMk e.left _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nf : X \u27f6 Y\ninst\u271d : Mono f\ng h : MonoOver X\ne : (map f).obj g \u27f6 (map f).obj h\n\u22a2 e.left \u226b arrow h = arrow g\n[PROOFSTEP]\nrw [\u2190 cancel_mono f, assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nf : X \u27f6 Y\ninst\u271d : Mono f\ng h : MonoOver X\ne : (map f).obj g \u27f6 (map f).obj h\n\u22a2 e.left \u226b arrow h \u226b f = arrow g \u226b f\n[PROOFSTEP]\napply w e\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nA B : C\ne : A \u2245 B\n\u22a2 map (e.hom \u226b e.inv) = map (\ud835\udfd9 A)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nA B : C\ne : A \u2245 B\n\u22a2 map (e.inv \u226b e.hom) = map (\ud835\udfd9 B)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : C \u224c D\nf : MonoOver X\n\u22a2 Mono ((Over.post e.functor).obj ((forget X).obj f)).hom\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : C \u224c D\nf : MonoOver X\n\u22a2 Mono (e.functor.map (arrow f))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : C \u224c D\nf : MonoOver (e.functor.obj X)\n\u22a2 Mono ((Over.post e.inverse).obj ((forget (e.functor.obj X)).obj f)).hom\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : C \u224c D\nf : MonoOver (e.functor.obj X)\n\u22a2 Mono (e.inverse.map (arrow f))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasImages C\nf g : Over X\nk : f \u27f6 g\n\u22a2 (fun f => imageMonoOver f.hom) f \u27f6 (fun f => imageMonoOver f.hom) g\n[PROOFSTEP]\napply (forget X).preimage _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasImages C\nf g : Over X\nk : f \u27f6 g\n\u22a2 (forget X).obj ((fun f => imageMonoOver f.hom) f) \u27f6 (forget X).obj ((fun f => imageMonoOver f.hom) g)\n[PROOFSTEP]\napply Over.homMk _ _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasImages C\nf g : Over X\nk : f \u27f6 g\n\u22a2 ((forget X).obj ((fun f => imageMonoOver f.hom) f)).left \u27f6 ((forget X).obj ((fun f => imageMonoOver f.hom) g)).left\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasImages C\nf g : Over X\nk : f \u27f6 g\n\u22a2 ?m.186308 \u226b ((forget X).obj ((fun f => imageMonoOver f.hom) g)).hom =\n ((forget X).obj ((fun f => imageMonoOver f.hom) f)).hom\n[PROOFSTEP]\nrefine'\n image.lift\n { I := Limits.image _\n m := image.\u03b9 g.hom\n e := k.left \u226b factorThruImage g.hom }\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasImages C\nf g : Over X\nk : f \u27f6 g\n\u22a2 image.lift (MonoFactorisation.mk (Limits.image g.hom) (image.\u03b9 g.hom) (k.left \u226b factorThruImage g.hom)) \u226b\n ((forget X).obj ((fun f => imageMonoOver f.hom) g)).hom =\n ((forget X).obj ((fun f => imageMonoOver f.hom) f)).hom\n[PROOFSTEP]\napply image.lift_fac\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasImages C\nf : Over X\ng : MonoOver X\nk : image.obj f \u27f6 g\n\u22a2 f \u27f6 (forget X).obj g\n[PROOFSTEP]\napply Over.homMk (factorThruImage f.hom \u226b k.left) _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasImages C\nf : Over X\ng : MonoOver X\nk : image.obj f \u27f6 g\n\u22a2 (factorThruImage f.hom \u226b k.left) \u226b g.obj.hom = f.hom\n[PROOFSTEP]\nchange (factorThruImage f.hom \u226b k.left) \u226b _ = f.hom\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasImages C\nf : Over X\ng : MonoOver X\nk : image.obj f \u27f6 g\n\u22a2 (factorThruImage f.hom \u226b k.left) \u226b g.obj.hom = f.hom\n[PROOFSTEP]\nrw [assoc, Over.w k]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasImages C\nf : Over X\ng : MonoOver X\nk : image.obj f \u27f6 g\n\u22a2 factorThruImage f.hom \u226b (image.obj f).obj.hom = f.hom\n[PROOFSTEP]\napply image.fac\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasImages C\nf : Over X\ng : MonoOver X\nk : f \u27f6 (forget X).obj g\n\u22a2 image.obj f \u27f6 g\n[PROOFSTEP]\nrefine' Over.homMk _ _\n[GOAL]\ncase refine'_1\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasImages C\nf : Over X\ng : MonoOver X\nk : f \u27f6 (forget X).obj g\n\u22a2 (image.obj f).obj.left \u27f6 g.obj.left\ncase refine'_2\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasImages C\nf : Over X\ng : MonoOver X\nk : f \u27f6 (forget X).obj g\n\u22a2 ?refine'_1 \u226b g.obj.hom = (image.obj f).obj.hom\n[PROOFSTEP]\nrefine'\n image.lift\n { I := g.obj.left\n m := g.arrow\n e := k.left\n fac := Over.w k }\n[GOAL]\ncase refine'_2\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasImages C\nf : Over X\ng : MonoOver X\nk : f \u27f6 (forget X).obj g\n\u22a2 image.lift (MonoFactorisation.mk g.obj.left (arrow g) k.left) \u226b g.obj.hom = (image.obj f).obj.hom\n[PROOFSTEP]\napply image.lift_fac\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasImages C\nf : Over X\ng : MonoOver X\nk : f \u27f6 (forget X).obj g\n\u22a2 (fun k => Over.homMk (factorThruImage f.hom \u226b k.left))\n ((fun k => Over.homMk (image.lift (MonoFactorisation.mk g.obj.left (arrow g) k.left))) k) =\n k\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasImages C\nf : Over X\ng : MonoOver X\nk : f \u27f6 (forget X).obj g\n\u22a2 ((fun k => Over.homMk (factorThruImage f.hom \u226b k.left))\n ((fun k => Over.homMk (image.lift (MonoFactorisation.mk g.obj.left (arrow g) k.left))) k)).left =\n k.left\n[PROOFSTEP]\nchange factorThruImage _ \u226b image.lift _ = _\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasImages C\nf : Over X\ng : MonoOver X\nk : f \u27f6 (forget X).obj g\n\u22a2 factorThruImage f.hom \u226b image.lift (MonoFactorisation.mk g.obj.left (arrow g) k.left) = k.left\n[PROOFSTEP]\nrw [\u2190 cancel_mono g.arrow, assoc, image.lift_fac, image.fac f.hom]\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasImages C\nf : Over X\ng : MonoOver X\nk : f \u27f6 (forget X).obj g\n\u22a2 f.hom = k.left \u226b arrow g\n[PROOFSTEP]\nexact (Over.w k).symm\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : HasImages C\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 (X_1 : MonoOver X) \u2192 (exists f).obj X_1 \u2245 (map f).obj X_1\n[PROOFSTEP]\nintro Z\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX Y Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : HasImages C\nf : X \u27f6 Y\ninst\u271d : Mono f\nZ : MonoOver X\n\u22a2 (exists f).obj Z \u2245 (map f).obj Z\n[PROOFSTEP]\nsuffices : (forget _).obj ((exists f).obj Z) \u2245 (forget _).obj ((map f).obj Z)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX Y Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : HasImages C\nf : X \u27f6 Y\ninst\u271d : Mono f\nZ : MonoOver X\nthis : (forget Y).obj ((exists f).obj Z) \u2245 (forget Y).obj ((map f).obj Z)\n\u22a2 (exists f).obj Z \u2245 (map f).obj Z\ncase this\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX Y Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : HasImages C\nf : X \u27f6 Y\ninst\u271d : Mono f\nZ : MonoOver X\n\u22a2 (forget Y).obj ((exists f).obj Z) \u2245 (forget Y).obj ((map f).obj Z)\n[PROOFSTEP]\napply (forget _).preimageIso this\n[GOAL]\ncase this\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX Y Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : HasImages C\nf : X \u27f6 Y\ninst\u271d : Mono f\nZ : MonoOver X\n\u22a2 (forget Y).obj ((exists f).obj Z) \u2245 (forget Y).obj ((map f).obj Z)\n[PROOFSTEP]\napply Over.isoMk _ _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX Y Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : HasImages C\nf : X \u27f6 Y\ninst\u271d : Mono f\nZ : MonoOver X\n\u22a2 ((forget Y).obj ((exists f).obj Z)).left \u2245 ((forget Y).obj ((map f).obj Z)).left\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX Y Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : HasImages C\nf : X \u27f6 Y\ninst\u271d : Mono f\nZ : MonoOver X\n\u22a2 ?m.226355.hom \u226b ((forget Y).obj ((map f).obj Z)).hom = ((forget Y).obj ((exists f).obj Z)).hom\n[PROOFSTEP]\napply imageMonoIsoSource (Z.arrow \u226b f)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX Y Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : HasImages C\nf : X \u27f6 Y\ninst\u271d : Mono f\nZ : MonoOver X\n\u22a2 (imageMonoIsoSource (arrow Z \u226b f)).hom \u226b ((forget Y).obj ((map f).obj Z)).hom =\n ((forget Y).obj ((exists f).obj Z)).hom\n[PROOFSTEP]\napply imageMonoIsoSource_hom_self\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Subobject.MonoOver", "llama_tokens": 5525, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.44939263446475963, "lm_q2_score": 0.029312232553284526, "lm_q1q2_score": 0.013172701409164222}} {"text": "[GOAL]\n\u03b1 : Sort u_1\na : \u03b1\n\u22a2 out (mk a) = a\n[PROOFSTEP]\nlet h := (mk a).2\n[GOAL]\n\u03b1 : Sort u_1\na : \u03b1\nh : \u2203 a_1, (fun b => a_1 = b) = (mk a).fst := (mk a).snd\n\u22a2 out (mk a) = a\n[PROOFSTEP]\nshow Classical.choose h = a\n[GOAL]\n\u03b1 : Sort u_1\na : \u03b1\nh : \u2203 a_1, (fun b => a_1 = b) = (mk a).fst := (mk a).snd\n\u22a2 Classical.choose h = a\n[PROOFSTEP]\nhave := Classical.choose_spec h\n[GOAL]\n\u03b1 : Sort u_1\na : \u03b1\nh : \u2203 a_1, (fun b => a_1 = b) = (mk a).fst := (mk a).snd\nthis : (fun b => Classical.choose h = b) = (mk a).fst\n\u22a2 Classical.choose h = a\n[PROOFSTEP]\nexact cast (congr_fun this a).symm rfl\n[GOAL]\n\u03b1 : Sort u_1\ns : \u03b1 \u2192 Prop\nh : \u2203 a, (fun b => a = b) = s\n\u22a2 mk (out { fst := s, snd := h }) = { fst := s, snd := h }\n[PROOFSTEP]\nsimp [mk]\n[GOAL]\n\u03b1 : Sort u_1\ns : \u03b1 \u2192 Prop\nh : \u2203 a, (fun b => a = b) = s\n\u22a2 { fst := fun b => out { fst := s, snd := h } = b,\n snd := (_ : \u2203 a, (fun b => a = b) = fun b => out { fst := s, snd := h } = b) } =\n { fst := s, snd := h }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_fst\n\u03b1 : Sort u_1\ns : \u03b1 \u2192 Prop\nh : \u2203 a, (fun b => a = b) = s\n\u22a2 (fun b => out { fst := s, snd := h } = b) = s\n[PROOFSTEP]\nexact Classical.choose_spec h\n[GOAL]\n\u03b1 : Sort u_1\na b : Erased \u03b1\nh : out a = out b\n\u22a2 a = b\n[PROOFSTEP]\nsimpa using congr_arg mk h\n[GOAL]\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\na : Erased \u03b1\n\u22a2 out (map f a) = f (out a)\n[PROOFSTEP]\nsimp [map]\n[GOAL]\nsrc\u271d : Monad Erased := Erased.Monad\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.5529}, Functor.mapConst = Functor.map \u2218 Function.const \u03b2\n[PROOFSTEP]\nintros\n[GOAL]\nsrc\u271d : Monad Erased := Erased.Monad\n\u03b1\u271d \u03b2\u271d : Type ?u.5529\n\u22a2 Functor.mapConst = Functor.map \u2218 Function.const \u03b2\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h.h.h\nsrc\u271d : Monad Erased := Erased.Monad\n\u03b1\u271d \u03b2\u271d : Type ?u.5529\nx\u271d\u00b9 : \u03b1\u271d\nx\u271d : Erased \u03b2\u271d\n\u22a2 out (Functor.mapConst x\u271d\u00b9 x\u271d) = out ((Functor.map \u2218 Function.const \u03b2\u271d) x\u271d\u00b9 x\u271d)\n[PROOFSTEP]\nsimp [Functor.mapConst]\n[GOAL]\nsrc\u271d : Monad Erased := Erased.Monad\n\u22a2 \u2200 {\u03b1 : Type ?u.5529} (x : Erased \u03b1), id <$> x = x\n[PROOFSTEP]\nintros\n[GOAL]\nsrc\u271d : Monad Erased := Erased.Monad\n\u03b1\u271d : Type ?u.5529\nx\u271d : Erased \u03b1\u271d\n\u22a2 id <$> x\u271d = x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nsrc\u271d : Monad Erased := Erased.Monad\n\u03b1\u271d : Type ?u.5529\nx\u271d : Erased \u03b1\u271d\n\u22a2 out (id <$> x\u271d) = out x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nsrc\u271d : Monad Erased := Erased.Monad\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.5529} (x : Erased \u03b1) (y : Erased \u03b2),\n (SeqLeft.seqLeft x fun x => y) = Seq.seq (Function.const \u03b2 <$> x) fun x => y\n[PROOFSTEP]\nintros\n[GOAL]\nsrc\u271d : Monad Erased := Erased.Monad\n\u03b1\u271d \u03b2\u271d : Type ?u.5529\nx\u271d : Erased \u03b1\u271d\ny\u271d : Erased \u03b2\u271d\n\u22a2 (SeqLeft.seqLeft x\u271d fun x => y\u271d) = Seq.seq (Function.const \u03b2\u271d <$> x\u271d) fun x => y\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nsrc\u271d : Monad Erased := Erased.Monad\n\u03b1\u271d \u03b2\u271d : Type ?u.5529\nx\u271d : Erased \u03b1\u271d\ny\u271d : Erased \u03b2\u271d\n\u22a2 out (SeqLeft.seqLeft x\u271d fun x => y\u271d) = out (Seq.seq (Function.const \u03b2\u271d <$> x\u271d) fun x => y\u271d)\n[PROOFSTEP]\nsimp [Seq.seq, Functor.mapConst, SeqLeft.seqLeft]\n[GOAL]\nsrc\u271d : Monad Erased := Erased.Monad\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.5529} (x : Erased \u03b1) (y : Erased \u03b2),\n (SeqRight.seqRight x fun x => y) = Seq.seq (Function.const \u03b1 id <$> x) fun x => y\n[PROOFSTEP]\nintros\n[GOAL]\nsrc\u271d : Monad Erased := Erased.Monad\n\u03b1\u271d \u03b2\u271d : Type ?u.5529\nx\u271d : Erased \u03b1\u271d\ny\u271d : Erased \u03b2\u271d\n\u22a2 (SeqRight.seqRight x\u271d fun x => y\u271d) = Seq.seq (Function.const \u03b1\u271d id <$> x\u271d) fun x => y\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nsrc\u271d : Monad Erased := Erased.Monad\n\u03b1\u271d \u03b2\u271d : Type ?u.5529\nx\u271d : Erased \u03b1\u271d\ny\u271d : Erased \u03b2\u271d\n\u22a2 out (SeqRight.seqRight x\u271d fun x => y\u271d) = out (Seq.seq (Function.const \u03b1\u271d id <$> x\u271d) fun x => y\u271d)\n[PROOFSTEP]\nsimp [Seq.seq, Functor.mapConst, SeqRight.seqRight]\n[GOAL]\nsrc\u271d : Monad Erased := Erased.Monad\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.5529} (g : \u03b1 \u2192 \u03b2) (x : Erased \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\n[PROOFSTEP]\nintros\n[GOAL]\nsrc\u271d : Monad Erased := Erased.Monad\n\u03b1\u271d \u03b2\u271d : Type ?u.5529\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : Erased \u03b1\u271d\n\u22a2 (Seq.seq (pure g\u271d) fun x => x\u271d) = g\u271d <$> x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nsrc\u271d : Monad Erased := Erased.Monad\n\u03b1\u271d \u03b2\u271d : Type ?u.5529\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : Erased \u03b1\u271d\n\u22a2 out (Seq.seq (pure g\u271d) fun x => x\u271d) = out (g\u271d <$> x\u271d)\n[PROOFSTEP]\nsimp [Seq.seq, Functor.mapConst, SeqRight.seqRight]\n[GOAL]\nsrc\u271d : Monad Erased := Erased.Monad\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.5529} (f : \u03b1 \u2192 \u03b2) (x : Erased \u03b1),\n (do\n let a \u2190 x\n pure (f a)) =\n f <$> x\n[PROOFSTEP]\nintros\n[GOAL]\nsrc\u271d : Monad Erased := Erased.Monad\n\u03b1\u271d \u03b2\u271d : Type ?u.5529\nf\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : Erased \u03b1\u271d\n\u22a2 (do\n let a \u2190 x\u271d\n pure (f\u271d a)) =\n f\u271d <$> x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nsrc\u271d : Monad Erased := Erased.Monad\n\u03b1\u271d \u03b2\u271d : Type ?u.5529\nf\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : Erased \u03b1\u271d\n\u22a2 (out do\n let a \u2190 x\u271d\n pure (f\u271d a)) =\n out (f\u271d <$> x\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nsrc\u271d : Monad Erased := Erased.Monad\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.5529} (f : Erased (\u03b1 \u2192 \u03b2)) (x : Erased \u03b1),\n (do\n let x_1 \u2190 f\n x_1 <$> x) =\n Seq.seq f fun x_1 => x\n[PROOFSTEP]\nintros\n[GOAL]\nsrc\u271d : Monad Erased := Erased.Monad\n\u03b1\u271d \u03b2\u271d : Type ?u.5529\nf\u271d : Erased (\u03b1\u271d \u2192 \u03b2\u271d)\nx\u271d : Erased \u03b1\u271d\n\u22a2 (do\n let x \u2190 f\u271d\n x <$> x\u271d) =\n Seq.seq f\u271d fun x => x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nsrc\u271d : Monad Erased := Erased.Monad\n\u03b1\u271d \u03b2\u271d : Type ?u.5529\nf\u271d : Erased (\u03b1\u271d \u2192 \u03b2\u271d)\nx\u271d : Erased \u03b1\u271d\n\u22a2 (out do\n let x \u2190 f\u271d\n x <$> x\u271d) =\n out (Seq.seq f\u271d fun x => x\u271d)\n[PROOFSTEP]\nsimp [Seq.seq]\n[GOAL]\nsrc\u271d : Monad Erased := Erased.Monad\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.5529} (x : \u03b1) (f : \u03b1 \u2192 Erased \u03b2), pure x >>= f = f x\n[PROOFSTEP]\nintros\n[GOAL]\nsrc\u271d : Monad Erased := Erased.Monad\n\u03b1\u271d \u03b2\u271d : Type ?u.5529\nx\u271d : \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 Erased \u03b2\u271d\n\u22a2 pure x\u271d >>= f\u271d = f\u271d x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nsrc\u271d : Monad Erased := Erased.Monad\n\u03b1\u271d \u03b2\u271d : Type ?u.5529\nx\u271d : \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 Erased \u03b2\u271d\n\u22a2 out (pure x\u271d >>= f\u271d) = out (f\u271d x\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nsrc\u271d : Monad Erased := Erased.Monad\n\u22a2 \u2200 {\u03b1 \u03b2 \u03b3 : Type ?u.5529} (x : Erased \u03b1) (f : \u03b1 \u2192 Erased \u03b2) (g : \u03b2 \u2192 Erased \u03b3),\n x >>= f >>= g = x >>= fun x => f x >>= g\n[PROOFSTEP]\nintros\n[GOAL]\nsrc\u271d : Monad Erased := Erased.Monad\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.5529\nx\u271d : Erased \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 Erased \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 Erased \u03b3\u271d\n\u22a2 x\u271d >>= f\u271d >>= g\u271d = x\u271d >>= fun x => f\u271d x >>= g\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nsrc\u271d : Monad Erased := Erased.Monad\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.5529\nx\u271d : Erased \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 Erased \u03b2\u271d\ng\u271d : \u03b2\u271d \u2192 Erased \u03b3\u271d\n\u22a2 out (x\u271d >>= f\u271d >>= g\u271d) = out (x\u271d >>= fun x => f\u271d x >>= g\u271d)\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Data.Erased", "llama_tokens": 3582, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.4455295350395727, "lm_q2_score": 0.029312228525445152, "lm_q1q2_score": 0.013059463545915279}} {"text": "[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : FG I\n\u22a2 IsIdempotentElem I \u2194 \u2203 e, IsIdempotentElem e \u2227 I = Submodule.span R {e}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : FG I\n\u22a2 IsIdempotentElem I \u2192 \u2203 e, IsIdempotentElem e \u2227 I = Submodule.span R {e}\n[PROOFSTEP]\nintro e\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : FG I\ne : IsIdempotentElem I\n\u22a2 \u2203 e, IsIdempotentElem e \u2227 I = Submodule.span R {e}\n[PROOFSTEP]\nobtain \u27e8r, hr, hr'\u27e9 :=\n Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I I h\n (by\n rw [smul_eq_mul]\n exact e.ge)\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : FG I\ne : IsIdempotentElem I\n\u22a2 I \u2264 I \u2022 I\n[PROOFSTEP]\nrw [smul_eq_mul]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : FG I\ne : IsIdempotentElem I\n\u22a2 I \u2264 I * I\n[PROOFSTEP]\nexact e.ge\n[GOAL]\ncase mp.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : FG I\ne : IsIdempotentElem I\nr : R\nhr : r \u2208 I\nhr' : \u2200 (n : R), n \u2208 I \u2192 r \u2022 n = n\n\u22a2 \u2203 e, IsIdempotentElem e \u2227 I = Submodule.span R {e}\n[PROOFSTEP]\nsimp_rw [smul_eq_mul] at hr' \n[GOAL]\ncase mp.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : FG I\ne : IsIdempotentElem I\nr : R\nhr : r \u2208 I\nhr' : \u2200 (n : R), n \u2208 I \u2192 r * n = n\n\u22a2 \u2203 e, IsIdempotentElem e \u2227 I = Submodule.span R {e}\n[PROOFSTEP]\nrefine' \u27e8r, hr' r hr, antisymm _ ((Submodule.span_singleton_le_iff_mem _ _).mpr hr)\u27e9\n[GOAL]\ncase mp.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : FG I\ne : IsIdempotentElem I\nr : R\nhr : r \u2208 I\nhr' : \u2200 (n : R), n \u2208 I \u2192 r * n = n\n\u22a2 I \u2264 Submodule.span R {r}\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase mp.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : FG I\ne : IsIdempotentElem I\nr : R\nhr : r \u2208 I\nhr' : \u2200 (n : R), n \u2208 I \u2192 r * n = n\nx : R\nhx : x \u2208 I\n\u22a2 x \u2208 Submodule.span R {r}\n[PROOFSTEP]\nrw [\u2190 hr' x hx]\n[GOAL]\ncase mp.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : FG I\ne : IsIdempotentElem I\nr : R\nhr : r \u2208 I\nhr' : \u2200 (n : R), n \u2208 I \u2192 r * n = n\nx : R\nhx : x \u2208 I\n\u22a2 r * x \u2208 Submodule.span R {r}\n[PROOFSTEP]\nexact Ideal.mem_span_singleton'.mpr \u27e8_, mul_comm _ _\u27e9\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : FG I\n\u22a2 (\u2203 e, IsIdempotentElem e \u2227 I = Submodule.span R {e}) \u2192 IsIdempotentElem I\n[PROOFSTEP]\nrintro \u27e8e, he, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\ne : R\nhe : IsIdempotentElem e\nh : FG (Submodule.span R {e})\n\u22a2 IsIdempotentElem (Submodule.span R {e})\n[PROOFSTEP]\nsimp [IsIdempotentElem, Ideal.span_singleton_mul_span_singleton, he.eq]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nI : Ideal R\nh : FG I\n\u22a2 IsIdempotentElem I \u2194 I = \u22a5 \u2228 I = \u22a4\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nI : Ideal R\nh : FG I\n\u22a2 IsIdempotentElem I \u2192 I = \u22a5 \u2228 I = \u22a4\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nI : Ideal R\nh : FG I\nH : IsIdempotentElem I\n\u22a2 I = \u22a5 \u2228 I = \u22a4\n[PROOFSTEP]\nobtain \u27e8e, he, rfl\u27e9 := (I.isIdempotentElem_iff_of_fg h).mp H\n[GOAL]\ncase mp.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\ne : R\nhe : IsIdempotentElem e\nh : FG (Submodule.span R {e})\nH : IsIdempotentElem (Submodule.span R {e})\n\u22a2 Submodule.span R {e} = \u22a5 \u2228 Submodule.span R {e} = \u22a4\n[PROOFSTEP]\nsimp only [Ideal.submodule_span_eq, Ideal.span_singleton_eq_bot]\n[GOAL]\ncase mp.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\ne : R\nhe : IsIdempotentElem e\nh : FG (Submodule.span R {e})\nH : IsIdempotentElem (Submodule.span R {e})\n\u22a2 e = 0 \u2228 span {e} = \u22a4\n[PROOFSTEP]\napply Or.imp id _ (IsIdempotentElem.iff_eq_zero_or_one.mp he)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\ne : R\nhe : IsIdempotentElem e\nh : FG (Submodule.span R {e})\nH : IsIdempotentElem (Submodule.span R {e})\n\u22a2 e = 1 \u2192 span {e} = \u22a4\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nhe : IsIdempotentElem 1\nh : FG (Submodule.span R {1})\nH : IsIdempotentElem (Submodule.span R {1})\n\u22a2 span {1} = \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nI : Ideal R\nh : FG I\n\u22a2 I = \u22a5 \u2228 I = \u22a4 \u2192 IsIdempotentElem I\n[PROOFSTEP]\nrintro (rfl | rfl)\n[GOAL]\ncase mpr.inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nh : FG \u22a5\n\u22a2 IsIdempotentElem \u22a5\n[PROOFSTEP]\nsimp [IsIdempotentElem]\n[GOAL]\ncase mpr.inr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nh : FG \u22a4\n\u22a2 IsIdempotentElem \u22a4\n[PROOFSTEP]\nsimp [IsIdempotentElem]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Ideal.IdempotentFG", "llama_tokens": 2341, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.4378234991142019, "lm_q2_score": 0.029312231281335196, "lm_q1q2_score": 0.012833583666438941}} {"text": "[GOAL]\nX : Type u_1\nY Z : CompHaus\nf : of (Ultrafilter X) \u27f6 Z\ng : Y \u27f6 Z\nhg : Epi g\n\u22a2 \u2203 f', f' \u226b g = f\n[PROOFSTEP]\nrw [epi_iff_surjective] at hg \n[GOAL]\nX : Type u_1\nY Z : CompHaus\nf : of (Ultrafilter X) \u27f6 Z\ng : Y \u27f6 Z\nhg : Surjective \u2191g\n\u22a2 \u2203 f', f' \u226b g = f\n[PROOFSTEP]\nobtain \u27e8g', hg'\u27e9 := hg.hasRightInverse\n[GOAL]\ncase intro\nX : Type u_1\nY Z : CompHaus\nf : of (Ultrafilter X) \u27f6 Z\ng : Y \u27f6 Z\nhg : Surjective \u2191g\ng' : (forget CompHaus).obj Z \u2192 (forget CompHaus).obj Y\nhg' : Function.RightInverse g' \u2191g\n\u22a2 \u2203 f', f' \u226b g = f\n[PROOFSTEP]\nlet t : X \u2192 Y := g' \u2218 f \u2218 (pure : X \u2192 Ultrafilter X)\n[GOAL]\ncase intro\nX : Type u_1\nY Z : CompHaus\nf : of (Ultrafilter X) \u27f6 Z\ng : Y \u27f6 Z\nhg : Surjective \u2191g\ng' : (forget CompHaus).obj Z \u2192 (forget CompHaus).obj Y\nhg' : Function.RightInverse g' \u2191g\nt : X \u2192 \u2191Y.toTop := g' \u2218 \u2191f \u2218 pure\n\u22a2 \u2203 f', f' \u226b g = f\n[PROOFSTEP]\nlet h : Ultrafilter X \u2192 Y := Ultrafilter.extend t\n[GOAL]\ncase intro\nX : Type u_1\nY Z : CompHaus\nf : of (Ultrafilter X) \u27f6 Z\ng : Y \u27f6 Z\nhg : Surjective \u2191g\ng' : (forget CompHaus).obj Z \u2192 (forget CompHaus).obj Y\nhg' : Function.RightInverse g' \u2191g\nt : X \u2192 \u2191Y.toTop := g' \u2218 \u2191f \u2218 pure\nh : Ultrafilter X \u2192 \u2191Y.toTop := Ultrafilter.extend t\n\u22a2 \u2203 f', f' \u226b g = f\n[PROOFSTEP]\nhave hh : Continuous h := continuous_ultrafilter_extend _\n[GOAL]\ncase intro\nX : Type u_1\nY Z : CompHaus\nf : of (Ultrafilter X) \u27f6 Z\ng : Y \u27f6 Z\nhg : Surjective \u2191g\ng' : (forget CompHaus).obj Z \u2192 (forget CompHaus).obj Y\nhg' : Function.RightInverse g' \u2191g\nt : X \u2192 \u2191Y.toTop := g' \u2218 \u2191f \u2218 pure\nh : Ultrafilter X \u2192 \u2191Y.toTop := Ultrafilter.extend t\nhh : Continuous h\n\u22a2 \u2203 f', f' \u226b g = f\n[PROOFSTEP]\nuse\u27e8h, hh\u27e9\n[GOAL]\ncase h\nX : Type u_1\nY Z : CompHaus\nf : of (Ultrafilter X) \u27f6 Z\ng : Y \u27f6 Z\nhg : Surjective \u2191g\ng' : (forget CompHaus).obj Z \u2192 (forget CompHaus).obj Y\nhg' : Function.RightInverse g' \u2191g\nt : X \u2192 \u2191Y.toTop := g' \u2218 \u2191f \u2218 pure\nh : Ultrafilter X \u2192 \u2191Y.toTop := Ultrafilter.extend t\nhh : Continuous h\n\u22a2 ContinuousMap.mk h \u226b g = f\n[PROOFSTEP]\napply Faithful.map_injective (F := forget CompHaus)\n[GOAL]\ncase h.a\nX : Type u_1\nY Z : CompHaus\nf : of (Ultrafilter X) \u27f6 Z\ng : Y \u27f6 Z\nhg : Surjective \u2191g\ng' : (forget CompHaus).obj Z \u2192 (forget CompHaus).obj Y\nhg' : Function.RightInverse g' \u2191g\nt : X \u2192 \u2191Y.toTop := g' \u2218 \u2191f \u2218 pure\nh : Ultrafilter X \u2192 \u2191Y.toTop := Ultrafilter.extend t\nhh : Continuous h\n\u22a2 (forget CompHaus).map (ContinuousMap.mk h \u226b g) = (forget CompHaus).map f\n[PROOFSTEP]\nsimp only [Functor.map_comp, ContinuousMap.coe_mk, coe_comp]\n[GOAL]\ncase h.a\nX : Type u_1\nY Z : CompHaus\nf : of (Ultrafilter X) \u27f6 Z\ng : Y \u27f6 Z\nhg : Surjective \u2191g\ng' : (forget CompHaus).obj Z \u2192 (forget CompHaus).obj Y\nhg' : Function.RightInverse g' \u2191g\nt : X \u2192 \u2191Y.toTop := g' \u2218 \u2191f \u2218 pure\nh : Ultrafilter X \u2192 \u2191Y.toTop := Ultrafilter.extend t\nhh : Continuous h\n\u22a2 (forget CompHaus).map (ContinuousMap.mk (Ultrafilter.extend (g' \u2218 \u2191f \u2218 pure))) \u226b (forget CompHaus).map g =\n (forget CompHaus).map f\n[PROOFSTEP]\nconvert\n denseRange_pure.equalizer (g.continuous.comp hh) f.continuous\n _\n -- Porting note: We need to get the coercions to functions under control.\n -- The next two lines should not be needed.\n[GOAL]\ncase h.a\nX : Type u_1\nY Z : CompHaus\nf : of (Ultrafilter X) \u27f6 Z\ng : Y \u27f6 Z\nhg : Surjective \u2191g\ng' : (forget CompHaus).obj Z \u2192 (forget CompHaus).obj Y\nhg' : Function.RightInverse g' \u2191g\nt : X \u2192 \u2191Y.toTop := g' \u2218 \u2191f \u2218 pure\nh : Ultrafilter X \u2192 \u2191Y.toTop := Ultrafilter.extend t\nhh : Continuous h\n\u22a2 (\u2191g \u2218 h) \u2218 pure = \u2191f \u2218 pure\n[PROOFSTEP]\nlet g'' : ContinuousMap Y Z := g\n[GOAL]\ncase h.a\nX : Type u_1\nY Z : CompHaus\nf : of (Ultrafilter X) \u27f6 Z\ng : Y \u27f6 Z\nhg : Surjective \u2191g\ng' : (forget CompHaus).obj Z \u2192 (forget CompHaus).obj Y\nhg' : Function.RightInverse g' \u2191g\nt : X \u2192 \u2191Y.toTop := g' \u2218 \u2191f \u2218 pure\nh : Ultrafilter X \u2192 \u2191Y.toTop := Ultrafilter.extend t\nhh : Continuous h\ng'' : C(\u2191Y.toTop, \u2191Z.toTop) := g\n\u22a2 (\u2191g \u2218 h) \u2218 pure = \u2191f \u2218 pure\n[PROOFSTEP]\nhave : g'' \u2218 g' = id := hg'.comp_eq_id\n[GOAL]\ncase h.a\nX : Type u_1\nY Z : CompHaus\nf : of (Ultrafilter X) \u27f6 Z\ng : Y \u27f6 Z\nhg : Surjective \u2191g\ng' : (forget CompHaus).obj Z \u2192 (forget CompHaus).obj Y\nhg' : Function.RightInverse g' \u2191g\nt : X \u2192 \u2191Y.toTop := g' \u2218 \u2191f \u2218 pure\nh : Ultrafilter X \u2192 \u2191Y.toTop := Ultrafilter.extend t\nhh : Continuous h\ng'' : C(\u2191Y.toTop, \u2191Z.toTop) := g\nthis : \u2191g'' \u2218 g' = id\n\u22a2 (\u2191g \u2218 h) \u2218 pure = \u2191f \u2218 pure\n[PROOFSTEP]\nrw [comp.assoc, ultrafilter_extend_extends, \u2190 comp.assoc, this, comp.left_id]\n", "meta": {"mathlib_filename": "Mathlib.Topology.Category.CompHaus.Projective", "llama_tokens": 2071, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.43782349911420193, "lm_q2_score": 0.028870907468524765, "lm_q1q2_score": 0.01264036173047186}} {"text": "[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.73, u_1} C\ninst\u271d : MonoidalCategory C\n\u22a2 Inhabited (MonoidalSingleObj C)\n[PROOFSTEP]\nunfold MonoidalSingleObj\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.73, u_1} C\ninst\u271d : MonoidalCategory C\n\u22a2 Inhabited PUnit\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.162, u_1} C\ninst\u271d : MonoidalCategory C\na\u271d b\u271d : MonoidalSingleObj C\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nx\u271d : f\u271d \u27f6 g\u271d\n\u22a2 (fun {a b c} X Y Z f => \ud835\udfd9 X \u2297 f) (\ud835\udfd9 a\u271d) f\u271d g\u271d x\u271d =\n ((fun {a b} X => \u03bb_ X) f\u271d).hom \u226b x\u271d \u226b ((fun {a b} X => \u03bb_ X) g\u271d).inv\n[PROOFSTEP]\nsimp_rw [leftUnitor_inv_naturality, Iso.hom_inv_id_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.162, u_1} C\ninst\u271d : MonoidalCategory C\na\u271d b\u271d c\u271d d\u271d : MonoidalSingleObj C\nx\u271d\u2074 : a\u271d \u27f6 b\u271d\nx\u271d\u00b3 : b\u271d \u27f6 c\u271d\nx\u271d\u00b2 x\u271d\u00b9 : c\u271d \u27f6 d\u271d\nx\u271d : x\u271d\u00b2 \u27f6 x\u271d\u00b9\n\u22a2 (fun {a b c} X Y Z f => \ud835\udfd9 X \u2297 f) (x\u271d\u2074 \u226b x\u271d\u00b3) x\u271d\u00b2 x\u271d\u00b9 x\u271d =\n ((fun {a b c d} X Y Z => \u03b1_ X Y Z) x\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2).hom \u226b\n (fun {a b c} X Y Z f => \ud835\udfd9 X \u2297 f) x\u271d\u2074 (x\u271d\u00b3 \u226b x\u271d\u00b2) (x\u271d\u00b3 \u226b x\u271d\u00b9) ((fun {a b c} X Y Z f => \ud835\udfd9 X \u2297 f) x\u271d\u00b3 x\u271d\u00b2 x\u271d\u00b9 x\u271d) \u226b\n ((fun {a b c d} X Y Z => \u03b1_ X Y Z) x\u271d\u2074 x\u271d\u00b3 x\u271d\u00b9).inv\n[PROOFSTEP]\nsimp_rw [associator_inv_naturality, Iso.hom_inv_id_assoc, tensor_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.162, u_1} C\ninst\u271d : MonoidalCategory C\na\u271d b\u271d : MonoidalSingleObj C\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nx\u271d : f\u271d \u27f6 g\u271d\n\u22a2 (fun {a b c} {f g} f_1 Z => f_1 \u2297 \ud835\udfd9 Z) x\u271d (\ud835\udfd9 b\u271d) =\n ((fun {a b} X => \u03c1_ X) f\u271d).hom \u226b x\u271d \u226b ((fun {a b} X => \u03c1_ X) g\u271d).inv\n[PROOFSTEP]\nsimp_rw [rightUnitor_inv_naturality, Iso.hom_inv_id_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.162, u_1} C\ninst\u271d : MonoidalCategory C\na\u271d b\u271d c\u271d d\u271d : MonoidalSingleObj C\nf\u271d f'\u271d : a\u271d \u27f6 b\u271d\nx\u271d\u00b2 : f\u271d \u27f6 f'\u271d\nx\u271d\u00b9 : b\u271d \u27f6 c\u271d\nx\u271d : c\u271d \u27f6 d\u271d\n\u22a2 (fun {a b c} {f g} f_1 Z => f_1 \u2297 \ud835\udfd9 Z) x\u271d\u00b2 (x\u271d\u00b9 \u226b x\u271d) =\n ((fun {a b c d} X Y Z => \u03b1_ X Y Z) f\u271d x\u271d\u00b9 x\u271d).inv \u226b\n (fun {a b c} {f g} f_1 Z => f_1 \u2297 \ud835\udfd9 Z) ((fun {a b c} {f g} f_1 Z => f_1 \u2297 \ud835\udfd9 Z) x\u271d\u00b2 x\u271d\u00b9) x\u271d \u226b\n ((fun {a b c d} X Y Z => \u03b1_ X Y Z) f'\u271d x\u271d\u00b9 x\u271d).hom\n[PROOFSTEP]\nsimp_rw [\u2190 tensor_id, associator_naturality, Iso.inv_hom_id_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.162, u_1} C\ninst\u271d : MonoidalCategory C\na\u271d b\u271d c\u271d d\u271d : MonoidalSingleObj C\nx\u271d\u2074 : a\u271d \u27f6 b\u271d\nx\u271d\u00b3 x\u271d\u00b2 : b\u271d \u27f6 c\u271d\nx\u271d\u00b9 : x\u271d\u00b3 \u27f6 x\u271d\u00b2\nx\u271d : c\u271d \u27f6 d\u271d\n\u22a2 (fun {a b c} {f g} f_1 Z => f_1 \u2297 \ud835\udfd9 Z) ((fun {a b c} X Y Z f => \ud835\udfd9 X \u2297 f) x\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 x\u271d\u00b9) x\u271d =\n ((fun {a b c d} X Y Z => \u03b1_ X Y Z) x\u271d\u2074 x\u271d\u00b3 x\u271d).hom \u226b\n (fun {a b c} X Y Z f => \ud835\udfd9 X \u2297 f) x\u271d\u2074 (x\u271d\u00b3 \u226b x\u271d) (x\u271d\u00b2 \u226b x\u271d) ((fun {a b c} {f g} f_1 Z => f_1 \u2297 \ud835\udfd9 Z) x\u271d\u00b9 x\u271d) \u226b\n ((fun {a b c d} X Y Z => \u03b1_ X Y Z) x\u271d\u2074 x\u271d\u00b2 x\u271d).inv\n[PROOFSTEP]\nsimp_rw [associator_inv_naturality, Iso.hom_inv_id_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.162, u_1} C\ninst\u271d : MonoidalCategory C\na\u271d b\u271d c\u271d d\u271d e\u271d : MonoidalSingleObj C\nx\u271d\u00b3 : a\u271d \u27f6 b\u271d\nx\u271d\u00b2 : b\u271d \u27f6 c\u271d\nx\u271d\u00b9 : c\u271d \u27f6 d\u271d\nx\u271d : d\u271d \u27f6 e\u271d\n\u22a2 (fun {a b c} {f g} f_1 Z => f_1 \u2297 \ud835\udfd9 Z) ((fun {a b c d} X Y Z => \u03b1_ X Y Z) x\u271d\u00b3 x\u271d\u00b2 x\u271d\u00b9).hom x\u271d \u226b\n ((fun {a b c d} X Y Z => \u03b1_ X Y Z) x\u271d\u00b3 (x\u271d\u00b2 \u226b x\u271d\u00b9) x\u271d).hom \u226b\n (fun {a b c} X Y Z f => \ud835\udfd9 X \u2297 f) x\u271d\u00b3 ((x\u271d\u00b2 \u226b x\u271d\u00b9) \u226b x\u271d) (x\u271d\u00b2 \u226b x\u271d\u00b9 \u226b x\u271d)\n ((fun {a b c d} X Y Z => \u03b1_ X Y Z) x\u271d\u00b2 x\u271d\u00b9 x\u271d).hom =\n ((fun {a b c d} X Y Z => \u03b1_ X Y Z) (x\u271d\u00b3 \u226b x\u271d\u00b2) x\u271d\u00b9 x\u271d).hom \u226b\n ((fun {a b c d} X Y Z => \u03b1_ X Y Z) x\u271d\u00b3 x\u271d\u00b2 (x\u271d\u00b9 \u226b x\u271d)).hom\n[PROOFSTEP]\nsimp_rw [pentagon]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.14692, u_1} C\ninst\u271d : MonoidalCategory C\nX\u271d Y\u271d X'\u271d Y'\u271d : EndMonoidal (MonoidalSingleObj.star C)\nf : X\u271d \u27f6 Y\u271d\ng : X'\u271d \u27f6 Y'\u271d\n\u22a2 ((Functor.mk { obj := fun X => X, map := fun {X Y} f => f }).map f \u2297\n (Functor.mk { obj := fun X => X, map := fun {X Y} f => f }).map g) \u226b\n (fun X Y =>\n \ud835\udfd9\n ((Functor.mk { obj := fun X => X, map := fun {X Y} f => f }).obj X \u2297\n (Functor.mk { obj := fun X => X, map := fun {X Y} f => f }).obj Y))\n Y\u271d Y'\u271d =\n (fun X Y =>\n \ud835\udfd9\n ((Functor.mk { obj := fun X => X, map := fun {X Y} f => f }).obj X \u2297\n (Functor.mk { obj := fun X => X, map := fun {X Y} f => f }).obj Y))\n X\u271d X'\u271d \u226b\n (Functor.mk { obj := fun X => X, map := fun {X Y} f => f }).map (f \u2297 g)\n[PROOFSTEP]\nsimp_rw [Category.id_comp, Category.comp_id]\n -- Should we provide further simp lemmas so this goal becomes visible?\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{?u.14692, u_1} C\ninst\u271d : MonoidalCategory C\nX\u271d Y\u271d X'\u271d Y'\u271d : EndMonoidal (MonoidalSingleObj.star C)\nf : X\u271d \u27f6 Y\u271d\ng : X'\u271d \u27f6 Y'\u271d\n\u22a2 f \u2297 g = f \u2297 g\n[PROOFSTEP]\nexact (tensor_id_comp_id_tensor _ _).symm\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Bicategory.SingleObj", "llama_tokens": 2815, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.44552952031526044, "lm_q2_score": 0.0280075200989184, "lm_q1q2_score": 0.01247817699489113}} {"text": "[GOAL]\nX : CompHaus\ninst\u271d : Projective X\n\u22a2 ExtremallyDisconnected \u2191X.toTop\n[PROOFSTEP]\napply CompactT2.Projective.extremallyDisconnected\n[GOAL]\ncase h\nX : CompHaus\ninst\u271d : Projective X\n\u22a2 CompactT2.Projective \u2191X.toTop\n[PROOFSTEP]\nintro A B _ _ _ _ _ _ f g hf hg hsurj\n[GOAL]\ncase h\nX : CompHaus\ninst\u271d\u2076 : Projective X\nA B : Type u\ninst\u271d\u2075 : TopologicalSpace A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : CompactSpace A\ninst\u271d\u00b2 : T2Space A\ninst\u271d\u00b9 : CompactSpace B\ninst\u271d : T2Space B\nf : \u2191X.toTop \u2192 B\ng : A \u2192 B\nhf : Continuous f\nhg : Continuous g\nhsurj : Function.Surjective g\n\u22a2 \u2203 h, Continuous h \u2227 g \u2218 h = f\n[PROOFSTEP]\nhave : CompactSpace (TopCat.of A) := by assumption\n[GOAL]\nX : CompHaus\ninst\u271d\u2076 : Projective X\nA B : Type u\ninst\u271d\u2075 : TopologicalSpace A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : CompactSpace A\ninst\u271d\u00b2 : T2Space A\ninst\u271d\u00b9 : CompactSpace B\ninst\u271d : T2Space B\nf : \u2191X.toTop \u2192 B\ng : A \u2192 B\nhf : Continuous f\nhg : Continuous g\nhsurj : Function.Surjective g\n\u22a2 CompactSpace \u2191(TopCat.of A)\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h\nX : CompHaus\ninst\u271d\u2076 : Projective X\nA B : Type u\ninst\u271d\u2075 : TopologicalSpace A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : CompactSpace A\ninst\u271d\u00b2 : T2Space A\ninst\u271d\u00b9 : CompactSpace B\ninst\u271d : T2Space B\nf : \u2191X.toTop \u2192 B\ng : A \u2192 B\nhf : Continuous f\nhg : Continuous g\nhsurj : Function.Surjective g\nthis : CompactSpace \u2191(TopCat.of A)\n\u22a2 \u2203 h, Continuous h \u2227 g \u2218 h = f\n[PROOFSTEP]\nhave : T2Space (TopCat.of A) := by assumption\n[GOAL]\nX : CompHaus\ninst\u271d\u2076 : Projective X\nA B : Type u\ninst\u271d\u2075 : TopologicalSpace A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : CompactSpace A\ninst\u271d\u00b2 : T2Space A\ninst\u271d\u00b9 : CompactSpace B\ninst\u271d : T2Space B\nf : \u2191X.toTop \u2192 B\ng : A \u2192 B\nhf : Continuous f\nhg : Continuous g\nhsurj : Function.Surjective g\nthis : CompactSpace \u2191(TopCat.of A)\n\u22a2 T2Space \u2191(TopCat.of A)\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h\nX : CompHaus\ninst\u271d\u2076 : Projective X\nA B : Type u\ninst\u271d\u2075 : TopologicalSpace A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : CompactSpace A\ninst\u271d\u00b2 : T2Space A\ninst\u271d\u00b9 : CompactSpace B\ninst\u271d : T2Space B\nf : \u2191X.toTop \u2192 B\ng : A \u2192 B\nhf : Continuous f\nhg : Continuous g\nhsurj : Function.Surjective g\nthis\u271d : CompactSpace \u2191(TopCat.of A)\nthis : T2Space \u2191(TopCat.of A)\n\u22a2 \u2203 h, Continuous h \u2227 g \u2218 h = f\n[PROOFSTEP]\nhave : CompactSpace (TopCat.of B) := by assumption\n[GOAL]\nX : CompHaus\ninst\u271d\u2076 : Projective X\nA B : Type u\ninst\u271d\u2075 : TopologicalSpace A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : CompactSpace A\ninst\u271d\u00b2 : T2Space A\ninst\u271d\u00b9 : CompactSpace B\ninst\u271d : T2Space B\nf : \u2191X.toTop \u2192 B\ng : A \u2192 B\nhf : Continuous f\nhg : Continuous g\nhsurj : Function.Surjective g\nthis\u271d : CompactSpace \u2191(TopCat.of A)\nthis : T2Space \u2191(TopCat.of A)\n\u22a2 CompactSpace \u2191(TopCat.of B)\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h\nX : CompHaus\ninst\u271d\u2076 : Projective X\nA B : Type u\ninst\u271d\u2075 : TopologicalSpace A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : CompactSpace A\ninst\u271d\u00b2 : T2Space A\ninst\u271d\u00b9 : CompactSpace B\ninst\u271d : T2Space B\nf : \u2191X.toTop \u2192 B\ng : A \u2192 B\nhf : Continuous f\nhg : Continuous g\nhsurj : Function.Surjective g\nthis\u271d\u00b9 : CompactSpace \u2191(TopCat.of A)\nthis\u271d : T2Space \u2191(TopCat.of A)\nthis : CompactSpace \u2191(TopCat.of B)\n\u22a2 \u2203 h, Continuous h \u2227 g \u2218 h = f\n[PROOFSTEP]\nhave : T2Space (TopCat.of B) := by assumption\n[GOAL]\nX : CompHaus\ninst\u271d\u2076 : Projective X\nA B : Type u\ninst\u271d\u2075 : TopologicalSpace A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : CompactSpace A\ninst\u271d\u00b2 : T2Space A\ninst\u271d\u00b9 : CompactSpace B\ninst\u271d : T2Space B\nf : \u2191X.toTop \u2192 B\ng : A \u2192 B\nhf : Continuous f\nhg : Continuous g\nhsurj : Function.Surjective g\nthis\u271d\u00b9 : CompactSpace \u2191(TopCat.of A)\nthis\u271d : T2Space \u2191(TopCat.of A)\nthis : CompactSpace \u2191(TopCat.of B)\n\u22a2 T2Space \u2191(TopCat.of B)\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h\nX : CompHaus\ninst\u271d\u2076 : Projective X\nA B : Type u\ninst\u271d\u2075 : TopologicalSpace A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : CompactSpace A\ninst\u271d\u00b2 : T2Space A\ninst\u271d\u00b9 : CompactSpace B\ninst\u271d : T2Space B\nf : \u2191X.toTop \u2192 B\ng : A \u2192 B\nhf : Continuous f\nhg : Continuous g\nhsurj : Function.Surjective g\nthis\u271d\u00b2 : CompactSpace \u2191(TopCat.of A)\nthis\u271d\u00b9 : T2Space \u2191(TopCat.of A)\nthis\u271d : CompactSpace \u2191(TopCat.of B)\nthis : T2Space \u2191(TopCat.of B)\n\u22a2 \u2203 h, Continuous h \u2227 g \u2218 h = f\n[PROOFSTEP]\nlet A' : CompHaus := \u27e8TopCat.of A\u27e9\n[GOAL]\ncase h\nX : CompHaus\ninst\u271d\u2076 : Projective X\nA B : Type u\ninst\u271d\u2075 : TopologicalSpace A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : CompactSpace A\ninst\u271d\u00b2 : T2Space A\ninst\u271d\u00b9 : CompactSpace B\ninst\u271d : T2Space B\nf : \u2191X.toTop \u2192 B\ng : A \u2192 B\nhf : Continuous f\nhg : Continuous g\nhsurj : Function.Surjective g\nthis\u271d\u00b2 : CompactSpace \u2191(TopCat.of A)\nthis\u271d\u00b9 : T2Space \u2191(TopCat.of A)\nthis\u271d : CompactSpace \u2191(TopCat.of B)\nthis : T2Space \u2191(TopCat.of B)\nA' : CompHaus := mk (TopCat.of A)\n\u22a2 \u2203 h, Continuous h \u2227 g \u2218 h = f\n[PROOFSTEP]\nlet B' : CompHaus := \u27e8TopCat.of B\u27e9\n[GOAL]\ncase h\nX : CompHaus\ninst\u271d\u2076 : Projective X\nA B : Type u\ninst\u271d\u2075 : TopologicalSpace A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : CompactSpace A\ninst\u271d\u00b2 : T2Space A\ninst\u271d\u00b9 : CompactSpace B\ninst\u271d : T2Space B\nf : \u2191X.toTop \u2192 B\ng : A \u2192 B\nhf : Continuous f\nhg : Continuous g\nhsurj : Function.Surjective g\nthis\u271d\u00b2 : CompactSpace \u2191(TopCat.of A)\nthis\u271d\u00b9 : T2Space \u2191(TopCat.of A)\nthis\u271d : CompactSpace \u2191(TopCat.of B)\nthis : T2Space \u2191(TopCat.of B)\nA' : CompHaus := mk (TopCat.of A)\nB' : CompHaus := mk (TopCat.of B)\n\u22a2 \u2203 h, Continuous h \u2227 g \u2218 h = f\n[PROOFSTEP]\nlet f' : X \u27f6 B' := \u27e8f, hf\u27e9\n[GOAL]\ncase h\nX : CompHaus\ninst\u271d\u2076 : Projective X\nA B : Type u\ninst\u271d\u2075 : TopologicalSpace A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : CompactSpace A\ninst\u271d\u00b2 : T2Space A\ninst\u271d\u00b9 : CompactSpace B\ninst\u271d : T2Space B\nf : \u2191X.toTop \u2192 B\ng : A \u2192 B\nhf : Continuous f\nhg : Continuous g\nhsurj : Function.Surjective g\nthis\u271d\u00b2 : CompactSpace \u2191(TopCat.of A)\nthis\u271d\u00b9 : T2Space \u2191(TopCat.of A)\nthis\u271d : CompactSpace \u2191(TopCat.of B)\nthis : T2Space \u2191(TopCat.of B)\nA' : CompHaus := mk (TopCat.of A)\nB' : CompHaus := mk (TopCat.of B)\nf' : X \u27f6 B' := ContinuousMap.mk f\n\u22a2 \u2203 h, Continuous h \u2227 g \u2218 h = f\n[PROOFSTEP]\nlet g' : A' \u27f6 B' := \u27e8g, hg\u27e9\n[GOAL]\ncase h\nX : CompHaus\ninst\u271d\u2076 : Projective X\nA B : Type u\ninst\u271d\u2075 : TopologicalSpace A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : CompactSpace A\ninst\u271d\u00b2 : T2Space A\ninst\u271d\u00b9 : CompactSpace B\ninst\u271d : T2Space B\nf : \u2191X.toTop \u2192 B\ng : A \u2192 B\nhf : Continuous f\nhg : Continuous g\nhsurj : Function.Surjective g\nthis\u271d\u00b2 : CompactSpace \u2191(TopCat.of A)\nthis\u271d\u00b9 : T2Space \u2191(TopCat.of A)\nthis\u271d : CompactSpace \u2191(TopCat.of B)\nthis : T2Space \u2191(TopCat.of B)\nA' : CompHaus := mk (TopCat.of A)\nB' : CompHaus := mk (TopCat.of B)\nf' : X \u27f6 B' := ContinuousMap.mk f\ng' : A' \u27f6 B' := ContinuousMap.mk g\n\u22a2 \u2203 h, Continuous h \u2227 g \u2218 h = f\n[PROOFSTEP]\nhave : Epi g' := by\n rw [CompHaus.epi_iff_surjective]\n assumption\n[GOAL]\nX : CompHaus\ninst\u271d\u2076 : Projective X\nA B : Type u\ninst\u271d\u2075 : TopologicalSpace A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : CompactSpace A\ninst\u271d\u00b2 : T2Space A\ninst\u271d\u00b9 : CompactSpace B\ninst\u271d : T2Space B\nf : \u2191X.toTop \u2192 B\ng : A \u2192 B\nhf : Continuous f\nhg : Continuous g\nhsurj : Function.Surjective g\nthis\u271d\u00b2 : CompactSpace \u2191(TopCat.of A)\nthis\u271d\u00b9 : T2Space \u2191(TopCat.of A)\nthis\u271d : CompactSpace \u2191(TopCat.of B)\nthis : T2Space \u2191(TopCat.of B)\nA' : CompHaus := mk (TopCat.of A)\nB' : CompHaus := mk (TopCat.of B)\nf' : X \u27f6 B' := ContinuousMap.mk f\ng' : A' \u27f6 B' := ContinuousMap.mk g\n\u22a2 Epi g'\n[PROOFSTEP]\nrw [CompHaus.epi_iff_surjective]\n[GOAL]\nX : CompHaus\ninst\u271d\u2076 : Projective X\nA B : Type u\ninst\u271d\u2075 : TopologicalSpace A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : CompactSpace A\ninst\u271d\u00b2 : T2Space A\ninst\u271d\u00b9 : CompactSpace B\ninst\u271d : T2Space B\nf : \u2191X.toTop \u2192 B\ng : A \u2192 B\nhf : Continuous f\nhg : Continuous g\nhsurj : Function.Surjective g\nthis\u271d\u00b2 : CompactSpace \u2191(TopCat.of A)\nthis\u271d\u00b9 : T2Space \u2191(TopCat.of A)\nthis\u271d : CompactSpace \u2191(TopCat.of B)\nthis : T2Space \u2191(TopCat.of B)\nA' : CompHaus := mk (TopCat.of A)\nB' : CompHaus := mk (TopCat.of B)\nf' : X \u27f6 B' := ContinuousMap.mk f\ng' : A' \u27f6 B' := ContinuousMap.mk g\n\u22a2 Function.Surjective \u2191g'\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h\nX : CompHaus\ninst\u271d\u2076 : Projective X\nA B : Type u\ninst\u271d\u2075 : TopologicalSpace A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : CompactSpace A\ninst\u271d\u00b2 : T2Space A\ninst\u271d\u00b9 : CompactSpace B\ninst\u271d : T2Space B\nf : \u2191X.toTop \u2192 B\ng : A \u2192 B\nhf : Continuous f\nhg : Continuous g\nhsurj : Function.Surjective g\nthis\u271d\u00b3 : CompactSpace \u2191(TopCat.of A)\nthis\u271d\u00b2 : T2Space \u2191(TopCat.of A)\nthis\u271d\u00b9 : CompactSpace \u2191(TopCat.of B)\nthis\u271d : T2Space \u2191(TopCat.of B)\nA' : CompHaus := mk (TopCat.of A)\nB' : CompHaus := mk (TopCat.of B)\nf' : X \u27f6 B' := ContinuousMap.mk f\ng' : A' \u27f6 B' := ContinuousMap.mk g\nthis : Epi g'\n\u22a2 \u2203 h, Continuous h \u2227 g \u2218 h = f\n[PROOFSTEP]\nobtain \u27e8h, hh\u27e9 := Projective.factors f' g'\n[GOAL]\ncase h.intro\nX : CompHaus\ninst\u271d\u2076 : Projective X\nA B : Type u\ninst\u271d\u2075 : TopologicalSpace A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : CompactSpace A\ninst\u271d\u00b2 : T2Space A\ninst\u271d\u00b9 : CompactSpace B\ninst\u271d : T2Space B\nf : \u2191X.toTop \u2192 B\ng : A \u2192 B\nhf : Continuous f\nhg : Continuous g\nhsurj : Function.Surjective g\nthis\u271d\u00b3 : CompactSpace \u2191(TopCat.of A)\nthis\u271d\u00b2 : T2Space \u2191(TopCat.of A)\nthis\u271d\u00b9 : CompactSpace \u2191(TopCat.of B)\nthis\u271d : T2Space \u2191(TopCat.of B)\nA' : CompHaus := mk (TopCat.of A)\nB' : CompHaus := mk (TopCat.of B)\nf' : X \u27f6 B' := ContinuousMap.mk f\ng' : A' \u27f6 B' := ContinuousMap.mk g\nthis : Epi g'\nh : X \u27f6 A'\nhh : h \u226b g' = f'\n\u22a2 \u2203 h, Continuous h \u2227 g \u2218 h = f\n[PROOFSTEP]\nrefine \u27e8h, h.2, ?_\u27e9\n[GOAL]\ncase h.intro\nX : CompHaus\ninst\u271d\u2076 : Projective X\nA B : Type u\ninst\u271d\u2075 : TopologicalSpace A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : CompactSpace A\ninst\u271d\u00b2 : T2Space A\ninst\u271d\u00b9 : CompactSpace B\ninst\u271d : T2Space B\nf : \u2191X.toTop \u2192 B\ng : A \u2192 B\nhf : Continuous f\nhg : Continuous g\nhsurj : Function.Surjective g\nthis\u271d\u00b3 : CompactSpace \u2191(TopCat.of A)\nthis\u271d\u00b2 : T2Space \u2191(TopCat.of A)\nthis\u271d\u00b9 : CompactSpace \u2191(TopCat.of B)\nthis\u271d : T2Space \u2191(TopCat.of B)\nA' : CompHaus := mk (TopCat.of A)\nB' : CompHaus := mk (TopCat.of B)\nf' : X \u27f6 B' := ContinuousMap.mk f\ng' : A' \u27f6 B' := ContinuousMap.mk g\nthis : Epi g'\nh : X \u27f6 A'\nhh : h \u226b g' = f'\n\u22a2 g \u2218 \u2191h = f\n[PROOFSTEP]\next t\n[GOAL]\ncase h.intro.h\nX : CompHaus\ninst\u271d\u2076 : Projective X\nA B : Type u\ninst\u271d\u2075 : TopologicalSpace A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : CompactSpace A\ninst\u271d\u00b2 : T2Space A\ninst\u271d\u00b9 : CompactSpace B\ninst\u271d : T2Space B\nf : \u2191X.toTop \u2192 B\ng : A \u2192 B\nhf : Continuous f\nhg : Continuous g\nhsurj : Function.Surjective g\nthis\u271d\u00b3 : CompactSpace \u2191(TopCat.of A)\nthis\u271d\u00b2 : T2Space \u2191(TopCat.of A)\nthis\u271d\u00b9 : CompactSpace \u2191(TopCat.of B)\nthis\u271d : T2Space \u2191(TopCat.of B)\nA' : CompHaus := mk (TopCat.of A)\nB' : CompHaus := mk (TopCat.of B)\nf' : X \u27f6 B' := ContinuousMap.mk f\ng' : A' \u27f6 B' := ContinuousMap.mk g\nthis : Epi g'\nh : X \u27f6 A'\nhh : h \u226b g' = f'\nt : \u2191X.toTop\n\u22a2 (g \u2218 \u2191h) t = f t\n[PROOFSTEP]\napply_fun (fun e => e t) at hh \n[GOAL]\ncase h.intro.h\nX : CompHaus\ninst\u271d\u2076 : Projective X\nA B : Type u\ninst\u271d\u2075 : TopologicalSpace A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : CompactSpace A\ninst\u271d\u00b2 : T2Space A\ninst\u271d\u00b9 : CompactSpace B\ninst\u271d : T2Space B\nf : \u2191X.toTop \u2192 B\ng : A \u2192 B\nhf : Continuous f\nhg : Continuous g\nhsurj : Function.Surjective g\nthis\u271d\u00b3 : CompactSpace \u2191(TopCat.of A)\nthis\u271d\u00b2 : T2Space \u2191(TopCat.of A)\nthis\u271d\u00b9 : CompactSpace \u2191(TopCat.of B)\nthis\u271d : T2Space \u2191(TopCat.of B)\nA' : CompHaus := mk (TopCat.of A)\nB' : CompHaus := mk (TopCat.of B)\nf' : X \u27f6 B' := ContinuousMap.mk f\ng' : A' \u27f6 B' := ContinuousMap.mk g\nthis : Epi g'\nh : X \u27f6 A'\nt : \u2191X.toTop\nhh : \u2191(h \u226b g') t = \u2191f' t\n\u22a2 (g \u2218 \u2191h) t = f t\n[PROOFSTEP]\nexact hh\n", "meta": {"mathlib_filename": "Mathlib.Topology.Category.Stonean.Basic", "llama_tokens": 5413, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.480478678047907, "lm_q2_score": 0.02517884051192404, "lm_q1q2_score": 0.012097896003948349}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\n\u22a2 CompleteLattice (Presieve X)\n[PROOFSTEP]\ndsimp [Presieve]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\n\u22a2 CompleteLattice (\u2983Y : C\u2984 \u2192 Set (Y \u27f6 X))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d f g : Y \u27f6 X\n\u22a2 singleton f g \u2194 f = g\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d f g : Y \u27f6 X\n\u22a2 singleton f g \u2192 f = g\n[PROOFSTEP]\nrintro \u27e8a, rfl\u27e9\n[GOAL]\ncase mp.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z : C\nf\u271d f : Y\u271d \u27f6 X\nY : C\n\u22a2 f = f\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d f g : Y \u27f6 X\n\u22a2 f = g \u2192 singleton f g\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d f : Y \u27f6 X\n\u22a2 singleton f f\n[PROOFSTEP]\napply singleton.mk\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\ninst\u271d : HasPullbacks C\ng : Z \u27f6 X\n\u22a2 pullbackArrows f (singleton g) = singleton pullback.snd\n[PROOFSTEP]\nfunext W\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\ninst\u271d : HasPullbacks C\ng : Z \u27f6 X\nW : C\n\u22a2 pullbackArrows f (singleton g) = singleton pullback.snd\n[PROOFSTEP]\next h\n[GOAL]\ncase h.h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\ninst\u271d : HasPullbacks C\ng : Z \u27f6 X\nW : C\nh : W \u27f6 Y\n\u22a2 h \u2208 pullbackArrows f (singleton g) \u2194 h \u2208 singleton pullback.snd\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.h.mp\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\ninst\u271d : HasPullbacks C\ng : Z \u27f6 X\nW : C\nh : W \u27f6 Y\n\u22a2 h \u2208 pullbackArrows f (singleton g) \u2192 h \u2208 singleton pullback.snd\n[PROOFSTEP]\nrintro \u27e8W, _, _, _\u27e9\n[GOAL]\ncase h.h.mp.mk.mk\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d\u00b9 Z : C\nf : Y\u271d\u00b9 \u27f6 X\ninst\u271d : HasPullbacks C\ng : Z \u27f6 X\nY\u271d Y : C\n\u22a2 pullback.snd \u2208 singleton pullback.snd\n[PROOFSTEP]\nexact singleton.mk\n[GOAL]\ncase h.h.mpr\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\ninst\u271d : HasPullbacks C\ng : Z \u27f6 X\nW : C\nh : W \u27f6 Y\n\u22a2 h \u2208 singleton pullback.snd \u2192 h \u2208 pullbackArrows f (singleton g)\n[PROOFSTEP]\nrintro \u27e8_\u27e9\n[GOAL]\ncase h.h.mpr.mk\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\ninst\u271d : HasPullbacks C\ng : Z \u27f6 X\nY : C\n\u22a2 pullback.snd \u2208 pullbackArrows f (singleton g)\n[PROOFSTEP]\nexact pullbackArrows.mk Z g singleton.mk\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\n\u22a2 (ofArrows (fun x => Y) fun x => f) = singleton f\n[PROOFSTEP]\nfunext Y\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nY : C\n\u22a2 (ofArrows (fun x => Y\u271d) fun x => f) = singleton f\n[PROOFSTEP]\next g\n[GOAL]\ncase h.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nY : C\ng : Y \u27f6 X\n\u22a2 (g \u2208 ofArrows (fun x => Y\u271d) fun x => f) \u2194 g \u2208 singleton f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.h.mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nY : C\ng : Y \u27f6 X\n\u22a2 (g \u2208 ofArrows (fun x => Y\u271d) fun x => f) \u2192 g \u2208 singleton f\n[PROOFSTEP]\nrintro \u27e8_\u27e9\n[GOAL]\ncase h.h.mp.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nY : C\ni\u271d : PUnit\n\u22a2 f \u2208 singleton f\n[PROOFSTEP]\napply singleton.mk\n[GOAL]\ncase h.h.mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nY : C\ng : Y \u27f6 X\n\u22a2 g \u2208 singleton f \u2192 g \u2208 ofArrows (fun x => Y\u271d) fun x => f\n[PROOFSTEP]\nrintro \u27e8_\u27e9\n[GOAL]\ncase h.h.mpr.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nY : C\n\u22a2 f \u2208 ofArrows (fun x => Y\u271d) fun x => f\n[PROOFSTEP]\nexact ofArrows.mk PUnit.unit\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z\u271d : C\nf : Y \u27f6 X\ninst\u271d : HasPullbacks C\n\u03b9 : Type u_1\nZ : \u03b9 \u2192 C\ng : (i : \u03b9) \u2192 Z i \u27f6 X\n\u22a2 (ofArrows (fun i => pullback (g i) f) fun i => pullback.snd) = pullbackArrows f (ofArrows Z g)\n[PROOFSTEP]\nfunext T\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z\u271d : C\nf : Y \u27f6 X\ninst\u271d : HasPullbacks C\n\u03b9 : Type u_1\nZ : \u03b9 \u2192 C\ng : (i : \u03b9) \u2192 Z i \u27f6 X\nT : C\n\u22a2 (ofArrows (fun i => pullback (g i) f) fun i => pullback.snd) = pullbackArrows f (ofArrows Z g)\n[PROOFSTEP]\next h\n[GOAL]\ncase h.h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z\u271d : C\nf : Y \u27f6 X\ninst\u271d : HasPullbacks C\n\u03b9 : Type u_1\nZ : \u03b9 \u2192 C\ng : (i : \u03b9) \u2192 Z i \u27f6 X\nT : C\nh : T \u27f6 Y\n\u22a2 (h \u2208 ofArrows (fun i => pullback (g i) f) fun i => pullback.snd) \u2194 h \u2208 pullbackArrows f (ofArrows Z g)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.h.mp\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z\u271d : C\nf : Y \u27f6 X\ninst\u271d : HasPullbacks C\n\u03b9 : Type u_1\nZ : \u03b9 \u2192 C\ng : (i : \u03b9) \u2192 Z i \u27f6 X\nT : C\nh : T \u27f6 Y\n\u22a2 (h \u2208 ofArrows (fun i => pullback (g i) f) fun i => pullback.snd) \u2192 h \u2208 pullbackArrows f (ofArrows Z g)\n[PROOFSTEP]\nrintro \u27e8hk\u27e9\n[GOAL]\ncase h.h.mp.mk\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z\u271d : C\nf : Y\u271d \u27f6 X\ninst\u271d : HasPullbacks C\n\u03b9 : Type u_1\nZ : \u03b9 \u2192 C\ng : (i : \u03b9) \u2192 Z i \u27f6 X\nY : C\nhk : \u03b9\n\u22a2 pullback.snd \u2208 pullbackArrows f (ofArrows Z g)\n[PROOFSTEP]\nexact pullbackArrows.mk _ _ (ofArrows.mk hk)\n[GOAL]\ncase h.h.mpr\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z\u271d : C\nf : Y \u27f6 X\ninst\u271d : HasPullbacks C\n\u03b9 : Type u_1\nZ : \u03b9 \u2192 C\ng : (i : \u03b9) \u2192 Z i \u27f6 X\nT : C\nh : T \u27f6 Y\n\u22a2 h \u2208 pullbackArrows f (ofArrows Z g) \u2192 h \u2208 ofArrows (fun i => pullback (g i) f) fun i => pullback.snd\n[PROOFSTEP]\nrintro \u27e8W, k, hk\u2081\u27e9\n[GOAL]\ncase h.h.mpr.mk\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z\u271d : C\nf : Y\u271d \u27f6 X\ninst\u271d : HasPullbacks C\n\u03b9 : Type u_1\nZ : \u03b9 \u2192 C\ng : (i : \u03b9) \u2192 Z i \u27f6 X\nY W : C\nk : W \u27f6 X\nhk\u2081 : ofArrows Z g k\n\u22a2 pullback.snd \u2208 ofArrows (fun i => pullback (g i) f) fun i => pullback.snd\n[PROOFSTEP]\ncases' hk\u2081 with i hi\n[GOAL]\ncase h.h.mpr.mk.mk\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z\u271d : C\nf : Y\u271d \u27f6 X\ninst\u271d : HasPullbacks C\n\u03b9 : Type u_1\nZ : \u03b9 \u2192 C\ng : (i : \u03b9) \u2192 Z i \u27f6 X\nY : C\ni : \u03b9\n\u22a2 pullback.snd \u2208 ofArrows (fun i => pullback (g i) f) fun i => pullback.snd\n[PROOFSTEP]\napply ofArrows.mk\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z\u271d : C\nf : Y \u27f6 X\n\u03b9 : Type u_1\nZ : \u03b9 \u2192 C\ng : (i : \u03b9) \u2192 Z i \u27f6 X\nj : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 ofArrows Z g f \u2192 Type u_2\nW : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 (H : ofArrows Z g f) \u2192 j f H \u2192 C\nk : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 (H : ofArrows Z g f) \u2192 (i : j f H) \u2192 W f H i \u27f6 Y\n\u22a2 (bind (ofArrows Z g) fun Y f H => ofArrows (W f H) (k f H)) =\n ofArrows (fun i => W (g i.fst) (_ : ofArrows Z g (g i.fst)) i.snd) fun ij =>\n k (g ij.fst) (_ : ofArrows Z g (g ij.fst)) ij.snd \u226b g ij.fst\n[PROOFSTEP]\nfunext Y\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z\u271d : C\nf : Y\u271d \u27f6 X\n\u03b9 : Type u_1\nZ : \u03b9 \u2192 C\ng : (i : \u03b9) \u2192 Z i \u27f6 X\nj : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 ofArrows Z g f \u2192 Type u_2\nW : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 (H : ofArrows Z g f) \u2192 j f H \u2192 C\nk : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 (H : ofArrows Z g f) \u2192 (i : j f H) \u2192 W f H i \u27f6 Y\nY : C\n\u22a2 (bind (ofArrows Z g) fun Y f H => ofArrows (W f H) (k f H)) =\n ofArrows (fun i => W (g i.fst) (_ : ofArrows Z g (g i.fst)) i.snd) fun ij =>\n k (g ij.fst) (_ : ofArrows Z g (g ij.fst)) ij.snd \u226b g ij.fst\n[PROOFSTEP]\next f\n[GOAL]\ncase h.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 X\n\u03b9 : Type u_1\nZ : \u03b9 \u2192 C\ng : (i : \u03b9) \u2192 Z i \u27f6 X\nj : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 ofArrows Z g f \u2192 Type u_2\nW : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 (H : ofArrows Z g f) \u2192 j f H \u2192 C\nk : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 (H : ofArrows Z g f) \u2192 (i : j f H) \u2192 W f H i \u27f6 Y\nY : C\nf : Y \u27f6 X\n\u22a2 (f \u2208 bind (ofArrows Z g) fun Y f H => ofArrows (W f H) (k f H)) \u2194\n f \u2208\n ofArrows (fun i => W (g i.fst) (_ : ofArrows Z g (g i.fst)) i.snd) fun ij =>\n k (g ij.fst) (_ : ofArrows Z g (g ij.fst)) ij.snd \u226b g ij.fst\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.h.mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 X\n\u03b9 : Type u_1\nZ : \u03b9 \u2192 C\ng : (i : \u03b9) \u2192 Z i \u27f6 X\nj : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 ofArrows Z g f \u2192 Type u_2\nW : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 (H : ofArrows Z g f) \u2192 j f H \u2192 C\nk : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 (H : ofArrows Z g f) \u2192 (i : j f H) \u2192 W f H i \u27f6 Y\nY : C\nf : Y \u27f6 X\n\u22a2 (f \u2208 bind (ofArrows Z g) fun Y f H => ofArrows (W f H) (k f H)) \u2192\n f \u2208\n ofArrows (fun i => W (g i.fst) (_ : ofArrows Z g (g i.fst)) i.snd) fun ij =>\n k (g ij.fst) (_ : ofArrows Z g (g ij.fst)) ij.snd \u226b g ij.fst\n[PROOFSTEP]\nrintro \u27e8_, _, _, \u27e8i\u27e9, \u27e8i'\u27e9, rfl\u27e9\n[GOAL]\ncase h.h.mp.intro.intro.intro.intro.mk.intro.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d\u00b9 Z\u271d : C\nf : Y\u271d\u00b9 \u27f6 X\n\u03b9 : Type u_1\nZ : \u03b9 \u2192 C\ng : (i : \u03b9) \u2192 Z i \u27f6 X\nj : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 ofArrows Z g f \u2192 Type u_2\nW : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 (H : ofArrows Z g f) \u2192 j f H \u2192 C\nk : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 (H : ofArrows Z g f) \u2192 (i : j f H) \u2192 W f H i \u27f6 Y\nY\u271d : C\ni : \u03b9\nY : C\ni' : j (g i) (_ : ofArrows Z g (g i))\n\u22a2 k (g i) (_ : ofArrows Z g (g i)) i' \u226b g i \u2208\n ofArrows (fun i => W (g i.fst) (_ : ofArrows Z g (g i.fst)) i.snd) fun ij =>\n k (g ij.fst) (_ : ofArrows Z g (g ij.fst)) ij.snd \u226b g ij.fst\n[PROOFSTEP]\nexact ofArrows.mk (Sigma.mk _ _)\n[GOAL]\ncase h.h.mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 X\n\u03b9 : Type u_1\nZ : \u03b9 \u2192 C\ng : (i : \u03b9) \u2192 Z i \u27f6 X\nj : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 ofArrows Z g f \u2192 Type u_2\nW : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 (H : ofArrows Z g f) \u2192 j f H \u2192 C\nk : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 (H : ofArrows Z g f) \u2192 (i : j f H) \u2192 W f H i \u27f6 Y\nY : C\nf : Y \u27f6 X\n\u22a2 (f \u2208\n ofArrows (fun i => W (g i.fst) (_ : ofArrows Z g (g i.fst)) i.snd) fun ij =>\n k (g ij.fst) (_ : ofArrows Z g (g ij.fst)) ij.snd \u226b g ij.fst) \u2192\n f \u2208 bind (ofArrows Z g) fun Y f H => ofArrows (W f H) (k f H)\n[PROOFSTEP]\nrintro \u27e8i\u27e9\n[GOAL]\ncase h.h.mpr.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z\u271d : C\nf : Y\u271d \u27f6 X\n\u03b9 : Type u_1\nZ : \u03b9 \u2192 C\ng : (i : \u03b9) \u2192 Z i \u27f6 X\nj : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 ofArrows Z g f \u2192 Type u_2\nW : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 (H : ofArrows Z g f) \u2192 j f H \u2192 C\nk : \u2983Y : C\u2984 \u2192 (f : Y \u27f6 X) \u2192 (H : ofArrows Z g f) \u2192 (i : j f H) \u2192 W f H i \u27f6 Y\nY : C\ni : (i : \u03b9) \u00d7 j (g i) (_ : ofArrows Z g (g i))\n\u22a2 k (g i.fst) (_ : ofArrows Z g (g i.fst)) i.snd \u226b g i.fst \u2208 bind (ofArrows Z g) fun Y f H => ofArrows (W f H) (k f H)\n[PROOFSTEP]\nexact bind_comp _ (ofArrows.mk _) (ofArrows.mk _)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF\u271d : C \u2964 D\nX Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nF : C \u2964 D\nS : Presieve X\nY : D\nf : Y \u27f6 F.obj X\nh : functorPushforward F S f\n\u22a2 FunctorPushforwardStructure F S f\n[PROOFSTEP]\nchoose Z f' g h\u2081 h using h\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF\u271d : C \u2964 D\nX Y\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nF : C \u2964 D\nS : Presieve X\nY : D\nf : Y \u27f6 F.obj X\nZ : C\nf' : Z \u27f6 X\ng : Y \u27f6 F.obj Z\nh\u2081 : S f'\nh : f = g \u226b F.map f'\n\u22a2 FunctorPushforwardStructure F S f\n[PROOFSTEP]\nexact \u27e8Z, f', g, h\u2081, h\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Presieve X\n\u22a2 functorPushforward (F \u22d9 G) R = functorPushforward G (functorPushforward F R)\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Presieve X\nx : E\n\u22a2 functorPushforward (F \u22d9 G) R = functorPushforward G (functorPushforward F R)\n[PROOFSTEP]\next f\n[GOAL]\ncase h.h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Presieve X\nx : E\nf : x \u27f6 (F \u22d9 G).obj X\n\u22a2 f \u2208 functorPushforward (F \u22d9 G) R \u2194 f \u2208 functorPushforward G (functorPushforward F R)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.h.mp\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Presieve X\nx : E\nf : x \u27f6 (F \u22d9 G).obj X\n\u22a2 f \u2208 functorPushforward (F \u22d9 G) R \u2192 f \u2208 functorPushforward G (functorPushforward F R)\n[PROOFSTEP]\nrintro \u27e8X, f\u2081, g\u2081, h\u2081, rfl\u27e9\n[GOAL]\ncase h.h.mp.intro.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Presieve X\u271d\nx : E\nX : C\nf\u2081 : X \u27f6 X\u271d\ng\u2081 : x \u27f6 (F \u22d9 G).obj X\nh\u2081 : R f\u2081\n\u22a2 g\u2081 \u226b (F \u22d9 G).map f\u2081 \u2208 functorPushforward G (functorPushforward F R)\n[PROOFSTEP]\nexact \u27e8F.obj X, F.map f\u2081, g\u2081, \u27e8X, f\u2081, \ud835\udfd9 _, h\u2081, by simp\u27e9, rfl\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Presieve X\u271d\nx : E\nX : C\nf\u2081 : X \u27f6 X\u271d\ng\u2081 : x \u27f6 (F \u22d9 G).obj X\nh\u2081 : R f\u2081\n\u22a2 F.map f\u2081 = \ud835\udfd9 (F.obj X) \u226b F.map f\u2081\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.h.mpr\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Presieve X\nx : E\nf : x \u27f6 (F \u22d9 G).obj X\n\u22a2 f \u2208 functorPushforward G (functorPushforward F R) \u2192 f \u2208 functorPushforward (F \u22d9 G) R\n[PROOFSTEP]\nrintro \u27e8X, f\u2081, g\u2081, \u27e8X', f\u2082, g\u2082, h\u2081, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase h.h.mpr.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Presieve X\u271d\nx : E\nX : D\ng\u2081 : x \u27f6 G.obj X\nX' : C\nf\u2082 : X' \u27f6 X\u271d\ng\u2082 : X \u27f6 F.obj X'\nh\u2081 : R f\u2082\n\u22a2 g\u2081 \u226b G.map (g\u2082 \u226b F.map f\u2082) \u2208 functorPushforward (F \u22d9 G) R\n[PROOFSTEP]\nexact \u27e8X', f\u2082, g\u2081 \u226b G.map g\u2082, h\u2081, by simp\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Presieve X\u271d\nx : E\nX : D\ng\u2081 : x \u27f6 G.obj X\nX' : C\nf\u2082 : X' \u27f6 X\u271d\ng\u2082 : X \u27f6 F.obj X'\nh\u2081 : R f\u2082\n\u22a2 g\u2081 \u226b G.map (g\u2082 \u226b F.map f\u2082) = (g\u2081 \u226b G.map g\u2082) \u226b (F \u22d9 G).map f\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Presieve X\nf : Y \u27f6 X\nh : R f\n\u22a2 F.map f = \ud835\udfd9 (F.obj Y) \u226b F.map f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R : Sieve X\n\u22a2 \u2200 {R S : Sieve X}, R.arrows = S.arrows \u2192 R = S\n[PROOFSTEP]\nrintro \u27e8_, _\u27e9 \u27e8_, _\u27e9 rfl\n[GOAL]\ncase mk.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R : Sieve X\narrows\u271d : Presieve X\ndownward_closed\u271d\u00b9 : \u2200 {Y Z : C} {f : Y \u27f6 X}, arrows\u271d f \u2192 \u2200 (g : Z \u27f6 Y), arrows\u271d (g \u226b f)\ndownward_closed\u271d :\n \u2200 {Y Z : C} {f : Y \u27f6 X},\n { arrows := arrows\u271d, downward_closed := downward_closed\u271d\u00b9 }.arrows f \u2192\n \u2200 (g : Z \u27f6 Y), { arrows := arrows\u271d, downward_closed := downward_closed\u271d\u00b9 }.arrows (g \u226b f)\n\u22a2 { arrows := arrows\u271d, downward_closed := downward_closed\u271d\u00b9 } =\n { arrows := { arrows := arrows\u271d, downward_closed := downward_closed\u271d\u00b9 }.arrows,\n downward_closed := downward_closed\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R : Sieve X\n\ud835\udcae : Set (Sieve X)\nx\u271d\u00b2 x\u271d\u00b9 : C\nf : x\u271d\u00b2 \u27f6 X\nhf : (fun Y => {f | \u2203 S, S \u2208 \ud835\udcae \u2227 S.arrows f}) x\u271d\u00b2 f\nx\u271d : x\u271d\u00b9 \u27f6 x\u271d\u00b2\n\u22a2 (fun Y => {f | \u2203 S, S \u2208 \ud835\udcae \u2227 S.arrows f}) x\u271d\u00b9 (x\u271d \u226b f)\n[PROOFSTEP]\nobtain \u27e8S, hS, hf\u27e9 := hf\n[GOAL]\ncase intro.intro\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS\u271d R : Sieve X\n\ud835\udcae : Set (Sieve X)\nx\u271d\u00b2 x\u271d\u00b9 : C\nf : x\u271d\u00b2 \u27f6 X\nx\u271d : x\u271d\u00b9 \u27f6 x\u271d\u00b2\nS : Sieve X\nhS : S \u2208 \ud835\udcae\nhf : S.arrows f\n\u22a2 setOf (fun f => \u2203 S, S \u2208 \ud835\udcae \u2227 S.arrows f) (x\u271d \u226b f)\n[PROOFSTEP]\nexact \u27e8S, hS, S.downward_closed hf _\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS\u271d R\u271d S R : Sieve X\n\u22a2 \u2200 {Y Z : C} {f : Y \u27f6 X},\n (fun Y f => S.arrows f \u2228 R.arrows f) Y f \u2192 \u2200 (g : Z \u27f6 Y), (fun Y f => S.arrows f \u2228 R.arrows f) Z (g \u226b f)\n[PROOFSTEP]\nrintro _ _ _ (h | h) g\n[GOAL]\ncase inl\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS\u271d R\u271d S R : Sieve X\nY\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 X\nh : S.arrows f\u271d\ng : Z\u271d \u27f6 Y\u271d\n\u22a2 S.arrows (g \u226b f\u271d) \u2228 R.arrows (g \u226b f\u271d)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase inr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS\u271d R\u271d S R : Sieve X\nY\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 X\nh : R.arrows f\u271d\ng : Z\u271d \u27f6 Y\u271d\n\u22a2 S.arrows (g \u226b f\u271d) \u2228 R.arrows (g \u226b f\u271d)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS\u271d R\u271d S R : Sieve X\n\u22a2 \u2200 {Y Z : C} {f : Y \u27f6 X},\n (fun Y f => S.arrows f \u2227 R.arrows f) Y f \u2192 \u2200 (g : Z \u27f6 Y), (fun Y f => S.arrows f \u2227 R.arrows f) Z (g \u226b f)\n[PROOFSTEP]\nrintro _ _ _ \u27e8h\u2081, h\u2082\u27e9 g\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS\u271d R\u271d S R : Sieve X\nY\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 X\nh\u2081 : S.arrows f\u271d\nh\u2082 : R.arrows f\u271d\ng : Z\u271d \u27f6 Y\u271d\n\u22a2 S.arrows (g \u226b f\u271d) \u2227 R.arrows (g \u226b f\u271d)\n[PROOFSTEP]\nsimp [h\u2081, h\u2082]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R x\u271d\u00b3 x\u271d\u00b2 x\u271d\u00b9 : Sieve X\nh\u2081 : x\u271d\u00b3 \u2264 x\u271d\u00b9\nh\u2082 : x\u271d\u00b2 \u2264 x\u271d\u00b9\nx\u271d : C\nf : x\u271d \u27f6 X\n\u22a2 (x\u271d\u00b3 \u2294 x\u271d\u00b2).arrows f \u2192 x\u271d\u00b9.arrows f\n[PROOFSTEP]\nrintro (hf | hf)\n[GOAL]\ncase inl\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R x\u271d\u00b3 x\u271d\u00b2 x\u271d\u00b9 : Sieve X\nh\u2081 : x\u271d\u00b3 \u2264 x\u271d\u00b9\nh\u2082 : x\u271d\u00b2 \u2264 x\u271d\u00b9\nx\u271d : C\nf : x\u271d \u27f6 X\nhf : x\u271d\u00b3.arrows f\n\u22a2 x\u271d\u00b9.arrows f\n[PROOFSTEP]\nexact h\u2081 _ hf\n[GOAL]\ncase inr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R x\u271d\u00b3 x\u271d\u00b2 x\u271d\u00b9 : Sieve X\nh\u2081 : x\u271d\u00b3 \u2264 x\u271d\u00b9\nh\u2082 : x\u271d\u00b2 \u2264 x\u271d\u00b9\nx\u271d : C\nf : x\u271d \u27f6 X\nhf : x\u271d\u00b2.arrows f\n\u22a2 x\u271d\u00b9.arrows f\n[PROOFSTEP]\nexact h\u2082 _ hf\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\nS R : Sieve X\nSs : Set (Sieve X)\nY : C\nf : Y \u27f6 X\n\u22a2 (sSup Ss).arrows f \u2194 \u2203 S x, S.arrows f\n[PROOFSTEP]\nsimp [sSup, Sieve.sup, setOf]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R\u271d : Sieve X\nR : Presieve X\n\u22a2 \u2200 {Y Z : C} {f : Y \u27f6 X},\n (fun Z f => \u2203 Y h g, R g \u2227 h \u226b g = f) Y f \u2192 \u2200 (g : Z \u27f6 Y), (fun Z f => \u2203 Y h g, R g \u2227 h \u226b g = f) Z (g \u226b f)\n[PROOFSTEP]\nrintro Y Z _ \u27e8W, g, f, hf, rfl\u27e9 h\n[GOAL]\ncase intro.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 X\nS R\u271d : Sieve X\nR : Presieve X\nY Z W : C\ng : Y \u27f6 W\nf : W \u27f6 X\nhf : R f\nh : Z \u27f6 Y\n\u22a2 \u2203 Y_1 h_1 g_1, R g_1 \u2227 h_1 \u226b g_1 = h \u226b g \u226b f\n[PROOFSTEP]\nexact \u27e8_, h \u226b g, _, hf, by simp\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 X\nS R\u271d : Sieve X\nR : Presieve X\nY Z W : C\ng : Y \u27f6 W\nf : W \u27f6 X\nhf : R f\nh : Z \u27f6 Y\n\u22a2 (h \u226b g) \u226b f = h \u226b g \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS\u271d R\u271d : Sieve X\nS : Presieve X\nR : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 S f \u2192 Sieve Y\n\u22a2 \u2200 {Y Z : C} {f : Y \u27f6 X},\n Presieve.bind S (fun Y f h => (R h).arrows) f \u2192 \u2200 (g : Z \u27f6 Y), Presieve.bind S (fun Y f h => (R h).arrows) (g \u226b f)\n[PROOFSTEP]\nrintro Y Z f \u27e8W, f, h, hh, hf, rfl\u27e9 g\n[GOAL]\ncase intro.intro.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 X\nS\u271d R\u271d : Sieve X\nS : Presieve X\nR : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 S f \u2192 Sieve Y\nY Z W : C\nf : Y \u27f6 W\nh : W \u27f6 X\nhh : S h\nhf : (R hh).arrows f\ng : Z \u27f6 Y\n\u22a2 Presieve.bind S (fun Y f h => (R h).arrows) (g \u226b f \u226b h)\n[PROOFSTEP]\nexact \u27e8_, g \u226b f, _, hh, by simp [hf]\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 X\nS\u271d R\u271d : Sieve X\nS : Presieve X\nR : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 S f \u2192 Sieve Y\nY Z W : C\nf : Y \u27f6 W\nh : W \u27f6 X\nhh : S h\nhf : (R hh).arrows f\ng : Z \u27f6 Y\n\u22a2 (fun Y f h => (R h).arrows) W h hh (g \u226b f) \u2227 (g \u226b f) \u226b h = g \u226b f \u226b h\n[PROOFSTEP]\nsimp [hf]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\nS\u271d R\u271d : Sieve X\nR : Presieve X\nS : Sieve X\nss : R \u2264 S.arrows\nY : C\nf : Y \u27f6 X\n\u22a2 (generate R).arrows f \u2192 S.arrows f\n[PROOFSTEP]\nrintro \u27e8Z, f, g, hg, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 X\nS\u271d R\u271d : Sieve X\nR : Presieve X\nS : Sieve X\nss : R \u2264 S.arrows\nY Z : C\nf : Y \u27f6 Z\ng : Z \u27f6 X\nhg : R g\n\u22a2 S.arrows (f \u226b g)\n[PROOFSTEP]\nexact S.downward_closed (ss Z hg) f\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\nS R : Sieve X\nh : S.arrows (\ud835\udfd9 X)\nY : C\nf : Y \u27f6 X\nx\u271d : \u22a4.arrows f\n\u22a2 S.arrows f\n[PROOFSTEP]\nsimpa using downward_closed _ h f\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R\u271d : Sieve X\nR : Presieve X\nf : Y \u27f6 X\ninst\u271d : IsSplitEpi f\nhf : R f\n\u22a2 generate R = \u22a4\n[PROOFSTEP]\nrw [\u2190 id_mem_iff_eq_top]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R\u271d : Sieve X\nR : Presieve X\nf : Y \u27f6 X\ninst\u271d : IsSplitEpi f\nhf : R f\n\u22a2 (generate R).arrows (\ud835\udfd9 X)\n[PROOFSTEP]\nexact \u27e8_, section_ f, f, hf, by simp\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R\u271d : Sieve X\nR : Presieve X\nf : Y \u27f6 X\ninst\u271d : IsSplitEpi f\nhf : R f\n\u22a2 section_ f \u226b f = \ud835\udfd9 X\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS\u271d R : Sieve X\nh : Y \u27f6 X\nS : Sieve X\nY\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 Y\ng : (fun Y_1 sl => S.arrows (sl \u226b h)) Y\u271d f\u271d\n\u22a2 \u2200 (g : Z\u271d \u27f6 Y\u271d), (fun Y_1 sl => S.arrows (sl \u226b h)) Z\u271d (g \u226b f\u271d)\n[PROOFSTEP]\nsimp [g]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R : Sieve X\n\u22a2 pullback (\ud835\udfd9 X) S = S\n[PROOFSTEP]\nsimp [Sieve.ext_iff]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS\u271d R : Sieve X\nf : Y \u27f6 X\ng : Z \u27f6 Y\nS : Sieve X\n\u22a2 pullback (g \u226b f) S = pullback g (pullback f S)\n[PROOFSTEP]\nsimp [Sieve.ext_iff]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS\u271d R\u271d : Sieve X\nf : Y \u27f6 X\nS R : Sieve X\n\u22a2 pullback f (S \u2293 R) = pullback f S \u2293 pullback f R\n[PROOFSTEP]\nsimp [Sieve.ext_iff]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R : Sieve X\nf : Y \u27f6 X\n\u22a2 S.arrows f \u2194 pullback f S = \u22a4\n[PROOFSTEP]\nrw [\u2190 id_mem_iff_eq_top, pullback_apply, id_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d\u00b9 : Y \u27f6 X\nS R\u271d : Sieve X\nf : Y \u27f6 X\nR : Sieve Y\nY\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 X\nx\u271d : (fun Z gf => \u2203 g, g \u226b f = gf \u2227 R.arrows g) Y\u271d f\u271d\nh : Z\u271d \u27f6 Y\u271d\nj : Y\u271d \u27f6 Y\nk : j \u226b f = f\u271d\nz : R.arrows j\n\u22a2 (h \u226b j) \u226b f = h \u226b f\u271d\n[PROOFSTEP]\nsimp [k]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d\u00b9 : Y \u27f6 X\nS R\u271d : Sieve X\nf : Y \u27f6 X\nR : Sieve Y\nY\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 X\nx\u271d : (fun Z gf => \u2203 g, g \u226b f = gf \u2227 R.arrows g) Y\u271d f\u271d\nh : Z\u271d \u27f6 Y\u271d\nj : Y\u271d \u27f6 Y\nk : j \u226b f = f\u271d\nz : R.arrows j\n\u22a2 R.arrows (h \u226b j)\n[PROOFSTEP]\nsimp [z]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R\u271d : Sieve X\nf : Y \u27f6 X\ng : Z \u27f6 Y\nR : Sieve Z\nW : C\nh : W \u27f6 X\nx\u271d : (pushforward (g \u226b f) R).arrows h\nf\u2081 : W \u27f6 Z\nhq : f\u2081 \u226b g \u226b f = h\nhf\u2081 : R.arrows f\u2081\n\u22a2 (f\u2081 \u226b g) \u226b f = h\n[PROOFSTEP]\nsimpa\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R\u271d : Sieve X\nf : Y \u27f6 X\ng : Z \u27f6 Y\nR : Sieve Z\nW : C\nh : W \u27f6 X\nx\u271d : (pushforward f (pushforward g R)).arrows h\ny : W \u27f6 Y\nhy : y \u226b f = h\nz : W \u27f6 Z\nhR : z \u226b g = y\nhz : R.arrows z\n\u22a2 z \u226b g \u226b f = h \u2227 R.arrows z\n[PROOFSTEP]\nrw [\u2190 Category.assoc, hR]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R\u271d : Sieve X\nf : Y \u27f6 X\ng : Z \u27f6 Y\nR : Sieve Z\nW : C\nh : W \u27f6 X\nx\u271d : (pushforward f (pushforward g R)).arrows h\ny : W \u27f6 Y\nhy : y \u226b f = h\nz : W \u27f6 Z\nhR : z \u226b g = y\nhz : R.arrows z\n\u22a2 y \u226b f = h \u2227 R.arrows z\n[PROOFSTEP]\ntauto\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS\u271d R\u271d : Sieve X\nS : Presieve X\nR : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 S f \u2192 Sieve Y\nf : Y \u27f6 X\nh : S f\n\u22a2 pushforward f (R h) \u2264 bind S R\n[PROOFSTEP]\nrintro Z _ \u27e8g, rfl, hg\u27e9\n[GOAL]\ncase intro.intro\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z\u271d : C\nf\u271d : Y \u27f6 X\nS\u271d R\u271d : Sieve X\nS : Presieve X\nR : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 S f \u2192 Sieve Y\nf : Y \u27f6 X\nh : S f\nZ : C\ng : Z \u27f6 Y\nhg : (R h).arrows g\n\u22a2 (bind S R).arrows (g \u226b f)\n[PROOFSTEP]\nexact \u27e8_, g, f, h, hg, rfl\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS\u271d R\u271d : Sieve X\nS : Presieve X\nR : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 S f \u2192 Sieve Y\nf : Y \u27f6 X\nh : S f\n\u22a2 R h \u2264 pullback f (bind S R)\n[PROOFSTEP]\nrw [\u2190 galoisConnection f]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS\u271d R\u271d : Sieve X\nS : Presieve X\nR : \u2983Y : C\u2984 \u2192 \u2983f : Y \u27f6 X\u2984 \u2192 S f \u2192 Sieve Y\nf : Y \u27f6 X\nh : S f\n\u22a2 pushforward f (R h) \u2264 bind S R\n[PROOFSTEP]\napply pushforward_le_bind_of_mem\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R : Sieve X\nf : Y \u27f6 X\ninst\u271d : Mono f\n\u22a2 GaloisCoinsertion (pushforward f) (pullback f)\n[PROOFSTEP]\napply (galoisConnection f).toGaloisCoinsertion\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R : Sieve X\nf : Y \u27f6 X\ninst\u271d : Mono f\n\u22a2 \u2200 (a : Sieve Y), pullback f (pushforward f a) \u2264 a\n[PROOFSTEP]\nrintro S Z g \u27e8g\u2081, hf, hg\u2081\u27e9\n[GOAL]\ncase intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z\u271d : C\nf\u271d : Y \u27f6 X\nS\u271d R : Sieve X\nf : Y \u27f6 X\ninst\u271d : Mono f\nS : Sieve Y\nZ : C\ng g\u2081 : Z \u27f6 Y\nhf : g\u2081 \u226b f = g \u226b f\nhg\u2081 : S.arrows g\u2081\n\u22a2 S.arrows g\n[PROOFSTEP]\nrw [cancel_mono f] at hf \n[GOAL]\ncase intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z\u271d : C\nf\u271d : Y \u27f6 X\nS\u271d R : Sieve X\nf : Y \u27f6 X\ninst\u271d : Mono f\nS : Sieve Y\nZ : C\ng g\u2081 : Z \u27f6 Y\nhf : g\u2081 = g\nhg\u2081 : S.arrows g\u2081\n\u22a2 S.arrows g\n[PROOFSTEP]\nrwa [\u2190 hf]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R : Sieve X\nf : Y \u27f6 X\ninst\u271d : IsSplitEpi f\n\u22a2 GaloisInsertion (pushforward f) (pullback f)\n[PROOFSTEP]\napply (galoisConnection f).toGaloisInsertion\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R : Sieve X\nf : Y \u27f6 X\ninst\u271d : IsSplitEpi f\n\u22a2 \u2200 (b : Sieve X), b \u2264 pushforward f (pullback f b)\n[PROOFSTEP]\nintro S Z g hg\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z\u271d : C\nf\u271d : Y \u27f6 X\nS\u271d R : Sieve X\nf : Y \u27f6 X\ninst\u271d : IsSplitEpi f\nS : Sieve X\nZ : C\ng : Z \u27f6 X\nhg : S.arrows g\n\u22a2 (pushforward f (pullback f S)).arrows g\n[PROOFSTEP]\nrefine' \u27e8g \u226b section_ f, by simpa\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z\u271d : C\nf\u271d : Y \u27f6 X\nS\u271d R : Sieve X\nf : Y \u27f6 X\ninst\u271d : IsSplitEpi f\nS : Sieve X\nZ : C\ng : Z \u27f6 X\nhg : S.arrows g\n\u22a2 (g \u226b section_ f) \u226b f = g \u2227 (pullback f S).arrows (g \u226b section_ f)\n[PROOFSTEP]\nsimpa\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\u271d\nS R\u271d : Sieve X\u271d\ninst\u271d : HasPullbacks C\nX Y : C\nf : Y \u27f6 X\nR : Presieve X\n\u22a2 generate (Presieve.pullbackArrows f R) = pullback f (generate R)\n[PROOFSTEP]\next W g\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\u271d\nS R\u271d : Sieve X\u271d\ninst\u271d : HasPullbacks C\nX Y : C\nf : Y \u27f6 X\nR : Presieve X\nW : C\ng : W \u27f6 Y\n\u22a2 (generate (Presieve.pullbackArrows f R)).arrows g \u2194 (pullback f (generate R)).arrows g\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\u271d\nS R\u271d : Sieve X\u271d\ninst\u271d : HasPullbacks C\nX Y : C\nf : Y \u27f6 X\nR : Presieve X\nW : C\ng : W \u27f6 Y\n\u22a2 (generate (Presieve.pullbackArrows f R)).arrows g \u2192 (pullback f (generate R)).arrows g\n[PROOFSTEP]\nrintro \u27e8_, h, k, hk, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\u271d\nS R\u271d : Sieve X\u271d\ninst\u271d : HasPullbacks C\nX Y : C\nf : Y \u27f6 X\nR : Presieve X\nW w\u271d : C\nh : W \u27f6 w\u271d\nk : w\u271d \u27f6 Y\nhk : Presieve.pullbackArrows f R k\n\u22a2 (pullback f (generate R)).arrows (h \u226b k)\n[PROOFSTEP]\ncases' hk with W g hg\n[GOAL]\ncase h.mp.intro.intro.intro.intro.mk\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\u271d\nS R\u271d : Sieve X\u271d\ninst\u271d : HasPullbacks C\nX Y : C\nf : Y \u27f6 X\nR : Presieve X\nW\u271d W : C\ng : W \u27f6 X\nhg : R g\nh : W\u271d \u27f6 Limits.pullback g f\n\u22a2 (pullback f (generate R)).arrows (h \u226b pullback.snd)\n[PROOFSTEP]\nchange (Sieve.generate R).pullback f (h \u226b pullback.snd)\n[GOAL]\ncase h.mp.intro.intro.intro.intro.mk\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\u271d\nS R\u271d : Sieve X\u271d\ninst\u271d : HasPullbacks C\nX Y : C\nf : Y \u27f6 X\nR : Presieve X\nW\u271d W : C\ng : W \u27f6 X\nhg : R g\nh : W\u271d \u27f6 Limits.pullback g f\n\u22a2 (pullback f (generate R)).arrows (h \u226b pullback.snd)\n[PROOFSTEP]\nrw [Sieve.pullback_apply, assoc, \u2190 pullback.condition, \u2190 assoc]\n[GOAL]\ncase h.mp.intro.intro.intro.intro.mk\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\u271d\nS R\u271d : Sieve X\u271d\ninst\u271d : HasPullbacks C\nX Y : C\nf : Y \u27f6 X\nR : Presieve X\nW\u271d W : C\ng : W \u27f6 X\nhg : R g\nh : W\u271d \u27f6 Limits.pullback g f\n\u22a2 (generate R).arrows ((h \u226b pullback.fst) \u226b g)\n[PROOFSTEP]\nexact Sieve.downward_closed _ (by exact Sieve.le_generate R W hg) (h \u226b pullback.fst)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\u271d\nS R\u271d : Sieve X\u271d\ninst\u271d : HasPullbacks C\nX Y : C\nf : Y \u27f6 X\nR : Presieve X\nW\u271d W : C\ng : W \u27f6 X\nhg : R g\nh : W\u271d \u27f6 Limits.pullback g f\n\u22a2 (generate R).arrows g\n[PROOFSTEP]\nexact Sieve.le_generate R W hg\n[GOAL]\ncase h.mpr\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\u271d\nS R\u271d : Sieve X\u271d\ninst\u271d : HasPullbacks C\nX Y : C\nf : Y \u27f6 X\nR : Presieve X\nW : C\ng : W \u27f6 Y\n\u22a2 (pullback f (generate R)).arrows g \u2192 (generate (Presieve.pullbackArrows f R)).arrows g\n[PROOFSTEP]\nrintro \u27e8W, h, k, hk, comm\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\u271d\nS R\u271d : Sieve X\u271d\ninst\u271d : HasPullbacks C\nX Y : C\nf : Y \u27f6 X\nR : Presieve X\nW\u271d : C\ng : W\u271d \u27f6 Y\nW : C\nh : W\u271d \u27f6 W\nk : W \u27f6 X\nhk : R k\ncomm : h \u226b k = g \u226b f\n\u22a2 (generate (Presieve.pullbackArrows f R)).arrows g\n[PROOFSTEP]\nexact \u27e8_, _, _, Presieve.pullbackArrows.mk _ _ hk, pullback.lift_snd _ _ comm\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve (F.obj X)\n\u22a2 \u2200 {Y Z : C} {f : Y \u27f6 X},\n Presieve.functorPullback F R.arrows f \u2192 \u2200 (g : Z \u27f6 Y), Presieve.functorPullback F R.arrows (g \u226b f)\n[PROOFSTEP]\nintro _ _ f hf g\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve (F.obj X)\nY\u271d Z\u271d : C\nf : Y\u271d \u27f6 X\nhf : Presieve.functorPullback F R.arrows f\ng : Z\u271d \u27f6 Y\u271d\n\u22a2 Presieve.functorPullback F R.arrows (g \u226b f)\n[PROOFSTEP]\nunfold Presieve.functorPullback\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve (F.obj X)\nY\u271d Z\u271d : C\nf : Y\u271d \u27f6 X\nhf : Presieve.functorPullback F R.arrows f\ng : Z\u271d \u27f6 Y\u271d\n\u22a2 R.arrows (F.map (g \u226b f))\n[PROOFSTEP]\nrw [F.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve (F.obj X)\nY\u271d Z\u271d : C\nf : Y\u271d \u27f6 X\nhf : Presieve.functorPullback F R.arrows f\ng : Z\u271d \u27f6 Y\u271d\n\u22a2 R.arrows (F.map g \u226b F.map f)\n[PROOFSTEP]\nexact R.downward_closed hf (F.map g)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve X\n\u22a2 functorPullback (\ud835\udfed C) R = R\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve X\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\n\u22a2 (functorPullback (\ud835\udfed C) R).arrows f\u271d \u2194 R.arrows f\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve ((F \u22d9 G).obj X)\n\u22a2 functorPullback (F \u22d9 G) R = functorPullback F (functorPullback G R)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve ((F \u22d9 G).obj X)\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\n\u22a2 (functorPullback (F \u22d9 G) R).arrows f\u271d \u2194 (functorPullback F (functorPullback G R)).arrows f\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Presieve X\n\u22a2 Presieve.functorPushforward F (generate R).arrows = Presieve.functorPushforward F R\n[PROOFSTEP]\nfunext Y\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Presieve X\nY : D\n\u22a2 Presieve.functorPushforward F (generate R).arrows = Presieve.functorPushforward F R\n[PROOFSTEP]\next f\n[GOAL]\ncase h.h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Presieve X\nY : D\nf : Y \u27f6 F.obj X\n\u22a2 f \u2208 Presieve.functorPushforward F (generate R).arrows \u2194 f \u2208 Presieve.functorPushforward F R\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.h.mp\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Presieve X\nY : D\nf : Y \u27f6 F.obj X\n\u22a2 f \u2208 Presieve.functorPushforward F (generate R).arrows \u2192 f \u2208 Presieve.functorPushforward F R\n[PROOFSTEP]\nrintro \u27e8X', g, f', \u27e8X'', g', f'', h\u2081, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase h.h.mp.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Presieve X\nY : D\nX' : C\nf' : Y \u27f6 F.obj X'\nX'' : C\ng' : X' \u27f6 X''\nf'' : X'' \u27f6 X\nh\u2081 : R f''\n\u22a2 f' \u226b F.map (g' \u226b f'') \u2208 Presieve.functorPushforward F R\n[PROOFSTEP]\nexact \u27e8X'', f'', f' \u226b F.map g', h\u2081, by simp\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Presieve X\nY : D\nX' : C\nf' : Y \u27f6 F.obj X'\nX'' : C\ng' : X' \u27f6 X''\nf'' : X'' \u27f6 X\nh\u2081 : R f''\n\u22a2 f' \u226b F.map (g' \u226b f'') = (f' \u226b F.map g') \u226b F.map f''\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.h.mpr\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Presieve X\nY : D\nf : Y \u27f6 F.obj X\n\u22a2 f \u2208 Presieve.functorPushforward F R \u2192 f \u2208 Presieve.functorPushforward F (generate R).arrows\n[PROOFSTEP]\nrintro \u27e8X', g, f', h\u2081, h\u2082\u27e9\n[GOAL]\ncase h.h.mpr.intro.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Presieve X\nY : D\nf : Y \u27f6 F.obj X\nX' : C\ng : X' \u27f6 X\nf' : Y \u27f6 F.obj X'\nh\u2081 : R g\nh\u2082 : f = f' \u226b F.map g\n\u22a2 f \u2208 Presieve.functorPushforward F (generate R).arrows\n[PROOFSTEP]\nexact \u27e8X', g, f', le_generate R _ h\u2081, h\u2082\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve X\n\u22a2 \u2200 {Y Z : D} {f : Y \u27f6 F.obj X},\n Presieve.functorPushforward F R.arrows f \u2192 \u2200 (g : Z \u27f6 Y), Presieve.functorPushforward F R.arrows (g \u226b f)\n[PROOFSTEP]\nintro _ _ f h g\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve X\nY\u271d Z\u271d : D\nf : Y\u271d \u27f6 F.obj X\nh : Presieve.functorPushforward F R.arrows f\ng : Z\u271d \u27f6 Y\u271d\n\u22a2 Presieve.functorPushforward F R.arrows (g \u226b f)\n[PROOFSTEP]\nobtain \u27e8X, \u03b1, \u03b2, h\u03b1, rfl\u27e9 := h\n[GOAL]\ncase intro.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS R\u271d : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve X\u271d\nY\u271d Z\u271d : D\ng : Z\u271d \u27f6 Y\u271d\nX : C\n\u03b1 : X \u27f6 X\u271d\n\u03b2 : Y\u271d \u27f6 F.obj X\nh\u03b1 : R.arrows \u03b1\n\u22a2 Presieve.functorPushforward F R.arrows (g \u226b \u03b2 \u226b F.map \u03b1)\n[PROOFSTEP]\nexact \u27e8X, \u03b1, g \u226b \u03b2, h\u03b1, by simp\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS R\u271d : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve X\u271d\nY\u271d Z\u271d : D\ng : Z\u271d \u27f6 Y\u271d\nX : C\n\u03b1 : X \u27f6 X\u271d\n\u03b2 : Y\u271d \u27f6 F.obj X\nh\u03b1 : R.arrows \u03b1\n\u22a2 g \u226b \u03b2 \u226b F.map \u03b1 = (g \u226b \u03b2) \u226b F.map \u03b1\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve X\n\u22a2 functorPushforward (\ud835\udfed C) R = R\n[PROOFSTEP]\next X f\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf\u271d : Y \u27f6 X\u271d\nS R\u271d : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve X\u271d\nX : C\nf : X \u27f6 (\ud835\udfed C).obj X\u271d\n\u22a2 (functorPushforward (\ud835\udfed C) R).arrows f \u2194 R.arrows f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf\u271d : Y \u27f6 X\u271d\nS R\u271d : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve X\u271d\nX : C\nf : X \u27f6 (\ud835\udfed C).obj X\u271d\n\u22a2 (functorPushforward (\ud835\udfed C) R).arrows f \u2192 R.arrows f\n[PROOFSTEP]\nintro hf\n[GOAL]\ncase h.mp\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf\u271d : Y \u27f6 X\u271d\nS R\u271d : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve X\u271d\nX : C\nf : X \u27f6 (\ud835\udfed C).obj X\u271d\nhf : (functorPushforward (\ud835\udfed C) R).arrows f\n\u22a2 R.arrows f\n[PROOFSTEP]\nobtain \u27e8X, g, h, hg, rfl\u27e9 := hf\n[GOAL]\ncase h.mp.intro.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d\u00b9 Y Z : C\nf : Y \u27f6 X\u271d\u00b9\nS R\u271d : Sieve X\u271d\u00b9\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve X\u271d\u00b9\nX\u271d X : C\ng : X \u27f6 X\u271d\u00b9\nh : X\u271d \u27f6 (\ud835\udfed C).obj X\nhg : R.arrows g\n\u22a2 R.arrows (h \u226b (\ud835\udfed C).map g)\n[PROOFSTEP]\nexact R.downward_closed hg h\n[GOAL]\ncase h.mpr\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf\u271d : Y \u27f6 X\u271d\nS R\u271d : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve X\u271d\nX : C\nf : X \u27f6 (\ud835\udfed C).obj X\u271d\n\u22a2 R.arrows f \u2192 (functorPushforward (\ud835\udfed C) R).arrows f\n[PROOFSTEP]\nintro hf\n[GOAL]\ncase h.mpr\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf\u271d : Y \u27f6 X\u271d\nS R\u271d : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve X\u271d\nX : C\nf : X \u27f6 (\ud835\udfed C).obj X\u271d\nhf : R.arrows f\n\u22a2 (functorPushforward (\ud835\udfed C) R).arrows f\n[PROOFSTEP]\nexact \u27e8X, f, \ud835\udfd9 _, hf, by simp\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf\u271d : Y \u27f6 X\u271d\nS R\u271d : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve X\u271d\nX : C\nf : X \u27f6 (\ud835\udfed C).obj X\u271d\nhf : R.arrows f\n\u22a2 f = \ud835\udfd9 X \u226b (\ud835\udfed C).map f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve X\n\u22a2 functorPushforward (F \u22d9 G) R = functorPushforward G (functorPushforward F R)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve X\nY\u271d : E\nf\u271d : Y\u271d \u27f6 (F \u22d9 G).obj X\n\u22a2 (functorPushforward (F \u22d9 G) R).arrows f\u271d \u2194 (functorPushforward G (functorPushforward F R)).arrows f\u271d\n[PROOFSTEP]\nsimp [R.arrows.functorPushforward_comp F G]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS R : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX : C\n\u22a2 GaloisConnection (functorPushforward F) (functorPullback F)\n[PROOFSTEP]\nintro R S\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS\u271d R\u271d : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX : C\nR : Sieve X\nS : Sieve (F.obj X)\n\u22a2 functorPushforward F R \u2264 S \u2194 R \u2264 functorPullback F S\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS\u271d R\u271d : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX : C\nR : Sieve X\nS : Sieve (F.obj X)\n\u22a2 functorPushforward F R \u2264 S \u2192 R \u2264 functorPullback F S\n[PROOFSTEP]\nintro hle X f hf\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d\u00b9 Y Z : C\nf\u271d : Y \u27f6 X\u271d\u00b9\nS\u271d R\u271d : Sieve X\u271d\u00b9\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX\u271d : C\nR : Sieve X\u271d\nS : Sieve (F.obj X\u271d)\nhle : functorPushforward F R \u2264 S\nX : C\nf : X \u27f6 X\u271d\nhf : R.arrows f\n\u22a2 (functorPullback F S).arrows f\n[PROOFSTEP]\napply hle\n[GOAL]\ncase mp.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d\u00b9 Y Z : C\nf\u271d : Y \u27f6 X\u271d\u00b9\nS\u271d R\u271d : Sieve X\u271d\u00b9\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX\u271d : C\nR : Sieve X\u271d\nS : Sieve (F.obj X\u271d)\nhle : functorPushforward F R \u2264 S\nX : C\nf : X \u27f6 X\u271d\nhf : R.arrows f\n\u22a2 (functorPushforward F R).arrows (F.map f)\n[PROOFSTEP]\nrefine' \u27e8X, f, \ud835\udfd9 _, hf, _\u27e9\n[GOAL]\ncase mp.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d\u00b9 Y Z : C\nf\u271d : Y \u27f6 X\u271d\u00b9\nS\u271d R\u271d : Sieve X\u271d\u00b9\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX\u271d : C\nR : Sieve X\u271d\nS : Sieve (F.obj X\u271d)\nhle : functorPushforward F R \u2264 S\nX : C\nf : X \u27f6 X\u271d\nhf : R.arrows f\n\u22a2 F.map f = \ud835\udfd9 (F.obj X) \u226b F.map f\n[PROOFSTEP]\nrw [id_comp]\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS\u271d R\u271d : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX : C\nR : Sieve X\nS : Sieve (F.obj X)\n\u22a2 R \u2264 functorPullback F S \u2192 functorPushforward F R \u2264 S\n[PROOFSTEP]\nrintro hle Y f \u27e8X, g, h, hg, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d\u00b9 Y\u271d Z : C\nf : Y\u271d \u27f6 X\u271d\u00b9\nS\u271d R\u271d : Sieve X\u271d\u00b9\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX\u271d : C\nR : Sieve X\u271d\nS : Sieve (F.obj X\u271d)\nhle : R \u2264 functorPullback F S\nY : D\nX : C\ng : X \u27f6 X\u271d\nh : Y \u27f6 F.obj X\nhg : R.arrows g\n\u22a2 S.arrows (h \u226b F.map g)\n[PROOFSTEP]\napply Sieve.downward_closed S\n[GOAL]\ncase mpr.intro.intro.intro.intro.x\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d\u00b9 Y\u271d Z : C\nf : Y\u271d \u27f6 X\u271d\u00b9\nS\u271d R\u271d : Sieve X\u271d\u00b9\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX\u271d : C\nR : Sieve X\u271d\nS : Sieve (F.obj X\u271d)\nhle : R \u2264 functorPullback F S\nY : D\nX : C\ng : X \u27f6 X\u271d\nh : Y \u27f6 F.obj X\nhg : R.arrows g\n\u22a2 S.arrows (F.map g)\n[PROOFSTEP]\nexact hle g hg\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF\u271d : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS R : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nF : C \u2964 D\nX : C\n\u22a2 functorPushforward F \u22a4 = \u22a4\n[PROOFSTEP]\nrefine' (generate_sieve _).symm.trans _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF\u271d : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS R : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nF : C \u2964 D\nX : C\n\u22a2 generate (functorPushforward F \u22a4).arrows = \u22a4\n[PROOFSTEP]\napply generate_of_contains_isSplitEpi (\ud835\udfd9 (F.obj X))\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF\u271d : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS R : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nF : C \u2964 D\nX : C\n\u22a2 (functorPushforward F \u22a4).arrows (\ud835\udfd9 (F.obj X))\n[PROOFSTEP]\nrefine' \u27e8X, \ud835\udfd9 _, \ud835\udfd9 _, trivial, by simp\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF\u271d : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS R : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nF : C \u2964 D\nX : C\n\u22a2 \ud835\udfd9 (F.obj X) = \ud835\udfd9 (F.obj X) \u226b F.map (\ud835\udfd9 X)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R\u271d : Sieve X\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nR : Sieve X\nV : C\nf : V \u27f6 X\nh : R.arrows f\n\u22a2 F.map f = \ud835\udfd9 (F.obj V) \u226b F.map f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS R : Sieve X\u271d\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\ninst\u271d\u00b9 : EssSurj F\ninst\u271d : Full F\nX : C\n\u22a2 GaloisInsertion (functorPushforward F) (functorPullback F)\n[PROOFSTEP]\napply (functor_galoisConnection F X).toGaloisInsertion\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS R : Sieve X\u271d\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\ninst\u271d\u00b9 : EssSurj F\ninst\u271d : Full F\nX : C\n\u22a2 \u2200 (b : Sieve (F.obj X)), b \u2264 functorPushforward F (functorPullback F b)\n[PROOFSTEP]\nintro S Y f hf\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\u271d\nS\u271d R : Sieve X\u271d\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\ninst\u271d\u00b9 : EssSurj F\ninst\u271d : Full F\nX : C\nS : Sieve (F.obj X)\nY : D\nf : Y \u27f6 F.obj X\nhf : S.arrows f\n\u22a2 (functorPushforward F (functorPullback F S)).arrows f\n[PROOFSTEP]\nrefine' \u27e8_, F.preimage ((F.objObjPreimageIso Y).hom \u226b f), (F.objObjPreimageIso Y).inv, _\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\u271d\nS\u271d R : Sieve X\u271d\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\ninst\u271d\u00b9 : EssSurj F\ninst\u271d : Full F\nX : C\nS : Sieve (F.obj X)\nY : D\nf : Y \u27f6 F.obj X\nhf : S.arrows f\n\u22a2 (functorPullback F S).arrows (F.preimage ((Functor.objObjPreimageIso F Y).hom \u226b f)) \u2227\n f = (Functor.objObjPreimageIso F Y).inv \u226b F.map (F.preimage ((Functor.objObjPreimageIso F Y).hom \u226b f))\n[PROOFSTEP]\nsimpa using S.downward_closed hf _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS R : Sieve X\u271d\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nX : C\n\u22a2 GaloisCoinsertion (functorPushforward F) (functorPullback F)\n[PROOFSTEP]\napply (functor_galoisConnection F X).toGaloisCoinsertion\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS R : Sieve X\u271d\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nX : C\n\u22a2 \u2200 (a : Sieve X), functorPullback F (functorPushforward F a) \u2264 a\n[PROOFSTEP]\nrintro S Y f \u27e8Z, g, h, h\u2081, h\u2082\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 X\u271d\nS\u271d R : Sieve X\u271d\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nX : C\nS : Sieve X\nY : C\nf : Y \u27f6 X\nZ : C\ng : Z \u27f6 X\nh : F.obj Y \u27f6 F.obj Z\nh\u2081 : S.arrows g\nh\u2082 : F.map f = h \u226b F.map g\n\u22a2 S.arrows f\n[PROOFSTEP]\nrw [\u2190 F.image_preimage h, \u2190 F.map_comp] at h\u2082 \n[GOAL]\ncase intro.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 X\u271d\nS\u271d R : Sieve X\u271d\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nX : C\nS : Sieve X\nY : C\nf : Y \u27f6 X\nZ : C\ng : Z \u27f6 X\nh : F.obj Y \u27f6 F.obj Z\nh\u2081 : S.arrows g\nh\u2082 : F.map f = F.map (F.preimage h \u226b g)\n\u22a2 S.arrows f\n[PROOFSTEP]\nrw [F.map_injective h\u2082]\n[GOAL]\ncase intro.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 X\u271d\nS\u271d R : Sieve X\u271d\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\nX : C\nS : Sieve X\nY : C\nf : Y \u27f6 X\nZ : C\ng : Z \u27f6 X\nh : F.obj Y \u27f6 F.obj Z\nh\u2081 : S.arrows g\nh\u2082 : F.map f = F.map (F.preimage h \u226b g)\n\u22a2 S.arrows (F.preimage h \u226b g)\n[PROOFSTEP]\nexact S.downward_closed h\u2081 _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R : Sieve X\nZ\u271d : C\u1d52\u1d56 \u2964 Type v\u2081\nf g : Z\u271d \u27f6 functor S\nh : f \u226b functorInclusion S = g \u226b functorInclusion S\n\u22a2 f = g\n[PROOFSTEP]\next Y y\n[GOAL]\ncase w.h.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z : C\nf\u271d : Y\u271d \u27f6 X\nS R : Sieve X\nZ\u271d : C\u1d52\u1d56 \u2964 Type v\u2081\nf g : Z\u271d \u27f6 functor S\nh : f \u226b functorInclusion S = g \u226b functorInclusion S\nY : C\u1d52\u1d56\ny : Z\u271d.obj Y\n\u22a2 NatTrans.app f Y y = NatTrans.app g Y y\n[PROOFSTEP]\nsimpa [Subtype.ext_iff_val] using congr_fun (NatTrans.congr_app h Y) y\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R\u271d : Sieve X\nR : C\u1d52\u1d56 \u2964 Type v\u2081\nf : R \u27f6 yoneda.obj X\n\u22a2 \u2200 {Y Z : C} {f_1 : Y \u27f6 X},\n (fun Y g => \u2203 t, NatTrans.app f (Opposite.op Y) t = g) Y f_1 \u2192\n \u2200 (g : Z \u27f6 Y), (fun Y g => \u2203 t, NatTrans.app f (Opposite.op Y) t = g) Z (g \u226b f_1)\n[PROOFSTEP]\nrintro Y Z _ \u27e8t, rfl\u27e9 g\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 X\nS R\u271d : Sieve X\nR : C\u1d52\u1d56 \u2964 Type v\u2081\nf : R \u27f6 yoneda.obj X\nY Z : C\nt : R.obj (Opposite.op Y)\ng : Z \u27f6 Y\n\u22a2 \u2203 t_1, NatTrans.app f (Opposite.op Z) t_1 = g \u226b NatTrans.app f (Opposite.op Y) t\n[PROOFSTEP]\nrefine' \u27e8R.map g.op t, _\u27e9\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 X\nS R\u271d : Sieve X\nR : C\u1d52\u1d56 \u2964 Type v\u2081\nf : R \u27f6 yoneda.obj X\nY Z : C\nt : R.obj (Opposite.op Y)\ng : Z \u27f6 Y\n\u22a2 NatTrans.app f (Opposite.op Z) (R.map g.op t) = g \u226b NatTrans.app f (Opposite.op Y) t\n[PROOFSTEP]\nrw [FunctorToTypes.naturality _ _ f]\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y\u271d Z\u271d : C\nf\u271d : Y\u271d \u27f6 X\nS R\u271d : Sieve X\nR : C\u1d52\u1d56 \u2964 Type v\u2081\nf : R \u27f6 yoneda.obj X\nY Z : C\nt : R.obj (Opposite.op Y)\ng : Z \u27f6 Y\n\u22a2 (yoneda.obj X).map g.op (NatTrans.app f (Opposite.op Y) t) = g \u226b NatTrans.app f (Opposite.op Y) t\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R : Sieve X\n\u22a2 sieveOfSubfunctor (functorInclusion S) = S\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R : Sieve X\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\n\u22a2 (sieveOfSubfunctor (functorInclusion S)).arrows f\u271d \u2194 S.arrows f\u271d\n[PROOFSTEP]\nsimp only [functorInclusion_app, sieveOfSubfunctor_apply]\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R : Sieve X\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\n\u22a2 (\u2203 t, \u2191t = f\u271d) \u2194 S.arrows f\u271d\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R : Sieve X\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\n\u22a2 (\u2203 t, \u2191t = f\u271d) \u2192 S.arrows f\u271d\n[PROOFSTEP]\nrintro \u27e8\u27e8f, hf\u27e9, rfl\u27e9\n[GOAL]\ncase h.mp.intro.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R : Sieve X\nY\u271d : C\nf : (Opposite.op Y\u271d).unop \u27f6 X\nhf : S.arrows f\n\u22a2 S.arrows \u2191{ val := f, property := hf }\n[PROOFSTEP]\nexact hf\n[GOAL]\ncase h.mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R : Sieve X\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\n\u22a2 S.arrows f\u271d \u2192 \u2203 t, \u2191t = f\u271d\n[PROOFSTEP]\nintro hf\n[GOAL]\ncase h.mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R : Sieve X\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\nhf : S.arrows f\u271d\n\u22a2 \u2203 t, \u2191t = f\u271d\n[PROOFSTEP]\nexact \u27e8\u27e8_, hf\u27e9, rfl\u27e9\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Sites.Sieves", "llama_tokens": 32831, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.4455295350395727, "lm_q2_score": 0.026355353706463396, "lm_q1q2_score": 0.011742088482644117}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b4 a\ns : Finset \u03b1\nhs : \u00aca \u2208 s\ne\u2081 e\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\neq : cons s a b e\u2081 = cons s a b e\u2082\ne : \u03b1\nh : e \u2208 a ::\u2098 s.val\n\u22a2 e \u2208 insert a s\n[PROOFSTEP]\nsimpa only [Multiset.mem_cons, mem_insert] using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b4 a\ns : Finset \u03b1\nhs : \u00aca \u2208 s\ne\u2081 e\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\neq : cons s a b e\u2081 = cons s a b e\u2082\ne : \u03b1\nh : e \u2208 a ::\u2098 s.val\n\u22a2 e \u2208 insert a s\n[PROOFSTEP]\nsimpa only [Multiset.mem_cons, mem_insert] using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b4 a\ns : Finset \u03b1\nhs : \u00aca \u2208 s\ne\u2081 e\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\neq : cons s a b e\u2081 = cons s a b e\u2082\ne : \u03b1\nh : e \u2208 a ::\u2098 s.val\n\u22a2 cons s a b e\u2081 e (_ : e \u2208 insert a s) = cons s a b e\u2082 e (_ : e \u2208 insert a s)\n[PROOFSTEP]\nrw [eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b2 a)\ns : Finset \u03b1\nt : (a : \u03b1) \u2192 Finset (\u03b2 a)\na : \u03b1\nha : \u00aca \u2208 s\n\u22a2 pi (insert a s) t = Finset.biUnion (t a) fun b => image (Pi.cons s a b) (pi s t)\n[PROOFSTEP]\napply eq_of_veq\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b2 a)\ns : Finset \u03b1\nt : (a : \u03b1) \u2192 Finset (\u03b2 a)\na : \u03b1\nha : \u00aca \u2208 s\n\u22a2 (pi (insert a s) t).val = (Finset.biUnion (t a) fun b => image (Pi.cons s a b) (pi s t)).val\n[PROOFSTEP]\nrw [\u2190 (pi (insert a s) t).2.dedup]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b2 a)\ns : Finset \u03b1\nt : (a : \u03b1) \u2192 Finset (\u03b2 a)\na : \u03b1\nha : \u00aca \u2208 s\n\u22a2 dedup (pi (insert a s) t).val = (Finset.biUnion (t a) fun b => image (Pi.cons s a b) (pi s t)).val\n[PROOFSTEP]\nrefine'\n (fun s' (h : s' = a ::\u2098 s.1) =>\n (_ :\n dedup (Multiset.pi s' fun a => (t a).1) =\n dedup\n ((t a).1.bind fun b =>\n dedup <|\n (Multiset.pi s.1 fun a : \u03b1 => (t a).val).map fun f a' h' => Multiset.Pi.cons s.1 a b f a' (h \u25b8 h'))))\n _ (insert_val_of_not_mem ha)\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b2 a)\ns : Finset \u03b1\nt : (a : \u03b1) \u2192 Finset (\u03b2 a)\na : \u03b1\nha : \u00aca \u2208 s\ns' : Multiset \u03b1\nh : s' = a ::\u2098 s.val\n\u22a2 dedup (Multiset.pi s' fun a => (t a).val) =\n dedup\n (Multiset.bind (t a).val fun b =>\n dedup\n (Multiset.map (fun f a' h' => Multiset.Pi.cons s.val a b f a' (_ : a' \u2208 a ::\u2098 s.val))\n (Multiset.pi s.val fun a => (t a).val)))\n[PROOFSTEP]\nsubst s'\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b2 a)\ns : Finset \u03b1\nt : (a : \u03b1) \u2192 Finset (\u03b2 a)\na : \u03b1\nha : \u00aca \u2208 s\n\u22a2 dedup (Multiset.pi (a ::\u2098 s.val) fun a => (t a).val) =\n dedup\n (Multiset.bind (t a).val fun b =>\n dedup\n (Multiset.map (fun f a' h' => Multiset.Pi.cons s.val a b f a' (_ : a' \u2208 a ::\u2098 s.val))\n (Multiset.pi s.val fun a => (t a).val)))\n[PROOFSTEP]\nrw [pi_cons]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b2 a)\ns : Finset \u03b1\nt : (a : \u03b1) \u2192 Finset (\u03b2 a)\na : \u03b1\nha : \u00aca \u2208 s\n\u22a2 dedup\n (Multiset.bind (t a).val fun b =>\n Multiset.map (Multiset.Pi.cons s.val a b) (Multiset.pi s.val fun a => (t a).val)) =\n dedup\n (Multiset.bind (t a).val fun b =>\n dedup\n (Multiset.map (fun f a' h' => Multiset.Pi.cons s.val a b f a' (_ : a' \u2208 a ::\u2098 s.val))\n (Multiset.pi s.val fun a => (t a).val)))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.e_s.e_f\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b2 a)\ns : Finset \u03b1\nt : (a : \u03b1) \u2192 Finset (\u03b2 a)\na : \u03b1\nha : \u00aca \u2208 s\n\u22a2 (fun b => Multiset.map (Multiset.Pi.cons s.val a b) (Multiset.pi s.val fun a => (t a).val)) = fun b =>\n dedup\n (Multiset.map (fun f a' h' => Multiset.Pi.cons s.val a b f a' (_ : a' \u2208 a ::\u2098 s.val))\n (Multiset.pi s.val fun a => (t a).val))\n[PROOFSTEP]\nfunext b\n[GOAL]\ncase a.e_s.e_f.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b2 a)\ns : Finset \u03b1\nt : (a : \u03b1) \u2192 Finset (\u03b2 a)\na : \u03b1\nha : \u00aca \u2208 s\nb : \u03b2 a\n\u22a2 Multiset.map (Multiset.Pi.cons s.val a b) (Multiset.pi s.val fun a => (t a).val) =\n dedup\n (Multiset.map (fun f a' h' => Multiset.Pi.cons s.val a b f a' (_ : a' \u2208 a ::\u2098 s.val))\n (Multiset.pi s.val fun a => (t a).val))\n[PROOFSTEP]\nexact ((pi s t).nodup.map <| Multiset.Pi.cons_injective ha).dedup.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ninst\u271d : DecidableEq \u03b1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 (pi s fun a => {f a}) = {fun a x => f a}\n[PROOFSTEP]\nrw [eq_singleton_iff_unique_mem]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ninst\u271d : DecidableEq \u03b1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 ((fun a x => f a) \u2208 pi s fun a => {f a}) \u2227 \u2200 (x : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2), (x \u2208 pi s fun a => {f a}) \u2192 x = fun a x => f a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1 : Type u_1\n\u03b2\u271d : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ninst\u271d : DecidableEq \u03b1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 (fun a x => f a) \u2208 pi s fun a => {f a}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2\u271d : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ninst\u271d : DecidableEq \u03b1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (x : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2), (x \u2208 pi s fun a => {f a}) \u2192 x = fun a x => f a\n[PROOFSTEP]\nintro a ha\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2\u271d : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ninst\u271d : DecidableEq \u03b1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\na : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nha : a \u2208 pi s fun a => {f a}\n\u22a2 a = fun a x => f a\n[PROOFSTEP]\next i hi\n[GOAL]\ncase right.h.h\n\u03b1 : Type u_1\n\u03b2\u271d : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ninst\u271d : DecidableEq \u03b1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\na : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nha : a \u2208 pi s fun a => {f a}\ni : \u03b1\nhi : i \u2208 s\n\u22a2 a i hi = f i\n[PROOFSTEP]\nrw [mem_pi] at ha \n[GOAL]\ncase right.h.h\n\u03b1 : Type u_1\n\u03b2\u271d : \u03b1 \u2192 Type u\n\u03b4 : \u03b1 \u2192 Sort v\ninst\u271d : DecidableEq \u03b1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\na : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b2\nha : \u2200 (a_1 : \u03b1) (h : a_1 \u2208 s), a a_1 h \u2208 {f a_1}\ni : \u03b1\nhi : i \u2208 s\n\u22a2 a i hi = f i\n[PROOFSTEP]\nsimpa using ha i hi\n", "meta": {"mathlib_filename": "Mathlib.Data.Finset.Pi", "llama_tokens": 3113, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.4339814648038985, "lm_q2_score": 0.02595735682104226, "lm_q1q2_score": 0.011265011735633386}} {"text": "[GOAL]\nR : Type u_1\nc\u2081 c\u2082 : R\nf : Fin 4 \u2192 R\n\u22a2 (fun a => ![a.re, a.imI, a.imJ, a.imK]) ((fun a => { re := a 0, imI := a 1, imJ := a 2, imK := a 3 }) f) = f\n[PROOFSTEP]\next \u27e8_, _ | _ | _ | _ | _ | \u27e8\u27e9\u27e9\n[GOAL]\ncase h.mk.refl\nR : Type u_1\nc\u2081 c\u2082 : R\nf : Fin 4 \u2192 R\n\u22a2 (fun a => ![a.re, a.imI, a.imJ, a.imK]) ((fun a => { re := a 0, imI := a 1, imJ := a 2, imK := a 3 }) f)\n { val := 3, isLt := (_ : Nat.le (Nat.succ 3) (Nat.succ 3)) } =\n f { val := 3, isLt := (_ : Nat.le (Nat.succ 3) (Nat.succ 3)) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.mk.step.refl\nR : Type u_1\nc\u2081 c\u2082 : R\nf : Fin 4 \u2192 R\n\u22a2 (fun a => ![a.re, a.imI, a.imJ, a.imK]) ((fun a => { re := a 0, imI := a 1, imJ := a 2, imK := a 3 }) f)\n { val := 2, isLt := (_ : Nat.le (Nat.succ 2) (Nat.succ 3)) } =\n f { val := 2, isLt := (_ : Nat.le (Nat.succ 2) (Nat.succ 3)) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.mk.step.step.refl\nR : Type u_1\nc\u2081 c\u2082 : R\nf : Fin 4 \u2192 R\n\u22a2 (fun a => ![a.re, a.imI, a.imJ, a.imK]) ((fun a => { re := a 0, imI := a 1, imJ := a 2, imK := a 3 }) f)\n { val := 1, isLt := (_ : Nat.le (Nat.succ 1) (Nat.succ 3)) } =\n f { val := 1, isLt := (_ : Nat.le (Nat.succ 1) (Nat.succ 3)) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.mk.step.step.step.refl\nR : Type u_1\nc\u2081 c\u2082 : R\nf : Fin 4 \u2192 R\n\u22a2 (fun a => ![a.re, a.imI, a.imJ, a.imK]) ((fun a => { re := a 0, imI := a 1, imJ := a 2, imK := a 3 }) f)\n { val := 0, isLt := (_ : Nat.le (Nat.succ 0) (Nat.succ 3)) } =\n f { val := 0, isLt := (_ : Nat.le (Nat.succ 0) (Nat.succ 3)) }\n[PROOFSTEP]\nrfl\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 \u2191(x + y) = \u2191x + \u2191y\n[PROOFSTEP]\next\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (\u2191(x + y)).re = (\u2191x + \u2191y).re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (\u2191(x + y)).imI = (\u2191x + \u2191y).imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (\u2191(x + y)).imJ = (\u2191x + \u2191y).imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (\u2191(x + y)).imK = (\u2191x + \u2191y).imK\n[PROOFSTEP]\nsimp\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 \u2191(-x) = -\u2191x\n[PROOFSTEP]\next\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (\u2191(-x)).re = (-\u2191x).re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (\u2191(-x)).imI = (-\u2191x).imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (\u2191(-x)).imJ = (-\u2191x).imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (\u2191(-x)).imK = (-\u2191x).imK\n[PROOFSTEP]\nsimp\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u2074 : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b3 : SMul S R\ninst\u271d\u00b2 : SMul T R\ns\u271d : S\ninst\u271d\u00b9 : SMul S T\ninst\u271d : IsScalarTower S T R\ns : S\nt : T\nx : \u210d[R,c\u2081,c\u2082]\n\u22a2 (s \u2022 t) \u2022 x = s \u2022 t \u2022 x\n[PROOFSTEP]\next\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u2074 : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b3 : SMul S R\ninst\u271d\u00b2 : SMul T R\ns\u271d : S\ninst\u271d\u00b9 : SMul S T\ninst\u271d : IsScalarTower S T R\ns : S\nt : T\nx : \u210d[R,c\u2081,c\u2082]\n\u22a2 ((s \u2022 t) \u2022 x).re = (s \u2022 t \u2022 x).re\n[PROOFSTEP]\nexact smul_assoc _ _ _\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u2074 : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b3 : SMul S R\ninst\u271d\u00b2 : SMul T R\ns\u271d : S\ninst\u271d\u00b9 : SMul S T\ninst\u271d : IsScalarTower S T R\ns : S\nt : T\nx : \u210d[R,c\u2081,c\u2082]\n\u22a2 ((s \u2022 t) \u2022 x).imI = (s \u2022 t \u2022 x).imI\n[PROOFSTEP]\nexact smul_assoc _ _ _\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u2074 : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b3 : SMul S R\ninst\u271d\u00b2 : SMul T R\ns\u271d : S\ninst\u271d\u00b9 : SMul S T\ninst\u271d : IsScalarTower S T R\ns : S\nt : T\nx : \u210d[R,c\u2081,c\u2082]\n\u22a2 ((s \u2022 t) \u2022 x).imJ = (s \u2022 t \u2022 x).imJ\n[PROOFSTEP]\nexact smul_assoc _ _ _\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u2074 : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b3 : SMul S R\ninst\u271d\u00b2 : SMul T R\ns\u271d : S\ninst\u271d\u00b9 : SMul S T\ninst\u271d : IsScalarTower S T R\ns : S\nt : T\nx : \u210d[R,c\u2081,c\u2082]\n\u22a2 ((s \u2022 t) \u2022 x).imK = (s \u2022 t \u2022 x).imK\n[PROOFSTEP]\nexact smul_assoc _ _ _\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b2 : SMul S R\ninst\u271d\u00b9 : SMul T R\ns\u271d : S\ninst\u271d : SMulCommClass S T R\ns : S\nt : T\nx : \u210d[R,c\u2081,c\u2082]\n\u22a2 s \u2022 t \u2022 x = t \u2022 s \u2022 x\n[PROOFSTEP]\next\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b2 : SMul S R\ninst\u271d\u00b9 : SMul T R\ns\u271d : S\ninst\u271d : SMulCommClass S T R\ns : S\nt : T\nx : \u210d[R,c\u2081,c\u2082]\n\u22a2 (s \u2022 t \u2022 x).re = (t \u2022 s \u2022 x).re\n[PROOFSTEP]\nexact smul_comm _ _ _\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b2 : SMul S R\ninst\u271d\u00b9 : SMul T R\ns\u271d : S\ninst\u271d : SMulCommClass S T R\ns : S\nt : T\nx : \u210d[R,c\u2081,c\u2082]\n\u22a2 (s \u2022 t \u2022 x).imI = (t \u2022 s \u2022 x).imI\n[PROOFSTEP]\nexact smul_comm _ _ _\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b2 : SMul S R\ninst\u271d\u00b9 : SMul T R\ns\u271d : S\ninst\u271d : SMulCommClass S T R\ns : S\nt : T\nx : \u210d[R,c\u2081,c\u2082]\n\u22a2 (s \u2022 t \u2022 x).imJ = (t \u2022 s \u2022 x).imJ\n[PROOFSTEP]\nexact smul_comm _ _ _\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b2 : SMul S R\ninst\u271d\u00b9 : SMul T R\ns\u271d : S\ninst\u271d : SMulCommClass S T R\ns : S\nt : T\nx : \u210d[R,c\u2081,c\u2082]\n\u22a2 (s \u2022 t \u2022 x).imK = (t \u2022 s \u2022 x).imK\n[PROOFSTEP]\nexact smul_comm _ _ _\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 NatCast.natCast 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 \u2200 (n : \u2115), NatCast.natCast (n + 1) = NatCast.natCast n + 1\n[PROOFSTEP]\nsimp\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nn : \u2115\n\u22a2 IntCast.intCast (Int.negSucc n) = -\u2191(n + 1)\n[PROOFSTEP]\nchange coe _ = -coe _\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nn : \u2115\n\u22a2 \u2191\u2191(Int.negSucc n) = -\u2191\u2191(n + 1)\n[PROOFSTEP]\nrw [Int.cast_negSucc, coe_neg]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 x\u271d\u00b2 * (x\u271d\u00b9 + x\u271d) = x\u271d\u00b2 * x\u271d\u00b9 + x\u271d\u00b2 * x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d\u00b2 * (x\u271d\u00b9 + x\u271d)).re = (x\u271d\u00b2 * x\u271d\u00b9 + x\u271d\u00b2 * x\u271d).re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d\u00b2 * (x\u271d\u00b9 + x\u271d)).imI = (x\u271d\u00b2 * x\u271d\u00b9 + x\u271d\u00b2 * x\u271d).imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d\u00b2 * (x\u271d\u00b9 + x\u271d)).imJ = (x\u271d\u00b2 * x\u271d\u00b9 + x\u271d\u00b2 * x\u271d).imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d\u00b2 * (x\u271d\u00b9 + x\u271d)).imK = (x\u271d\u00b2 * x\u271d\u00b9 + x\u271d\u00b2 * x\u271d).imK\n[PROOFSTEP]\nsimp\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 x\u271d\u00b2.re * (x\u271d\u00b9.re + x\u271d.re) + c\u2081 * x\u271d\u00b2.imI * (x\u271d\u00b9.imI + x\u271d.imI) + c\u2082 * x\u271d\u00b2.imJ * (x\u271d\u00b9.imJ + x\u271d.imJ) -\n c\u2081 * c\u2082 * x\u271d\u00b2.imK * (x\u271d\u00b9.imK + x\u271d.imK) =\n x\u271d\u00b2.re * x\u271d\u00b9.re + c\u2081 * x\u271d\u00b2.imI * x\u271d\u00b9.imI + c\u2082 * x\u271d\u00b2.imJ * x\u271d\u00b9.imJ - c\u2081 * c\u2082 * x\u271d\u00b2.imK * x\u271d\u00b9.imK +\n (x\u271d\u00b2.re * x\u271d.re + c\u2081 * x\u271d\u00b2.imI * x\u271d.imI + c\u2082 * x\u271d\u00b2.imJ * x\u271d.imJ - c\u2081 * c\u2082 * x\u271d\u00b2.imK * x\u271d.imK)\n[PROOFSTEP]\nring\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 x\u271d\u00b2.re * (x\u271d\u00b9.imI + x\u271d.imI) + x\u271d\u00b2.imI * (x\u271d\u00b9.re + x\u271d.re) - c\u2082 * x\u271d\u00b2.imJ * (x\u271d\u00b9.imK + x\u271d.imK) +\n c\u2082 * x\u271d\u00b2.imK * (x\u271d\u00b9.imJ + x\u271d.imJ) =\n x\u271d\u00b2.re * x\u271d\u00b9.imI + x\u271d\u00b2.imI * x\u271d\u00b9.re - c\u2082 * x\u271d\u00b2.imJ * x\u271d\u00b9.imK + c\u2082 * x\u271d\u00b2.imK * x\u271d\u00b9.imJ +\n (x\u271d\u00b2.re * x\u271d.imI + x\u271d\u00b2.imI * x\u271d.re - c\u2082 * x\u271d\u00b2.imJ * x\u271d.imK + c\u2082 * x\u271d\u00b2.imK * x\u271d.imJ)\n[PROOFSTEP]\nring\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 x\u271d\u00b2.re * (x\u271d\u00b9.imJ + x\u271d.imJ) + c\u2081 * x\u271d\u00b2.imI * (x\u271d\u00b9.imK + x\u271d.imK) + x\u271d\u00b2.imJ * (x\u271d\u00b9.re + x\u271d.re) -\n c\u2081 * x\u271d\u00b2.imK * (x\u271d\u00b9.imI + x\u271d.imI) =\n x\u271d\u00b2.re * x\u271d\u00b9.imJ + c\u2081 * x\u271d\u00b2.imI * x\u271d\u00b9.imK + x\u271d\u00b2.imJ * x\u271d\u00b9.re - c\u2081 * x\u271d\u00b2.imK * x\u271d\u00b9.imI +\n (x\u271d\u00b2.re * x\u271d.imJ + c\u2081 * x\u271d\u00b2.imI * x\u271d.imK + x\u271d\u00b2.imJ * x\u271d.re - c\u2081 * x\u271d\u00b2.imK * x\u271d.imI)\n[PROOFSTEP]\nring\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 x\u271d\u00b2.re * (x\u271d\u00b9.imK + x\u271d.imK) + x\u271d\u00b2.imI * (x\u271d\u00b9.imJ + x\u271d.imJ) - x\u271d\u00b2.imJ * (x\u271d\u00b9.imI + x\u271d.imI) +\n x\u271d\u00b2.imK * (x\u271d\u00b9.re + x\u271d.re) =\n x\u271d\u00b2.re * x\u271d\u00b9.imK + x\u271d\u00b2.imI * x\u271d\u00b9.imJ - x\u271d\u00b2.imJ * x\u271d\u00b9.imI + x\u271d\u00b2.imK * x\u271d\u00b9.re +\n (x\u271d\u00b2.re * x\u271d.imK + x\u271d\u00b2.imI * x\u271d.imJ - x\u271d\u00b2.imJ * x\u271d.imI + x\u271d\u00b2.imK * x\u271d.re)\n[PROOFSTEP]\nring\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d\u00b2 + x\u271d\u00b9) * x\u271d = x\u271d\u00b2 * x\u271d + x\u271d\u00b9 * x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 ((x\u271d\u00b2 + x\u271d\u00b9) * x\u271d).re = (x\u271d\u00b2 * x\u271d + x\u271d\u00b9 * x\u271d).re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 ((x\u271d\u00b2 + x\u271d\u00b9) * x\u271d).imI = (x\u271d\u00b2 * x\u271d + x\u271d\u00b9 * x\u271d).imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 ((x\u271d\u00b2 + x\u271d\u00b9) * x\u271d).imJ = (x\u271d\u00b2 * x\u271d + x\u271d\u00b9 * x\u271d).imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 ((x\u271d\u00b2 + x\u271d\u00b9) * x\u271d).imK = (x\u271d\u00b2 * x\u271d + x\u271d\u00b9 * x\u271d).imK\n[PROOFSTEP]\nsimp\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d\u00b2.re + x\u271d\u00b9.re) * x\u271d.re + c\u2081 * (x\u271d\u00b2.imI + x\u271d\u00b9.imI) * x\u271d.imI + c\u2082 * (x\u271d\u00b2.imJ + x\u271d\u00b9.imJ) * x\u271d.imJ -\n c\u2081 * c\u2082 * (x\u271d\u00b2.imK + x\u271d\u00b9.imK) * x\u271d.imK =\n x\u271d\u00b2.re * x\u271d.re + c\u2081 * x\u271d\u00b2.imI * x\u271d.imI + c\u2082 * x\u271d\u00b2.imJ * x\u271d.imJ - c\u2081 * c\u2082 * x\u271d\u00b2.imK * x\u271d.imK +\n (x\u271d\u00b9.re * x\u271d.re + c\u2081 * x\u271d\u00b9.imI * x\u271d.imI + c\u2082 * x\u271d\u00b9.imJ * x\u271d.imJ - c\u2081 * c\u2082 * x\u271d\u00b9.imK * x\u271d.imK)\n[PROOFSTEP]\nring\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d\u00b2.re + x\u271d\u00b9.re) * x\u271d.imI + (x\u271d\u00b2.imI + x\u271d\u00b9.imI) * x\u271d.re - c\u2082 * (x\u271d\u00b2.imJ + x\u271d\u00b9.imJ) * x\u271d.imK +\n c\u2082 * (x\u271d\u00b2.imK + x\u271d\u00b9.imK) * x\u271d.imJ =\n x\u271d\u00b2.re * x\u271d.imI + x\u271d\u00b2.imI * x\u271d.re - c\u2082 * x\u271d\u00b2.imJ * x\u271d.imK + c\u2082 * x\u271d\u00b2.imK * x\u271d.imJ +\n (x\u271d\u00b9.re * x\u271d.imI + x\u271d\u00b9.imI * x\u271d.re - c\u2082 * x\u271d\u00b9.imJ * x\u271d.imK + c\u2082 * x\u271d\u00b9.imK * x\u271d.imJ)\n[PROOFSTEP]\nring\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d\u00b2.re + x\u271d\u00b9.re) * x\u271d.imJ + c\u2081 * (x\u271d\u00b2.imI + x\u271d\u00b9.imI) * x\u271d.imK + (x\u271d\u00b2.imJ + x\u271d\u00b9.imJ) * x\u271d.re -\n c\u2081 * (x\u271d\u00b2.imK + x\u271d\u00b9.imK) * x\u271d.imI =\n x\u271d\u00b2.re * x\u271d.imJ + c\u2081 * x\u271d\u00b2.imI * x\u271d.imK + x\u271d\u00b2.imJ * x\u271d.re - c\u2081 * x\u271d\u00b2.imK * x\u271d.imI +\n (x\u271d\u00b9.re * x\u271d.imJ + c\u2081 * x\u271d\u00b9.imI * x\u271d.imK + x\u271d\u00b9.imJ * x\u271d.re - c\u2081 * x\u271d\u00b9.imK * x\u271d.imI)\n[PROOFSTEP]\nring\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d\u00b2.re + x\u271d\u00b9.re) * x\u271d.imK + (x\u271d\u00b2.imI + x\u271d\u00b9.imI) * x\u271d.imJ - (x\u271d\u00b2.imJ + x\u271d\u00b9.imJ) * x\u271d.imI +\n (x\u271d\u00b2.imK + x\u271d\u00b9.imK) * x\u271d.re =\n x\u271d\u00b2.re * x\u271d.imK + x\u271d\u00b2.imI * x\u271d.imJ - x\u271d\u00b2.imJ * x\u271d.imI + x\u271d\u00b2.imK * x\u271d.re +\n (x\u271d\u00b9.re * x\u271d.imK + x\u271d\u00b9.imI * x\u271d.imJ - x\u271d\u00b9.imJ * x\u271d.imI + x\u271d\u00b9.imK * x\u271d.re)\n[PROOFSTEP]\nring\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 0 * x\u271d = 0\n[PROOFSTEP]\next\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (0 * x\u271d).re = 0.re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (0 * x\u271d).imI = 0.imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (0 * x\u271d).imJ = 0.imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (0 * x\u271d).imK = 0.imK\n[PROOFSTEP]\nsimp\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 x\u271d * 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d * 0).re = 0.re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d * 0).imI = 0.imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d * 0).imJ = 0.imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d * 0).imK = 0.imK\n[PROOFSTEP]\nsimp\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 x\u271d\u00b2 * x\u271d\u00b9 * x\u271d = x\u271d\u00b2 * (x\u271d\u00b9 * x\u271d)\n[PROOFSTEP]\next\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d\u00b2 * x\u271d\u00b9 * x\u271d).re = (x\u271d\u00b2 * (x\u271d\u00b9 * x\u271d)).re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d\u00b2 * x\u271d\u00b9 * x\u271d).imI = (x\u271d\u00b2 * (x\u271d\u00b9 * x\u271d)).imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d\u00b2 * x\u271d\u00b9 * x\u271d).imJ = (x\u271d\u00b2 * (x\u271d\u00b9 * x\u271d)).imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d\u00b2 * x\u271d\u00b9 * x\u271d).imK = (x\u271d\u00b2 * (x\u271d\u00b9 * x\u271d)).imK\n[PROOFSTEP]\nsimp\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d\u00b2.re * x\u271d\u00b9.re + c\u2081 * x\u271d\u00b2.imI * x\u271d\u00b9.imI + c\u2082 * x\u271d\u00b2.imJ * x\u271d\u00b9.imJ - c\u2081 * c\u2082 * x\u271d\u00b2.imK * x\u271d\u00b9.imK) * x\u271d.re +\n c\u2081 * (x\u271d\u00b2.re * x\u271d\u00b9.imI + x\u271d\u00b2.imI * x\u271d\u00b9.re - c\u2082 * x\u271d\u00b2.imJ * x\u271d\u00b9.imK + c\u2082 * x\u271d\u00b2.imK * x\u271d\u00b9.imJ) * x\u271d.imI +\n c\u2082 * (x\u271d\u00b2.re * x\u271d\u00b9.imJ + c\u2081 * x\u271d\u00b2.imI * x\u271d\u00b9.imK + x\u271d\u00b2.imJ * x\u271d\u00b9.re - c\u2081 * x\u271d\u00b2.imK * x\u271d\u00b9.imI) * x\u271d.imJ -\n c\u2081 * c\u2082 * (x\u271d\u00b2.re * x\u271d\u00b9.imK + x\u271d\u00b2.imI * x\u271d\u00b9.imJ - x\u271d\u00b2.imJ * x\u271d\u00b9.imI + x\u271d\u00b2.imK * x\u271d\u00b9.re) * x\u271d.imK =\n x\u271d\u00b2.re * (x\u271d\u00b9.re * x\u271d.re + c\u2081 * x\u271d\u00b9.imI * x\u271d.imI + c\u2082 * x\u271d\u00b9.imJ * x\u271d.imJ - c\u2081 * c\u2082 * x\u271d\u00b9.imK * x\u271d.imK) +\n c\u2081 * x\u271d\u00b2.imI * (x\u271d\u00b9.re * x\u271d.imI + x\u271d\u00b9.imI * x\u271d.re - c\u2082 * x\u271d\u00b9.imJ * x\u271d.imK + c\u2082 * x\u271d\u00b9.imK * x\u271d.imJ) +\n c\u2082 * x\u271d\u00b2.imJ * (x\u271d\u00b9.re * x\u271d.imJ + c\u2081 * x\u271d\u00b9.imI * x\u271d.imK + x\u271d\u00b9.imJ * x\u271d.re - c\u2081 * x\u271d\u00b9.imK * x\u271d.imI) -\n c\u2081 * c\u2082 * x\u271d\u00b2.imK * (x\u271d\u00b9.re * x\u271d.imK + x\u271d\u00b9.imI * x\u271d.imJ - x\u271d\u00b9.imJ * x\u271d.imI + x\u271d\u00b9.imK * x\u271d.re)\n[PROOFSTEP]\nring\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d\u00b2.re * x\u271d\u00b9.re + c\u2081 * x\u271d\u00b2.imI * x\u271d\u00b9.imI + c\u2082 * x\u271d\u00b2.imJ * x\u271d\u00b9.imJ - c\u2081 * c\u2082 * x\u271d\u00b2.imK * x\u271d\u00b9.imK) * x\u271d.imI +\n (x\u271d\u00b2.re * x\u271d\u00b9.imI + x\u271d\u00b2.imI * x\u271d\u00b9.re - c\u2082 * x\u271d\u00b2.imJ * x\u271d\u00b9.imK + c\u2082 * x\u271d\u00b2.imK * x\u271d\u00b9.imJ) * x\u271d.re -\n c\u2082 * (x\u271d\u00b2.re * x\u271d\u00b9.imJ + c\u2081 * x\u271d\u00b2.imI * x\u271d\u00b9.imK + x\u271d\u00b2.imJ * x\u271d\u00b9.re - c\u2081 * x\u271d\u00b2.imK * x\u271d\u00b9.imI) * x\u271d.imK +\n c\u2082 * (x\u271d\u00b2.re * x\u271d\u00b9.imK + x\u271d\u00b2.imI * x\u271d\u00b9.imJ - x\u271d\u00b2.imJ * x\u271d\u00b9.imI + x\u271d\u00b2.imK * x\u271d\u00b9.re) * x\u271d.imJ =\n x\u271d\u00b2.re * (x\u271d\u00b9.re * x\u271d.imI + x\u271d\u00b9.imI * x\u271d.re - c\u2082 * x\u271d\u00b9.imJ * x\u271d.imK + c\u2082 * x\u271d\u00b9.imK * x\u271d.imJ) +\n x\u271d\u00b2.imI * (x\u271d\u00b9.re * x\u271d.re + c\u2081 * x\u271d\u00b9.imI * x\u271d.imI + c\u2082 * x\u271d\u00b9.imJ * x\u271d.imJ - c\u2081 * c\u2082 * x\u271d\u00b9.imK * x\u271d.imK) -\n c\u2082 * x\u271d\u00b2.imJ * (x\u271d\u00b9.re * x\u271d.imK + x\u271d\u00b9.imI * x\u271d.imJ - x\u271d\u00b9.imJ * x\u271d.imI + x\u271d\u00b9.imK * x\u271d.re) +\n c\u2082 * x\u271d\u00b2.imK * (x\u271d\u00b9.re * x\u271d.imJ + c\u2081 * x\u271d\u00b9.imI * x\u271d.imK + x\u271d\u00b9.imJ * x\u271d.re - c\u2081 * x\u271d\u00b9.imK * x\u271d.imI)\n[PROOFSTEP]\nring\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d\u00b2.re * x\u271d\u00b9.re + c\u2081 * x\u271d\u00b2.imI * x\u271d\u00b9.imI + c\u2082 * x\u271d\u00b2.imJ * x\u271d\u00b9.imJ - c\u2081 * c\u2082 * x\u271d\u00b2.imK * x\u271d\u00b9.imK) * x\u271d.imJ +\n c\u2081 * (x\u271d\u00b2.re * x\u271d\u00b9.imI + x\u271d\u00b2.imI * x\u271d\u00b9.re - c\u2082 * x\u271d\u00b2.imJ * x\u271d\u00b9.imK + c\u2082 * x\u271d\u00b2.imK * x\u271d\u00b9.imJ) * x\u271d.imK +\n (x\u271d\u00b2.re * x\u271d\u00b9.imJ + c\u2081 * x\u271d\u00b2.imI * x\u271d\u00b9.imK + x\u271d\u00b2.imJ * x\u271d\u00b9.re - c\u2081 * x\u271d\u00b2.imK * x\u271d\u00b9.imI) * x\u271d.re -\n c\u2081 * (x\u271d\u00b2.re * x\u271d\u00b9.imK + x\u271d\u00b2.imI * x\u271d\u00b9.imJ - x\u271d\u00b2.imJ * x\u271d\u00b9.imI + x\u271d\u00b2.imK * x\u271d\u00b9.re) * x\u271d.imI =\n x\u271d\u00b2.re * (x\u271d\u00b9.re * x\u271d.imJ + c\u2081 * x\u271d\u00b9.imI * x\u271d.imK + x\u271d\u00b9.imJ * x\u271d.re - c\u2081 * x\u271d\u00b9.imK * x\u271d.imI) +\n c\u2081 * x\u271d\u00b2.imI * (x\u271d\u00b9.re * x\u271d.imK + x\u271d\u00b9.imI * x\u271d.imJ - x\u271d\u00b9.imJ * x\u271d.imI + x\u271d\u00b9.imK * x\u271d.re) +\n x\u271d\u00b2.imJ * (x\u271d\u00b9.re * x\u271d.re + c\u2081 * x\u271d\u00b9.imI * x\u271d.imI + c\u2082 * x\u271d\u00b9.imJ * x\u271d.imJ - c\u2081 * c\u2082 * x\u271d\u00b9.imK * x\u271d.imK) -\n c\u2081 * x\u271d\u00b2.imK * (x\u271d\u00b9.re * x\u271d.imI + x\u271d\u00b9.imI * x\u271d.re - c\u2082 * x\u271d\u00b9.imJ * x\u271d.imK + c\u2082 * x\u271d\u00b9.imK * x\u271d.imJ)\n[PROOFSTEP]\nring\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d\u00b2.re * x\u271d\u00b9.re + c\u2081 * x\u271d\u00b2.imI * x\u271d\u00b9.imI + c\u2082 * x\u271d\u00b2.imJ * x\u271d\u00b9.imJ - c\u2081 * c\u2082 * x\u271d\u00b2.imK * x\u271d\u00b9.imK) * x\u271d.imK +\n (x\u271d\u00b2.re * x\u271d\u00b9.imI + x\u271d\u00b2.imI * x\u271d\u00b9.re - c\u2082 * x\u271d\u00b2.imJ * x\u271d\u00b9.imK + c\u2082 * x\u271d\u00b2.imK * x\u271d\u00b9.imJ) * x\u271d.imJ -\n (x\u271d\u00b2.re * x\u271d\u00b9.imJ + c\u2081 * x\u271d\u00b2.imI * x\u271d\u00b9.imK + x\u271d\u00b2.imJ * x\u271d\u00b9.re - c\u2081 * x\u271d\u00b2.imK * x\u271d\u00b9.imI) * x\u271d.imI +\n (x\u271d\u00b2.re * x\u271d\u00b9.imK + x\u271d\u00b2.imI * x\u271d\u00b9.imJ - x\u271d\u00b2.imJ * x\u271d\u00b9.imI + x\u271d\u00b2.imK * x\u271d\u00b9.re) * x\u271d.re =\n x\u271d\u00b2.re * (x\u271d\u00b9.re * x\u271d.imK + x\u271d\u00b9.imI * x\u271d.imJ - x\u271d\u00b9.imJ * x\u271d.imI + x\u271d\u00b9.imK * x\u271d.re) +\n x\u271d\u00b2.imI * (x\u271d\u00b9.re * x\u271d.imJ + c\u2081 * x\u271d\u00b9.imI * x\u271d.imK + x\u271d\u00b9.imJ * x\u271d.re - c\u2081 * x\u271d\u00b9.imK * x\u271d.imI) -\n x\u271d\u00b2.imJ * (x\u271d\u00b9.re * x\u271d.imI + x\u271d\u00b9.imI * x\u271d.re - c\u2082 * x\u271d\u00b9.imJ * x\u271d.imK + c\u2082 * x\u271d\u00b9.imK * x\u271d.imJ) +\n x\u271d\u00b2.imK * (x\u271d\u00b9.re * x\u271d.re + c\u2081 * x\u271d\u00b9.imI * x\u271d.imI + c\u2082 * x\u271d\u00b9.imJ * x\u271d.imJ - c\u2081 * c\u2082 * x\u271d\u00b9.imK * x\u271d.imK)\n[PROOFSTEP]\nring\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 1 * x\u271d = x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (1 * x\u271d).re = x\u271d.re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (1 * x\u271d).imI = x\u271d.imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (1 * x\u271d).imJ = x\u271d.imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (1 * x\u271d).imK = x\u271d.imK\n[PROOFSTEP]\nsimp\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 x\u271d * 1 = x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d * 1).re = x\u271d.re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d * 1).imI = x\u271d.imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d * 1).imJ = x\u271d.imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : AddCommGroupWithOne \u210d[R,c\u2081,c\u2082] := inferInstanceAs (AddCommGroupWithOne \u210d[R,c\u2081,c\u2082])\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x\u271d * 1).imK = x\u271d.imK\n[PROOFSTEP]\nsimp\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 \u2191(x * y) = \u2191x * \u2191y\n[PROOFSTEP]\next\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (\u2191(x * y)).re = (\u2191x * \u2191y).re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (\u2191(x * y)).imI = (\u2191x * \u2191y).imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (\u2191(x * y)).imJ = (\u2191x * \u2191y).imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (\u2191(x * y)).imK = (\u2191x * \u2191y).imK\n[PROOFSTEP]\nsimp\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\n\u22a2 (fun s => \u2191(\u2191(algebraMap S R) s)) 1 = 1\n[PROOFSTEP]\nsimp only [map_one, coe_one]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\nc\u2081 c\u2082 r x\u271d y\u271d z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\nx y : S\n\u22a2 OneHom.toFun { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) } (x * y) =\n OneHom.toFun { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) } x *\n OneHom.toFun { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) } y\n[PROOFSTEP]\nsimp only [map_mul, coe_mul]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\n\u22a2 OneHom.toFun\n (\u2191{ toOneHom := { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x * y)) = \u2191(\u2191(algebraMap S R) x) * \u2191(\u2191(algebraMap S R) y)) })\n 0 =\n 0\n[PROOFSTEP]\nsimp only [map_zero, coe_zero]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\nc\u2081 c\u2082 r x\u271d y\u271d z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\nx y : S\n\u22a2 OneHom.toFun\n (\u2191{ toOneHom := { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x * y)) = \u2191(\u2191(algebraMap S R) x) * \u2191(\u2191(algebraMap S R) y)) })\n (x + y) =\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x * y)) = \u2191(\u2191(algebraMap S R) x) * \u2191(\u2191(algebraMap S R) y)) })\n x +\n OneHom.toFun\n (\u2191{ toOneHom := { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x * y)) = \u2191(\u2191(algebraMap S R) x) * \u2191(\u2191(algebraMap S R) y)) })\n y\n[PROOFSTEP]\nsimp only [map_add, coe_add]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\ns : S\nx : (fun x => \u210d[R,c\u2081,c\u2082]) s\n\u22a2 \u2191{\n toMonoidHom :=\n { toOneHom := { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x * y)) = \u2191(\u2191(algebraMap S R) x) * \u2191(\u2191(algebraMap S R) y)) },\n map_zero' := (_ : \u2191(\u2191(algebraMap S R) 0) = 0),\n map_add' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x + y)) = \u2191(\u2191(algebraMap S R) x) + \u2191(\u2191(algebraMap S R) y)) }\n s *\n x =\n x *\n \u2191{\n toMonoidHom :=\n { toOneHom := { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x * y)) = \u2191(\u2191(algebraMap S R) x) * \u2191(\u2191(algebraMap S R) y)) },\n map_zero' := (_ : \u2191(\u2191(algebraMap S R) 0) = 0),\n map_add' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x + y)) = \u2191(\u2191(algebraMap S R) x) + \u2191(\u2191(algebraMap S R) y)) }\n s\n[PROOFSTEP]\next\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\ns : S\nx : (fun x => \u210d[R,c\u2081,c\u2082]) s\n\u22a2 (\u2191{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x * y)) = \u2191(\u2191(algebraMap S R) x) * \u2191(\u2191(algebraMap S R) y)) },\n map_zero' := (_ : \u2191(\u2191(algebraMap S R) 0) = 0),\n map_add' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x + y)) = \u2191(\u2191(algebraMap S R) x) + \u2191(\u2191(algebraMap S R) y)) }\n s *\n x).re =\n (x *\n \u2191{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x * y)) = \u2191(\u2191(algebraMap S R) x) * \u2191(\u2191(algebraMap S R) y)) },\n map_zero' := (_ : \u2191(\u2191(algebraMap S R) 0) = 0),\n map_add' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x + y)) = \u2191(\u2191(algebraMap S R) x) + \u2191(\u2191(algebraMap S R) y)) }\n s).re\n[PROOFSTEP]\nsimp [Algebra.commutes]\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\ns : S\nx : (fun x => \u210d[R,c\u2081,c\u2082]) s\n\u22a2 (\u2191{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x * y)) = \u2191(\u2191(algebraMap S R) x) * \u2191(\u2191(algebraMap S R) y)) },\n map_zero' := (_ : \u2191(\u2191(algebraMap S R) 0) = 0),\n map_add' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x + y)) = \u2191(\u2191(algebraMap S R) x) + \u2191(\u2191(algebraMap S R) y)) }\n s *\n x).imI =\n (x *\n \u2191{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x * y)) = \u2191(\u2191(algebraMap S R) x) * \u2191(\u2191(algebraMap S R) y)) },\n map_zero' := (_ : \u2191(\u2191(algebraMap S R) 0) = 0),\n map_add' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x + y)) = \u2191(\u2191(algebraMap S R) x) + \u2191(\u2191(algebraMap S R) y)) }\n s).imI\n[PROOFSTEP]\nsimp [Algebra.commutes]\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\ns : S\nx : (fun x => \u210d[R,c\u2081,c\u2082]) s\n\u22a2 (\u2191{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x * y)) = \u2191(\u2191(algebraMap S R) x) * \u2191(\u2191(algebraMap S R) y)) },\n map_zero' := (_ : \u2191(\u2191(algebraMap S R) 0) = 0),\n map_add' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x + y)) = \u2191(\u2191(algebraMap S R) x) + \u2191(\u2191(algebraMap S R) y)) }\n s *\n x).imJ =\n (x *\n \u2191{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x * y)) = \u2191(\u2191(algebraMap S R) x) * \u2191(\u2191(algebraMap S R) y)) },\n map_zero' := (_ : \u2191(\u2191(algebraMap S R) 0) = 0),\n map_add' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x + y)) = \u2191(\u2191(algebraMap S R) x) + \u2191(\u2191(algebraMap S R) y)) }\n s).imJ\n[PROOFSTEP]\nsimp [Algebra.commutes]\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\ns : S\nx : (fun x => \u210d[R,c\u2081,c\u2082]) s\n\u22a2 (\u2191{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x * y)) = \u2191(\u2191(algebraMap S R) x) * \u2191(\u2191(algebraMap S R) y)) },\n map_zero' := (_ : \u2191(\u2191(algebraMap S R) 0) = 0),\n map_add' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x + y)) = \u2191(\u2191(algebraMap S R) x) + \u2191(\u2191(algebraMap S R) y)) }\n s *\n x).imK =\n (x *\n \u2191{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x * y)) = \u2191(\u2191(algebraMap S R) x) * \u2191(\u2191(algebraMap S R) y)) },\n map_zero' := (_ : \u2191(\u2191(algebraMap S R) 0) = 0),\n map_add' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x + y)) = \u2191(\u2191(algebraMap S R) x) + \u2191(\u2191(algebraMap S R) y)) }\n s).imK\n[PROOFSTEP]\nsimp [Algebra.commutes]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\ns : S\nx : (fun x => \u210d[R,c\u2081,c\u2082]) s\n\u22a2 s \u2022 x =\n \u2191{\n toMonoidHom :=\n { toOneHom := { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x * y)) = \u2191(\u2191(algebraMap S R) x) * \u2191(\u2191(algebraMap S R) y)) },\n map_zero' := (_ : \u2191(\u2191(algebraMap S R) 0) = 0),\n map_add' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x + y)) = \u2191(\u2191(algebraMap S R) x) + \u2191(\u2191(algebraMap S R) y)) }\n s *\n x\n[PROOFSTEP]\next\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\ns : S\nx : (fun x => \u210d[R,c\u2081,c\u2082]) s\n\u22a2 (s \u2022 x).re =\n (\u2191{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x * y)) = \u2191(\u2191(algebraMap S R) x) * \u2191(\u2191(algebraMap S R) y)) },\n map_zero' := (_ : \u2191(\u2191(algebraMap S R) 0) = 0),\n map_add' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x + y)) = \u2191(\u2191(algebraMap S R) x) + \u2191(\u2191(algebraMap S R) y)) }\n s *\n x).re\n[PROOFSTEP]\nsimp [Algebra.smul_def]\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\ns : S\nx : (fun x => \u210d[R,c\u2081,c\u2082]) s\n\u22a2 (s \u2022 x).imI =\n (\u2191{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x * y)) = \u2191(\u2191(algebraMap S R) x) * \u2191(\u2191(algebraMap S R) y)) },\n map_zero' := (_ : \u2191(\u2191(algebraMap S R) 0) = 0),\n map_add' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x + y)) = \u2191(\u2191(algebraMap S R) x) + \u2191(\u2191(algebraMap S R) y)) }\n s *\n x).imI\n[PROOFSTEP]\nsimp [Algebra.smul_def]\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\ns : S\nx : (fun x => \u210d[R,c\u2081,c\u2082]) s\n\u22a2 (s \u2022 x).imJ =\n (\u2191{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x * y)) = \u2191(\u2191(algebraMap S R) x) * \u2191(\u2191(algebraMap S R) y)) },\n map_zero' := (_ : \u2191(\u2191(algebraMap S R) 0) = 0),\n map_add' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x + y)) = \u2191(\u2191(algebraMap S R) x) + \u2191(\u2191(algebraMap S R) y)) }\n s *\n x).imJ\n[PROOFSTEP]\nsimp [Algebra.smul_def]\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S R\ns : S\nx : (fun x => \u210d[R,c\u2081,c\u2082]) s\n\u22a2 (s \u2022 x).imK =\n (\u2191{\n toMonoidHom :=\n {\n toOneHom :=\n { toFun := fun s => \u2191(\u2191(algebraMap S R) s), map_one' := (_ : \u2191(\u2191(algebraMap S R) 1) = 1) },\n map_mul' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x * y)) = \u2191(\u2191(algebraMap S R) x) * \u2191(\u2191(algebraMap S R) y)) },\n map_zero' := (_ : \u2191(\u2191(algebraMap S R) 0) = 0),\n map_add' :=\n (_ : \u2200 (x y : S), \u2191(\u2191(algebraMap S R) (x + y)) = \u2191(\u2191(algebraMap S R) x) + \u2191(\u2191(algebraMap S R) y)) }\n s *\n x).imK\n[PROOFSTEP]\nsimp [Algebra.smul_def]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d : StrongRankCondition R\n\u22a2 Module.rank R \u210d[R,c\u2081,c\u2082] = 4\n[PROOFSTEP]\nrw [rank_eq_card_basis (basisOneIJK c\u2081 c\u2082), Fintype.card_fin]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d : StrongRankCondition R\n\u22a2 \u21914 = 4\n[PROOFSTEP]\nnorm_num\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\ninst\u271d : StrongRankCondition R\n\u22a2 FiniteDimensional.finrank R \u210d[R,c\u2081,c\u2082] = 4\n[PROOFSTEP]\nrw [FiniteDimensional.finrank, rank_eq_four, Cardinal.toNat_ofNat]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 a * \u2191r = r \u2022 a\n[PROOFSTEP]\nrw [\u2190 coe_commutes, coe_mul_eq_smul]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 x \u2022 \u2191y = \u2191(x * y)\n[PROOFSTEP]\nrw [coe_mul, coe_mul_eq_smul]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na b c x : \u210d[R,c\u2081,c\u2082]\n\u22a2 star (star x) = x\n[PROOFSTEP]\nsimp [Star.star]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na\u271d b\u271d c a b : \u210d[R,c\u2081,c\u2082]\n\u22a2 star (a * b) = star b * star a\n[PROOFSTEP]\next\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na\u271d b\u271d c a b : \u210d[R,c\u2081,c\u2082]\n\u22a2 (star (a * b)).re = (star b * star a).re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na\u271d b\u271d c a b : \u210d[R,c\u2081,c\u2082]\n\u22a2 (star (a * b)).imI = (star b * star a).imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na\u271d b\u271d c a b : \u210d[R,c\u2081,c\u2082]\n\u22a2 (star (a * b)).imJ = (star b * star a).imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na\u271d b\u271d c a b : \u210d[R,c\u2081,c\u2082]\n\u22a2 (star (a * b)).imK = (star b * star a).imK\n[PROOFSTEP]\nsimp\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na\u271d b\u271d c a b : \u210d[R,c\u2081,c\u2082]\n\u22a2 a.re * b.re + c\u2081 * a.imI * b.imI + c\u2082 * a.imJ * b.imJ - c\u2081 * c\u2082 * a.imK * b.imK =\n b.re * a.re + c\u2081 * b.imI * a.imI + c\u2082 * b.imJ * a.imJ - c\u2081 * c\u2082 * b.imK * a.imK\n[PROOFSTEP]\nring\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na\u271d b\u271d c a b : \u210d[R,c\u2081,c\u2082]\n\u22a2 -(c\u2082 * a.imK * b.imJ) + (c\u2082 * a.imJ * b.imK - (a.re * b.imI + a.imI * b.re)) =\n -(b.re * a.imI) + -(b.imI * a.re) - c\u2082 * b.imJ * a.imK + c\u2082 * b.imK * a.imJ\n[PROOFSTEP]\nring\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na\u271d b\u271d c a b : \u210d[R,c\u2081,c\u2082]\n\u22a2 c\u2081 * a.imK * b.imI - (a.re * b.imJ + c\u2081 * a.imI * b.imK + a.imJ * b.re) =\n -(b.re * a.imJ) + c\u2081 * b.imI * a.imK + -(b.imJ * a.re) - c\u2081 * b.imK * a.imI\n[PROOFSTEP]\nring\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na\u271d b\u271d c a b : \u210d[R,c\u2081,c\u2082]\n\u22a2 -(a.imK * b.re) + (a.imJ * b.imI - (a.re * b.imK + a.imI * b.imJ)) =\n -(b.re * a.imK) + b.imI * a.imJ - b.imJ * a.imI + -(b.imK * a.re)\n[PROOFSTEP]\nring\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na\u271d b\u271d c a b : \u210d[R,c\u2081,c\u2082]\n\u22a2 star (a + b) = star a + star b\n[PROOFSTEP]\next\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na\u271d b\u271d c a b : \u210d[R,c\u2081,c\u2082]\n\u22a2 (star (a + b)).re = (star a + star b).re\n[PROOFSTEP]\nsimp [add_comm]\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na\u271d b\u271d c a b : \u210d[R,c\u2081,c\u2082]\n\u22a2 (star (a + b)).imI = (star a + star b).imI\n[PROOFSTEP]\nsimp [add_comm]\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na\u271d b\u271d c a b : \u210d[R,c\u2081,c\u2082]\n\u22a2 (star (a + b)).imJ = (star a + star b).imJ\n[PROOFSTEP]\nsimp [add_comm]\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na\u271d b\u271d c a b : \u210d[R,c\u2081,c\u2082]\n\u22a2 (star (a + b)).imK = (star a + star b).imK\n[PROOFSTEP]\nsimp [add_comm]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 a + star a = \u2191(2 * a.re)\n[PROOFSTEP]\next\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (a + star a).re = (\u2191(2 * a.re)).re\n[PROOFSTEP]\nsimp [two_mul]\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (a + star a).imI = (\u2191(2 * a.re)).imI\n[PROOFSTEP]\nsimp [two_mul]\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (a + star a).imJ = (\u2191(2 * a.re)).imJ\n[PROOFSTEP]\nsimp [two_mul]\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (a + star a).imK = (\u2191(2 * a.re)).imK\n[PROOFSTEP]\nsimp [two_mul]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 a + star a = 2 * \u2191a.re\n[PROOFSTEP]\nsimp only [self_add_star', two_mul, coe_add]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 star a + a = \u2191(2 * a.re)\n[PROOFSTEP]\nrw [add_comm, self_add_star']\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 star a + a = 2 * \u2191a.re\n[PROOFSTEP]\nrw [add_comm, self_add_star]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 Commute (star a) a\n[PROOFSTEP]\nrw [a.star_eq_two_re_sub]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 Commute (\u2191(2 * a.re) - a) a\n[PROOFSTEP]\nexact (coe_commute (2 * a.re) a).sub_left (Commute.refl a)\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 star \u2191x = \u2191x\n[PROOFSTEP]\next\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (star \u2191x).re = (\u2191x).re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (star \u2191x).imI = (\u2191x).imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (star \u2191x).imJ = (\u2191x).imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (star \u2191x).imK = (\u2191x).imK\n[PROOFSTEP]\nsimp\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x\u271d y z : R\na\u271d b c a : \u210d[R,c\u2081,c\u2082]\nx : R\nh : a = \u2191x\n\u22a2 a = \u2191a.re\n[PROOFSTEP]\nrw [h, coe_re]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\nc\u2081\u271d c\u2082\u271d r x y z : R\na\u271d b c : \u210d[R,c\u2081\u271d,c\u2082\u271d]\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\nc\u2081 c\u2082 : R\na : \u210d[R,c\u2081,c\u2082]\n\u22a2 star a = a \u2194 a = \u2191a.re\n[PROOFSTEP]\nsimp [QuaternionAlgebra.ext_iff, neg_eq_iff_add_eq_zero, add_self_eq_zero]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\nc\u2081\u271d c\u2082\u271d r x y z : R\na\u271d b c : \u210d[R,c\u2081\u271d,c\u2082\u271d]\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\nc\u2081 c\u2082 : R\na : \u210d[R,c\u2081,c\u2082]\n\u22a2 star a = -a \u2194 a.re = 0\n[PROOFSTEP]\nsimp [QuaternionAlgebra.ext_iff, eq_neg_iff_add_eq_zero]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 star a * a = \u2191(star a * a).re\n[PROOFSTEP]\next\n[GOAL]\ncase re\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (star a * a).re = (\u2191(star a * a).re).re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (star a * a).imI = (\u2191(star a * a).re).imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (star a * a).imJ = (\u2191(star a * a).re).imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 (star a * a).imK = (\u2191(star a * a).re).imK\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 a.re * a.imI + -(a.imI * a.re) + c\u2082 * a.imJ * a.imK + -(c\u2082 * a.imK * a.imJ) = 0\n[PROOFSTEP]\nring\n[GOAL]\ncase imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 a.re * a.imJ + -(c\u2081 * a.imI * a.imK) + -(a.imJ * a.re) + c\u2081 * a.imK * a.imI = 0\n[PROOFSTEP]\nring\n[GOAL]\ncase imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 a.re * a.imK + -(a.imI * a.imJ) + a.imJ * a.imI + -(a.imK * a.re) = 0\n[PROOFSTEP]\nring\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 a * star a = \u2191(a * star a).re\n[PROOFSTEP]\nrw [\u2190 star_comm_self']\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 star a * a = \u2191(star a * a).re\n[PROOFSTEP]\nexact a.star_mul_eq_coe\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x\u271d y\u271d z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : \u210d[R,c\u2081,c\u2082] \u2243+ \u210d[R,c\u2081,c\u2082]\u1d50\u1d52\u1d56 := AddEquiv.trans starAddEquiv opAddEquiv\nx y : \u210d[R,c\u2081,c\u2082]\n\u22a2 Equiv.toFun\n { toFun := op \u2218 star, invFun := star \u2218 unop, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (x * y) =\n Equiv.toFun\n { toFun := op \u2218 star, invFun := star \u2218 unop, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n x *\n Equiv.toFun\n { toFun := op \u2218 star, invFun := star \u2218 unop, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n y\n[PROOFSTEP]\nsimp\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r\u271d x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\nsrc\u271d : \u210d[R,c\u2081,c\u2082] \u2243+ \u210d[R,c\u2081,c\u2082]\u1d50\u1d52\u1d56 := AddEquiv.trans starAddEquiv opAddEquiv\nr : R\n\u22a2 Equiv.toFun\n { toFun := op \u2218 star, invFun := star \u2218 unop, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n (\u2191(algebraMap R \u210d[R,c\u2081,c\u2082]) r) =\n \u2191(algebraMap R \u210d[R,c\u2081,c\u2082]\u1d50\u1d52\u1d56) r\n[PROOFSTEP]\nsimp\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x y z : R\na b c : \u210d[R]\n\u22a2 a.re * b.re + -1 * a.imI * b.imI + -1 * a.imJ * b.imJ - -1 * -1 * a.imK * b.imK =\n a.re * b.re - a.imI * b.imI - a.imJ * b.imJ - a.imK * b.imK\n[PROOFSTEP]\nsimp only [one_mul, neg_mul, sub_eq_add_neg, neg_neg]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x y z : R\na b c : \u210d[R]\n\u22a2 a.re * b.imI + a.imI * b.re - -1 * a.imJ * b.imK + -1 * a.imK * b.imJ =\n a.re * b.imI + a.imI * b.re + a.imJ * b.imK - a.imK * b.imJ\n[PROOFSTEP]\nsimp only [one_mul, neg_mul, sub_eq_add_neg, neg_neg]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x y z : R\na b c : \u210d[R]\n\u22a2 a.re * b.imJ + -1 * a.imI * b.imK + a.imJ * b.re - -1 * a.imK * b.imI =\n a.re * b.imJ - a.imI * b.imK + a.imJ * b.re + a.imK * b.imI\n[PROOFSTEP]\nsimp only [one_mul, neg_mul, sub_eq_add_neg, neg_neg]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x y z : R\na b c : \u210d[R]\n\u22a2 a.re * b.imK + a.imI * b.imJ - a.imJ * b.imI + a.imK * b.re =\n a.re * b.imK + a.imI * b.imJ - a.imJ * b.imI + a.imK * b.re\n[PROOFSTEP]\nsimp only [one_mul, neg_mul, sub_eq_add_neg, neg_neg]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x y z : R\na b c : \u210d[R]\n\u22a2 (fun a => (a * star a).re) 0 = 0\n[PROOFSTEP]\nsimp only [star_zero, zero_mul, zero_re]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x y z : R\na b c : \u210d[R]\n\u22a2 ZeroHom.toFun { toFun := fun a => (a * star a).re, map_zero' := (_ : (0 * star 0).re = 0) } 1 = 1\n[PROOFSTEP]\nsimp only [star_one, one_mul, one_re]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x\u271d y\u271d z : R\na b c x y : \u210d[R]\n\u22a2 \u2191(ZeroHom.toFun { toFun := fun a => (a * star a).re, map_zero' := (_ : (0 * star 0).re = 0) } (x * y)) =\n \u2191(ZeroHom.toFun { toFun := fun a => (a * star a).re, map_zero' := (_ : (0 * star 0).re = 0) } x *\n ZeroHom.toFun { toFun := fun a => (a * star a).re, map_zero' := (_ : (0 * star 0).re = 0) } y)\n[PROOFSTEP]\nconv_lhs =>\n rw [\u2190 mul_star_eq_coe, star_mul, mul_assoc, \u2190 mul_assoc y, y.mul_star_eq_coe, coe_commutes, \u2190 mul_assoc,\n x.mul_star_eq_coe, \u2190 coe_mul]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x\u271d y\u271d z : R\na b c x y : \u210d[R]\n| \u2191(ZeroHom.toFun { toFun := fun a => (a * star a).re, map_zero' := (_ : (0 * star 0).re = 0) } (x * y))\n[PROOFSTEP]\nrw [\u2190 mul_star_eq_coe, star_mul, mul_assoc, \u2190 mul_assoc y, y.mul_star_eq_coe, coe_commutes, \u2190 mul_assoc,\n x.mul_star_eq_coe, \u2190 coe_mul]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x\u271d y\u271d z : R\na b c x y : \u210d[R]\n| \u2191(ZeroHom.toFun { toFun := fun a => (a * star a).re, map_zero' := (_ : (0 * star 0).re = 0) } (x * y))\n[PROOFSTEP]\nrw [\u2190 mul_star_eq_coe, star_mul, mul_assoc, \u2190 mul_assoc y, y.mul_star_eq_coe, coe_commutes, \u2190 mul_assoc,\n x.mul_star_eq_coe, \u2190 coe_mul]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x\u271d y\u271d z : R\na b c x y : \u210d[R]\n| \u2191(ZeroHom.toFun { toFun := fun a => (a * star a).re, map_zero' := (_ : (0 * star 0).re = 0) } (x * y))\n[PROOFSTEP]\nrw [\u2190 mul_star_eq_coe, star_mul, mul_assoc, \u2190 mul_assoc y, y.mul_star_eq_coe, coe_commutes, \u2190 mul_assoc,\n x.mul_star_eq_coe, \u2190 coe_mul]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x y z : R\na b c : \u210d[R]\n\u22a2 \u2191normSq a = a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2\n[PROOFSTEP]\nsimp only [normSq_def, sq, mul_neg, sub_neg_eq_add, mul_re, star_re, star_imI, star_imJ, star_imK]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x y z : R\na b c : \u210d[R]\n\u22a2 \u2191normSq \u2191x = x ^ 2\n[PROOFSTEP]\nrw [normSq_def, star_coe, \u2190 coe_mul, coe_re, sq]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x y z : R\na b c : \u210d[R]\n\u22a2 \u2191normSq (star a) = \u2191normSq a\n[PROOFSTEP]\nsimp [normSq_def']\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x y z : R\na b c : \u210d[R]\nn : \u2115\n\u22a2 \u2191normSq \u2191n = \u2191n ^ 2\n[PROOFSTEP]\nrw [\u2190 coe_nat_cast, normSq_coe]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x y z\u271d : R\na b c : \u210d[R]\nz : \u2124\n\u22a2 \u2191normSq \u2191z = \u2191z ^ 2\n[PROOFSTEP]\nrw [\u2190 coe_int_cast, normSq_coe]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x y z : R\na b c : \u210d[R]\n\u22a2 \u2191normSq (-a) = \u2191normSq a\n[PROOFSTEP]\nsimp only [normSq_def, star_neg, neg_mul_neg]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x y z : R\na b c : \u210d[R]\n\u22a2 a * star a = \u2191(\u2191normSq a)\n[PROOFSTEP]\nrw [mul_star_eq_coe, normSq_def]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x y z : R\na b c : \u210d[R]\n\u22a2 star a * a = \u2191(\u2191normSq a)\n[PROOFSTEP]\nrw [star_comm_self, self_mul_star]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x y z : R\na b c : \u210d[R]\n\u22a2 im a ^ 2 = -\u2191(\u2191normSq (im a))\n[PROOFSTEP]\nsimp_rw [sq, \u2190 star_mul_self, im_star, neg_mul, neg_neg]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x y z : R\na b c : \u210d[R]\n\u22a2 \u2191(\u2191normSq (a + b)) = \u2191(\u2191normSq a) + a * star b + b * star a + \u2191(\u2191normSq b)\n[PROOFSTEP]\nsimp only [star_add, \u2190 self_mul_star, mul_add, add_mul, add_assoc, add_left_comm]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr\u271d x y z : R\na b c : \u210d[R]\nr : R\nq : \u210d[R]\n\u22a2 \u2191normSq (r \u2022 q) = r ^ 2 * \u2191normSq q\n[PROOFSTEP]\nsimp only [normSq_def', smul_re, smul_imI, smul_imJ, smul_imK, mul_pow, mul_add, smul_eq_mul]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x y z : R\na\u271d b\u271d c a b : \u210d[R]\n\u22a2 \u2191normSq (a + b) = \u2191normSq a + (a * star b).re + ((b * star a).re + \u2191normSq b)\n[PROOFSTEP]\nsimp_rw [normSq_def, star_add, add_mul, mul_add, add_re]\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x y z : R\na\u271d b\u271d c a b : \u210d[R]\n\u22a2 \u2191normSq a + (a * star b).re + ((b * star a).re + \u2191normSq b) =\n \u2191normSq a + \u2191normSq b + ((a * star b).re + (b * star a).re)\n[PROOFSTEP]\nabel\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x y z : R\na\u271d b\u271d c a b : \u210d[R]\n\u22a2 \u2191normSq a + (a * star b).re + ((b * star a).re + \u2191normSq b) =\n \u2191normSq a + \u2191normSq b + ((a * star b).re + (b * star a).re)\n[PROOFSTEP]\nabel\n[GOAL]\nS : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nr x y z : R\na\u271d b\u271d c a b : \u210d[R]\n\u22a2 \u2191normSq a + \u2191normSq b + ((a * star b).re + (b * star a).re) = \u2191normSq a + \u2191normSq b + 2 * (a * star b).re\n[PROOFSTEP]\nrw [\u2190 add_re, \u2190 star_mul_star a b, self_add_star', coe_re]\n[GOAL]\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\n\u22a2 \u2191normSq a = 0 \u2194 a = 0\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => h.symm \u25b8 normSq.map_zero\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : \u2191normSq a = 0\n\u22a2 a = 0\n[PROOFSTEP]\nrw [normSq_def', add_eq_zero_iff', add_eq_zero_iff', add_eq_zero_iff'] at h \n[GOAL]\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : ((a.re ^ 2 = 0 \u2227 a.imI ^ 2 = 0) \u2227 a.imJ ^ 2 = 0) \u2227 a.imK ^ 2 = 0\n\u22a2 a = 0\ncase ha\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : (a.re ^ 2 + a.imI ^ 2 = 0 \u2227 a.imJ ^ 2 = 0) \u2227 a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.re ^ 2\ncase hb\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : (a.re ^ 2 + a.imI ^ 2 = 0 \u2227 a.imJ ^ 2 = 0) \u2227 a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.imI ^ 2\ncase ha\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 = 0 \u2227 a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.re ^ 2 + a.imI ^ 2\ncase hb\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 = 0 \u2227 a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.imJ ^ 2\ncase ha\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2\ncase hb\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.imK ^ 2\n[PROOFSTEP]\nexact ext a 0 (pow_eq_zero h.1.1.1) (pow_eq_zero h.1.1.2) (pow_eq_zero h.1.2) (pow_eq_zero h.2)\n[GOAL]\ncase ha\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : (a.re ^ 2 + a.imI ^ 2 = 0 \u2227 a.imJ ^ 2 = 0) \u2227 a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.re ^ 2\ncase hb\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : (a.re ^ 2 + a.imI ^ 2 = 0 \u2227 a.imJ ^ 2 = 0) \u2227 a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.imI ^ 2\ncase ha\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 = 0 \u2227 a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.re ^ 2 + a.imI ^ 2\ncase hb\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 = 0 \u2227 a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.imJ ^ 2\ncase ha\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2\ncase hb\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.imK ^ 2\n[PROOFSTEP]\nall_goals apply_rules [sq_nonneg, add_nonneg]\n[GOAL]\ncase ha\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : (a.re ^ 2 + a.imI ^ 2 = 0 \u2227 a.imJ ^ 2 = 0) \u2227 a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.re ^ 2\n[PROOFSTEP]\napply_rules [sq_nonneg, add_nonneg]\n[GOAL]\ncase hb\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : (a.re ^ 2 + a.imI ^ 2 = 0 \u2227 a.imJ ^ 2 = 0) \u2227 a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.imI ^ 2\n[PROOFSTEP]\napply_rules [sq_nonneg, add_nonneg]\n[GOAL]\ncase ha\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 = 0 \u2227 a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.re ^ 2 + a.imI ^ 2\n[PROOFSTEP]\napply_rules [sq_nonneg, add_nonneg]\n[GOAL]\ncase hb\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 = 0 \u2227 a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.imJ ^ 2\n[PROOFSTEP]\napply_rules [sq_nonneg, add_nonneg]\n[GOAL]\ncase ha\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2\n[PROOFSTEP]\napply_rules [sq_nonneg, add_nonneg]\n[GOAL]\ncase hb\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.imK ^ 2\n[PROOFSTEP]\napply_rules [sq_nonneg, add_nonneg]\n[GOAL]\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\n\u22a2 0 \u2264 \u2191normSq a\n[PROOFSTEP]\nrw [normSq_def']\n[GOAL]\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\n\u22a2 0 \u2264 a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2\n[PROOFSTEP]\napply_rules [sq_nonneg, add_nonneg]\n[GOAL]\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na\u271d a b : \u210d[R]\nhab : a * b = 0\n\u22a2 \u2191normSq a * \u2191normSq b = 0\n[PROOFSTEP]\nrwa [\u2190 map_mul, normSq_eq_zero]\n[GOAL]\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\n\u22a2 a ^ 2 = \u2191(\u2191normSq a) \u2194 a = \u2191a.re\n[PROOFSTEP]\nrw [\u2190 star_eq_self, \u2190 star_mul_self, sq, mul_eq_mul_right_iff, eq_comm]\n[GOAL]\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\n\u22a2 star a = a \u2228 a = 0 \u2194 star a = a\n[PROOFSTEP]\nexact or_iff_left_of_imp fun ha \u21a6 ha.symm \u25b8 star_zero _\n[GOAL]\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\n\u22a2 a ^ 2 = -\u2191(\u2191normSq a) \u2194 a.re = 0\n[PROOFSTEP]\nsimp_rw [\u2190 star_eq_neg]\n[GOAL]\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\n\u22a2 a ^ 2 = -\u2191(\u2191normSq a) \u2194 star a = -a\n[PROOFSTEP]\nobtain rfl | hq0 := eq_or_ne a 0\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\n\u22a2 0 ^ 2 = -\u2191(\u2191normSq 0) \u2194 star 0 = -0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nhq0 : a \u2260 0\n\u22a2 a ^ 2 = -\u2191(\u2191normSq a) \u2194 star a = -a\n[PROOFSTEP]\nrw [\u2190 star_mul_self, \u2190 mul_neg, \u2190 neg_sq, sq, mul_left_inj' (neg_ne_zero.mpr hq0), eq_comm]\n[GOAL]\nR : Type u_1\ninst\u271d : LinearOrderedField R\na b : \u210d[R]\nsrc\u271d : Nontrivial \u210d[R] := instNontrivial\n\u22a2 MonoidWithZero \u210d[R]\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u_1\ninst\u271d : LinearOrderedField R\na b : \u210d[R]\nsrc\u271d\u00b9 : Nontrivial \u210d[R] := instNontrivial\nsrc\u271d : MonoidWithZero \u210d[R] := inferInstance\n\u22a2 0\u207b\u00b9 = 0\n[PROOFSTEP]\nrw [instInv_inv, star_zero, smul_zero]\n[GOAL]\nR : Type u_1\ninst\u271d : LinearOrderedField R\na\u271d b : \u210d[R]\nsrc\u271d\u00b9 : Nontrivial \u210d[R] := instNontrivial\nsrc\u271d : MonoidWithZero \u210d[R] := inferInstance\na : \u210d[R]\nha : a \u2260 0\n\u22a2 a * a\u207b\u00b9 = 1\n[PROOFSTEP]\nletI : Semiring \u210d[R] := inferInstanceAs (Semiring \u210d[R,-1,-1])\n[GOAL]\nR : Type u_1\ninst\u271d : LinearOrderedField R\na\u271d b : \u210d[R]\nsrc\u271d\u00b9 : Nontrivial \u210d[R] := instNontrivial\nsrc\u271d : MonoidWithZero \u210d[R] := inferInstance\na : \u210d[R]\nha : a \u2260 0\nthis : Semiring \u210d[R] := inferInstanceAs (Semiring \u210d[R,-1,-1])\n\u22a2 a * a\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [instInv_inv, Algebra.mul_smul_comm (normSq a)\u207b\u00b9 a (star a), self_mul_star, smul_coe,\n inv_mul_cancel (normSq_ne_zero.2 ha), coe_one]\n[GOAL]\nR : Type u_1\ninst\u271d : LinearOrderedField R\na b : \u210d[R]\nsrc\u271d\u00b9 : GroupWithZero \u210d[R] := instGroupWithZero\nsrc\u271d : Ring \u210d[R] := instRing\nn : \u2124\nd : \u2115\nhd : d \u2260 0\nh : Nat.coprime (Int.natAbs n) d\n\u22a2 \u2191(Rat.mk' n d) = \u2191n * (\u2191d)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 coe_rat_cast, Rat.cast_mk', coe_mul, coe_int_cast, coe_inv, coe_nat_cast]\n[GOAL]\nR : Type u_1\ninst\u271d : LinearOrderedField R\na b : \u210d[R]\nsrc\u271d\u00b9 : GroupWithZero \u210d[R] := instGroupWithZero\nsrc\u271d : Ring \u210d[R] := instRing\nq : \u211a\nx : \u210d[R]\n\u22a2 (fun x x_1 => x \u2022 x_1) q x = \u2191q * x\n[PROOFSTEP]\nrw [\u2190 coe_rat_cast, coe_mul_eq_smul]\n[GOAL]\nR : Type u_1\ninst\u271d : LinearOrderedField R\na b : \u210d[R]\nsrc\u271d\u00b9 : GroupWithZero \u210d[R] := instGroupWithZero\nsrc\u271d : Ring \u210d[R] := instRing\nq : \u211a\nx : \u210d[R]\n\u22a2 (fun x x_1 => x \u2022 x_1) q x = \u2191q \u2022 x\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d : LinearOrderedField R\na b : \u210d[R]\nsrc\u271d\u00b9 : GroupWithZero \u210d[R] := instGroupWithZero\nsrc\u271d : Ring \u210d[R] := instRing\nq : \u211a\nx : \u210d[R]\n\u22a2 ((fun x x_1 => x \u2022 x_1) q x).re = (\u2191q \u2022 x).re\n[PROOFSTEP]\nexact DivisionRing.qsmul_eq_mul' _ _\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d : LinearOrderedField R\na b : \u210d[R]\nsrc\u271d\u00b9 : GroupWithZero \u210d[R] := instGroupWithZero\nsrc\u271d : Ring \u210d[R] := instRing\nq : \u211a\nx : \u210d[R]\n\u22a2 ((fun x x_1 => x \u2022 x_1) q x).imI = (\u2191q \u2022 x).imI\n[PROOFSTEP]\nexact DivisionRing.qsmul_eq_mul' _ _\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d : LinearOrderedField R\na b : \u210d[R]\nsrc\u271d\u00b9 : GroupWithZero \u210d[R] := instGroupWithZero\nsrc\u271d : Ring \u210d[R] := instRing\nq : \u211a\nx : \u210d[R]\n\u22a2 ((fun x x_1 => x \u2022 x_1) q x).imJ = (\u2191q \u2022 x).imJ\n[PROOFSTEP]\nexact DivisionRing.qsmul_eq_mul' _ _\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d : LinearOrderedField R\na b : \u210d[R]\nsrc\u271d\u00b9 : GroupWithZero \u210d[R] := instGroupWithZero\nsrc\u271d : Ring \u210d[R] := instRing\nq : \u211a\nx : \u210d[R]\n\u22a2 ((fun x x_1 => x \u2022 x_1) q x).imK = (\u2191q \u2022 x).imK\n[PROOFSTEP]\nexact DivisionRing.qsmul_eq_mul' _ _\n[GOAL]\nR : Type u_1\ninst\u271d : LinearOrderedField R\na b : \u210d[R]\nq : \u211a\n\u22a2 \u2191(\u2191normSq \u2191q) = \u2191q ^ 2\n[PROOFSTEP]\nrw [\u2190 coe_rat_cast, normSq_coe, coe_pow]\n[GOAL]\nR : Type u_1\nc\u2081 c\u2082 : R\ninst\u271d : Infinite R\n\u22a2 1 \u2264 4\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nc\u2081 c\u2082 : R\n\u22a2 #\u210d[R,c\u2081,c\u2082] = #R ^ 4\n[PROOFSTEP]\nrw [mk_congr (QuaternionAlgebra.equivProd c\u2081 c\u2082)]\n[GOAL]\nR : Type u_1\nc\u2081 c\u2082 : R\n\u22a2 #(R \u00d7 R \u00d7 R \u00d7 R) = #R ^ 4\n[PROOFSTEP]\nsimp only [mk_prod, lift_id]\n[GOAL]\nR : Type u_1\nc\u2081 c\u2082 : R\n\u22a2 #R * (#R * (#R * #R)) = #R ^ 4\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nc\u2081 c\u2082 : R\ninst\u271d : Infinite R\n\u22a2 #\u210d[R,c\u2081,c\u2082] = #R\n[PROOFSTEP]\nrw [mk_quaternionAlgebra, pow_four]\n[GOAL]\nR : Type u_1\nc\u2081 c\u2082 : R\n\u22a2 #\u2191Set.univ = #R ^ 4\n[PROOFSTEP]\nrw [mk_univ, mk_quaternionAlgebra]\n[GOAL]\nR : Type u_1\nc\u2081 c\u2082 : R\ninst\u271d : Infinite R\n\u22a2 #\u2191Set.univ = #R\n[PROOFSTEP]\nrw [mk_univ_quaternionAlgebra, pow_four]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Quaternion", "llama_tokens": 38064, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.5, "lm_q2_score": 0.02228618310405795, "lm_q1q2_score": 0.011143091552028975}} {"text": "[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.243, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : HasZeroMorphisms C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 X.obj \u0394 \u27f6 N s (SimplexCategory.len A.fst.unop)\n[PROOFSTEP]\nrefine' (s.iso \u0394).inv \u226b Sigma.desc fun B => _\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.243, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : HasZeroMorphisms C\n\u0394 : SimplexCategory\u1d52\u1d56\nA B : IndexSet \u0394\n\u22a2 summand s.N \u0394 B \u27f6 N s (SimplexCategory.len A.fst.unop)\n[PROOFSTEP]\nby_cases B = A\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.243, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : HasZeroMorphisms C\n\u0394 : SimplexCategory\u1d52\u1d56\nA B : IndexSet \u0394\n\u22a2 summand s.N \u0394 B \u27f6 N s (SimplexCategory.len A.fst.unop)\n[PROOFSTEP]\nby_cases B = A\n[GOAL]\ncase pos\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.243, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : HasZeroMorphisms C\n\u0394 : SimplexCategory\u1d52\u1d56\nA B : IndexSet \u0394\nh : B = A\n\u22a2 summand s.N \u0394 B \u27f6 N s (SimplexCategory.len A.fst.unop)\n[PROOFSTEP]\nexact eqToHom (by subst h; rfl)\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.243, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : HasZeroMorphisms C\n\u0394 : SimplexCategory\u1d52\u1d56\nA B : IndexSet \u0394\nh : B = A\n\u22a2 summand s.N \u0394 B = N s (SimplexCategory.len A.fst.unop)\n[PROOFSTEP]\nsubst h\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.243, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : HasZeroMorphisms C\n\u0394 : SimplexCategory\u1d52\u1d56\nB : IndexSet \u0394\n\u22a2 summand s.N \u0394 B = N s (SimplexCategory.len B.fst.unop)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.243, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : HasZeroMorphisms C\n\u0394 : SimplexCategory\u1d52\u1d56\nA B : IndexSet \u0394\nh : \u00acB = A\n\u22a2 summand s.N \u0394 B \u27f6 N s (SimplexCategory.len A.fst.unop)\n[PROOFSTEP]\nexact 0\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : HasZeroMorphisms C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 \u03b9Summand s A \u226b \u03c0Summand s A = \ud835\udfd9 (N s (SimplexCategory.len A.fst.unop))\n[PROOFSTEP]\ndsimp only [\u03b9Summand, iso_hom, \u03c0Summand, iso_inv, summand]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : HasZeroMorphisms C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 ((\u03b9Coprod s.N A \u226b map X s.\u03b9 \u0394) \u226b\n inv (map X s.\u03b9 \u0394) \u226b\n Sigma.desc fun B =>\n if h : B = A then eqToHom (_ : summand s.N \u0394 B = N s (SimplexCategory.len A.fst.unop)) else 0) =\n \ud835\udfd9 (N s (SimplexCategory.len A.fst.unop))\n[PROOFSTEP]\nsimp only [summand, assoc, IsIso.hom_inv_id_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : HasZeroMorphisms C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 (\u03b9Coprod s.N A \u226b\n Sigma.desc fun B =>\n if h : B = A then eqToHom (_ : summand s.N \u0394 B = N s (SimplexCategory.len A.fst.unop)) else 0) =\n \ud835\udfd9 (N s (SimplexCategory.len A.fst.unop))\n[PROOFSTEP]\nerw [colimit.\u03b9_desc, Cofan.mk_\u03b9_app]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : HasZeroMorphisms C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 (if h : { as := A }.as = A then eqToHom (_ : summand s.N \u0394 { as := A }.as = N s (SimplexCategory.len A.fst.unop))\n else 0) =\n \ud835\udfd9 (N s (SimplexCategory.len A.fst.unop))\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : HasZeroMorphisms C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 (if h : A = A then \ud835\udfd9 (N s (SimplexCategory.len A.fst.unop)) else 0) = \ud835\udfd9 (N s (SimplexCategory.len A.fst.unop))\n[PROOFSTEP]\nsimp only [dite_eq_ite, ite_true]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : HasZeroMorphisms C\n\u0394 : SimplexCategory\u1d52\u1d56\nA B : IndexSet \u0394\nh : B \u2260 A\n\u22a2 \u03b9Summand s A \u226b \u03c0Summand s B = 0\n[PROOFSTEP]\ndsimp only [\u03b9Summand, iso_hom, \u03c0Summand, iso_inv, summand]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : HasZeroMorphisms C\n\u0394 : SimplexCategory\u1d52\u1d56\nA B : IndexSet \u0394\nh : B \u2260 A\n\u22a2 ((\u03b9Coprod s.N A \u226b map X s.\u03b9 \u0394) \u226b\n inv (map X s.\u03b9 \u0394) \u226b\n Sigma.desc fun B_1 =>\n if h : B_1 = B then eqToHom (_ : summand s.N \u0394 B_1 = N s (SimplexCategory.len B.fst.unop)) else 0) =\n 0\n[PROOFSTEP]\nsimp only [summand, assoc, IsIso.hom_inv_id_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : HasZeroMorphisms C\n\u0394 : SimplexCategory\u1d52\u1d56\nA B : IndexSet \u0394\nh : B \u2260 A\n\u22a2 (\u03b9Coprod s.N A \u226b\n Sigma.desc fun B_1 =>\n if h : B_1 = B then eqToHom (_ : summand s.N \u0394 B_1 = N s (SimplexCategory.len B.fst.unop)) else 0) =\n 0\n[PROOFSTEP]\nerw [colimit.\u03b9_desc, Cofan.mk_\u03b9_app]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : HasZeroMorphisms C\n\u0394 : SimplexCategory\u1d52\u1d56\nA B : IndexSet \u0394\nh : B \u2260 A\n\u22a2 (if h : { as := A }.as = B then eqToHom (_ : summand s.N \u0394 { as := A }.as = N s (SimplexCategory.len B.fst.unop))\n else 0) =\n 0\n[PROOFSTEP]\nexact dif_neg h.symm\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\n\u0394 : SimplexCategory\u1d52\u1d56\n\u22a2 \ud835\udfd9 (X.obj \u0394) = \u2211 A : IndexSet \u0394, \u03c0Summand s A \u226b \u03b9Summand s A\n[PROOFSTEP]\napply s.hom_ext'\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\n\u0394 : SimplexCategory\u1d52\u1d56\n\u22a2 \u2200 (A : IndexSet \u0394), \u03b9Summand s A \u226b \ud835\udfd9 (X.obj \u0394) = \u03b9Summand s A \u226b \u2211 A : IndexSet \u0394, \u03c0Summand s A \u226b \u03b9Summand s A\n[PROOFSTEP]\nintro A\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 \u03b9Summand s A \u226b \ud835\udfd9 (X.obj \u0394) = \u03b9Summand s A \u226b \u2211 A : IndexSet \u0394, \u03c0Summand s A \u226b \u03b9Summand s A\n[PROOFSTEP]\nrw [comp_id, comp_sum, Finset.sum_eq_single A, \u03b9_\u03c0Summand_eq_id_assoc]\n[GOAL]\ncase h.h\u2080\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 \u2200 (b : IndexSet \u0394), b \u2208 Finset.univ \u2192 b \u2260 A \u2192 \u03b9Summand s A \u226b \u03c0Summand s b \u226b \u03b9Summand s b = 0\n[PROOFSTEP]\nintro B _ h\u2082\n[GOAL]\ncase h.h\u2080\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\n\u0394 : SimplexCategory\u1d52\u1d56\nA B : IndexSet \u0394\na\u271d : B \u2208 Finset.univ\nh\u2082 : B \u2260 A\n\u22a2 \u03b9Summand s A \u226b \u03c0Summand s B \u226b \u03b9Summand s B = 0\n[PROOFSTEP]\nrw [s.\u03b9_\u03c0Summand_eq_zero_assoc _ _ h\u2082, zero_comp]\n[GOAL]\ncase h.h\u2081\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\n\u0394 : SimplexCategory\u1d52\u1d56\nA : IndexSet \u0394\n\u22a2 \u00acA \u2208 Finset.univ \u2192 \u03b9Summand s A \u226b \u03c0Summand s A \u226b \u03b9Summand s A = 0\n[PROOFSTEP]\nsimp only [Finset.mem_univ, not_true, IsEmpty.forall_iff]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\ni : Fin (n + 1)\n\u22a2 SimplicialObject.\u03c3 X i \u226b \u03c0Summand s (IndexSet.id (op [n + 1])) = 0\n[PROOFSTEP]\napply s.hom_ext'\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\ni : Fin (n + 1)\n\u22a2 \u2200 (A : IndexSet (op [n])),\n \u03b9Summand s A \u226b SimplicialObject.\u03c3 X i \u226b \u03c0Summand s (IndexSet.id (op [n + 1])) = \u03b9Summand s A \u226b 0\n[PROOFSTEP]\nintro A\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\ni : Fin (n + 1)\nA : IndexSet (op [n])\n\u22a2 \u03b9Summand s A \u226b SimplicialObject.\u03c3 X i \u226b \u03c0Summand s (IndexSet.id (op [n + 1])) = \u03b9Summand s A \u226b 0\n[PROOFSTEP]\ndsimp only [SimplicialObject.\u03c3]\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\ni : Fin (n + 1)\nA : IndexSet (op [n])\n\u22a2 \u03b9Summand s A \u226b X.map (SimplexCategory.\u03c3 i).op \u226b \u03c0Summand s (IndexSet.id (op [n + 1])) = \u03b9Summand s A \u226b 0\n[PROOFSTEP]\nrw [comp_zero, s.\u03b9Summand_epi_naturality_assoc A (SimplexCategory.\u03c3 i).op, \u03b9_\u03c0Summand_eq_zero]\n[GOAL]\ncase h.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\ni : Fin (n + 1)\nA : IndexSet (op [n])\n\u22a2 IndexSet.id (op [n + 1]) \u2260 IndexSet.epiComp A (SimplexCategory.\u03c3 i).op\n[PROOFSTEP]\nrw [ne_comm]\n[GOAL]\ncase h.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\ni : Fin (n + 1)\nA : IndexSet (op [n])\n\u22a2 IndexSet.epiComp A (SimplexCategory.\u03c3 i).op \u2260 IndexSet.id (op [n + 1])\n[PROOFSTEP]\nchange \u00ac(A.epiComp (SimplexCategory.\u03c3 i).op).EqId\n[GOAL]\ncase h.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\ni : Fin (n + 1)\nA : IndexSet (op [n])\n\u22a2 \u00acIndexSet.EqId (IndexSet.epiComp A (SimplexCategory.\u03c3 i).op)\n[PROOFSTEP]\nrw [IndexSet.eqId_iff_len_eq]\n[GOAL]\ncase h.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\ni : Fin (n + 1)\nA : IndexSet (op [n])\n\u22a2 \u00acSimplexCategory.len (IndexSet.epiComp A (SimplexCategory.\u03c3 i).op).fst.unop = SimplexCategory.len (op [n + 1]).unop\n[PROOFSTEP]\nhave h := SimplexCategory.len_le_of_epi (inferInstance : Epi A.e)\n[GOAL]\ncase h.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\ni : Fin (n + 1)\nA : IndexSet (op [n])\nh : SimplexCategory.len A.fst.unop \u2264 SimplexCategory.len (op [n]).unop\n\u22a2 \u00acSimplexCategory.len (IndexSet.epiComp A (SimplexCategory.\u03c3 i).op).fst.unop = SimplexCategory.len (op [n + 1]).unop\n[PROOFSTEP]\ndsimp at h \u22a2\n[GOAL]\ncase h.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\ni : Fin (n + 1)\nA : IndexSet (op [n])\nh : SimplexCategory.len A.fst.unop \u2264 n\n\u22a2 \u00acSimplexCategory.len A.fst.unop = n + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX\u271d : SimplicialObject C\ns\u271d : Splitting X\u271d\ninst\u271d : Preadditive C\nX : SimplicialObject C\ns : Splitting X\nn : \u2115\nA : IndexSet (op [n])\nhA : \u00acIndexSet.EqId A\n\u22a2 \u03b9Summand s A \u226b HomologicalComplex.Hom.f PInfty n = 0\n[PROOFSTEP]\nrw [SimplicialObject.Splitting.IndexSet.eqId_iff_mono] at hA \n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX\u271d : SimplicialObject C\ns\u271d : Splitting X\u271d\ninst\u271d : Preadditive C\nX : SimplicialObject C\ns : Splitting X\nn : \u2115\nA : IndexSet (op [n])\nhA : \u00acMono (IndexSet.e A)\n\u22a2 \u03b9Summand s A \u226b HomologicalComplex.Hom.f PInfty n = 0\n[PROOFSTEP]\nrw [SimplicialObject.Splitting.\u03b9Summand_eq, assoc, degeneracy_comp_PInfty X n A.e hA, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nZ : C\nn : \u2115\nf : Z \u27f6 X.obj (op [n])\n\u22a2 f \u226b HomologicalComplex.Hom.f PInfty n = 0 \u2194 f \u226b \u03c0Summand s (IndexSet.id (op [n])) = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nZ : C\nn : \u2115\nf : Z \u27f6 X.obj (op [n])\n\u22a2 f \u226b HomologicalComplex.Hom.f PInfty n = 0 \u2192 f \u226b \u03c0Summand s (IndexSet.id (op [n])) = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nZ : C\nn : \u2115\nf : Z \u27f6 X.obj (op [n])\nh : f \u226b HomologicalComplex.Hom.f PInfty n = 0\n\u22a2 f \u226b \u03c0Summand s (IndexSet.id (op [n])) = 0\n[PROOFSTEP]\nrcases n with _ | n\n[GOAL]\ncase mp.zero\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nZ : C\nf : Z \u27f6 X.obj (op [Nat.zero])\nh : f \u226b HomologicalComplex.Hom.f PInfty Nat.zero = 0\n\u22a2 f \u226b \u03c0Summand s (IndexSet.id (op [Nat.zero])) = 0\n[PROOFSTEP]\ndsimp at h \n[GOAL]\ncase mp.zero\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nZ : C\nf : Z \u27f6 X.obj (op [Nat.zero])\nh : f \u226b \ud835\udfd9 (X.obj (op [0])) = 0\n\u22a2 f \u226b \u03c0Summand s (IndexSet.id (op [Nat.zero])) = 0\n[PROOFSTEP]\nrw [comp_id] at h \n[GOAL]\ncase mp.zero\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nZ : C\nf : Z \u27f6 X.obj (op [Nat.zero])\nh : f = 0\n\u22a2 f \u226b \u03c0Summand s (IndexSet.id (op [Nat.zero])) = 0\n[PROOFSTEP]\nrw [h, zero_comp]\n[GOAL]\ncase mp.succ\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nZ : C\nn : \u2115\nf : Z \u27f6 X.obj (op [Nat.succ n])\nh : f \u226b HomologicalComplex.Hom.f PInfty (Nat.succ n) = 0\n\u22a2 f \u226b \u03c0Summand s (IndexSet.id (op [Nat.succ n])) = 0\n[PROOFSTEP]\nhave h' := f \u226b= PInfty_f_add_QInfty_f (n + 1)\n[GOAL]\ncase mp.succ\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nZ : C\nn : \u2115\nf : Z \u27f6 X.obj (op [Nat.succ n])\nh : f \u226b HomologicalComplex.Hom.f PInfty (Nat.succ n) = 0\nh' :\n f \u226b (HomologicalComplex.Hom.f PInfty (n + 1) + HomologicalComplex.Hom.f QInfty (n + 1)) =\n f \u226b \ud835\udfd9 (HomologicalComplex.X K[X] (n + 1))\n\u22a2 f \u226b \u03c0Summand s (IndexSet.id (op [Nat.succ n])) = 0\n[PROOFSTEP]\ndsimp at h' \n[GOAL]\ncase mp.succ\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nZ : C\nn : \u2115\nf : Z \u27f6 X.obj (op [Nat.succ n])\nh : f \u226b HomologicalComplex.Hom.f PInfty (Nat.succ n) = 0\nh' :\n f \u226b (HomologicalComplex.Hom.f PInfty (n + 1) + HomologicalComplex.Hom.f QInfty (n + 1)) = f \u226b \ud835\udfd9 (X.obj (op [n + 1]))\n\u22a2 f \u226b \u03c0Summand s (IndexSet.id (op [Nat.succ n])) = 0\n[PROOFSTEP]\nrw [comp_id, comp_add, h, zero_add] at h' \n[GOAL]\ncase mp.succ\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nZ : C\nn : \u2115\nf : Z \u27f6 X.obj (op [Nat.succ n])\nh : f \u226b HomologicalComplex.Hom.f PInfty (Nat.succ n) = 0\nh' : f \u226b HomologicalComplex.Hom.f QInfty (n + 1) = f\n\u22a2 f \u226b \u03c0Summand s (IndexSet.id (op [Nat.succ n])) = 0\n[PROOFSTEP]\nrw [\u2190 h', assoc, QInfty_f, decomposition_Q, Preadditive.sum_comp, Preadditive.comp_sum, Finset.sum_eq_zero]\n[GOAL]\ncase mp.succ\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nZ : C\nn : \u2115\nf : Z \u27f6 X.obj (op [Nat.succ n])\nh : f \u226b HomologicalComplex.Hom.f PInfty (Nat.succ n) = 0\nh' : f \u226b HomologicalComplex.Hom.f QInfty (n + 1) = f\n\u22a2 \u2200 (x : Fin (n + 1)),\n x \u2208 Finset.filter (fun i => \u2191i < n + 1) Finset.univ \u2192\n f \u226b\n (HomologicalComplex.Hom.f (P \u2191x) (n + 1) \u226b\n SimplicialObject.\u03b4 X (Fin.succ (\u2191Fin.revPerm x)) \u226b SimplicialObject.\u03c3 X (\u2191Fin.revPerm x)) \u226b\n \u03c0Summand s (IndexSet.id (op [Nat.succ n])) =\n 0\n[PROOFSTEP]\nintro i _\n[GOAL]\ncase mp.succ\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nZ : C\nn : \u2115\nf : Z \u27f6 X.obj (op [Nat.succ n])\nh : f \u226b HomologicalComplex.Hom.f PInfty (Nat.succ n) = 0\nh' : f \u226b HomologicalComplex.Hom.f QInfty (n + 1) = f\ni : Fin (n + 1)\na\u271d : i \u2208 Finset.filter (fun i => \u2191i < n + 1) Finset.univ\n\u22a2 f \u226b\n (HomologicalComplex.Hom.f (P \u2191i) (n + 1) \u226b\n SimplicialObject.\u03b4 X (Fin.succ (\u2191Fin.revPerm i)) \u226b SimplicialObject.\u03c3 X (\u2191Fin.revPerm i)) \u226b\n \u03c0Summand s (IndexSet.id (op [Nat.succ n])) =\n 0\n[PROOFSTEP]\nsimp only [assoc, \u03c3_comp_\u03c0Summand_id_eq_zero, comp_zero]\n[GOAL]\ncase mpr\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nZ : C\nn : \u2115\nf : Z \u27f6 X.obj (op [n])\n\u22a2 f \u226b \u03c0Summand s (IndexSet.id (op [n])) = 0 \u2192 f \u226b HomologicalComplex.Hom.f PInfty n = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nZ : C\nn : \u2115\nf : Z \u27f6 X.obj (op [n])\nh : f \u226b \u03c0Summand s (IndexSet.id (op [n])) = 0\n\u22a2 f \u226b HomologicalComplex.Hom.f PInfty n = 0\n[PROOFSTEP]\nrw [\u2190 comp_id f, assoc, s.decomposition_id, Preadditive.sum_comp, Preadditive.comp_sum, Fintype.sum_eq_zero]\n[GOAL]\ncase mpr.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nZ : C\nn : \u2115\nf : Z \u27f6 X.obj (op [n])\nh : f \u226b \u03c0Summand s (IndexSet.id (op [n])) = 0\n\u22a2 \u2200 (a : IndexSet (op [n])), f \u226b (\u03c0Summand s a \u226b \u03b9Summand s a) \u226b HomologicalComplex.Hom.f PInfty n = 0\n[PROOFSTEP]\nintro A\n[GOAL]\ncase mpr.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nZ : C\nn : \u2115\nf : Z \u27f6 X.obj (op [n])\nh : f \u226b \u03c0Summand s (IndexSet.id (op [n])) = 0\nA : IndexSet (op [n])\n\u22a2 f \u226b (\u03c0Summand s A \u226b \u03b9Summand s A) \u226b HomologicalComplex.Hom.f PInfty n = 0\n[PROOFSTEP]\nby_cases hA : A.EqId\n[GOAL]\ncase pos\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nZ : C\nn : \u2115\nf : Z \u27f6 X.obj (op [n])\nh : f \u226b \u03c0Summand s (IndexSet.id (op [n])) = 0\nA : IndexSet (op [n])\nhA : IndexSet.EqId A\n\u22a2 f \u226b (\u03c0Summand s A \u226b \u03b9Summand s A) \u226b HomologicalComplex.Hom.f PInfty n = 0\n[PROOFSTEP]\ndsimp at hA \n[GOAL]\ncase pos\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nZ : C\nn : \u2115\nf : Z \u27f6 X.obj (op [n])\nh : f \u226b \u03c0Summand s (IndexSet.id (op [n])) = 0\nA : IndexSet (op [n])\nhA : A = IndexSet.id (op [n])\n\u22a2 f \u226b (\u03c0Summand s A \u226b \u03b9Summand s A) \u226b HomologicalComplex.Hom.f PInfty n = 0\n[PROOFSTEP]\nsubst hA\n[GOAL]\ncase pos\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nZ : C\nn : \u2115\nf : Z \u27f6 X.obj (op [n])\nh : f \u226b \u03c0Summand s (IndexSet.id (op [n])) = 0\n\u22a2 f \u226b (\u03c0Summand s (IndexSet.id (op [n])) \u226b \u03b9Summand s (IndexSet.id (op [n]))) \u226b HomologicalComplex.Hom.f PInfty n = 0\n[PROOFSTEP]\nrw [assoc, reassoc_of% h, zero_comp]\n[GOAL]\ncase neg\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nZ : C\nn : \u2115\nf : Z \u27f6 X.obj (op [n])\nh : f \u226b \u03c0Summand s (IndexSet.id (op [n])) = 0\nA : IndexSet (op [n])\nhA : \u00acIndexSet.EqId A\n\u22a2 f \u226b (\u03c0Summand s A \u226b \u03b9Summand s A) \u226b HomologicalComplex.Hom.f PInfty n = 0\n[PROOFSTEP]\nsimp only [assoc, s.\u03b9Summand_comp_PInfty_eq_zero A hA, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f PInfty n \u226b \u03c0Summand s (IndexSet.id (op [n])) = \u03c0Summand s (IndexSet.id (op [n]))\n[PROOFSTEP]\nconv_rhs => rw [\u2190 id_comp (s.\u03c0Summand _)]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\n| \u03c0Summand s (IndexSet.id (op [n]))\n[PROOFSTEP]\nrw [\u2190 id_comp (s.\u03c0Summand _)]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\n| \u03c0Summand s (IndexSet.id (op [n]))\n[PROOFSTEP]\nrw [\u2190 id_comp (s.\u03c0Summand _)]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\n| \u03c0Summand s (IndexSet.id (op [n]))\n[PROOFSTEP]\nrw [\u2190 id_comp (s.\u03c0Summand _)]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f PInfty n \u226b \u03c0Summand s (IndexSet.id (op [n])) =\n \ud835\udfd9 (X.obj (op [n])) \u226b \u03c0Summand s (IndexSet.id (op [n]))\n[PROOFSTEP]\nsymm\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\n\u22a2 \ud835\udfd9 (X.obj (op [n])) \u226b \u03c0Summand s (IndexSet.id (op [n])) =\n HomologicalComplex.Hom.f PInfty n \u226b \u03c0Summand s (IndexSet.id (op [n]))\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 sub_comp, \u2190 comp_PInfty_eq_zero_iff, sub_comp, id_comp, PInfty_f_idem, sub_self]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\n\u22a2 \u03c0Summand s (IndexSet.id (op [n])) \u226b \u03b9Summand s (IndexSet.id (op [n])) \u226b HomologicalComplex.Hom.f PInfty n =\n HomologicalComplex.Hom.f PInfty n\n[PROOFSTEP]\nconv_rhs => rw [\u2190 id_comp (PInfty.f n)]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\n| HomologicalComplex.Hom.f PInfty n\n[PROOFSTEP]\nrw [\u2190 id_comp (PInfty.f n)]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\n| HomologicalComplex.Hom.f PInfty n\n[PROOFSTEP]\nrw [\u2190 id_comp (PInfty.f n)]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\n| HomologicalComplex.Hom.f PInfty n\n[PROOFSTEP]\nrw [\u2190 id_comp (PInfty.f n)]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\n\u22a2 \u03c0Summand s (IndexSet.id (op [n])) \u226b \u03b9Summand s (IndexSet.id (op [n])) \u226b HomologicalComplex.Hom.f PInfty n =\n \ud835\udfd9 (HomologicalComplex.X K[X] n) \u226b HomologicalComplex.Hom.f PInfty n\n[PROOFSTEP]\nerw [s.decomposition_id, Preadditive.sum_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\n\u22a2 \u03c0Summand s (IndexSet.id (op [n])) \u226b \u03b9Summand s (IndexSet.id (op [n])) \u226b HomologicalComplex.Hom.f PInfty n =\n \u2211 j : IndexSet (op [n]), (\u03c0Summand s j \u226b \u03b9Summand s j) \u226b HomologicalComplex.Hom.f PInfty n\n[PROOFSTEP]\nrw [Fintype.sum_eq_single (IndexSet.id (op [n])), assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\n\u22a2 \u2200 (x : IndexSet (op [n])),\n x \u2260 IndexSet.id (op [n]) \u2192 (\u03c0Summand s x \u226b \u03b9Summand s x) \u226b HomologicalComplex.Hom.f PInfty n = 0\n[PROOFSTEP]\nrintro A (hA : \u00acA.EqId)\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\nA : IndexSet (op [n])\nhA : \u00acIndexSet.EqId A\n\u22a2 (\u03c0Summand s A \u226b \u03b9Summand s A) \u226b HomologicalComplex.Hom.f PInfty n = 0\n[PROOFSTEP]\nrw [assoc, s.\u03b9Summand_comp_PInfty_eq_zero A hA, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nj k : \u2115\nA : IndexSet (op [j])\nhA : \u00acIndexSet.EqId A\n\u22a2 \u03b9Summand s A \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) = 0\n[PROOFSTEP]\nrw [A.eqId_iff_mono] at hA \n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nj k : \u2115\nA : IndexSet (op [j])\nhA : \u00acMono (IndexSet.e A)\n\u22a2 \u03b9Summand s A \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) = 0\n[PROOFSTEP]\nrw [\u2190 assoc, \u2190 s.comp_PInfty_eq_zero_iff, assoc, \u2190 PInfty.comm j k, s.\u03b9Summand_eq, assoc,\n degeneracy_comp_PInfty_assoc X j A.e hA, zero_comp, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.276754, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j : \u2115\nhij : \u00acComplexShape.Rel (ComplexShape.down \u2115) i j\n\u22a2 d s i j = 0\n[PROOFSTEP]\nsimp only [d, K[X].shape i j hij, zero_comp, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.276754, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j k : \u2115\nx\u271d\u00b9 : ComplexShape.Rel (ComplexShape.down \u2115) i j\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) j k\n\u22a2 d s i j \u226b d s j k = 0\n[PROOFSTEP]\nsimp only [d, assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.276754, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j k : \u2115\nx\u271d\u00b9 : ComplexShape.Rel (ComplexShape.down \u2115) i j\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) j k\n\u22a2 \u03b9Summand s (IndexSet.id (op [i])) \u226b\n HomologicalComplex.d K[X] i j \u226b\n \u03c0Summand s (IndexSet.id (op [j])) \u226b\n \u03b9Summand s (IndexSet.id (op [j])) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) =\n 0\n[PROOFSTEP]\nhave eq : K[X].d i j \u226b \ud835\udfd9 (X.obj (op [j])) \u226b K[X].d j k \u226b s.\u03c0Summand (IndexSet.id (op [k])) = 0 := by\n erw [id_comp, HomologicalComplex.d_comp_d_assoc, zero_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.276754, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j k : \u2115\nx\u271d\u00b9 : ComplexShape.Rel (ComplexShape.down \u2115) i j\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) j k\n\u22a2 HomologicalComplex.d K[X] i j \u226b\n \ud835\udfd9 (X.obj (op [j])) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) =\n 0\n[PROOFSTEP]\nerw [id_comp, HomologicalComplex.d_comp_d_assoc, zero_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.276754, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j k : \u2115\nx\u271d\u00b9 : ComplexShape.Rel (ComplexShape.down \u2115) i j\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) j k\neq :\n HomologicalComplex.d K[X] i j \u226b\n \ud835\udfd9 (X.obj (op [j])) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) =\n 0\n\u22a2 \u03b9Summand s (IndexSet.id (op [i])) \u226b\n HomologicalComplex.d K[X] i j \u226b\n \u03c0Summand s (IndexSet.id (op [j])) \u226b\n \u03b9Summand s (IndexSet.id (op [j])) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) =\n 0\n[PROOFSTEP]\nrw [s.decomposition_id] at eq \n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.276754, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j k : \u2115\nx\u271d\u00b9 : ComplexShape.Rel (ComplexShape.down \u2115) i j\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) j k\neq :\n HomologicalComplex.d K[X] i j \u226b\n (\u2211 A : IndexSet (op [j]), \u03c0Summand s A \u226b \u03b9Summand s A) \u226b\n HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) =\n 0\n\u22a2 \u03b9Summand s (IndexSet.id (op [i])) \u226b\n HomologicalComplex.d K[X] i j \u226b\n \u03c0Summand s (IndexSet.id (op [j])) \u226b\n \u03b9Summand s (IndexSet.id (op [j])) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) =\n 0\n[PROOFSTEP]\nclassical\nrw [Fintype.sum_eq_add_sum_compl (IndexSet.id (op [j])), add_comp, comp_add, assoc, Preadditive.sum_comp,\n Preadditive.comp_sum, Finset.sum_eq_zero, add_zero] at eq \nswap\n\u00b7 intro A hA\n simp only [Finset.mem_compl, Finset.mem_singleton] at hA \n simp only [assoc, \u03b9Summand_comp_d_comp_\u03c0Summand_eq_zero _ _ _ _ hA, comp_zero]\nrw [eq, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.276754, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j k : \u2115\nx\u271d\u00b9 : ComplexShape.Rel (ComplexShape.down \u2115) i j\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) j k\neq :\n HomologicalComplex.d K[X] i j \u226b\n (\u2211 A : IndexSet (op [j]), \u03c0Summand s A \u226b \u03b9Summand s A) \u226b\n HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) =\n 0\n\u22a2 \u03b9Summand s (IndexSet.id (op [i])) \u226b\n HomologicalComplex.d K[X] i j \u226b\n \u03c0Summand s (IndexSet.id (op [j])) \u226b\n \u03b9Summand s (IndexSet.id (op [j])) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) =\n 0\n[PROOFSTEP]\nrw [Fintype.sum_eq_add_sum_compl (IndexSet.id (op [j])), add_comp, comp_add, assoc, Preadditive.sum_comp,\n Preadditive.comp_sum, Finset.sum_eq_zero, add_zero] at eq \n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.276754, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j k : \u2115\nx\u271d\u00b9 : ComplexShape.Rel (ComplexShape.down \u2115) i j\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) j k\neq :\n HomologicalComplex.d K[X] i j \u226b\n \u03c0Summand s (IndexSet.id (op [j])) \u226b\n \u03b9Summand s (IndexSet.id (op [j])) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) =\n 0\n\u22a2 \u03b9Summand s (IndexSet.id (op [i])) \u226b\n HomologicalComplex.d K[X] i j \u226b\n \u03c0Summand s (IndexSet.id (op [j])) \u226b\n \u03b9Summand s (IndexSet.id (op [j])) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) =\n 0\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.276754, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j k : \u2115\nx\u271d\u00b9 : ComplexShape.Rel (ComplexShape.down \u2115) i j\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) j k\neq :\n HomologicalComplex.d K[X] i j \u226b\n \u03c0Summand s (IndexSet.id (op [j])) \u226b\n \u03b9Summand s (IndexSet.id (op [j])) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) +\n \u2211 j_1 in {IndexSet.id (op [j])}\u1d9c,\n HomologicalComplex.d K[X] i j \u226b\n (\u03c0Summand s j_1 \u226b \u03b9Summand s j_1) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) =\n 0\n\u22a2 \u2200 (x : IndexSet (op [j])),\n x \u2208 {IndexSet.id (op [j])}\u1d9c \u2192\n HomologicalComplex.d K[X] i j \u226b\n (\u03c0Summand s x \u226b \u03b9Summand s x) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) =\n 0\n[PROOFSTEP]\nswap\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.276754, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j k : \u2115\nx\u271d\u00b9 : ComplexShape.Rel (ComplexShape.down \u2115) i j\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) j k\neq :\n HomologicalComplex.d K[X] i j \u226b\n \u03c0Summand s (IndexSet.id (op [j])) \u226b\n \u03b9Summand s (IndexSet.id (op [j])) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) +\n \u2211 j_1 in {IndexSet.id (op [j])}\u1d9c,\n HomologicalComplex.d K[X] i j \u226b\n (\u03c0Summand s j_1 \u226b \u03b9Summand s j_1) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) =\n 0\n\u22a2 \u2200 (x : IndexSet (op [j])),\n x \u2208 {IndexSet.id (op [j])}\u1d9c \u2192\n HomologicalComplex.d K[X] i j \u226b\n (\u03c0Summand s x \u226b \u03b9Summand s x) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) =\n 0\n[PROOFSTEP]\nintro A hA\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.276754, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j k : \u2115\nx\u271d\u00b9 : ComplexShape.Rel (ComplexShape.down \u2115) i j\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) j k\neq :\n HomologicalComplex.d K[X] i j \u226b\n \u03c0Summand s (IndexSet.id (op [j])) \u226b\n \u03b9Summand s (IndexSet.id (op [j])) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) +\n \u2211 j_1 in {IndexSet.id (op [j])}\u1d9c,\n HomologicalComplex.d K[X] i j \u226b\n (\u03c0Summand s j_1 \u226b \u03b9Summand s j_1) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) =\n 0\nA : IndexSet (op [j])\nhA : A \u2208 {IndexSet.id (op [j])}\u1d9c\n\u22a2 HomologicalComplex.d K[X] i j \u226b\n (\u03c0Summand s A \u226b \u03b9Summand s A) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) =\n 0\n[PROOFSTEP]\nsimp only [Finset.mem_compl, Finset.mem_singleton] at hA \n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.276754, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j k : \u2115\nx\u271d\u00b9 : ComplexShape.Rel (ComplexShape.down \u2115) i j\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) j k\neq :\n HomologicalComplex.d K[X] i j \u226b\n \u03c0Summand s (IndexSet.id (op [j])) \u226b\n \u03b9Summand s (IndexSet.id (op [j])) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) +\n \u2211 j_1 in {IndexSet.id (op [j])}\u1d9c,\n HomologicalComplex.d K[X] i j \u226b\n (\u03c0Summand s j_1 \u226b \u03b9Summand s j_1) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) =\n 0\nA : IndexSet (op [j])\nhA : \u00acA = IndexSet.id (op [j])\n\u22a2 HomologicalComplex.d K[X] i j \u226b\n (\u03c0Summand s A \u226b \u03b9Summand s A) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) =\n 0\n[PROOFSTEP]\nsimp only [assoc, \u03b9Summand_comp_d_comp_\u03c0Summand_eq_zero _ _ _ _ hA, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.276754, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j k : \u2115\nx\u271d\u00b9 : ComplexShape.Rel (ComplexShape.down \u2115) i j\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) j k\neq :\n HomologicalComplex.d K[X] i j \u226b\n \u03c0Summand s (IndexSet.id (op [j])) \u226b\n \u03b9Summand s (IndexSet.id (op [j])) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) =\n 0\n\u22a2 \u03b9Summand s (IndexSet.id (op [i])) \u226b\n HomologicalComplex.d K[X] i j \u226b\n \u03c0Summand s (IndexSet.id (op [j])) \u226b\n \u03b9Summand s (IndexSet.id (op [j])) \u226b HomologicalComplex.d K[X] j k \u226b \u03c0Summand s (IndexSet.id (op [k])) =\n 0\n[PROOFSTEP]\nrw [eq, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n\u22a2 (fun n => \u03b9Summand s (IndexSet.id (op [n])) \u226b HomologicalComplex.Hom.f PInfty n) i \u226b\n HomologicalComplex.d (N\u2081.obj X).X i j =\n HomologicalComplex.d ((toKaroubi (ChainComplex C \u2115)).obj (nondegComplex s)).X i j \u226b\n (fun n => \u03b9Summand s (IndexSet.id (op [n])) \u226b HomologicalComplex.Hom.f PInfty n) j\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n\u22a2 (\u03b9Summand s (IndexSet.id (op [i])) \u226b HomologicalComplex.Hom.f PInfty i) \u226b HomologicalComplex.d K[X] i j =\n (\u03b9Summand s (IndexSet.id (op [i])) \u226b HomologicalComplex.d K[X] i j \u226b \u03c0Summand s (IndexSet.id (op [j]))) \u226b\n \u03b9Summand s (IndexSet.id (op [j])) \u226b HomologicalComplex.Hom.f PInfty j\n[PROOFSTEP]\nrw [assoc, assoc, assoc, \u03c0Summand_comp_\u03b9Summand_comp_PInfty_eq_PInfty, HomologicalComplex.Hom.comm]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\n\u22a2 (HomologicalComplex.Hom.mk fun n => \u03b9Summand s (IndexSet.id (op [n])) \u226b HomologicalComplex.Hom.f PInfty n) =\n ((toKaroubi (ChainComplex C \u2115)).obj (nondegComplex s)).p \u226b\n (HomologicalComplex.Hom.mk fun n => \u03b9Summand s (IndexSet.id (op [n])) \u226b HomologicalComplex.Hom.f PInfty n) \u226b\n (N\u2081.obj X).p\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f\n (HomologicalComplex.Hom.mk fun n => \u03b9Summand s (IndexSet.id (op [n])) \u226b HomologicalComplex.Hom.f PInfty n) n =\n HomologicalComplex.Hom.f\n (((toKaroubi (ChainComplex C \u2115)).obj (nondegComplex s)).p \u226b\n (HomologicalComplex.Hom.mk fun n => \u03b9Summand s (IndexSet.id (op [n])) \u226b HomologicalComplex.Hom.f PInfty n) \u226b\n (N\u2081.obj X).p)\n n\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\n\u22a2 \u03b9Summand s (IndexSet.id (op [n])) \u226b HomologicalComplex.Hom.f PInfty n =\n \ud835\udfd9 (N s n) \u226b\n (\u03b9Summand s (IndexSet.id (op [n])) \u226b HomologicalComplex.Hom.f PInfty n) \u226b HomologicalComplex.Hom.f PInfty n\n[PROOFSTEP]\nrw [id_comp, assoc, PInfty_f_idem]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n\u22a2 (fun n => \u03c0Summand s (IndexSet.id (op [n]))) i \u226b\n HomologicalComplex.d ((toKaroubi (ChainComplex C \u2115)).obj (nondegComplex s)).X i j =\n HomologicalComplex.d (N\u2081.obj X).X i j \u226b (fun n => \u03c0Summand s (IndexSet.id (op [n]))) j\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n\u22a2 \u03c0Summand s (IndexSet.id (op [i])) \u226b\n \u03b9Summand s (IndexSet.id (op [i])) \u226b HomologicalComplex.d K[X] i j \u226b \u03c0Summand s (IndexSet.id (op [j])) =\n HomologicalComplex.d K[X] i j \u226b \u03c0Summand s (IndexSet.id (op [j]))\n[PROOFSTEP]\nslice_rhs 1 1 => rw [\u2190 id_comp (K[X].d i j)]\n[GOAL]\ncase a\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n| HomologicalComplex.d K[X] i j\ncase a\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n| \u03c0Summand s (IndexSet.id (op [j]))\n[PROOFSTEP]\nrw [\u2190 id_comp (K[X].d i j)]\n[GOAL]\ncase a\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n| HomologicalComplex.d K[X] i j\ncase a\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n| \u03c0Summand s (IndexSet.id (op [j]))\n[PROOFSTEP]\nrw [\u2190 id_comp (K[X].d i j)]\n[GOAL]\ncase a\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n| HomologicalComplex.d K[X] i j\ncase a\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n| \u03c0Summand s (IndexSet.id (op [j]))\n[PROOFSTEP]\nrw [\u2190 id_comp (K[X].d i j)]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n\u22a2 \u03c0Summand s (IndexSet.id (op [i])) \u226b\n \u03b9Summand s (IndexSet.id (op [i])) \u226b HomologicalComplex.d K[X] i j \u226b \u03c0Summand s (IndexSet.id (op [j])) =\n (\ud835\udfd9 (HomologicalComplex.X K[X] i) \u226b HomologicalComplex.d K[X] i j) \u226b \u03c0Summand s (IndexSet.id (op [j]))\n[PROOFSTEP]\nerw [s.decomposition_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n\u22a2 \u03c0Summand s (IndexSet.id (op [i])) \u226b\n \u03b9Summand s (IndexSet.id (op [i])) \u226b HomologicalComplex.d K[X] i j \u226b \u03c0Summand s (IndexSet.id (op [j])) =\n ((\u2211 A : IndexSet (op [i]), \u03c0Summand s A \u226b \u03b9Summand s A) \u226b HomologicalComplex.d K[X] i j) \u226b\n \u03c0Summand s (IndexSet.id (op [j]))\n[PROOFSTEP]\nrw [sum_comp, sum_comp, Finset.sum_eq_single (IndexSet.id (op [i])), assoc, assoc]\n[GOAL]\ncase h\u2080\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n\u22a2 \u2200 (b : IndexSet (op [i])),\n b \u2208 Finset.univ \u2192\n b \u2260 IndexSet.id (op [i]) \u2192\n ((\u03c0Summand s b \u226b \u03b9Summand s b) \u226b HomologicalComplex.d K[X] i j) \u226b \u03c0Summand s (IndexSet.id (op [j])) = 0\n[PROOFSTEP]\nintro A _ hA\n[GOAL]\ncase h\u2080\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\nA : IndexSet (op [i])\na\u271d : A \u2208 Finset.univ\nhA : A \u2260 IndexSet.id (op [i])\n\u22a2 ((\u03c0Summand s A \u226b \u03b9Summand s A) \u226b HomologicalComplex.d K[X] i j) \u226b \u03c0Summand s (IndexSet.id (op [j])) = 0\n[PROOFSTEP]\nsimp only [assoc, s.\u03b9Summand_comp_d_comp_\u03c0Summand_eq_zero _ _ _ hA, comp_zero]\n[GOAL]\ncase h\u2081\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n\u22a2 \u00acIndexSet.id (op [i]) \u2208 Finset.univ \u2192\n ((\u03c0Summand s (IndexSet.id (op [i])) \u226b \u03b9Summand s (IndexSet.id (op [i]))) \u226b HomologicalComplex.d K[X] i j) \u226b\n \u03c0Summand s (IndexSet.id (op [j])) =\n 0\n[PROOFSTEP]\nsimp only [Finset.mem_univ, not_true, IsEmpty.forall_iff]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\n\u22a2 (HomologicalComplex.Hom.mk fun n => \u03c0Summand s (IndexSet.id (op [n]))) =\n (N\u2081.obj X).p \u226b\n (HomologicalComplex.Hom.mk fun n => \u03c0Summand s (IndexSet.id (op [n]))) \u226b\n ((toKaroubi (ChainComplex C \u2115)).obj (nondegComplex s)).p\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f (HomologicalComplex.Hom.mk fun n => \u03c0Summand s (IndexSet.id (op [n]))) n =\n HomologicalComplex.Hom.f\n ((N\u2081.obj X).p \u226b\n (HomologicalComplex.Hom.mk fun n => \u03c0Summand s (IndexSet.id (op [n]))) \u226b\n ((toKaroubi (ChainComplex C \u2115)).obj (nondegComplex s)).p)\n n\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\n\u22a2 \u03c0Summand s (IndexSet.id (op [n])) = HomologicalComplex.Hom.f PInfty n \u226b \u03c0Summand s (IndexSet.id (op [n])) \u226b \ud835\udfd9 (N s n)\n[PROOFSTEP]\nsimp only [comp_id, PInfty_comp_\u03c0Summand_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\n\u22a2 Karoubi.Hom.mk\n (HomologicalComplex.Hom.mk fun n => \u03b9Summand s (IndexSet.id (op [n])) \u226b HomologicalComplex.Hom.f PInfty n) \u226b\n Karoubi.Hom.mk (HomologicalComplex.Hom.mk fun n => \u03c0Summand s (IndexSet.id (op [n]))) =\n \ud835\udfd9 ((toKaroubi (ChainComplex C \u2115)).obj (nondegComplex s))\n[PROOFSTEP]\next n\n[GOAL]\ncase h.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f\n (Karoubi.Hom.mk\n (HomologicalComplex.Hom.mk fun n => \u03b9Summand s (IndexSet.id (op [n])) \u226b HomologicalComplex.Hom.f PInfty n) \u226b\n Karoubi.Hom.mk (HomologicalComplex.Hom.mk fun n => \u03c0Summand s (IndexSet.id (op [n])))).f\n n =\n HomologicalComplex.Hom.f (\ud835\udfd9 ((toKaroubi (ChainComplex C \u2115)).obj (nondegComplex s))).f n\n[PROOFSTEP]\nsimp only [assoc, PInfty_comp_\u03c0Summand_id, Karoubi.comp_f, HomologicalComplex.comp_f, \u03b9_\u03c0Summand_eq_id]\n[GOAL]\ncase h.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\n\u22a2 \ud835\udfd9 (N s (SimplexCategory.len (IndexSet.id (op [n])).fst.unop)) =\n HomologicalComplex.Hom.f (\ud835\udfd9 ((toKaroubi (ChainComplex C \u2115)).obj (nondegComplex s))).f n\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\n\u22a2 Karoubi.Hom.mk (HomologicalComplex.Hom.mk fun n => \u03c0Summand s (IndexSet.id (op [n]))) \u226b\n Karoubi.Hom.mk\n (HomologicalComplex.Hom.mk fun n => \u03b9Summand s (IndexSet.id (op [n])) \u226b HomologicalComplex.Hom.f PInfty n) =\n \ud835\udfd9 (N\u2081.obj X)\n[PROOFSTEP]\next n\n[GOAL]\ncase h.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.336016, u_1} C\ninst\u271d\u00b9 : HasFiniteCoproducts C\nX : SimplicialObject C\ns : Splitting X\ninst\u271d : Preadditive C\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f\n (Karoubi.Hom.mk (HomologicalComplex.Hom.mk fun n => \u03c0Summand s (IndexSet.id (op [n]))) \u226b\n Karoubi.Hom.mk\n (HomologicalComplex.Hom.mk fun n =>\n \u03b9Summand s (IndexSet.id (op [n])) \u226b HomologicalComplex.Hom.f PInfty n)).f\n n =\n HomologicalComplex.Hom.f (\ud835\udfd9 (N\u2081.obj X)).f n\n[PROOFSTEP]\nsimp only [\u03c0Summand_comp_\u03b9Summand_comp_PInfty_eq_PInfty, Karoubi.comp_f, HomologicalComplex.comp_f, N\u2081_obj_p,\n Karoubi.id_eq]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.406470, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6 : S\u2081 \u27f6 S\u2082\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n\u22a2 Hom.f \u03a6 i \u226b HomologicalComplex.d ((fun S => Splitting.nondegComplex S.s) S\u2082) i j =\n HomologicalComplex.d ((fun S => Splitting.nondegComplex S.s) S\u2081) i j \u226b Hom.f \u03a6 j\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.406470, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6 : S\u2081 \u27f6 S\u2082\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n\u22a2 Hom.f \u03a6 i \u226b\n Splitting.\u03b9Summand S\u2082.s (Splitting.IndexSet.id (op [i])) \u226b\n HomologicalComplex.d K[S\u2082.X] i j \u226b Splitting.\u03c0Summand S\u2082.s (Splitting.IndexSet.id (op [j])) =\n (Splitting.\u03b9Summand S\u2081.s (Splitting.IndexSet.id (op [i])) \u226b\n HomologicalComplex.d K[S\u2081.X] i j \u226b Splitting.\u03c0Summand S\u2081.s (Splitting.IndexSet.id (op [j]))) \u226b\n Hom.f \u03a6 j\n[PROOFSTEP]\nerw [\u2190 \u03b9Summand_naturality_symm_assoc \u03a6 (Splitting.IndexSet.id (op [i])),\n ((alternatingFaceMapComplex C).map \u03a6.F).comm_assoc i j]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.406470, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6 : S\u2081 \u27f6 S\u2082\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n\u22a2 Splitting.\u03b9Summand S\u2081.s (Splitting.IndexSet.id (op [i])) \u226b\n HomologicalComplex.d ((alternatingFaceMapComplex C).obj S\u2081.X) i j \u226b\n HomologicalComplex.Hom.f ((alternatingFaceMapComplex C).map \u03a6.F) j \u226b\n Splitting.\u03c0Summand S\u2082.s (Splitting.IndexSet.id (op [j])) =\n (Splitting.\u03b9Summand S\u2081.s (Splitting.IndexSet.id (op [i])) \u226b\n HomologicalComplex.d K[S\u2081.X] i j \u226b Splitting.\u03c0Summand S\u2081.s (Splitting.IndexSet.id (op [j]))) \u226b\n Hom.f \u03a6 j\n[PROOFSTEP]\nsimp only [assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.406470, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6 : S\u2081 \u27f6 S\u2082\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n\u22a2 Splitting.\u03b9Summand S\u2081.s (Splitting.IndexSet.id (op [i])) \u226b\n HomologicalComplex.d ((alternatingFaceMapComplex C).obj S\u2081.X) i j \u226b\n HomologicalComplex.Hom.f ((alternatingFaceMapComplex C).map \u03a6.F) j \u226b\n Splitting.\u03c0Summand S\u2082.s (Splitting.IndexSet.id (op [j])) =\n Splitting.\u03b9Summand S\u2081.s (Splitting.IndexSet.id (op [i])) \u226b\n HomologicalComplex.d K[S\u2081.X] i j \u226b Splitting.\u03c0Summand S\u2081.s (Splitting.IndexSet.id (op [j])) \u226b Hom.f \u03a6 j\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_a.e_a\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.406470, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6 : S\u2081 \u27f6 S\u2082\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n\u22a2 HomologicalComplex.Hom.f ((alternatingFaceMapComplex C).map \u03a6.F) j \u226b\n Splitting.\u03c0Summand S\u2082.s (Splitting.IndexSet.id (op [j])) =\n Splitting.\u03c0Summand S\u2081.s (Splitting.IndexSet.id (op [j])) \u226b Hom.f \u03a6 j\n[PROOFSTEP]\napply S\u2081.s.hom_ext'\n[GOAL]\ncase e_a.e_a.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.406470, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6 : S\u2081 \u27f6 S\u2082\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n\u22a2 \u2200 (A : Splitting.IndexSet (op [j])),\n Splitting.\u03b9Summand S\u2081.s A \u226b\n HomologicalComplex.Hom.f ((alternatingFaceMapComplex C).map \u03a6.F) j \u226b\n Splitting.\u03c0Summand S\u2082.s (Splitting.IndexSet.id (op [j])) =\n Splitting.\u03b9Summand S\u2081.s A \u226b Splitting.\u03c0Summand S\u2081.s (Splitting.IndexSet.id (op [j])) \u226b Hom.f \u03a6 j\n[PROOFSTEP]\nintro A\n[GOAL]\ncase e_a.e_a.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.406470, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6 : S\u2081 \u27f6 S\u2082\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\nA : Splitting.IndexSet (op [j])\n\u22a2 Splitting.\u03b9Summand S\u2081.s A \u226b\n HomologicalComplex.Hom.f ((alternatingFaceMapComplex C).map \u03a6.F) j \u226b\n Splitting.\u03c0Summand S\u2082.s (Splitting.IndexSet.id (op [j])) =\n Splitting.\u03b9Summand S\u2081.s A \u226b Splitting.\u03c0Summand S\u2081.s (Splitting.IndexSet.id (op [j])) \u226b Hom.f \u03a6 j\n[PROOFSTEP]\ndsimp [alternatingFaceMapComplex]\n[GOAL]\ncase e_a.e_a.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.406470, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6 : S\u2081 \u27f6 S\u2082\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\nA : Splitting.IndexSet (op [j])\n\u22a2 Splitting.\u03b9Summand S\u2081.s A \u226b NatTrans.app \u03a6.F (op [j]) \u226b Splitting.\u03c0Summand S\u2082.s (Splitting.IndexSet.id (op [j])) =\n Splitting.\u03b9Summand S\u2081.s A \u226b Splitting.\u03c0Summand S\u2081.s (Splitting.IndexSet.id (op [j])) \u226b Hom.f \u03a6 j\n[PROOFSTEP]\nerw [\u03b9Summand_naturality_symm_assoc \u03a6 A]\n[GOAL]\ncase e_a.e_a.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.406470, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6 : S\u2081 \u27f6 S\u2082\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\nA : Splitting.IndexSet (op [j])\n\u22a2 Hom.f \u03a6 (SimplexCategory.len A.fst.unop) \u226b\n Splitting.\u03b9Summand S\u2082.s A \u226b Splitting.\u03c0Summand S\u2082.s (Splitting.IndexSet.id (op [j])) =\n Splitting.\u03b9Summand S\u2081.s A \u226b Splitting.\u03c0Summand S\u2081.s (Splitting.IndexSet.id (op [j])) \u226b Hom.f \u03a6 j\n[PROOFSTEP]\nby_cases A.EqId\n[GOAL]\ncase e_a.e_a.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.406470, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6 : S\u2081 \u27f6 S\u2082\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\nA : Splitting.IndexSet (op [j])\n\u22a2 Hom.f \u03a6 (SimplexCategory.len A.fst.unop) \u226b\n Splitting.\u03b9Summand S\u2082.s A \u226b Splitting.\u03c0Summand S\u2082.s (Splitting.IndexSet.id (op [j])) =\n Splitting.\u03b9Summand S\u2081.s A \u226b Splitting.\u03c0Summand S\u2081.s (Splitting.IndexSet.id (op [j])) \u226b Hom.f \u03a6 j\n[PROOFSTEP]\nby_cases A.EqId\n[GOAL]\ncase pos\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.406470, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6 : S\u2081 \u27f6 S\u2082\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\nA : Splitting.IndexSet (op [j])\nh : Splitting.IndexSet.EqId A\n\u22a2 Hom.f \u03a6 (SimplexCategory.len A.fst.unop) \u226b\n Splitting.\u03b9Summand S\u2082.s A \u226b Splitting.\u03c0Summand S\u2082.s (Splitting.IndexSet.id (op [j])) =\n Splitting.\u03b9Summand S\u2081.s A \u226b Splitting.\u03c0Summand S\u2081.s (Splitting.IndexSet.id (op [j])) \u226b Hom.f \u03a6 j\n[PROOFSTEP]\ndsimp at h \n[GOAL]\ncase pos\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.406470, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6 : S\u2081 \u27f6 S\u2082\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\nA : Splitting.IndexSet (op [j])\nh : A = Splitting.IndexSet.id (op [j])\n\u22a2 Hom.f \u03a6 (SimplexCategory.len A.fst.unop) \u226b\n Splitting.\u03b9Summand S\u2082.s A \u226b Splitting.\u03c0Summand S\u2082.s (Splitting.IndexSet.id (op [j])) =\n Splitting.\u03b9Summand S\u2081.s A \u226b Splitting.\u03c0Summand S\u2081.s (Splitting.IndexSet.id (op [j])) \u226b Hom.f \u03a6 j\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.406470, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6 : S\u2081 \u27f6 S\u2082\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n\u22a2 Hom.f \u03a6 (SimplexCategory.len (Splitting.IndexSet.id (op [j])).fst.unop) \u226b\n Splitting.\u03b9Summand S\u2082.s (Splitting.IndexSet.id (op [j])) \u226b\n Splitting.\u03c0Summand S\u2082.s (Splitting.IndexSet.id (op [j])) =\n Splitting.\u03b9Summand S\u2081.s (Splitting.IndexSet.id (op [j])) \u226b\n Splitting.\u03c0Summand S\u2081.s (Splitting.IndexSet.id (op [j])) \u226b Hom.f \u03a6 j\n[PROOFSTEP]\nsimp only [Splitting.\u03b9_\u03c0Summand_eq_id, comp_id, Splitting.\u03b9_\u03c0Summand_eq_id_assoc]\n[GOAL]\ncase pos\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.406470, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6 : S\u2081 \u27f6 S\u2082\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\n\u22a2 Hom.f \u03a6 (SimplexCategory.len (Splitting.IndexSet.id (op [j])).fst.unop) = Hom.f \u03a6 j\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.406470, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6 : S\u2081 \u27f6 S\u2082\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\nA : Splitting.IndexSet (op [j])\nh : \u00acSplitting.IndexSet.EqId A\n\u22a2 Hom.f \u03a6 (SimplexCategory.len A.fst.unop) \u226b\n Splitting.\u03b9Summand S\u2082.s A \u226b Splitting.\u03c0Summand S\u2082.s (Splitting.IndexSet.id (op [j])) =\n Splitting.\u03b9Summand S\u2081.s A \u226b Splitting.\u03c0Summand S\u2081.s (Splitting.IndexSet.id (op [j])) \u226b Hom.f \u03a6 j\n[PROOFSTEP]\nhave h' : Splitting.IndexSet.id (op [j]) \u2260 A := by\n rw [ne_comm]\n exact h\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.406470, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6 : S\u2081 \u27f6 S\u2082\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\nA : Splitting.IndexSet (op [j])\nh : \u00acSplitting.IndexSet.EqId A\n\u22a2 Splitting.IndexSet.id (op [j]) \u2260 A\n[PROOFSTEP]\nrw [ne_comm]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.406470, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6 : S\u2081 \u27f6 S\u2082\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\nA : Splitting.IndexSet (op [j])\nh : \u00acSplitting.IndexSet.EqId A\n\u22a2 A \u2260 Splitting.IndexSet.id (op [j])\n[PROOFSTEP]\nexact h\n[GOAL]\ncase neg\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.406470, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6 : S\u2081 \u27f6 S\u2082\ni j : \u2115\nx\u271d : ComplexShape.Rel (ComplexShape.down \u2115) i j\nA : Splitting.IndexSet (op [j])\nh : \u00acSplitting.IndexSet.EqId A\nh' : Splitting.IndexSet.id (op [j]) \u2260 A\n\u22a2 Hom.f \u03a6 (SimplexCategory.len A.fst.unop) \u226b\n Splitting.\u03b9Summand S\u2082.s A \u226b Splitting.\u03c0Summand S\u2082.s (Splitting.IndexSet.id (op [j])) =\n Splitting.\u03b9Summand S\u2081.s A \u226b Splitting.\u03c0Summand S\u2081.s (Splitting.IndexSet.id (op [j])) \u226b Hom.f \u03a6 j\n[PROOFSTEP]\nrw [S\u2081.s.\u03b9_\u03c0Summand_eq_zero_assoc _ _ h', S\u2082.s.\u03b9_\u03c0Summand_eq_zero _ _ h', zero_comp, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.449707, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nX\u271d Y\u271d : Split C\n\u03a6 : X\u271d \u27f6 Y\u271d\n\u22a2 (nondegComplexFunctor \u22d9 toKaroubi (ChainComplex C \u2115)).map \u03a6 \u226b\n ((fun S => Splitting.toKaroubiNondegComplexIsoN\u2081 S.s) Y\u271d).hom =\n ((fun S => Splitting.toKaroubiNondegComplexIsoN\u2081 S.s) X\u271d).hom \u226b (forget C \u22d9 N\u2081).map \u03a6\n[PROOFSTEP]\next n\n[GOAL]\ncase h.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.449707, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nX\u271d Y\u271d : Split C\n\u03a6 : X\u271d \u27f6 Y\u271d\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f\n ((nondegComplexFunctor \u22d9 toKaroubi (ChainComplex C \u2115)).map \u03a6 \u226b\n ((fun S => Splitting.toKaroubiNondegComplexIsoN\u2081 S.s) Y\u271d).hom).f\n n =\n HomologicalComplex.Hom.f (((fun S => Splitting.toKaroubiNondegComplexIsoN\u2081 S.s) X\u271d).hom \u226b (forget C \u22d9 N\u2081).map \u03a6).f n\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.449707, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nX\u271d Y\u271d : Split C\n\u03a6 : X\u271d \u27f6 Y\u271d\nn : \u2115\n\u22a2 Hom.f \u03a6 n \u226b Splitting.\u03b9Summand Y\u271d.s (Splitting.IndexSet.id (op [n])) \u226b HomologicalComplex.Hom.f PInfty n =\n (Splitting.\u03b9Summand X\u271d.s (Splitting.IndexSet.id (op [n])) \u226b HomologicalComplex.Hom.f PInfty n) \u226b\n HomologicalComplex.Hom.f PInfty n \u226b NatTrans.app \u03a6.F (op [n])\n[PROOFSTEP]\nsimp only [Karoubi.comp_f, toKaroubi_map_f, HomologicalComplex.comp_f, nondegComplexFunctor_map_f,\n Splitting.toKaroubiNondegComplexIsoN\u2081_hom_f_f, N\u2081_map_f, AlternatingFaceMapComplex.map_f, assoc, PInfty_f_idem_assoc]\n[GOAL]\ncase h.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.449707, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nX\u271d Y\u271d : Split C\n\u03a6 : X\u271d \u27f6 Y\u271d\nn : \u2115\n\u22a2 Hom.f \u03a6 n \u226b Splitting.\u03b9Summand Y\u271d.s (Splitting.IndexSet.id (op [n])) \u226b HomologicalComplex.Hom.f PInfty n =\n Splitting.\u03b9Summand X\u271d.s (Splitting.IndexSet.id (op [n])) \u226b\n HomologicalComplex.Hom.f PInfty n \u226b NatTrans.app \u03a6.F (op [n])\n[PROOFSTEP]\nerw [\u2190 Split.\u03b9Summand_naturality_symm_assoc \u03a6 (Splitting.IndexSet.id (op [n]))]\n[GOAL]\ncase h.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{?u.449707, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nX\u271d Y\u271d : Split C\n\u03a6 : X\u271d \u27f6 Y\u271d\nn : \u2115\n\u22a2 Splitting.\u03b9Summand X\u271d.s (Splitting.IndexSet.id (op [n])) \u226b\n NatTrans.app \u03a6.F (op [n]) \u226b HomologicalComplex.Hom.f PInfty n =\n Splitting.\u03b9Summand X\u271d.s (Splitting.IndexSet.id (op [n])) \u226b\n HomologicalComplex.Hom.f PInfty n \u226b NatTrans.app \u03a6.F (op [n])\n[PROOFSTEP]\nrw [PInfty_f_naturality]\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject", "llama_tokens": 28454, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.48438008427698437, "lm_q2_score": 0.022977370286363187, "lm_q1q2_score": 0.011129780555772077}} {"text": "[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\n\u22a2 Category.{?u.61, max u v} (SimplicialObject C)\n[PROOFSTEP]\ndsimp only [SimplicialObject]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\n\u22a2 Category.{?u.61, max u v} (SimplexCategory\u1d52\u1d56 \u2964 C)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : HasLimitsOfShape J C\n\u22a2 HasLimitsOfShape J (SimplicialObject C)\n[PROOFSTEP]\ndsimp [SimplicialObject]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : HasLimitsOfShape J C\n\u22a2 HasLimitsOfShape J (SimplexCategory\u1d52\u1d56 \u2964 C)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : HasColimitsOfShape J C\n\u22a2 HasColimitsOfShape J (SimplicialObject C)\n[PROOFSTEP]\ndsimp [SimplicialObject]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : HasColimitsOfShape J C\n\u22a2 HasColimitsOfShape J (SimplexCategory\u1d52\u1d56 \u2964 C)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : SimplicialObject C\nf g : X \u27f6 Y\nh : \u2200 (n : SimplexCategory\u1d52\u1d56), NatTrans.app f n = NatTrans.app g n\n\u22a2 f.app = g.app\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : SimplicialObject C\nf g : X \u27f6 Y\nh : \u2200 (n : SimplexCategory\u1d52\u1d56), NatTrans.app f n = NatTrans.app g n\nx\u271d : SimplexCategory\u1d52\u1d56\n\u22a2 NatTrans.app f x\u271d = NatTrans.app g x\u271d\n[PROOFSTEP]\napply h\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn m : \u2115\nh : n = m\n\u22a2 op [n] = op [m]\n[PROOFSTEP]\ncongr\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\nh : n = n\n\u22a2 eqToIso X h = Iso.refl (X.obj (op [n]))\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\nh : n = n\n\u22a2 (eqToIso X h).hom = (Iso.refl (X.obj (op [n]))).hom\n[PROOFSTEP]\nsimp [eqToIso]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni j : Fin (n + 2)\nH : i \u2264 j\n\u22a2 \u03b4 X (Fin.succ j) \u226b \u03b4 X i = \u03b4 X (Fin.castSucc i) \u226b \u03b4 X j\n[PROOFSTEP]\ndsimp [\u03b4]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni j : Fin (n + 2)\nH : i \u2264 j\n\u22a2 X.map (SimplexCategory.\u03b4 (Fin.succ j)).op \u226b X.map (SimplexCategory.\u03b4 i).op =\n X.map (SimplexCategory.\u03b4 (Fin.castSucc i)).op \u226b X.map (SimplexCategory.\u03b4 j).op\n[PROOFSTEP]\nsimp only [\u2190 X.map_comp, \u2190 op_comp, SimplexCategory.\u03b4_comp_\u03b4 H]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 2)\nj : Fin (n + 3)\nH : Fin.castSucc i < j\nhj : j = 0\n\u22a2 False\n[PROOFSTEP]\nsimp [hj, Fin.not_lt_zero] at H \n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 2)\nj : Fin (n + 3)\nH : Fin.castSucc i < j\n\u22a2 \u03b4 X j \u226b \u03b4 X i = \u03b4 X (Fin.castSucc i) \u226b \u03b4 X (Fin.pred j (_ : j = 0 \u2192 False))\n[PROOFSTEP]\ndsimp [\u03b4]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 2)\nj : Fin (n + 3)\nH : Fin.castSucc i < j\n\u22a2 X.map (SimplexCategory.\u03b4 j).op \u226b X.map (SimplexCategory.\u03b4 i).op =\n X.map (SimplexCategory.\u03b4 (Fin.castSucc i)).op \u226b X.map (SimplexCategory.\u03b4 (Fin.pred j (_ : j = 0 \u2192 False))).op\n[PROOFSTEP]\nsimp only [\u2190 X.map_comp, \u2190 op_comp, SimplexCategory.\u03b4_comp_\u03b4' H]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : i \u2264 Fin.castSucc j\n\u22a2 \u03b4 X (Fin.succ j) \u226b \u03b4 X (Fin.castLT i (_ : \u2191i < n + 2)) = \u03b4 X i \u226b \u03b4 X j\n[PROOFSTEP]\ndsimp [\u03b4]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : i \u2264 Fin.castSucc j\n\u22a2 X.map (SimplexCategory.\u03b4 (Fin.succ j)).op \u226b X.map (SimplexCategory.\u03b4 (Fin.castLT i (_ : \u2191i < n + 2))).op =\n X.map (SimplexCategory.\u03b4 i).op \u226b X.map (SimplexCategory.\u03b4 j).op\n[PROOFSTEP]\nsimp only [\u2190 X.map_comp, \u2190 op_comp, SimplexCategory.\u03b4_comp_\u03b4'' H]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 2)\n\u22a2 \u03b4 X (Fin.castSucc i) \u226b \u03b4 X i = \u03b4 X (Fin.succ i) \u226b \u03b4 X i\n[PROOFSTEP]\ndsimp [\u03b4]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 2)\n\u22a2 X.map (SimplexCategory.\u03b4 (Fin.castSucc i)).op \u226b X.map (SimplexCategory.\u03b4 i).op =\n X.map (SimplexCategory.\u03b4 (Fin.succ i)).op \u226b X.map (SimplexCategory.\u03b4 i).op\n[PROOFSTEP]\nsimp only [\u2190 X.map_comp, \u2190 op_comp, SimplexCategory.\u03b4_comp_\u03b4_self]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\nj : Fin (n + 3)\ni : Fin (n + 2)\nH : j = Fin.castSucc i\n\u22a2 \u03b4 X j \u226b \u03b4 X i = \u03b4 X (Fin.succ i) \u226b \u03b4 X i\n[PROOFSTEP]\nsubst H\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 2)\n\u22a2 \u03b4 X (Fin.castSucc i) \u226b \u03b4 X i = \u03b4 X (Fin.succ i) \u226b \u03b4 X i\n[PROOFSTEP]\nrw [\u03b4_comp_\u03b4_self]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : i \u2264 Fin.castSucc j\n\u22a2 \u03c3 X (Fin.succ j) \u226b \u03b4 X (Fin.castSucc i) = \u03b4 X i \u226b \u03c3 X j\n[PROOFSTEP]\ndsimp [\u03b4, \u03c3]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : i \u2264 Fin.castSucc j\n\u22a2 X.map (SimplexCategory.\u03c3 (Fin.succ j)).op \u226b X.map (SimplexCategory.\u03b4 (Fin.castSucc i)).op =\n X.map (SimplexCategory.\u03b4 i).op \u226b X.map (SimplexCategory.\u03c3 j).op\n[PROOFSTEP]\nsimp only [\u2190 X.map_comp, \u2190 op_comp, SimplexCategory.\u03b4_comp_\u03c3_of_le H]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 1)\n\u22a2 \u03c3 X i \u226b \u03b4 X (Fin.castSucc i) = \ud835\udfd9 (X.obj (op [n]))\n[PROOFSTEP]\ndsimp [\u03b4, \u03c3]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 1)\n\u22a2 X.map (SimplexCategory.\u03c3 i).op \u226b X.map (SimplexCategory.\u03b4 (Fin.castSucc i)).op = \ud835\udfd9 (X.obj (op [n]))\n[PROOFSTEP]\nsimp only [\u2190 X.map_comp, \u2190 op_comp, SimplexCategory.\u03b4_comp_\u03c3_self, op_id, X.map_id]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\nj : Fin (n + 2)\ni : Fin (n + 1)\nH : j = Fin.castSucc i\n\u22a2 \u03c3 X i \u226b \u03b4 X j = \ud835\udfd9 (X.obj (op [n]))\n[PROOFSTEP]\nsubst H\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 1)\n\u22a2 \u03c3 X i \u226b \u03b4 X (Fin.castSucc i) = \ud835\udfd9 (X.obj (op [n]))\n[PROOFSTEP]\nrw [\u03b4_comp_\u03c3_self]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 1)\n\u22a2 \u03c3 X i \u226b \u03b4 X (Fin.succ i) = \ud835\udfd9 (X.obj (op [n]))\n[PROOFSTEP]\ndsimp [\u03b4, \u03c3]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 1)\n\u22a2 X.map (SimplexCategory.\u03c3 i).op \u226b X.map (SimplexCategory.\u03b4 (Fin.succ i)).op = \ud835\udfd9 (X.obj (op [n]))\n[PROOFSTEP]\nsimp only [\u2190 X.map_comp, \u2190 op_comp, SimplexCategory.\u03b4_comp_\u03c3_succ, op_id, X.map_id]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\nj : Fin (n + 2)\ni : Fin (n + 1)\nH : j = Fin.succ i\n\u22a2 \u03c3 X i \u226b \u03b4 X j = \ud835\udfd9 (X.obj (op [n]))\n[PROOFSTEP]\nsubst H\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 1)\n\u22a2 \u03c3 X i \u226b \u03b4 X (Fin.succ i) = \ud835\udfd9 (X.obj (op [n]))\n[PROOFSTEP]\nrw [\u03b4_comp_\u03c3_succ]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : Fin.castSucc j < i\n\u22a2 \u03c3 X (Fin.castSucc j) \u226b \u03b4 X (Fin.succ i) = \u03b4 X i \u226b \u03c3 X j\n[PROOFSTEP]\ndsimp [\u03b4, \u03c3]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : Fin.castSucc j < i\n\u22a2 X.map (SimplexCategory.\u03c3 (Fin.castSucc j)).op \u226b X.map (SimplexCategory.\u03b4 (Fin.succ i)).op =\n X.map (SimplexCategory.\u03b4 i).op \u226b X.map (SimplexCategory.\u03c3 j).op\n[PROOFSTEP]\nsimp only [\u2190 X.map_comp, \u2190 op_comp, SimplexCategory.\u03b4_comp_\u03c3_of_gt H]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : Fin.succ j < i\nhi : i = 0\n\u22a2 False\n[PROOFSTEP]\nsimp only [Fin.not_lt_zero, hi] at H \n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : Fin.succ j < i\n\u22a2 \u03c3 X j \u226b \u03b4 X i = \u03b4 X (Fin.pred i (_ : i = 0 \u2192 False)) \u226b \u03c3 X (Fin.castLT j (_ : \u2191j < n + 1))\n[PROOFSTEP]\ndsimp [\u03b4, \u03c3]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : Fin.succ j < i\n\u22a2 X.map (SimplexCategory.\u03c3 j).op \u226b X.map (SimplexCategory.\u03b4 i).op =\n X.map (SimplexCategory.\u03b4 (Fin.pred i (_ : i = 0 \u2192 False))).op \u226b\n X.map (SimplexCategory.\u03c3 (Fin.castLT j (_ : \u2191j < n + 1))).op\n[PROOFSTEP]\nsimp only [\u2190 X.map_comp, \u2190 op_comp, SimplexCategory.\u03b4_comp_\u03c3_of_gt' H]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni j : Fin (n + 1)\nH : i \u2264 j\n\u22a2 \u03c3 X j \u226b \u03c3 X (Fin.castSucc i) = \u03c3 X i \u226b \u03c3 X (Fin.succ j)\n[PROOFSTEP]\ndsimp [\u03b4, \u03c3]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\ni j : Fin (n + 1)\nH : i \u2264 j\n\u22a2 X.map (SimplexCategory.\u03c3 j).op \u226b X.map (SimplexCategory.\u03c3 (Fin.castSucc i)).op =\n X.map (SimplexCategory.\u03c3 i).op \u226b X.map (SimplexCategory.\u03c3 (Fin.succ j)).op\n[PROOFSTEP]\nsimp only [\u2190 X.map_comp, \u2190 op_comp, SimplexCategory.\u03c3_comp_\u03c3 H]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\n\u22a2 Category.{?u.68325, max u v} (Truncated C n)\n[PROOFSTEP]\ndsimp [Truncated]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\n\u22a2 Category.{?u.68325, max u v} ((SimplexCategory.Truncated n)\u1d52\u1d56 \u2964 C)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : HasLimitsOfShape J C\n\u22a2 HasLimitsOfShape J (Truncated C n)\n[PROOFSTEP]\ndsimp [Truncated]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : HasLimitsOfShape J C\n\u22a2 HasLimitsOfShape J ((SimplexCategory.Truncated n)\u1d52\u1d56 \u2964 C)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : HasColimitsOfShape J C\n\u22a2 HasColimitsOfShape J (Truncated C n)\n[PROOFSTEP]\ndsimp [Truncated]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX : SimplicialObject C\nn : \u2115\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : HasColimitsOfShape J C\n\u22a2 HasColimitsOfShape J ((SimplexCategory.Truncated n)\u1d52\u1d56 \u2964 C)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\n\u22a2 Category.{?u.76503, max u v} (Augmented C)\n[PROOFSTEP]\ndsimp only [Augmented]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\n\u22a2 Category.{?u.76503, max u v} (Comma (\ud835\udfed (SimplicialObject C)) (const C))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nX\u271d Y\u271d : Augmented C\n\u03b7 : X\u271d \u27f6 Y\u271d\n\u22a2 (\ud835\udfed C).map (NatTrans.app (drop.map \u03b7) (op [0])) \u226b\n ((fun X => { left := (drop.obj X).obj (op [0]), right := point.obj X, hom := NatTrans.app X.hom (op [0]) })\n Y\u271d).hom =\n ((fun X => { left := (drop.obj X).obj (op [0]), right := point.obj X, hom := NatTrans.app X.hom (op [0]) })\n X\u271d).hom \u226b\n (\ud835\udfed C).map (point.map \u03b7)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nX\u271d Y\u271d : Augmented C\n\u03b7 : X\u271d \u27f6 Y\u271d\n\u22a2 NatTrans.app \u03b7.left (op [0]) \u226b NatTrans.app Y\u271d.hom (op [0]) = NatTrans.app X\u271d.hom (op [0]) \u226b \u03b7.right\n[PROOFSTEP]\nrw [\u2190 NatTrans.comp_app]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nX\u271d Y\u271d : Augmented C\n\u03b7 : X\u271d \u27f6 Y\u271d\n\u22a2 NatTrans.app (\u03b7.left \u226b Y\u271d.hom) (op [0]) = NatTrans.app X\u271d.hom (op [0]) \u226b \u03b7.right\n[PROOFSTEP]\nerw [\u03b7.w]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : SimplicialObject C\nX\u271d Y\u271d : Augmented C\n\u03b7 : X\u271d \u27f6 Y\u271d\n\u22a2 NatTrans.app (X\u271d.hom \u226b (const C).map \u03b7.right) (op [0]) = NatTrans.app X\u271d.hom (op [0]) \u226b \u03b7.right\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d : SimplicialObject C\nX Y : Augmented C\nf : X \u27f6 Y\n\u22a2 NatTrans.app (drop.map f) (op [0]) \u226b NatTrans.app Y.hom (op [0]) = NatTrans.app X.hom (op [0]) \u226b point.map f\n[PROOFSTEP]\nconvert congr_app f.w (op (SimplexCategory.mk 0))\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : SimplicialObject C\nD : Type u_1\ninst\u271d : Category.{?u.105136, u_1} D\nF : C \u2964 D\nX\u271d Y\u271d : Augmented C\n\u03b7 : X\u271d \u27f6 Y\u271d\n\u22a2 (\ud835\udfed (SimplicialObject D)).map (whiskerRight \u03b7.left F) \u226b\n ((fun X =>\n { left := ((whiskering C D).obj F).obj (drop.obj X), right := F.obj (point.obj X),\n hom := whiskerRight X.hom F \u226b (Functor.constComp SimplexCategory\u1d52\u1d56 X.right F).hom })\n Y\u271d).hom =\n ((fun X =>\n { left := ((whiskering C D).obj F).obj (drop.obj X), right := F.obj (point.obj X),\n hom := whiskerRight X.hom F \u226b (Functor.constComp SimplexCategory\u1d52\u1d56 X.right F).hom })\n X\u271d).hom \u226b\n (const D).map (F.map \u03b7.right)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : SimplicialObject C\nD : Type u_1\ninst\u271d : Category.{?u.105136, u_1} D\nF : C \u2964 D\nX\u271d Y\u271d : Augmented C\n\u03b7 : X\u271d \u27f6 Y\u271d\nn\u271d : SimplexCategory\u1d52\u1d56\n\u22a2 NatTrans.app\n ((\ud835\udfed (SimplicialObject D)).map (whiskerRight \u03b7.left F) \u226b\n ((fun X =>\n { left := ((whiskering C D).obj F).obj (drop.obj X), right := F.obj (point.obj X),\n hom := whiskerRight X.hom F \u226b (Functor.constComp SimplexCategory\u1d52\u1d56 X.right F).hom })\n Y\u271d).hom)\n n\u271d =\n NatTrans.app\n (((fun X =>\n { left := ((whiskering C D).obj F).obj (drop.obj X), right := F.obj (point.obj X),\n hom := whiskerRight X.hom F \u226b (Functor.constComp SimplexCategory\u1d52\u1d56 X.right F).hom })\n X\u271d).hom \u226b\n (const D).map (F.map \u03b7.right))\n n\u271d\n[PROOFSTEP]\ndsimp [whiskerRight]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : SimplicialObject C\nD : Type u_1\ninst\u271d : Category.{?u.105136, u_1} D\nF : C \u2964 D\nX\u271d Y\u271d : Augmented C\n\u03b7 : X\u271d \u27f6 Y\u271d\nn\u271d : SimplexCategory\u1d52\u1d56\n\u22a2 F.map (NatTrans.app \u03b7.left n\u271d) \u226b F.map (NatTrans.app Y\u271d.hom n\u271d) \u226b \ud835\udfd9 (F.obj Y\u271d.right) =\n (F.map (NatTrans.app X\u271d.hom n\u271d) \u226b \ud835\udfd9 (F.obj X\u271d.right)) \u226b F.map \u03b7.right\n[PROOFSTEP]\nsimp only [Category.comp_id, \u2190 F.map_comp, \u2190 NatTrans.comp_app]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : SimplicialObject C\nD : Type u_1\ninst\u271d : Category.{?u.105136, u_1} D\nF : C \u2964 D\nX\u271d Y\u271d : Augmented C\n\u03b7 : X\u271d \u27f6 Y\u271d\nn\u271d : SimplexCategory\u1d52\u1d56\n\u22a2 F.map (NatTrans.app (\u03b7.left \u226b Y\u271d.hom) n\u271d) = F.map (NatTrans.app X\u271d.hom n\u271d \u226b \u03b7.right)\n[PROOFSTEP]\nerw [\u03b7.w]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : SimplicialObject C\nD : Type u_1\ninst\u271d : Category.{?u.105136, u_1} D\nF : C \u2964 D\nX\u271d Y\u271d : Augmented C\n\u03b7 : X\u271d \u27f6 Y\u271d\nn\u271d : SimplexCategory\u1d52\u1d56\n\u22a2 F.map (NatTrans.app (X\u271d.hom \u226b (const C).map \u03b7.right) n\u271d) = F.map (NatTrans.app X\u271d.hom n\u271d \u226b \u03b7.right)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : SimplicialObject C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nX\u271d Y\u271d : C \u2964 D\n\u03b7 : X\u271d \u27f6 Y\u271d\nA : Augmented C\n\u22a2 (\ud835\udfed (SimplicialObject D)).map (whiskerLeft (drop.obj A) \u03b7) \u226b ((whiskeringObj C D Y\u271d).obj A).hom =\n ((whiskeringObj C D X\u271d).obj A).hom \u226b (const D).map (NatTrans.app \u03b7 (point.obj A))\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : SimplicialObject C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nX\u271d Y\u271d : C \u2964 D\n\u03b7 : X\u271d \u27f6 Y\u271d\nA : Augmented C\nn : SimplexCategory\u1d52\u1d56\n\u22a2 NatTrans.app ((\ud835\udfed (SimplicialObject D)).map (whiskerLeft (drop.obj A) \u03b7) \u226b ((whiskeringObj C D Y\u271d).obj A).hom) n =\n NatTrans.app (((whiskeringObj C D X\u271d).obj A).hom \u226b (const D).map (NatTrans.app \u03b7 (point.obj A))) n\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : SimplicialObject C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nX\u271d Y\u271d : C \u2964 D\n\u03b7 : X\u271d \u27f6 Y\u271d\nA : Augmented C\nn : SimplexCategory\u1d52\u1d56\n\u22a2 NatTrans.app \u03b7 (A.left.obj n) \u226b Y\u271d.map (NatTrans.app A.hom n) \u226b \ud835\udfd9 (Y\u271d.obj A.right) =\n (X\u271d.map (NatTrans.app A.hom n) \u226b \ud835\udfd9 (X\u271d.obj A.right)) \u226b NatTrans.app \u03b7 A.right\n[PROOFSTEP]\nrw [Category.comp_id, Category.comp_id, \u03b7.naturality]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : SimplicialObject C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nX\u271d Y\u271d Z\u271d : C \u2964 D\nx\u271d\u00b9 : X\u271d \u27f6 Y\u271d\nx\u271d : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := whiskeringObj C D,\n map := fun {X Y} \u03b7 =>\n NatTrans.mk fun A => CommaMorphism.mk (whiskerLeft (drop.obj A) \u03b7) (NatTrans.app \u03b7 (point.obj A)) }.map\n (x\u271d\u00b9 \u226b x\u271d) =\n { obj := whiskeringObj C D,\n map := fun {X Y} \u03b7 =>\n NatTrans.mk fun A => CommaMorphism.mk (whiskerLeft (drop.obj A) \u03b7) (NatTrans.app \u03b7 (point.obj A)) }.map\n x\u271d\u00b9 \u226b\n { obj := whiskeringObj C D,\n map := fun {X Y} \u03b7 =>\n NatTrans.mk fun A => CommaMorphism.mk (whiskerLeft (drop.obj A) \u03b7) (NatTrans.app \u03b7 (point.obj A)) }.map\n x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w.h.h\u2081.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : SimplicialObject C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nX\u271d Y\u271d Z\u271d : C \u2964 D\nx\u271d\u00b2 : X\u271d \u27f6 Y\u271d\nx\u271d\u00b9 : Y\u271d \u27f6 Z\u271d\nx\u271d : Augmented C\nn\u271d : SimplexCategory\u1d52\u1d56\n\u22a2 NatTrans.app\n (NatTrans.app\n ({ obj := whiskeringObj C D,\n map := fun {X Y} \u03b7 =>\n NatTrans.mk fun A =>\n CommaMorphism.mk (whiskerLeft (drop.obj A) \u03b7) (NatTrans.app \u03b7 (point.obj A)) }.map\n (x\u271d\u00b2 \u226b x\u271d\u00b9))\n x\u271d).left\n n\u271d =\n NatTrans.app\n (NatTrans.app\n ({ obj := whiskeringObj C D,\n map := fun {X Y} \u03b7 =>\n NatTrans.mk fun A =>\n CommaMorphism.mk (whiskerLeft (drop.obj A) \u03b7) (NatTrans.app \u03b7 (point.obj A)) }.map\n x\u271d\u00b2 \u226b\n { obj := whiskeringObj C D,\n map := fun {X Y} \u03b7 =>\n NatTrans.mk fun A =>\n CommaMorphism.mk (whiskerLeft (drop.obj A) \u03b7) (NatTrans.app \u03b7 (point.obj A)) }.map\n x\u271d\u00b9)\n x\u271d).left\n n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase w.h.h\u2082\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : SimplicialObject C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nX\u271d Y\u271d Z\u271d : C \u2964 D\nx\u271d\u00b2 : X\u271d \u27f6 Y\u271d\nx\u271d\u00b9 : Y\u271d \u27f6 Z\u271d\nx\u271d : Augmented C\n\u22a2 (NatTrans.app\n ({ obj := whiskeringObj C D,\n map := fun {X Y} \u03b7 =>\n NatTrans.mk fun A => CommaMorphism.mk (whiskerLeft (drop.obj A) \u03b7) (NatTrans.app \u03b7 (point.obj A)) }.map\n (x\u271d\u00b2 \u226b x\u271d\u00b9))\n x\u271d).right =\n (NatTrans.app\n ({ obj := whiskeringObj C D,\n map := fun {X Y} \u03b7 =>\n NatTrans.mk fun A =>\n CommaMorphism.mk (whiskerLeft (drop.obj A) \u03b7) (NatTrans.app \u03b7 (point.obj A)) }.map\n x\u271d\u00b2 \u226b\n { obj := whiskeringObj C D,\n map := fun {X Y} \u03b7 =>\n NatTrans.mk fun A =>\n CommaMorphism.mk (whiskerLeft (drop.obj A) \u03b7) (NatTrans.app \u03b7 (point.obj A)) }.map\n x\u271d\u00b9)\n x\u271d).right\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d X : SimplicialObject C\nX\u2080 : C\nf : X.obj (op [0]) \u27f6 X\u2080\nw : \u2200 (i : SimplexCategory) (g\u2081 g\u2082 : [0] \u27f6 i), X.map g\u2081.op \u226b f = X.map g\u2082.op \u226b f\n\u22a2 \u2200 \u2983X_1 Y : SimplexCategory\u1d52\u1d56\u2984 (f_1 : X_1 \u27f6 Y),\n ((\ud835\udfed (SimplicialObject C)).obj X).map f_1 \u226b (fun i => X.map (SimplexCategory.const i.unop 0).op \u226b f) Y =\n (fun i => X.map (SimplexCategory.const i.unop 0).op \u226b f) X_1 \u226b ((const C).obj X\u2080).map f_1\n[PROOFSTEP]\nintro i j g\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d X : SimplicialObject C\nX\u2080 : C\nf : X.obj (op [0]) \u27f6 X\u2080\nw : \u2200 (i : SimplexCategory) (g\u2081 g\u2082 : [0] \u27f6 i), X.map g\u2081.op \u226b f = X.map g\u2082.op \u226b f\ni j : SimplexCategory\u1d52\u1d56\ng : i \u27f6 j\n\u22a2 ((\ud835\udfed (SimplicialObject C)).obj X).map g \u226b (fun i => X.map (SimplexCategory.const i.unop 0).op \u226b f) j =\n (fun i => X.map (SimplexCategory.const i.unop 0).op \u226b f) i \u226b ((const C).obj X\u2080).map g\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d X : SimplicialObject C\nX\u2080 : C\nf : X.obj (op [0]) \u27f6 X\u2080\nw : \u2200 (i : SimplexCategory) (g\u2081 g\u2082 : [0] \u27f6 i), X.map g\u2081.op \u226b f = X.map g\u2082.op \u226b f\ni j : SimplexCategory\u1d52\u1d56\ng : i \u27f6 j\n\u22a2 X.map g \u226b X.map (SimplexCategory.const j.unop 0).op \u226b f = (X.map (SimplexCategory.const i.unop 0).op \u226b f) \u226b \ud835\udfd9 X\u2080\n[PROOFSTEP]\nrw [\u2190 g.op_unop]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d X : SimplicialObject C\nX\u2080 : C\nf : X.obj (op [0]) \u27f6 X\u2080\nw : \u2200 (i : SimplexCategory) (g\u2081 g\u2082 : [0] \u27f6 i), X.map g\u2081.op \u226b f = X.map g\u2082.op \u226b f\ni j : SimplexCategory\u1d52\u1d56\ng : i \u27f6 j\n\u22a2 X.map g.unop.op \u226b X.map (SimplexCategory.const j.unop 0).op \u226b f =\n (X.map (SimplexCategory.const i.unop 0).op \u226b f) \u226b \ud835\udfd9 X\u2080\n[PROOFSTEP]\nsimpa only [\u2190 X.map_comp, \u2190 Category.assoc, Category.comp_id, \u2190 op_comp] using w _ _ _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d X : SimplicialObject C\nX\u2080 : C\nf : X.obj (op [0]) \u27f6 X\u2080\nw : \u2200 (i : SimplexCategory) (g\u2081 g\u2082 : [0] \u27f6 i), X.map g\u2081.op \u226b f = X.map g\u2082.op \u226b f\n\u22a2 NatTrans.app (augment X X\u2080 f w).hom (op [0]) = f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d X : SimplicialObject C\nX\u2080 : C\nf : X.obj (op [0]) \u27f6 X\u2080\nw : \u2200 (i : SimplexCategory) (g\u2081 g\u2082 : [0] \u27f6 i), X.map g\u2081.op \u226b f = X.map g\u2082.op \u226b f\n\u22a2 X.map (SimplexCategory.const [0] 0).op \u226b f = f\n[PROOFSTEP]\nrw [SimplexCategory.hom_zero_zero ([0].const 0), op_id, X.map_id, Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\n\u22a2 Category.{?u.208707, max u v} (CosimplicialObject C)\n[PROOFSTEP]\ndsimp only [CosimplicialObject]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\n\u22a2 Category.{?u.208707, max u v} (SimplexCategory \u2964 C)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : HasLimitsOfShape J C\n\u22a2 HasLimitsOfShape J (CosimplicialObject C)\n[PROOFSTEP]\ndsimp [CosimplicialObject]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : HasLimitsOfShape J C\n\u22a2 HasLimitsOfShape J (SimplexCategory \u2964 C)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : HasColimitsOfShape J C\n\u22a2 HasColimitsOfShape J (CosimplicialObject C)\n[PROOFSTEP]\ndsimp [CosimplicialObject]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : HasColimitsOfShape J C\n\u22a2 HasColimitsOfShape J (SimplexCategory \u2964 C)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : CosimplicialObject C\nf g : X \u27f6 Y\nh : \u2200 (n : SimplexCategory), NatTrans.app f n = NatTrans.app g n\n\u22a2 f.app = g.app\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : CosimplicialObject C\nf g : X \u27f6 Y\nh : \u2200 (n : SimplexCategory), NatTrans.app f n = NatTrans.app g n\nx\u271d : SimplexCategory\n\u22a2 NatTrans.app f x\u271d = NatTrans.app g x\u271d\n[PROOFSTEP]\napply h\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn m : \u2115\nh : n = m\n\u22a2 [n] = [m]\n[PROOFSTEP]\nrw [h]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\nh : n = n\n\u22a2 eqToIso X h = Iso.refl (X.obj [n])\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\nh : n = n\n\u22a2 (eqToIso X h).hom = (Iso.refl (X.obj [n])).hom\n[PROOFSTEP]\nsimp [eqToIso]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni j : Fin (n + 2)\nH : i \u2264 j\n\u22a2 \u03b4 X i \u226b \u03b4 X (Fin.succ j) = \u03b4 X j \u226b \u03b4 X (Fin.castSucc i)\n[PROOFSTEP]\ndsimp [\u03b4]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni j : Fin (n + 2)\nH : i \u2264 j\n\u22a2 X.map (SimplexCategory.\u03b4 i) \u226b X.map (SimplexCategory.\u03b4 (Fin.succ j)) =\n X.map (SimplexCategory.\u03b4 j) \u226b X.map (SimplexCategory.\u03b4 (Fin.castSucc i))\n[PROOFSTEP]\nsimp only [\u2190 X.map_comp, SimplexCategory.\u03b4_comp_\u03b4 H]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 2)\nj : Fin (n + 3)\nH : Fin.castSucc i < j\nhj : j = 0\n\u22a2 False\n[PROOFSTEP]\nsimp only [hj, Fin.not_lt_zero] at H \n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 2)\nj : Fin (n + 3)\nH : Fin.castSucc i < j\n\u22a2 \u03b4 X i \u226b \u03b4 X j = \u03b4 X (Fin.pred j (_ : j = 0 \u2192 False)) \u226b \u03b4 X (Fin.castSucc i)\n[PROOFSTEP]\ndsimp [\u03b4]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 2)\nj : Fin (n + 3)\nH : Fin.castSucc i < j\n\u22a2 X.map (SimplexCategory.\u03b4 i) \u226b X.map (SimplexCategory.\u03b4 j) =\n X.map (SimplexCategory.\u03b4 (Fin.pred j (_ : j = 0 \u2192 False))) \u226b X.map (SimplexCategory.\u03b4 (Fin.castSucc i))\n[PROOFSTEP]\nsimp only [\u2190 X.map_comp, \u2190 op_comp, SimplexCategory.\u03b4_comp_\u03b4' H]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : i \u2264 Fin.castSucc j\n\u22a2 \u03b4 X (Fin.castLT i (_ : \u2191i < n + 2)) \u226b \u03b4 X (Fin.succ j) = \u03b4 X j \u226b \u03b4 X i\n[PROOFSTEP]\ndsimp [\u03b4]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : i \u2264 Fin.castSucc j\n\u22a2 X.map (SimplexCategory.\u03b4 (Fin.castLT i (_ : \u2191i < n + 2))) \u226b X.map (SimplexCategory.\u03b4 (Fin.succ j)) =\n X.map (SimplexCategory.\u03b4 j) \u226b X.map (SimplexCategory.\u03b4 i)\n[PROOFSTEP]\nsimp only [\u2190 X.map_comp, \u2190 op_comp, SimplexCategory.\u03b4_comp_\u03b4'' H]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 2)\n\u22a2 \u03b4 X i \u226b \u03b4 X (Fin.castSucc i) = \u03b4 X i \u226b \u03b4 X (Fin.succ i)\n[PROOFSTEP]\ndsimp [\u03b4]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 2)\n\u22a2 X.map (SimplexCategory.\u03b4 i) \u226b X.map (SimplexCategory.\u03b4 (Fin.castSucc i)) =\n X.map (SimplexCategory.\u03b4 i) \u226b X.map (SimplexCategory.\u03b4 (Fin.succ i))\n[PROOFSTEP]\nsimp only [\u2190 X.map_comp, SimplexCategory.\u03b4_comp_\u03b4_self]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 2)\nj : Fin (n + 3)\nH : j = Fin.castSucc i\n\u22a2 \u03b4 X i \u226b \u03b4 X j = \u03b4 X i \u226b \u03b4 X (Fin.succ i)\n[PROOFSTEP]\nsubst H\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 2)\n\u22a2 \u03b4 X i \u226b \u03b4 X (Fin.castSucc i) = \u03b4 X i \u226b \u03b4 X (Fin.succ i)\n[PROOFSTEP]\nrw [\u03b4_comp_\u03b4_self]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : i \u2264 Fin.castSucc j\n\u22a2 \u03b4 X (Fin.castSucc i) \u226b \u03c3 X (Fin.succ j) = \u03c3 X j \u226b \u03b4 X i\n[PROOFSTEP]\ndsimp [\u03b4, \u03c3]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : i \u2264 Fin.castSucc j\n\u22a2 X.map (SimplexCategory.\u03b4 (Fin.castSucc i)) \u226b X.map (SimplexCategory.\u03c3 (Fin.succ j)) =\n X.map (SimplexCategory.\u03c3 j) \u226b X.map (SimplexCategory.\u03b4 i)\n[PROOFSTEP]\nsimp only [\u2190 X.map_comp, SimplexCategory.\u03b4_comp_\u03c3_of_le H]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 1)\n\u22a2 \u03b4 X (Fin.castSucc i) \u226b \u03c3 X i = \ud835\udfd9 (X.obj [n])\n[PROOFSTEP]\ndsimp [\u03b4, \u03c3]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 1)\n\u22a2 X.map (SimplexCategory.\u03b4 (Fin.castSucc i)) \u226b X.map (SimplexCategory.\u03c3 i) = \ud835\udfd9 (X.obj [n])\n[PROOFSTEP]\nsimp only [\u2190 X.map_comp, SimplexCategory.\u03b4_comp_\u03c3_self, X.map_id]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\nj : Fin (n + 2)\ni : Fin (n + 1)\nH : j = Fin.castSucc i\n\u22a2 \u03b4 X j \u226b \u03c3 X i = \ud835\udfd9 (X.obj [n])\n[PROOFSTEP]\nsubst H\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 1)\n\u22a2 \u03b4 X (Fin.castSucc i) \u226b \u03c3 X i = \ud835\udfd9 (X.obj [n])\n[PROOFSTEP]\nrw [\u03b4_comp_\u03c3_self]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 1)\n\u22a2 \u03b4 X (Fin.succ i) \u226b \u03c3 X i = \ud835\udfd9 (X.obj [n])\n[PROOFSTEP]\ndsimp [\u03b4, \u03c3]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 1)\n\u22a2 X.map (SimplexCategory.\u03b4 (Fin.succ i)) \u226b X.map (SimplexCategory.\u03c3 i) = \ud835\udfd9 (X.obj [n])\n[PROOFSTEP]\nsimp only [\u2190 X.map_comp, SimplexCategory.\u03b4_comp_\u03c3_succ, X.map_id]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\nj : Fin (n + 2)\ni : Fin (n + 1)\nH : j = Fin.succ i\n\u22a2 \u03b4 X j \u226b \u03c3 X i = \ud835\udfd9 (X.obj [n])\n[PROOFSTEP]\nsubst H\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 1)\n\u22a2 \u03b4 X (Fin.succ i) \u226b \u03c3 X i = \ud835\udfd9 (X.obj [n])\n[PROOFSTEP]\nrw [\u03b4_comp_\u03c3_succ]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : Fin.castSucc j < i\n\u22a2 \u03b4 X (Fin.succ i) \u226b \u03c3 X (Fin.castSucc j) = \u03c3 X j \u226b \u03b4 X i\n[PROOFSTEP]\ndsimp [\u03b4, \u03c3]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : Fin.castSucc j < i\n\u22a2 X.map (SimplexCategory.\u03b4 (Fin.succ i)) \u226b X.map (SimplexCategory.\u03c3 (Fin.castSucc j)) =\n X.map (SimplexCategory.\u03c3 j) \u226b X.map (SimplexCategory.\u03b4 i)\n[PROOFSTEP]\nsimp only [\u2190 X.map_comp, SimplexCategory.\u03b4_comp_\u03c3_of_gt H]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : Fin.succ j < i\nhi : i = 0\n\u22a2 False\n[PROOFSTEP]\nsimp only [Fin.not_lt_zero, hi] at H \n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : Fin.succ j < i\n\u22a2 \u03b4 X i \u226b \u03c3 X j = \u03c3 X (Fin.castLT j (_ : \u2191j < n + 1)) \u226b \u03b4 X (Fin.pred i (_ : i = 0 \u2192 False))\n[PROOFSTEP]\ndsimp [\u03b4, \u03c3]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : Fin.succ j < i\n\u22a2 X.map (SimplexCategory.\u03b4 i) \u226b X.map (SimplexCategory.\u03c3 j) =\n X.map (SimplexCategory.\u03c3 (Fin.castLT j (_ : \u2191j < n + 1))) \u226b\n X.map (SimplexCategory.\u03b4 (Fin.pred i (_ : i = 0 \u2192 False)))\n[PROOFSTEP]\nsimp only [\u2190 X.map_comp, \u2190 op_comp, SimplexCategory.\u03b4_comp_\u03c3_of_gt' H]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni j : Fin (n + 1)\nH : i \u2264 j\n\u22a2 \u03c3 X (Fin.castSucc i) \u226b \u03c3 X j = \u03c3 X (Fin.succ j) \u226b \u03c3 X i\n[PROOFSTEP]\ndsimp [\u03b4, \u03c3]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni j : Fin (n + 1)\nH : i \u2264 j\n\u22a2 X.map (SimplexCategory.\u03c3 (Fin.castSucc i)) \u226b X.map (SimplexCategory.\u03c3 j) =\n X.map (SimplexCategory.\u03c3 (Fin.succ j)) \u226b X.map (SimplexCategory.\u03c3 i)\n[PROOFSTEP]\nsimp only [\u2190 X.map_comp, SimplexCategory.\u03c3_comp_\u03c3 H]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\n\u22a2 Category.{?u.295628, max u v} (Truncated C n)\n[PROOFSTEP]\ndsimp [Truncated]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\n\u22a2 Category.{?u.295628, max u v} (SimplexCategory.Truncated n \u2964 C)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : HasLimitsOfShape J C\n\u22a2 HasLimitsOfShape J (Truncated C n)\n[PROOFSTEP]\ndsimp [Truncated]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : HasLimitsOfShape J C\n\u22a2 HasLimitsOfShape J (SimplexCategory.Truncated n \u2964 C)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : HasColimitsOfShape J C\n\u22a2 HasColimitsOfShape J (Truncated C n)\n[PROOFSTEP]\ndsimp [Truncated]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : HasColimitsOfShape J C\n\u22a2 HasColimitsOfShape J (SimplexCategory.Truncated n \u2964 C)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\n\u22a2 Category.{?u.303800, max u v} (Augmented C)\n[PROOFSTEP]\ndsimp only [Augmented]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\n\u22a2 Category.{?u.303800, max u v} (Comma (const C) (\ud835\udfed (CosimplicialObject C)))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nX\u271d Y\u271d : Augmented C\n\u03b7 : X\u271d \u27f6 Y\u271d\n\u22a2 (\ud835\udfed C).map (point.map \u03b7) \u226b\n ((fun X => { left := point.obj X, right := (drop.obj X).obj [0], hom := NatTrans.app X.hom [0] }) Y\u271d).hom =\n ((fun X => { left := point.obj X, right := (drop.obj X).obj [0], hom := NatTrans.app X.hom [0] }) X\u271d).hom \u226b\n (\ud835\udfed C).map (NatTrans.app (drop.map \u03b7) [0])\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nX\u271d Y\u271d : Augmented C\n\u03b7 : X\u271d \u27f6 Y\u271d\n\u22a2 \u03b7.left \u226b NatTrans.app Y\u271d.hom [0] = NatTrans.app X\u271d.hom [0] \u226b NatTrans.app \u03b7.right [0]\n[PROOFSTEP]\nrw [\u2190 NatTrans.comp_app]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nX\u271d Y\u271d : Augmented C\n\u03b7 : X\u271d \u27f6 Y\u271d\n\u22a2 \u03b7.left \u226b NatTrans.app Y\u271d.hom [0] = NatTrans.app (X\u271d.hom \u226b \u03b7.right) [0]\n[PROOFSTEP]\nerw [\u2190 \u03b7.w]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nX\u271d Y\u271d : Augmented C\n\u03b7 : X\u271d \u27f6 Y\u271d\n\u22a2 \u03b7.left \u226b NatTrans.app Y\u271d.hom [0] = NatTrans.app ((const C).map \u03b7.left \u226b Y\u271d.hom) [0]\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u_1\ninst\u271d : Category.{?u.334362, u_1} D\nF : C \u2964 D\nX\u271d Y\u271d : Augmented C\n\u03b7 : X\u271d \u27f6 Y\u271d\n\u22a2 (const D).map (F.map \u03b7.left) \u226b\n ((fun X =>\n { left := F.obj (point.obj X), right := ((whiskering C D).obj F).obj (drop.obj X),\n hom := (Functor.constComp SimplexCategory (point.obj X) F).inv \u226b whiskerRight X.hom F })\n Y\u271d).hom =\n ((fun X =>\n { left := F.obj (point.obj X), right := ((whiskering C D).obj F).obj (drop.obj X),\n hom := (Functor.constComp SimplexCategory (point.obj X) F).inv \u226b whiskerRight X.hom F })\n X\u271d).hom \u226b\n (\ud835\udfed (CosimplicialObject D)).map (whiskerRight \u03b7.right F)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u_1\ninst\u271d : Category.{?u.334362, u_1} D\nF : C \u2964 D\nX\u271d Y\u271d : Augmented C\n\u03b7 : X\u271d \u27f6 Y\u271d\nn\u271d : SimplexCategory\n\u22a2 NatTrans.app\n ((const D).map (F.map \u03b7.left) \u226b\n ((fun X =>\n { left := F.obj (point.obj X), right := ((whiskering C D).obj F).obj (drop.obj X),\n hom := (Functor.constComp SimplexCategory (point.obj X) F).inv \u226b whiskerRight X.hom F })\n Y\u271d).hom)\n n\u271d =\n NatTrans.app\n (((fun X =>\n { left := F.obj (point.obj X), right := ((whiskering C D).obj F).obj (drop.obj X),\n hom := (Functor.constComp SimplexCategory (point.obj X) F).inv \u226b whiskerRight X.hom F })\n X\u271d).hom \u226b\n (\ud835\udfed (CosimplicialObject D)).map (whiskerRight \u03b7.right F))\n n\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u_1\ninst\u271d : Category.{?u.334362, u_1} D\nF : C \u2964 D\nX\u271d Y\u271d : Augmented C\n\u03b7 : X\u271d \u27f6 Y\u271d\nn\u271d : SimplexCategory\n\u22a2 F.map \u03b7.left \u226b \ud835\udfd9 (F.obj Y\u271d.left) \u226b F.map (NatTrans.app Y\u271d.hom n\u271d) =\n (\ud835\udfd9 (F.obj X\u271d.left) \u226b F.map (NatTrans.app X\u271d.hom n\u271d)) \u226b F.map (NatTrans.app \u03b7.right n\u271d)\n[PROOFSTEP]\nrw [Category.id_comp, Category.id_comp, \u2190 F.map_comp, \u2190 F.map_comp, \u2190 NatTrans.comp_app]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u_1\ninst\u271d : Category.{?u.334362, u_1} D\nF : C \u2964 D\nX\u271d Y\u271d : Augmented C\n\u03b7 : X\u271d \u27f6 Y\u271d\nn\u271d : SimplexCategory\n\u22a2 F.map (\u03b7.left \u226b NatTrans.app Y\u271d.hom n\u271d) = F.map (NatTrans.app (X\u271d.hom \u226b \u03b7.right) n\u271d)\n[PROOFSTEP]\nerw [\u2190 \u03b7.w]\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u_1\ninst\u271d : Category.{?u.334362, u_1} D\nF : C \u2964 D\nX\u271d Y\u271d : Augmented C\n\u03b7 : X\u271d \u27f6 Y\u271d\nn\u271d : SimplexCategory\n\u22a2 F.map (\u03b7.left \u226b NatTrans.app Y\u271d.hom n\u271d) = F.map (NatTrans.app ((const C).map \u03b7.left \u226b Y\u271d.hom) n\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nX\u271d Y\u271d : C \u2964 D\n\u03b7 : X\u271d \u27f6 Y\u271d\nA : Augmented C\n\u22a2 (const D).map (NatTrans.app \u03b7 (point.obj A)) \u226b ((whiskeringObj C D Y\u271d).obj A).hom =\n ((whiskeringObj C D X\u271d).obj A).hom \u226b (\ud835\udfed (CosimplicialObject D)).map (whiskerLeft (drop.obj A) \u03b7)\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nX\u271d Y\u271d : C \u2964 D\n\u03b7 : X\u271d \u27f6 Y\u271d\nA : Augmented C\nn : SimplexCategory\n\u22a2 NatTrans.app ((const D).map (NatTrans.app \u03b7 (point.obj A)) \u226b ((whiskeringObj C D Y\u271d).obj A).hom) n =\n NatTrans.app (((whiskeringObj C D X\u271d).obj A).hom \u226b (\ud835\udfed (CosimplicialObject D)).map (whiskerLeft (drop.obj A) \u03b7)) n\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nX\u271d Y\u271d : C \u2964 D\n\u03b7 : X\u271d \u27f6 Y\u271d\nA : Augmented C\nn : SimplexCategory\n\u22a2 NatTrans.app \u03b7 A.left \u226b \ud835\udfd9 (Y\u271d.obj A.left) \u226b Y\u271d.map (NatTrans.app A.hom n) =\n (\ud835\udfd9 (X\u271d.obj A.left) \u226b X\u271d.map (NatTrans.app A.hom n)) \u226b NatTrans.app \u03b7 (A.right.obj n)\n[PROOFSTEP]\nrw [Category.id_comp, Category.id_comp, \u03b7.naturality]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nX\u271d Y\u271d : C \u2964 D\n\u03b7 : X\u271d \u27f6 Y\u271d\nx\u271d\u00b9 x\u271d : Augmented C\nf : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 (whiskeringObj C D X\u271d).map f \u226b\n (fun A => CommaMorphism.mk (NatTrans.app \u03b7 (point.obj A)) (whiskerLeft (drop.obj A) \u03b7)) x\u271d =\n (fun A => CommaMorphism.mk (NatTrans.app \u03b7 (point.obj A)) (whiskerLeft (drop.obj A) \u03b7)) x\u271d\u00b9 \u226b\n (whiskeringObj C D Y\u271d).map f\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nX\u271d Y\u271d : C \u2964 D\n\u03b7 : X\u271d \u27f6 Y\u271d\nx\u271d\u00b9 x\u271d : Augmented C\nf : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 ((whiskeringObj C D X\u271d).map f \u226b\n (fun A => CommaMorphism.mk (NatTrans.app \u03b7 (point.obj A)) (whiskerLeft (drop.obj A) \u03b7)) x\u271d).left =\n ((fun A => CommaMorphism.mk (NatTrans.app \u03b7 (point.obj A)) (whiskerLeft (drop.obj A) \u03b7)) x\u271d\u00b9 \u226b\n (whiskeringObj C D Y\u271d).map f).left\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2082.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nX\u271d Y\u271d : C \u2964 D\n\u03b7 : X\u271d \u27f6 Y\u271d\nx\u271d\u00b9 x\u271d : Augmented C\nf : x\u271d\u00b9 \u27f6 x\u271d\nn\u271d : SimplexCategory\n\u22a2 NatTrans.app\n ((whiskeringObj C D X\u271d).map f \u226b\n (fun A => CommaMorphism.mk (NatTrans.app \u03b7 (point.obj A)) (whiskerLeft (drop.obj A) \u03b7)) x\u271d).right\n n\u271d =\n NatTrans.app\n ((fun A => CommaMorphism.mk (NatTrans.app \u03b7 (point.obj A)) (whiskerLeft (drop.obj A) \u03b7)) x\u271d\u00b9 \u226b\n (whiskeringObj C D Y\u271d).map f).right\n n\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nX\u271d Y\u271d : C \u2964 D\n\u03b7 : X\u271d \u27f6 Y\u271d\nx\u271d\u00b9 x\u271d : Augmented C\nf : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 X\u271d.map f.left \u226b NatTrans.app \u03b7 x\u271d.left = NatTrans.app \u03b7 x\u271d\u00b9.left \u226b Y\u271d.map f.left\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX : CosimplicialObject C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nX\u271d Y\u271d : C \u2964 D\n\u03b7 : X\u271d \u27f6 Y\u271d\nx\u271d\u00b9 x\u271d : Augmented C\nf : x\u271d\u00b9 \u27f6 x\u271d\nn\u271d : SimplexCategory\n\u22a2 X\u271d.map (NatTrans.app f.right n\u271d) \u226b NatTrans.app \u03b7 (x\u271d.right.obj n\u271d) =\n NatTrans.app \u03b7 (x\u271d\u00b9.right.obj n\u271d) \u226b Y\u271d.map (NatTrans.app f.right n\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d X : CosimplicialObject C\nX\u2080 : C\nf : X\u2080 \u27f6 X.obj [0]\nw : \u2200 (i : SimplexCategory) (g\u2081 g\u2082 : [0] \u27f6 i), f \u226b X.map g\u2081 = f \u226b X.map g\u2082\n\u22a2 \u2200 \u2983X_1 Y : SimplexCategory\u2984 (f_1 : X_1 \u27f6 Y),\n ((const C).obj X\u2080).map f_1 \u226b (fun i => f \u226b X.map (SimplexCategory.const i 0)) Y =\n (fun i => f \u226b X.map (SimplexCategory.const i 0)) X_1 \u226b ((\ud835\udfed (CosimplicialObject C)).obj X).map f_1\n[PROOFSTEP]\nintro i j g\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d X : CosimplicialObject C\nX\u2080 : C\nf : X\u2080 \u27f6 X.obj [0]\nw : \u2200 (i : SimplexCategory) (g\u2081 g\u2082 : [0] \u27f6 i), f \u226b X.map g\u2081 = f \u226b X.map g\u2082\ni j : SimplexCategory\ng : i \u27f6 j\n\u22a2 ((const C).obj X\u2080).map g \u226b (fun i => f \u226b X.map (SimplexCategory.const i 0)) j =\n (fun i => f \u226b X.map (SimplexCategory.const i 0)) i \u226b ((\ud835\udfed (CosimplicialObject C)).obj X).map g\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d X : CosimplicialObject C\nX\u2080 : C\nf : X\u2080 \u27f6 X.obj [0]\nw : \u2200 (i : SimplexCategory) (g\u2081 g\u2082 : [0] \u27f6 i), f \u226b X.map g\u2081 = f \u226b X.map g\u2082\ni j : SimplexCategory\ng : i \u27f6 j\n\u22a2 \ud835\udfd9 X\u2080 \u226b f \u226b X.map (SimplexCategory.const j 0) = (f \u226b X.map (SimplexCategory.const i 0)) \u226b X.map g\n[PROOFSTEP]\nsimpa [\u2190 X.map_comp] using w _ _ _\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d X : CosimplicialObject C\nX\u2080 : C\nf : X\u2080 \u27f6 X.obj [0]\nw : \u2200 (i : SimplexCategory) (g\u2081 g\u2082 : [0] \u27f6 i), f \u226b X.map g\u2081 = f \u226b X.map g\u2082\n\u22a2 NatTrans.app (augment X X\u2080 f w).hom [0] = f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d X : CosimplicialObject C\nX\u2080 : C\nf : X\u2080 \u27f6 X.obj [0]\nw : \u2200 (i : SimplexCategory) (g\u2081 g\u2082 : [0] \u27f6 i), f \u226b X.map g\u2081 = f \u226b X.map g\u2082\n\u22a2 f \u226b X.map (SimplexCategory.const [0] 0) = f\n[PROOFSTEP]\nrw [SimplexCategory.hom_zero_zero ([0].const 0), X.map_id, Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Augmented C\n\u22a2 (CosimplicialObject.Augmented.leftOp (rightOp X)).right = X.right\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX : Augmented C\u1d52\u1d56\n\u22a2 (SimplicialObject.Augmented.rightOp (leftOp X)).left = X.left\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : (SimplicialObject.Augmented C)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (CosimplicialObject.const C\u1d52\u1d56).map f.unop.right.op \u226b ((fun X => SimplicialObject.Augmented.rightOp X.unop) Y\u271d).hom =\n ((fun X => SimplicialObject.Augmented.rightOp X.unop) X\u271d).hom \u226b\n (\ud835\udfed (CosimplicialObject C\u1d52\u1d56)).map (NatTrans.rightOp f.unop.left)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : (SimplicialObject.Augmented C)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\nx : SimplexCategory\n\u22a2 NatTrans.app\n ((CosimplicialObject.const C\u1d52\u1d56).map f.unop.right.op \u226b\n ((fun X => SimplicialObject.Augmented.rightOp X.unop) Y\u271d).hom)\n x =\n NatTrans.app\n (((fun X => SimplicialObject.Augmented.rightOp X.unop) X\u271d).hom \u226b\n (\ud835\udfed (CosimplicialObject C\u1d52\u1d56)).map (NatTrans.rightOp f.unop.left))\n x\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : (SimplicialObject.Augmented C)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\nx : SimplexCategory\n\u22a2 f.unop.right.op \u226b (NatTrans.app Y\u271d.unop.hom (op x)).op =\n (NatTrans.app X\u271d.unop.hom (op x)).op \u226b (NatTrans.app f.unop.left (op x)).op\n[PROOFSTEP]\nsimp_rw [\u2190 op_comp]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : (SimplicialObject.Augmented C)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\nx : SimplexCategory\n\u22a2 (NatTrans.app Y\u271d.unop.hom (op x) \u226b f.unop.right).op =\n (NatTrans.app f.unop.left (op x) \u226b NatTrans.app X\u271d.unop.hom (op x)).op\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e_f\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : (SimplicialObject.Augmented C)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\nx : SimplexCategory\n\u22a2 NatTrans.app Y\u271d.unop.hom (op x) \u226b f.unop.right = NatTrans.app f.unop.left (op x) \u226b NatTrans.app X\u271d.unop.hom (op x)\n[PROOFSTEP]\nexact (congr_app f.unop.w (op x)).symm\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : CosimplicialObject.Augmented C\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (\ud835\udfed (SimplicialObject C)).map (NatTrans.leftOp f.right) \u226b (CosimplicialObject.Augmented.leftOp X\u271d).hom =\n (CosimplicialObject.Augmented.leftOp Y\u271d).hom \u226b (SimplicialObject.const C).map f.left.unop\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : CosimplicialObject.Augmented C\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\nx : SimplexCategory\u1d52\u1d56\n\u22a2 NatTrans.app ((\ud835\udfed (SimplicialObject C)).map (NatTrans.leftOp f.right) \u226b (CosimplicialObject.Augmented.leftOp X\u271d).hom)\n x =\n NatTrans.app ((CosimplicialObject.Augmented.leftOp Y\u271d).hom \u226b (SimplicialObject.const C).map f.left.unop) x\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : CosimplicialObject.Augmented C\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\nx : SimplexCategory\u1d52\u1d56\n\u22a2 (NatTrans.app f.right x.unop).unop \u226b (NatTrans.app X\u271d.hom x.unop).unop =\n (NatTrans.app Y\u271d.hom x.unop).unop \u226b f.left.unop\n[PROOFSTEP]\nsimp_rw [\u2190 unop_comp]\n[GOAL]\ncase h\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : CosimplicialObject.Augmented C\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\nx : SimplexCategory\u1d52\u1d56\n\u22a2 (NatTrans.app X\u271d.hom x.unop \u226b NatTrans.app f.right x.unop).unop = (f.left \u226b NatTrans.app Y\u271d.hom x.unop).unop\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e_f\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : CosimplicialObject.Augmented C\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\nx : SimplexCategory\u1d52\u1d56\n\u22a2 NatTrans.app X\u271d.hom x.unop \u226b NatTrans.app f.right x.unop = f.left \u226b NatTrans.app Y\u271d.hom x.unop\n[PROOFSTEP]\nexact (congr_app f.w (unop x)).symm\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : (SimplicialObject.Augmented C)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (\ud835\udfed (SimplicialObject.Augmented C)\u1d52\u1d56).map f \u226b\n ((fun X => Iso.op (SimplicialObject.Augmented.rightOpLeftOpIso X.unop)) Y\u271d).hom =\n ((fun X => Iso.op (SimplicialObject.Augmented.rightOpLeftOpIso X.unop)) X\u271d).hom \u226b\n (simplicialToCosimplicialAugmented C \u22d9 cosimplicialToSimplicialAugmented C).map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : (SimplicialObject.Augmented C)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\n\u22a2 f \u226b (SimplicialObject.Augmented.rightOpLeftOpIso Y\u271d.unop).hom.op =\n (SimplicialObject.Augmented.rightOpLeftOpIso X\u271d.unop).hom.op \u226b\n (CommaMorphism.mk (NatTrans.leftOp (NatTrans.rightOp f.unop.left)) f.unop.right).op\n[PROOFSTEP]\nrw [\u2190 f.op_unop]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : (SimplicialObject.Augmented C)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\n\u22a2 f.unop.op \u226b (SimplicialObject.Augmented.rightOpLeftOpIso Y\u271d.unop).hom.op =\n (SimplicialObject.Augmented.rightOpLeftOpIso X\u271d.unop).hom.op \u226b\n (CommaMorphism.mk (NatTrans.leftOp (NatTrans.rightOp f.unop.op.unop.left)) f.unop.op.unop.right).op\n[PROOFSTEP]\nsimp_rw [\u2190 op_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : (SimplicialObject.Augmented C)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\n\u22a2 ((SimplicialObject.Augmented.rightOpLeftOpIso Y\u271d.unop).hom \u226b f.unop).op =\n (CommaMorphism.mk (NatTrans.leftOp (NatTrans.rightOp f.unop.op.unop.left)) f.unop.op.unop.right \u226b\n (SimplicialObject.Augmented.rightOpLeftOpIso X\u271d.unop).hom).op\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_f\nC : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : (SimplicialObject.Augmented C)\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (SimplicialObject.Augmented.rightOpLeftOpIso Y\u271d.unop).hom \u226b f.unop =\n CommaMorphism.mk (NatTrans.leftOp (NatTrans.rightOp f.unop.op.unop.left)) f.unop.op.unop.right \u226b\n (SimplicialObject.Augmented.rightOpLeftOpIso X\u271d.unop).hom\n[PROOFSTEP]\naesop_cat\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicTopology.SimplicialObject", "llama_tokens": 23094, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.44939263446475963, "lm_q2_score": 0.024423087568798994, "lm_q1q2_score": 0.010975555664306101}} {"text": "[GOAL]\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\nx : Option \u03b1\n\u22a2 Option.traverse pure x = x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase none\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\n\u22a2 Option.traverse pure none = none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\nval\u271d : \u03b1\n\u22a2 Option.traverse pure (some val\u271d) = some val\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\n\u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\nx : Option \u03b1\n\u22a2 Option.traverse (Comp.mk \u2218 (fun x x_1 => x <$> x_1) f \u2218 g) x = Comp.mk (Option.traverse f <$> Option.traverse g x)\n[PROOFSTEP]\ncases x\n[GOAL]\ncase none\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\n\u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\n\u22a2 Option.traverse (Comp.mk \u2218 (fun x x_1 => x <$> x_1) f \u2218 g) none =\n Comp.mk (Option.traverse f <$> Option.traverse g none)\n[PROOFSTEP]\nsimp! [functor_norm]\n[GOAL]\ncase some\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\n\u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\nval\u271d : \u03b1\n\u22a2 Option.traverse (Comp.mk \u2218 (fun x x_1 => x <$> x_1) f \u2218 g) (some val\u271d) =\n Comp.mk (Option.traverse f <$> Option.traverse g (some val\u271d))\n[PROOFSTEP]\nsimp! [functor_norm]\n[GOAL]\ncase none\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\n\u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\n\u22a2 pure none = Comp.mk (pure (pure none))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\n\u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\nval\u271d : \u03b1\n\u22a2 Comp.mk (((fun x => some <$> x) \u2218 f) <$> g val\u271d) = Comp.mk ((Option.traverse f \u2218 some) <$> g val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b2\nx : Option \u03b1\n\u22a2 Option.traverse (pure \u2218 f) x = pure (f <$> x)\n[PROOFSTEP]\ncases x\n[GOAL]\ncase none\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b2\n\u22a2 Option.traverse (pure \u2218 f) none = pure (f <$> none)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b2\nval\u271d : \u03b1\n\u22a2 Option.traverse (pure \u2218 f) (some val\u271d) = pure (f <$> some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 : Type u_1\n\u03b2 : Type u\nf : \u03b1 \u2192 F \u03b2\nx : Option \u03b1\n\u22a2 (fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) (Option.traverse f x) =\n Option.traverse ((fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) \u2218 f) x\n[PROOFSTEP]\ncases' x with x\n[GOAL]\ncase none\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 : Type u_1\n\u03b2 : Type u\nf : \u03b1 \u2192 F \u03b2\n\u22a2 (fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) (Option.traverse f none) =\n Option.traverse ((fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) \u2218 f) none\n[PROOFSTEP]\nsimp! [*, functor_norm, ApplicativeTransformation.preserves_map, ApplicativeTransformation.preserves_seq,\n ApplicativeTransformation.preserves_pure]\n[GOAL]\ncase some\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 : Type u_1\n\u03b2 : Type u\nf : \u03b1 \u2192 F \u03b2\nx : \u03b1\n\u22a2 (fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) (Option.traverse f (some x)) =\n Option.traverse ((fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) \u2218 f) (some x)\n[PROOFSTEP]\nsimp! [*, functor_norm, ApplicativeTransformation.preserves_map, ApplicativeTransformation.preserves_seq,\n ApplicativeTransformation.preserves_pure]\n[GOAL]\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\nxs : List \u03b1\n\u22a2 List.traverse pure xs = xs\n[PROOFSTEP]\ninduction xs\n[GOAL]\ncase nil\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\n\u22a2 List.traverse pure [] = []\n[PROOFSTEP]\nsimp! [*, List.traverse, functor_norm]\n[GOAL]\ncase cons\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : List.traverse pure tail\u271d = tail\u271d\n\u22a2 List.traverse pure (head\u271d :: tail\u271d) = head\u271d :: tail\u271d\n[PROOFSTEP]\nsimp! [*, List.traverse, functor_norm]\n[GOAL]\ncase cons\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : List.traverse pure tail\u271d = tail\u271d\n\u22a2 (Seq.seq (cons head\u271d) fun x => tail\u271d) = head\u271d :: tail\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\n\u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\nx : List \u03b1\n\u22a2 List.traverse (Comp.mk \u2218 (fun x x_1 => x <$> x_1) f \u2218 g) x = Comp.mk (List.traverse f <$> List.traverse g x)\n[PROOFSTEP]\ninduction x\n[GOAL]\ncase nil\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\n\u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\n\u22a2 List.traverse (Comp.mk \u2218 (fun x x_1 => x <$> x_1) f \u2218 g) [] = Comp.mk (List.traverse f <$> List.traverse g [])\n[PROOFSTEP]\nsimp! [*, functor_norm]\n[GOAL]\ncase cons\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\n\u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d :\n List.traverse (Comp.mk \u2218 (fun x x_1 => x <$> x_1) f \u2218 g) tail\u271d = Comp.mk (List.traverse f <$> List.traverse g tail\u271d)\n\u22a2 List.traverse (Comp.mk \u2218 (fun x x_1 => x <$> x_1) f \u2218 g) (head\u271d :: tail\u271d) =\n Comp.mk (List.traverse f <$> List.traverse g (head\u271d :: tail\u271d))\n[PROOFSTEP]\nsimp! [*, functor_norm]\n[GOAL]\ncase nil\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\n\u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\n\u22a2 pure [] = Comp.mk (pure (pure []))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\n\u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d :\n List.traverse (Comp.mk \u2218 (fun x x_1 => x <$> x_1) f \u2218 g) tail\u271d = Comp.mk (List.traverse f <$> List.traverse g tail\u271d)\n\u22a2 Comp.mk\n (Seq.seq\n (((fun x => x \u2218 List.traverse f) \u2218 (fun x x_1 => Seq.seq x fun x => x_1) \u2218 (fun x => cons <$> x) \u2218 f) <$>\n g head\u271d)\n fun x => List.traverse g tail\u271d) =\n Comp.mk (Seq.seq (((fun x => List.traverse f \u2218 x) \u2218 cons) <$> g head\u271d) fun x => List.traverse g tail\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b2\nx : List \u03b1\n\u22a2 List.traverse (pure \u2218 f) x = pure (f <$> x)\n[PROOFSTEP]\ninduction x\n[GOAL]\ncase nil\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b2\n\u22a2 List.traverse (pure \u2218 f) [] = pure (f <$> [])\n[PROOFSTEP]\nsimp! [*, functor_norm]\n[GOAL]\ncase cons\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b2\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : List.traverse (pure \u2218 f) tail\u271d = pure (f <$> tail\u271d)\n\u22a2 List.traverse (pure \u2218 f) (head\u271d :: tail\u271d) = pure (f <$> (head\u271d :: tail\u271d))\n[PROOFSTEP]\nsimp! [*, functor_norm]\n[GOAL]\ncase cons\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 : Type u_1\nf : \u03b1 \u2192 \u03b2\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : List.traverse (pure \u2218 f) tail\u271d = pure (f <$> tail\u271d)\n\u22a2 (Seq.seq (cons (f head\u271d)) fun x => map f tail\u271d) = f head\u271d :: map f tail\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 : Type u_1\n\u03b2 : Type u\nf : \u03b1 \u2192 F \u03b2\nx : List \u03b1\n\u22a2 (fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) (List.traverse f x) =\n List.traverse ((fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) \u2218 f) x\n[PROOFSTEP]\ninduction x\n[GOAL]\ncase nil\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 : Type u_1\n\u03b2 : Type u\nf : \u03b1 \u2192 F \u03b2\n\u22a2 (fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) (List.traverse f []) =\n List.traverse ((fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) \u2218 f) []\n[PROOFSTEP]\nsimp! [*, functor_norm, ApplicativeTransformation.preserves_map, ApplicativeTransformation.preserves_seq,\n ApplicativeTransformation.preserves_pure]\n[GOAL]\ncase cons\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 : Type u_1\n\u03b2 : Type u\nf : \u03b1 \u2192 F \u03b2\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d :\n (fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) (List.traverse f tail\u271d) =\n List.traverse ((fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) \u2218 f) tail\u271d\n\u22a2 (fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) (List.traverse f (head\u271d :: tail\u271d)) =\n List.traverse ((fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) \u2218 f) (head\u271d :: tail\u271d)\n[PROOFSTEP]\nsimp! [*, functor_norm, ApplicativeTransformation.preserves_map, ApplicativeTransformation.preserves_seq,\n ApplicativeTransformation.preserves_pure]\n[GOAL]\nF G : Type u \u2192 Type u\ninst\u271d\u00b2 : Applicative F\ninst\u271d\u00b9 : Applicative G\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\ninst\u271d : LawfulApplicative F\nbs : List \u03b1'\n\u22a2 traverse f ([] ++ bs) = Seq.seq ((fun x x_1 => x ++ x_1) <$> traverse f []) fun x => traverse f bs\n[PROOFSTEP]\nsimp [functor_norm]\n[GOAL]\nF G : Type u \u2192 Type u\ninst\u271d\u00b2 : Applicative F\ninst\u271d\u00b9 : Applicative G\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\ninst\u271d : LawfulApplicative F\na : \u03b1'\nas bs : List \u03b1'\n\u22a2 traverse f (a :: as ++ bs) = Seq.seq ((fun x x_1 => x ++ x_1) <$> traverse f (a :: as)) fun x => traverse f bs\n[PROOFSTEP]\nsimp [traverse_append as bs, functor_norm]\n[GOAL]\nF G : Type u \u2192 Type u\ninst\u271d\u00b2 : Applicative F\ninst\u271d\u00b9 : Applicative G\n\u03b1' \u03b2' : Type u\nf : \u03b1' \u2192 F \u03b2'\ninst\u271d : LawfulApplicative F\na : \u03b1'\nas bs : List \u03b1'\n\u22a2 (Seq.seq\n (Seq.seq (((fun x => x \u2218 fun x x_1 => x ++ x_1) \u2218 Function.comp \u2218 fun x x_1 => x :: x_1) <$> f a) fun x =>\n traverse f as)\n fun x => traverse f bs) =\n Seq.seq (Seq.seq (((fun x => (fun x x_1 => x ++ x_1) \u2218 x) \u2218 fun x x_1 => x :: x_1) <$> f a) fun x => traverse f as)\n fun x => traverse f bs\n[PROOFSTEP]\ncongr\n[GOAL]\nF G : Type u \u2192 Type u\ninst\u271d\u00b2 : Applicative F\ninst\u271d\u00b9 : Applicative G\n\u03b1' \u03b2' : Type u\nf\u271d : \u03b1' \u2192 F \u03b2'\ninst\u271d : LawfulApplicative F\nf : \u03b1' \u2192 Set \u03b2'\n\u22a2 [] \u2208 traverse f [] \u2194 Forall\u2082 (fun b a => b \u2208 f a) [] []\n[PROOFSTEP]\nsimp\n[GOAL]\nF G : Type u \u2192 Type u\ninst\u271d\u00b2 : Applicative F\ninst\u271d\u00b9 : Applicative G\n\u03b1' \u03b2' : Type u\nf\u271d : \u03b1' \u2192 F \u03b2'\ninst\u271d : LawfulApplicative F\nf : \u03b1' \u2192 Set \u03b2'\na : \u03b1'\nas : List \u03b1'\n\u22a2 [] \u2208 traverse f (a :: as) \u2194 Forall\u2082 (fun b a => b \u2208 f a) [] (a :: as)\n[PROOFSTEP]\nsimp\n[GOAL]\nF G : Type u \u2192 Type u\ninst\u271d\u00b2 : Applicative F\ninst\u271d\u00b9 : Applicative G\n\u03b1' \u03b2' : Type u\nf\u271d : \u03b1' \u2192 F \u03b2'\ninst\u271d : LawfulApplicative F\nf : \u03b1' \u2192 Set \u03b2'\nb : \u03b2'\nbs : List \u03b2'\n\u22a2 b :: bs \u2208 traverse f [] \u2194 Forall\u2082 (fun b a => b \u2208 f a) (b :: bs) []\n[PROOFSTEP]\nsimp\n[GOAL]\nF G : Type u \u2192 Type u\ninst\u271d\u00b2 : Applicative F\ninst\u271d\u00b9 : Applicative G\n\u03b1' \u03b2' : Type u\nf\u271d : \u03b1' \u2192 F \u03b2'\ninst\u271d : LawfulApplicative F\nf : \u03b1' \u2192 Set \u03b2'\na : \u03b1'\nas : List \u03b1'\nb : \u03b2'\nbs : List \u03b2'\n\u22a2 b :: bs \u2208 traverse f (a :: as) \u2194 Forall\u2082 (fun b a => b \u2208 f a) (b :: bs) (a :: as)\n[PROOFSTEP]\nsimp [mem_traverse as bs]\n[GOAL]\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : Applicative G\n\u03b1 \u03b2 \u03b3 : Type u\ng : \u03b1 \u2192 \u03b2\nf : \u03b2 \u2192 G \u03b3\nx : \u03c3 \u2295 \u03b1\n\u22a2 Sum.traverse f (g <$> x) = Sum.traverse (f \u2218 g) x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : Applicative G\n\u03b1 \u03b2 \u03b3 : Type u\ng : \u03b1 \u2192 \u03b2\nf : \u03b2 \u2192 G \u03b3\nval\u271d : \u03c3\n\u22a2 Sum.traverse f (g <$> inl val\u271d) = Sum.traverse (f \u2218 g) (inl val\u271d)\n[PROOFSTEP]\nsimp [Sum.traverse, id_map, functor_norm]\n[GOAL]\ncase inr\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : Applicative G\n\u03b1 \u03b2 \u03b3 : Type u\ng : \u03b1 \u2192 \u03b2\nf : \u03b2 \u2192 G \u03b3\nval\u271d : \u03b1\n\u22a2 Sum.traverse f (g <$> inr val\u271d) = Sum.traverse (f \u2218 g) (inr val\u271d)\n[PROOFSTEP]\nsimp [Sum.traverse, id_map, functor_norm]\n[GOAL]\ncase inl\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : Applicative G\n\u03b1 \u03b2 \u03b3 : Type u\ng : \u03b1 \u2192 \u03b2\nf : \u03b2 \u2192 G \u03b3\nval\u271d : \u03c3\n\u22a2 (match g <$> inl val\u271d with\n | inl x => pure (inl x)\n | inr x => inr <$> f x) =\n pure (inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b9 : Applicative F\ninst\u271d : Applicative G\n\u03b1 \u03b2 \u03b3 : Type u\ng : \u03b1 \u2192 \u03b2\nf : \u03b2 \u2192 G \u03b3\nval\u271d : \u03b1\n\u22a2 (match g <$> inr val\u271d with\n | inl x => pure (inl x)\n | inr x => inr <$> f x) =\n inr <$> f (g val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03c3\u271d : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03c3 \u03b1 : Type u_1\nx : \u03c3 \u2295 \u03b1\n\u22a2 Sum.traverse pure x = x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03c3\u271d : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03c3 \u03b1 : Type u_1\nval\u271d : \u03c3\n\u22a2 Sum.traverse pure (inl val\u271d) = inl val\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03c3\u271d : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03c3 \u03b1 : Type u_1\nval\u271d : \u03b1\n\u22a2 Sum.traverse pure (inr val\u271d) = inr val\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\nx : \u03c3 \u2295 \u03b1\n\u22a2 Sum.traverse (Comp.mk \u2218 (fun x x_1 => x <$> x_1) f \u2218 g) x = Comp.mk (Sum.traverse f <$> Sum.traverse g x)\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\nval\u271d : \u03c3\n\u22a2 Sum.traverse (Comp.mk \u2218 (fun x x_1 => x <$> x_1) f \u2218 g) (inl val\u271d) =\n Comp.mk (Sum.traverse f <$> Sum.traverse g (inl val\u271d))\n[PROOFSTEP]\nsimp! [Sum.traverse, map_id, functor_norm]\n[GOAL]\ncase inr\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\nval\u271d : \u03b1\n\u22a2 Sum.traverse (Comp.mk \u2218 (fun x x_1 => x <$> x_1) f \u2218 g) (inr val\u271d) =\n Comp.mk (Sum.traverse f <$> Sum.traverse g (inr val\u271d))\n[PROOFSTEP]\nsimp! [Sum.traverse, map_id, functor_norm]\n[GOAL]\ncase inl\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\nval\u271d : \u03c3\n\u22a2 pure (inl val\u271d) = Comp.mk (pure (pure (inl val\u271d)))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b2 \u2192 F \u03b3\ng : \u03b1 \u2192 G \u03b2\nval\u271d : \u03b1\n\u22a2 Comp.mk (((fun x => inr <$> x) \u2218 f) <$> g val\u271d) = Comp.mk ((Sum.traverse f \u2218 inr) <$> g val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nx : \u03c3 \u2295 \u03b1\n\u22a2 Sum.traverse (pure \u2218 f) x = pure (f <$> x)\n[PROOFSTEP]\ninduction x\n[GOAL]\ncase inl\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nval\u271d : \u03c3\n\u22a2 Sum.traverse (pure \u2218 f) (inl val\u271d) = pure (f <$> inl val\u271d)\n[PROOFSTEP]\nsimp! [*, functor_norm]\n[GOAL]\ncase inr\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nval\u271d : \u03b1\n\u22a2 Sum.traverse (pure \u2218 f) (inr val\u271d) = pure (f <$> inr val\u271d)\n[PROOFSTEP]\nsimp! [*, functor_norm]\n[GOAL]\ncase inl\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nval\u271d : \u03c3\n\u22a2 inl val\u271d = f <$> inl val\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 : Type u\nf : \u03b1 \u2192 \u03b2\nval\u271d : \u03b1\n\u22a2 inr (f val\u271d) = f <$> inr val\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\n\u03b2 \u03b3 : Type u\ng : \u03b1 \u2192 G \u03b2\nf : \u03b2 \u2192 \u03b3\nx : \u03c3 \u2295 \u03b1\n\u22a2 (fun x x_1 => x <$> x_1) f <$> Sum.traverse g x = Sum.traverse ((fun x x_1 => x <$> x_1) f \u2218 g) x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\n\u03b2 \u03b3 : Type u\ng : \u03b1 \u2192 G \u03b2\nf : \u03b2 \u2192 \u03b3\nval\u271d : \u03c3\n\u22a2 (fun x x_1 => x <$> x_1) f <$> Sum.traverse g (inl val\u271d) = Sum.traverse ((fun x x_1 => x <$> x_1) f \u2218 g) (inl val\u271d)\n[PROOFSTEP]\nsimp [Sum.traverse, id_map, functor_norm]\n[GOAL]\ncase inr\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\n\u03b2 \u03b3 : Type u\ng : \u03b1 \u2192 G \u03b2\nf : \u03b2 \u2192 \u03b3\nval\u271d : \u03b1\n\u22a2 (fun x x_1 => x <$> x_1) f <$> Sum.traverse g (inr val\u271d) = Sum.traverse ((fun x x_1 => x <$> x_1) f \u2218 g) (inr val\u271d)\n[PROOFSTEP]\nsimp [Sum.traverse, id_map, functor_norm]\n[GOAL]\ncase inl\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\n\u03b2 \u03b3 : Type u\ng : \u03b1 \u2192 G \u03b2\nf : \u03b2 \u2192 \u03b3\nval\u271d : \u03c3\n\u22a2 pure (f <$> inl val\u271d) = pure (inl val\u271d)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase inr\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 : Type u_1\n\u03b2 \u03b3 : Type u\ng : \u03b1 \u2192 G \u03b2\nf : \u03b2 \u2192 \u03b3\nval\u271d : \u03b1\n\u22a2 ((fun x => f <$> x) \u2218 inr) <$> g val\u271d = (inr \u2218 f) <$> g val\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 : Type u_1\n\u03b2 : Type u\nf : \u03b1 \u2192 F \u03b2\nx : \u03c3 \u2295 \u03b1\n\u22a2 (fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) (Sum.traverse f x) =\n Sum.traverse ((fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) \u2218 f) x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase inl\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 : Type u_1\n\u03b2 : Type u\nf : \u03b1 \u2192 F \u03b2\nval\u271d : \u03c3\n\u22a2 (fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) (Sum.traverse f (inl val\u271d)) =\n Sum.traverse ((fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) \u2218 f) (inl val\u271d)\n[PROOFSTEP]\nsimp! [Sum.traverse, functor_norm, ApplicativeTransformation.preserves_map, ApplicativeTransformation.preserves_seq,\n ApplicativeTransformation.preserves_pure]\n[GOAL]\ncase inr\n\u03c3 : Type u\nF G : Type u \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b7 : ApplicativeTransformation F G\n\u03b1 : Type u_1\n\u03b2 : Type u\nf : \u03b1 \u2192 F \u03b2\nval\u271d : \u03b1\n\u22a2 (fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) (Sum.traverse f (inr val\u271d)) =\n Sum.traverse ((fun {\u03b1} => ApplicativeTransformation.app \u03b7 \u03b1) \u2218 f) (inr val\u271d)\n[PROOFSTEP]\nsimp! [Sum.traverse, functor_norm, ApplicativeTransformation.preserves_map, ApplicativeTransformation.preserves_seq,\n ApplicativeTransformation.preserves_pure]\n", "meta": {"mathlib_filename": "Mathlib.Control.Traversable.Instances", "llama_tokens": 9933, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.48438008427698437, "lm_q2_score": 0.022629201804480158, "lm_q1q2_score": 0.010961134677174985}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\n\u22a2 (X Y : Discrete PEmpty) \u2192 Zero (X \u27f6 Y)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nX Y : Discrete PUnit\n\u22a2 Zero (X \u27f6 Y)\n[PROOFSTEP]\nrepeat (constructor)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nX Y : Discrete PUnit\n\u22a2 Zero (X \u27f6 Y)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase zero\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nX Y : Discrete PUnit\n\u22a2 X \u27f6 Y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase zero.down\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nX Y : Discrete PUnit\n\u22a2 PLift (X.as = Y.as)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase zero.down.down\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nX Y : Discrete PUnit\n\u22a2 X.as = Y.as\n[PROOFSTEP]\nconstructor\n[GOAL]\n\n[PROOFSTEP]\nconstructor\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nI J : HasZeroMorphisms C\nw : \u2200 (X Y : C), Zero.zero = Zero.zero\n\u22a2 I = J\n[PROOFSTEP]\nhave : I.Zero = J.Zero := by\n funext X Y\n specialize w X Y\n apply congrArg Zero.mk w\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nI J : HasZeroMorphisms C\nw : \u2200 (X Y : C), Zero.zero = Zero.zero\n\u22a2 Zero = Zero\n[PROOFSTEP]\nfunext X Y\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nI J : HasZeroMorphisms C\nw : \u2200 (X Y : C), Zero.zero = Zero.zero\nX Y : C\n\u22a2 Zero X Y = Zero X Y\n[PROOFSTEP]\nspecialize w X Y\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nI J : HasZeroMorphisms C\nX Y : C\nw : Zero.zero = Zero.zero\n\u22a2 Zero X Y = Zero X Y\n[PROOFSTEP]\napply congrArg Zero.mk w\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nI J : HasZeroMorphisms C\nw : \u2200 (X Y : C), Zero.zero = Zero.zero\nthis : Zero = Zero\n\u22a2 I = J\n[PROOFSTEP]\ncases I\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nJ : HasZeroMorphisms C\nZero\u271d : (X Y : C) \u2192 _root_.Zero (X \u27f6 Y)\ncomp_zero\u271d : \u2200 {X Y : C} (f : X \u27f6 Y) (Z : C), f \u226b 0 = 0\nzero_comp\u271d : \u2200 (X : C) {Y Z : C} (f : Y \u27f6 Z), 0 \u226b f = 0\nw : \u2200 (X Y : C), Zero.zero = Zero.zero\nthis : Zero = Zero\n\u22a2 mk = J\n[PROOFSTEP]\ncases J\n[GOAL]\ncase mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nZero\u271d\u00b9 : (X Y : C) \u2192 _root_.Zero (X \u27f6 Y)\ncomp_zero\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y) (Z : C), f \u226b 0 = 0\nzero_comp\u271d\u00b9 : \u2200 (X : C) {Y Z : C} (f : Y \u27f6 Z), 0 \u226b f = 0\nZero\u271d : (X Y : C) \u2192 _root_.Zero (X \u27f6 Y)\ncomp_zero\u271d : \u2200 {X Y : C} (f : X \u27f6 Y) (Z : C), f \u226b 0 = 0\nzero_comp\u271d : \u2200 (X : C) {Y Z : C} (f : Y \u27f6 Z), 0 \u226b f = 0\nw : \u2200 (X Y : C), Zero.zero = Zero.zero\nthis : Zero = Zero\n\u22a2 mk = mk\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.h.e_4\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nZero\u271d\u00b9 : (X Y : C) \u2192 _root_.Zero (X \u27f6 Y)\ncomp_zero\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y) (Z : C), f \u226b 0 = 0\nzero_comp\u271d\u00b9 : \u2200 (X : C) {Y Z : C} (f : Y \u27f6 Z), 0 \u226b f = 0\nZero\u271d : (X Y : C) \u2192 _root_.Zero (X \u27f6 Y)\ncomp_zero\u271d : \u2200 {X Y : C} (f : X \u27f6 Y) (Z : C), f \u226b 0 = 0\nzero_comp\u271d : \u2200 (X : C) {Y Z : C} (f : Y \u27f6 Z), 0 \u226b f = 0\nw : \u2200 (X Y : C), Zero.zero = Zero.zero\nthis : Zero = Zero\n\u22a2 HEq comp_zero\u271d\u00b9 comp_zero\u271d\n[PROOFSTEP]\napply proof_irrel_heq\n[GOAL]\ncase mk.mk.h.e_5\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nZero\u271d\u00b9 : (X Y : C) \u2192 _root_.Zero (X \u27f6 Y)\ncomp_zero\u271d\u00b9 : \u2200 {X Y : C} (f : X \u27f6 Y) (Z : C), f \u226b 0 = 0\nzero_comp\u271d\u00b9 : \u2200 (X : C) {Y Z : C} (f : Y \u27f6 Z), 0 \u226b f = 0\nZero\u271d : (X Y : C) \u2192 _root_.Zero (X \u27f6 Y)\ncomp_zero\u271d : \u2200 {X Y : C} (f : X \u27f6 Y) (Z : C), f \u226b 0 = 0\nzero_comp\u271d : \u2200 (X : C) {Y Z : C} (f : Y \u27f6 Z), 0 \u226b f = 0\nw : \u2200 (X Y : C), Zero.zero = Zero.zero\nthis : Zero = Zero\n\u22a2 HEq zero_comp\u271d\u00b9 zero_comp\u271d\n[PROOFSTEP]\napply proof_irrel_heq\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nI J : HasZeroMorphisms C\n\u22a2 I = J\n[PROOFSTEP]\napply ext_aux\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nI J : HasZeroMorphisms C\n\u22a2 \u2200 (X Y : C), Zero.zero = Zero.zero\n[PROOFSTEP]\nintro X Y\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nI J : HasZeroMorphisms C\nX Y : C\n\u22a2 Zero.zero = Zero.zero\n[PROOFSTEP]\nhave : (I.Zero X Y).zero \u226b (J.Zero Y Y).zero = (I.Zero X Y).zero := by apply I.zero_comp X (J.Zero Y Y).zero\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nI J : HasZeroMorphisms C\nX Y : C\n\u22a2 Zero.zero \u226b Zero.zero = Zero.zero\n[PROOFSTEP]\napply I.zero_comp X (J.Zero Y Y).zero\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nI J : HasZeroMorphisms C\nX Y : C\nthis : Zero.zero \u226b Zero.zero = Zero.zero\n\u22a2 Zero.zero = Zero.zero\n[PROOFSTEP]\nhave that : (I.Zero X Y).zero \u226b (J.Zero Y Y).zero = (J.Zero X Y).zero := by apply J.comp_zero (I.Zero X Y).zero Y\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nI J : HasZeroMorphisms C\nX Y : C\nthis : Zero.zero \u226b Zero.zero = Zero.zero\n\u22a2 Zero.zero \u226b Zero.zero = Zero.zero\n[PROOFSTEP]\napply J.comp_zero (I.Zero X Y).zero Y\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nI J : HasZeroMorphisms C\nX Y : C\nthis : Zero.zero \u226b Zero.zero = Zero.zero\nthat : Zero.zero \u226b Zero.zero = Zero.zero\n\u22a2 Zero.zero = Zero.zero\n[PROOFSTEP]\nrw [\u2190 this, \u2190 that]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d : Mono g\nh : f \u226b g = 0\n\u22a2 f = 0\n[PROOFSTEP]\nrw [\u2190 zero_comp, cancel_mono] at h \n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d : Mono g\nh\u271d : f \u226b g = 0\nh : f = 0\n\u22a2 f = 0\n[PROOFSTEP]\nexact h\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d : Epi f\nh : f \u226b g = 0\n\u22a2 g = 0\n[PROOFSTEP]\nrw [\u2190 comp_zero, cancel_epi] at h \n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d : Epi f\nh\u271d : f \u226b g = 0\nh : g = 0\n\u22a2 g = 0\n[PROOFSTEP]\nexact h\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : HasImage f\nw : image.\u03b9 f = 0\n\u22a2 f = 0\n[PROOFSTEP]\nrw [\u2190 image.fac f, w, HasZeroMorphisms.comp_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroMorphisms D\nX\u271d Y\u271d : C \u2964 D\n\u03b7 : X\u271d \u27f6 Y\u271d\nH : C \u2964 D\n\u22a2 \u03b7 \u226b 0 = 0\n[PROOFSTEP]\next X\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroMorphisms D\nX\u271d Y\u271d : C \u2964 D\n\u03b7 : X\u271d \u27f6 Y\u271d\nH : C \u2964 D\nX : C\n\u22a2 NatTrans.app (\u03b7 \u226b 0) X = NatTrans.app 0 X\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroMorphisms D\nX\u271d Y\u271d : C \u2964 D\n\u03b7 : X\u271d \u27f6 Y\u271d\nH : C \u2964 D\nX : C\n\u22a2 NatTrans.app \u03b7 X \u226b NatTrans.app 0 X = NatTrans.app 0 X\n[PROOFSTEP]\napply comp_zero\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroMorphisms D\nF G H : C \u2964 D\n\u03b7 : G \u27f6 H\n\u22a2 0 \u226b \u03b7 = 0\n[PROOFSTEP]\next X\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroMorphisms D\nF G H : C \u2964 D\n\u03b7 : G \u27f6 H\nX : C\n\u22a2 NatTrans.app (0 \u226b \u03b7) X = NatTrans.app 0 X\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroMorphisms D\nF G H : C \u2964 D\n\u03b7 : G \u27f6 H\nX : C\n\u22a2 NatTrans.app 0 X \u226b NatTrans.app \u03b7 X = NatTrans.app 0 X\n[PROOFSTEP]\napply zero_comp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroMorphisms C\nX : C\nh : \ud835\udfd9 X = 0\nY : C\nf : X \u27f6 Y\n\u22a2 f = default\n[PROOFSTEP]\nrw [\u2190 id_comp f, \u2190 id_comp (0 : X \u27f6 Y), h, zero_comp, zero_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroMorphisms C\nX : C\nh : \ud835\udfd9 X = 0\nY : C\nf : X \u27f6 Y\n\u22a2 0 = default\n[PROOFSTEP]\nsimp only\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroMorphisms C\nX : C\nh : \ud835\udfd9 X = 0\nY : C\nf : Y \u27f6 X\n\u22a2 f = default\n[PROOFSTEP]\nrw [\u2190 comp_id f, \u2190 comp_id (0 : Y \u27f6 X), h, comp_zero, comp_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroMorphisms C\nX : C\nh : \ud835\udfd9 X = 0\nY : C\nf : Y \u27f6 X\n\u22a2 0 = default\n[PROOFSTEP]\nsimp only\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\ninst\u271d : Mono 0\n\u22a2 \ud835\udfd9 X \u226b 0 = 0 \u226b 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\ninst\u271d : Epi 0\n\u22a2 0 \u226b \ud835\udfd9 Y = 0 \u226b 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : Mono f\nh : f = 0\n\u22a2 IsZero X\n[PROOFSTEP]\nsubst h\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\ninst\u271d : Mono 0\n\u22a2 IsZero X\n[PROOFSTEP]\napply of_mono_zero X Y\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : Epi f\nh : f = 0\n\u22a2 IsZero Y\n[PROOFSTEP]\nsubst h\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\ninst\u271d : Epi 0\n\u22a2 IsZero Y\n[PROOFSTEP]\napply of_epi_zero X Y\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsSplitMono f\n\u22a2 IsZero X \u2194 f = 0\n[PROOFSTEP]\nrw [iff_id_eq_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsSplitMono f\n\u22a2 \ud835\udfd9 X = 0 \u2194 f = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsSplitMono f\n\u22a2 \ud835\udfd9 X = 0 \u2192 f = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsSplitMono f\nh : \ud835\udfd9 X = 0\n\u22a2 f = 0\n[PROOFSTEP]\nrw [\u2190 Category.id_comp f, h, zero_comp]\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsSplitMono f\n\u22a2 f = 0 \u2192 \ud835\udfd9 X = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsSplitMono f\nh : f = 0\n\u22a2 \ud835\udfd9 X = 0\n[PROOFSTEP]\nrw [\u2190 IsSplitMono.id f]\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsSplitMono f\nh : f = 0\n\u22a2 f \u226b retraction f = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsSplitEpi f\n\u22a2 IsZero Y \u2194 f = 0\n[PROOFSTEP]\nrw [iff_id_eq_zero]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsSplitEpi f\n\u22a2 \ud835\udfd9 Y = 0 \u2194 f = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsSplitEpi f\n\u22a2 \ud835\udfd9 Y = 0 \u2192 f = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsSplitEpi f\nh : \ud835\udfd9 Y = 0\n\u22a2 f = 0\n[PROOFSTEP]\nrw [\u2190 Category.comp_id f, h, comp_zero]\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsSplitEpi f\n\u22a2 f = 0 \u2192 \ud835\udfd9 Y = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsSplitEpi f\nh : f = 0\n\u22a2 \ud835\udfd9 Y = 0\n[PROOFSTEP]\nrw [\u2190 IsSplitEpi.id f]\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsSplitEpi f\nh : f = 0\n\u22a2 section_ f \u226b f = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : Mono f\ni : IsZero Y\n\u22a2 IsZero X\n[PROOFSTEP]\nhave hf := i.eq_zero_of_tgt f\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : Mono f\ni : IsZero Y\nhf : f = 0\n\u22a2 IsZero X\n[PROOFSTEP]\nsubst hf\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\ni : IsZero Y\ninst\u271d : Mono 0\n\u22a2 IsZero X\n[PROOFSTEP]\nexact IsZero.of_mono_zero X Y\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : Epi f\ni : IsZero X\n\u22a2 IsZero Y\n[PROOFSTEP]\nhave hf := i.eq_zero_of_src f\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : Epi f\ni : IsZero X\nhf : f = 0\n\u22a2 IsZero Y\n[PROOFSTEP]\nsubst hf\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\ni : IsZero X\ninst\u271d : Epi 0\n\u22a2 IsZero Y\n[PROOFSTEP]\nexact IsZero.of_epi_zero X Y\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nO : C\nhO : IsZero O\nX Y : C\nf : X \u27f6 Y\nZ : C\n\u22a2 f \u226b 0 = 0\n[PROOFSTEP]\nchange f \u226b (hO.from_ Y \u226b hO.to_ Z) = hO.from_ X \u226b hO.to_ Z\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nO : C\nhO : IsZero O\nX Y : C\nf : X \u27f6 Y\nZ : C\n\u22a2 f \u226b IsZero.from_ hO Y \u226b IsZero.to_ hO Z = IsZero.from_ hO X \u226b IsZero.to_ hO Z\n[PROOFSTEP]\nrw [\u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nO : C\nhO : IsZero O\nX Y : C\nf : X \u27f6 Y\nZ : C\n\u22a2 (f \u226b IsZero.from_ hO Y) \u226b IsZero.to_ hO Z = IsZero.from_ hO X \u226b IsZero.to_ hO Z\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nO : C\nhO : IsZero O\nX Y : C\nf : X \u27f6 Y\nZ : C\n\u22a2 f \u226b IsZero.from_ hO Y = IsZero.from_ hO X\n[PROOFSTEP]\napply hO.eq_of_tgt\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nO : C\nhO : IsZero O\nX Y Z : C\nf : Y \u27f6 Z\n\u22a2 0 \u226b f = 0\n[PROOFSTEP]\nchange (hO.from_ X \u226b hO.to_ Y) \u226b f = hO.from_ X \u226b hO.to_ Z\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nO : C\nhO : IsZero O\nX Y Z : C\nf : Y \u27f6 Z\n\u22a2 (IsZero.from_ hO X \u226b IsZero.to_ hO Y) \u226b f = IsZero.from_ hO X \u226b IsZero.to_ hO Z\n[PROOFSTEP]\nrw [Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nO : C\nhO : IsZero O\nX Y Z : C\nf : Y \u27f6 Z\n\u22a2 IsZero.from_ hO X \u226b IsZero.to_ hO Y \u226b f = IsZero.from_ hO X \u226b IsZero.to_ hO Z\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u'\ninst\u271d : Category.{v', u'} D\nO : C\nhO : IsZero O\nX Y Z : C\nf : Y \u27f6 Z\n\u22a2 IsZero.to_ hO Y \u226b f = IsZero.to_ hO Z\n[PROOFSTEP]\napply hO.eq_of_src\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroObject C\nX Y : C\nf : X \u27f6 Y\nZ : C\n\u22a2 f \u226b 0 = 0\n[PROOFSTEP]\nchange f \u226b (default : Y \u27f6 0) \u226b default = (default : X \u27f6 0) \u226b default\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroObject C\nX Y : C\nf : X \u27f6 Y\nZ : C\n\u22a2 f \u226b default \u226b default = default \u226b default\n[PROOFSTEP]\nrw [\u2190 Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroObject C\nX Y : C\nf : X \u27f6 Y\nZ : C\n\u22a2 (f \u226b default) \u226b default = default \u226b default\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroObject C\nX Y : C\nf : X \u27f6 Y\nZ : C\n\u22a2 f \u226b default = default\n[PROOFSTEP]\nsimp only [eq_iff_true_of_subsingleton]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroObject C\nX Y Z : C\nf : Y \u27f6 Z\n\u22a2 0 \u226b f = 0\n[PROOFSTEP]\nchange ((default : X \u27f6 0) \u226b default) \u226b f = (default : X \u27f6 0) \u226b default\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroObject C\nX Y Z : C\nf : Y \u27f6 Z\n\u22a2 (default \u226b default) \u226b f = default \u226b default\n[PROOFSTEP]\nrw [Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroObject C\nX Y Z : C\nf : Y \u27f6 Z\n\u22a2 default \u226b default \u226b f = default \u226b default\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroObject C\nX Y Z : C\nf : Y \u27f6 Z\n\u22a2 default \u226b f = default\n[PROOFSTEP]\nsimp only [eq_iff_true_of_subsingleton]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\nt : IsInitial X\n\u22a2 (zeroIsoIsInitial t).hom = 0\n[PROOFSTEP]\next\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\nt : IsInitial X\n\u22a2 (zeroIsoIsInitial t).inv = 0\n[PROOFSTEP]\next\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\nt : IsTerminal X\n\u22a2 (zeroIsoIsTerminal t).hom = 0\n[PROOFSTEP]\next\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\nt : IsTerminal X\n\u22a2 (zeroIsoIsTerminal t).inv = 0\n[PROOFSTEP]\next\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\ninst\u271d\u00b2 : HasZeroObject C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasInitial C\n\u22a2 zeroIsoInitial.hom = 0\n[PROOFSTEP]\next\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\ninst\u271d\u00b2 : HasZeroObject C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasInitial C\n\u22a2 zeroIsoInitial.inv = 0\n[PROOFSTEP]\next\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\ninst\u271d\u00b2 : HasZeroObject C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasTerminal C\n\u22a2 zeroIsoTerminal.hom = 0\n[PROOFSTEP]\next\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\ninst\u271d\u00b2 : HasZeroObject C\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasTerminal C\n\u22a2 zeroIsoTerminal.inv = 0\n[PROOFSTEP]\next\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\n\u22a2 \ud835\udfd9 0 = 0\n[PROOFSTEP]\napply HasZeroObject.from_zero_ext\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\nf : X \u27f6 0\n\u22a2 f = 0\n[PROOFSTEP]\next\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ni : Y \u2245 0\n\u22a2 f = 0\n[PROOFSTEP]\nhave h : f = f \u226b i.hom \u226b \ud835\udfd9 0 \u226b i.inv := by simp only [Iso.hom_inv_id, id_comp, comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ni : Y \u2245 0\n\u22a2 f = f \u226b i.hom \u226b \ud835\udfd9 0 \u226b i.inv\n[PROOFSTEP]\nsimp only [Iso.hom_inv_id, id_comp, comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ni : Y \u2245 0\nh : f = f \u226b i.hom \u226b \ud835\udfd9 0 \u226b i.inv\n\u22a2 f = 0\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\nf : 0 \u27f6 X\n\u22a2 f = 0\n[PROOFSTEP]\next\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ni : X \u2245 0\n\u22a2 f = 0\n[PROOFSTEP]\nhave h : f = i.hom \u226b \ud835\udfd9 0 \u226b i.inv \u226b f := by simp only [Iso.hom_inv_id_assoc, id_comp, comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ni : X \u2245 0\n\u22a2 f = i.hom \u226b \ud835\udfd9 0 \u226b i.inv \u226b f\n[PROOFSTEP]\nsimp only [Iso.hom_inv_id_assoc, id_comp, comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ni : X \u2245 0\nh : f = i.hom \u226b \ud835\udfd9 0 \u226b i.inv \u226b f\n\u22a2 f = 0\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ni : X \u2245 0\nZ : C\ng h : Z \u27f6 X\nx\u271d : g \u226b f = h \u226b f\n\u22a2 g = h\n[PROOFSTEP]\nrw [zero_of_target_iso_zero g i, zero_of_target_iso_zero h i]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ni : Y \u2245 0\nZ : C\ng h : Y \u27f6 Z\nx\u271d : f \u226b g = f \u226b h\n\u22a2 g = h\n[PROOFSTEP]\nrw [zero_of_source_iso_zero g i, zero_of_source_iso_zero h i]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\n\u22a2 Function.LeftInverse (_ : (X \u2245 0) \u2192 \ud835\udfd9 X = 0) fun h => Iso.mk 0 0\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\n\u22a2 Function.RightInverse (_ : (X \u2245 0) \u2192 \ud835\udfd9 X = 0) fun h => Iso.mk 0 0\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nh : Mono 0\n\u22a2 (0 \u226b 0) \u226b 0 = \ud835\udfd9 X \u226b 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nh : Epi 0\n\u22a2 0 \u226b 0 \u226b 0 = 0 \u226b \ud835\udfd9 Y\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\ninst\u271d\u00b2 : HasZeroObject C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : Mono f\nh : f = 0\n\u22a2 X \u2245 0\n[PROOFSTEP]\nsubst h\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\ninst\u271d\u00b2 : HasZeroObject C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\ninst\u271d : Mono 0\n\u22a2 X \u2245 0\n[PROOFSTEP]\napply isoZeroOfMonoZero \u2039_\u203a\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\ninst\u271d\u00b2 : HasZeroObject C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ninst\u271d : Epi f\nh : f = 0\n\u22a2 Y \u2245 0\n[PROOFSTEP]\nsubst h\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\ninst\u271d\u00b2 : HasZeroObject C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\ninst\u271d : Epi 0\n\u22a2 Y \u2245 0\n[PROOFSTEP]\napply isoZeroOfEpiZero \u2039_\u203a\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\nP : IsIsomorphic X 0\n\u22a2 0 \u226b 0 = \ud835\udfd9 X\n[PROOFSTEP]\ncases' P with P\n[GOAL]\ncase intro\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\nP : X \u2245 0\n\u22a2 0 \u226b 0 = \ud835\udfd9 X\n[PROOFSTEP]\nrw [\u2190 P.hom_inv_id, \u2190 Category.id_comp P.inv]\n[GOAL]\ncase intro\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\nP : X \u2245 0\n\u22a2 0 \u226b 0 = P.hom \u226b \ud835\udfd9 0 \u226b P.inv\n[PROOFSTEP]\napply Eq.symm\n[GOAL]\ncase intro.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\nP : X \u2245 0\n\u22a2 P.hom \u226b \ud835\udfd9 0 \u226b P.inv = 0 \u226b 0\n[PROOFSTEP]\nsimp only [id_comp, Iso.hom_inv_id, comp_zero]\n[GOAL]\ncase intro.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\nP : X \u2245 0\n\u22a2 \ud835\udfd9 X = 0\n[PROOFSTEP]\napply (idZeroEquivIsoZero X).invFun P\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX : C\nP : IsIsomorphic X 0\n\u22a2 0 \u226b 0 = \ud835\udfd9 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroMorphisms C\nX Y : C\n\u22a2 IsIso 0 \u2192 \ud835\udfd9 X = 0 \u2227 \ud835\udfd9 Y = 0\n[PROOFSTEP]\nintro i\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroMorphisms C\nX Y : C\ni : IsIso 0\n\u22a2 \ud835\udfd9 X = 0 \u2227 \ud835\udfd9 Y = 0\n[PROOFSTEP]\nrw [\u2190 IsIso.hom_inv_id (0 : X \u27f6 Y)]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroMorphisms C\nX Y : C\ni : IsIso 0\n\u22a2 0 \u226b inv 0 = 0 \u2227 \ud835\udfd9 Y = 0\n[PROOFSTEP]\nrw [\u2190 IsIso.inv_hom_id (0 : X \u27f6 Y)]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroMorphisms C\nX Y : C\ni : IsIso 0\n\u22a2 0 \u226b inv 0 = 0 \u2227 inv 0 \u226b 0 = 0\n[PROOFSTEP]\nsimp only [eq_self_iff_true, comp_zero, and_self, zero_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroMorphisms C\nX Y : C\nh : \ud835\udfd9 X = 0 \u2227 \ud835\udfd9 Y = 0\n\u22a2 0 \u226b 0 = \ud835\udfd9 X \u2227 0 \u226b 0 = \ud835\udfd9 Y\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroMorphisms C\nX Y : C\n\u22a2 Function.LeftInverse (_ : \ud835\udfd9 X = 0 \u2227 \ud835\udfd9 Y = 0 \u2192 IsIso 0) (_ : IsIso 0 \u2192 \ud835\udfd9 X = 0 \u2227 \ud835\udfd9 Y = 0)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroMorphisms C\nX Y : C\n\u22a2 Function.RightInverse (_ : \ud835\udfd9 X = 0 \u2227 \ud835\udfd9 Y = 0 \u2192 IsIso 0) (_ : IsIso 0 \u2192 \ud835\udfd9 X = 0 \u2227 \ud835\udfd9 Y = 0)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b9 : Category.{v', u'} D\ninst\u271d : HasZeroMorphisms C\nX : C\n\u22a2 IsIso 0 \u2243 \ud835\udfd9 X = 0\n[PROOFSTEP]\nsimpa using isIsoZeroEquiv X X\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasZeroObject C\nX Y : C\n\u22a2 IsIso 0 \u2243 (X \u2245 0) \u00d7 (Y \u2245 0)\n[PROOFSTEP]\nrefine' (isIsoZeroEquiv X Y).trans _\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasZeroObject C\nX Y : C\n\u22a2 \ud835\udfd9 X = 0 \u2227 \ud835\udfd9 Y = 0 \u2243 (X \u2245 0) \u00d7 (Y \u2245 0)\n[PROOFSTEP]\nsymm\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasZeroObject C\nX Y : C\n\u22a2 (X \u2245 0) \u00d7 (Y \u2245 0) \u2243 \ud835\udfd9 X = 0 \u2227 \ud835\udfd9 Y = 0\n[PROOFSTEP]\nfconstructor\n[GOAL]\ncase toFun\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasZeroObject C\nX Y : C\n\u22a2 (X \u2245 0) \u00d7 (Y \u2245 0) \u2192 \ud835\udfd9 X = 0 \u2227 \ud835\udfd9 Y = 0\n[PROOFSTEP]\nrintro \u27e8eX, eY\u27e9\n[GOAL]\ncase toFun.mk\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasZeroObject C\nX Y : C\neX : X \u2245 0\neY : Y \u2245 0\n\u22a2 \ud835\udfd9 X = 0 \u2227 \ud835\udfd9 Y = 0\n[PROOFSTEP]\nfconstructor\n[GOAL]\ncase toFun.mk.left\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasZeroObject C\nX Y : C\neX : X \u2245 0\neY : Y \u2245 0\n\u22a2 \ud835\udfd9 X = 0\ncase toFun.mk.right\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasZeroObject C\nX Y : C\neX : X \u2245 0\neY : Y \u2245 0\n\u22a2 \ud835\udfd9 Y = 0\n[PROOFSTEP]\nexact (idZeroEquivIsoZero X).symm eX\n[GOAL]\ncase toFun.mk.right\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasZeroObject C\nX Y : C\neX : X \u2245 0\neY : Y \u2245 0\n\u22a2 \ud835\udfd9 Y = 0\n[PROOFSTEP]\nexact (idZeroEquivIsoZero Y).symm eY\n[GOAL]\ncase invFun\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasZeroObject C\nX Y : C\n\u22a2 \ud835\udfd9 X = 0 \u2227 \ud835\udfd9 Y = 0 \u2192 (X \u2245 0) \u00d7 (Y \u2245 0)\n[PROOFSTEP]\nrintro \u27e8hX, hY\u27e9\n[GOAL]\ncase invFun.intro\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasZeroObject C\nX Y : C\nhX : \ud835\udfd9 X = 0\nhY : \ud835\udfd9 Y = 0\n\u22a2 (X \u2245 0) \u00d7 (Y \u2245 0)\n[PROOFSTEP]\nfconstructor\n[GOAL]\ncase invFun.intro.fst\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasZeroObject C\nX Y : C\nhX : \ud835\udfd9 X = 0\nhY : \ud835\udfd9 Y = 0\n\u22a2 X \u2245 0\ncase invFun.intro.snd\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasZeroObject C\nX Y : C\nhX : \ud835\udfd9 X = 0\nhY : \ud835\udfd9 Y = 0\n\u22a2 Y \u2245 0\n[PROOFSTEP]\nexact (idZeroEquivIsoZero X) hX\n[GOAL]\ncase invFun.intro.snd\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasZeroObject C\nX Y : C\nhX : \ud835\udfd9 X = 0\nhY : \ud835\udfd9 Y = 0\n\u22a2 Y \u2245 0\n[PROOFSTEP]\nexact (idZeroEquivIsoZero Y) hY\n[GOAL]\ncase left_inv\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasZeroObject C\nX Y : C\n\u22a2 Function.LeftInverse (fun a => And.casesOn a fun hX hY => (\u2191(idZeroEquivIsoZero X) hX, \u2191(idZeroEquivIsoZero Y) hY))\n (_ : (X \u2245 0) \u00d7 (Y \u2245 0) \u2192 \ud835\udfd9 X = 0 \u2227 \ud835\udfd9 Y = 0)\n[PROOFSTEP]\naesop_cat\n[GOAL]\ncase right_inv\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasZeroObject C\nX Y : C\n\u22a2 Function.RightInverse (fun a => And.casesOn a fun hX hY => (\u2191(idZeroEquivIsoZero X) hX, \u2191(idZeroEquivIsoZero Y) hY))\n (_ : (X \u2245 0) \u00d7 (Y \u2245 0) \u2192 \ud835\udfd9 X = 0 \u2227 \ud835\udfd9 Y = 0)\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasZeroObject C\nX Y : C\nf : X \u27f6 Y\ni : X \u2245 0\nj : Y \u2245 0\n\u22a2 IsIso f\n[PROOFSTEP]\nrw [zero_of_source_iso_zero f i]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasZeroObject C\nX Y : C\nf : X \u27f6 Y\ni : X \u2245 0\nj : Y \u2245 0\n\u22a2 IsIso 0\n[PROOFSTEP]\nexact (isIsoZeroEquivIsoZero _ _).invFun \u27e8i, j\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasInitial C\n\u22a2 HasZeroObject C\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u22a5_ C, fun X => \u27e8\u27e8\u27e80\u27e9, by aesop_cat\u27e9\u27e9, fun X => \u27e8\u27e8\u27e80\u27e9, fun f => _\u27e9\u27e9\u27e9\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasInitial C\nX : C\n\u22a2 \u2200 (a : \u22a5_ C \u27f6 X), a = default\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasInitial C\nX : C\nf : X \u27f6 \u22a5_ C\n\u22a2 f = default\n[PROOFSTEP]\ncalc\n f = f \u226b \ud835\udfd9 _ := (Category.comp_id _).symm\n _ = f \u226b 0 := by congr!\n _ = 0 := HasZeroMorphisms.comp_zero _ _\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasInitial C\nX : C\nf : X \u27f6 \u22a5_ C\n\u22a2 f \u226b \ud835\udfd9 (\u22a5_ C) = f \u226b 0\n[PROOFSTEP]\ncongr!\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasTerminal C\n\u22a2 HasZeroObject C\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u22a4_ C, fun X => \u27e8\u27e8\u27e80\u27e9, fun f => _\u27e9\u27e9, fun X => \u27e8\u27e8\u27e80\u27e9, by aesop_cat\u27e9\u27e9\u27e9\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasTerminal C\nX : C\n\u22a2 \u2200 (a : X \u27f6 \u22a4_ C), a = default\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasTerminal C\nX : C\nf : \u22a4_ C \u27f6 X\n\u22a2 f = default\n[PROOFSTEP]\ncalc\n f = \ud835\udfd9 _ \u226b f := (Category.id_comp _).symm\n _ = 0 \u226b f := by congr!\n _ = 0 := zero_comp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasTerminal C\nX : C\nf : \u22a4_ C \u27f6 X\n\u22a2 \ud835\udfd9 (\u22a4_ C) \u226b f = 0 \u226b f\n[PROOFSTEP]\ncongr!\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\ninst\u271d\u00b2 : HasZeroMorphisms C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasImage f\ninst\u271d : Epi (factorThruImage f)\nh : f \u226b g = 0\n\u22a2 factorThruImage f \u226b image.\u03b9 f \u226b g = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d : HasImage g\nh : f \u226b g = 0\n\u22a2 (f \u226b factorThruImage g) \u226b image.\u03b9 g = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX Y : C\ninst\u271d : HasImage 0\n\u22a2 \u03b9 0 = 0\n[PROOFSTEP]\nrw [\u2190 image.lift_fac (monoFactorisationZero X Y)]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX Y : C\ninst\u271d : HasImage 0\n\u22a2 lift (monoFactorisationZero X Y) \u226b (monoFactorisationZero X Y).m = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasZeroObject C\ninst\u271d\u00b9 : HasEqualizers C\nX Y : C\nf : X \u27f6 Y\nh : f = 0\ninst\u271d : HasImage f\n\u22a2 \u03b9 f = 0\n[PROOFSTEP]\nrw [image.eq_fac h]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u'\ninst\u271d\u2074 : Category.{v', u'} D\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasZeroObject C\ninst\u271d\u00b9 : HasEqualizers C\nX Y : C\nf : X \u27f6 Y\nh : f = 0\ninst\u271d : HasImage f\n\u22a2 (eqToIso h).hom \u226b \u03b9 0 = 0\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms", "llama_tokens": 18389, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.4571367168274948, "lm_q2_score": 0.023330768511379073, "lm_q1q2_score": 0.010665350918354128}} {"text": "[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{v, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : HasFiniteBiproducts C\nJ : Type\ninst\u271d : Finite J\nF : J \u2192 Karoubi C\n\u22a2 ((biproduct.map fun j => (F j).p) \u226b biproduct.map fun j => (F j).p) = biproduct.map fun j => (F j).p\n[PROOFSTEP]\next\n[GOAL]\ncase w.w\nC : Type u_1\ninst\u271d\u00b3 : Category.{v, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : HasFiniteBiproducts C\nJ : Type\ninst\u271d : Finite J\nF : J \u2192 Karoubi C\nj\u271d\u00b9 j\u271d : J\n\u22a2 biproduct.\u03b9 (fun j => (F j).X) j\u271d \u226b\n ((biproduct.map fun j => (F j).p) \u226b biproduct.map fun j => (F j).p) \u226b biproduct.\u03c0 (fun j => (F j).X) j\u271d\u00b9 =\n biproduct.\u03b9 (fun j => (F j).X) j\u271d \u226b (biproduct.map fun j => (F j).p) \u226b biproduct.\u03c0 (fun j => (F j).X) j\u271d\u00b9\n[PROOFSTEP]\nsimp only [assoc, biproduct.map_\u03c0, biproduct.map_\u03c0_assoc, idem]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{v, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : HasFiniteBiproducts C\nJ : Type\ninst\u271d : Finite J\nF : J \u2192 Karoubi C\nj : J\n\u22a2 (biproduct.map fun j => (F j).p) \u226b Bicone.\u03c0 (biproduct.bicone fun j => (F j).X) j =\n (mk (\u2a01 fun j => (F j).X) (biproduct.map fun j => (F j).p)).p \u226b\n ((biproduct.map fun j => (F j).p) \u226b Bicone.\u03c0 (biproduct.bicone fun j => (F j).X) j) \u226b (F j).p\n[PROOFSTEP]\nsimp only [assoc, biproduct.bicone_\u03c0, biproduct.map_\u03c0, biproduct.map_\u03c0_assoc, (F j).idem]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{v, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : HasFiniteBiproducts C\nJ : Type\ninst\u271d : Finite J\nF : J \u2192 Karoubi C\nj : J\n\u22a2 (biproduct.\u03b9 (fun j => (F j).X) j \u226b biproduct.map fun j => (F j).p) =\n (F j).p \u226b\n (biproduct.\u03b9 (fun j => (F j).X) j \u226b biproduct.map fun j => (F j).p) \u226b\n (mk (\u2a01 fun j => (F j).X) (biproduct.map fun j => (F j).p)).p\n[PROOFSTEP]\nsimp only [biproduct.\u03b9_map, assoc, idem_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b3 : Category.{v, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : HasFiniteBiproducts C\nJ : Type\ninst\u271d : Finite J\nF : J \u2192 Karoubi C\nj j' : J\n\u22a2 (fun j => Hom.mk (biproduct.\u03b9 (fun j => (F j).X) j \u226b biproduct.map fun j => (F j).p)) j \u226b\n (fun j => Hom.mk ((biproduct.map fun j => (F j).p) \u226b Bicone.\u03c0 (biproduct.bicone fun j => (F j).X) j)) j' =\n if h : j = j' then eqToHom (_ : F j = F j') else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nC : Type u_1\ninst\u271d\u00b3 : Category.{v, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : HasFiniteBiproducts C\nJ : Type\ninst\u271d : Finite J\nF : J \u2192 Karoubi C\nj j' : J\nh : j = j'\n\u22a2 (fun j => Hom.mk (biproduct.\u03b9 (fun j => (F j).X) j \u226b biproduct.map fun j => (F j).p)) j \u226b\n (fun j => Hom.mk ((biproduct.map fun j => (F j).p) \u226b Bicone.\u03c0 (biproduct.bicone fun j => (F j).X) j)) j' =\n eqToHom (_ : F j = F j')\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\nC : Type u_1\ninst\u271d\u00b3 : Category.{v, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : HasFiniteBiproducts C\nJ : Type\ninst\u271d : Finite J\nF : J \u2192 Karoubi C\nj : J\n\u22a2 (fun j => Hom.mk (biproduct.\u03b9 (fun j => (F j).X) j \u226b biproduct.map fun j => (F j).p)) j \u226b\n (fun j => Hom.mk ((biproduct.map fun j => (F j).p) \u226b Bicone.\u03c0 (biproduct.bicone fun j => (F j).X) j)) j =\n eqToHom (_ : F j = F j)\n[PROOFSTEP]\nsimp only [biproduct.\u03b9_map, biproduct.bicone_\u03c0, biproduct.map_\u03c0, eqToHom_refl, id_eq, hom_ext_iff, comp_f, assoc,\n bicone_\u03b9_\u03c0_self_assoc, idem]\n[GOAL]\ncase neg\nC : Type u_1\ninst\u271d\u00b3 : Category.{v, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : HasFiniteBiproducts C\nJ : Type\ninst\u271d : Finite J\nF : J \u2192 Karoubi C\nj j' : J\nh : \u00acj = j'\n\u22a2 (fun j => Hom.mk (biproduct.\u03b9 (fun j => (F j).X) j \u226b biproduct.map fun j => (F j).p)) j \u226b\n (fun j => Hom.mk ((biproduct.map fun j => (F j).p) \u226b Bicone.\u03c0 (biproduct.bicone fun j => (F j).X) j)) j' =\n 0\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase neg\nC : Type u_1\ninst\u271d\u00b3 : Category.{v, u_1} C\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : HasFiniteBiproducts C\nJ : Type\ninst\u271d : Finite J\nF : J \u2192 Karoubi C\nj j' : J\nh : \u00acj = j'\n\u22a2 Hom.mk (biproduct.\u03b9 (fun j => (F j).X) j \u226b biproduct.map fun j => (F j).p) \u226b\n Hom.mk ((biproduct.map fun j => (F j).p) \u226b biproduct.\u03c0 (fun j => (F j).X) j') =\n Hom.mk 0\n[PROOFSTEP]\nsimp only [hom_ext_iff, biproduct.\u03b9_map, biproduct.map_\u03c0, comp_f, assoc, ne_eq, biproduct.\u03b9_\u03c0_ne_assoc _ h, comp_zero,\n zero_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{v, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteBiproducts C\nn : \u2115\nF : Fin n \u2192 Karoubi C\n\u22a2 HasBiproduct F\n[PROOFSTEP]\napply hasBiproduct_of_total (Biproducts.bicone F)\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{v, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteBiproducts C\nn : \u2115\nF : Fin n \u2192 Karoubi C\n\u22a2 (Finset.sum Finset.univ fun j => Bicone.\u03c0 (Biproducts.bicone F) j \u226b Bicone.\u03b9 (Biproducts.bicone F) j) =\n \ud835\udfd9 (Biproducts.bicone F).pt\n[PROOFSTEP]\nsimp only [hom_ext_iff]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{v, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteBiproducts C\nn : \u2115\nF : Fin n \u2192 Karoubi C\n\u22a2 (Finset.sum Finset.univ fun j => Bicone.\u03c0 (Biproducts.bicone F) j \u226b Bicone.\u03b9 (Biproducts.bicone F) j).f =\n (\ud835\udfd9 (Biproducts.bicone F).pt).f\n[PROOFSTEP]\nrefine' biproduct.hom_ext' _ _ (fun j => _)\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{v, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteBiproducts C\nn : \u2115\nF : Fin n \u2192 Karoubi C\nj : Fin n\n\u22a2 biproduct.\u03b9 (fun j => (F j).X) j \u226b\n (Finset.sum Finset.univ fun j => Bicone.\u03c0 (Biproducts.bicone F) j \u226b Bicone.\u03b9 (Biproducts.bicone F) j).f =\n biproduct.\u03b9 (fun j => (F j).X) j \u226b (\ud835\udfd9 (Biproducts.bicone F).pt).f\n[PROOFSTEP]\nsimp only [Biproducts.bicone_pt_X, sum_hom, comp_f, Biproducts.bicone_\u03c0_f, biproduct.bicone_\u03c0, biproduct.map_\u03c0,\n Biproducts.bicone_\u03b9_f, biproduct.\u03b9_map, assoc, idem_assoc, id_eq, Biproducts.bicone_pt_p, comp_sum]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{v, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteBiproducts C\nn : \u2115\nF : Fin n \u2192 Karoubi C\nj : Fin n\n\u22a2 (Finset.sum Finset.univ fun j_1 =>\n biproduct.\u03b9 (fun j => (F j).X) j \u226b\n biproduct.\u03c0 (fun b => (F b).X) j_1 \u226b (F j_1).p \u226b biproduct.\u03b9 (fun j => (F j).X) j_1) =\n (F j).p \u226b biproduct.\u03b9 (fun j => (F j).X) j\n[PROOFSTEP]\nrw [Finset.sum_eq_single j]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b2 : Category.{v, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteBiproducts C\nn : \u2115\nF : Fin n \u2192 Karoubi C\nj : Fin n\n\u22a2 biproduct.\u03b9 (fun j => (F j).X) j \u226b biproduct.\u03c0 (fun b => (F b).X) j \u226b (F j).p \u226b biproduct.\u03b9 (fun j => (F j).X) j =\n (F j).p \u226b biproduct.\u03b9 (fun j => (F j).X) j\n[PROOFSTEP]\nsimp only [bicone_\u03b9_\u03c0_self_assoc]\n[GOAL]\ncase h\u2080\nC : Type u_1\ninst\u271d\u00b2 : Category.{v, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteBiproducts C\nn : \u2115\nF : Fin n \u2192 Karoubi C\nj : Fin n\n\u22a2 \u2200 (b : Fin n),\n b \u2208 Finset.univ \u2192\n b \u2260 j \u2192\n biproduct.\u03b9 (fun j => (F j).X) j \u226b\n biproduct.\u03c0 (fun b => (F b).X) b \u226b (F b).p \u226b biproduct.\u03b9 (fun j => (F j).X) b =\n 0\n[PROOFSTEP]\nintro b _ hb\n[GOAL]\ncase h\u2080\nC : Type u_1\ninst\u271d\u00b2 : Category.{v, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteBiproducts C\nn : \u2115\nF : Fin n \u2192 Karoubi C\nj b : Fin n\na\u271d : b \u2208 Finset.univ\nhb : b \u2260 j\n\u22a2 biproduct.\u03b9 (fun j => (F j).X) j \u226b biproduct.\u03c0 (fun b => (F b).X) b \u226b (F b).p \u226b biproduct.\u03b9 (fun j => (F j).X) b = 0\n[PROOFSTEP]\nsimp only [biproduct.\u03b9_\u03c0_ne_assoc _ hb.symm, zero_comp]\n[GOAL]\ncase h\u2081\nC : Type u_1\ninst\u271d\u00b2 : Category.{v, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteBiproducts C\nn : \u2115\nF : Fin n \u2192 Karoubi C\nj : Fin n\n\u22a2 \u00acj \u2208 Finset.univ \u2192\n biproduct.\u03b9 (fun j => (F j).X) j \u226b biproduct.\u03c0 (fun b => (F b).X) j \u226b (F j).p \u226b biproduct.\u03b9 (fun j => (F j).X) j = 0\n[PROOFSTEP]\nintro hj\n[GOAL]\ncase h\u2081\nC : Type u_1\ninst\u271d\u00b2 : Category.{v, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteBiproducts C\nn : \u2115\nF : Fin n \u2192 Karoubi C\nj : Fin n\nhj : \u00acj \u2208 Finset.univ\n\u22a2 biproduct.\u03b9 (fun j => (F j).X) j \u226b biproduct.\u03c0 (fun b => (F b).X) j \u226b (F j).p \u226b biproduct.\u03b9 (fun j => (F j).X) j = 0\n[PROOFSTEP]\nsimp only [Finset.mem_univ, not_true] at hj \n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{v, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi C\n\u22a2 decompId_i P \u226b decompId_p (complement P) = 0\n[PROOFSTEP]\nsimp only [instAddCommGroupHom_zero, hom_ext_iff, complement_X, comp_f, decompId_i_f, decompId_p_f, complement_p,\n comp_sub, comp_id, idem, sub_self]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{v, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi C\n\u22a2 decompId_i (complement P) \u226b decompId_p P = 0\n[PROOFSTEP]\nsimp only [instAddCommGroupHom_zero, hom_ext_iff, complement_X, comp_f, decompId_i_f, complement_p, decompId_p_f,\n sub_comp, id_comp, idem, sub_self]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{v, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi C\n\u22a2 (BinaryBicone.mk (mk P.X (\ud835\udfd9 P.X)) (decompId_p P) (decompId_p (complement P)) (decompId_i P)\n (decompId_i (complement P))).fst \u226b\n (BinaryBicone.mk (mk P.X (\ud835\udfd9 P.X)) (decompId_p P) (decompId_p (complement P)) (decompId_i P)\n (decompId_i (complement P))).inl +\n (BinaryBicone.mk (mk P.X (\ud835\udfd9 P.X)) (decompId_p P) (decompId_p (complement P)) (decompId_i P)\n (decompId_i (complement P))).snd \u226b\n (BinaryBicone.mk (mk P.X (\ud835\udfd9 P.X)) (decompId_p P) (decompId_p (complement P)) (decompId_i P)\n (decompId_i (complement P))).inr =\n \ud835\udfd9\n (BinaryBicone.mk (mk P.X (\ud835\udfd9 P.X)) (decompId_p P) (decompId_p (complement P)) (decompId_i P)\n (decompId_i (complement P))).pt\n[PROOFSTEP]\nsimp only [id_eq, complement_X, comp_f, decompId_i_f, decompId_p_f, complement_p, instAddCommGroupHom_add, idem,\n comp_sub, comp_id, sub_comp, id_comp, sub_self, sub_zero, add_sub_cancel'_right]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{v, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi C\n\u22a2 biprod.desc (decompId_i P) (decompId_i (complement P)) \u226b biprod.lift (decompId_p P) (decompId_p (complement P)) =\n \ud835\udfd9 (P \u229e complement P)\n[PROOFSTEP]\napply biprod.hom_ext'\n[GOAL]\ncase h\u2080\nC : Type u_1\ninst\u271d\u00b9 : Category.{v, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi C\n\u22a2 biprod.inl \u226b\n biprod.desc (decompId_i P) (decompId_i (complement P)) \u226b biprod.lift (decompId_p P) (decompId_p (complement P)) =\n biprod.inl \u226b \ud835\udfd9 (P \u229e complement P)\n[PROOFSTEP]\nrw [biprod.inl_desc_assoc, comp_id, biprod.lift_eq, comp_add, \u2190 decompId_assoc, add_right_eq_self, \u2190 assoc]\n[GOAL]\ncase h\u2080\nC : Type u_1\ninst\u271d\u00b9 : Category.{v, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi C\n\u22a2 (decompId_i P \u226b decompId_p (complement P)) \u226b biprod.inr = 0\n[PROOFSTEP]\nrefine' (_ =\u226b _).trans zero_comp\n[GOAL]\ncase h\u2080\nC : Type u_1\ninst\u271d\u00b9 : Category.{v, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi C\n\u22a2 decompId_i P \u226b decompId_p (complement P) = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{v, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi C\n\u22a2 (decompId_i P \u226b decompId_p (complement P)).f = 0.f\n[PROOFSTEP]\nsimp only [comp_f, toKaroubi_obj_X, decompId_i_f, decompId_p_f, complement_p, comp_sub, comp_id, idem, sub_self,\n instAddCommGroupHom_zero]\n[GOAL]\ncase h\u2081\nC : Type u_1\ninst\u271d\u00b9 : Category.{v, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi C\n\u22a2 biprod.inr \u226b\n biprod.desc (decompId_i P) (decompId_i (complement P)) \u226b biprod.lift (decompId_p P) (decompId_p (complement P)) =\n biprod.inr \u226b \ud835\udfd9 (P \u229e complement P)\n[PROOFSTEP]\nrw [biprod.inr_desc_assoc, comp_id, biprod.lift_eq, comp_add, \u2190 decompId_assoc, add_left_eq_self, \u2190 assoc]\n[GOAL]\ncase h\u2081\nC : Type u_1\ninst\u271d\u00b9 : Category.{v, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi C\n\u22a2 (decompId_i (complement P) \u226b decompId_p P) \u226b biprod.inl = 0\n[PROOFSTEP]\nrefine' (_ =\u226b _).trans zero_comp\n[GOAL]\ncase h\u2081\nC : Type u_1\ninst\u271d\u00b9 : Category.{v, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi C\n\u22a2 decompId_i (complement P) \u226b decompId_p P = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081.h\nC : Type u_1\ninst\u271d\u00b9 : Category.{v, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi C\n\u22a2 (decompId_i (complement P) \u226b decompId_p P).f = 0.f\n[PROOFSTEP]\nsimp only [complement_X, comp_f, decompId_i_f, complement_p, decompId_p_f, sub_comp, id_comp, idem, sub_self,\n instAddCommGroupHom_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u00b9 : Category.{v, u_1} C\ninst\u271d : Preadditive C\nP : Karoubi C\n\u22a2 biprod.lift (decompId_p P) (decompId_p (complement P)) \u226b biprod.desc (decompId_i P) (decompId_i (complement P)) =\n \ud835\udfd9 ((toKaroubi C).obj P.X)\n[PROOFSTEP]\nsimp only [biprod.lift_desc, instAddCommGroupHom_add, toKaroubi_obj_X, comp_f, decompId_p_f, decompId_i_f, idem,\n complement_X, complement_p, comp_sub, comp_id, sub_comp, id_comp, sub_self, sub_zero, add_sub_cancel'_right, id_eq,\n toKaroubi_obj_p]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Idempotents.Biproducts", "llama_tokens": 6219, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.4610167793123159, "lm_q2_score": 0.022286185295708725, "lm_q1q2_score": 0.01027430536818513}} {"text": "[GOAL]\n\u03b1 : Type ?u.3173\n\u03b9 : Type ?u.3179\n\u03ba : Type ?u.3180\nC : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\n\u22a2 Inhabited (ColorFocused C)\n[PROOFSTEP]\nrefine' \u27e8\u27e80, fun _ => none, fun h => _, Multiset.nodup_zero\u27e9\u27e9\n[GOAL]\n\u03b1 : Type ?u.3173\n\u03b9 : Type ?u.3179\n\u03ba : Type ?u.3180\nC : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nh : AlmostMono C\n\u22a2 h \u2208 0 \u2192 (fun x i => Option.getD (idxFun h.line i) x) none = fun x => none\n[PROOFSTEP]\nsimp only [Multiset.not_mem_zero, IsEmpty.forall_iff]\n[GOAL]\n\u03b1 : Type ?u.3673\n\u03b1' : Type ?u.3677\n\u03b9 : Type ?u.3674\nf : \u03b1 \u2192 \u03b1'\nl : Line \u03b1 \u03b9\n\u22a2 (fun i => Option.map f (idxFun l i)) (Exists.choose (_ : \u2203 i, idxFun l i = none)) = none\n[PROOFSTEP]\nsimp only [l.proper.choose_spec, Option.map_none']\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nl : Line \u03b1 \u03b9\nx : \u03b1\ni : \u03b9\nh : idxFun l i = none\n\u22a2 (fun x i => Option.getD (idxFun l i) x) x i = x\n[PROOFSTEP]\nsimp only [Option.getD_none, h, l.apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nl : Line \u03b1 \u03b9\nx : \u03b1\ni : \u03b9\nh : idxFun l i \u2260 none\n\u22a2 some ((fun x i => Option.getD (idxFun l i) x) x i) = idxFun l i\n[PROOFSTEP]\nrw [l.apply, Option.getD_of_ne_none h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u03b1'\nl : Line \u03b1 \u03b9\nx : \u03b1\n\u22a2 (fun x i => Option.getD (idxFun (map f l) i) x) (f x) = f \u2218 (fun x i => Option.getD (idxFun l i) x) x\n[PROOFSTEP]\nsimp only [Line.apply, Line.map, Option.getD_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b9 : Type u_3\nf : \u03b1 \u2192 \u03b1'\nl : Line \u03b1 \u03b9\nx : \u03b1\n\u22a2 (fun i => f (Option.getD (idxFun l i) x)) = f \u2218 fun i => Option.getD (idxFun l i) x\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b9' : Type u_3\nv : \u03b9 \u2192 \u03b1\nl : Line \u03b1 \u03b9'\nx : \u03b1\n\u22a2 (fun x i => Option.getD (idxFun (vertical v l) i) x) x = Sum.elim v ((fun x i => Option.getD (idxFun l i) x) x)\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b9' : Type u_3\nv : \u03b9 \u2192 \u03b1\nl : Line \u03b1 \u03b9'\nx : \u03b1\ni : \u03b9 \u2295 \u03b9'\n\u22a2 (fun x i => Option.getD (idxFun (vertical v l) i) x) x i = Sum.elim v ((fun x i => Option.getD (idxFun l i) x) x) i\n[PROOFSTEP]\ncases i\n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b9' : Type u_3\nv : \u03b9 \u2192 \u03b1\nl : Line \u03b1 \u03b9'\nx : \u03b1\nval\u271d : \u03b9\n\u22a2 (fun x i => Option.getD (idxFun (vertical v l) i) x) x (Sum.inl val\u271d) =\n Sum.elim v ((fun x i => Option.getD (idxFun l i) x) x) (Sum.inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b9' : Type u_3\nv : \u03b9 \u2192 \u03b1\nl : Line \u03b1 \u03b9'\nx : \u03b1\nval\u271d : \u03b9'\n\u22a2 (fun x i => Option.getD (idxFun (vertical v l) i) x) x (Sum.inr val\u271d) =\n Sum.elim v ((fun x i => Option.getD (idxFun l i) x) x) (Sum.inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b9' : Type u_3\nl : Line \u03b1 \u03b9\nv : \u03b9' \u2192 \u03b1\nx : \u03b1\n\u22a2 (fun x i => Option.getD (idxFun (horizontal l v) i) x) x = Sum.elim ((fun x i => Option.getD (idxFun l i) x) x) v\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b9' : Type u_3\nl : Line \u03b1 \u03b9\nv : \u03b9' \u2192 \u03b1\nx : \u03b1\ni : \u03b9 \u2295 \u03b9'\n\u22a2 (fun x i => Option.getD (idxFun (horizontal l v) i) x) x i = Sum.elim ((fun x i => Option.getD (idxFun l i) x) x) v i\n[PROOFSTEP]\ncases i\n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b9' : Type u_3\nl : Line \u03b1 \u03b9\nv : \u03b9' \u2192 \u03b1\nx : \u03b1\nval\u271d : \u03b9\n\u22a2 (fun x i => Option.getD (idxFun (horizontal l v) i) x) x (Sum.inl val\u271d) =\n Sum.elim ((fun x i => Option.getD (idxFun l i) x) x) v (Sum.inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b9' : Type u_3\nl : Line \u03b1 \u03b9\nv : \u03b9' \u2192 \u03b1\nx : \u03b1\nval\u271d : \u03b9'\n\u22a2 (fun x i => Option.getD (idxFun (horizontal l v) i) x) x (Sum.inr val\u271d) =\n Sum.elim ((fun x i => Option.getD (idxFun l i) x) x) v (Sum.inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b9' : Type u_3\nl : Line \u03b1 \u03b9\nl' : Line \u03b1 \u03b9'\nx : \u03b1\n\u22a2 (fun x i => Option.getD (idxFun (prod l l') i) x) x =\n Sum.elim ((fun x i => Option.getD (idxFun l i) x) x) ((fun x i => Option.getD (idxFun l' i) x) x)\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b9' : Type u_3\nl : Line \u03b1 \u03b9\nl' : Line \u03b1 \u03b9'\nx : \u03b1\ni : \u03b9 \u2295 \u03b9'\n\u22a2 (fun x i => Option.getD (idxFun (prod l l') i) x) x i =\n Sum.elim ((fun x i => Option.getD (idxFun l i) x) x) ((fun x i => Option.getD (idxFun l' i) x) x) i\n[PROOFSTEP]\ncases i\n[GOAL]\ncase h.inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b9' : Type u_3\nl : Line \u03b1 \u03b9\nl' : Line \u03b1 \u03b9'\nx : \u03b1\nval\u271d : \u03b9\n\u22a2 (fun x i => Option.getD (idxFun (prod l l') i) x) x (Sum.inl val\u271d) =\n Sum.elim ((fun x i => Option.getD (idxFun l i) x) x) ((fun x i => Option.getD (idxFun l' i) x) x) (Sum.inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b9' : Type u_3\nl : Line \u03b1 \u03b9\nl' : Line \u03b1 \u03b9'\nx : \u03b1\nval\u271d : \u03b9'\n\u22a2 (fun x i => Option.getD (idxFun (prod l l') i) x) x (Sum.inr val\u271d) =\n Sum.elim ((fun x i => Option.getD (idxFun l i) x) x) ((fun x i => Option.getD (idxFun l' i) x) x) (Sum.inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Nonempty \u03b9\nx : \u03b1\n\u22a2 (fun x i => Option.getD (idxFun (diagonal \u03b1 \u03b9) i) x) x = fun x_1 => x\n[PROOFSTEP]\nsimp_rw [Line.diagonal, Option.getD_none]\n[GOAL]\n\u03b1 \u03b1' : Type u\ne : \u03b1 \u2243 \u03b1'\n\u03ba : Type (max v u)\nx\u271d\u00b9 : Finite \u03ba\n\u03b9 : Type\nx\u271d : Fintype \u03b9\nh : \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\nC : (\u03b9 \u2192 \u03b1') \u2192 \u03ba\nl : Line \u03b1 \u03b9\nc : \u03ba\nlc : \u2200 (x : \u03b1), (fun v => C (\u2191e \u2218 v)) ((fun x i => Option.getD (idxFun l i) x) x) = c\nx : \u03b1\n\u22a2 C ((fun x i => Option.getD (idxFun (map (\u2191e) l) i) x) (\u2191e x)) = c\n[PROOFSTEP]\nrw [\u2190 lc x, Line.map_apply]\n[GOAL]\n\u22a2 \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 PEmpty) \u2192 \u03ba), \u2203 l, IsMono C l\n[PROOFSTEP]\nintro \u03ba _\n[GOAL]\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\n\u22a2 \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 PEmpty) \u2192 \u03ba), \u2203 l, IsMono C l\n[PROOFSTEP]\nby_cases h : Nonempty \u03ba\n[GOAL]\ncase pos\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nh : Nonempty \u03ba\n\u22a2 \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 PEmpty) \u2192 \u03ba), \u2203 l, IsMono C l\n[PROOFSTEP]\nrefine' \u27e8Unit, inferInstance, fun C => \u27e8default, Classical.arbitrary _, PEmpty.rec\u27e9\u27e9\n[GOAL]\ncase neg\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nh : \u00acNonempty \u03ba\n\u22a2 \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 PEmpty) \u2192 \u03ba), \u2203 l, IsMono C l\n[PROOFSTEP]\nexact \u27e8Empty, inferInstance, fun C => (h \u27e8C (Empty.rec)\u27e9).elim\u27e9\n[GOAL]\n\u22a2 \u2200 {\u03b1 : Type u} [inst : Fintype \u03b1],\n (\u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l) \u2192\n \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n[PROOFSTEP]\nintro \u03b1 _ ih\u03b1 \u03ba _\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\n\u22a2 \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n[PROOFSTEP]\ncases nonempty_fintype \u03ba\n[GOAL]\ncase intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\n\u22a2 \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n[PROOFSTEP]\nby_cases h : Nonempty \u03b1\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\n\u22a2 \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\ncase neg\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : \u00acNonempty \u03b1\n\u22a2 \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n[PROOFSTEP]\ncase neg =>\n refine' \u27e8Unit, inferInstance, fun C => \u27e8diagonal _ Unit, C fun _ => none, ?_\u27e9\u27e9\n rintro (_ | \u27e8a\u27e9)\n \u00b7 rfl\n \u00b7 exact (h \u27e8a\u27e9).elim\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : \u00acNonempty \u03b1\n\u22a2 \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n[PROOFSTEP]\ncase neg =>\n refine' \u27e8Unit, inferInstance, fun C => \u27e8diagonal _ Unit, C fun _ => none, ?_\u27e9\u27e9\n rintro (_ | \u27e8a\u27e9)\n \u00b7 rfl\n \u00b7 exact (h \u27e8a\u27e9).elim\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : \u00acNonempty \u03b1\n\u22a2 \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n[PROOFSTEP]\nrefine' \u27e8Unit, inferInstance, fun C => \u27e8diagonal _ Unit, C fun _ => none, ?_\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : \u00acNonempty \u03b1\nC : (Unit \u2192 Option \u03b1) \u2192 \u03ba\n\u22a2 \u2200 (x : Option \u03b1), C ((fun x i => Option.getD (idxFun (diagonal (Option \u03b1) Unit) i) x) x) = C fun x => none\n[PROOFSTEP]\nrintro (_ | \u27e8a\u27e9)\n[GOAL]\ncase none\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : \u00acNonempty \u03b1\nC : (Unit \u2192 Option \u03b1) \u2192 \u03ba\n\u22a2 C ((fun x i => Option.getD (idxFun (diagonal (Option \u03b1) Unit) i) x) none) = C fun x => none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase some\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : \u00acNonempty \u03b1\nC : (Unit \u2192 Option \u03b1) \u2192 \u03ba\na : \u03b1\n\u22a2 C ((fun x i => Option.getD (idxFun (diagonal (Option \u03b1) Unit) i) x) (some a)) = C fun x => none\n[PROOFSTEP]\nexact (h \u27e8a\u27e9).elim\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\n\u22a2 \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n[PROOFSTEP]\nsuffices key :\n \u2200 r : \u2115,\n \u2203 (\u03b9 : Type) (_ : Fintype \u03b9),\n \u2200 C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba, (\u2203 s : ColorFocused C, Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nkey : \u2200 (r : \u2115), \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u22a2 \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n[PROOFSTEP]\nobtain \u27e8\u03b9, _inst, h\u03b9\u27e9 := key (Fintype.card \u03ba + 1)\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nkey : \u2200 (r : \u2115), \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u03b9 : Type\n_inst : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = Fintype.card \u03ba + 1) \u2228 \u2203 l, IsMono C l\n\u22a2 \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n[PROOFSTEP]\nrefine' \u27e8\u03b9, _inst, fun C => (h\u03b9 C).resolve_left _\u27e9\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nkey : \u2200 (r : \u2115), \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u03b9 : Type\n_inst : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = Fintype.card \u03ba + 1) \u2228 \u2203 l, IsMono C l\nC : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\n\u22a2 \u00ac\u2203 s, \u2191Multiset.card s.lines = Fintype.card \u03ba + 1\n[PROOFSTEP]\nrintro \u27e8s, sr\u27e9\n[GOAL]\ncase pos.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nkey : \u2200 (r : \u2115), \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u03b9 : Type\n_inst : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = Fintype.card \u03ba + 1) \u2228 \u2203 l, IsMono C l\nC : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\ns : ColorFocused C\nsr : \u2191Multiset.card s.lines = Fintype.card \u03ba + 1\n\u22a2 False\n[PROOFSTEP]\napply Nat.not_succ_le_self (Fintype.card \u03ba)\n[GOAL]\ncase pos.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nkey : \u2200 (r : \u2115), \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u03b9 : Type\n_inst : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = Fintype.card \u03ba + 1) \u2228 \u2203 l, IsMono C l\nC : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\ns : ColorFocused C\nsr : \u2191Multiset.card s.lines = Fintype.card \u03ba + 1\n\u22a2 Nat.succ (Fintype.card \u03ba) \u2264 Fintype.card \u03ba\n[PROOFSTEP]\nrw [\u2190 Nat.add_one, \u2190 sr, \u2190 Multiset.card_map, \u2190 Finset.card_mk]\n[GOAL]\ncase pos.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nkey : \u2200 (r : \u2115), \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u03b9 : Type\n_inst : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = Fintype.card \u03ba + 1) \u2228 \u2203 l, IsMono C l\nC : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\ns : ColorFocused C\nsr : \u2191Multiset.card s.lines = Fintype.card \u03ba + 1\n\u22a2 Finset.card { val := Multiset.map ?pos.intro.intro.intro.f\u271d s.lines, nodup := ?m.16664 } \u2264 Fintype.card \u03ba\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nkey : \u2200 (r : \u2115), \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u03b9 : Type\n_inst : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = Fintype.card \u03ba + 1) \u2228 \u2203 l, IsMono C l\nC : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\ns : ColorFocused C\nsr : \u2191Multiset.card s.lines = Fintype.card \u03ba + 1\n\u22a2 Type ?u.16661\ncase pos.intro.intro.intro.f\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nkey : \u2200 (r : \u2115), \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u03b9 : Type\n_inst : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = Fintype.card \u03ba + 1) \u2228 \u2203 l, IsMono C l\nC : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\ns : ColorFocused C\nsr : \u2191Multiset.card s.lines = Fintype.card \u03ba + 1\n\u22a2 AlmostMono C \u2192 ?m.16662\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nkey : \u2200 (r : \u2115), \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u03b9 : Type\n_inst : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = Fintype.card \u03ba + 1) \u2228 \u2203 l, IsMono C l\nC : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\ns : ColorFocused C\nsr : \u2191Multiset.card s.lines = Fintype.card \u03ba + 1\n\u22a2 Multiset.Nodup (Multiset.map ?pos.intro.intro.intro.f\u271d s.lines)\ncase pos.intro.intro.intro.f\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nkey : \u2200 (r : \u2115), \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u03b9 : Type\n_inst : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = Fintype.card \u03ba + 1) \u2228 \u2203 l, IsMono C l\nC : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\ns : ColorFocused C\nsr : \u2191Multiset.card s.lines = Fintype.card \u03ba + 1\n\u22a2 AlmostMono C \u2192 ?m.16662\n[PROOFSTEP]\nexact\n Finset.card_le_univ\n \u27e8_, s.distinct_colors\u27e9\n -- We now prove the key claim, by induction on `r`.\n[GOAL]\ncase key\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\n\u22a2 \u2200 (r : \u2115), \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n[PROOFSTEP]\nintro r\n[GOAL]\ncase key\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u22a2 \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n[PROOFSTEP]\ninduction' r with r ihr\n[GOAL]\ncase key.zero\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\n\u22a2 \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = Nat.zero) \u2228 \u2203 l, IsMono C l\n[PROOFSTEP]\nexact\n \u27e8Empty, inferInstance, fun C => Or.inl \u27e8default, Multiset.card_zero\u27e9\u27e9\n -- Supposing the key claim holds for `r`, we need to show it for `r+1`. First pick a high\n -- enough dimension `\u03b9` for `r`.\n[GOAL]\ncase key.succ\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\nihr : \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u22a2 \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = Nat.succ r) \u2228 \u2203 l, IsMono C l\n[PROOFSTEP]\nobtain \u27e8\u03b9, _inst, h\u03b9\u27e9 := ihr\n[GOAL]\ncase key.succ.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\nih\u03b1 : \u2200 (\u03ba : Type (max v u)) [inst : Finite \u03ba], \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u22a2 \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = Nat.succ r) \u2228 \u2203 l, IsMono C l\n[PROOFSTEP]\nspecialize ih\u03b1 ((\u03b9 \u2192 Option \u03b1) \u2192 \u03ba)\n[GOAL]\ncase key.succ.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\nih\u03b1 : \u2203 \u03b9_1 x, \u2200 (C : (\u03b9_1 \u2192 \u03b1) \u2192 (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u22a2 \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = Nat.succ r) \u2228 \u2203 l, IsMono C l\n[PROOFSTEP]\nobtain \u27e8\u03b9', _inst, h\u03b9'\u27e9 := ih\u03b1\n[GOAL]\ncase key.succ.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u03b9' : Type\n_inst : Fintype \u03b9'\nh\u03b9' : \u2200 (C : (\u03b9' \u2192 \u03b1) \u2192 (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u22a2 \u2203 \u03b9 x, \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = Nat.succ r) \u2228 \u2203 l, IsMono C l\n[PROOFSTEP]\nrefine' \u27e8Sum \u03b9 \u03b9', inferInstance, _\u27e9\n[GOAL]\ncase key.succ.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u03b9' : Type\n_inst : Fintype \u03b9'\nh\u03b9' : \u2200 (C : (\u03b9' \u2192 \u03b1) \u2192 (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\n\u22a2 \u2200 (C : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = Nat.succ r) \u2228 \u2203 l, IsMono C l\n[PROOFSTEP]\nintro C\n[GOAL]\ncase key.succ.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u03b9' : Type\n_inst : Fintype \u03b9'\nh\u03b9' : \u2200 (C : (\u03b9' \u2192 \u03b1) \u2192 (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), \u2203 l, IsMono C l\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\n\u22a2 (\u2203 s, \u2191Multiset.card s.lines = Nat.succ r) \u2228 \u2203 l, IsMono C l\n[PROOFSTEP]\nspecialize\n h\u03b9' fun v' v =>\n C\n (Sum.elim v (some \u2218 v'))\n -- By choice of `\u03b9'` this coloring has a monochromatic line `l'` with color class `C'`, where\n -- `C'` is a `\u03ba`-coloring of `\u03b9 \u2192 \u03b1`.\n[GOAL]\ncase key.succ.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nh\u03b9' : \u2203 l, IsMono (fun v' v => C (Sum.elim v (some \u2218 v'))) l\n\u22a2 (\u2203 s, \u2191Multiset.card s.lines = Nat.succ r) \u2228 \u2203 l, IsMono C l\n[PROOFSTEP]\nobtain \u27e8l', C', hl'\u27e9 := h\u03b9'\n[GOAL]\ncase key.succ.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\n\u22a2 (\u2203 s, \u2191Multiset.card s.lines = Nat.succ r) \u2228 \u2203 l, IsMono C l\n[PROOFSTEP]\nhave mono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l :=\n by\n rintro \u27e8l, c, hl\u27e9\n refine' \u27e8l.horizontal (some \u2218 l' (Classical.arbitrary \u03b1)), c, fun x => _\u27e9\n rw [Line.horizontal_apply, \u2190 hl, \u2190 hl']\n -- By choice of `\u03b9`, `C'` either has `r` color-focused lines or a monochromatic line.\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\n\u22a2 (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\n[PROOFSTEP]\nrintro \u27e8l, c, hl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nl : Line (Option \u03b1) \u03b9\nc : \u03ba\nhl : \u2200 (x : Option \u03b1), C' ((fun x i => Option.getD (idxFun l i) x) x) = c\n\u22a2 \u2203 l, IsMono C l\n[PROOFSTEP]\nrefine' \u27e8l.horizontal (some \u2218 l' (Classical.arbitrary \u03b1)), c, fun x => _\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nl : Line (Option \u03b1) \u03b9\nc : \u03ba\nhl : \u2200 (x : Option \u03b1), C' ((fun x i => Option.getD (idxFun l i) x) x) = c\nx : Option \u03b1\n\u22a2 C\n ((fun x i =>\n Option.getD\n (idxFun (horizontal l (some \u2218 (fun x i => Option.getD (idxFun l' i) x) (Classical.arbitrary \u03b1))) i) x)\n x) =\n c\n[PROOFSTEP]\nrw [Line.horizontal_apply, \u2190 hl, \u2190 hl']\n -- By choice of `\u03b9`, `C'` either has `r` color-focused lines or a monochromatic line.\n[GOAL]\ncase key.succ.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba), (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C l\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\n\u22a2 (\u2203 s, \u2191Multiset.card s.lines = Nat.succ r) \u2228 \u2203 l, IsMono C l\n[PROOFSTEP]\nspecialize h\u03b9 C'\n[GOAL]\ncase key.succ.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\nh\u03b9 : (\u2203 s, \u2191Multiset.card s.lines = r) \u2228 \u2203 l, IsMono C' l\n\u22a2 (\u2203 s, \u2191Multiset.card s.lines = Nat.succ r) \u2228 \u2203 l, IsMono C l\n[PROOFSTEP]\nrcases h\u03b9 with (\u27e8s, sr\u27e9 | h)\n[GOAL]\ncase key.succ.intro.intro.intro.intro.intro.intro.inl.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\ns : ColorFocused C'\nsr : \u2191Multiset.card s.lines = r\n\u22a2 (\u2203 s, \u2191Multiset.card s.lines = Nat.succ r) \u2228 \u2203 l, IsMono C l\ncase key.succ.intro.intro.intro.intro.intro.intro.inr\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh\u271d : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\nh : \u2203 l, IsMono C' l\n\u22a2 (\u2203 s, \u2191Multiset.card s.lines = Nat.succ r) \u2228 \u2203 l, IsMono C l\n[PROOFSTEP]\non_goal 2 =>\n exact\n Or.inr\n (mono_of_mono h)\n -- Here we assume `C'` has `r` color focused lines. We split into cases depending on whether\n -- one of these `r` lines has the same color as the focus point.\n[GOAL]\ncase key.succ.intro.intro.intro.intro.intro.intro.inl.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\ns : ColorFocused C'\nsr : \u2191Multiset.card s.lines = r\n\u22a2 (\u2203 s, \u2191Multiset.card s.lines = Nat.succ r) \u2228 \u2203 l, IsMono C l\ncase key.succ.intro.intro.intro.intro.intro.intro.inr\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh\u271d : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\nh : \u2203 l, IsMono C' l\n\u22a2 (\u2203 s, \u2191Multiset.card s.lines = Nat.succ r) \u2228 \u2203 l, IsMono C l\n[PROOFSTEP]\non_goal 2 =>\n exact\n Or.inr\n (mono_of_mono h)\n -- Here we assume `C'` has `r` color focused lines. We split into cases depending on whether\n -- one of these `r` lines has the same color as the focus point.\n[GOAL]\ncase key.succ.intro.intro.intro.intro.intro.intro.inr\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh\u271d : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\nh : \u2203 l, IsMono C' l\n\u22a2 (\u2203 s, \u2191Multiset.card s.lines = Nat.succ r) \u2228 \u2203 l, IsMono C l\n[PROOFSTEP]\nexact\n Or.inr\n (mono_of_mono h)\n -- Here we assume `C'` has `r` color focused lines. We split into cases depending on whether\n -- one of these `r` lines has the same color as the focus point.\n[GOAL]\ncase key.succ.intro.intro.intro.intro.intro.intro.inl.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\ns : ColorFocused C'\nsr : \u2191Multiset.card s.lines = r\n\u22a2 (\u2203 s, \u2191Multiset.card s.lines = Nat.succ r) \u2228 \u2203 l, IsMono C l\n[PROOFSTEP]\nby_cases h : \u2203 p \u2208 s.lines, (p : AlmostMono _).color = C' s.focus\n[GOAL]\ncase pos\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh\u271d : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\ns : ColorFocused C'\nsr : \u2191Multiset.card s.lines = r\nh : \u2203 p, p \u2208 s.lines \u2227 p.color = C' s.focus\n\u22a2 (\u2203 s, \u2191Multiset.card s.lines = Nat.succ r) \u2228 \u2203 l, IsMono C l\n[PROOFSTEP]\nobtain \u27e8p, p_mem, hp\u27e9 := h\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\ns : ColorFocused C'\nsr : \u2191Multiset.card s.lines = r\np : AlmostMono C'\np_mem : p \u2208 s.lines\nhp : p.color = C' s.focus\n\u22a2 (\u2203 s, \u2191Multiset.card s.lines = Nat.succ r) \u2228 \u2203 l, IsMono C l\n[PROOFSTEP]\nrefine' Or.inr (mono_of_mono \u27e8p.line, p.color, _\u27e9)\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\ns : ColorFocused C'\nsr : \u2191Multiset.card s.lines = r\np : AlmostMono C'\np_mem : p \u2208 s.lines\nhp : p.color = C' s.focus\n\u22a2 \u2200 (x : Option \u03b1), C' ((fun x i => Option.getD (idxFun p.line i) x) x) = p.color\n[PROOFSTEP]\nrintro (_ | _)\n[GOAL]\ncase pos.intro.intro.none\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\ns : ColorFocused C'\nsr : \u2191Multiset.card s.lines = r\np : AlmostMono C'\np_mem : p \u2208 s.lines\nhp : p.color = C' s.focus\n\u22a2 C' ((fun x i => Option.getD (idxFun p.line i) x) none) = p.color\ncase pos.intro.intro.some\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d\u00b9 : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\ns : ColorFocused C'\nsr : \u2191Multiset.card s.lines = r\np : AlmostMono C'\np_mem : p \u2208 s.lines\nhp : p.color = C' s.focus\nval\u271d : \u03b1\n\u22a2 C' ((fun x i => Option.getD (idxFun p.line i) x) (some val\u271d)) = p.color\n[PROOFSTEP]\nrw [hp, s.is_focused p p_mem]\n[GOAL]\ncase pos.intro.intro.some\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d\u00b9 : Fintype \u03ba\nh : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\ns : ColorFocused C'\nsr : \u2191Multiset.card s.lines = r\np : AlmostMono C'\np_mem : p \u2208 s.lines\nhp : p.color = C' s.focus\nval\u271d : \u03b1\n\u22a2 C' ((fun x i => Option.getD (idxFun p.line i) x) (some val\u271d)) = p.color\n[PROOFSTEP]\napply p.has_color\n[GOAL]\ncase neg\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh\u271d : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\ns : ColorFocused C'\nsr : \u2191Multiset.card s.lines = r\nh : \u00ac\u2203 p, p \u2208 s.lines \u2227 p.color = C' s.focus\n\u22a2 (\u2203 s, \u2191Multiset.card s.lines = Nat.succ r) \u2228 \u2203 l, IsMono C l\n[PROOFSTEP]\nrefine'\n Or.inl\n \u27e8\u27e8(s.lines.map _).cons \u27e8(l'.map some).vertical s.focus, C' s.focus, fun x => _\u27e9,\n Sum.elim s.focus (l'.map some none), _, _\u27e9,\n _\u27e9\n -- Porting note: Needed to reorder the following two goals\n -- The product lines are almost monochromatic.\n[GOAL]\ncase neg.refine'_1\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh\u271d : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\ns : ColorFocused C'\nsr : \u2191Multiset.card s.lines = r\nh : \u00ac\u2203 p, p \u2208 s.lines \u2227 p.color = C' s.focus\n\u22a2 AlmostMono C' \u2192 AlmostMono C\n[PROOFSTEP]\nrefine' fun p => \u27e8p.line.prod (l'.map some), p.color, fun x => _\u27e9\n[GOAL]\ncase neg.refine'_1\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh\u271d : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\ns : ColorFocused C'\nsr : \u2191Multiset.card s.lines = r\nh : \u00ac\u2203 p, p \u2208 s.lines \u2227 p.color = C' s.focus\np : AlmostMono C'\nx : \u03b1\n\u22a2 C ((fun x i => Option.getD (idxFun (prod p.line (map some l')) i) x) (some x)) = p.color\n[PROOFSTEP]\nrw [Line.prod_apply, Line.map_apply, \u2190 p.has_color, \u2190 congr_fun (hl' x)]\n -- The vertical line is almost monochromatic.\n[GOAL]\ncase neg.refine'_2\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh\u271d : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\ns : ColorFocused C'\nsr : \u2191Multiset.card s.lines = r\nh : \u00ac\u2203 p, p \u2208 s.lines \u2227 p.color = C' s.focus\nx : \u03b1\n\u22a2 C ((fun x i => Option.getD (idxFun (vertical s.focus (map some l')) i) x) (some x)) = C' s.focus\n[PROOFSTEP]\nrw [vertical_apply, \u2190 congr_fun (hl' x), Line.map_apply]\n -- Our `r+1` lines have the same endpoint.\n[GOAL]\ncase neg.refine'_3\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh\u271d : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\ns : ColorFocused C'\nsr : \u2191Multiset.card s.lines = r\nh : \u00ac\u2203 p, p \u2208 s.lines \u2227 p.color = C' s.focus\n\u22a2 \u2200 (p : AlmostMono C),\n p \u2208\n { line := vertical s.focus (map some l'), color := C' s.focus,\n has_color :=\n (_ :\n \u2200 (x : \u03b1),\n C ((fun x i => Option.getD (idxFun (vertical s.focus (map some l')) i) x) (some x)) =\n C' s.focus) } ::\u2098\n Multiset.map\n (fun p =>\n { line := prod p.line (map some l'), color := p.color,\n has_color :=\n (_ :\n \u2200 (x : \u03b1),\n C ((fun x i => Option.getD (idxFun (prod p.line (map some l')) i) x) (some x)) = p.color) })\n s.lines \u2192\n (fun x i => Option.getD (idxFun p.line i) x) none =\n Sum.elim s.focus ((fun x i => Option.getD (idxFun (map some l') i) x) none)\n[PROOFSTEP]\nsimp_rw [Multiset.mem_cons, Multiset.mem_map]\n[GOAL]\ncase neg.refine'_3\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh\u271d : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\ns : ColorFocused C'\nsr : \u2191Multiset.card s.lines = r\nh : \u00ac\u2203 p, p \u2208 s.lines \u2227 p.color = C' s.focus\n\u22a2 \u2200 (p : AlmostMono C),\n (p =\n { line := vertical s.focus (map some l'), color := C' s.focus,\n has_color :=\n (_ :\n \u2200 (x : \u03b1),\n C ((fun x i => Option.getD (idxFun (vertical s.focus (map some l')) i) x) (some x)) = C' s.focus) } \u2228\n \u2203 a,\n a \u2208 s.lines \u2227\n { line := prod a.line (map some l'), color := a.color,\n has_color :=\n (_ :\n \u2200 (x : \u03b1),\n C ((fun x i => Option.getD (idxFun (prod a.line (map some l')) i) x) (some x)) = a.color) } =\n p) \u2192\n (fun i => Option.getD (idxFun p.line i) none) =\n Sum.elim s.focus fun i => Option.getD (idxFun (map some l') i) none\n[PROOFSTEP]\nrintro _ (rfl | \u27e8q, hq, rfl\u27e9)\n[GOAL]\ncase neg.refine'_3.inl\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh\u271d : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\ns : ColorFocused C'\nsr : \u2191Multiset.card s.lines = r\nh : \u00ac\u2203 p, p \u2208 s.lines \u2227 p.color = C' s.focus\n\u22a2 (fun i =>\n Option.getD\n (idxFun\n { line := vertical s.focus (map some l'), color := C' s.focus,\n has_color :=\n (_ :\n \u2200 (x : \u03b1),\n C ((fun x i => Option.getD (idxFun (vertical s.focus (map some l')) i) x) (some x)) =\n C' s.focus) }.line\n i)\n none) =\n Sum.elim s.focus fun i => Option.getD (idxFun (map some l') i) none\n[PROOFSTEP]\nsimp only [vertical_apply]\n[GOAL]\ncase neg.refine'_3.inr.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh\u271d : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\ns : ColorFocused C'\nsr : \u2191Multiset.card s.lines = r\nh : \u00ac\u2203 p, p \u2208 s.lines \u2227 p.color = C' s.focus\nq : AlmostMono C'\nhq : q \u2208 s.lines\n\u22a2 (fun i =>\n Option.getD\n (idxFun\n { line := prod q.line (map some l'), color := q.color,\n has_color :=\n (_ :\n \u2200 (x : \u03b1),\n C ((fun x i => Option.getD (idxFun (prod q.line (map some l')) i) x) (some x)) = q.color) }.line\n i)\n none) =\n Sum.elim s.focus fun i => Option.getD (idxFun (map some l') i) none\n[PROOFSTEP]\nsimp only [prod_apply, s.is_focused q hq]\n -- Our `r+1` lines have distinct colors (this is why we needed to split into cases above).\n[GOAL]\ncase neg.refine'_4\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh\u271d : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\ns : ColorFocused C'\nsr : \u2191Multiset.card s.lines = r\nh : \u00ac\u2203 p, p \u2208 s.lines \u2227 p.color = C' s.focus\n\u22a2 Multiset.Nodup\n (Multiset.map AlmostMono.color\n ({ line := vertical s.focus (map some l'), color := C' s.focus,\n has_color :=\n (_ :\n \u2200 (x : \u03b1),\n C ((fun x i => Option.getD (idxFun (vertical s.focus (map some l')) i) x) (some x)) = C' s.focus) } ::\u2098\n Multiset.map\n (fun p =>\n { line := prod p.line (map some l'), color := p.color,\n has_color :=\n (_ :\n \u2200 (x : \u03b1),\n C ((fun x i => Option.getD (idxFun (prod p.line (map some l')) i) x) (some x)) = p.color) })\n s.lines))\n[PROOFSTEP]\nrw [Multiset.map_cons, Multiset.map_map, Multiset.nodup_cons, Multiset.mem_map]\n[GOAL]\ncase neg.refine'_4\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh\u271d : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\ns : ColorFocused C'\nsr : \u2191Multiset.card s.lines = r\nh : \u00ac\u2203 p, p \u2208 s.lines \u2227 p.color = C' s.focus\n\u22a2 (\u00ac\u2203 a,\n a \u2208 s.lines \u2227\n (AlmostMono.color \u2218 fun p =>\n { line := prod p.line (map some l'), color := p.color,\n has_color :=\n (_ :\n \u2200 (x : \u03b1),\n C ((fun x i => Option.getD (idxFun (prod p.line (map some l')) i) x) (some x)) = p.color) })\n a =\n { line := vertical s.focus (map some l'), color := C' s.focus,\n has_color :=\n (_ :\n \u2200 (x : \u03b1),\n C ((fun x i => Option.getD (idxFun (vertical s.focus (map some l')) i) x) (some x)) =\n C' s.focus) }.color) \u2227\n Multiset.Nodup\n (Multiset.map\n (AlmostMono.color \u2218 fun p =>\n { line := prod p.line (map some l'), color := p.color,\n has_color :=\n (_ :\n \u2200 (x : \u03b1), C ((fun x i => Option.getD (idxFun (prod p.line (map some l')) i) x) (some x)) = p.color) })\n s.lines)\n[PROOFSTEP]\nexact\n \u27e8fun \u27e8q, hq, he\u27e9 => h \u27e8q, hq, he\u27e9, s.distinct_colors\u27e9\n -- Finally, we really do have `r+1` lines!\n[GOAL]\ncase neg.refine'_5\n\u03b1 : Type u\ninst\u271d\u00b9 : Fintype \u03b1\n\u03ba : Type (max v u)\ninst\u271d : Finite \u03ba\nval\u271d : Fintype \u03ba\nh\u271d : Nonempty \u03b1\nr : \u2115\n\u03b9 : Type\n_inst\u271d : Fintype \u03b9\n\u03b9' : Type\n_inst : Fintype \u03b9'\nC : (\u03b9 \u2295 \u03b9' \u2192 Option \u03b1) \u2192 \u03ba\nl' : Line \u03b1 \u03b9'\nC' : (\u03b9 \u2192 Option \u03b1) \u2192 \u03ba\nhl' : \u2200 (x : \u03b1), (fun v' v => C (Sum.elim v (some \u2218 v'))) ((fun x i => Option.getD (idxFun l' i) x) x) = C'\nmono_of_mono : (\u2203 l, IsMono C' l) \u2192 \u2203 l, IsMono C l\ns : ColorFocused C'\nsr : \u2191Multiset.card s.lines = r\nh : \u00ac\u2203 p, p \u2208 s.lines \u2227 p.color = C' s.focus\n\u22a2 \u2191Multiset.card\n {\n lines :=\n { line := vertical s.focus (map some l'), color := C' s.focus,\n has_color :=\n (_ :\n \u2200 (x : \u03b1),\n C ((fun x i => Option.getD (idxFun (vertical s.focus (map some l')) i) x) (some x)) =\n C' s.focus) } ::\u2098\n Multiset.map\n (fun p =>\n { line := prod p.line (map some l'), color := p.color,\n has_color :=\n (_ :\n \u2200 (x : \u03b1),\n C ((fun x i => Option.getD (idxFun (prod p.line (map some l')) i) x) (some x)) = p.color) })\n s.lines,\n focus := Sum.elim s.focus ((fun x i => Option.getD (idxFun (map some l') i) x) none),\n is_focused :=\n (_ :\n \u2200 (p : AlmostMono C),\n p \u2208\n { line := vertical s.focus (map some l'), color := C' s.focus,\n has_color :=\n (_ :\n \u2200 (x : \u03b1),\n C ((fun x i => Option.getD (idxFun (vertical s.focus (map some l')) i) x) (some x)) =\n C' s.focus) } ::\u2098\n Multiset.map\n (fun p =>\n { line := prod p.line (map some l'), color := p.color,\n has_color :=\n (_ :\n \u2200 (x : \u03b1),\n C ((fun x i => Option.getD (idxFun (prod p.line (map some l')) i) x) (some x)) =\n p.color) })\n s.lines \u2192\n (fun x i => Option.getD (idxFun p.line i) x) none =\n Sum.elim s.focus ((fun x i => Option.getD (idxFun (map some l') i) x) none)),\n distinct_colors :=\n (_ :\n Multiset.Nodup\n (Multiset.map AlmostMono.color\n ({ line := vertical s.focus (map some l'), color := C' s.focus,\n has_color :=\n (_ :\n \u2200 (x : \u03b1),\n C ((fun x i => Option.getD (idxFun (vertical s.focus (map some l')) i) x) (some x)) =\n C' s.focus) } ::\u2098\n Multiset.map\n (fun p =>\n { line := prod p.line (map some l'), color := p.color,\n has_color :=\n (_ :\n \u2200 (x : \u03b1),\n C ((fun x i => Option.getD (idxFun (prod p.line (map some l')) i) x) (some x)) =\n p.color) })\n s.lines))) }.lines =\n Nat.succ r\n[PROOFSTEP]\nrw [Multiset.card_cons, Multiset.card_map, sr]\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : Finite \u03b1\n\u03ba : Type v\ninst\u271d : Finite \u03ba\n\u03b9 : Type\n\u03b9fin : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 \u03b1) \u2192 ULift \u03ba), \u2203 l, IsMono C l\nC : (\u03b9 \u2192 \u03b1) \u2192 \u03ba\nl : Line \u03b1 \u03b9\nc : ULift \u03ba\nhc : \u2200 (x : \u03b1), (ULift.up \u2218 C) ((fun x i => Option.getD (idxFun l i) x) x) = c\nx : \u03b1\n\u22a2 C ((fun x i => Option.getD (idxFun l i) x) x) = c.down\n[PROOFSTEP]\nrw [\u2190 hc x, Function.comp_apply]\n[GOAL]\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u22a2 \u2203 a, a > 0 \u2227 \u2203 b c, \u2200 (s : M), s \u2208 S \u2192 C (a \u2022 s + b) = c\n[PROOFSTEP]\nobtain \u27e8\u03b9, _inst, h\u03b9\u27e9 := Line.exists_mono_in_high_dimension S \u03ba\n[GOAL]\ncase intro.intro\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nh\u03b9 : \u2200 (C : (\u03b9 \u2192 { x // x \u2208 S }) \u2192 \u03ba), \u2203 l, Line.IsMono C l\n\u22a2 \u2203 a, a > 0 \u2227 \u2203 b c, \u2200 (s : M), s \u2208 S \u2192 C (a \u2022 s + b) = c\n[PROOFSTEP]\nspecialize h\u03b9 fun v => C <| \u2211 i, v i\n[GOAL]\ncase intro.intro\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nh\u03b9 : \u2203 l, Line.IsMono (fun v => C (\u2211 i : \u03b9, \u2191(v i))) l\n\u22a2 \u2203 a, a > 0 \u2227 \u2203 b c, \u2200 (s : M), s \u2208 S \u2192 C (a \u2022 s + b) = c\n[PROOFSTEP]\nobtain \u27e8l, c, hl\u27e9 := h\u03b9\n[GOAL]\ncase intro.intro.intro.intro\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\nhl : \u2200 (x : { x // x \u2208 S }), (fun v => C (\u2211 i : \u03b9, \u2191(v i))) ((fun x i => Option.getD (Line.idxFun l i) x) x) = c\n\u22a2 \u2203 a, a > 0 \u2227 \u2203 b c, \u2200 (s : M), s \u2208 S \u2192 C (a \u2022 s + b) = c\n[PROOFSTEP]\nset s : Finset \u03b9 := Finset.univ.filter (fun i => l.idxFun i = none) with hs\n[GOAL]\ncase intro.intro.intro.intro\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\nhl : \u2200 (x : { x // x \u2208 S }), (fun v => C (\u2211 i : \u03b9, \u2191(v i))) ((fun x i => Option.getD (Line.idxFun l i) x) x) = c\ns : Finset \u03b9 := Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nhs : s = Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\n\u22a2 \u2203 a, a > 0 \u2227 \u2203 b c, \u2200 (s : M), s \u2208 S \u2192 C (a \u2022 s + b) = c\n[PROOFSTEP]\nrefine' \u27e8s.card, Finset.card_pos.mpr \u27e8l.proper.choose, _\u27e9, \u2211 i in s\u1d9c, ((l.idxFun i).map _).getD 0, c, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\nhl : \u2200 (x : { x // x \u2208 S }), (fun v => C (\u2211 i : \u03b9, \u2191(v i))) ((fun x i => Option.getD (Line.idxFun l i) x) x) = c\ns : Finset \u03b9 := Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nhs : s = Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\n\u22a2 Exists.choose (_ : \u2203 i, Line.idxFun l i = none) \u2208 s\n[PROOFSTEP]\nrw [hs, Finset.mem_filter]\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\nhl : \u2200 (x : { x // x \u2208 S }), (fun v => C (\u2211 i : \u03b9, \u2191(v i))) ((fun x i => Option.getD (Line.idxFun l i) x) x) = c\ns : Finset \u03b9 := Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nhs : s = Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\n\u22a2 Exists.choose (_ : \u2203 i, Line.idxFun l i = none) \u2208 Finset.univ \u2227\n Line.idxFun l (Exists.choose (_ : \u2203 i, Line.idxFun l i = none)) = none\n[PROOFSTEP]\nexact \u27e8Finset.mem_univ _, l.proper.choose_spec\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\nhl : \u2200 (x : { x // x \u2208 S }), (fun v => C (\u2211 i : \u03b9, \u2191(v i))) ((fun x i => Option.getD (Line.idxFun l i) x) x) = c\ns : Finset \u03b9 := Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nhs : s = Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\ni : \u03b9\n\u22a2 { x // x \u2208 S } \u2192 M\n[PROOFSTEP]\nexact fun m => m\n[GOAL]\ncase intro.intro.intro.intro.refine'_3\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\nhl : \u2200 (x : { x // x \u2208 S }), (fun v => C (\u2211 i : \u03b9, \u2191(v i))) ((fun x i => Option.getD (Line.idxFun l i) x) x) = c\ns : Finset \u03b9 := Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nhs : s = Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\n\u22a2 \u2200 (s_1 : M),\n s_1 \u2208 S \u2192 C (Finset.card s \u2022 s_1 + \u2211 i in s\u1d9c, Option.getD (Option.map (fun m => \u2191m) (Line.idxFun l i)) 0) = c\n[PROOFSTEP]\nintro x xs\n[GOAL]\ncase intro.intro.intro.intro.refine'_3\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\nhl : \u2200 (x : { x // x \u2208 S }), (fun v => C (\u2211 i : \u03b9, \u2191(v i))) ((fun x i => Option.getD (Line.idxFun l i) x) x) = c\ns : Finset \u03b9 := Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nhs : s = Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nx : M\nxs : x \u2208 S\n\u22a2 C (Finset.card s \u2022 x + \u2211 i in s\u1d9c, Option.getD (Option.map (fun m => \u2191m) (Line.idxFun l i)) 0) = c\n[PROOFSTEP]\nrw [\u2190 hl \u27e8x, xs\u27e9]\n[GOAL]\ncase intro.intro.intro.intro.refine'_3\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\nhl : \u2200 (x : { x // x \u2208 S }), (fun v => C (\u2211 i : \u03b9, \u2191(v i))) ((fun x i => Option.getD (Line.idxFun l i) x) x) = c\ns : Finset \u03b9 := Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nhs : s = Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nx : M\nxs : x \u2208 S\n\u22a2 C (Finset.card s \u2022 x + \u2211 i in s\u1d9c, Option.getD (Option.map (fun m => \u2191m) (Line.idxFun l i)) 0) =\n (fun v => C (\u2211 i : \u03b9, \u2191(v i))) ((fun x i => Option.getD (Line.idxFun l i) x) { val := x, property := xs })\n[PROOFSTEP]\nclear hl\n[GOAL]\ncase intro.intro.intro.intro.refine'_3\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\ns : Finset \u03b9 := Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nhs : s = Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nx : M\nxs : x \u2208 S\n\u22a2 C (Finset.card s \u2022 x + \u2211 i in s\u1d9c, Option.getD (Option.map (fun m => \u2191m) (Line.idxFun l i)) 0) =\n (fun v => C (\u2211 i : \u03b9, \u2191(v i))) ((fun x i => Option.getD (Line.idxFun l i) x) { val := x, property := xs })\n[PROOFSTEP]\ncongr\n[GOAL]\ncase intro.intro.intro.intro.refine'_3.e_a\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\ns : Finset \u03b9 := Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nhs : s = Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nx : M\nxs : x \u2208 S\n\u22a2 Finset.card s \u2022 x + \u2211 i in s\u1d9c, Option.getD (Option.map (fun m => \u2191m) (Line.idxFun l i)) 0 =\n \u2211 i : \u03b9, \u2191((fun x i => Option.getD (Line.idxFun l i) x) { val := x, property := xs } i)\n[PROOFSTEP]\nrw [\u2190 Finset.sum_add_sum_compl s]\n[GOAL]\ncase intro.intro.intro.intro.refine'_3.e_a\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\ns : Finset \u03b9 := Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nhs : s = Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nx : M\nxs : x \u2208 S\n\u22a2 Finset.card s \u2022 x + \u2211 i in s\u1d9c, Option.getD (Option.map (fun m => \u2191m) (Line.idxFun l i)) 0 =\n \u2211 i in s, \u2191((fun x i => Option.getD (Line.idxFun l i) x) { val := x, property := xs } i) +\n \u2211 i in s\u1d9c, \u2191((fun x i => Option.getD (Line.idxFun l i) x) { val := x, property := xs } i)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase intro.intro.intro.intro.refine'_3.e_a.e_a\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\ns : Finset \u03b9 := Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nhs : s = Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nx : M\nxs : x \u2208 S\n\u22a2 Finset.card s \u2022 x = \u2211 i in s, \u2191((fun x i => Option.getD (Line.idxFun l i) x) { val := x, property := xs } i)\n[PROOFSTEP]\nrw [\u2190 Finset.sum_const]\n[GOAL]\ncase intro.intro.intro.intro.refine'_3.e_a.e_a\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\ns : Finset \u03b9 := Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nhs : s = Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nx : M\nxs : x \u2208 S\n\u22a2 \u2211 _x in s, x = \u2211 i in s, \u2191((fun x i => Option.getD (Line.idxFun l i) x) { val := x, property := xs } i)\n[PROOFSTEP]\napply Finset.sum_congr rfl\n[GOAL]\ncase intro.intro.intro.intro.refine'_3.e_a.e_a\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\ns : Finset \u03b9 := Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nhs : s = Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nx : M\nxs : x \u2208 S\n\u22a2 \u2200 (x_1 : \u03b9), x_1 \u2208 s \u2192 x = \u2191((fun x i => Option.getD (Line.idxFun l i) x) { val := x, property := xs } x_1)\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase intro.intro.intro.intro.refine'_3.e_a.e_a\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\ns : Finset \u03b9 := Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nhs : s = Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nx : M\nxs : x \u2208 S\ni : \u03b9\nhi : i \u2208 s\n\u22a2 x = \u2191((fun x i => Option.getD (Line.idxFun l i) x) { val := x, property := xs } i)\n[PROOFSTEP]\nrw [hs, Finset.mem_filter] at hi \n[GOAL]\ncase intro.intro.intro.intro.refine'_3.e_a.e_a\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\ns : Finset \u03b9 := Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nhs : s = Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nx : M\nxs : x \u2208 S\ni : \u03b9\nhi : i \u2208 Finset.univ \u2227 Line.idxFun l i = none\n\u22a2 x = \u2191((fun x i => Option.getD (Line.idxFun l i) x) { val := x, property := xs } i)\n[PROOFSTEP]\nrw [l.apply_none _ _ hi.right, Subtype.coe_mk]\n[GOAL]\ncase intro.intro.intro.intro.refine'_3.e_a.e_a\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\ns : Finset \u03b9 := Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nhs : s = Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nx : M\nxs : x \u2208 S\n\u22a2 \u2211 i in s\u1d9c, Option.getD (Option.map (fun m => \u2191m) (Line.idxFun l i)) 0 =\n \u2211 i in s\u1d9c, \u2191((fun x i => Option.getD (Line.idxFun l i) x) { val := x, property := xs } i)\n[PROOFSTEP]\napply Finset.sum_congr rfl\n[GOAL]\ncase intro.intro.intro.intro.refine'_3.e_a.e_a\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\ns : Finset \u03b9 := Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nhs : s = Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nx : M\nxs : x \u2208 S\n\u22a2 \u2200 (x_1 : \u03b9),\n x_1 \u2208 s\u1d9c \u2192\n Option.getD (Option.map (fun m => \u2191m) (Line.idxFun l x_1)) 0 =\n \u2191((fun x i => Option.getD (Line.idxFun l i) x) { val := x, property := xs } x_1)\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase intro.intro.intro.intro.refine'_3.e_a.e_a\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\ns : Finset \u03b9 := Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nhs : s = Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nx : M\nxs : x \u2208 S\ni : \u03b9\nhi : i \u2208 s\u1d9c\n\u22a2 Option.getD (Option.map (fun m => \u2191m) (Line.idxFun l i)) 0 =\n \u2191((fun x i => Option.getD (Line.idxFun l i) x) { val := x, property := xs } i)\n[PROOFSTEP]\nrw [hs, Finset.compl_filter, Finset.mem_filter] at hi \n[GOAL]\ncase intro.intro.intro.intro.refine'_3.e_a.e_a\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\ns : Finset \u03b9 := Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nhs : s = Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nx : M\nxs : x \u2208 S\ni : \u03b9\nhi : i \u2208 Finset.univ \u2227 \u00acLine.idxFun l i = none\n\u22a2 Option.getD (Option.map (fun m => \u2191m) (Line.idxFun l i)) 0 =\n \u2191((fun x i => Option.getD (Line.idxFun l i) x) { val := x, property := xs } i)\n[PROOFSTEP]\nobtain \u27e8y, hy\u27e9 := Option.ne_none_iff_exists.mp hi.right\n[GOAL]\ncase intro.intro.intro.intro.refine'_3.e_a.e_a.intro\nM : Type u_1\n\u03ba : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\nS : Finset M\ninst\u271d : Finite \u03ba\nC : M \u2192 \u03ba\n\u03b9 : Type\n_inst : Fintype \u03b9\nl : Line { x // x \u2208 S } \u03b9\nc : \u03ba\ns : Finset \u03b9 := Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nhs : s = Finset.filter (fun i => Line.idxFun l i = none) Finset.univ\nx : M\nxs : x \u2208 S\ni : \u03b9\nhi : i \u2208 Finset.univ \u2227 \u00acLine.idxFun l i = none\ny : { x // x \u2208 S }\nhy : some y = Line.idxFun l i\n\u22a2 Option.getD (Option.map (fun m => \u2191m) (Line.idxFun l i)) 0 =\n \u2191((fun x i => Option.getD (Line.idxFun l i) x) { val := x, property := xs } i)\n[PROOFSTEP]\nsimp_rw [\u2190 hy, Option.map_some', Option.getD]\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.HalesJewett", "llama_tokens": 28971, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.4687906266262438, "lm_q2_score": 0.02161533500795898, "lm_q1q2_score": 0.010133066443117275}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX Y : C\nhX : Closed X\nhY : Closed Y\n\u22a2 IsLeftAdjoint (tensorLeft (X \u2297 Y))\n[PROOFSTEP]\nhaveI := hX.isAdj\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX Y : C\nhX : Closed X\nhY : Closed Y\nthis : IsLeftAdjoint (tensorLeft X)\n\u22a2 IsLeftAdjoint (tensorLeft (X \u2297 Y))\n[PROOFSTEP]\nhaveI := hY.isAdj\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX Y : C\nhX : Closed X\nhY : Closed Y\nthis\u271d : IsLeftAdjoint (tensorLeft X)\nthis : IsLeftAdjoint (tensorLeft Y)\n\u22a2 IsLeftAdjoint (tensorLeft (X \u2297 Y))\n[PROOFSTEP]\nexact Adjunction.leftAdjointOfNatIso (MonoidalCategory.tensorLeftTensor _ _).symm\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX x\u271d : C\n\u22a2 Function.LeftInverse (fun a => (\u03bb_ X).hom \u226b a) fun a => (\u03bb_ X).inv \u226b a\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX x\u271d : C\n\u22a2 Function.RightInverse (fun a => (\u03bb_ X).hom \u226b a) fun a => (\u03bb_ X).inv \u226b a\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX'\u271d X\u271d Y\u271d : C\nf : X'\u271d \u27f6 X\u271d\ng : X\u271d \u27f6 (\ud835\udfed C).obj Y\u271d\n\u22a2 \u2191((fun X x =>\n { toFun := fun a => (\u03bb_ X).inv \u226b a, invFun := fun a => (\u03bb_ X).hom \u226b a,\n left_inv := (_ : \u2200 (x_1 : (tensorLeft (\ud835\udfd9_ C)).obj X \u27f6 x), (\u03bb_ X).hom \u226b (\u03bb_ X).inv \u226b x_1 = x_1),\n right_inv := (_ : \u2200 (x_1 : X \u27f6 (\ud835\udfed C).obj x), (\u03bb_ X).inv \u226b (\u03bb_ X).hom \u226b x_1 = x_1) })\n X'\u271d Y\u271d).symm\n (f \u226b g) =\n (tensorLeft (\ud835\udfd9_ C)).map f \u226b\n \u2191((fun X x =>\n { toFun := fun a => (\u03bb_ X).inv \u226b a, invFun := fun a => (\u03bb_ X).hom \u226b a,\n left_inv := (_ : \u2200 (x_1 : (tensorLeft (\ud835\udfd9_ C)).obj X \u27f6 x), (\u03bb_ X).hom \u226b (\u03bb_ X).inv \u226b x_1 = x_1),\n right_inv := (_ : \u2200 (x_1 : X \u27f6 (\ud835\udfed C).obj x), (\u03bb_ X).inv \u226b (\u03bb_ X).hom \u226b x_1 = x_1) })\n X\u271d Y\u271d).symm\n g\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX'\u271d X\u271d Y\u271d : C\nf : X'\u271d \u27f6 X\u271d\ng : X\u271d \u27f6 (\ud835\udfed C).obj Y\u271d\n\u22a2 (\u03bb_ X'\u271d).hom \u226b f \u226b g = (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 f) \u226b (\u03bb_ X\u271d).hom \u226b g\n[PROOFSTEP]\nrw [leftUnitor_naturality_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : MonoidalCategory C\nA B X X' Y Y' Z : C\ninst\u271d : Closed A\n\u22a2 uncurry (\ud835\udfd9 ((ihom A).obj X)) = NatTrans.app (ihom.ev A) X\n[PROOFSTEP]\nrw [uncurry_eq, tensor_id, id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : MonoidalCategory C\nA B X X' Y Y' Z : C\ninst\u271d : Closed A\n\u22a2 curry (\ud835\udfd9 (A \u2297 (\ud835\udfed C).obj X)) = NatTrans.app (ihom.coev A) X\n[PROOFSTEP]\nrw [curry_eq, (ihom A).map_id (A \u2297 _)]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : MonoidalCategory C\nA B X X' Y Y' Z : C\ninst\u271d : Closed A\n\u22a2 NatTrans.app (ihom.coev A) ((\ud835\udfed C).obj X) \u226b \ud835\udfd9 ((ihom A).obj (A \u2297 (\ud835\udfed C).obj X)) = NatTrans.app (ihom.coev A) X\n[PROOFSTEP]\napply comp_id\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : MonoidalCategory C\nA B X\u271d X' Y Y' Z : C\ninst\u271d\u00b9 : Closed A\ninst\u271d : Closed B\nf : B \u27f6 A\nX : C\n\u22a2 uncurry (NatTrans.app (pre f) X) = (f \u2297 \ud835\udfd9 ((ihom A).obj X)) \u226b NatTrans.app (ihom.ev A) X\n[PROOFSTEP]\nrw [uncurry_eq, id_tensor_pre_app_comp_ev]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : MonoidalCategory C\nA\u271d B X X' Y Y' Z : C\ninst\u271d\u00b2 : Closed A\u271d\ninst\u271d\u00b9 : Closed B\nA : C\ninst\u271d : Closed A\n\u22a2 pre (\ud835\udfd9 A) = \ud835\udfd9 (ihom A)\n[PROOFSTEP]\nsimp only [pre, Functor.map_id]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : MonoidalCategory C\nA\u271d B X X' Y Y' Z : C\ninst\u271d\u00b2 : Closed A\u271d\ninst\u271d\u00b9 : Closed B\nA : C\ninst\u271d : Closed A\n\u22a2 \u2191(transferNatTransSelf (ihom.adjunction A) (ihom.adjunction A)) (\ud835\udfd9 ((tensoringLeft C).obj A)) = \ud835\udfd9 (ihom A)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : MonoidalCategory C\nA\u271d B X X' Y Y' Z : C\ninst\u271d\u00b2 : Closed A\u271d\ninst\u271d\u00b9 : Closed B\nA : C\ninst\u271d : Closed A\n\u22a2 \u2191(transferNatTransSelf (ihom.adjunction A) (ihom.adjunction A)) (\ud835\udfd9 (tensorLeft A)) = \ud835\udfd9 (ihom A)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2076 : Category.{v, u} C\ninst\u271d\u2075 : MonoidalCategory C\nA B X X' Y Y' Z : C\ninst\u271d\u2074 : Closed A\ninst\u271d\u00b3 : Closed B\nA\u2081 A\u2082 A\u2083 : C\ninst\u271d\u00b2 : Closed A\u2081\ninst\u271d\u00b9 : Closed A\u2082\ninst\u271d : Closed A\u2083\nf : A\u2081 \u27f6 A\u2082\ng : A\u2082 \u27f6 A\u2083\n\u22a2 pre (f \u226b g) = pre g \u226b pre f\n[PROOFSTEP]\nrw [pre, pre, pre, transferNatTransSelf_comp, (tensoringLeft C).map_comp]\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nA B X\u271d X' Y\u271d Y' Z\u271d : C\ninst\u271d\u00b3 : Closed A\ninst\u271d\u00b2 : Closed B\nW X Y Z : C\ninst\u271d\u00b9 : Closed W\ninst\u271d : Closed X\nf : W \u27f6 X\ng : Y \u27f6 Z\n\u22a2 NatTrans.app (pre f) Y \u226b (ihom W).map g = (ihom X).map g \u226b NatTrans.app (pre f) Z\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nA B X\u271d X' Y Y' Z : C\ninst\u271d\u00b3 : Closed A\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\nh : MonoidalClosed D\nX : C\n\u22a2 IsLeftAdjoint (tensorLeft X)\n[PROOFSTEP]\nhaveI q : Closed (F.obj X) := inferInstance\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nA B X\u271d X' Y Y' Z : C\ninst\u271d\u00b3 : Closed A\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\nh : MonoidalClosed D\nX : C\nq : Closed (F.obj X)\n\u22a2 IsLeftAdjoint (tensorLeft X)\n[PROOFSTEP]\nhaveI : IsLeftAdjoint (tensorLeft (F.obj X)) := q.isAdj\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nA B X\u271d X' Y Y' Z : C\ninst\u271d\u00b3 : Closed A\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\nh : MonoidalClosed D\nX : C\nq : Closed (F.obj X)\nthis : IsLeftAdjoint (tensorLeft (F.obj X))\n\u22a2 IsLeftAdjoint (tensorLeft X)\n[PROOFSTEP]\nhave i := compInvIso (MonoidalFunctor.commTensorLeft F X)\n[GOAL]\nC : Type u\ninst\u271d\u2075 : Category.{v, u} C\ninst\u271d\u2074 : MonoidalCategory C\nA B X\u271d X' Y Y' Z : C\ninst\u271d\u00b3 : Closed A\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d : IsEquivalence F.toFunctor\nh : MonoidalClosed D\nX : C\nq : Closed (F.obj X)\nthis : IsLeftAdjoint (tensorLeft (F.obj X))\ni : (F.toFunctor \u22d9 tensorLeft (F.obj X)) \u22d9 Functor.inv F.toFunctor \u2245 tensorLeft X\n\u22a2 IsLeftAdjoint (tensorLeft X)\n[PROOFSTEP]\nexact Adjunction.leftAdjointOfNatIso i\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Closed.Monoidal", "llama_tokens": 3209, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.44552954976388515, "lm_q2_score": 0.02262920172208746, "lm_q1q2_score": 0.010081978054757761}} {"text": "[GOAL]\nX\u271d : LocallyRingedSpace\nX : LocallyRingedSpace\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 IsLocalRingHom (PresheafedSpace.stalkMap (\ud835\udfd9 X.toSheafedSpace) x)\n[PROOFSTEP]\nerw [PresheafedSpace.stalkMap.id]\n[GOAL]\nX\u271d : LocallyRingedSpace\nX : LocallyRingedSpace\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 IsLocalRingHom (\ud835\udfd9 (PresheafedSpace.stalk X.toPresheafedSpace x))\n[PROOFSTEP]\napply isLocalRingHom_id\n[GOAL]\nX\u271d : LocallyRingedSpace\nX Y Z : LocallyRingedSpace\nf : Hom X Y\ng : Hom Y Z\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 IsLocalRingHom (PresheafedSpace.stalkMap (f.val \u226b g.val) x)\n[PROOFSTEP]\nerw [PresheafedSpace.stalkMap.comp]\n[GOAL]\nX\u271d : LocallyRingedSpace\nX Y Z : LocallyRingedSpace\nf : Hom X Y\ng : Hom Y Z\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 IsLocalRingHom (PresheafedSpace.stalkMap g.val (\u2191f.val.base x) \u226b PresheafedSpace.stalkMap f.val x)\n[PROOFSTEP]\nexact @isLocalRingHom_comp _ _ _ _ _ _ _ _ (f.2 _) (g.2 _)\n[GOAL]\nX\u271d : LocallyRingedSpace\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\n\u22a2 (\ud835\udfd9 X \u226b f).val = f.val\n[PROOFSTEP]\nsimp [comp]\n[GOAL]\nX\u271d : LocallyRingedSpace\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\n\u22a2 (f \u226b \ud835\udfd9 Y).val = f.val\n[PROOFSTEP]\nsimp [comp]\n[GOAL]\nX : LocallyRingedSpace\nx\u271d\u00b3 x\u271d\u00b2 x\u271d\u00b9 x\u271d : LocallyRingedSpace\nf : x\u271d\u00b3 \u27f6 x\u271d\u00b2\ng : x\u271d\u00b2 \u27f6 x\u271d\u00b9\nh : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 ((f \u226b g) \u226b h).val = (f \u226b g \u226b h).val\n[PROOFSTEP]\nsimp [comp]\n[GOAL]\nX\u271d : LocallyRingedSpace\nX Y : LocallyRingedSpace\nf : X.toSheafedSpace \u27f6 Y.toSheafedSpace\ninst\u271d : IsIso f\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 IsLocalRingHom (PresheafedSpace.stalkMap (SheafedSpace.forgetToPresheafedSpace.map f) x)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX\u271d : LocallyRingedSpace\nU : TopCat\nX : LocallyRingedSpace\nf : U \u27f6 toTopCat X\nh : OpenEmbedding \u2191f\n\u22a2 \u2200 (x : \u2191\u2191(SheafedSpace.restrict X.toSheafedSpace h).toPresheafedSpace),\n LocalRing \u2191(Presheaf.stalk (SheafedSpace.restrict X.toSheafedSpace h).toPresheafedSpace.presheaf x)\n[PROOFSTEP]\nintro x\n[GOAL]\nX\u271d : LocallyRingedSpace\nU : TopCat\nX : LocallyRingedSpace\nf : U \u27f6 toTopCat X\nh : OpenEmbedding \u2191f\nx : \u2191\u2191(SheafedSpace.restrict X.toSheafedSpace h).toPresheafedSpace\n\u22a2 LocalRing \u2191(Presheaf.stalk (SheafedSpace.restrict X.toSheafedSpace h).toPresheafedSpace.presheaf x)\n[PROOFSTEP]\napply @RingEquiv.localRing _ _ _ (X.localRing (f x))\n[GOAL]\ncase e\nX\u271d : LocallyRingedSpace\nU : TopCat\nX : LocallyRingedSpace\nf : U \u27f6 toTopCat X\nh : OpenEmbedding \u2191f\nx : \u2191\u2191(SheafedSpace.restrict X.toSheafedSpace h).toPresheafedSpace\n\u22a2 \u2191(Presheaf.stalk X.presheaf (\u2191f x)) \u2243+*\n \u2191(Presheaf.stalk (SheafedSpace.restrict X.toSheafedSpace h).toPresheafedSpace.presheaf x)\n[PROOFSTEP]\nexact (X.restrictStalkIso h x).symm.commRingCatIsoToRingEquiv\n[GOAL]\nX\u271d : LocallyRingedSpace\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nU : Opens \u2191(toTopCat Y)\ns : \u2191(Y.presheaf.obj (op U))\n\u22a2 (Opens.map f.val.base).obj (RingedSpace.basicOpen (toRingedSpace Y) s) =\n RingedSpace.basicOpen (toRingedSpace X) (\u2191(NatTrans.app f.val.c (op U)) s)\n[PROOFSTEP]\next x\n[GOAL]\ncase h.h\nX\u271d : LocallyRingedSpace\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nU : Opens \u2191(toTopCat Y)\ns : \u2191(Y.presheaf.obj (op U))\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208 \u2191((Opens.map f.val.base).obj (RingedSpace.basicOpen (toRingedSpace Y) s)) \u2194\n x \u2208 \u2191(RingedSpace.basicOpen (toRingedSpace X) (\u2191(NatTrans.app f.val.c (op U)) s))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.h.mp\nX\u271d : LocallyRingedSpace\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nU : Opens \u2191(toTopCat Y)\ns : \u2191(Y.presheaf.obj (op U))\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208 \u2191((Opens.map f.val.base).obj (RingedSpace.basicOpen (toRingedSpace Y) s)) \u2192\n x \u2208 \u2191(RingedSpace.basicOpen (toRingedSpace X) (\u2191(NatTrans.app f.val.c (op U)) s))\n[PROOFSTEP]\nrintro \u27e8\u27e8y, hyU\u27e9, hy : IsUnit _, rfl : y = _\u27e9\n[GOAL]\ncase h.h.mp.intro.mk.intro\nX\u271d : LocallyRingedSpace\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nU : Opens \u2191(toTopCat Y)\ns : \u2191(Y.presheaf.obj (op U))\nx : \u2191\u2191X.toPresheafedSpace\nhyU : \u2191f.val.base x \u2208 U\nhy : IsUnit (\u2191(Presheaf.germ (toRingedSpace Y).toPresheafedSpace.presheaf { val := \u2191f.val.base x, property := hyU }) s)\n\u22a2 x \u2208 \u2191(RingedSpace.basicOpen (toRingedSpace X) (\u2191(NatTrans.app f.val.c (op U)) s))\n[PROOFSTEP]\nerw [RingedSpace.mem_basicOpen _ _ \u27e8x, show x \u2208 (Opens.map f.1.base).obj U from hyU\u27e9, \u2190\n PresheafedSpace.stalkMap_germ_apply]\n[GOAL]\ncase h.h.mp.intro.mk.intro\nX\u271d : LocallyRingedSpace\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nU : Opens \u2191(toTopCat Y)\ns : \u2191(Y.presheaf.obj (op U))\nx : \u2191\u2191X.toPresheafedSpace\nhyU : \u2191f.val.base x \u2208 U\nhy : IsUnit (\u2191(Presheaf.germ (toRingedSpace Y).toPresheafedSpace.presheaf { val := \u2191f.val.base x, property := hyU }) s)\n\u22a2 IsUnit\n (\u2191(PresheafedSpace.stalkMap f.val \u2191{ val := x, property := (_ : x \u2208 (Opens.map f.val.base).obj U) })\n (\u2191(Presheaf.germ Y.presheaf\n { val := \u2191f.val.base \u2191{ val := x, property := (_ : x \u2208 (Opens.map f.val.base).obj U) },\n property :=\n (_ :\n \u2191{ val := x, property := (_ : x \u2208 (Opens.map f.val.base).obj U) } \u2208 (Opens.map f.val.base).obj U) })\n s))\n[PROOFSTEP]\nexact (PresheafedSpace.stalkMap f.1 _).isUnit_map hy\n[GOAL]\ncase h.h.mpr\nX\u271d : LocallyRingedSpace\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nU : Opens \u2191(toTopCat Y)\ns : \u2191(Y.presheaf.obj (op U))\nx : \u2191\u2191X.toPresheafedSpace\n\u22a2 x \u2208 \u2191(RingedSpace.basicOpen (toRingedSpace X) (\u2191(NatTrans.app f.val.c (op U)) s)) \u2192\n x \u2208 \u2191((Opens.map f.val.base).obj (RingedSpace.basicOpen (toRingedSpace Y) s))\n[PROOFSTEP]\nrintro \u27e8y, hy : IsUnit _, rfl\u27e9\n[GOAL]\ncase h.h.mpr.intro.intro\nX\u271d : LocallyRingedSpace\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nU : Opens \u2191(toTopCat Y)\ns : \u2191(Y.presheaf.obj (op U))\ny : { x // x \u2208 (Opens.map f.val.base).obj U }\nhy : IsUnit (\u2191(Presheaf.germ (toRingedSpace X).toPresheafedSpace.presheaf y) (\u2191(NatTrans.app f.val.c (op U)) s))\n\u22a2 \u2191y \u2208 \u2191((Opens.map f.val.base).obj (RingedSpace.basicOpen (toRingedSpace Y) s))\n[PROOFSTEP]\nerw [RingedSpace.mem_basicOpen _ _ \u27e8f.1.base y.1, y.2\u27e9]\n[GOAL]\ncase h.h.mpr.intro.intro\nX\u271d : LocallyRingedSpace\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nU : Opens \u2191(toTopCat Y)\ns : \u2191(Y.presheaf.obj (op U))\ny : { x // x \u2208 (Opens.map f.val.base).obj U }\nhy : IsUnit (\u2191(Presheaf.germ (toRingedSpace X).toPresheafedSpace.presheaf y) (\u2191(NatTrans.app f.val.c (op U)) s))\n\u22a2 IsUnit (\u2191(Presheaf.germ Y.presheaf { val := \u2191f.val.base \u2191y, property := (_ : \u2191y \u2208 (Opens.map f.val.base).obj U) }) s)\n[PROOFSTEP]\nerw [\u2190 PresheafedSpace.stalkMap_germ_apply] at hy \n[GOAL]\ncase h.h.mpr.intro.intro\nX\u271d : LocallyRingedSpace\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nU : Opens \u2191(toTopCat Y)\ns : \u2191(Y.presheaf.obj (op U))\ny : { x // x \u2208 (Opens.map f.val.base).obj U }\nhy :\n IsUnit\n (\u2191(PresheafedSpace.stalkMap f.val \u2191y)\n (\u2191(Presheaf.germ Y.presheaf { val := \u2191f.val.base \u2191y, property := (_ : \u2191y \u2208 (Opens.map f.val.base).obj U) }) s))\n\u22a2 IsUnit (\u2191(Presheaf.germ Y.presheaf { val := \u2191f.val.base \u2191y, property := (_ : \u2191y \u2208 (Opens.map f.val.base).obj U) }) s)\n[PROOFSTEP]\nexact (isUnit_map_iff (PresheafedSpace.stalkMap f.1 _) _).mp hy\n[GOAL]\nX\u271d : LocallyRingedSpace\nX : LocallyRingedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\n\u22a2 RingedSpace.basicOpen (toRingedSpace X) 0 = \u22a5\n[PROOFSTEP]\next x\n[GOAL]\ncase h.h\nX\u271d : LocallyRingedSpace\nX : LocallyRingedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : \u2191\u2191(toRingedSpace X).toPresheafedSpace\n\u22a2 x \u2208 \u2191(RingedSpace.basicOpen (toRingedSpace X) 0) \u2194 x \u2208 \u2191\u22a5\n[PROOFSTEP]\nsimp only [RingedSpace.basicOpen, Opens.coe_mk, Set.mem_image, Set.mem_setOf_eq, Subtype.exists, exists_and_right,\n exists_eq_right, Opens.coe_bot, Set.mem_empty_iff_false, iff_false, not_exists]\n[GOAL]\ncase h.h\nX\u271d : LocallyRingedSpace\nX : LocallyRingedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : \u2191\u2191(toRingedSpace X).toPresheafedSpace\n\u22a2 \u2200 (x_1 : x \u2208 U),\n \u00acIsUnit (\u2191(Presheaf.germ (toRingedSpace X).toPresheafedSpace.presheaf { val := x, property := (_ : x \u2208 U) }) 0)\n[PROOFSTEP]\nintros hx\n[GOAL]\ncase h.h\nX\u271d : LocallyRingedSpace\nX : LocallyRingedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : \u2191\u2191(toRingedSpace X).toPresheafedSpace\nhx : x \u2208 U\n\u22a2 \u00acIsUnit (\u2191(Presheaf.germ (toRingedSpace X).toPresheafedSpace.presheaf { val := x, property := (_ : x \u2208 U) }) 0)\n[PROOFSTEP]\nrw [map_zero, isUnit_zero_iff]\n[GOAL]\ncase h.h\nX\u271d : LocallyRingedSpace\nX : LocallyRingedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : \u2191\u2191(toRingedSpace X).toPresheafedSpace\nhx : x \u2208 U\n\u22a2 \u00ac0 = 1\n[PROOFSTEP]\nchange (0 : X.stalk x) \u2260 (1 : X.stalk x)\n[GOAL]\ncase h.h\nX\u271d : LocallyRingedSpace\nX : LocallyRingedSpace\nU : Opens \u2191\u2191X.toPresheafedSpace\nx : \u2191\u2191(toRingedSpace X).toPresheafedSpace\nhx : x \u2208 U\n\u22a2 0 \u2260 1\n[PROOFSTEP]\nexact zero_ne_one\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.LocallyRingedSpace", "llama_tokens": 4091, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.4765796510636759, "lm_q2_score": 0.020023441654952852, "lm_q1q2_score": 0.009542764837011303}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX\u271d : C\nS : Sieve X\u271d\nR : Presieve X\u271d\nE : A\u1d52\u1d56\n\u03c0 : (Functor.cones ((diagram S.arrows).op \u22d9 P)).obj E\nX Y : C\nf : X \u27f6 X\u271d\ng : Y \u27f6 X\nhf : S.arrows f\n\u22a2 (fun Y f h => NatTrans.app \u03c0 (op { obj := Over.mk f, property := h })) Y (g \u226b f) (_ : S.arrows (g \u226b f)) =\n (P \u22d9 coyoneda.obj E).map g.op ((fun Y f h => NatTrans.app \u03c0 (op { obj := Over.mk f, property := h })) X f hf)\n[PROOFSTEP]\napply (id_comp _).symm.trans\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX\u271d : C\nS : Sieve X\u271d\nR : Presieve X\u271d\nE : A\u1d52\u1d56\n\u03c0 : (Functor.cones ((diagram S.arrows).op \u22d9 P)).obj E\nX Y : C\nf : X \u27f6 X\u271d\ng : Y \u27f6 X\nhf : S.arrows f\n\u22a2 \ud835\udfd9\n (((Functor.const (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56).op.obj E).unop.obj\n (op { obj := Over.mk (g \u226b f), property := (_ : S.arrows (g \u226b f)) })) \u226b\n (fun Y f h => NatTrans.app \u03c0 (op { obj := Over.mk f, property := h })) Y (g \u226b f) (_ : S.arrows (g \u226b f)) =\n (P \u22d9 coyoneda.obj E).map g.op ((fun Y f h => NatTrans.app \u03c0 (op { obj := Over.mk f, property := h })) X f hf)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX\u271d : C\nS : Sieve X\u271d\nR : Presieve X\u271d\nE : A\u1d52\u1d56\n\u03c0 : (Functor.cones ((diagram S.arrows).op \u22d9 P)).obj E\nX Y : C\nf : X \u27f6 X\u271d\ng : Y \u27f6 X\nhf : S.arrows f\n\u22a2 \ud835\udfd9 E.unop \u226b NatTrans.app \u03c0 (op { obj := Over.mk (g \u226b f), property := (_ : S.arrows (g \u226b f)) }) =\n NatTrans.app \u03c0 (op { obj := Over.mk f, property := hf }) \u226b P.map g.op\n[PROOFSTEP]\nexact \u03c0.naturality (Quiver.Hom.op (Over.homMk _ (by rfl)))\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX\u271d : C\nS : Sieve X\u271d\nR : Presieve X\u271d\nE : A\u1d52\u1d56\n\u03c0 : (Functor.cones ((diagram S.arrows).op \u22d9 P)).obj E\nX Y : C\nf : X \u27f6 X\u271d\ng : Y \u27f6 X\nhf : S.arrows f\n\u22a2 g \u226b { obj := Over.mk f, property := hf }.obj.hom =\n { obj := Over.mk (g \u226b f), property := (_ : S.arrows (g \u226b f)) }.obj.hom\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE : A\u1d52\u1d56\nx : { x // SieveCompatible x }\nf f' : (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56\ng : f \u27f6 f'\n\u22a2 ((Functor.const (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56).op.obj E).unop.map g \u226b\n (fun f => \u2191x f.unop.obj.hom (_ : S.arrows f.unop.obj.hom)) f' =\n (fun f => \u2191x f.unop.obj.hom (_ : S.arrows f.unop.obj.hom)) f \u226b ((diagram S.arrows).op \u22d9 P).map g\n[PROOFSTEP]\nrefine' Eq.trans _ (x.2 f.unop.1.hom g.unop.left f.unop.2)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE : A\u1d52\u1d56\nx : { x // SieveCompatible x }\nf f' : (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56\ng : f \u27f6 f'\n\u22a2 ((Functor.const (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56).op.obj E).unop.map g \u226b\n (fun f => \u2191x f.unop.obj.hom (_ : S.arrows f.unop.obj.hom)) f' =\n \u2191x (g.unop.left \u226b f.unop.obj.hom) (_ : S.arrows (g.unop.left \u226b f.unop.obj.hom))\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE : A\u1d52\u1d56\nx : { x // SieveCompatible x }\nf f' : (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56\ng : f \u27f6 f'\n\u22a2 \ud835\udfd9 E.unop \u226b \u2191x f'.unop.obj.hom (_ : S.arrows f'.unop.obj.hom) =\n \u2191x (g.unop.left \u226b f.unop.obj.hom) (_ : S.arrows (g.unop.left \u226b f.unop.obj.hom))\n[PROOFSTEP]\nrw [id_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE : A\u1d52\u1d56\nx : { x // SieveCompatible x }\nf f' : (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56\ng : f \u27f6 f'\n\u22a2 \u2191x f'.unop.obj.hom (_ : S.arrows f'.unop.obj.hom) =\n \u2191x (g.unop.left \u226b f.unop.obj.hom) (_ : S.arrows (g.unop.left \u226b f.unop.obj.hom))\n[PROOFSTEP]\nconvert rfl\n[GOAL]\ncase h.e'_3.h.e'_4\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE : A\u1d52\u1d56\nx : { x // SieveCompatible x }\nf f' : (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56\ng : f \u27f6 f'\n\u22a2 g.unop.left \u226b f.unop.obj.hom = f'.unop.obj.hom\n[PROOFSTEP]\nrw [Over.w]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nhx : SieveCompatible x\n\u22a2 Nonempty (IsLimit (P.mapCone (Cocone.op (cocone S.arrows)))) \u2194 \u2200 (E : A\u1d52\u1d56), IsSheafFor (P \u22d9 coyoneda.obj E) S.arrows\n[PROOFSTEP]\ndsimp [IsSheafFor]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nhx : SieveCompatible x\n\u22a2 Nonempty (IsLimit (P.mapCone (Cocone.op (cocone S.arrows)))) \u2194\n \u2200 (E : A\u1d52\u1d56) (x : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows), Compatible x \u2192 \u2203! t, IsAmalgamation x t\n[PROOFSTEP]\nsimp_rw [compatible_iff_sieveCompatible]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nhx : SieveCompatible x\n\u22a2 Nonempty (IsLimit (P.mapCone (Cocone.op (cocone S.arrows)))) \u2194\n \u2200 (E : A\u1d52\u1d56) (x : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows), SieveCompatible x \u2192 \u2203! t, IsAmalgamation x t\n[PROOFSTEP]\nrw [((Cone.isLimitEquivIsTerminal _).trans (isTerminalEquivUnique _ _)).nonempty_congr]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nhx : SieveCompatible x\n\u22a2 Nonempty ((X_1 : Cone ((diagram S.arrows).op \u22d9 P)) \u2192 Unique (X_1 \u27f6 P.mapCone (Cocone.op (cocone S.arrows)))) \u2194\n \u2200 (E : A\u1d52\u1d56) (x : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows), SieveCompatible x \u2192 \u2203! t, IsAmalgamation x t\n[PROOFSTEP]\nrw [Classical.nonempty_pi]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nhx : SieveCompatible x\n\u22a2 (\u2200 (i : Cone ((diagram S.arrows).op \u22d9 P)), Nonempty (Unique (i \u27f6 P.mapCone (Cocone.op (cocone S.arrows))))) \u2194\n \u2200 (E : A\u1d52\u1d56) (x : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows), SieveCompatible x \u2192 \u2203! t, IsAmalgamation x t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nhx : SieveCompatible x\n\u22a2 (\u2200 (i : Cone ((diagram S.arrows).op \u22d9 P)), Nonempty (Unique (i \u27f6 P.mapCone (Cocone.op (cocone S.arrows))))) \u2192\n \u2200 (E : A\u1d52\u1d56) (x : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows), SieveCompatible x \u2192 \u2203! t, IsAmalgamation x t\n[PROOFSTEP]\nintro hu E x hx\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx\u271d : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx\u271d : SieveCompatible x\u271d\nhu : \u2200 (i : Cone ((diagram S.arrows).op \u22d9 P)), Nonempty (Unique (i \u27f6 P.mapCone (Cocone.op (cocone S.arrows))))\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nhx : SieveCompatible x\n\u22a2 \u2203! t, IsAmalgamation x t\n[PROOFSTEP]\nspecialize hu hx.cone\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx\u271d : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx\u271d : SieveCompatible x\u271d\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nhx : SieveCompatible x\nhu : Nonempty (Unique (SieveCompatible.cone hx \u27f6 P.mapCone (Cocone.op (cocone S.arrows))))\n\u22a2 \u2203! t, IsAmalgamation x t\n[PROOFSTEP]\nerw [(homEquivAmalgamation hx).uniqueCongr.nonempty_congr] at hu \n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx\u271d : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx\u271d : SieveCompatible x\u271d\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nhx : SieveCompatible x\nhu : Nonempty (Unique { t // IsAmalgamation x t })\n\u22a2 \u2203! t, IsAmalgamation x t\n[PROOFSTEP]\nexact (unique_subtype_iff_exists_unique _).1 hu\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nhx : SieveCompatible x\n\u22a2 (\u2200 (E : A\u1d52\u1d56) (x : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows), SieveCompatible x \u2192 \u2203! t, IsAmalgamation x t) \u2192\n \u2200 (i : Cone ((diagram S.arrows).op \u22d9 P)), Nonempty (Unique (i \u27f6 P.mapCone (Cocone.op (cocone S.arrows))))\n[PROOFSTEP]\nrintro h \u27e8E, \u03c0\u27e9\n[GOAL]\ncase mpr.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx : SieveCompatible x\nh : \u2200 (E : A\u1d52\u1d56) (x : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows), SieveCompatible x \u2192 \u2203! t, IsAmalgamation x t\nE : A\n\u03c0 : (Functor.const (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56).obj E \u27f6 (diagram S.arrows).op \u22d9 P\n\u22a2 Nonempty (Unique ({ pt := E, \u03c0 := \u03c0 } \u27f6 P.mapCone (Cocone.op (cocone S.arrows))))\n[PROOFSTEP]\nlet eqv := conesEquivSieveCompatibleFamily P S (op E)\n[GOAL]\ncase mpr.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx : SieveCompatible x\nh : \u2200 (E : A\u1d52\u1d56) (x : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows), SieveCompatible x \u2192 \u2203! t, IsAmalgamation x t\nE : A\n\u03c0 : (Functor.const (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56).obj E \u27f6 (diagram S.arrows).op \u22d9 P\neqv : (Functor.cones ((diagram S.arrows).op \u22d9 P)).obj (op E) \u2243 { x // SieveCompatible x } :=\n conesEquivSieveCompatibleFamily P S (op E)\n\u22a2 Nonempty (Unique ({ pt := E, \u03c0 := \u03c0 } \u27f6 P.mapCone (Cocone.op (cocone S.arrows))))\n[PROOFSTEP]\nrw [\u2190 eqv.left_inv \u03c0]\n[GOAL]\ncase mpr.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx : SieveCompatible x\nh : \u2200 (E : A\u1d52\u1d56) (x : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows), SieveCompatible x \u2192 \u2203! t, IsAmalgamation x t\nE : A\n\u03c0 : (Functor.const (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56).obj E \u27f6 (diagram S.arrows).op \u22d9 P\neqv : (Functor.cones ((diagram S.arrows).op \u22d9 P)).obj (op E) \u2243 { x // SieveCompatible x } :=\n conesEquivSieveCompatibleFamily P S (op E)\n\u22a2 Nonempty (Unique ({ pt := E, \u03c0 := Equiv.invFun eqv (Equiv.toFun eqv \u03c0) } \u27f6 P.mapCone (Cocone.op (cocone S.arrows))))\n[PROOFSTEP]\nerw [(homEquivAmalgamation (eqv \u03c0).2).uniqueCongr.nonempty_congr]\n[GOAL]\ncase mpr.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx : SieveCompatible x\nh : \u2200 (E : A\u1d52\u1d56) (x : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows), SieveCompatible x \u2192 \u2203! t, IsAmalgamation x t\nE : A\n\u03c0 : (Functor.const (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56).obj E \u27f6 (diagram S.arrows).op \u22d9 P\neqv : (Functor.cones ((diagram S.arrows).op \u22d9 P)).obj (op E) \u2243 { x // SieveCompatible x } :=\n conesEquivSieveCompatibleFamily P S (op E)\n\u22a2 Nonempty (Unique { t // IsAmalgamation (\u2191(\u2191eqv \u03c0)) t })\n[PROOFSTEP]\nrw [unique_subtype_iff_exists_unique]\n[GOAL]\ncase mpr.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx : SieveCompatible x\nh : \u2200 (E : A\u1d52\u1d56) (x : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows), SieveCompatible x \u2192 \u2203! t, IsAmalgamation x t\nE : A\n\u03c0 : (Functor.const (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56).obj E \u27f6 (diagram S.arrows).op \u22d9 P\neqv : (Functor.cones ((diagram S.arrows).op \u22d9 P)).obj (op E) \u2243 { x // SieveCompatible x } :=\n conesEquivSieveCompatibleFamily P S (op E)\n\u22a2 \u2203! a, IsAmalgamation (\u2191(\u2191eqv \u03c0)) a\n[PROOFSTEP]\nexact h _ _ (eqv \u03c0).2\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nhx : SieveCompatible x\n\u22a2 (\u2200 (c : Cone ((diagram S.arrows).op \u22d9 P)), Subsingleton (c \u27f6 P.mapCone (Cocone.op (cocone S.arrows)))) \u2194\n \u2200 (E : A\u1d52\u1d56), IsSeparatedFor (P \u22d9 coyoneda.obj E) S.arrows\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nhx : SieveCompatible x\n\u22a2 (\u2200 (c : Cone ((diagram S.arrows).op \u22d9 P)), Subsingleton (c \u27f6 P.mapCone (Cocone.op (cocone S.arrows)))) \u2192\n \u2200 (E : A\u1d52\u1d56), IsSeparatedFor (P \u22d9 coyoneda.obj E) S.arrows\n[PROOFSTEP]\nintro hs E x t\u2081 t\u2082 h\u2081 h\u2082\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx\u271d : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx : SieveCompatible x\u271d\nhs : \u2200 (c : Cone ((diagram S.arrows).op \u22d9 P)), Subsingleton (c \u27f6 P.mapCone (Cocone.op (cocone S.arrows)))\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nt\u2081 t\u2082 : (P \u22d9 coyoneda.obj E).obj (op X)\nh\u2081 : IsAmalgamation x t\u2081\nh\u2082 : IsAmalgamation x t\u2082\n\u22a2 t\u2081 = t\u2082\n[PROOFSTEP]\nhave hx := is_compatible_of_exists_amalgamation x \u27e8t\u2081, h\u2081\u27e9\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx\u271d : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx\u271d : SieveCompatible x\u271d\nhs : \u2200 (c : Cone ((diagram S.arrows).op \u22d9 P)), Subsingleton (c \u27f6 P.mapCone (Cocone.op (cocone S.arrows)))\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nt\u2081 t\u2082 : (P \u22d9 coyoneda.obj E).obj (op X)\nh\u2081 : IsAmalgamation x t\u2081\nh\u2082 : IsAmalgamation x t\u2082\nhx : Compatible x\n\u22a2 t\u2081 = t\u2082\n[PROOFSTEP]\nrw [compatible_iff_sieveCompatible] at hx \n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx\u271d : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx\u271d : SieveCompatible x\u271d\nhs : \u2200 (c : Cone ((diagram S.arrows).op \u22d9 P)), Subsingleton (c \u27f6 P.mapCone (Cocone.op (cocone S.arrows)))\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nt\u2081 t\u2082 : (P \u22d9 coyoneda.obj E).obj (op X)\nh\u2081 : IsAmalgamation x t\u2081\nh\u2082 : IsAmalgamation x t\u2082\nhx : SieveCompatible x\n\u22a2 t\u2081 = t\u2082\n[PROOFSTEP]\nspecialize hs hx.cone\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx\u271d : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx\u271d : SieveCompatible x\u271d\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nt\u2081 t\u2082 : (P \u22d9 coyoneda.obj E).obj (op X)\nh\u2081 : IsAmalgamation x t\u2081\nh\u2082 : IsAmalgamation x t\u2082\nhx : SieveCompatible x\nhs : Subsingleton (SieveCompatible.cone hx \u27f6 P.mapCone (Cocone.op (cocone S.arrows)))\n\u22a2 t\u2081 = t\u2082\n[PROOFSTEP]\nrcases hs with \u27e8hs\u27e9\n[GOAL]\ncase mp.intro\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx\u271d : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx\u271d : SieveCompatible x\u271d\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nt\u2081 t\u2082 : (P \u22d9 coyoneda.obj E).obj (op X)\nh\u2081 : IsAmalgamation x t\u2081\nh\u2082 : IsAmalgamation x t\u2082\nhx : SieveCompatible x\nhs : \u2200 (a b : SieveCompatible.cone hx \u27f6 P.mapCone (Cocone.op (cocone S.arrows))), a = b\n\u22a2 t\u2081 = t\u2082\n[PROOFSTEP]\nsimpa only [Subtype.mk.injEq] using\n (show Subtype.mk t\u2081 h\u2081 = \u27e8t\u2082, h\u2082\u27e9 from (homEquivAmalgamation hx).symm.injective (hs _ _))\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nhx : SieveCompatible x\n\u22a2 (\u2200 (E : A\u1d52\u1d56), IsSeparatedFor (P \u22d9 coyoneda.obj E) S.arrows) \u2192\n \u2200 (c : Cone ((diagram S.arrows).op \u22d9 P)), Subsingleton (c \u27f6 P.mapCone (Cocone.op (cocone S.arrows)))\n[PROOFSTEP]\nrintro h \u27e8E, \u03c0\u27e9\n[GOAL]\ncase mpr.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx : SieveCompatible x\nh : \u2200 (E : A\u1d52\u1d56), IsSeparatedFor (P \u22d9 coyoneda.obj E) S.arrows\nE : A\n\u03c0 : (Functor.const (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56).obj E \u27f6 (diagram S.arrows).op \u22d9 P\n\u22a2 Subsingleton ({ pt := E, \u03c0 := \u03c0 } \u27f6 P.mapCone (Cocone.op (cocone S.arrows)))\n[PROOFSTEP]\nlet eqv := conesEquivSieveCompatibleFamily P S (op E)\n[GOAL]\ncase mpr.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx : SieveCompatible x\nh : \u2200 (E : A\u1d52\u1d56), IsSeparatedFor (P \u22d9 coyoneda.obj E) S.arrows\nE : A\n\u03c0 : (Functor.const (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56).obj E \u27f6 (diagram S.arrows).op \u22d9 P\neqv : (Functor.cones ((diagram S.arrows).op \u22d9 P)).obj (op E) \u2243 { x // SieveCompatible x } :=\n conesEquivSieveCompatibleFamily P S (op E)\n\u22a2 Subsingleton ({ pt := E, \u03c0 := \u03c0 } \u27f6 P.mapCone (Cocone.op (cocone S.arrows)))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.mk.allEq\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx : SieveCompatible x\nh : \u2200 (E : A\u1d52\u1d56), IsSeparatedFor (P \u22d9 coyoneda.obj E) S.arrows\nE : A\n\u03c0 : (Functor.const (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56).obj E \u27f6 (diagram S.arrows).op \u22d9 P\neqv : (Functor.cones ((diagram S.arrows).op \u22d9 P)).obj (op E) \u2243 { x // SieveCompatible x } :=\n conesEquivSieveCompatibleFamily P S (op E)\n\u22a2 \u2200 (a b : { pt := E, \u03c0 := \u03c0 } \u27f6 P.mapCone (Cocone.op (cocone S.arrows))), a = b\n[PROOFSTEP]\nrw [\u2190 eqv.left_inv \u03c0]\n[GOAL]\ncase mpr.mk.allEq\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx : SieveCompatible x\nh : \u2200 (E : A\u1d52\u1d56), IsSeparatedFor (P \u22d9 coyoneda.obj E) S.arrows\nE : A\n\u03c0 : (Functor.const (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56).obj E \u27f6 (diagram S.arrows).op \u22d9 P\neqv : (Functor.cones ((diagram S.arrows).op \u22d9 P)).obj (op E) \u2243 { x // SieveCompatible x } :=\n conesEquivSieveCompatibleFamily P S (op E)\n\u22a2 \u2200 (a b : { pt := E, \u03c0 := Equiv.invFun eqv (Equiv.toFun eqv \u03c0) } \u27f6 P.mapCone (Cocone.op (cocone S.arrows))), a = b\n[PROOFSTEP]\nintro f\u2081 f\u2082\n[GOAL]\ncase mpr.mk.allEq\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx : SieveCompatible x\nh : \u2200 (E : A\u1d52\u1d56), IsSeparatedFor (P \u22d9 coyoneda.obj E) S.arrows\nE : A\n\u03c0 : (Functor.const (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56).obj E \u27f6 (diagram S.arrows).op \u22d9 P\neqv : (Functor.cones ((diagram S.arrows).op \u22d9 P)).obj (op E) \u2243 { x // SieveCompatible x } :=\n conesEquivSieveCompatibleFamily P S (op E)\nf\u2081 f\u2082 : { pt := E, \u03c0 := Equiv.invFun eqv (Equiv.toFun eqv \u03c0) } \u27f6 P.mapCone (Cocone.op (cocone S.arrows))\n\u22a2 f\u2081 = f\u2082\n[PROOFSTEP]\nlet eqv' := homEquivAmalgamation (eqv \u03c0).2\n[GOAL]\ncase mpr.mk.allEq\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx : SieveCompatible x\nh : \u2200 (E : A\u1d52\u1d56), IsSeparatedFor (P \u22d9 coyoneda.obj E) S.arrows\nE : A\n\u03c0 : (Functor.const (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56).obj E \u27f6 (diagram S.arrows).op \u22d9 P\neqv : (Functor.cones ((diagram S.arrows).op \u22d9 P)).obj (op E) \u2243 { x // SieveCompatible x } :=\n conesEquivSieveCompatibleFamily P S (op E)\nf\u2081 f\u2082 : { pt := E, \u03c0 := Equiv.invFun eqv (Equiv.toFun eqv \u03c0) } \u27f6 P.mapCone (Cocone.op (cocone S.arrows))\neqv' : (SieveCompatible.cone (_ : SieveCompatible \u2191(\u2191eqv \u03c0)) \u27f6 P.mapCone (Cocone.op (cocone S.arrows))) \u2243\n { t // IsAmalgamation (\u2191(\u2191eqv \u03c0)) t } :=\n homEquivAmalgamation (_ : SieveCompatible \u2191(\u2191eqv \u03c0))\n\u22a2 f\u2081 = f\u2082\n[PROOFSTEP]\napply eqv'.injective\n[GOAL]\ncase mpr.mk.allEq.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx : SieveCompatible x\nh : \u2200 (E : A\u1d52\u1d56), IsSeparatedFor (P \u22d9 coyoneda.obj E) S.arrows\nE : A\n\u03c0 : (Functor.const (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56).obj E \u27f6 (diagram S.arrows).op \u22d9 P\neqv : (Functor.cones ((diagram S.arrows).op \u22d9 P)).obj (op E) \u2243 { x // SieveCompatible x } :=\n conesEquivSieveCompatibleFamily P S (op E)\nf\u2081 f\u2082 : { pt := E, \u03c0 := Equiv.invFun eqv (Equiv.toFun eqv \u03c0) } \u27f6 P.mapCone (Cocone.op (cocone S.arrows))\neqv' : (SieveCompatible.cone (_ : SieveCompatible \u2191(\u2191eqv \u03c0)) \u27f6 P.mapCone (Cocone.op (cocone S.arrows))) \u2243\n { t // IsAmalgamation (\u2191(\u2191eqv \u03c0)) t } :=\n homEquivAmalgamation (_ : SieveCompatible \u2191(\u2191eqv \u03c0))\n\u22a2 \u2191eqv' f\u2081 = \u2191eqv' f\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase mpr.mk.allEq.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx : SieveCompatible x\nh : \u2200 (E : A\u1d52\u1d56), IsSeparatedFor (P \u22d9 coyoneda.obj E) S.arrows\nE : A\n\u03c0 : (Functor.const (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56).obj E \u27f6 (diagram S.arrows).op \u22d9 P\neqv : (Functor.cones ((diagram S.arrows).op \u22d9 P)).obj (op E) \u2243 { x // SieveCompatible x } :=\n conesEquivSieveCompatibleFamily P S (op E)\nf\u2081 f\u2082 : { pt := E, \u03c0 := Equiv.invFun eqv (Equiv.toFun eqv \u03c0) } \u27f6 P.mapCone (Cocone.op (cocone S.arrows))\neqv' : (SieveCompatible.cone (_ : SieveCompatible \u2191(\u2191eqv \u03c0)) \u27f6 P.mapCone (Cocone.op (cocone S.arrows))) \u2243\n { t // IsAmalgamation (\u2191(\u2191eqv \u03c0)) t } :=\n homEquivAmalgamation (_ : SieveCompatible \u2191(\u2191eqv \u03c0))\n\u22a2 \u2191(\u2191eqv' f\u2081) = \u2191(\u2191eqv' f\u2082)\n[PROOFSTEP]\napply h _ (eqv \u03c0).1\n[GOAL]\ncase mpr.mk.allEq.a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx : SieveCompatible x\nh : \u2200 (E : A\u1d52\u1d56), IsSeparatedFor (P \u22d9 coyoneda.obj E) S.arrows\nE : A\n\u03c0 : (Functor.const (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56).obj E \u27f6 (diagram S.arrows).op \u22d9 P\neqv : (Functor.cones ((diagram S.arrows).op \u22d9 P)).obj (op E) \u2243 { x // SieveCompatible x } :=\n conesEquivSieveCompatibleFamily P S (op E)\nf\u2081 f\u2082 : { pt := E, \u03c0 := Equiv.invFun eqv (Equiv.toFun eqv \u03c0) } \u27f6 P.mapCone (Cocone.op (cocone S.arrows))\neqv' : (SieveCompatible.cone (_ : SieveCompatible \u2191(\u2191eqv \u03c0)) \u27f6 P.mapCone (Cocone.op (cocone S.arrows))) \u2243\n { t // IsAmalgamation (\u2191(\u2191eqv \u03c0)) t } :=\n homEquivAmalgamation (_ : SieveCompatible \u2191(\u2191eqv \u03c0))\n\u22a2 IsAmalgamation \u2191(\u2191eqv \u03c0) \u2191(\u2191eqv' f\u2081)\n[PROOFSTEP]\nexact (eqv' _).2\n[GOAL]\ncase mpr.mk.allEq.a.a.a\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE\u271d : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E\u271d) S.arrows\nhx : SieveCompatible x\nh : \u2200 (E : A\u1d52\u1d56), IsSeparatedFor (P \u22d9 coyoneda.obj E) S.arrows\nE : A\n\u03c0 : (Functor.const (FullSubcategory fun f => S.arrows f.hom)\u1d52\u1d56).obj E \u27f6 (diagram S.arrows).op \u22d9 P\neqv : (Functor.cones ((diagram S.arrows).op \u22d9 P)).obj (op E) \u2243 { x // SieveCompatible x } :=\n conesEquivSieveCompatibleFamily P S (op E)\nf\u2081 f\u2082 : { pt := E, \u03c0 := Equiv.invFun eqv (Equiv.toFun eqv \u03c0) } \u27f6 P.mapCone (Cocone.op (cocone S.arrows))\neqv' : (SieveCompatible.cone (_ : SieveCompatible \u2191(\u2191eqv \u03c0)) \u27f6 P.mapCone (Cocone.op (cocone S.arrows))) \u2243\n { t // IsAmalgamation (\u2191(\u2191eqv \u03c0)) t } :=\n homEquivAmalgamation (_ : SieveCompatible \u2191(\u2191eqv \u03c0))\n\u22a2 IsAmalgamation \u2191(\u2191eqv \u03c0) \u2191(\u2191eqv' f\u2082)\n[PROOFSTEP]\nexact (eqv' _).2\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nhx : SieveCompatible x\ninst\u271d : HasPullbacks C\nK : Pretopology C\n\u22a2 IsSheaf (Pretopology.toGrothendieck C K) P \u2194\n \u2200 \u2983X : C\u2984 (R : Presieve X),\n R \u2208 Pretopology.coverings K X \u2192 Nonempty (IsLimit (P.mapCone (Cocone.op (cocone (generate R).arrows))))\n[PROOFSTEP]\ndsimp [IsSheaf]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nhx : SieveCompatible x\ninst\u271d : HasPullbacks C\nK : Pretopology C\n\u22a2 (\u2200 (E : A), Presieve.IsSheaf (Pretopology.toGrothendieck C K) (P \u22d9 coyoneda.obj (op E))) \u2194\n \u2200 \u2983X : C\u2984 (R : Presieve X),\n R \u2208 Pretopology.coverings K X \u2192 Nonempty (IsLimit (P.mapCone (Cocone.op (cocone (generate R).arrows))))\n[PROOFSTEP]\nsimp_rw [isSheaf_pretopology]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nJ : GrothendieckTopology C\nP : C\u1d52\u1d56 \u2964 A\nX : C\nS : Sieve X\nR : Presieve X\nE : A\u1d52\u1d56\nx : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows\nhx : SieveCompatible x\ninst\u271d : HasPullbacks C\nK : Pretopology C\n\u22a2 (\u2200 (E : A) {X : C} (R : Presieve X), R \u2208 Pretopology.coverings K X \u2192 IsSheafFor (P \u22d9 coyoneda.obj (op E)) R) \u2194\n \u2200 \u2983X : C\u2984 (R : Presieve X),\n R \u2208 Pretopology.coverings K X \u2192 Nonempty (IsLimit (P.mapCone (Cocone.op (cocone (generate R).arrows))))\n[PROOFSTEP]\nexact\n \u27e8fun h X R hR => (isLimit_iff_isSheafFor_presieve P R).2 fun E => h E.unop R hR, fun h E X R hR =>\n (isLimit_iff_isSheafFor_presieve P R).1 (h R hR) (op E)\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA\u271d : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\u271d\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nE : A\nX : C\nP : C\u1d52\u1d56 \u2964 A\nhP : IsSheaf J P\nS : GrothendieckTopology.Cover J X\nx : (I : GrothendieckTopology.Cover.Arrow S) \u2192 E \u27f6 P.obj (op I.Y)\nhx :\n \u2200 (I : GrothendieckTopology.Cover.Relation S),\n x (GrothendieckTopology.Cover.Relation.fst I) \u226b P.map I.g\u2081.op =\n x (GrothendieckTopology.Cover.Relation.snd I) \u226b P.map I.g\u2082.op\nI : GrothendieckTopology.Cover.Arrow S\n\u22a2 amalgamate hP S x hx \u226b P.map I.f.op = x I\n[PROOFSTEP]\nrcases I with \u27e8Y, f, hf\u27e9\n[GOAL]\ncase mk\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA\u271d : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\u271d\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nE : A\nX : C\nP : C\u1d52\u1d56 \u2964 A\nhP : IsSheaf J P\nS : GrothendieckTopology.Cover J X\nx : (I : GrothendieckTopology.Cover.Arrow S) \u2192 E \u27f6 P.obj (op I.Y)\nhx :\n \u2200 (I : GrothendieckTopology.Cover.Relation S),\n x (GrothendieckTopology.Cover.Relation.fst I) \u226b P.map I.g\u2081.op =\n x (GrothendieckTopology.Cover.Relation.snd I) \u226b P.map I.g\u2082.op\nY : C\nf : Y \u27f6 X\nhf : (GrothendieckTopology.Cover.sieve S).arrows f\n\u22a2 amalgamate hP S x hx \u226b P.map { Y := Y, f := f, hf := hf }.f.op = x { Y := Y, f := f, hf := hf }\n[PROOFSTEP]\napply\n @Presieve.IsSheafFor.valid_glue _ _ _ _ _ _ (hP _ _ S.condition) (fun Y f hf => x \u27e8Y, f, hf\u27e9)\n (fun Y\u2081 Y\u2082 Z g\u2081 g\u2082 f\u2081 f\u2082 h\u2081 h\u2082 w => hx \u27e8Y\u2081, Y\u2082, Z, g\u2081, g\u2082, f\u2081, f\u2082, h\u2081, h\u2082, w\u27e9) f hf\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nP : C\u1d52\u1d56 \u2964 Type w\n\u22a2 Presheaf.IsSheaf J P \u2194 Presieve.IsSheaf J P\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nP : C\u1d52\u1d56 \u2964 Type w\n\u22a2 Presheaf.IsSheaf J P \u2192 Presieve.IsSheaf J P\n[PROOFSTEP]\nintro hP\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nP : C\u1d52\u1d56 \u2964 Type w\nhP : Presheaf.IsSheaf J P\n\u22a2 Presieve.IsSheaf J P\n[PROOFSTEP]\nrefine' Presieve.isSheaf_iso J _ (hP PUnit)\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nP : C\u1d52\u1d56 \u2964 Type w\nhP : Presheaf.IsSheaf J P\n\u22a2 P \u22d9 coyoneda.obj (op PUnit) \u2245 P\n[PROOFSTEP]\nrefine' isoWhiskerLeft _ Coyoneda.punitIso \u226a\u226b P.rightUnitor\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nP : C\u1d52\u1d56 \u2964 Type w\n\u22a2 Presieve.IsSheaf J P \u2192 Presheaf.IsSheaf J P\n[PROOFSTEP]\nintro hP X Y S hS z hz\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nP : C\u1d52\u1d56 \u2964 Type w\nhP : Presieve.IsSheaf J P\nX : Type w\nY : C\nS : Sieve Y\nhS : S \u2208 GrothendieckTopology.sieves J Y\nz : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op X)) S.arrows\nhz : Presieve.FamilyOfElements.Compatible z\n\u22a2 \u2203! t, Presieve.FamilyOfElements.IsAmalgamation z t\n[PROOFSTEP]\nrefine' \u27e8fun x => (hP S hS).amalgamate (fun Z f hf => z f hf x) _, _, _\u27e9\n[GOAL]\ncase mpr.refine'_1\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nP : C\u1d52\u1d56 \u2964 Type w\nhP : Presieve.IsSheaf J P\nX : Type w\nY : C\nS : Sieve Y\nhS : S \u2208 GrothendieckTopology.sieves J Y\nz : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op X)) S.arrows\nhz : Presieve.FamilyOfElements.Compatible z\nx : (op X).unop\n\u22a2 Presieve.FamilyOfElements.Compatible fun Z f hf => z f hf x\n[PROOFSTEP]\nintro Y\u2081 Y\u2082 Z g\u2081 g\u2082 f\u2081 f\u2082 hf\u2081 hf\u2082 h\n[GOAL]\ncase mpr.refine'_1\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nP : C\u1d52\u1d56 \u2964 Type w\nhP : Presieve.IsSheaf J P\nX : Type w\nY : C\nS : Sieve Y\nhS : S \u2208 GrothendieckTopology.sieves J Y\nz : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op X)) S.arrows\nhz : Presieve.FamilyOfElements.Compatible z\nx : (op X).unop\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 Y\nf\u2082 : Y\u2082 \u27f6 Y\nhf\u2081 : S.arrows f\u2081\nhf\u2082 : S.arrows f\u2082\nh : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 P.map g\u2081.op ((fun Z f hf => z f hf x) Y\u2081 f\u2081 hf\u2081) = P.map g\u2082.op ((fun Z f hf => z f hf x) Y\u2082 f\u2082 hf\u2082)\n[PROOFSTEP]\nexact congr_fun (hz g\u2081 g\u2082 hf\u2081 hf\u2082 h) x\n[GOAL]\ncase mpr.refine'_2\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nP : C\u1d52\u1d56 \u2964 Type w\nhP : Presieve.IsSheaf J P\nX : Type w\nY : C\nS : Sieve Y\nhS : S \u2208 GrothendieckTopology.sieves J Y\nz : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op X)) S.arrows\nhz : Presieve.FamilyOfElements.Compatible z\n\u22a2 (fun t => Presieve.FamilyOfElements.IsAmalgamation z t) fun x =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P S.arrows) (fun Z f hf => z f hf x)\n (_ :\n \u2200 \u2983Y\u2081 Y\u2082 Z : C\u2984 (g\u2081 : Z \u27f6 Y\u2081) (g\u2082 : Z \u27f6 Y\u2082) \u2983f\u2081 : Y\u2081 \u27f6 Y\u2984 \u2983f\u2082 : Y\u2082 \u27f6 Y\u2984 (hf\u2081 : S.arrows f\u2081) (hf\u2082 : S.arrows f\u2082),\n g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082 \u2192\n (P \u22d9 coyoneda.obj (op X)).map g\u2081.op (z f\u2081 hf\u2081) x = (P \u22d9 coyoneda.obj (op X)).map g\u2082.op (z f\u2082 hf\u2082) x)\n[PROOFSTEP]\nintro Z f hf\n[GOAL]\ncase mpr.refine'_2\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nP : C\u1d52\u1d56 \u2964 Type w\nhP : Presieve.IsSheaf J P\nX : Type w\nY : C\nS : Sieve Y\nhS : S \u2208 GrothendieckTopology.sieves J Y\nz : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op X)) S.arrows\nhz : Presieve.FamilyOfElements.Compatible z\nZ : C\nf : Z \u27f6 Y\nhf : S.arrows f\n\u22a2 ((P \u22d9 coyoneda.obj (op X)).map f.op fun x =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P S.arrows) (fun Z f hf => z f hf x)\n (_ :\n \u2200 \u2983Y\u2081 Y\u2082 Z : C\u2984 (g\u2081 : Z \u27f6 Y\u2081) (g\u2082 : Z \u27f6 Y\u2082) \u2983f\u2081 : Y\u2081 \u27f6 Y\u2984 \u2983f\u2082 : Y\u2082 \u27f6 Y\u2984 (hf\u2081 : S.arrows f\u2081)\n (hf\u2082 : S.arrows f\u2082),\n g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082 \u2192\n (P \u22d9 coyoneda.obj (op X)).map g\u2081.op (z f\u2081 hf\u2081) x = (P \u22d9 coyoneda.obj (op X)).map g\u2082.op (z f\u2082 hf\u2082) x)) =\n z f hf\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase mpr.refine'_2.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nP : C\u1d52\u1d56 \u2964 Type w\nhP : Presieve.IsSheaf J P\nX : Type w\nY : C\nS : Sieve Y\nhS : S \u2208 GrothendieckTopology.sieves J Y\nz : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op X)) S.arrows\nhz : Presieve.FamilyOfElements.Compatible z\nZ : C\nf : Z \u27f6 Y\nhf : S.arrows f\nx : (op X).unop\n\u22a2 (P \u22d9 coyoneda.obj (op X)).map f.op\n (fun x =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P S.arrows) (fun Z f hf => z f hf x)\n (_ :\n \u2200 \u2983Y\u2081 Y\u2082 Z : C\u2984 (g\u2081 : Z \u27f6 Y\u2081) (g\u2082 : Z \u27f6 Y\u2082) \u2983f\u2081 : Y\u2081 \u27f6 Y\u2984 \u2983f\u2082 : Y\u2082 \u27f6 Y\u2984 (hf\u2081 : S.arrows f\u2081)\n (hf\u2082 : S.arrows f\u2082),\n g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082 \u2192\n (P \u22d9 coyoneda.obj (op X)).map g\u2081.op (z f\u2081 hf\u2081) x = (P \u22d9 coyoneda.obj (op X)).map g\u2082.op (z f\u2082 hf\u2082) x))\n x =\n z f hf x\n[PROOFSTEP]\napply Presieve.IsSheafFor.valid_glue\n[GOAL]\ncase mpr.refine'_3\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nP : C\u1d52\u1d56 \u2964 Type w\nhP : Presieve.IsSheaf J P\nX : Type w\nY : C\nS : Sieve Y\nhS : S \u2208 GrothendieckTopology.sieves J Y\nz : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op X)) S.arrows\nhz : Presieve.FamilyOfElements.Compatible z\n\u22a2 \u2200 (y : (P \u22d9 coyoneda.obj (op X)).obj (op Y)),\n (fun t => Presieve.FamilyOfElements.IsAmalgamation z t) y \u2192\n y = fun x =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P S.arrows) (fun Z f hf => z f hf x)\n (_ :\n \u2200 \u2983Y\u2081 Y\u2082 Z : C\u2984 (g\u2081 : Z \u27f6 Y\u2081) (g\u2082 : Z \u27f6 Y\u2082) \u2983f\u2081 : Y\u2081 \u27f6 Y\u2984 \u2983f\u2082 : Y\u2082 \u27f6 Y\u2984 (hf\u2081 : S.arrows f\u2081)\n (hf\u2082 : S.arrows f\u2082),\n g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082 \u2192\n (P \u22d9 coyoneda.obj (op X)).map g\u2081.op (z f\u2081 hf\u2081) x = (P \u22d9 coyoneda.obj (op X)).map g\u2082.op (z f\u2082 hf\u2082) x)\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase mpr.refine'_3\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nP : C\u1d52\u1d56 \u2964 Type w\nhP : Presieve.IsSheaf J P\nX : Type w\nY : C\nS : Sieve Y\nhS : S \u2208 GrothendieckTopology.sieves J Y\nz : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op X)) S.arrows\nhz : Presieve.FamilyOfElements.Compatible z\ny : (P \u22d9 coyoneda.obj (op X)).obj (op Y)\nhy : Presieve.FamilyOfElements.IsAmalgamation z y\n\u22a2 y = fun x =>\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P S.arrows) (fun Z f hf => z f hf x)\n (_ :\n \u2200 \u2983Y\u2081 Y\u2082 Z : C\u2984 (g\u2081 : Z \u27f6 Y\u2081) (g\u2082 : Z \u27f6 Y\u2082) \u2983f\u2081 : Y\u2081 \u27f6 Y\u2984 \u2983f\u2082 : Y\u2082 \u27f6 Y\u2984 (hf\u2081 : S.arrows f\u2081) (hf\u2082 : S.arrows f\u2082),\n g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082 \u2192\n (P \u22d9 coyoneda.obj (op X)).map g\u2081.op (z f\u2081 hf\u2081) x = (P \u22d9 coyoneda.obj (op X)).map g\u2082.op (z f\u2082 hf\u2082) x)\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase mpr.refine'_3.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nP : C\u1d52\u1d56 \u2964 Type w\nhP : Presieve.IsSheaf J P\nX : Type w\nY : C\nS : Sieve Y\nhS : S \u2208 GrothendieckTopology.sieves J Y\nz : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op X)) S.arrows\nhz : Presieve.FamilyOfElements.Compatible z\ny : (P \u22d9 coyoneda.obj (op X)).obj (op Y)\nhy : Presieve.FamilyOfElements.IsAmalgamation z y\nx : (op X).unop\n\u22a2 y x =\n Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P S.arrows) (fun Z f hf => z f hf x)\n (_ :\n \u2200 \u2983Y\u2081 Y\u2082 Z : C\u2984 (g\u2081 : Z \u27f6 Y\u2081) (g\u2082 : Z \u27f6 Y\u2082) \u2983f\u2081 : Y\u2081 \u27f6 Y\u2984 \u2983f\u2082 : Y\u2082 \u27f6 Y\u2984 (hf\u2081 : S.arrows f\u2081) (hf\u2082 : S.arrows f\u2082),\n g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082 \u2192\n (P \u22d9 coyoneda.obj (op X)).map g\u2081.op (z f\u2081 hf\u2081) x = (P \u22d9 coyoneda.obj (op X)).map g\u2082.op (z f\u2082 hf\u2082) x)\n[PROOFSTEP]\napply (hP S hS).isSeparatedFor.ext\n[GOAL]\ncase mpr.refine'_3.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nP : C\u1d52\u1d56 \u2964 Type w\nhP : Presieve.IsSheaf J P\nX : Type w\nY : C\nS : Sieve Y\nhS : S \u2208 GrothendieckTopology.sieves J Y\nz : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op X)) S.arrows\nhz : Presieve.FamilyOfElements.Compatible z\ny : (P \u22d9 coyoneda.obj (op X)).obj (op Y)\nhy : Presieve.FamilyOfElements.IsAmalgamation z y\nx : (op X).unop\n\u22a2 \u2200 \u2983Y_1 : C\u2984 \u2983f : Y_1 \u27f6 Y\u2984,\n S.arrows f \u2192\n P.map f.op (y x) =\n P.map f.op\n (Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P S.arrows) (fun Z f hf => z f hf x)\n (_ :\n \u2200 \u2983Y\u2081 Y\u2082 Z : C\u2984 (g\u2081 : Z \u27f6 Y\u2081) (g\u2082 : Z \u27f6 Y\u2082) \u2983f\u2081 : Y\u2081 \u27f6 Y\u2984 \u2983f\u2082 : Y\u2082 \u27f6 Y\u2984 (hf\u2081 : S.arrows f\u2081)\n (hf\u2082 : S.arrows f\u2082),\n g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082 \u2192\n (P \u22d9 coyoneda.obj (op X)).map g\u2081.op (z f\u2081 hf\u2081) x = (P \u22d9 coyoneda.obj (op X)).map g\u2082.op (z f\u2082 hf\u2082) x))\n[PROOFSTEP]\nintro Y' f hf\n[GOAL]\ncase mpr.refine'_3.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nP : C\u1d52\u1d56 \u2964 Type w\nhP : Presieve.IsSheaf J P\nX : Type w\nY : C\nS : Sieve Y\nhS : S \u2208 GrothendieckTopology.sieves J Y\nz : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op X)) S.arrows\nhz : Presieve.FamilyOfElements.Compatible z\ny : (P \u22d9 coyoneda.obj (op X)).obj (op Y)\nhy : Presieve.FamilyOfElements.IsAmalgamation z y\nx : (op X).unop\nY' : C\nf : Y' \u27f6 Y\nhf : S.arrows f\n\u22a2 P.map f.op (y x) =\n P.map f.op\n (Presieve.IsSheafFor.amalgamate (_ : Presieve.IsSheafFor P S.arrows) (fun Z f hf => z f hf x)\n (_ :\n \u2200 \u2983Y\u2081 Y\u2082 Z : C\u2984 (g\u2081 : Z \u27f6 Y\u2081) (g\u2082 : Z \u27f6 Y\u2082) \u2983f\u2081 : Y\u2081 \u27f6 Y\u2984 \u2983f\u2082 : Y\u2082 \u27f6 Y\u2984 (hf\u2081 : S.arrows f\u2081)\n (hf\u2082 : S.arrows f\u2082),\n g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082 \u2192\n (P \u22d9 coyoneda.obj (op X)).map g\u2081.op (z f\u2081 hf\u2081) x = (P \u22d9 coyoneda.obj (op X)).map g\u2082.op (z f\u2082 hf\u2082) x))\n[PROOFSTEP]\nrw [Presieve.IsSheafFor.valid_glue _ _ _ hf, \u2190 hy _ hf]\n[GOAL]\ncase mpr.refine'_3.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nP : C\u1d52\u1d56 \u2964 Type w\nhP : Presieve.IsSheaf J P\nX : Type w\nY : C\nS : Sieve Y\nhS : S \u2208 GrothendieckTopology.sieves J Y\nz : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op X)) S.arrows\nhz : Presieve.FamilyOfElements.Compatible z\ny : (P \u22d9 coyoneda.obj (op X)).obj (op Y)\nhy : Presieve.FamilyOfElements.IsAmalgamation z y\nx : (op X).unop\nY' : C\nf : Y' \u27f6 Y\nhf : S.arrows f\n\u22a2 P.map f.op (y x) = (P \u22d9 coyoneda.obj (op X)).map f.op y x\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nF : Sheaf J A\nX : C\nH : \u22a5 \u2208 GrothendieckTopology.sieves J X\n\u22a2 IsTerminal (F.val.obj (op X))\n[PROOFSTEP]\nrefine' @IsTerminal.ofUnique _ _ _ ?_\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nF : Sheaf J A\nX : C\nH : \u22a5 \u2208 GrothendieckTopology.sieves J X\n\u22a2 (X_1 : A) \u2192 Unique (X_1 \u27f6 F.val.obj (op X))\n[PROOFSTEP]\nintro Y\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nF : Sheaf J A\nX : C\nH : \u22a5 \u2208 GrothendieckTopology.sieves J X\nY : A\n\u22a2 Unique (Y \u27f6 F.val.obj (op X))\n[PROOFSTEP]\nchoose t h using F.2 Y _ H (by tauto) (by tauto)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nF : Sheaf J A\nX : C\nH : \u22a5 \u2208 GrothendieckTopology.sieves J X\nY : A\n\u22a2 Presieve.FamilyOfElements (F.val \u22d9 coyoneda.obj (op Y)) \u22a5.arrows\n[PROOFSTEP]\ntauto\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nF : Sheaf J A\nX : C\nH : \u22a5 \u2208 GrothendieckTopology.sieves J X\nY : A\n\u22a2 Presieve.FamilyOfElements.Compatible\n (let_fun em := Classical.propDecidable;\n fun \u2983Y_1\u2984 f a => False.elim (_ : False))\n[PROOFSTEP]\ntauto\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nF : Sheaf J A\nX : C\nH : \u22a5 \u2208 GrothendieckTopology.sieves J X\nY : A\nt : (F.val \u22d9 coyoneda.obj (op Y)).obj (op X)\nh :\n (fun t =>\n Presieve.FamilyOfElements.IsAmalgamation\n (let_fun em := Classical.propDecidable;\n fun \u2983Y_1\u2984 f a => False.elim (_ : False))\n t)\n t \u2227\n \u2200 (y : (F.val \u22d9 coyoneda.obj (op Y)).obj (op X)),\n (fun t =>\n Presieve.FamilyOfElements.IsAmalgamation\n (let_fun em := Classical.propDecidable;\n fun \u2983Y_1\u2984 f a => False.elim (_ : False))\n t)\n y \u2192\n y = t\n\u22a2 Unique (Y \u27f6 F.val.obj (op X))\n[PROOFSTEP]\nexact \u27e8\u27e8t\u27e9, fun a => h.2 a (by tauto)\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} A\nF : Sheaf J A\nX : C\nH : \u22a5 \u2208 GrothendieckTopology.sieves J X\nY : A\nt : (F.val \u22d9 coyoneda.obj (op Y)).obj (op X)\nh :\n (fun t =>\n Presieve.FamilyOfElements.IsAmalgamation\n (let_fun em := Classical.propDecidable;\n fun \u2983Y_1\u2984 f a => False.elim (_ : False))\n t)\n t \u2227\n \u2200 (y : (F.val \u22d9 coyoneda.obj (op Y)).obj (op X)),\n (fun t =>\n Presieve.FamilyOfElements.IsAmalgamation\n (let_fun em := Classical.propDecidable;\n fun \u2983Y_1\u2984 f a => False.elim (_ : False))\n t)\n y \u2192\n y = t\na : Y \u27f6 F.val.obj (op X)\n\u22a2 (fun t =>\n Presieve.FamilyOfElements.IsAmalgamation\n (let_fun em := Classical.propDecidable;\n fun \u2983Y_1\u2984 f a => False.elim (_ : False))\n t)\n a\n[PROOFSTEP]\ntauto\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\ninst\u271d : Preadditive A\nP Q : Sheaf J A\nn : \u2124\nf : P \u27f6 Q\nU V : C\u1d52\u1d56\ni : U \u27f6 V\n\u22a2 P.val.map i \u226b (fun U => n \u2022 NatTrans.app f.val U) V = (fun U => n \u2022 NatTrans.app f.val U) U \u226b Q.val.map i\n[PROOFSTEP]\ninduction' n using Int.induction_on with n ih n ih\n[GOAL]\ncase hz\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\ninst\u271d : Preadditive A\nP Q : Sheaf J A\nf : P \u27f6 Q\nU V : C\u1d52\u1d56\ni : U \u27f6 V\n\u22a2 P.val.map i \u226b (fun U => 0 \u2022 NatTrans.app f.val U) V = (fun U => 0 \u2022 NatTrans.app f.val U) U \u226b Q.val.map i\n[PROOFSTEP]\nsimp only [zero_smul, comp_zero, zero_comp]\n[GOAL]\ncase hp\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\ninst\u271d : Preadditive A\nP Q : Sheaf J A\nf : P \u27f6 Q\nU V : C\u1d52\u1d56\ni : U \u27f6 V\nn : \u2115\nih : P.val.map i \u226b (fun U => \u2191n \u2022 NatTrans.app f.val U) V = (fun U => \u2191n \u2022 NatTrans.app f.val U) U \u226b Q.val.map i\n\u22a2 P.val.map i \u226b (fun U => (\u2191n + 1) \u2022 NatTrans.app f.val U) V =\n (fun U => (\u2191n + 1) \u2022 NatTrans.app f.val U) U \u226b Q.val.map i\n[PROOFSTEP]\nsimpa only [add_zsmul, one_zsmul, comp_add, NatTrans.naturality, add_comp, add_left_inj]\n[GOAL]\ncase hn\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\ninst\u271d : Preadditive A\nP Q : Sheaf J A\nf : P \u27f6 Q\nU V : C\u1d52\u1d56\ni : U \u27f6 V\nn : \u2115\nih : P.val.map i \u226b (fun U => -\u2191n \u2022 NatTrans.app f.val U) V = (fun U => -\u2191n \u2022 NatTrans.app f.val U) U \u226b Q.val.map i\n\u22a2 P.val.map i \u226b (fun U => (-\u2191n - 1) \u2022 NatTrans.app f.val U) V =\n (fun U => (-\u2191n - 1) \u2022 NatTrans.app f.val U) U \u226b Q.val.map i\n[PROOFSTEP]\nsimpa only [sub_smul, one_zsmul, comp_sub, NatTrans.naturality, sub_comp, sub_left_inj] using ih\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\ninst\u271d : Preadditive A\nP Q : Sheaf J A\nn : \u2115\nf : P \u27f6 Q\nU V : C\u1d52\u1d56\ni : U \u27f6 V\n\u22a2 P.val.map i \u226b (fun U => n \u2022 NatTrans.app f.val U) V = (fun U => n \u2022 NatTrans.app f.val U) U \u226b Q.val.map i\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\ninst\u271d : Preadditive A\nP Q : Sheaf J A\nf : P \u27f6 Q\nU V : C\u1d52\u1d56\ni : U \u27f6 V\n\u22a2 P.val.map i \u226b (fun U => Nat.zero \u2022 NatTrans.app f.val U) V =\n (fun U => Nat.zero \u2022 NatTrans.app f.val U) U \u226b Q.val.map i\n[PROOFSTEP]\nsimp only [zero_smul, comp_zero, zero_comp, Nat.zero_eq]\n[GOAL]\ncase succ\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\ninst\u271d : Preadditive A\nP Q : Sheaf J A\nf : P \u27f6 Q\nU V : C\u1d52\u1d56\ni : U \u27f6 V\nn : \u2115\nih : P.val.map i \u226b (fun U => n \u2022 NatTrans.app f.val U) V = (fun U => n \u2022 NatTrans.app f.val U) U \u226b Q.val.map i\n\u22a2 P.val.map i \u226b (fun U => Nat.succ n \u2022 NatTrans.app f.val U) V =\n (fun U => Nat.succ n \u2022 NatTrans.app f.val U) U \u226b Q.val.map i\n[PROOFSTEP]\nsimp only [Nat.succ_eq_add_one, add_smul, ih, one_nsmul, comp_add, NatTrans.naturality, add_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\ninst\u271d : Preadditive A\nP Q : Sheaf J A\nx\u271d\u00b9 : P \u27f6 Q\nx\u271d : \u2115\n\u22a2 (fun f => f.val) (x\u271d \u2022 x\u271d\u00b9) = x\u271d \u2022 (fun f => f.val) x\u271d\u00b9\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\ninst\u271d : Preadditive A\nP Q : Sheaf J A\nx\u271d\u00b9 : P \u27f6 Q\nx\u271d : \u2124\n\u22a2 (fun f => f.val) (x\u271d \u2022 x\u271d\u00b9) = x\u271d \u2022 (fun f => f.val) x\u271d\u00b9\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nX : C\nS : GrothendieckTopology.Cover J X\nhP : IsSheaf J P\n\u22a2 \u2200 (s : Cone (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)))\n (j : WalkingMulticospan (GrothendieckTopology.Cover.index S P).fstTo (GrothendieckTopology.Cover.index S P).sndTo),\n (fun E =>\n IsSheaf.amalgamate hP S (fun I => Multifork.\u03b9 E I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation S),\n Multifork.\u03b9 E (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index S P) I =\n Multifork.\u03b9 E (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index S P) I))\n s \u226b\n NatTrans.app (GrothendieckTopology.Cover.multifork S P).\u03c0 j =\n NatTrans.app s.\u03c0 j\n[PROOFSTEP]\nrintro (E : Multifork _) (a | b)\n[GOAL]\ncase left\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nX : C\nS : GrothendieckTopology.Cover J X\nhP : IsSheaf J P\nE : Multifork (GrothendieckTopology.Cover.index S P)\na : (GrothendieckTopology.Cover.index S P).L\n\u22a2 (fun E =>\n IsSheaf.amalgamate hP S (fun I => Multifork.\u03b9 E I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation S),\n Multifork.\u03b9 E (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index S P) I =\n Multifork.\u03b9 E (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index S P) I))\n E \u226b\n NatTrans.app (GrothendieckTopology.Cover.multifork S P).\u03c0 (WalkingMulticospan.left a) =\n NatTrans.app E.\u03c0 (WalkingMulticospan.left a)\n[PROOFSTEP]\napply hP.amalgamate_map\n[GOAL]\ncase right\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nX : C\nS : GrothendieckTopology.Cover J X\nhP : IsSheaf J P\nE : Multifork (GrothendieckTopology.Cover.index S P)\nb : (GrothendieckTopology.Cover.index S P).R\n\u22a2 (fun E =>\n IsSheaf.amalgamate hP S (fun I => Multifork.\u03b9 E I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation S),\n Multifork.\u03b9 E (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index S P) I =\n Multifork.\u03b9 E (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index S P) I))\n E \u226b\n NatTrans.app (GrothendieckTopology.Cover.multifork S P).\u03c0 (WalkingMulticospan.right b) =\n NatTrans.app E.\u03c0 (WalkingMulticospan.right b)\n[PROOFSTEP]\nrw [\u2190 E.w (WalkingMulticospan.Hom.fst b), \u2190 (S.multifork P).w (WalkingMulticospan.Hom.fst b), \u2190 assoc]\n[GOAL]\ncase right\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nX : C\nS : GrothendieckTopology.Cover J X\nhP : IsSheaf J P\nE : Multifork (GrothendieckTopology.Cover.index S P)\nb : (GrothendieckTopology.Cover.index S P).R\n\u22a2 ((fun E =>\n IsSheaf.amalgamate hP S (fun I => Multifork.\u03b9 E I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation S),\n Multifork.\u03b9 E (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index S P) I =\n Multifork.\u03b9 E (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index S P) I))\n E \u226b\n NatTrans.app (GrothendieckTopology.Cover.multifork S P).\u03c0\n (WalkingMulticospan.left (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index S P) b))) \u226b\n (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)).map (WalkingMulticospan.Hom.fst b) =\n NatTrans.app E.\u03c0 (WalkingMulticospan.left (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index S P) b)) \u226b\n (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)).map (WalkingMulticospan.Hom.fst b)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase right.e_a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nX : C\nS : GrothendieckTopology.Cover J X\nhP : IsSheaf J P\nE : Multifork (GrothendieckTopology.Cover.index S P)\nb : (GrothendieckTopology.Cover.index S P).R\n\u22a2 (fun E =>\n IsSheaf.amalgamate hP S (fun I => Multifork.\u03b9 E I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation S),\n Multifork.\u03b9 E (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index S P) I =\n Multifork.\u03b9 E (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index S P) I))\n E \u226b\n NatTrans.app (GrothendieckTopology.Cover.multifork S P).\u03c0\n (WalkingMulticospan.left (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index S P) b)) =\n NatTrans.app E.\u03c0 (WalkingMulticospan.left (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index S P) b))\n[PROOFSTEP]\napply hP.amalgamate_map\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nX : C\nS : GrothendieckTopology.Cover J X\nhP : IsSheaf J P\n\u22a2 \u2200 (s : Cone (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)))\n (m : s.pt \u27f6 (GrothendieckTopology.Cover.multifork S P).pt),\n (\u2200\n (j :\n WalkingMulticospan (GrothendieckTopology.Cover.index S P).fstTo (GrothendieckTopology.Cover.index S P).sndTo),\n m \u226b NatTrans.app (GrothendieckTopology.Cover.multifork S P).\u03c0 j = NatTrans.app s.\u03c0 j) \u2192\n m =\n (fun E =>\n IsSheaf.amalgamate hP S (fun I => Multifork.\u03b9 E I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation S),\n Multifork.\u03b9 E (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index S P) I =\n Multifork.\u03b9 E (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index S P) I))\n s\n[PROOFSTEP]\nrintro (E : Multifork _) m hm\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nX : C\nS : GrothendieckTopology.Cover J X\nhP : IsSheaf J P\nE : Multifork (GrothendieckTopology.Cover.index S P)\nm : E.pt \u27f6 (GrothendieckTopology.Cover.multifork S P).pt\nhm :\n \u2200 (j : WalkingMulticospan (GrothendieckTopology.Cover.index S P).fstTo (GrothendieckTopology.Cover.index S P).sndTo),\n m \u226b NatTrans.app (GrothendieckTopology.Cover.multifork S P).\u03c0 j = NatTrans.app E.\u03c0 j\n\u22a2 m =\n (fun E =>\n IsSheaf.amalgamate hP S (fun I => Multifork.\u03b9 E I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation S),\n Multifork.\u03b9 E (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index S P) I =\n Multifork.\u03b9 E (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index S P) I))\n E\n[PROOFSTEP]\napply hP.hom_ext S\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nX : C\nS : GrothendieckTopology.Cover J X\nhP : IsSheaf J P\nE : Multifork (GrothendieckTopology.Cover.index S P)\nm : E.pt \u27f6 (GrothendieckTopology.Cover.multifork S P).pt\nhm :\n \u2200 (j : WalkingMulticospan (GrothendieckTopology.Cover.index S P).fstTo (GrothendieckTopology.Cover.index S P).sndTo),\n m \u226b NatTrans.app (GrothendieckTopology.Cover.multifork S P).\u03c0 j = NatTrans.app E.\u03c0 j\n\u22a2 \u2200 (I : GrothendieckTopology.Cover.Arrow S),\n m \u226b P.map I.f.op =\n (fun E =>\n IsSheaf.amalgamate hP S (fun I => Multifork.\u03b9 E I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation S),\n Multifork.\u03b9 E (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index S P) I =\n Multifork.\u03b9 E (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index S P) I))\n E \u226b\n P.map I.f.op\n[PROOFSTEP]\nintro I\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nX : C\nS : GrothendieckTopology.Cover J X\nhP : IsSheaf J P\nE : Multifork (GrothendieckTopology.Cover.index S P)\nm : E.pt \u27f6 (GrothendieckTopology.Cover.multifork S P).pt\nhm :\n \u2200 (j : WalkingMulticospan (GrothendieckTopology.Cover.index S P).fstTo (GrothendieckTopology.Cover.index S P).sndTo),\n m \u226b NatTrans.app (GrothendieckTopology.Cover.multifork S P).\u03c0 j = NatTrans.app E.\u03c0 j\nI : GrothendieckTopology.Cover.Arrow S\n\u22a2 m \u226b P.map I.f.op =\n (fun E =>\n IsSheaf.amalgamate hP S (fun I => Multifork.\u03b9 E I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation S),\n Multifork.\u03b9 E (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index S P) I =\n Multifork.\u03b9 E (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index S P) I))\n E \u226b\n P.map I.f.op\n[PROOFSTEP]\nerw [hm (WalkingMulticospan.left I)]\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nX : C\nS : GrothendieckTopology.Cover J X\nhP : IsSheaf J P\nE : Multifork (GrothendieckTopology.Cover.index S P)\nm : E.pt \u27f6 (GrothendieckTopology.Cover.multifork S P).pt\nhm :\n \u2200 (j : WalkingMulticospan (GrothendieckTopology.Cover.index S P).fstTo (GrothendieckTopology.Cover.index S P).sndTo),\n m \u226b NatTrans.app (GrothendieckTopology.Cover.multifork S P).\u03c0 j = NatTrans.app E.\u03c0 j\nI : GrothendieckTopology.Cover.Arrow S\n\u22a2 NatTrans.app E.\u03c0 (WalkingMulticospan.left I) =\n (fun E =>\n IsSheaf.amalgamate hP S (fun I => Multifork.\u03b9 E I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation S),\n Multifork.\u03b9 E (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index S P) I =\n Multifork.\u03b9 E (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index S P) I))\n E \u226b\n P.map I.f.op\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nX : C\nS : GrothendieckTopology.Cover J X\nhP : IsSheaf J P\nE : Multifork (GrothendieckTopology.Cover.index S P)\nm : E.pt \u27f6 (GrothendieckTopology.Cover.multifork S P).pt\nhm :\n \u2200 (j : WalkingMulticospan (GrothendieckTopology.Cover.index S P).fstTo (GrothendieckTopology.Cover.index S P).sndTo),\n m \u226b NatTrans.app (GrothendieckTopology.Cover.multifork S P).\u03c0 j = NatTrans.app E.\u03c0 j\nI : GrothendieckTopology.Cover.Arrow S\n\u22a2 (fun E =>\n IsSheaf.amalgamate hP S (fun I => Multifork.\u03b9 E I)\n (_ :\n \u2200 (I : GrothendieckTopology.Cover.Relation S),\n Multifork.\u03b9 E (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index S P) I =\n Multifork.\u03b9 E (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index S P) I) \u226b\n MulticospanIndex.snd (GrothendieckTopology.Cover.index S P) I))\n E \u226b\n P.map I.f.op =\n NatTrans.app E.\u03c0 (WalkingMulticospan.left I)\n[PROOFSTEP]\napply hP.amalgamate_map\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\n\u22a2 IsSheaf J P \u2194\n \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\n[PROOFSTEP]\nrefine' \u27e8fun hP X S => \u27e8isLimitOfIsSheaf _ _ _ hP\u27e9, _\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\n\u22a2 (\u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))) \u2192\n IsSheaf J P\n[PROOFSTEP]\nintro h E X S hS x hx\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nh : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nE : A\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\nx : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op E)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\n\u22a2 \u2203! t, Presieve.FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nlet T : J.Cover X := \u27e8S, hS\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nh : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nE : A\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\nx : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op E)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nT : GrothendieckTopology.Cover J X := { val := S, property := hS }\n\u22a2 \u2203! t, Presieve.FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nobtain \u27e8hh\u27e9 := h _ T\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nh : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nE : A\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\nx : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op E)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nT : GrothendieckTopology.Cover J X := { val := S, property := hS }\nhh : IsLimit (GrothendieckTopology.Cover.multifork T P)\n\u22a2 \u2203! t, Presieve.FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nlet K : Multifork (T.index P) := Multifork.of\u03b9 _ E (fun I => x I.f I.hf) fun I => hx _ _ _ _ I.w\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nh : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nE : A\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\nx : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op E)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nT : GrothendieckTopology.Cover J X := { val := S, property := hS }\nhh : IsLimit (GrothendieckTopology.Cover.multifork T P)\nK : Multifork (GrothendieckTopology.Cover.index T P) :=\n Multifork.of\u03b9 (GrothendieckTopology.Cover.index T P) E\n (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve T).arrows I.f))\n (_ :\n \u2200 (I : (GrothendieckTopology.Cover.index T P).R),\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2081.op\n (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f)) =\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2082.op\n (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f)))\n\u22a2 \u2203! t, Presieve.FamilyOfElements.IsAmalgamation x t\n[PROOFSTEP]\nuse hh.lift K\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nh : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nE : A\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\nx : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op E)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nT : GrothendieckTopology.Cover J X := { val := S, property := hS }\nhh : IsLimit (GrothendieckTopology.Cover.multifork T P)\nK : Multifork (GrothendieckTopology.Cover.index T P) :=\n Multifork.of\u03b9 (GrothendieckTopology.Cover.index T P) E\n (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve T).arrows I.f))\n (_ :\n \u2200 (I : (GrothendieckTopology.Cover.index T P).R),\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2081.op\n (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f)) =\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2082.op\n (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f)))\n\u22a2 (fun t => Presieve.FamilyOfElements.IsAmalgamation x t) (IsLimit.lift hh K) \u2227\n \u2200 (y : (P \u22d9 coyoneda.obj (op E)).obj (op X)),\n (fun t => Presieve.FamilyOfElements.IsAmalgamation x t) y \u2192 y = IsLimit.lift hh K\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nh : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nE : A\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\nx : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op E)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nT : GrothendieckTopology.Cover J X := { val := S, property := hS }\nhh : IsLimit (GrothendieckTopology.Cover.multifork T P)\nK : Multifork (GrothendieckTopology.Cover.index T P) :=\n Multifork.of\u03b9 (GrothendieckTopology.Cover.index T P) E\n (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve T).arrows I.f))\n (_ :\n \u2200 (I : (GrothendieckTopology.Cover.index T P).R),\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2081.op\n (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f)) =\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2082.op\n (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f)))\n\u22a2 Presieve.FamilyOfElements.IsAmalgamation x\n (IsLimit.lift hh\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index { val := S, property := hS } P) E\n (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows I.f))\n (_ :\n \u2200 (I : (GrothendieckTopology.Cover.index { val := S, property := hS } P).R),\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2081.op\n (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index { val := S, property := hS } P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index { val := S, property := hS } P)\n I).f)) =\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2082.op\n (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index { val := S, property := hS } P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index { val := S, property := hS } P)\n I).f))))) \u2227\n \u2200 (y : E \u27f6 P.obj (op X)),\n Presieve.FamilyOfElements.IsAmalgamation x y \u2192\n y =\n IsLimit.lift hh\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index { val := S, property := hS } P) E\n (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows I.f))\n (_ :\n \u2200 (I : (GrothendieckTopology.Cover.index { val := S, property := hS } P).R),\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2081.op\n (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index { val := S, property := hS } P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index { val := S, property := hS } P)\n I).f)) =\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2082.op\n (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index { val := S, property := hS } P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index { val := S, property := hS } P)\n I).f))))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nh : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nE : A\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\nx : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op E)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nT : GrothendieckTopology.Cover J X := { val := S, property := hS }\nhh : IsLimit (GrothendieckTopology.Cover.multifork T P)\nK : Multifork (GrothendieckTopology.Cover.index T P) :=\n Multifork.of\u03b9 (GrothendieckTopology.Cover.index T P) E\n (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve T).arrows I.f))\n (_ :\n \u2200 (I : (GrothendieckTopology.Cover.index T P).R),\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2081.op\n (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f)) =\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2082.op\n (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f)))\n\u22a2 Presieve.FamilyOfElements.IsAmalgamation x\n (IsLimit.lift hh\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index { val := S, property := hS } P) E\n (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows I.f))\n (_ :\n \u2200 (I : (GrothendieckTopology.Cover.index { val := S, property := hS } P).R),\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2081.op\n (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index { val := S, property := hS } P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index { val := S, property := hS } P) I).f)) =\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2082.op\n (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index { val := S, property := hS } P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index { val := S, property := hS } P)\n I).f)))))\n[PROOFSTEP]\nintro Y f hf\n[GOAL]\ncase h.left\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nh : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nE : A\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\nx : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op E)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nT : GrothendieckTopology.Cover J X := { val := S, property := hS }\nhh : IsLimit (GrothendieckTopology.Cover.multifork T P)\nK : Multifork (GrothendieckTopology.Cover.index T P) :=\n Multifork.of\u03b9 (GrothendieckTopology.Cover.index T P) E\n (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve T).arrows I.f))\n (_ :\n \u2200 (I : (GrothendieckTopology.Cover.index T P).R),\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2081.op\n (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f)) =\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2082.op\n (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f)))\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\n\u22a2 (P \u22d9 coyoneda.obj (op E)).map f.op\n (IsLimit.lift hh\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index { val := S, property := hS } P) E\n (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows I.f))\n (_ :\n \u2200 (I : (GrothendieckTopology.Cover.index { val := S, property := hS } P).R),\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2081.op\n (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index { val := S, property := hS } P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index { val := S, property := hS } P)\n I).f)) =\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2082.op\n (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index { val := S, property := hS } P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index { val := S, property := hS } P)\n I).f))))) =\n x f hf\n[PROOFSTEP]\napply hh.fac K (WalkingMulticospan.left \u27e8Y, f, hf\u27e9)\n[GOAL]\ncase h.right\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nh : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nE : A\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\nx : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op E)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nT : GrothendieckTopology.Cover J X := { val := S, property := hS }\nhh : IsLimit (GrothendieckTopology.Cover.multifork T P)\nK : Multifork (GrothendieckTopology.Cover.index T P) :=\n Multifork.of\u03b9 (GrothendieckTopology.Cover.index T P) E\n (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve T).arrows I.f))\n (_ :\n \u2200 (I : (GrothendieckTopology.Cover.index T P).R),\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2081.op\n (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f)) =\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2082.op\n (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f)))\n\u22a2 \u2200 (y : E \u27f6 P.obj (op X)),\n Presieve.FamilyOfElements.IsAmalgamation x y \u2192\n y =\n IsLimit.lift hh\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index { val := S, property := hS } P) E\n (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows I.f))\n (_ :\n \u2200 (I : (GrothendieckTopology.Cover.index { val := S, property := hS } P).R),\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2081.op\n (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index { val := S, property := hS } P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index { val := S, property := hS } P)\n I).f)) =\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2082.op\n (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index { val := S, property := hS } P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index { val := S, property := hS } P)\n I).f))))\n[PROOFSTEP]\nintro e he\n[GOAL]\ncase h.right\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nh : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nE : A\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\nx : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op E)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nT : GrothendieckTopology.Cover J X := { val := S, property := hS }\nhh : IsLimit (GrothendieckTopology.Cover.multifork T P)\nK : Multifork (GrothendieckTopology.Cover.index T P) :=\n Multifork.of\u03b9 (GrothendieckTopology.Cover.index T P) E\n (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve T).arrows I.f))\n (_ :\n \u2200 (I : (GrothendieckTopology.Cover.index T P).R),\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2081.op\n (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f)) =\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2082.op\n (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f)))\ne : E \u27f6 P.obj (op X)\nhe : Presieve.FamilyOfElements.IsAmalgamation x e\n\u22a2 e =\n IsLimit.lift hh\n (Multifork.of\u03b9 (GrothendieckTopology.Cover.index { val := S, property := hS } P) E\n (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows I.f))\n (_ :\n \u2200 (I : (GrothendieckTopology.Cover.index { val := S, property := hS } P).R),\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2081.op\n (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index { val := S, property := hS } P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index { val := S, property := hS } P) I).f)) =\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2082.op\n (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index { val := S, property := hS } P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve { val := S, property := hS }).arrows\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index { val := S, property := hS } P) I).f))))\n[PROOFSTEP]\napply hh.uniq K\n[GOAL]\ncase h.right.x\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nh : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nE : A\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\nx : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op E)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nT : GrothendieckTopology.Cover J X := { val := S, property := hS }\nhh : IsLimit (GrothendieckTopology.Cover.multifork T P)\nK : Multifork (GrothendieckTopology.Cover.index T P) :=\n Multifork.of\u03b9 (GrothendieckTopology.Cover.index T P) E\n (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve T).arrows I.f))\n (_ :\n \u2200 (I : (GrothendieckTopology.Cover.index T P).R),\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2081.op\n (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f)) =\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2082.op\n (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f)))\ne : E \u27f6 P.obj (op X)\nhe : Presieve.FamilyOfElements.IsAmalgamation x e\n\u22a2 \u2200 (j : WalkingMulticospan (GrothendieckTopology.Cover.index T P).fstTo (GrothendieckTopology.Cover.index T P).sndTo),\n e \u226b NatTrans.app (GrothendieckTopology.Cover.multifork T P).\u03c0 j = NatTrans.app K.\u03c0 j\n[PROOFSTEP]\nrintro (a | b)\n[GOAL]\ncase h.right.x.left\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nh : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nE : A\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\nx : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op E)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nT : GrothendieckTopology.Cover J X := { val := S, property := hS }\nhh : IsLimit (GrothendieckTopology.Cover.multifork T P)\nK : Multifork (GrothendieckTopology.Cover.index T P) :=\n Multifork.of\u03b9 (GrothendieckTopology.Cover.index T P) E\n (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve T).arrows I.f))\n (_ :\n \u2200 (I : (GrothendieckTopology.Cover.index T P).R),\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2081.op\n (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f)) =\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2082.op\n (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f)))\ne : E \u27f6 P.obj (op X)\nhe : Presieve.FamilyOfElements.IsAmalgamation x e\na : (GrothendieckTopology.Cover.index T P).L\n\u22a2 e \u226b NatTrans.app (GrothendieckTopology.Cover.multifork T P).\u03c0 (WalkingMulticospan.left a) =\n NatTrans.app K.\u03c0 (WalkingMulticospan.left a)\n[PROOFSTEP]\napply he\n[GOAL]\ncase h.right.x.right\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nh : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nE : A\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\nx : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op E)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nT : GrothendieckTopology.Cover J X := { val := S, property := hS }\nhh : IsLimit (GrothendieckTopology.Cover.multifork T P)\nK : Multifork (GrothendieckTopology.Cover.index T P) :=\n Multifork.of\u03b9 (GrothendieckTopology.Cover.index T P) E\n (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve T).arrows I.f))\n (_ :\n \u2200 (I : (GrothendieckTopology.Cover.index T P).R),\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2081.op\n (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f)) =\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2082.op\n (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f)))\ne : E \u27f6 P.obj (op X)\nhe : Presieve.FamilyOfElements.IsAmalgamation x e\nb : (GrothendieckTopology.Cover.index T P).R\n\u22a2 e \u226b NatTrans.app (GrothendieckTopology.Cover.multifork T P).\u03c0 (WalkingMulticospan.right b) =\n NatTrans.app K.\u03c0 (WalkingMulticospan.right b)\n[PROOFSTEP]\nrw [\u2190 K.w (WalkingMulticospan.Hom.fst b), \u2190 (T.multifork P).w (WalkingMulticospan.Hom.fst b), \u2190 assoc]\n[GOAL]\ncase h.right.x.right\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nh : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nE : A\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\nx : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op E)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nT : GrothendieckTopology.Cover J X := { val := S, property := hS }\nhh : IsLimit (GrothendieckTopology.Cover.multifork T P)\nK : Multifork (GrothendieckTopology.Cover.index T P) :=\n Multifork.of\u03b9 (GrothendieckTopology.Cover.index T P) E\n (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve T).arrows I.f))\n (_ :\n \u2200 (I : (GrothendieckTopology.Cover.index T P).R),\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2081.op\n (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f)) =\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2082.op\n (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f)))\ne : E \u27f6 P.obj (op X)\nhe : Presieve.FamilyOfElements.IsAmalgamation x e\nb : (GrothendieckTopology.Cover.index T P).R\n\u22a2 (e \u226b\n NatTrans.app (GrothendieckTopology.Cover.multifork T P).\u03c0\n (WalkingMulticospan.left (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) b))) \u226b\n (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index T P)).map (WalkingMulticospan.Hom.fst b) =\n NatTrans.app K.\u03c0 (WalkingMulticospan.left (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) b)) \u226b\n (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index T P)).map (WalkingMulticospan.Hom.fst b)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.right.x.right.e_a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\nh : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\nE : A\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\nx : Presieve.FamilyOfElements (P \u22d9 coyoneda.obj (op E)) S.arrows\nhx : Presieve.FamilyOfElements.Compatible x\nT : GrothendieckTopology.Cover J X := { val := S, property := hS }\nhh : IsLimit (GrothendieckTopology.Cover.multifork T P)\nK : Multifork (GrothendieckTopology.Cover.index T P) :=\n Multifork.of\u03b9 (GrothendieckTopology.Cover.index T P) E\n (fun I => x I.f (_ : (GrothendieckTopology.Cover.sieve T).arrows I.f))\n (_ :\n \u2200 (I : (GrothendieckTopology.Cover.index T P).R),\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2081.op\n (x (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) I).f)) =\n (P \u22d9 coyoneda.obj (op E)).map I.g\u2082.op\n (x (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f\n (_ :\n (GrothendieckTopology.Cover.sieve T).arrows\n (MulticospanIndex.sndTo (GrothendieckTopology.Cover.index T P) I).f)))\ne : E \u27f6 P.obj (op X)\nhe : Presieve.FamilyOfElements.IsAmalgamation x e\nb : (GrothendieckTopology.Cover.index T P).R\n\u22a2 e \u226b\n NatTrans.app (GrothendieckTopology.Cover.multifork T P).\u03c0\n (WalkingMulticospan.left (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) b)) =\n NatTrans.app K.\u03c0 (WalkingMulticospan.left (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index T P) b))\n[PROOFSTEP]\napply he\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b9 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\n\u22a2 IsSheaf J P \u2194 \u2200 (X : C) (S : GrothendieckTopology.Cover J X), IsIso (GrothendieckTopology.Cover.toMultiequalizer S P)\n[PROOFSTEP]\nrw [isSheaf_iff_multifork]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b9 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\n\u22a2 (\u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))) \u2194\n \u2200 (X : C) (S : GrothendieckTopology.Cover J X), IsIso (GrothendieckTopology.Cover.toMultiequalizer S P)\n[PROOFSTEP]\nrefine' forall\u2082_congr fun X S => \u27e8_, _\u27e9\n[GOAL]\ncase refine'_1\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b9 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\nX : C\nS : GrothendieckTopology.Cover J X\n\u22a2 Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P)) \u2192\n IsIso (GrothendieckTopology.Cover.toMultiequalizer S P)\n[PROOFSTEP]\nrintro \u27e8h\u27e9\n[GOAL]\ncase refine'_1.intro\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b9 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\nX : C\nS : GrothendieckTopology.Cover J X\nh : IsLimit (GrothendieckTopology.Cover.multifork S P)\n\u22a2 IsIso (GrothendieckTopology.Cover.toMultiequalizer S P)\n[PROOFSTEP]\nlet e : P.obj (op X) \u2245 multiequalizer (S.index P) := h.conePointUniqueUpToIso (limit.isLimit _)\n[GOAL]\ncase refine'_1.intro\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b9 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\nX : C\nS : GrothendieckTopology.Cover J X\nh : IsLimit (GrothendieckTopology.Cover.multifork S P)\ne : P.obj (op X) \u2245 multiequalizer (GrothendieckTopology.Cover.index S P) :=\n IsLimit.conePointUniqueUpToIso h (limit.isLimit (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)))\n\u22a2 IsIso (GrothendieckTopology.Cover.toMultiequalizer S P)\n[PROOFSTEP]\nexact (inferInstance : IsIso e.hom)\n[GOAL]\ncase refine'_2\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b9 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\nX : C\nS : GrothendieckTopology.Cover J X\n\u22a2 IsIso (GrothendieckTopology.Cover.toMultiequalizer S P) \u2192\n Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\n[PROOFSTEP]\nintro h\n[GOAL]\ncase refine'_2\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b9 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\nX : C\nS : GrothendieckTopology.Cover J X\nh : IsIso (GrothendieckTopology.Cover.toMultiequalizer S P)\n\u22a2 Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S P))\n[PROOFSTEP]\nrefine' \u27e8IsLimit.ofIsoLimit (limit.isLimit _) (Cones.ext _ _)\u27e9\n[GOAL]\ncase refine'_2.refine'_1\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b9 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\nX : C\nS : GrothendieckTopology.Cover J X\nh : IsIso (GrothendieckTopology.Cover.toMultiequalizer S P)\n\u22a2 (limit.cone (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P))).pt \u2245\n (GrothendieckTopology.Cover.multifork S P).pt\n[PROOFSTEP]\napply (@asIso _ _ _ _ _ h).symm\n[GOAL]\ncase refine'_2.refine'_2\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b9 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\nX : C\nS : GrothendieckTopology.Cover J X\nh : IsIso (GrothendieckTopology.Cover.toMultiequalizer S P)\n\u22a2 \u2200 (j : WalkingMulticospan (GrothendieckTopology.Cover.index S P).fstTo (GrothendieckTopology.Cover.index S P).sndTo),\n NatTrans.app (limit.cone (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P))).\u03c0 j =\n (asIso (GrothendieckTopology.Cover.toMultiequalizer S P)).symm.hom \u226b\n NatTrans.app (GrothendieckTopology.Cover.multifork S P).\u03c0 j\n[PROOFSTEP]\nintro a\n[GOAL]\ncase refine'_2.refine'_2\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b9 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\nX : C\nS : GrothendieckTopology.Cover J X\nh : IsIso (GrothendieckTopology.Cover.toMultiequalizer S P)\na : WalkingMulticospan (GrothendieckTopology.Cover.index S P).fstTo (GrothendieckTopology.Cover.index S P).sndTo\n\u22a2 NatTrans.app (limit.cone (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P))).\u03c0 a =\n (asIso (GrothendieckTopology.Cover.toMultiequalizer S P)).symm.hom \u226b\n NatTrans.app (GrothendieckTopology.Cover.multifork S P).\u03c0 a\n[PROOFSTEP]\nsymm\n[GOAL]\ncase refine'_2.refine'_2\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b9 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\nX : C\nS : GrothendieckTopology.Cover J X\nh : IsIso (GrothendieckTopology.Cover.toMultiequalizer S P)\na : WalkingMulticospan (GrothendieckTopology.Cover.index S P).fstTo (GrothendieckTopology.Cover.index S P).sndTo\n\u22a2 (asIso (GrothendieckTopology.Cover.toMultiequalizer S P)).symm.hom \u226b\n NatTrans.app (GrothendieckTopology.Cover.multifork S P).\u03c0 a =\n NatTrans.app (limit.cone (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P))).\u03c0 a\n[PROOFSTEP]\nerw [IsIso.inv_comp_eq]\n[GOAL]\ncase refine'_2.refine'_2\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b9 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\nX : C\nS : GrothendieckTopology.Cover J X\nh : IsIso (GrothendieckTopology.Cover.toMultiequalizer S P)\na : WalkingMulticospan (GrothendieckTopology.Cover.index S P).fstTo (GrothendieckTopology.Cover.index S P).sndTo\n\u22a2 NatTrans.app (GrothendieckTopology.Cover.multifork S P).\u03c0 a =\n GrothendieckTopology.Cover.toMultiequalizer S P \u226b\n NatTrans.app (limit.cone (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P))).\u03c0 a\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_2.refine'_2\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b9 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d : \u2200 (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\nX : C\nS : GrothendieckTopology.Cover J X\nh : IsIso (GrothendieckTopology.Cover.toMultiequalizer S P)\na : WalkingMulticospan (GrothendieckTopology.Cover.index S P).fstTo (GrothendieckTopology.Cover.index S P).sndTo\n\u22a2 (match a with\n | WalkingMulticospan.left a => P.map a.f.op\n | WalkingMulticospan.right b =>\n P.map (MulticospanIndex.fstTo (GrothendieckTopology.Cover.index S P) b).f.op \u226b\n MulticospanIndex.fst (GrothendieckTopology.Cover.index S P) b) =\n GrothendieckTopology.Cover.toMultiequalizer S P \u226b\n limit.\u03c0 (MulticospanIndex.multicospan (GrothendieckTopology.Cover.index S P)) a\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\n\u22a2 forkMap R P \u226b firstMap R P = forkMap R P \u226b secondMap R P\n[PROOFSTEP]\napply limit.hom_ext\n[GOAL]\ncase w\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\n\u22a2 \u2200 (j : Discrete (((V : C) \u00d7 { f // R f }) \u00d7 (W : C) \u00d7 { g // R g })),\n (forkMap R P \u226b firstMap R P) \u226b\n limit.\u03c0 (Discrete.functor fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd))) j =\n (forkMap R P \u226b secondMap R P) \u226b\n limit.\u03c0 (Discrete.functor fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd))) j\n[PROOFSTEP]\nrintro \u27e8\u27e8Y, f, hf\u27e9, \u27e8Z, g, hg\u27e9\u27e9\n[GOAL]\ncase w.mk.mk.mk.mk.mk.mk\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\nY : C\nf : Y \u27f6 U\nhf : R f\nZ : C\ng : Z \u27f6 U\nhg : R g\n\u22a2 (forkMap R P \u226b firstMap R P) \u226b\n limit.\u03c0 (Discrete.functor fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))\n {\n as :=\n ({ fst := Y, snd := { val := f, property := hf } }, { fst := Z, snd := { val := g, property := hg } }) } =\n (forkMap R P \u226b secondMap R P) \u226b\n limit.\u03c0 (Discrete.functor fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))\n { as := ({ fst := Y, snd := { val := f, property := hf } }, { fst := Z, snd := { val := g, property := hg } }) }\n[PROOFSTEP]\nsimp only [firstMap, secondMap, forkMap, limit.lift_\u03c0, limit.lift_\u03c0_assoc, assoc, Fan.mk_\u03c0_app, Subtype.coe_mk]\n[GOAL]\ncase w.mk.mk.mk.mk.mk.mk\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\nY : C\nf : Y \u27f6 U\nhf : R f\nZ : C\ng : Z \u27f6 U\nhg : R g\n\u22a2 P.map f.op \u226b P.map pullback.fst.op = P.map g.op \u226b P.map pullback.snd.op\n[PROOFSTEP]\nrw [\u2190 P.map_comp, \u2190 op_comp, pullback.condition]\n[GOAL]\ncase w.mk.mk.mk.mk.mk.mk\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\nY : C\nf : Y \u27f6 U\nhf : R f\nZ : C\ng : Z \u27f6 U\nhg : R g\n\u22a2 P.map (pullback.snd \u226b g).op = P.map g.op \u226b P.map pullback.snd.op\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP\u271d : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasProducts A\ninst\u271d\u00b2 : HasProducts A'\ninst\u271d\u00b9 : HasPullbacks C\nP : C\u1d52\u1d56 \u2964 A\ns : A \u2964 Type (max v\u2081 u\u2081)\ninst\u271d : (J : Type (max v\u2081 u\u2081)) \u2192 PreservesLimitsOfShape (Discrete J) s\nU : C\nR : Presieve U\n\u22a2 IsLimit (s.mapCone (Fork.of\u03b9 (forkMap R P) (_ : forkMap R P \u226b firstMap R P = forkMap R P \u226b secondMap R P))) \u2243\n IsLimit\n (Fork.of\u03b9 (Equalizer.forkMap (P \u22d9 s) R)\n (_ :\n Equalizer.forkMap (P \u22d9 s) R \u226b Equalizer.Presieve.firstMap (P \u22d9 s) R =\n Equalizer.forkMap (P \u22d9 s) R \u226b Equalizer.Presieve.secondMap (P \u22d9 s) R))\n[PROOFSTEP]\napply Equiv.trans (isLimitMapConeForkEquiv _ _) _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP\u271d : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasProducts A\ninst\u271d\u00b2 : HasProducts A'\ninst\u271d\u00b9 : HasPullbacks C\nP : C\u1d52\u1d56 \u2964 A\ns : A \u2964 Type (max v\u2081 u\u2081)\ninst\u271d : (J : Type (max v\u2081 u\u2081)) \u2192 PreservesLimitsOfShape (Discrete J) s\nU : C\nR : Presieve U\n\u22a2 IsLimit\n (Fork.of\u03b9 (s.map (forkMap R P))\n (_ : s.map (forkMap R P) \u226b s.map (firstMap R P) = s.map (forkMap R P) \u226b s.map (secondMap R P))) \u2243\n IsLimit\n (Fork.of\u03b9 (Equalizer.forkMap (P \u22d9 s) R)\n (_ :\n Equalizer.forkMap (P \u22d9 s) R \u226b Equalizer.Presieve.firstMap (P \u22d9 s) R =\n Equalizer.forkMap (P \u22d9 s) R \u226b Equalizer.Presieve.secondMap (P \u22d9 s) R))\n[PROOFSTEP]\napply (IsLimit.postcomposeHomEquiv _ _).symm.trans (IsLimit.equivIsoLimit _)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP\u271d : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasProducts A\ninst\u271d\u00b2 : HasProducts A'\ninst\u271d\u00b9 : HasPullbacks C\nP : C\u1d52\u1d56 \u2964 A\ns : A \u2964 Type (max v\u2081 u\u2081)\ninst\u271d : (J : Type (max v\u2081 u\u2081)) \u2192 PreservesLimitsOfShape (Discrete J) s\nU : C\nR : Presieve U\n\u22a2 parallelPair (s.map (firstMap R P)) (s.map (secondMap R P)) \u2245\n parallelPair (Equalizer.Presieve.firstMap (P \u22d9 s) R) (Equalizer.Presieve.secondMap (P \u22d9 s) R)\n[PROOFSTEP]\napply NatIso.ofComponents _ _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP\u271d : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasProducts A\ninst\u271d\u00b2 : HasProducts A'\ninst\u271d\u00b9 : HasPullbacks C\nP : C\u1d52\u1d56 \u2964 A\ns : A \u2964 Type (max v\u2081 u\u2081)\ninst\u271d : (J : Type (max v\u2081 u\u2081)) \u2192 PreservesLimitsOfShape (Discrete J) s\nU : C\nR : Presieve U\n\u22a2 (X : WalkingParallelPair) \u2192\n (parallelPair (s.map (firstMap R P)) (s.map (secondMap R P))).obj X \u2245\n (parallelPair (Equalizer.Presieve.firstMap (P \u22d9 s) R) (Equalizer.Presieve.secondMap (P \u22d9 s) R)).obj X\n[PROOFSTEP]\nrintro (_ | _)\n[GOAL]\ncase zero\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP\u271d : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasProducts A\ninst\u271d\u00b2 : HasProducts A'\ninst\u271d\u00b9 : HasPullbacks C\nP : C\u1d52\u1d56 \u2964 A\ns : A \u2964 Type (max v\u2081 u\u2081)\ninst\u271d : (J : Type (max v\u2081 u\u2081)) \u2192 PreservesLimitsOfShape (Discrete J) s\nU : C\nR : Presieve U\n\u22a2 (parallelPair (s.map (firstMap R P)) (s.map (secondMap R P))).obj WalkingParallelPair.zero \u2245\n (parallelPair (Equalizer.Presieve.firstMap (P \u22d9 s) R) (Equalizer.Presieve.secondMap (P \u22d9 s) R)).obj\n WalkingParallelPair.zero\n[PROOFSTEP]\napply PreservesProduct.iso s\n[GOAL]\ncase one\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP\u271d : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasProducts A\ninst\u271d\u00b2 : HasProducts A'\ninst\u271d\u00b9 : HasPullbacks C\nP : C\u1d52\u1d56 \u2964 A\ns : A \u2964 Type (max v\u2081 u\u2081)\ninst\u271d : (J : Type (max v\u2081 u\u2081)) \u2192 PreservesLimitsOfShape (Discrete J) s\nU : C\nR : Presieve U\n\u22a2 (parallelPair (s.map (firstMap R P)) (s.map (secondMap R P))).obj WalkingParallelPair.one \u2245\n (parallelPair (Equalizer.Presieve.firstMap (P \u22d9 s) R) (Equalizer.Presieve.secondMap (P \u22d9 s) R)).obj\n WalkingParallelPair.one\n[PROOFSTEP]\napply PreservesProduct.iso s\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP\u271d : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasProducts A\ninst\u271d\u00b2 : HasProducts A'\ninst\u271d\u00b9 : HasPullbacks C\nP : C\u1d52\u1d56 \u2964 A\ns : A \u2964 Type (max v\u2081 u\u2081)\ninst\u271d : (J : Type (max v\u2081 u\u2081)) \u2192 PreservesLimitsOfShape (Discrete J) s\nU : C\nR : Presieve U\n\u22a2 \u2200 {X Y : WalkingParallelPair} (f : X \u27f6 Y),\n (parallelPair (s.map (firstMap R P)) (s.map (secondMap R P))).map f \u226b\n (WalkingParallelPair.casesOn Y (PreservesProduct.iso s fun f => P.obj (op f.fst))\n (PreservesProduct.iso s fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))).hom =\n (WalkingParallelPair.casesOn X (PreservesProduct.iso s fun f => P.obj (op f.fst))\n (PreservesProduct.iso s fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))).hom \u226b\n (parallelPair (Equalizer.Presieve.firstMap (P \u22d9 s) R) (Equalizer.Presieve.secondMap (P \u22d9 s) R)).map f\n[PROOFSTEP]\nrintro _ _ (_ | _)\n[GOAL]\ncase left\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP\u271d : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasProducts A\ninst\u271d\u00b2 : HasProducts A'\ninst\u271d\u00b9 : HasPullbacks C\nP : C\u1d52\u1d56 \u2964 A\ns : A \u2964 Type (max v\u2081 u\u2081)\ninst\u271d : (J : Type (max v\u2081 u\u2081)) \u2192 PreservesLimitsOfShape (Discrete J) s\nU : C\nR : Presieve U\n\u22a2 (parallelPair (s.map (firstMap R P)) (s.map (secondMap R P))).map WalkingParallelPairHom.left \u226b\n (WalkingParallelPair.casesOn WalkingParallelPair.one (PreservesProduct.iso s fun f => P.obj (op f.fst))\n (PreservesProduct.iso s fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))).hom =\n (WalkingParallelPair.casesOn WalkingParallelPair.zero (PreservesProduct.iso s fun f => P.obj (op f.fst))\n (PreservesProduct.iso s fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))).hom \u226b\n (parallelPair (Equalizer.Presieve.firstMap (P \u22d9 s) R) (Equalizer.Presieve.secondMap (P \u22d9 s) R)).map\n WalkingParallelPairHom.left\n[PROOFSTEP]\nrefine' limit.hom_ext (fun j => _)\n[GOAL]\ncase left\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP\u271d : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasProducts A\ninst\u271d\u00b2 : HasProducts A'\ninst\u271d\u00b9 : HasPullbacks C\nP : C\u1d52\u1d56 \u2964 A\ns : A \u2964 Type (max v\u2081 u\u2081)\ninst\u271d : (J : Type (max v\u2081 u\u2081)) \u2192 PreservesLimitsOfShape (Discrete J) s\nU : C\nR : Presieve U\nj : Discrete (((Y : C) \u00d7 { f // R f }) \u00d7 (Z : C) \u00d7 { g // R g })\n\u22a2 ((parallelPair (s.map (firstMap R P)) (s.map (secondMap R P))).map WalkingParallelPairHom.left \u226b\n (WalkingParallelPair.casesOn WalkingParallelPair.one (PreservesProduct.iso s fun f => P.obj (op f.fst))\n (PreservesProduct.iso s fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))).hom) \u226b\n limit.\u03c0 (Discrete.functor fun fg => (P \u22d9 s).obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd))) j =\n ((WalkingParallelPair.casesOn WalkingParallelPair.zero (PreservesProduct.iso s fun f => P.obj (op f.fst))\n (PreservesProduct.iso s fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))).hom \u226b\n (parallelPair (Equalizer.Presieve.firstMap (P \u22d9 s) R) (Equalizer.Presieve.secondMap (P \u22d9 s) R)).map\n WalkingParallelPairHom.left) \u226b\n limit.\u03c0 (Discrete.functor fun fg => (P \u22d9 s).obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd))) j\n[PROOFSTEP]\ndsimp [Equalizer.Presieve.firstMap, firstMap]\n[GOAL]\ncase left\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP\u271d : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasProducts A\ninst\u271d\u00b2 : HasProducts A'\ninst\u271d\u00b9 : HasPullbacks C\nP : C\u1d52\u1d56 \u2964 A\ns : A \u2964 Type (max v\u2081 u\u2081)\ninst\u271d : (J : Type (max v\u2081 u\u2081)) \u2192 PreservesLimitsOfShape (Discrete J) s\nU : C\nR : Presieve U\nj : Discrete (((Y : C) \u00d7 { f // R f }) \u00d7 (Z : C) \u00d7 { g // R g })\n\u22a2 (s.map (Pi.lift fun x => Pi.\u03c0 (fun f => P.obj (op f.fst)) x.fst \u226b P.map pullback.fst.op) \u226b\n piComparison s fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd))) \u226b\n limit.\u03c0 (Discrete.functor fun fg => s.obj (P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))) j =\n ((piComparison s fun f => P.obj (op f.fst)) \u226b\n Pi.lift fun x => Pi.\u03c0 (fun f => s.obj (P.obj (op f.fst))) x.fst \u226b s.map (P.map pullback.fst.op)) \u226b\n limit.\u03c0 (Discrete.functor fun fg => s.obj (P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))) j\n[PROOFSTEP]\nsimp only [limit.lift_\u03c0, map_lift_piComparison, assoc, Fan.mk_\u03c0_app, Functor.map_comp]\n[GOAL]\ncase left\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP\u271d : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasProducts A\ninst\u271d\u00b2 : HasProducts A'\ninst\u271d\u00b9 : HasPullbacks C\nP : C\u1d52\u1d56 \u2964 A\ns : A \u2964 Type (max v\u2081 u\u2081)\ninst\u271d : (J : Type (max v\u2081 u\u2081)) \u2192 PreservesLimitsOfShape (Discrete J) s\nU : C\nR : Presieve U\nj : Discrete (((Y : C) \u00d7 { f // R f }) \u00d7 (Z : C) \u00d7 { g // R g })\n\u22a2 s.map (Pi.\u03c0 (fun f => P.obj (op f.fst)) j.as.fst) \u226b s.map (P.map pullback.fst.op) =\n (piComparison s fun f => P.obj (op f.fst)) \u226b\n Pi.\u03c0 (fun f => s.obj (P.obj (op f.fst))) j.as.fst \u226b s.map (P.map pullback.fst.op)\n[PROOFSTEP]\nrw [piComparison_comp_\u03c0_assoc]\n[GOAL]\ncase right\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP\u271d : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasProducts A\ninst\u271d\u00b2 : HasProducts A'\ninst\u271d\u00b9 : HasPullbacks C\nP : C\u1d52\u1d56 \u2964 A\ns : A \u2964 Type (max v\u2081 u\u2081)\ninst\u271d : (J : Type (max v\u2081 u\u2081)) \u2192 PreservesLimitsOfShape (Discrete J) s\nU : C\nR : Presieve U\n\u22a2 (parallelPair (s.map (firstMap R P)) (s.map (secondMap R P))).map WalkingParallelPairHom.right \u226b\n (WalkingParallelPair.casesOn WalkingParallelPair.one (PreservesProduct.iso s fun f => P.obj (op f.fst))\n (PreservesProduct.iso s fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))).hom =\n (WalkingParallelPair.casesOn WalkingParallelPair.zero (PreservesProduct.iso s fun f => P.obj (op f.fst))\n (PreservesProduct.iso s fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))).hom \u226b\n (parallelPair (Equalizer.Presieve.firstMap (P \u22d9 s) R) (Equalizer.Presieve.secondMap (P \u22d9 s) R)).map\n WalkingParallelPairHom.right\n[PROOFSTEP]\nrefine' limit.hom_ext (fun j => _)\n[GOAL]\ncase right\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP\u271d : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasProducts A\ninst\u271d\u00b2 : HasProducts A'\ninst\u271d\u00b9 : HasPullbacks C\nP : C\u1d52\u1d56 \u2964 A\ns : A \u2964 Type (max v\u2081 u\u2081)\ninst\u271d : (J : Type (max v\u2081 u\u2081)) \u2192 PreservesLimitsOfShape (Discrete J) s\nU : C\nR : Presieve U\nj : Discrete (((Y : C) \u00d7 { f // R f }) \u00d7 (Z : C) \u00d7 { g // R g })\n\u22a2 ((parallelPair (s.map (firstMap R P)) (s.map (secondMap R P))).map WalkingParallelPairHom.right \u226b\n (WalkingParallelPair.casesOn WalkingParallelPair.one (PreservesProduct.iso s fun f => P.obj (op f.fst))\n (PreservesProduct.iso s fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))).hom) \u226b\n limit.\u03c0 (Discrete.functor fun fg => (P \u22d9 s).obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd))) j =\n ((WalkingParallelPair.casesOn WalkingParallelPair.zero (PreservesProduct.iso s fun f => P.obj (op f.fst))\n (PreservesProduct.iso s fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))).hom \u226b\n (parallelPair (Equalizer.Presieve.firstMap (P \u22d9 s) R) (Equalizer.Presieve.secondMap (P \u22d9 s) R)).map\n WalkingParallelPairHom.right) \u226b\n limit.\u03c0 (Discrete.functor fun fg => (P \u22d9 s).obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd))) j\n[PROOFSTEP]\ndsimp [Equalizer.Presieve.secondMap, secondMap]\n[GOAL]\ncase right\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP\u271d : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasProducts A\ninst\u271d\u00b2 : HasProducts A'\ninst\u271d\u00b9 : HasPullbacks C\nP : C\u1d52\u1d56 \u2964 A\ns : A \u2964 Type (max v\u2081 u\u2081)\ninst\u271d : (J : Type (max v\u2081 u\u2081)) \u2192 PreservesLimitsOfShape (Discrete J) s\nU : C\nR : Presieve U\nj : Discrete (((Y : C) \u00d7 { f // R f }) \u00d7 (Z : C) \u00d7 { g // R g })\n\u22a2 (s.map (Pi.lift fun x => Pi.\u03c0 (fun f => P.obj (op f.fst)) x.snd \u226b P.map pullback.snd.op) \u226b\n piComparison s fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd))) \u226b\n limit.\u03c0 (Discrete.functor fun fg => s.obj (P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))) j =\n ((piComparison s fun f => P.obj (op f.fst)) \u226b\n Pi.lift fun x => Pi.\u03c0 (fun f => s.obj (P.obj (op f.fst))) x.snd \u226b s.map (P.map pullback.snd.op)) \u226b\n limit.\u03c0 (Discrete.functor fun fg => s.obj (P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))) j\n[PROOFSTEP]\nsimp only [limit.lift_\u03c0, map_lift_piComparison, assoc, Fan.mk_\u03c0_app, Functor.map_comp]\n[GOAL]\ncase right\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP\u271d : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasProducts A\ninst\u271d\u00b2 : HasProducts A'\ninst\u271d\u00b9 : HasPullbacks C\nP : C\u1d52\u1d56 \u2964 A\ns : A \u2964 Type (max v\u2081 u\u2081)\ninst\u271d : (J : Type (max v\u2081 u\u2081)) \u2192 PreservesLimitsOfShape (Discrete J) s\nU : C\nR : Presieve U\nj : Discrete (((Y : C) \u00d7 { f // R f }) \u00d7 (Z : C) \u00d7 { g // R g })\n\u22a2 s.map (Pi.\u03c0 (fun f => P.obj (op f.fst)) j.as.snd) \u226b s.map (P.map pullback.snd.op) =\n (piComparison s fun f => P.obj (op f.fst)) \u226b\n Pi.\u03c0 (fun f => s.obj (P.obj (op f.fst))) j.as.snd \u226b s.map (P.map pullback.snd.op)\n[PROOFSTEP]\nrw [piComparison_comp_\u03c0_assoc]\n[GOAL]\ncase id\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP\u271d : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasProducts A\ninst\u271d\u00b2 : HasProducts A'\ninst\u271d\u00b9 : HasPullbacks C\nP : C\u1d52\u1d56 \u2964 A\ns : A \u2964 Type (max v\u2081 u\u2081)\ninst\u271d : (J : Type (max v\u2081 u\u2081)) \u2192 PreservesLimitsOfShape (Discrete J) s\nU : C\nR : Presieve U\nX\u271d : WalkingParallelPair\n\u22a2 (parallelPair (s.map (firstMap R P)) (s.map (secondMap R P))).map (WalkingParallelPairHom.id X\u271d) \u226b\n (WalkingParallelPair.casesOn X\u271d (PreservesProduct.iso s fun f => P.obj (op f.fst))\n (PreservesProduct.iso s fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))).hom =\n (WalkingParallelPair.casesOn X\u271d (PreservesProduct.iso s fun f => P.obj (op f.fst))\n (PreservesProduct.iso s fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))).hom \u226b\n (parallelPair (Equalizer.Presieve.firstMap (P \u22d9 s) R) (Equalizer.Presieve.secondMap (P \u22d9 s) R)).map\n (WalkingParallelPairHom.id X\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase id\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP\u271d : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasProducts A\ninst\u271d\u00b2 : HasProducts A'\ninst\u271d\u00b9 : HasPullbacks C\nP : C\u1d52\u1d56 \u2964 A\ns : A \u2964 Type (max v\u2081 u\u2081)\ninst\u271d : (J : Type (max v\u2081 u\u2081)) \u2192 PreservesLimitsOfShape (Discrete J) s\nU : C\nR : Presieve U\nX\u271d : WalkingParallelPair\n\u22a2 (parallelPair (s.map (firstMap R P)) (s.map (secondMap R P))).map (\ud835\udfd9 X\u271d) \u226b\n (WalkingParallelPair.rec (PreservesProduct.iso s fun f => P.obj (op f.fst))\n (PreservesProduct.iso s fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd))) X\u271d).hom =\n (WalkingParallelPair.rec (PreservesProduct.iso s fun f => P.obj (op f.fst))\n (PreservesProduct.iso s fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd))) X\u271d).hom \u226b\n (parallelPair (Equalizer.Presieve.firstMap (P \u22d9 s) R) (Equalizer.Presieve.secondMap (P \u22d9 s) R)).map (\ud835\udfd9 X\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP\u271d : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasProducts A\ninst\u271d\u00b2 : HasProducts A'\ninst\u271d\u00b9 : HasPullbacks C\nP : C\u1d52\u1d56 \u2964 A\ns : A \u2964 Type (max v\u2081 u\u2081)\ninst\u271d : (J : Type (max v\u2081 u\u2081)) \u2192 PreservesLimitsOfShape (Discrete J) s\nU : C\nR : Presieve U\n\u22a2 (Cones.postcompose\n (NatIso.ofComponents fun X =>\n WalkingParallelPair.casesOn X (PreservesProduct.iso s fun f => P.obj (op f.fst))\n (PreservesProduct.iso s fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))).hom).obj\n (Fork.of\u03b9 (s.map (forkMap R P))\n (_ : s.map (forkMap R P) \u226b s.map (firstMap R P) = s.map (forkMap R P) \u226b s.map (secondMap R P))) \u2245\n Fork.of\u03b9 (Equalizer.forkMap (P \u22d9 s) R)\n (_ :\n Equalizer.forkMap (P \u22d9 s) R \u226b Equalizer.Presieve.firstMap (P \u22d9 s) R =\n Equalizer.forkMap (P \u22d9 s) R \u226b Equalizer.Presieve.secondMap (P \u22d9 s) R)\n[PROOFSTEP]\nrefine' Fork.ext (Iso.refl _) _\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP\u271d : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasProducts A\ninst\u271d\u00b2 : HasProducts A'\ninst\u271d\u00b9 : HasPullbacks C\nP : C\u1d52\u1d56 \u2964 A\ns : A \u2964 Type (max v\u2081 u\u2081)\ninst\u271d : (J : Type (max v\u2081 u\u2081)) \u2192 PreservesLimitsOfShape (Discrete J) s\nU : C\nR : Presieve U\n\u22a2 (Iso.refl\n ((Cones.postcompose\n (NatIso.ofComponents fun X =>\n WalkingParallelPair.casesOn X (PreservesProduct.iso s fun f => P.obj (op f.fst))\n (PreservesProduct.iso s fun fg =>\n P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))).hom).obj\n (Fork.of\u03b9 (s.map (forkMap R P))\n (_ :\n s.map (forkMap R P) \u226b s.map (firstMap R P) = s.map (forkMap R P) \u226b s.map (secondMap R P)))).pt).hom \u226b\n Fork.\u03b9\n (Fork.of\u03b9 (Equalizer.forkMap (P \u22d9 s) R)\n (_ :\n Equalizer.forkMap (P \u22d9 s) R \u226b Equalizer.Presieve.firstMap (P \u22d9 s) R =\n Equalizer.forkMap (P \u22d9 s) R \u226b Equalizer.Presieve.secondMap (P \u22d9 s) R)) =\n Fork.\u03b9\n ((Cones.postcompose\n (NatIso.ofComponents fun X =>\n WalkingParallelPair.casesOn X (PreservesProduct.iso s fun f => P.obj (op f.fst))\n (PreservesProduct.iso s fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd)))).hom).obj\n (Fork.of\u03b9 (s.map (forkMap R P))\n (_ : s.map (forkMap R P) \u226b s.map (firstMap R P) = s.map (forkMap R P) \u226b s.map (secondMap R P))))\n[PROOFSTEP]\ndsimp [Equalizer.forkMap, forkMap]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP\u271d : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasProducts A\ninst\u271d\u00b2 : HasProducts A'\ninst\u271d\u00b9 : HasPullbacks C\nP : C\u1d52\u1d56 \u2964 A\ns : A \u2964 Type (max v\u2081 u\u2081)\ninst\u271d : (J : Type (max v\u2081 u\u2081)) \u2192 PreservesLimitsOfShape (Discrete J) s\nU : C\nR : Presieve U\n\u22a2 (\ud835\udfd9 (s.obj (P.obj (op U))) \u226b Pi.lift fun f => s.map (P.map (\u2191f.snd).op)) =\n Fork.\u03b9\n ((Cones.postcompose\n (NatIso.ofComponents fun X =>\n WalkingParallelPair.rec (PreservesProduct.iso s fun f => P.obj (op f.fst))\n (PreservesProduct.iso s fun fg => P.obj (op (Limits.pullback \u2191fg.fst.snd \u2191fg.snd.snd))) X).hom).obj\n (Fork.of\u03b9 (s.map (Pi.lift fun f => P.map (\u2191f.snd).op))\n (_ :\n s.map (Pi.lift fun f => P.map (\u2191f.snd).op) \u226b s.map (firstMap R P) =\n s.map (Pi.lift fun f => P.map (\u2191f.snd).op) \u226b s.map (secondMap R P))))\n[PROOFSTEP]\nsimp [Fork.\u03b9]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\n\u22a2 IsSheaf J P' \u2194 IsSheaf' J P'\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\n\u22a2 IsSheaf J P' \u2192 IsSheaf' J P'\n[PROOFSTEP]\nintro h U R hR\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\nh : IsSheaf J P'\nU : C\nR : Presieve U\nhR : generate R \u2208 GrothendieckTopology.sieves J U\n\u22a2 Nonempty (IsLimit (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P')))\n[PROOFSTEP]\nrefine' \u27e8_\u27e9\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\nh : IsSheaf J P'\nU : C\nR : Presieve U\nhR : generate R \u2208 GrothendieckTopology.sieves J U\n\u22a2 IsLimit (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P'))\n[PROOFSTEP]\napply coyonedaJointlyReflectsLimits\n[GOAL]\ncase mp.t\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\nh : IsSheaf J P'\nU : C\nR : Presieve U\nhR : generate R \u2208 GrothendieckTopology.sieves J U\n\u22a2 (X : A'\u1d52\u1d56) \u2192\n IsLimit\n ((coyoneda.obj X).mapCone\n (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P')))\n[PROOFSTEP]\nintro X\n[GOAL]\ncase mp.t\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\nh : IsSheaf J P'\nU : C\nR : Presieve U\nhR : generate R \u2208 GrothendieckTopology.sieves J U\nX : A'\u1d52\u1d56\n\u22a2 IsLimit\n ((coyoneda.obj X).mapCone\n (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P')))\n[PROOFSTEP]\nhave q : Presieve.IsSheafFor (P' \u22d9 coyoneda.obj X) _ := h X.unop _ hR\n[GOAL]\ncase mp.t\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\nh : IsSheaf J P'\nU : C\nR : Presieve U\nhR : generate R \u2208 GrothendieckTopology.sieves J U\nX : A'\u1d52\u1d56\nq : Presieve.IsSheafFor (P' \u22d9 coyoneda.obj X) (generate R).arrows\n\u22a2 IsLimit\n ((coyoneda.obj X).mapCone\n (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P')))\n[PROOFSTEP]\nrw [\u2190 Presieve.isSheafFor_iff_generate] at q \n[GOAL]\ncase mp.t\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\nh : IsSheaf J P'\nU : C\nR : Presieve U\nhR : generate R \u2208 GrothendieckTopology.sieves J U\nX : A'\u1d52\u1d56\nq : Presieve.IsSheafFor (P' \u22d9 coyoneda.obj X) R\n\u22a2 IsLimit\n ((coyoneda.obj X).mapCone\n (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P')))\n[PROOFSTEP]\nrw [Equalizer.Presieve.sheaf_condition] at q \n[GOAL]\ncase mp.t\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\nh : IsSheaf J P'\nU : C\nR : Presieve U\nhR : generate R \u2208 GrothendieckTopology.sieves J U\nX : A'\u1d52\u1d56\nq :\n Nonempty\n (IsLimit\n (Fork.of\u03b9 (Equalizer.forkMap (P' \u22d9 coyoneda.obj X) R)\n (_ :\n Equalizer.forkMap (P' \u22d9 coyoneda.obj X) R \u226b Equalizer.Presieve.firstMap (P' \u22d9 coyoneda.obj X) R =\n Equalizer.forkMap (P' \u22d9 coyoneda.obj X) R \u226b Equalizer.Presieve.secondMap (P' \u22d9 coyoneda.obj X) R)))\n\u22a2 IsLimit\n ((coyoneda.obj X).mapCone\n (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P')))\n[PROOFSTEP]\nreplace q := Classical.choice q\n[GOAL]\ncase mp.t\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\nh : IsSheaf J P'\nU : C\nR : Presieve U\nhR : generate R \u2208 GrothendieckTopology.sieves J U\nX : A'\u1d52\u1d56\nq :\n IsLimit\n (Fork.of\u03b9 (Equalizer.forkMap (P' \u22d9 coyoneda.obj X) R)\n (_ :\n Equalizer.forkMap (P' \u22d9 coyoneda.obj X) R \u226b Equalizer.Presieve.firstMap (P' \u22d9 coyoneda.obj X) R =\n Equalizer.forkMap (P' \u22d9 coyoneda.obj X) R \u226b Equalizer.Presieve.secondMap (P' \u22d9 coyoneda.obj X) R))\n\u22a2 IsLimit\n ((coyoneda.obj X).mapCone\n (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P')))\n[PROOFSTEP]\napply (isSheafForIsSheafFor' _ _ _ _).symm q\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\n\u22a2 IsSheaf' J P' \u2192 IsSheaf J P'\n[PROOFSTEP]\nintro h U X S hS\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\nh : IsSheaf' J P'\nU : A'\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\n\u22a2 Presieve.IsSheafFor (P' \u22d9 coyoneda.obj (op U)) S.arrows\n[PROOFSTEP]\nrw [Equalizer.Presieve.sheaf_condition]\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\nh : IsSheaf' J P'\nU : A'\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\n\u22a2 Nonempty\n (IsLimit\n (Fork.of\u03b9 (Equalizer.forkMap (P' \u22d9 coyoneda.obj (op U)) S.arrows)\n (_ :\n Equalizer.forkMap (P' \u22d9 coyoneda.obj (op U)) S.arrows \u226b\n Equalizer.Presieve.firstMap (P' \u22d9 coyoneda.obj (op U)) S.arrows =\n Equalizer.forkMap (P' \u22d9 coyoneda.obj (op U)) S.arrows \u226b\n Equalizer.Presieve.secondMap (P' \u22d9 coyoneda.obj (op U)) S.arrows)))\n[PROOFSTEP]\nrefine' \u27e8_\u27e9\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\nh : IsSheaf' J P'\nU : A'\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\n\u22a2 IsLimit\n (Fork.of\u03b9 (Equalizer.forkMap (P' \u22d9 coyoneda.obj (op U)) S.arrows)\n (_ :\n Equalizer.forkMap (P' \u22d9 coyoneda.obj (op U)) S.arrows \u226b\n Equalizer.Presieve.firstMap (P' \u22d9 coyoneda.obj (op U)) S.arrows =\n Equalizer.forkMap (P' \u22d9 coyoneda.obj (op U)) S.arrows \u226b\n Equalizer.Presieve.secondMap (P' \u22d9 coyoneda.obj (op U)) S.arrows))\n[PROOFSTEP]\nrefine' isSheafForIsSheafFor' _ _ _ _ _\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\nh : IsSheaf' J P'\nU : A'\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\n\u22a2 IsLimit\n ((coyoneda.obj (op U)).mapCone\n (Fork.of\u03b9 (forkMap S.arrows P')\n (_ : forkMap S.arrows P' \u226b firstMap S.arrows P' = forkMap S.arrows P' \u226b secondMap S.arrows P')))\n[PROOFSTEP]\nletI := preservesSmallestLimitsOfPreservesLimits (coyoneda.obj (op U))\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\nh : IsSheaf' J P'\nU : A'\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\nthis : PreservesLimitsOfSize.{0, 0, max u\u2081 v\u2081, max u\u2081 v\u2081, u\u2082, (max u\u2081 v\u2081) + 1} (coyoneda.obj (op U)) :=\n preservesSmallestLimitsOfPreservesLimits (coyoneda.obj (op U))\n\u22a2 IsLimit\n ((coyoneda.obj (op U)).mapCone\n (Fork.of\u03b9 (forkMap S.arrows P')\n (_ : forkMap S.arrows P' \u226b firstMap S.arrows P' = forkMap S.arrows P' \u226b secondMap S.arrows P')))\n[PROOFSTEP]\napply isLimitOfPreserves\n[GOAL]\ncase mpr.t\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\nh : IsSheaf' J P'\nU : A'\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\nthis : PreservesLimitsOfSize.{0, 0, max u\u2081 v\u2081, max u\u2081 v\u2081, u\u2082, (max u\u2081 v\u2081) + 1} (coyoneda.obj (op U)) :=\n preservesSmallestLimitsOfPreservesLimits (coyoneda.obj (op U))\n\u22a2 IsLimit\n (Fork.of\u03b9 (forkMap S.arrows P')\n (_ : forkMap S.arrows P' \u226b firstMap S.arrows P' = forkMap S.arrows P' \u226b secondMap S.arrows P'))\n[PROOFSTEP]\napply Classical.choice (h _ S.arrows _)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u00b3 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b2 : HasProducts A\ninst\u271d\u00b9 : HasProducts A'\ninst\u271d : HasPullbacks C\nh : IsSheaf' J P'\nU : A'\nX : C\nS : Sieve X\nhS : S \u2208 GrothendieckTopology.sieves J X\nthis : PreservesLimitsOfSize.{0, 0, max u\u2081 v\u2081, max u\u2081 v\u2081, u\u2082, (max u\u2081 v\u2081) + 1} (coyoneda.obj (op U)) :=\n preservesSmallestLimitsOfPreservesLimits (coyoneda.obj (op U))\n\u22a2 generate S.arrows \u2208 GrothendieckTopology.sieves J X\n[PROOFSTEP]\nsimpa\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasPullbacks C\ns : A' \u2964 Type (max v\u2081 u\u2081)\ninst\u271d\u00b2 : HasLimits A'\ninst\u271d\u00b9 : PreservesLimits s\ninst\u271d : ReflectsIsomorphisms s\n\u22a2 IsSheaf J P' \u2194 IsSheaf J (P' \u22d9 s)\n[PROOFSTEP]\nrw [isSheaf_iff_isSheaf', isSheaf_iff_isSheaf']\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasPullbacks C\ns : A' \u2964 Type (max v\u2081 u\u2081)\ninst\u271d\u00b2 : HasLimits A'\ninst\u271d\u00b9 : PreservesLimits s\ninst\u271d : ReflectsIsomorphisms s\n\u22a2 IsSheaf' J P' \u2194 IsSheaf' J (P' \u22d9 s)\n[PROOFSTEP]\nrefine' forall_congr' (fun U => ball_congr (fun R _ => _))\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasPullbacks C\ns : A' \u2964 Type (max v\u2081 u\u2081)\ninst\u271d\u00b2 : HasLimits A'\ninst\u271d\u00b9 : PreservesLimits s\ninst\u271d : ReflectsIsomorphisms s\nU : C\nR : Presieve U\nx\u271d : generate R \u2208 GrothendieckTopology.sieves J U\n\u22a2 Nonempty (IsLimit (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P'))) \u2194\n Nonempty\n (IsLimit\n (Fork.of\u03b9 (forkMap R (P' \u22d9 s))\n (_ : forkMap R (P' \u22d9 s) \u226b firstMap R (P' \u22d9 s) = forkMap R (P' \u22d9 s) \u226b secondMap R (P' \u22d9 s))))\n[PROOFSTEP]\nletI : ReflectsLimits s := reflectsLimitsOfReflectsIsomorphisms\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasPullbacks C\ns : A' \u2964 Type (max v\u2081 u\u2081)\ninst\u271d\u00b2 : HasLimits A'\ninst\u271d\u00b9 : PreservesLimits s\ninst\u271d : ReflectsIsomorphisms s\nU : C\nR : Presieve U\nx\u271d : generate R \u2208 GrothendieckTopology.sieves J U\nthis : ReflectsLimits s := reflectsLimitsOfReflectsIsomorphisms\n\u22a2 Nonempty (IsLimit (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P'))) \u2194\n Nonempty\n (IsLimit\n (Fork.of\u03b9 (forkMap R (P' \u22d9 s))\n (_ : forkMap R (P' \u22d9 s) \u226b firstMap R (P' \u22d9 s) = forkMap R (P' \u22d9 s) \u226b secondMap R (P' \u22d9 s))))\n[PROOFSTEP]\nhave : IsLimit (s.mapCone (Fork.of\u03b9 _ (w R P'))) \u2243 IsLimit (Fork.of\u03b9 _ (w R (P' \u22d9 s))) := isSheafForIsSheafFor' P' s U R\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasPullbacks C\ns : A' \u2964 Type (max v\u2081 u\u2081)\ninst\u271d\u00b2 : HasLimits A'\ninst\u271d\u00b9 : PreservesLimits s\ninst\u271d : ReflectsIsomorphisms s\nU : C\nR : Presieve U\nx\u271d : generate R \u2208 GrothendieckTopology.sieves J U\nthis\u271d : ReflectsLimits s := reflectsLimitsOfReflectsIsomorphisms\nthis :\n IsLimit (s.mapCone (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P'))) \u2243\n IsLimit\n (Fork.of\u03b9 (forkMap R (P' \u22d9 s))\n (_ : forkMap R (P' \u22d9 s) \u226b firstMap R (P' \u22d9 s) = forkMap R (P' \u22d9 s) \u226b secondMap R (P' \u22d9 s)))\n\u22a2 Nonempty (IsLimit (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P'))) \u2194\n Nonempty\n (IsLimit\n (Fork.of\u03b9 (forkMap R (P' \u22d9 s))\n (_ : forkMap R (P' \u22d9 s) \u226b firstMap R (P' \u22d9 s) = forkMap R (P' \u22d9 s) \u226b secondMap R (P' \u22d9 s))))\n[PROOFSTEP]\nrw [\u2190 Equiv.nonempty_congr this]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasPullbacks C\ns : A' \u2964 Type (max v\u2081 u\u2081)\ninst\u271d\u00b2 : HasLimits A'\ninst\u271d\u00b9 : PreservesLimits s\ninst\u271d : ReflectsIsomorphisms s\nU : C\nR : Presieve U\nx\u271d : generate R \u2208 GrothendieckTopology.sieves J U\nthis\u271d : ReflectsLimits s := reflectsLimitsOfReflectsIsomorphisms\nthis :\n IsLimit (s.mapCone (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P'))) \u2243\n IsLimit\n (Fork.of\u03b9 (forkMap R (P' \u22d9 s))\n (_ : forkMap R (P' \u22d9 s) \u226b firstMap R (P' \u22d9 s) = forkMap R (P' \u22d9 s) \u226b secondMap R (P' \u22d9 s)))\n\u22a2 Nonempty (IsLimit (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P'))) \u2194\n Nonempty\n (IsLimit (s.mapCone (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P'))))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasPullbacks C\ns : A' \u2964 Type (max v\u2081 u\u2081)\ninst\u271d\u00b2 : HasLimits A'\ninst\u271d\u00b9 : PreservesLimits s\ninst\u271d : ReflectsIsomorphisms s\nU : C\nR : Presieve U\nx\u271d : generate R \u2208 GrothendieckTopology.sieves J U\nthis\u271d : ReflectsLimits s := reflectsLimitsOfReflectsIsomorphisms\nthis :\n IsLimit (s.mapCone (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P'))) \u2243\n IsLimit\n (Fork.of\u03b9 (forkMap R (P' \u22d9 s))\n (_ : forkMap R (P' \u22d9 s) \u226b firstMap R (P' \u22d9 s) = forkMap R (P' \u22d9 s) \u226b secondMap R (P' \u22d9 s)))\n\u22a2 Nonempty (IsLimit (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P'))) \u2192\n Nonempty\n (IsLimit (s.mapCone (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P'))))\n[PROOFSTEP]\nhaveI := preservesSmallestLimitsOfPreservesLimits s\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasPullbacks C\ns : A' \u2964 Type (max v\u2081 u\u2081)\ninst\u271d\u00b2 : HasLimits A'\ninst\u271d\u00b9 : PreservesLimits s\ninst\u271d : ReflectsIsomorphisms s\nU : C\nR : Presieve U\nx\u271d : generate R \u2208 GrothendieckTopology.sieves J U\nthis\u271d\u00b9 : ReflectsLimits s := reflectsLimitsOfReflectsIsomorphisms\nthis\u271d :\n IsLimit (s.mapCone (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P'))) \u2243\n IsLimit\n (Fork.of\u03b9 (forkMap R (P' \u22d9 s))\n (_ : forkMap R (P' \u22d9 s) \u226b firstMap R (P' \u22d9 s) = forkMap R (P' \u22d9 s) \u226b secondMap R (P' \u22d9 s)))\nthis : PreservesLimitsOfSize.{0, 0, max u\u2081 v\u2081, max u\u2081 v\u2081, u\u2082, max (u\u2081 + 1) (v\u2081 + 1)} s\n\u22a2 Nonempty (IsLimit (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P'))) \u2192\n Nonempty\n (IsLimit (s.mapCone (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P'))))\n[PROOFSTEP]\nexact Nonempty.map fun t => isLimitOfPreserves s t\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasPullbacks C\ns : A' \u2964 Type (max v\u2081 u\u2081)\ninst\u271d\u00b2 : HasLimits A'\ninst\u271d\u00b9 : PreservesLimits s\ninst\u271d : ReflectsIsomorphisms s\nU : C\nR : Presieve U\nx\u271d : generate R \u2208 GrothendieckTopology.sieves J U\nthis\u271d : ReflectsLimits s := reflectsLimitsOfReflectsIsomorphisms\nthis :\n IsLimit (s.mapCone (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P'))) \u2243\n IsLimit\n (Fork.of\u03b9 (forkMap R (P' \u22d9 s))\n (_ : forkMap R (P' \u22d9 s) \u226b firstMap R (P' \u22d9 s) = forkMap R (P' \u22d9 s) \u226b secondMap R (P' \u22d9 s)))\n\u22a2 Nonempty\n (IsLimit\n (s.mapCone (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P')))) \u2192\n Nonempty (IsLimit (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P')))\n[PROOFSTEP]\nhaveI := reflectsSmallestLimitsOfReflectsLimits s\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nJ : GrothendieckTopology C\nU\u271d : C\nR\u271d : Presieve U\u271d\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ninst\u271d\u00b3 : HasPullbacks C\ns : A' \u2964 Type (max v\u2081 u\u2081)\ninst\u271d\u00b2 : HasLimits A'\ninst\u271d\u00b9 : PreservesLimits s\ninst\u271d : ReflectsIsomorphisms s\nU : C\nR : Presieve U\nx\u271d : generate R \u2208 GrothendieckTopology.sieves J U\nthis\u271d\u00b9 : ReflectsLimits s := reflectsLimitsOfReflectsIsomorphisms\nthis\u271d :\n IsLimit (s.mapCone (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P'))) \u2243\n IsLimit\n (Fork.of\u03b9 (forkMap R (P' \u22d9 s))\n (_ : forkMap R (P' \u22d9 s) \u226b firstMap R (P' \u22d9 s) = forkMap R (P' \u22d9 s) \u226b secondMap R (P' \u22d9 s)))\nthis : ReflectsLimitsOfSize.{0, 0, max u\u2081 v\u2081, max u\u2081 v\u2081, u\u2082, max (u\u2081 + 1) (v\u2081 + 1)} s\n\u22a2 Nonempty\n (IsLimit\n (s.mapCone (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P')))) \u2192\n Nonempty (IsLimit (Fork.of\u03b9 (forkMap R P') (_ : forkMap R P' \u226b firstMap R P' = forkMap R P' \u226b secondMap R P')))\n[PROOFSTEP]\nexact Nonempty.map fun t => isLimitOfReflects s t\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Sites.Sheaf", "llama_tokens": 69130, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.45713671682749474, "lm_q2_score": 0.02064593051324596, "lm_q1q2_score": 0.009438012890673851}} {"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\no : Part \u03b1\ninst\u271d : Decidable o.Dom\nx : \u03b1\n\u22a2 x \u2208 toFinset o \u2194 x \u2208 o\n[PROOFSTEP]\nsimp [toFinset]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Decidable none.Dom\n\u22a2 toFinset none = \u2205\n[PROOFSTEP]\nsimp [toFinset]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na : \u03b1\ninst\u271d : Decidable (some a).Dom\n\u22a2 toFinset (some a) = {a}\n[PROOFSTEP]\nsimp [toFinset]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\nf g : \u03b1 \u2192. \u03b2\ninst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (f x).Dom\ninst\u271d : (x : \u03b1) \u2192 Decidable (g x).Dom\ns t : Finset \u03b1\nb : \u03b2\n\u22a2 b \u2208 pimage f s \u2194 \u2203 a, a \u2208 s \u2227 b \u2208 f a\n[PROOFSTEP]\nsimp [pimage]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DecidableEq \u03b2\nf\u271d g : \u03b1 \u2192. \u03b2\ninst\u271d\u00b2 : (x : \u03b1) \u2192 Decidable (f\u271d x).Dom\ninst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (g x).Dom\ns\u271d t : Finset \u03b1\nb : \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : (x : \u03b1) \u2192 Decidable (Part.some (f x)).Dom\n\u22a2 pimage (fun x => Part.some (f x)) s = image f s\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DecidableEq \u03b2\nf\u271d g : \u03b1 \u2192. \u03b2\ninst\u271d\u00b2 : (x : \u03b1) \u2192 Decidable (f\u271d x).Dom\ninst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (g x).Dom\ns\u271d t : Finset \u03b1\nb : \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : (x : \u03b1) \u2192 Decidable (Part.some (f x)).Dom\na\u271d : \u03b2\n\u22a2 a\u271d \u2208 pimage (fun x => Part.some (f x)) s \u2194 a\u271d \u2208 image f s\n[PROOFSTEP]\nsimp [eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\nf g : \u03b1 \u2192. \u03b2\ninst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (f x).Dom\ninst\u271d : (x : \u03b1) \u2192 Decidable (g x).Dom\ns t : Finset \u03b1\nb : \u03b2\nh\u2081 : s = t\nh\u2082 : \u2200 (x : \u03b1), x \u2208 t \u2192 f x = g x\n\u22a2 pimage f s = pimage g t\n[PROOFSTEP]\nsubst s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\nf g : \u03b1 \u2192. \u03b2\ninst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (f x).Dom\ninst\u271d : (x : \u03b1) \u2192 Decidable (g x).Dom\nt : Finset \u03b1\nb : \u03b2\nh\u2082 : \u2200 (x : \u03b1), x \u2208 t \u2192 f x = g x\n\u22a2 pimage f t = pimage g t\n[PROOFSTEP]\next y\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\nf g : \u03b1 \u2192. \u03b2\ninst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (f x).Dom\ninst\u271d : (x : \u03b1) \u2192 Decidable (g x).Dom\nt : Finset \u03b1\nb : \u03b2\nh\u2082 : \u2200 (x : \u03b1), x \u2208 t \u2192 f x = g x\ny : \u03b2\n\u22a2 y \u2208 pimage f t \u2194 y \u2208 pimage g t\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [mem_pimage, \u2190 exists_prop, h\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\nf g : \u03b1 \u2192. \u03b2\ninst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (f x).Dom\ninst\u271d : (x : \u03b1) \u2192 Decidable (g x).Dom\ns t : Finset \u03b1\nb : \u03b2\n\u22a2 pimage f s = image (fun x => Part.get (f \u2191x) (_ : (f \u2191x).Dom)) (attach (filter (fun x => (f x).Dom) s))\n[PROOFSTEP]\next x\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\nf g : \u03b1 \u2192. \u03b2\ninst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (f x).Dom\ninst\u271d : (x : \u03b1) \u2192 Decidable (g x).Dom\ns t : Finset \u03b1\nb x : \u03b2\n\u22a2 x \u2208 pimage f s \u2194 x \u2208 image (fun x => Part.get (f \u2191x) (_ : (f \u2191x).Dom)) (attach (filter (fun x => (f x).Dom) s))\n[PROOFSTEP]\nsimp [Part.mem_eq, And.exists]\n -- Porting note: `\u2190exists_prop` required because `\u2203 x \u2208 s, p x` is defined differently\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\nf g : \u03b1 \u2192. \u03b2\ninst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (f x).Dom\ninst\u271d : (x : \u03b1) \u2192 Decidable (g x).Dom\ns t : Finset \u03b1\nb x : \u03b2\n\u22a2 (\u2203 a, a \u2208 s \u2227 \u2203 h, Part.get (f a) h = x) \u2194\n \u2203 a hp hq, Part.get (f a) (_ : (f \u2191{ val := a, property := (_ : a \u2208 filter (fun x => (f x).Dom) s) }).Dom) = x\n[PROOFSTEP]\nsimp only [\u2190 exists_prop]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DecidableEq \u03b2\nf g : \u03b1 \u2192. \u03b2\ninst\u271d\u00b2 : (x : \u03b1) \u2192 Decidable (f x).Dom\ninst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (g x).Dom\ns t : Finset \u03b1\nb : \u03b2\ninst\u271d : DecidableEq \u03b1\n\u22a2 \u2191(pimage f (s \u222a t)) = \u2191(pimage f s \u222a pimage f t)\n[PROOFSTEP]\nsimp only [coe_pimage, coe_union, \u2190 PFun.image_union]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\nf g : \u03b1 \u2192. \u03b2\ninst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (f x).Dom\ninst\u271d : (x : \u03b1) \u2192 Decidable (g x).Dom\ns t : Finset \u03b1\nb : \u03b2\n\u22a2 pimage f \u2205 = \u2205\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\nf g : \u03b1 \u2192. \u03b2\ninst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (f x).Dom\ninst\u271d : (x : \u03b1) \u2192 Decidable (g x).Dom\ns t : Finset \u03b1\nb a\u271d : \u03b2\n\u22a2 a\u271d \u2208 pimage f \u2205 \u2194 a\u271d \u2208 \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\nf g : \u03b1 \u2192. \u03b2\ninst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (f x).Dom\ninst\u271d : (x : \u03b1) \u2192 Decidable (g x).Dom\ns t\u271d : Finset \u03b1\nb : \u03b2\nt : Finset \u03b2\n\u22a2 pimage f s \u2286 t \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b2), y \u2208 f x \u2192 y \u2208 t\n[PROOFSTEP]\nsimp [subset_iff, @forall_swap _ \u03b2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : DecidableEq \u03b2\nf g : \u03b1 \u2192. \u03b2\ninst\u271d\u00b2 : (x : \u03b1) \u2192 Decidable (f x).Dom\ninst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (g x).Dom\ns t : Finset \u03b1\nb : \u03b2\ninst\u271d : DecidableEq \u03b1\n\u22a2 pimage f (s \u2229 t) \u2286 pimage f s \u2229 pimage f t\n[PROOFSTEP]\nsimp only [\u2190 coe_subset, coe_pimage, coe_inter, PFun.image_inter]\n", "meta": {"mathlib_filename": "Mathlib.Data.Finset.PImage", "llama_tokens": 2430, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.43014733397551624, "lm_q2_score": 0.021615334614044697, "lm_q1q2_score": 0.009297778557220021}} {"text": "[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u22a2 \u22a5 = span R {0}\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nK : Type u\ninst\u271d : DivisionRing K\nS : Ideal K\n\u22a2 IsPrincipal S\n[PROOFSTEP]\nrcases Ideal.eq_bot_or_top S with (rfl | rfl)\n[GOAL]\ncase inl\nR : Type u\nM : Type v\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nK : Type u\ninst\u271d : DivisionRing K\n\u22a2 IsPrincipal \u22a5\ncase inr\nR : Type u\nM : Type v\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nK : Type u\ninst\u271d : DivisionRing K\n\u22a2 IsPrincipal \u22a4\n[PROOFSTEP]\napply bot_isPrincipal\n[GOAL]\ncase inr\nR : Type u\nM : Type v\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nK : Type u\ninst\u271d : DivisionRing K\n\u22a2 IsPrincipal \u22a4\n[PROOFSTEP]\napply top_isPrincipal\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Module R M\nS : Submodule R M\ninst\u271d : IsPrincipal S\n\u22a2 generator S \u2208 S\n[PROOFSTEP]\nconv_rhs => rw [\u2190 span_singleton_generator S]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Module R M\nS : Submodule R M\ninst\u271d : IsPrincipal S\n| S\n[PROOFSTEP]\nrw [\u2190 span_singleton_generator S]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Module R M\nS : Submodule R M\ninst\u271d : IsPrincipal S\n| S\n[PROOFSTEP]\nrw [\u2190 span_singleton_generator S]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Module R M\nS : Submodule R M\ninst\u271d : IsPrincipal S\n| S\n[PROOFSTEP]\nrw [\u2190 span_singleton_generator S]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Module R M\nS : Submodule R M\ninst\u271d : IsPrincipal S\n\u22a2 generator S \u2208 span R {generator S}\n[PROOFSTEP]\nexact subset_span (mem_singleton _)\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Module R M\nS : Submodule R M\ninst\u271d : IsPrincipal S\nx : M\n\u22a2 x \u2208 S \u2194 \u2203 s, x = s \u2022 generator S\n[PROOFSTEP]\nsimp_rw [@eq_comm _ x, \u2190 mem_span_singleton, span_singleton_generator]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Module R M\nS : Submodule R M\ninst\u271d : IsPrincipal S\n\u22a2 S = \u22a5 \u2194 generator S = 0\n[PROOFSTEP]\nrw [\u2190 @span_singleton_eq_bot R M, span_singleton_generator]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nS : Ideal R\ninst\u271d : IsPrincipal S\nx a : R\n\u22a2 x = a \u2022 generator S \u2194 x = generator S * a\n[PROOFSTEP]\nsimp only [mul_comm, smul_eq_mul]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nS : Ideal R\ninst\u271d : IsPrincipal S\nis_prime : Ideal.IsPrime S\nne_bot : S \u2260 \u22a5\nx\u271d\u00b9 x\u271d : R\n\u22a2 generator S \u2223 x\u271d\u00b9 * x\u271d \u2192 generator S \u2223 x\u271d\u00b9 \u2228 generator S \u2223 x\u271d\n[PROOFSTEP]\nsimpa only [\u2190 mem_iff_generator_dvd S] using is_prime.2\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nN : Submodule R M\n\u03d5 : M \u2192\u2097[R] R\ninst\u271d : IsPrincipal (map \u03d5 N)\nx : M\nhx : x \u2208 N\n\u22a2 generator (map \u03d5 N) \u2223 \u2191\u03d5 x\n[PROOFSTEP]\nrw [\u2190 mem_iff_generator_dvd, Submodule.mem_map]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nN : Submodule R M\n\u03d5 : M \u2192\u2097[R] R\ninst\u271d : IsPrincipal (map \u03d5 N)\nx : M\nhx : x \u2208 N\n\u22a2 \u2203 y, y \u2208 N \u2227 \u2191\u03d5 y = \u2191\u03d5 x\n[PROOFSTEP]\nexact \u27e8x, hx, rfl\u27e9\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\nx : M\nhx : x \u2208 N\n\u22a2 generator (LinearMap.submoduleImage \u03d5 N) \u2223 \u2191\u03d5 { val := x, property := (_ : x \u2208 O) }\n[PROOFSTEP]\nrw [\u2190 mem_iff_generator_dvd, LinearMap.mem_submoduleImage_of_le hNO]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nN O : Submodule R M\nhNO : N \u2264 O\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] R\ninst\u271d : IsPrincipal (LinearMap.submoduleImage \u03d5 N)\nx : M\nhx : x \u2208 N\n\u22a2 \u2203 y yN, \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = \u2191\u03d5 { val := x, property := (_ : x \u2208 O) }\n[PROOFSTEP]\nexact \u27e8x, hx, rfl\u27e9\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\nS : Ideal R\nhpi : IsPrime S\nhS : S \u2260 \u22a5\n\u22a2 \u2200 (J : Ideal R) (x : R), S \u2264 J \u2192 \u00acx \u2208 S \u2192 x \u2208 J \u2192 1 \u2208 J\n[PROOFSTEP]\nintro T x hST hxS hxT\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\nS : Ideal R\nhpi : IsPrime S\nhS : S \u2260 \u22a5\nT : Ideal R\nx : R\nhST : S \u2264 T\nhxS : \u00acx \u2208 S\nhxT : x \u2208 T\n\u22a2 1 \u2208 T\n[PROOFSTEP]\ncases' (mem_iff_generator_dvd _).1 (hST <| generator_mem S) with z hz\n[GOAL]\ncase intro\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\nS : Ideal R\nhpi : IsPrime S\nhS : S \u2260 \u22a5\nT : Ideal R\nx : R\nhST : S \u2264 T\nhxS : \u00acx \u2208 S\nhxT : x \u2208 T\nz : R\nhz : generator S = generator T * z\n\u22a2 1 \u2208 T\n[PROOFSTEP]\ncases hpi.mem_or_mem (show generator T * z \u2208 S from hz \u25b8 generator_mem S)\n[GOAL]\ncase intro.inl\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\nS : Ideal R\nhpi : IsPrime S\nhS : S \u2260 \u22a5\nT : Ideal R\nx : R\nhST : S \u2264 T\nhxS : \u00acx \u2208 S\nhxT : x \u2208 T\nz : R\nhz : generator S = generator T * z\nh\u271d : generator T \u2208 S\n\u22a2 1 \u2208 T\ncase intro.inr\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\nS : Ideal R\nhpi : IsPrime S\nhS : S \u2260 \u22a5\nT : Ideal R\nx : R\nhST : S \u2264 T\nhxS : \u00acx \u2208 S\nhxT : x \u2208 T\nz : R\nhz : generator S = generator T * z\nh\u271d : z \u2208 S\n\u22a2 1 \u2208 T\n[PROOFSTEP]\ncase inl h =>\n have hTS : T \u2264 S\n rwa [\u2190 T.span_singleton_generator, Ideal.span_le, singleton_subset_iff]\n exact (hxS <| hTS hxT).elim\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\nS : Ideal R\nhpi : IsPrime S\nhS : S \u2260 \u22a5\nT : Ideal R\nx : R\nhST : S \u2264 T\nhxS : \u00acx \u2208 S\nhxT : x \u2208 T\nz : R\nhz : generator S = generator T * z\nh : generator T \u2208 S\n\u22a2 1 \u2208 T\n[PROOFSTEP]\ncase inl h =>\n have hTS : T \u2264 S\n rwa [\u2190 T.span_singleton_generator, Ideal.span_le, singleton_subset_iff]\n exact (hxS <| hTS hxT).elim\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\nS : Ideal R\nhpi : IsPrime S\nhS : S \u2260 \u22a5\nT : Ideal R\nx : R\nhST : S \u2264 T\nhxS : \u00acx \u2208 S\nhxT : x \u2208 T\nz : R\nhz : generator S = generator T * z\nh : generator T \u2208 S\n\u22a2 1 \u2208 T\n[PROOFSTEP]\nhave hTS : T \u2264 S\n[GOAL]\ncase hTS\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\nS : Ideal R\nhpi : IsPrime S\nhS : S \u2260 \u22a5\nT : Ideal R\nx : R\nhST : S \u2264 T\nhxS : \u00acx \u2208 S\nhxT : x \u2208 T\nz : R\nhz : generator S = generator T * z\nh : generator T \u2208 S\n\u22a2 T \u2264 S\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\nS : Ideal R\nhpi : IsPrime S\nhS : S \u2260 \u22a5\nT : Ideal R\nx : R\nhST : S \u2264 T\nhxS : \u00acx \u2208 S\nhxT : x \u2208 T\nz : R\nhz : generator S = generator T * z\nh : generator T \u2208 S\nhTS : T \u2264 S\n\u22a2 1 \u2208 T\n[PROOFSTEP]\nrwa [\u2190 T.span_singleton_generator, Ideal.span_le, singleton_subset_iff]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\nS : Ideal R\nhpi : IsPrime S\nhS : S \u2260 \u22a5\nT : Ideal R\nx : R\nhST : S \u2264 T\nhxS : \u00acx \u2208 S\nhxT : x \u2208 T\nz : R\nhz : generator S = generator T * z\nh : generator T \u2208 S\nhTS : T \u2264 S\n\u22a2 1 \u2208 T\n[PROOFSTEP]\nexact (hxS <| hTS hxT).elim\n[GOAL]\ncase intro.inr\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\nS : Ideal R\nhpi : IsPrime S\nhS : S \u2260 \u22a5\nT : Ideal R\nx : R\nhST : S \u2264 T\nhxS : \u00acx \u2208 S\nhxT : x \u2208 T\nz : R\nhz : generator S = generator T * z\nh\u271d : z \u2208 S\n\u22a2 1 \u2208 T\n[PROOFSTEP]\ncase inr h =>\n cases' (mem_iff_generator_dvd _).1 h with y hy\n have : generator S \u2260 0 := mt (eq_bot_iff_generator_eq_zero _).2 hS\n rw [\u2190 mul_one (generator S), hy, mul_left_comm, mul_right_inj' this] at hz \n exact hz.symm \u25b8 T.mul_mem_right _ (generator_mem T)\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\nS : Ideal R\nhpi : IsPrime S\nhS : S \u2260 \u22a5\nT : Ideal R\nx : R\nhST : S \u2264 T\nhxS : \u00acx \u2208 S\nhxT : x \u2208 T\nz : R\nhz : generator S = generator T * z\nh : z \u2208 S\n\u22a2 1 \u2208 T\n[PROOFSTEP]\ncase inr h =>\n cases' (mem_iff_generator_dvd _).1 h with y hy\n have : generator S \u2260 0 := mt (eq_bot_iff_generator_eq_zero _).2 hS\n rw [\u2190 mul_one (generator S), hy, mul_left_comm, mul_right_inj' this] at hz \n exact hz.symm \u25b8 T.mul_mem_right _ (generator_mem T)\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\nS : Ideal R\nhpi : IsPrime S\nhS : S \u2260 \u22a5\nT : Ideal R\nx : R\nhST : S \u2264 T\nhxS : \u00acx \u2208 S\nhxT : x \u2208 T\nz : R\nhz : generator S = generator T * z\nh : z \u2208 S\n\u22a2 1 \u2208 T\n[PROOFSTEP]\ncases' (mem_iff_generator_dvd _).1 h with y hy\n[GOAL]\ncase intro\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\nS : Ideal R\nhpi : IsPrime S\nhS : S \u2260 \u22a5\nT : Ideal R\nx : R\nhST : S \u2264 T\nhxS : \u00acx \u2208 S\nhxT : x \u2208 T\nz : R\nhz : generator S = generator T * z\nh : z \u2208 S\ny : R\nhy : z = generator S * y\n\u22a2 1 \u2208 T\n[PROOFSTEP]\nhave : generator S \u2260 0 := mt (eq_bot_iff_generator_eq_zero _).2 hS\n[GOAL]\ncase intro\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\nS : Ideal R\nhpi : IsPrime S\nhS : S \u2260 \u22a5\nT : Ideal R\nx : R\nhST : S \u2264 T\nhxS : \u00acx \u2208 S\nhxT : x \u2208 T\nz : R\nhz : generator S = generator T * z\nh : z \u2208 S\ny : R\nhy : z = generator S * y\nthis : generator S \u2260 0\n\u22a2 1 \u2208 T\n[PROOFSTEP]\nrw [\u2190 mul_one (generator S), hy, mul_left_comm, mul_right_inj' this] at hz \n[GOAL]\ncase intro\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\nS : Ideal R\nhpi : IsPrime S\nhS : S \u2260 \u22a5\nT : Ideal R\nx : R\nhST : S \u2264 T\nhxS : \u00acx \u2208 S\nhxT : x \u2208 T\nz : R\nh : z \u2208 S\ny : R\nhz : 1 = generator T * y\nhy : z = generator S * y\nthis : generator S \u2260 0\n\u22a2 1 \u2208 T\n[PROOFSTEP]\nexact hz.symm \u25b8 T.mul_mem_right _ (generator_mem T)\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d : EuclideanDomain R\nS : Ideal R\nh : Set.Nonempty {x | x \u2208 S \u2227 x \u2260 0}\nwf : WellFounded EuclideanDomain.r\nhmin : WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h \u2208 S \u2227 WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h \u2260 0\nx : R\nhx : x \u2208 S\n\u22a2 WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h \u2223 x % WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h\n[PROOFSTEP]\nhave : x % WellFounded.min wf {x : R | x \u2208 S \u2227 x \u2260 0} h \u2209 {x : R | x \u2208 S \u2227 x \u2260 0} := fun h\u2081 =>\n WellFounded.not_lt_min wf _ h h\u2081 (mod_lt x hmin.2)\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d : EuclideanDomain R\nS : Ideal R\nh : Set.Nonempty {x | x \u2208 S \u2227 x \u2260 0}\nwf : WellFounded EuclideanDomain.r\nhmin : WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h \u2208 S \u2227 WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h \u2260 0\nx : R\nhx : x \u2208 S\nthis : \u00acx % WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h \u2208 {x | x \u2208 S \u2227 x \u2260 0}\n\u22a2 WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h \u2223 x % WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h\n[PROOFSTEP]\nhave : x % WellFounded.min wf {x : R | x \u2208 S \u2227 x \u2260 0} h = 0 :=\n by\n simp only [not_and_or, Set.mem_setOf_eq, not_ne_iff] at this \n exact this.neg_resolve_left <| (mod_mem_iff hmin.1).2 hx\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d : EuclideanDomain R\nS : Ideal R\nh : Set.Nonempty {x | x \u2208 S \u2227 x \u2260 0}\nwf : WellFounded EuclideanDomain.r\nhmin : WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h \u2208 S \u2227 WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h \u2260 0\nx : R\nhx : x \u2208 S\nthis : \u00acx % WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h \u2208 {x | x \u2208 S \u2227 x \u2260 0}\n\u22a2 x % WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h = 0\n[PROOFSTEP]\nsimp only [not_and_or, Set.mem_setOf_eq, not_ne_iff] at this \n[GOAL]\nR : Type u\nM : Type v\ninst\u271d : EuclideanDomain R\nS : Ideal R\nh : Set.Nonempty {x | x \u2208 S \u2227 x \u2260 0}\nwf : WellFounded EuclideanDomain.r\nhmin : WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h \u2208 S \u2227 WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h \u2260 0\nx : R\nhx : x \u2208 S\nthis : \u00acx % WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h \u2208 S \u2228 x % WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h = 0\n\u22a2 x % WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h = 0\n[PROOFSTEP]\nexact this.neg_resolve_left <| (mod_mem_iff hmin.1).2 hx\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d : EuclideanDomain R\nS : Ideal R\nh : Set.Nonempty {x | x \u2208 S \u2227 x \u2260 0}\nwf : WellFounded EuclideanDomain.r\nhmin : WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h \u2208 S \u2227 WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h \u2260 0\nx : R\nhx : x \u2208 S\nthis\u271d : \u00acx % WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h \u2208 {x | x \u2208 S \u2227 x \u2260 0}\nthis : x % WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h = 0\n\u22a2 WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h \u2223 x % WellFounded.min wf {x | x \u2208 S \u2227 x \u2260 0} h\n[PROOFSTEP]\nsimp [*]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d : EuclideanDomain R\nS : Ideal R\nh : \u00acSet.Nonempty {x | x \u2208 S \u2227 x \u2260 0}\na : R\n\u22a2 a \u2208 S \u2194 a \u2208 span R {0}\n[PROOFSTEP]\nrw [\u2190 @Submodule.bot_coe R R _ _ _, span_eq, Submodule.mem_bot]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d : EuclideanDomain R\nS : Ideal R\nh : \u00acSet.Nonempty {x | x \u2208 S \u2227 x \u2260 0}\na : R\n\u22a2 a \u2208 S \u2194 a = 0\n[PROOFSTEP]\nexact \u27e8fun haS => by_contra fun ha0 => h \u27e8a, \u27e8haS, ha0\u27e9\u27e9, fun h\u2081 => h\u2081.symm \u25b8 S.zero_mem\u27e9\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsPrincipalIdealRing R\ns : Ideal R\n\u22a2 FG s\n[PROOFSTEP]\nrcases(IsPrincipalIdealRing.principal s).principal with \u27e8a, rfl\u27e9\n[GOAL]\ncase intro\nR : Type u\nM : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsPrincipalIdealRing R\na : R\n\u22a2 FG (span R {a})\n[PROOFSTEP]\nrw [\u2190 Finset.coe_singleton]\n[GOAL]\ncase intro\nR : Type u\nM : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsPrincipalIdealRing R\na : R\n\u22a2 FG (span R \u2191{a})\n[PROOFSTEP]\nexact \u27e8{ a }, SetLike.coe_injective rfl\u27e9\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\nI : Ideal R\nhI : span R {p} < I\n\u22a2 I = \u22a4\n[PROOFSTEP]\nrcases principal I with \u27e8a, rfl\u27e9\n[GOAL]\ncase mk.intro\nR : Type u\nM : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\na : R\nhI : span R {p} < span R {a}\n\u22a2 span R {a} = \u22a4\n[PROOFSTEP]\nerw [Ideal.span_singleton_eq_top]\n[GOAL]\ncase mk.intro\nR : Type u\nM : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\na : R\nhI : span R {p} < span R {a}\n\u22a2 IsUnit a\n[PROOFSTEP]\nrcases Ideal.span_singleton_le_span_singleton.1 (le_of_lt hI) with \u27e8b, rfl\u27e9\n[GOAL]\ncase mk.intro.intro\nR : Type u\nM : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsPrincipalIdealRing R\na b : R\nhp : Irreducible (a * b)\nhI : span R {a * b} < span R {a}\n\u22a2 IsUnit a\n[PROOFSTEP]\nrefine' (of_irreducible_mul hp).resolve_right (mt (fun hb => _) (not_le_of_lt hI))\n[GOAL]\ncase mk.intro.intro\nR : Type u\nM : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsPrincipalIdealRing R\na b : R\nhp : Irreducible (a * b)\nhI : span R {a * b} < span R {a}\nhb : IsUnit b\n\u22a2 span R {a} \u2264 span R {a * b}\n[PROOFSTEP]\nerw [Ideal.span_singleton_le_span_singleton, IsUnit.mul_right_dvd hb]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\na : R\nh : a \u2260 0\n\u22a2 (\u2200 (b : R), b \u2208 factors a \u2192 Irreducible b) \u2227 Associated (Multiset.prod (factors a)) a\n[PROOFSTEP]\nunfold factors\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\na : R\nh : a \u2260 0\n\u22a2 (\u2200 (b : R),\n (b \u2208\n if h : a = 0 then \u2205\n else choose (_ : \u2203 f, (\u2200 (b : R), b \u2208 f \u2192 Irreducible b) \u2227 Associated (Multiset.prod f) a)) \u2192\n Irreducible b) \u2227\n Associated\n (Multiset.prod\n (if h : a = 0 then \u2205\n else choose (_ : \u2203 f, (\u2200 (b : R), b \u2208 f \u2192 Irreducible b) \u2227 Associated (Multiset.prod f) a)))\n a\n[PROOFSTEP]\nrw [dif_neg h]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\na : R\nh : a \u2260 0\n\u22a2 (\u2200 (b : R),\n b \u2208 choose (_ : \u2203 f, (\u2200 (b : R), b \u2208 f \u2192 Irreducible b) \u2227 Associated (Multiset.prod f) a) \u2192 Irreducible b) \u2227\n Associated (Multiset.prod (choose (_ : \u2203 f, (\u2200 (b : R), b \u2208 f \u2192 Irreducible b) \u2227 Associated (Multiset.prod f) a))) a\n[PROOFSTEP]\nexact Classical.choose_spec (WfDvdMonoid.exists_factors a h)\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\ns : Submonoid R\na : R\nha : a \u2260 0\nhfac : \u2200 (b : R), b \u2208 factors a \u2192 b \u2208 s\nhunit : \u2200 (c : R\u02e3), \u2191c \u2208 s\n\u22a2 a \u2208 s\n[PROOFSTEP]\nrcases(factors_spec a ha).2 with \u27e8c, hc\u27e9\n[GOAL]\ncase intro\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\ns : Submonoid R\na : R\nha : a \u2260 0\nhfac : \u2200 (b : R), b \u2208 factors a \u2192 b \u2208 s\nhunit : \u2200 (c : R\u02e3), \u2191c \u2208 s\nc : R\u02e3\nhc : Multiset.prod (factors a) * \u2191c = a\n\u22a2 a \u2208 s\n[PROOFSTEP]\nrw [\u2190 hc]\n[GOAL]\ncase intro\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsPrincipalIdealRing R\ns : Submonoid R\na : R\nha : a \u2260 0\nhfac : \u2200 (b : R), b \u2208 factors a \u2192 b \u2208 s\nhunit : \u2200 (c : R\u02e3), \u2191c \u2208 s\nc : R\u02e3\nhc : Multiset.prod (factors a) * \u2191c = a\n\u22a2 Multiset.prod (factors a) * \u2191c \u2208 s\n[PROOFSTEP]\nexact mul_mem (multiset_prod_mem _ hfac) (hunit _)\n[GOAL]\nR : Type u\nM : Type v\nS\u271d : Type u_1\nN : Type u_2\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Ring S\u271d\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nf : M \u2192\u2097[R] N\nhf : Surjective \u2191f\nS : Submodule R N\nhI : IsPrincipal (comap f S)\n\u22a2 S = span R {\u2191f (generator (comap f S))}\n[PROOFSTEP]\nrw [\u2190 Set.image_singleton, \u2190 Submodule.map_span, IsPrincipal.span_singleton_generator,\n Submodule.map_comap_eq_of_surjective hf]\n[GOAL]\nR : Type u\nM : Type v\nS : Type u_1\nN : Type u_2\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Ring S\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nf : R \u2192+* S\nhf : Surjective \u2191f\nI : Ideal S\nhI : IsPrincipal (comap f I)\n\u22a2 I = Submodule.span S {\u2191f (IsPrincipal.generator (comap f I))}\n[PROOFSTEP]\nrw [Ideal.submodule_span_eq, \u2190 Set.image_singleton, \u2190 Ideal.map_span, Ideal.span_singleton_generator,\n Ideal.map_comap_of_surjective f hf]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y : R\n\u22a2 Ideal.span {gcd x y} = Ideal.span {x, y}\n[PROOFSTEP]\nobtain \u27e8d, hd\u27e9 := IsPrincipalIdealRing.principal (span ({ x, y } : Set R))\n[GOAL]\ncase mk.intro\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y d : R\nhd : Ideal.span {x, y} = Submodule.span R {d}\n\u22a2 Ideal.span {gcd x y} = Ideal.span {x, y}\n[PROOFSTEP]\nrw [submodule_span_eq] at hd \n[GOAL]\ncase mk.intro\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y d : R\nhd : Ideal.span {x, y} = Ideal.span {d}\n\u22a2 Ideal.span {gcd x y} = Ideal.span {x, y}\n[PROOFSTEP]\nrw [hd]\n[GOAL]\ncase mk.intro\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y d : R\nhd : Ideal.span {x, y} = Ideal.span {d}\n\u22a2 Ideal.span {gcd x y} = Ideal.span {d}\n[PROOFSTEP]\nsuffices Associated d (gcd x y) by\n obtain \u27e8D, HD\u27e9 := this\n rw [\u2190 HD]\n exact span_singleton_mul_right_unit D.isUnit _\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y d : R\nhd : Ideal.span {x, y} = Ideal.span {d}\nthis : Associated d (gcd x y)\n\u22a2 Ideal.span {gcd x y} = Ideal.span {d}\n[PROOFSTEP]\nobtain \u27e8D, HD\u27e9 := this\n[GOAL]\ncase intro\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y d : R\nhd : Ideal.span {x, y} = Ideal.span {d}\nD : R\u02e3\nHD : d * \u2191D = gcd x y\n\u22a2 Ideal.span {gcd x y} = Ideal.span {d}\n[PROOFSTEP]\nrw [\u2190 HD]\n[GOAL]\ncase intro\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y d : R\nhd : Ideal.span {x, y} = Ideal.span {d}\nD : R\u02e3\nHD : d * \u2191D = gcd x y\n\u22a2 Ideal.span {d * \u2191D} = Ideal.span {d}\n[PROOFSTEP]\nexact span_singleton_mul_right_unit D.isUnit _\n[GOAL]\ncase mk.intro\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y d : R\nhd : Ideal.span {x, y} = Ideal.span {d}\n\u22a2 Associated d (gcd x y)\n[PROOFSTEP]\napply associated_of_dvd_dvd\n[GOAL]\ncase mk.intro.hab\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y d : R\nhd : Ideal.span {x, y} = Ideal.span {d}\n\u22a2 d \u2223 gcd x y\n[PROOFSTEP]\nrw [dvd_gcd_iff]\n[GOAL]\ncase mk.intro.hab\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y d : R\nhd : Ideal.span {x, y} = Ideal.span {d}\n\u22a2 d \u2223 x \u2227 d \u2223 y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mk.intro.hab.left\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y d : R\nhd : Ideal.span {x, y} = Ideal.span {d}\n\u22a2 d \u2223 x\n[PROOFSTEP]\nrw [\u2190 Ideal.mem_span_singleton, \u2190 hd, Ideal.mem_span_pair]\n[GOAL]\ncase mk.intro.hab.right\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y d : R\nhd : Ideal.span {x, y} = Ideal.span {d}\n\u22a2 d \u2223 y\n[PROOFSTEP]\nrw [\u2190 Ideal.mem_span_singleton, \u2190 hd, Ideal.mem_span_pair]\n[GOAL]\ncase mk.intro.hab.left\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y d : R\nhd : Ideal.span {x, y} = Ideal.span {d}\n\u22a2 \u2203 a b, a * x + b * y = x\n[PROOFSTEP]\nuse 1, 0\n[GOAL]\ncase h\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y d : R\nhd : Ideal.span {x, y} = Ideal.span {d}\n\u22a2 1 * x + 0 * y = x\n[PROOFSTEP]\nrw [one_mul, zero_mul, add_zero]\n[GOAL]\ncase mk.intro.hab.right\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y d : R\nhd : Ideal.span {x, y} = Ideal.span {d}\n\u22a2 \u2203 a b, a * x + b * y = y\n[PROOFSTEP]\nuse 0, 1\n[GOAL]\ncase h\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y d : R\nhd : Ideal.span {x, y} = Ideal.span {d}\n\u22a2 0 * x + 1 * y = y\n[PROOFSTEP]\nrw [one_mul, zero_mul, zero_add]\n[GOAL]\ncase mk.intro.hba\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y d : R\nhd : Ideal.span {x, y} = Ideal.span {d}\n\u22a2 gcd x y \u2223 d\n[PROOFSTEP]\nobtain \u27e8r, s, rfl\u27e9 : \u2203 r s, r * x + s * y = d := by rw [\u2190 Ideal.mem_span_pair, hd, Ideal.mem_span_singleton]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y d : R\nhd : Ideal.span {x, y} = Ideal.span {d}\n\u22a2 \u2203 r s, r * x + s * y = d\n[PROOFSTEP]\nrw [\u2190 Ideal.mem_span_pair, hd, Ideal.mem_span_singleton]\n[GOAL]\ncase mk.intro.hba.intro.intro\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y r s : R\nhd : Ideal.span {x, y} = Ideal.span {r * x + s * y}\n\u22a2 gcd x y \u2223 r * x + s * y\n[PROOFSTEP]\napply dvd_add\n[GOAL]\ncase mk.intro.hba.intro.intro.h\u2081\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y r s : R\nhd : Ideal.span {x, y} = Ideal.span {r * x + s * y}\n\u22a2 gcd x y \u2223 r * x\n[PROOFSTEP]\napply dvd_mul_of_dvd_right\n[GOAL]\ncase mk.intro.hba.intro.intro.h\u2082\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y r s : R\nhd : Ideal.span {x, y} = Ideal.span {r * x + s * y}\n\u22a2 gcd x y \u2223 s * y\n[PROOFSTEP]\napply dvd_mul_of_dvd_right\n[GOAL]\ncase mk.intro.hba.intro.intro.h\u2081.h\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y r s : R\nhd : Ideal.span {x, y} = Ideal.span {r * x + s * y}\n\u22a2 gcd x y \u2223 x\ncase mk.intro.hba.intro.intro.h\u2082.h\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y r s : R\nhd : Ideal.span {x, y} = Ideal.span {r * x + s * y}\n\u22a2 gcd x y \u2223 y\n[PROOFSTEP]\nexacts [gcd_dvd_left x y, gcd_dvd_right x y]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\na b z : R\n\u22a2 gcd a b \u2223 z \u2194 \u2203 x y, z = a * x + b * y\n[PROOFSTEP]\nsimp_rw [mul_comm a, mul_comm b, @eq_comm _ z, \u2190 Ideal.mem_span_pair, \u2190 span_gcd, Ideal.mem_span_singleton]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\na b : R\n\u22a2 \u2203 x y, gcd a b = a * x + b * y\n[PROOFSTEP]\nrw [\u2190 gcd_dvd_iff_exists]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y : R\n\u22a2 IsUnit (gcd x y) \u2194 IsCoprime x y\n[PROOFSTEP]\nrw [IsCoprime, \u2190 Ideal.mem_span_pair, \u2190 span_gcd, \u2190 span_singleton_eq_top, eq_top_iff_one]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y : R\nnonzero : \u00ac(x = 0 \u2227 y = 0)\nH : \u2200 (z : R), z \u2208 nonunits R \u2192 z \u2260 0 \u2192 z \u2223 x \u2192 \u00acz \u2223 y\n\u22a2 IsCoprime x y\n[PROOFSTEP]\nrw [\u2190 gcd_isUnit_iff]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y : R\nnonzero : \u00ac(x = 0 \u2227 y = 0)\nH : \u2200 (z : R), z \u2208 nonunits R \u2192 z \u2260 0 \u2192 z \u2223 x \u2192 \u00acz \u2223 y\n\u22a2 IsUnit (gcd x y)\n[PROOFSTEP]\nby_contra h\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y : R\nnonzero : \u00ac(x = 0 \u2227 y = 0)\nH : \u2200 (z : R), z \u2208 nonunits R \u2192 z \u2260 0 \u2192 z \u2223 x \u2192 \u00acz \u2223 y\nh : \u00acIsUnit (gcd x y)\n\u22a2 False\n[PROOFSTEP]\nrefine' H _ h _ (gcd_dvd_left _ _) (gcd_dvd_right _ _)\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y : R\nnonzero : \u00ac(x = 0 \u2227 y = 0)\nH : \u2200 (z : R), z \u2208 nonunits R \u2192 z \u2260 0 \u2192 z \u2223 x \u2192 \u00acz \u2223 y\nh : \u00acIsUnit (gcd x y)\n\u22a2 gcd x y \u2260 0\n[PROOFSTEP]\nrwa [Ne, gcd_eq_zero_iff]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y : R\nh : Irreducible x\n\u22a2 x \u2223 y \u2228 IsCoprime x y\n[PROOFSTEP]\nrefine' or_iff_not_imp_left.2 fun h' => _\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y : R\nh : Irreducible x\nh' : \u00acx \u2223 y\n\u22a2 IsCoprime x y\n[PROOFSTEP]\napply isCoprime_of_dvd\n[GOAL]\ncase nonzero\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y : R\nh : Irreducible x\nh' : \u00acx \u2223 y\n\u22a2 \u00ac(x = 0 \u2227 y = 0)\n[PROOFSTEP]\nrintro \u27e8rfl, rfl\u27e9\n[GOAL]\ncase nonzero.intro\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nh : Irreducible 0\nh' : \u00ac0 \u2223 0\n\u22a2 False\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase H\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y : R\nh : Irreducible x\nh' : \u00acx \u2223 y\n\u22a2 \u2200 (z : R), z \u2208 nonunits R \u2192 z \u2260 0 \u2192 z \u2223 x \u2192 \u00acz \u2223 y\n[PROOFSTEP]\nrintro z nu - \u27e8w, rfl\u27e9 dy\n[GOAL]\ncase H.intro\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\ny z : R\nnu : z \u2208 nonunits R\nw : R\nh : Irreducible (z * w)\nh' : \u00acz * w \u2223 y\ndy : z \u2223 y\n\u22a2 False\n[PROOFSTEP]\nrefine' h' (dvd_trans _ dy)\n[GOAL]\ncase H.intro\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\ny z : R\nnu : z \u2208 nonunits R\nw : R\nh : Irreducible (z * w)\nh' : \u00acz * w \u2223 y\ndy : z \u2223 y\n\u22a2 z * w \u2223 z\n[PROOFSTEP]\nsimpa using mul_dvd_mul_left z (isUnit_iff_dvd_one.1 <| (of_irreducible_mul h).resolve_left nu)\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y : R\nnonzero : \u00ac(x = 0 \u2227 y = 0)\nH : \u2200 (z : R), Irreducible z \u2192 z \u2223 x \u2192 \u00acz \u2223 y\n\u22a2 IsCoprime x y\n[PROOFSTEP]\napply isCoprime_of_dvd x y nonzero\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y : R\nnonzero : \u00ac(x = 0 \u2227 y = 0)\nH : \u2200 (z : R), Irreducible z \u2192 z \u2223 x \u2192 \u00acz \u2223 y\n\u22a2 \u2200 (z : R), z \u2208 nonunits R \u2192 z \u2260 0 \u2192 z \u2223 x \u2192 \u00acz \u2223 y\n[PROOFSTEP]\nintro z znu znz zx zy\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y : R\nnonzero : \u00ac(x = 0 \u2227 y = 0)\nH : \u2200 (z : R), Irreducible z \u2192 z \u2223 x \u2192 \u00acz \u2223 y\nz : R\nznu : z \u2208 nonunits R\nznz : z \u2260 0\nzx : z \u2223 x\nzy : z \u2223 y\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8i, h1, h2\u27e9 := WfDvdMonoid.exists_irreducible_factor znu znz\n[GOAL]\ncase intro.intro\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y : R\nnonzero : \u00ac(x = 0 \u2227 y = 0)\nH : \u2200 (z : R), Irreducible z \u2192 z \u2223 x \u2192 \u00acz \u2223 y\nz : R\nznu : z \u2208 nonunits R\nznz : z \u2260 0\nzx : z \u2223 x\nzy : z \u2223 y\ni : R\nh1 : Irreducible i\nh2 : i \u2223 z\n\u22a2 False\n[PROOFSTEP]\napply H i h1\n[GOAL]\ncase intro.intro.a\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y : R\nnonzero : \u00ac(x = 0 \u2227 y = 0)\nH : \u2200 (z : R), Irreducible z \u2192 z \u2223 x \u2192 \u00acz \u2223 y\nz : R\nznu : z \u2208 nonunits R\nznz : z \u2260 0\nzx : z \u2223 x\nzy : z \u2223 y\ni : R\nh1 : Irreducible i\nh2 : i \u2223 z\n\u22a2 i \u2223 x\n[PROOFSTEP]\napply dvd_trans h2\n[GOAL]\ncase intro.intro.a\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y : R\nnonzero : \u00ac(x = 0 \u2227 y = 0)\nH : \u2200 (z : R), Irreducible z \u2192 z \u2223 x \u2192 \u00acz \u2223 y\nz : R\nznu : z \u2208 nonunits R\nznz : z \u2260 0\nzx : z \u2223 x\nzy : z \u2223 y\ni : R\nh1 : Irreducible i\nh2 : i \u2223 z\n\u22a2 z \u2223 x\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.a\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y : R\nnonzero : \u00ac(x = 0 \u2227 y = 0)\nH : \u2200 (z : R), Irreducible z \u2192 z \u2223 x \u2192 \u00acz \u2223 y\nz : R\nznu : z \u2208 nonunits R\nznz : z \u2260 0\nzx : z \u2223 x\nzy : z \u2223 y\ni : R\nh1 : Irreducible i\nh2 : i \u2223 z\n\u22a2 i \u2223 y\n[PROOFSTEP]\napply dvd_trans h2\n[GOAL]\ncase intro.intro.a\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\nx y : R\nnonzero : \u00ac(x = 0 \u2227 y = 0)\nH : \u2200 (z : R), Irreducible z \u2192 z \u2223 x \u2192 \u00acz \u2223 y\nz : R\nznu : z \u2208 nonunits R\nznz : z \u2260 0\nzx : z \u2223 x\nzy : z \u2223 y\ni : R\nh1 : Irreducible i\nh2 : i \u2223 z\n\u22a2 z \u2223 y\n[PROOFSTEP]\nassumption\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\np n : R\npp : Irreducible p\n\u22a2 IsCoprime p n \u2194 \u00acp \u2223 n\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\np n : R\npp : Irreducible p\n\u22a2 IsCoprime p n \u2192 \u00acp \u2223 n\n[PROOFSTEP]\nintro co H\n[GOAL]\ncase mp\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\np n : R\npp : Irreducible p\nco : IsCoprime p n\nH : p \u2223 n\n\u22a2 False\n[PROOFSTEP]\napply pp.not_unit\n[GOAL]\ncase mp\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\np n : R\npp : Irreducible p\nco : IsCoprime p n\nH : p \u2223 n\n\u22a2 IsUnit p\n[PROOFSTEP]\nrw [isUnit_iff_dvd_one]\n[GOAL]\ncase mp\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\np n : R\npp : Irreducible p\nco : IsCoprime p n\nH : p \u2223 n\n\u22a2 p \u2223 1\n[PROOFSTEP]\napply IsCoprime.dvd_of_dvd_mul_left co\n[GOAL]\ncase mp\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\np n : R\npp : Irreducible p\nco : IsCoprime p n\nH : p \u2223 n\n\u22a2 p \u2223 n * 1\n[PROOFSTEP]\nrw [mul_one n]\n[GOAL]\ncase mp\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\np n : R\npp : Irreducible p\nco : IsCoprime p n\nH : p \u2223 n\n\u22a2 p \u2223 n\n[PROOFSTEP]\nexact H\n[GOAL]\ncase mpr\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\np n : R\npp : Irreducible p\n\u22a2 \u00acp \u2223 n \u2192 IsCoprime p n\n[PROOFSTEP]\nintro nd\n[GOAL]\ncase mpr\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\np n : R\npp : Irreducible p\nnd : \u00acp \u2223 n\n\u22a2 IsCoprime p n\n[PROOFSTEP]\napply isCoprime_of_irreducible_dvd\n[GOAL]\ncase mpr.nonzero\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\np n : R\npp : Irreducible p\nnd : \u00acp \u2223 n\n\u22a2 \u00ac(p = 0 \u2227 n = 0)\n[PROOFSTEP]\nrintro \u27e8hp, -\u27e9\n[GOAL]\ncase mpr.nonzero.intro\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\np n : R\npp : Irreducible p\nnd : \u00acp \u2223 n\nhp : p = 0\n\u22a2 False\n[PROOFSTEP]\nexact pp.ne_zero hp\n[GOAL]\ncase mpr.H\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\np n : R\npp : Irreducible p\nnd : \u00acp \u2223 n\n\u22a2 \u2200 (z : R), Irreducible z \u2192 z \u2223 p \u2192 \u00acz \u2223 n\n[PROOFSTEP]\nrintro z zi zp zn\n[GOAL]\ncase mpr.H\nR : Type u\nM : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : GCDMonoid R\np n : R\npp : Irreducible p\nnd : \u00acp \u2223 n\nz : R\nzi : Irreducible z\nzp : z \u2223 p\nzn : z \u2223 n\n\u22a2 False\n[PROOFSTEP]\nexact nd ((zi.associated_of_dvd pp zp).symm.dvd.trans zn)\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d : CommRing R\n\u22a2 nonPrincipals R = \u2205 \u2194 IsPrincipalIdealRing R\n[PROOFSTEP]\nsimp [Set.eq_empty_iff_forall_not_mem, isPrincipalIdealRing_iff, nonPrincipals_def]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nc : Set (Ideal R)\nhs : c \u2286 nonPrincipals R\nhchain : IsChain (fun x x_1 => x \u2264 x_1) c\nK : Ideal R\nhKmem : K \u2208 c\n\u22a2 \u2203 I, I \u2208 nonPrincipals R \u2227 \u2200 (J : Ideal R), J \u2208 c \u2192 J \u2264 I\n[PROOFSTEP]\nrefine' \u27e8sSup c, _, fun J hJ => le_sSup hJ\u27e9\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nc : Set (Ideal R)\nhs : c \u2286 nonPrincipals R\nhchain : IsChain (fun x x_1 => x \u2264 x_1) c\nK : Ideal R\nhKmem : K \u2208 c\n\u22a2 sSup c \u2208 nonPrincipals R\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9\n[GOAL]\ncase mk.intro\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nc : Set (Ideal R)\nhs : c \u2286 nonPrincipals R\nhchain : IsChain (fun x x_1 => x \u2264 x_1) c\nK : Ideal R\nhKmem : K \u2208 c\nx : R\nhx : sSup c = Submodule.span R {x}\n\u22a2 False\n[PROOFSTEP]\nhave hxmem : x \u2208 sSup c := hx.symm \u25b8 Submodule.mem_span_singleton_self x\n[GOAL]\ncase mk.intro\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nc : Set (Ideal R)\nhs : c \u2286 nonPrincipals R\nhchain : IsChain (fun x x_1 => x \u2264 x_1) c\nK : Ideal R\nhKmem : K \u2208 c\nx : R\nhx : sSup c = Submodule.span R {x}\nhxmem : x \u2208 sSup c\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8J, hJc, hxJ\u27e9 := (Submodule.mem_sSup_of_directed \u27e8K, hKmem\u27e9 hchain.directedOn).1 hxmem\n[GOAL]\ncase mk.intro.intro.intro\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nc : Set (Ideal R)\nhs : c \u2286 nonPrincipals R\nhchain : IsChain (fun x x_1 => x \u2264 x_1) c\nK : Ideal R\nhKmem : K \u2208 c\nx : R\nhx : sSup c = Submodule.span R {x}\nhxmem : x \u2208 sSup c\nJ : Submodule R R\nhJc : J \u2208 c\nhxJ : x \u2208 J\n\u22a2 False\n[PROOFSTEP]\nhave hsSupJ : sSup c = J := le_antisymm (by simp [hx, Ideal.span_le, hxJ]) (le_sSup hJc)\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nc : Set (Ideal R)\nhs : c \u2286 nonPrincipals R\nhchain : IsChain (fun x x_1 => x \u2264 x_1) c\nK : Ideal R\nhKmem : K \u2208 c\nx : R\nhx : sSup c = Submodule.span R {x}\nhxmem : x \u2208 sSup c\nJ : Submodule R R\nhJc : J \u2208 c\nhxJ : x \u2208 J\n\u22a2 sSup c \u2264 J\n[PROOFSTEP]\nsimp [hx, Ideal.span_le, hxJ]\n[GOAL]\ncase mk.intro.intro.intro\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nc : Set (Ideal R)\nhs : c \u2286 nonPrincipals R\nhchain : IsChain (fun x x_1 => x \u2264 x_1) c\nK : Ideal R\nhKmem : K \u2208 c\nx : R\nhx : sSup c = Submodule.span R {x}\nhxmem : x \u2208 sSup c\nJ : Submodule R R\nhJc : J \u2208 c\nhxJ : x \u2208 J\nhsSupJ : sSup c = J\n\u22a2 False\n[PROOFSTEP]\nspecialize hs hJc\n[GOAL]\ncase mk.intro.intro.intro\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nc : Set (Ideal R)\nhchain : IsChain (fun x x_1 => x \u2264 x_1) c\nK : Ideal R\nhKmem : K \u2208 c\nx : R\nhx : sSup c = Submodule.span R {x}\nhxmem : x \u2208 sSup c\nJ : Submodule R R\nhJc : J \u2208 c\nhxJ : x \u2208 J\nhsSupJ : sSup c = J\nhs : J \u2208 nonPrincipals R\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 hsSupJ, hx, nonPrincipals_def] at hs \n[GOAL]\ncase mk.intro.intro.intro\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nc : Set (Ideal R)\nhchain : IsChain (fun x x_1 => x \u2264 x_1) c\nK : Ideal R\nhKmem : K \u2208 c\nx : R\nhx : sSup c = Submodule.span R {x}\nhxmem : x \u2208 sSup c\nJ : Submodule R R\nhJc : J \u2208 c\nhxJ : x \u2208 J\nhsSupJ : sSup c = J\nhs : \u00acIsPrincipal (Submodule.span R {x})\n\u22a2 False\n[PROOFSTEP]\nexact hs \u27e8\u27e8x, rfl\u27e9\u27e9\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\n\u22a2 IsPrincipalIdealRing R\n[PROOFSTEP]\nrw [\u2190 nonPrincipals_eq_empty_iff, Set.eq_empty_iff_forall_not_mem]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\n\u22a2 \u2200 (x : Ideal R), \u00acx \u2208 nonPrincipals R\n[PROOFSTEP]\nintro J hJ\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8I, Ibad, -, Imax\u27e9 := zorn_nonempty_partialOrder\u2080 (nonPrincipals R) nonPrincipals_zorn _ hJ\n[GOAL]\ncase intro.intro.intro\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\n\u22a2 False\n[PROOFSTEP]\nhave Imax' : \u2200 {J}, I < J \u2192 J.IsPrincipal := by\n intro J hJ\n by_contra He\n exact hJ.ne (Imax _ ((nonPrincipals_def R).2 He) hJ.le).symm\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\n\u22a2 \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\n[PROOFSTEP]\nintro J hJ\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ\u271d : Ideal R\nhJ\u271d : J\u271d \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nJ : Ideal R\nhJ : I < J\n\u22a2 IsPrincipal J\n[PROOFSTEP]\nby_contra He\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ\u271d : Ideal R\nhJ\u271d : J\u271d \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nJ : Ideal R\nhJ : I < J\nHe : \u00acIsPrincipal J\n\u22a2 False\n[PROOFSTEP]\nexact hJ.ne (Imax _ ((nonPrincipals_def R).2 He) hJ.le).symm\n[GOAL]\ncase intro.intro.intro\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\n\u22a2 False\n[PROOFSTEP]\nby_cases hI1 : I = \u22a4\n[GOAL]\ncase pos\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : I = \u22a4\n\u22a2 False\n[PROOFSTEP]\nsubst hI1\n[GOAL]\ncase pos\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nIbad : \u22a4 \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 \u22a4 \u2264 z \u2192 z = \u22a4\nImax' : \u2200 {J : Ideal R}, \u22a4 < J \u2192 IsPrincipal J\n\u22a2 False\n[PROOFSTEP]\nexact Ibad top_isPrincipal\n[GOAL]\ncase neg\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\n\u22a2 False\n[PROOFSTEP]\nrefine' Ibad (H I \u27e8hI1, fun {x y} hxy => or_iff_not_imp_right.mpr fun hy => _\u27e9)\n[GOAL]\ncase neg\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx y : R\nhxy : x * y \u2208 I\nhy : \u00acy \u2208 I\n\u22a2 x \u2208 I\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 : (I \u2294 span { y }).IsPrincipal :=\n Imax'\n (left_lt_sup.mpr (mt I.span_singleton_le_iff_mem.mp hy))\n -- Then `x \u2208 I.colon (span {y})`, which is equal to `I` if it's not principal.\n[GOAL]\ncase neg.mk.intro\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx y : R\nhxy : x * y \u2208 I\nhy : \u00acy \u2208 I\na : R\nha : I \u2294 Ideal.span {y} = Submodule.span R {a}\n\u22a2 x \u2208 I\n[PROOFSTEP]\nsuffices He : \u00ac(I.colon (span { y })).IsPrincipal\n[GOAL]\ncase neg.mk.intro\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx y : R\nhxy : x * y \u2208 I\nhy : \u00acy \u2208 I\na : R\nha : I \u2294 Ideal.span {y} = Submodule.span R {a}\nHe : \u00acIsPrincipal (colon I (Ideal.span {y}))\n\u22a2 x \u2208 I\n[PROOFSTEP]\nrw [\u2190 Imax _ ((nonPrincipals_def R).2 He) fun a ha => Ideal.mem_colon_singleton.2 (mul_mem_right _ _ ha)]\n[GOAL]\ncase neg.mk.intro\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx y : R\nhxy : x * y \u2208 I\nhy : \u00acy \u2208 I\na : R\nha : I \u2294 Ideal.span {y} = Submodule.span R {a}\nHe : \u00acIsPrincipal (colon I (Ideal.span {y}))\n\u22a2 x \u2208 colon I (Ideal.span {y})\n[PROOFSTEP]\nexact Ideal.mem_colon_singleton.2 hxy\n[GOAL]\ncase He\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx y : R\nhxy : x * y \u2208 I\nhy : \u00acy \u2208 I\na : R\nha : I \u2294 Ideal.span {y} = Submodule.span R {a}\n\u22a2 \u00acIsPrincipal (colon I (Ideal.span {y}))\n[PROOFSTEP]\nrintro\n \u27e8b, hb\u27e9\n -- We will show `I` is generated by `a * b`.\n[GOAL]\ncase He.mk.intro\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx y : R\nhxy : x * y \u2208 I\nhy : \u00acy \u2208 I\na : R\nha : I \u2294 Ideal.span {y} = Submodule.span R {a}\nb : R\nhb : colon I (Ideal.span {y}) = Submodule.span R {b}\n\u22a2 False\n[PROOFSTEP]\nrefine (nonPrincipals_def _).1 Ibad \u27e8a * b, ?_\u27e9\n[GOAL]\ncase He.mk.intro\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx y : R\nhxy : x * y \u2208 I\nhy : \u00acy \u2208 I\na : R\nha : I \u2294 Ideal.span {y} = Submodule.span R {a}\nb : R\nhb : colon I (Ideal.span {y}) = Submodule.span R {b}\n\u22a2 I = Submodule.span R {a * b}\n[PROOFSTEP]\nrefine' le_antisymm (\u03b1 := Ideal R) (fun i hi => _) <| (span_singleton_mul_span_singleton a b).ge.trans _\n[GOAL]\ncase He.mk.intro.refine'_1\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx y : R\nhxy : x * y \u2208 I\nhy : \u00acy \u2208 I\na : R\nha : I \u2294 Ideal.span {y} = Submodule.span R {a}\nb : R\nhb : colon I (Ideal.span {y}) = Submodule.span R {b}\ni : R\nhi : i \u2208 I\n\u22a2 i \u2208 Submodule.span R {a * b}\n[PROOFSTEP]\nhave hisup : i \u2208 I \u2294 span { y } := Ideal.mem_sup_left hi\n[GOAL]\ncase He.mk.intro.refine'_1\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx y : R\nhxy : x * y \u2208 I\nhy : \u00acy \u2208 I\na : R\nha : I \u2294 Ideal.span {y} = Submodule.span R {a}\nb : R\nhb : colon I (Ideal.span {y}) = Submodule.span R {b}\ni : R\nhi : i \u2208 I\nhisup : i \u2208 I \u2294 Ideal.span {y}\n\u22a2 i \u2208 Submodule.span R {a * b}\n[PROOFSTEP]\nhave : y \u2208 I \u2294 span { y } := Ideal.mem_sup_right (Ideal.mem_span_singleton_self y)\n[GOAL]\ncase He.mk.intro.refine'_1\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx y : R\nhxy : x * y \u2208 I\nhy : \u00acy \u2208 I\na : R\nha : I \u2294 Ideal.span {y} = Submodule.span R {a}\nb : R\nhb : colon I (Ideal.span {y}) = Submodule.span R {b}\ni : R\nhi : i \u2208 I\nhisup : i \u2208 I \u2294 Ideal.span {y}\nthis : y \u2208 I \u2294 Ideal.span {y}\n\u22a2 i \u2208 Submodule.span R {a * b}\n[PROOFSTEP]\nerw [ha, mem_span_singleton'] at hisup this \n[GOAL]\ncase He.mk.intro.refine'_1\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx y : R\nhxy : x * y \u2208 I\nhy : \u00acy \u2208 I\na : R\nha : I \u2294 Ideal.span {y} = Submodule.span R {a}\nb : R\nhb : colon I (Ideal.span {y}) = Submodule.span R {b}\ni : R\nhi : i \u2208 I\nhisup : \u2203 a_1, a_1 * a = i\nthis : \u2203 a_1, a_1 * a = y\n\u22a2 i \u2208 Submodule.span R {a * b}\n[PROOFSTEP]\nobtain \u27e8v, rfl\u27e9 := this\n[GOAL]\ncase He.mk.intro.refine'_1.intro\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx a b i : R\nhi : i \u2208 I\nhisup : \u2203 a_1, a_1 * a = i\nv : R\nhxy : x * (v * a) \u2208 I\nhy : \u00acv * a \u2208 I\nha : I \u2294 Ideal.span {v * a} = Submodule.span R {a}\nhb : colon I (Ideal.span {v * a}) = Submodule.span R {b}\n\u22a2 i \u2208 Submodule.span R {a * b}\n[PROOFSTEP]\nobtain \u27e8u, rfl\u27e9 := hisup\n[GOAL]\ncase He.mk.intro.refine'_1.intro.intro\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx a b v : R\nhxy : x * (v * a) \u2208 I\nhy : \u00acv * a \u2208 I\nha : I \u2294 Ideal.span {v * a} = Submodule.span R {a}\nhb : colon I (Ideal.span {v * a}) = Submodule.span R {b}\nu : R\nhi : u * a \u2208 I\n\u22a2 u * a \u2208 Submodule.span R {a * b}\n[PROOFSTEP]\nhave hucolon : u \u2208 I.colon (span {v * a}) :=\n by\n rw [Ideal.mem_colon_singleton, mul_comm v, \u2190 mul_assoc]\n exact mul_mem_right _ _ hi\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx a b v : R\nhxy : x * (v * a) \u2208 I\nhy : \u00acv * a \u2208 I\nha : I \u2294 Ideal.span {v * a} = Submodule.span R {a}\nhb : colon I (Ideal.span {v * a}) = Submodule.span R {b}\nu : R\nhi : u * a \u2208 I\n\u22a2 u \u2208 colon I (Ideal.span {v * a})\n[PROOFSTEP]\nrw [Ideal.mem_colon_singleton, mul_comm v, \u2190 mul_assoc]\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx a b v : R\nhxy : x * (v * a) \u2208 I\nhy : \u00acv * a \u2208 I\nha : I \u2294 Ideal.span {v * a} = Submodule.span R {a}\nhb : colon I (Ideal.span {v * a}) = Submodule.span R {b}\nu : R\nhi : u * a \u2208 I\n\u22a2 u * a * v \u2208 I\n[PROOFSTEP]\nexact mul_mem_right _ _ hi\n[GOAL]\ncase He.mk.intro.refine'_1.intro.intro\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx a b v : R\nhxy : x * (v * a) \u2208 I\nhy : \u00acv * a \u2208 I\nha : I \u2294 Ideal.span {v * a} = Submodule.span R {a}\nhb : colon I (Ideal.span {v * a}) = Submodule.span R {b}\nu : R\nhi : u * a \u2208 I\nhucolon : u \u2208 colon I (Ideal.span {v * a})\n\u22a2 u * a \u2208 Submodule.span R {a * b}\n[PROOFSTEP]\nerw [hb, mem_span_singleton'] at hucolon \n[GOAL]\ncase He.mk.intro.refine'_1.intro.intro\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx a b v : R\nhxy : x * (v * a) \u2208 I\nhy : \u00acv * a \u2208 I\nha : I \u2294 Ideal.span {v * a} = Submodule.span R {a}\nhb : colon I (Ideal.span {v * a}) = Submodule.span R {b}\nu : R\nhi : u * a \u2208 I\nhucolon : \u2203 a, a * b = u\n\u22a2 u * a \u2208 Submodule.span R {a * b}\n[PROOFSTEP]\nobtain \u27e8z, rfl\u27e9 := hucolon\n[GOAL]\ncase He.mk.intro.refine'_1.intro.intro.intro\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx a b v : R\nhxy : x * (v * a) \u2208 I\nhy : \u00acv * a \u2208 I\nha : I \u2294 Ideal.span {v * a} = Submodule.span R {a}\nhb : colon I (Ideal.span {v * a}) = Submodule.span R {b}\nz : R\nhi : z * b * a \u2208 I\n\u22a2 z * b * a \u2208 Submodule.span R {a * b}\n[PROOFSTEP]\nexact mem_span_singleton'.2 \u27e8z, by ring\u27e9\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx a b v : R\nhxy : x * (v * a) \u2208 I\nhy : \u00acv * a \u2208 I\nha : I \u2294 Ideal.span {v * a} = Submodule.span R {a}\nhb : colon I (Ideal.span {v * a}) = Submodule.span R {b}\nz : R\nhi : z * b * a \u2208 I\n\u22a2 z * (a * b) = z * b * a\n[PROOFSTEP]\nring\n[GOAL]\ncase He.mk.intro.refine'_2\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx y : R\nhxy : x * y \u2208 I\nhy : \u00acy \u2208 I\na : R\nha : I \u2294 Ideal.span {y} = Submodule.span R {a}\nb : R\nhb : colon I (Ideal.span {y}) = Submodule.span R {b}\n\u22a2 Ideal.span {a} * Ideal.span {b} \u2264 I\n[PROOFSTEP]\nrw [\u2190 Ideal.submodule_span_eq, \u2190 ha, Ideal.sup_mul, sup_le_iff, span_singleton_mul_span_singleton, mul_comm y,\n Ideal.span_singleton_le_iff_mem]\n[GOAL]\ncase He.mk.intro.refine'_2\nR : Type u\nM : Type v\ninst\u271d : CommRing R\nH : \u2200 (P : Ideal R), IsPrime P \u2192 IsPrincipal P\nJ : Ideal R\nhJ : J \u2208 nonPrincipals R\nI : Ideal R\nIbad : I \u2208 nonPrincipals R\nImax : \u2200 (z : Ideal R), z \u2208 nonPrincipals R \u2192 I \u2264 z \u2192 z = I\nImax' : \u2200 {J : Ideal R}, I < J \u2192 IsPrincipal J\nhI1 : \u00acI = \u22a4\nx y : R\nhxy : x * y \u2208 I\nhy : \u00acy \u2208 I\na : R\nha : I \u2294 Ideal.span {y} = Submodule.span R {a}\nb : R\nhb : colon I (Ideal.span {y}) = Submodule.span R {b}\n\u22a2 I * Ideal.span {b} \u2264 I \u2227 b * y \u2208 I\n[PROOFSTEP]\nexact \u27e8mul_le_right, Ideal.mem_colon_singleton.1 <| hb.symm \u25b8 Ideal.mem_span_singleton_self b\u27e9\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.PrincipalIdealDomain", "llama_tokens": 26572, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.43014734858584286, "lm_q2_score": 0.02096424207972777, "lm_q1q2_score": 0.009017713145706657}} {"text": "[GOAL]\nB : Type u\ninst\u271d : Quiver B\na b : B\nf g : Discrete (Path a b)\n\u03b7 : f \u27f6 g\n\u22a2 PrelaxFunctor.map\u2082 (preinclusion B) \u03b7 = eqToHom (_ : (\u2191(preinclusion B)).map f = (\u2191(preinclusion B)).map g)\n[PROOFSTEP]\nrcases \u03b7 with \u27e8\u27e8\u27e9\u27e9\n[GOAL]\ncase up.up\nB : Type u\ninst\u271d : Quiver B\na b : B\nf g : Discrete (Path a b)\ndown\u271d : f.as = g.as\n\u22a2 PrelaxFunctor.map\u2082 (preinclusion B) { down := { down := down\u271d } } =\n eqToHom (_ : (\u2191(preinclusion B)).map f = (\u2191(preinclusion B)).map g)\n[PROOFSTEP]\ncases Discrete.ext _ _ (by assumption)\n[GOAL]\nB : Type u\ninst\u271d : Quiver B\na b : B\nf g : Discrete (Path a b)\ndown\u271d : f.as = g.as\n\u22a2 ?m.4748.as = ?m.4749.as\n[PROOFSTEP]\nassumption\n[GOAL]\ncase up.up.refl\nB : Type u\ninst\u271d : Quiver B\na b : B\nf : Discrete (Path a b)\ndown\u271d : f.as = f.as\n\u22a2 PrelaxFunctor.map\u2082 (preinclusion B) { down := { down := down\u271d } } =\n eqToHom (_ : (\u2191(preinclusion B)).map f = (\u2191(preinclusion B)).map f)\n[PROOFSTEP]\nconvert (inclusionPath a b).map_id _\n[GOAL]\nB : Type u\ninst\u271d : Quiver B\na b c : B\np : Path a b\nf g : Hom b c\n\u03b7 : f \u27f6 g\n\u22a2 normalizeAux p f = normalizeAux p g\n[PROOFSTEP]\nrcases \u03b7 with \u27e8\u03b7'\u27e9\n[GOAL]\ncase mk\nB : Type u\ninst\u271d : Quiver B\na b c : B\np : Path a b\nf g : Hom b c\n\u03b7 : f \u27f6 g\n\u03b7' : Hom\u2082 f g\n\u22a2 normalizeAux p f = normalizeAux p g\n[PROOFSTEP]\napply @congr_fun _ _ fun p => normalizeAux p f\n[GOAL]\ncase mk.h\nB : Type u\ninst\u271d : Quiver B\na b c : B\np : Path a b\nf g : Hom b c\n\u03b7 : f \u27f6 g\n\u03b7' : Hom\u2082 f g\n\u22a2 (fun p => normalizeAux p f) = fun p => normalizeAux p g\n[PROOFSTEP]\nclear p \u03b7\n[GOAL]\ncase mk.h\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\n\u03b7' : Hom\u2082 f g\n\u22a2 (fun p => normalizeAux p f) = fun p => normalizeAux p g\n[PROOFSTEP]\ninduction \u03b7' with\n| vcomp _ _ _ _ => apply Eq.trans <;> assumption\n| whisker_left _ _ ih => funext; apply congr_fun ih\n| whisker_right _ _ ih => funext; apply congr_arg\u2082 _ (congr_fun ih _) rfl\n| _ => funext; rfl\n[GOAL]\ncase mk.h\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\n\u03b7' : Hom\u2082 f g\n\u22a2 (fun p => normalizeAux p f) = fun p => normalizeAux p g\n[PROOFSTEP]\ninduction \u03b7' with\n| vcomp _ _ _ _ => apply Eq.trans <;> assumption\n| whisker_left _ _ ih => funext; apply congr_fun ih\n| whisker_right _ _ ih => funext; apply congr_arg\u2082 _ (congr_fun ih _) rfl\n| _ => funext; rfl\n[GOAL]\ncase mk.h.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7\u271d : Hom\u2082 f\u271d g\u271d\n\u03b8\u271d : Hom\u2082 g\u271d h\u271d\n\u03b7_ih\u271d : (fun p => normalizeAux p f\u271d) = fun p => normalizeAux p g\u271d\n\u03b8_ih\u271d : (fun p => normalizeAux p g\u271d) = fun p => normalizeAux p h\u271d\n\u22a2 (fun p => normalizeAux p f\u271d) = fun p => normalizeAux p h\u271d\n[PROOFSTEP]\n\n| vcomp _ _ _ _ => apply Eq.trans <;> assumption\n[GOAL]\ncase mk.h.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7\u271d : Hom\u2082 f\u271d g\u271d\n\u03b8\u271d : Hom\u2082 g\u271d h\u271d\n\u03b7_ih\u271d : (fun p => normalizeAux p f\u271d) = fun p => normalizeAux p g\u271d\n\u03b8_ih\u271d : (fun p => normalizeAux p g\u271d) = fun p => normalizeAux p h\u271d\n\u22a2 (fun p => normalizeAux p f\u271d) = fun p => normalizeAux p h\u271d\n[PROOFSTEP]\napply Eq.trans\n[GOAL]\ncase mk.h.vcomp.h\u2081\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7\u271d : Hom\u2082 f\u271d g\u271d\n\u03b8\u271d : Hom\u2082 g\u271d h\u271d\n\u03b7_ih\u271d : (fun p => normalizeAux p f\u271d) = fun p => normalizeAux p g\u271d\n\u03b8_ih\u271d : (fun p => normalizeAux p g\u271d) = fun p => normalizeAux p h\u271d\n\u22a2 (fun p => normalizeAux p f\u271d) = ?mk.h.vcomp.b\n[PROOFSTEP]\nassumption\n[GOAL]\ncase mk.h.vcomp.h\u2082\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7\u271d : Hom\u2082 f\u271d g\u271d\n\u03b8\u271d : Hom\u2082 g\u271d h\u271d\n\u03b7_ih\u271d : (fun p => normalizeAux p f\u271d) = fun p => normalizeAux p g\u271d\n\u03b8_ih\u271d : (fun p => normalizeAux p g\u271d) = fun p => normalizeAux p h\u271d\n\u22a2 (fun p => normalizeAux p g\u271d) = fun p => normalizeAux p h\u271d\n[PROOFSTEP]\nassumption\n[GOAL]\ncase mk.h.whisker_left\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d h\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : Hom\u2082 g\u271d h\u271d\nih : (fun p => normalizeAux p g\u271d) = fun p => normalizeAux p h\u271d\n\u22a2 (fun p => normalizeAux p (f\u271d \u226b g\u271d)) = fun p => normalizeAux p (f\u271d \u226b h\u271d)\n[PROOFSTEP]\n\n| whisker_left _ _ ih => funext; apply congr_fun ih\n[GOAL]\ncase mk.h.whisker_left\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d h\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : Hom\u2082 g\u271d h\u271d\nih : (fun p => normalizeAux p g\u271d) = fun p => normalizeAux p h\u271d\n\u22a2 (fun p => normalizeAux p (f\u271d \u226b g\u271d)) = fun p => normalizeAux p (f\u271d \u226b h\u271d)\n[PROOFSTEP]\nfunext\n[GOAL]\ncase mk.h.whisker_left.h\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d h\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : Hom\u2082 g\u271d h\u271d\nih : (fun p => normalizeAux p g\u271d) = fun p => normalizeAux p h\u271d\nx\u271d : Path a a\u271d\n\u22a2 normalizeAux x\u271d (f\u271d \u226b g\u271d) = normalizeAux x\u271d (f\u271d \u226b h\u271d)\n[PROOFSTEP]\napply congr_fun ih\n[GOAL]\ncase mk.h.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : Hom\u2082 f\u271d g\u271d\nih : (fun p => normalizeAux p f\u271d) = fun p => normalizeAux p g\u271d\n\u22a2 (fun p => normalizeAux p (Hom.comp f\u271d h\u271d)) = fun p => normalizeAux p (Hom.comp g\u271d h\u271d)\n[PROOFSTEP]\n\n| whisker_right _ _ ih => funext; apply congr_arg\u2082 _ (congr_fun ih _) rfl\n[GOAL]\ncase mk.h.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : Hom\u2082 f\u271d g\u271d\nih : (fun p => normalizeAux p f\u271d) = fun p => normalizeAux p g\u271d\n\u22a2 (fun p => normalizeAux p (Hom.comp f\u271d h\u271d)) = fun p => normalizeAux p (Hom.comp g\u271d h\u271d)\n[PROOFSTEP]\nfunext\n[GOAL]\ncase mk.h.whisker_right.h\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : Hom\u2082 f\u271d g\u271d\nih : (fun p => normalizeAux p f\u271d) = fun p => normalizeAux p g\u271d\nx\u271d : Path a a\u271d\n\u22a2 normalizeAux x\u271d (Hom.comp f\u271d h\u271d) = normalizeAux x\u271d (Hom.comp g\u271d h\u271d)\n[PROOFSTEP]\napply congr_arg\u2082 _ (congr_fun ih _) rfl\n[GOAL]\ncase mk.h.id\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\n\u22a2 (fun p => normalizeAux p f\u271d) = fun p => normalizeAux p f\u271d\n[PROOFSTEP]\n\n| _ => funext; rfl\n[GOAL]\ncase mk.h.id\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\n\u22a2 (fun p => normalizeAux p f\u271d) = fun p => normalizeAux p f\u271d\n[PROOFSTEP]\nfunext\n[GOAL]\ncase mk.h.id.h\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\nx\u271d : Path a a\u271d\n\u22a2 normalizeAux x\u271d f\u271d = normalizeAux x\u271d f\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.h.associator\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d d\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d : b\u271d \u27f6 c\u271d\nh\u271d : c\u271d \u27f6 d\u271d\n\u22a2 (fun p => normalizeAux p ((f\u271d \u226b g\u271d) \u226b h\u271d)) = fun p => normalizeAux p (f\u271d \u226b g\u271d \u226b h\u271d)\n[PROOFSTEP]\n\n| _ => funext; rfl\n[GOAL]\ncase mk.h.associator\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d d\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d : b\u271d \u27f6 c\u271d\nh\u271d : c\u271d \u27f6 d\u271d\n\u22a2 (fun p => normalizeAux p ((f\u271d \u226b g\u271d) \u226b h\u271d)) = fun p => normalizeAux p (f\u271d \u226b g\u271d \u226b h\u271d)\n[PROOFSTEP]\nfunext\n[GOAL]\ncase mk.h.associator.h\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d d\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d : b\u271d \u27f6 c\u271d\nh\u271d : c\u271d \u27f6 d\u271d\nx\u271d : Path a a\u271d\n\u22a2 normalizeAux x\u271d ((f\u271d \u226b g\u271d) \u226b h\u271d) = normalizeAux x\u271d (f\u271d \u226b g\u271d \u226b h\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.h.associator_inv\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d d\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d : b\u271d \u27f6 c\u271d\nh\u271d : c\u271d \u27f6 d\u271d\n\u22a2 (fun p => normalizeAux p (f\u271d \u226b g\u271d \u226b h\u271d)) = fun p => normalizeAux p ((f\u271d \u226b g\u271d) \u226b h\u271d)\n[PROOFSTEP]\n\n| _ => funext; rfl\n[GOAL]\ncase mk.h.associator_inv\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d d\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d : b\u271d \u27f6 c\u271d\nh\u271d : c\u271d \u27f6 d\u271d\n\u22a2 (fun p => normalizeAux p (f\u271d \u226b g\u271d \u226b h\u271d)) = fun p => normalizeAux p ((f\u271d \u226b g\u271d) \u226b h\u271d)\n[PROOFSTEP]\nfunext\n[GOAL]\ncase mk.h.associator_inv.h\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d d\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d : b\u271d \u27f6 c\u271d\nh\u271d : c\u271d \u27f6 d\u271d\nx\u271d : Path a a\u271d\n\u22a2 normalizeAux x\u271d (f\u271d \u226b g\u271d \u226b h\u271d) = normalizeAux x\u271d ((f\u271d \u226b g\u271d) \u226b h\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.h.right_unitor\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\n\u22a2 (fun p => normalizeAux p (f\u271d \u226b \ud835\udfd9 b\u271d)) = fun p => normalizeAux p f\u271d\n[PROOFSTEP]\n\n| _ => funext; rfl\n[GOAL]\ncase mk.h.right_unitor\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\n\u22a2 (fun p => normalizeAux p (f\u271d \u226b \ud835\udfd9 b\u271d)) = fun p => normalizeAux p f\u271d\n[PROOFSTEP]\nfunext\n[GOAL]\ncase mk.h.right_unitor.h\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\nx\u271d : Path a a\u271d\n\u22a2 normalizeAux x\u271d (f\u271d \u226b \ud835\udfd9 b\u271d) = normalizeAux x\u271d f\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.h.right_unitor_inv\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\n\u22a2 (fun p => normalizeAux p f\u271d) = fun p => normalizeAux p (f\u271d \u226b \ud835\udfd9 b\u271d)\n[PROOFSTEP]\n\n| _ => funext; rfl\n[GOAL]\ncase mk.h.right_unitor_inv\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\n\u22a2 (fun p => normalizeAux p f\u271d) = fun p => normalizeAux p (f\u271d \u226b \ud835\udfd9 b\u271d)\n[PROOFSTEP]\nfunext\n[GOAL]\ncase mk.h.right_unitor_inv.h\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\nx\u271d : Path a a\u271d\n\u22a2 normalizeAux x\u271d f\u271d = normalizeAux x\u271d (f\u271d \u226b \ud835\udfd9 b\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.h.left_unitor\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\n\u22a2 (fun p => normalizeAux p (\ud835\udfd9 a\u271d \u226b f\u271d)) = fun p => normalizeAux p f\u271d\n[PROOFSTEP]\n\n| _ => funext; rfl\n[GOAL]\ncase mk.h.left_unitor\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\n\u22a2 (fun p => normalizeAux p (\ud835\udfd9 a\u271d \u226b f\u271d)) = fun p => normalizeAux p f\u271d\n[PROOFSTEP]\nfunext\n[GOAL]\ncase mk.h.left_unitor.h\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\nx\u271d : Path a a\u271d\n\u22a2 normalizeAux x\u271d (\ud835\udfd9 a\u271d \u226b f\u271d) = normalizeAux x\u271d f\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.h.left_unitor_inv\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\n\u22a2 (fun p => normalizeAux p f\u271d) = fun p => normalizeAux p (\ud835\udfd9 a\u271d \u226b f\u271d)\n[PROOFSTEP]\n\n| _ => funext; rfl\n[GOAL]\ncase mk.h.left_unitor_inv\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\n\u22a2 (fun p => normalizeAux p f\u271d) = fun p => normalizeAux p (\ud835\udfd9 a\u271d \u226b f\u271d)\n[PROOFSTEP]\nfunext\n[GOAL]\ncase mk.h.left_unitor_inv.h\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\nx\u271d : Path a a\u271d\n\u22a2 normalizeAux x\u271d f\u271d = normalizeAux x\u271d (\ud835\udfd9 a\u271d \u226b f\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nB : Type u\ninst\u271d : Quiver B\na b c : B\np : Path a b\nf g : Hom b c\n\u03b7 : f \u27f6 g\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 \u03b7 \u226b (normalizeIso p g).hom =\n (normalizeIso p f).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f } = { as := normalizeAux p g }))\n[PROOFSTEP]\nrcases \u03b7 with \u27e8\u03b7'\u27e9\n[GOAL]\ncase mk\nB : Type u\ninst\u271d : Quiver B\na b c : B\np : Path a b\nf g : Hom b c\n\u03b7 : f \u27f6 g\n\u03b7' : Hom\u2082 f g\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g).hom =\n (normalizeIso p f).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f } = { as := normalizeAux p g }))\n[PROOFSTEP]\nclear \u03b7\n[GOAL]\ncase mk\nB : Type u\ninst\u271d : Quiver B\na b c : B\np : Path a b\nf g : Hom b c\n\u03b7' : Hom\u2082 f g\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g).hom =\n (normalizeIso p f).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f } = { as := normalizeAux p g }))\n[PROOFSTEP]\ninduction \u03b7' with\n| id => simp\n| vcomp \u03b7 \u03b8 ihf ihg =>\n simp only [mk_vcomp, Bicategory.whiskerLeft_comp]\n slice_lhs 2 3 => rw [ihg]\n slice_lhs 1 2 => rw [ihf]\n simp\n -- p \u2260 nil required! See the docstring of `normalizeAux`.\n| whisker_left _ _ ih =>\n dsimp\n rw [associator_inv_naturality_right_assoc, whisker_exchange_assoc, ih]\n simp\n| whisker_right h \u03b7' ih =>\n dsimp\n rw [associator_inv_naturality_middle_assoc, \u2190 comp_whiskerRight_assoc, ih, comp_whiskerRight]\n have := dcongr_arg (fun x => (normalizeIso x h).hom) (normalizeAux_congr p (Quot.mk _ \u03b7'))\n dsimp at this ; simp [this]\n| _ => simp\n[GOAL]\ncase mk\nB : Type u\ninst\u271d : Quiver B\na b c : B\np : Path a b\nf g : Hom b c\n\u03b7' : Hom\u2082 f g\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g).hom =\n (normalizeIso p f).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f } = { as := normalizeAux p g }))\n[PROOFSTEP]\ninduction \u03b7' with\n| id => simp\n| vcomp \u03b7 \u03b8 ihf ihg =>\n simp only [mk_vcomp, Bicategory.whiskerLeft_comp]\n slice_lhs 2 3 => rw [ihg]\n slice_lhs 1 2 => rw [ihf]\n simp\n -- p \u2260 nil required! See the docstring of `normalizeAux`.\n| whisker_left _ _ ih =>\n dsimp\n rw [associator_inv_naturality_right_assoc, whisker_exchange_assoc, ih]\n simp\n| whisker_right h \u03b7' ih =>\n dsimp\n rw [associator_inv_naturality_middle_assoc, \u2190 comp_whiskerRight_assoc, ih, comp_whiskerRight]\n have := dcongr_arg (fun x => (normalizeIso x h).hom) (normalizeAux_congr p (Quot.mk _ \u03b7'))\n dsimp at this ; simp [this]\n| _ => simp\n[GOAL]\ncase mk.id\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.id f\u271d) \u226b (normalizeIso p f\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p f\u271d }))\n[PROOFSTEP]\n\n| id => simp\n[GOAL]\ncase mk.id\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.id f\u271d) \u226b (normalizeIso p f\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p f\u271d }))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.vcomp \u03b7 \u03b8) \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p h\u271d }))\n[PROOFSTEP]\n\n| vcomp \u03b7 \u03b8 ihf ihg =>\n simp only [mk_vcomp, Bicategory.whiskerLeft_comp]\n slice_lhs 2 3 => rw [ihg]\n slice_lhs 1 2 => rw [ihf]\n simp\n -- p \u2260 nil required! See the docstring of `normalizeAux`.\n[GOAL]\ncase mk.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.vcomp \u03b7 \u03b8) \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p h\u271d }))\n[PROOFSTEP]\nsimp only [mk_vcomp, Bicategory.whiskerLeft_comp]\n[GOAL]\ncase mk.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 ((\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b7 \u226b (\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b8) \u226b\n (normalizeIso p h\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p h\u271d }))\n[PROOFSTEP]\nslice_lhs 2 3 => rw [ihg]\n[GOAL]\ncase a\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n| (\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b8 \u226b (normalizeIso p h\u271d).hom\ncase a\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n| (\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b7\n[PROOFSTEP]\nrw [ihg]\n[GOAL]\ncase a\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n| (\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b8 \u226b (normalizeIso p h\u271d).hom\ncase a\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n| (\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b7\n[PROOFSTEP]\nrw [ihg]\n[GOAL]\ncase a\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n| (\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b8 \u226b (normalizeIso p h\u271d).hom\ncase a\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n| (\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b7\n[PROOFSTEP]\nrw [ihg]\n[GOAL]\ncase mk.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b7 \u226b\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d })) =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p h\u271d }))\n[PROOFSTEP]\nslice_lhs 1 2 => rw [ihf]\n[GOAL]\ncase a\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n| (\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b7 \u226b (normalizeIso p g\u271d).hom\ncase a\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n| PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\n[PROOFSTEP]\nrw [ihf]\n[GOAL]\ncase a\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n| (\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b7 \u226b (normalizeIso p g\u271d).hom\ncase a\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n| PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\n[PROOFSTEP]\nrw [ihf]\n[GOAL]\ncase a\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n| (\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b7 \u226b (normalizeIso p g\u271d).hom\ncase a\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n| PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\n[PROOFSTEP]\nrw [ihf]\n[GOAL]\ncase mk.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 ((normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))) \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d })) =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p h\u271d }))\n[PROOFSTEP]\nsimp\n -- p \u2260 nil required! See the docstring of `normalizeAux`.\n[GOAL]\ncase mk.whisker_left\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d h\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : Hom\u2082 g\u271d h\u271d\nih :\n \u2200 (p : Path a b\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7\u271d \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.whisker_left f\u271d \u03b7\u271d) \u226b (normalizeIso p (f\u271d \u226b h\u271d)).hom =\n (normalizeIso p (f\u271d \u226b g\u271d)).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux p (f\u271d \u226b g\u271d) } = { as := normalizeAux p (f\u271d \u226b h\u271d) }))\n[PROOFSTEP]\n\n| whisker_left _ _ ih =>\n dsimp\n rw [associator_inv_naturality_right_assoc, whisker_exchange_assoc, ih]\n simp\n[GOAL]\ncase mk.whisker_left\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d h\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : Hom\u2082 g\u271d h\u271d\nih :\n \u2200 (p : Path a b\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7\u271d \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.whisker_left f\u271d \u03b7\u271d) \u226b (normalizeIso p (f\u271d \u226b h\u271d)).hom =\n (normalizeIso p (f\u271d \u226b g\u271d)).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux p (f\u271d \u226b g\u271d) } = { as := normalizeAux p (f\u271d \u226b h\u271d) }))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.whisker_left\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d h\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : Hom\u2082 g\u271d h\u271d\nih :\n \u2200 (p : Path a b\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7\u271d \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 f\u271d \u25c1 Hom\u2082.mk \u03b7\u271d \u226b\n (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h\u271d).inv \u226b\n (normalizeIso p f\u271d).hom \u25b7 h\u271d \u226b (normalizeIso (normalizeAux p f\u271d) h\u271d).hom =\n ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d g\u271d).inv \u226b\n (normalizeIso p f\u271d).hom \u25b7 g\u271d \u226b (normalizeIso (normalizeAux p f\u271d) g\u271d).hom) \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux (normalizeAux p f\u271d) g\u271d } = { as := normalizeAux (normalizeAux p f\u271d) h\u271d }))\n[PROOFSTEP]\nrw [associator_inv_naturality_right_assoc, whisker_exchange_assoc, ih]\n[GOAL]\ncase mk.whisker_left\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d h\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : Hom\u2082 g\u271d h\u271d\nih :\n \u2200 (p : Path a b\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7\u271d \u226b (normalizeIso p h\u271d).hom =\n (normalizeIso p g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d g\u271d).inv \u226b\n (normalizeIso p f\u271d).hom \u25b7 g\u271d \u226b\n (normalizeIso (normalizeAux p f\u271d) g\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom\n (_ : { as := normalizeAux (normalizeAux p f\u271d) g\u271d } = { as := normalizeAux (normalizeAux p f\u271d) h\u271d })) =\n ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d g\u271d).inv \u226b\n (normalizeIso p f\u271d).hom \u25b7 g\u271d \u226b (normalizeIso (normalizeAux p f\u271d) g\u271d).hom) \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux (normalizeAux p f\u271d) g\u271d } = { as := normalizeAux (normalizeAux p f\u271d) h\u271d }))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.whisker_right h \u03b7') \u226b (normalizeIso p (Hom.comp g\u271d h)).hom =\n (normalizeIso p (Hom.comp f\u271d h)).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux p (Hom.comp f\u271d h) } = { as := normalizeAux p (Hom.comp g\u271d h) }))\n[PROOFSTEP]\n\n| whisker_right h \u03b7' ih =>\n dsimp\n rw [associator_inv_naturality_middle_assoc, \u2190 comp_whiskerRight_assoc, ih, comp_whiskerRight]\n have := dcongr_arg (fun x => (normalizeIso x h).hom) (normalizeAux_congr p (Quot.mk _ \u03b7'))\n dsimp at this ; simp [this]\n[GOAL]\ncase mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.whisker_right h \u03b7') \u226b (normalizeIso p (Hom.comp g\u271d h)).hom =\n (normalizeIso p (Hom.comp f\u271d h)).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux p (Hom.comp f\u271d h) } = { as := normalizeAux p (Hom.comp g\u271d h) }))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b7' \u25b7 h \u226b\n (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) g\u271d h).inv \u226b\n (normalizeIso p g\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p g\u271d) h).hom =\n ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n (normalizeIso p f\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p f\u271d) h).hom) \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux (normalizeAux p f\u271d) h } = { as := normalizeAux (normalizeAux p g\u271d) h }))\n[PROOFSTEP]\nrw [associator_inv_naturality_middle_assoc, \u2190 comp_whiskerRight_assoc, ih, comp_whiskerRight]\n[GOAL]\ncase mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\np : Path a a\u271d\n\u22a2 (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n ((normalizeIso p f\u271d).hom \u25b7 h \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d })) \u25b7\n h) \u226b\n (normalizeIso (normalizeAux p g\u271d) h).hom =\n ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n (normalizeIso p f\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p f\u271d) h).hom) \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux (normalizeAux p f\u271d) h } = { as := normalizeAux (normalizeAux p g\u271d) h }))\n[PROOFSTEP]\nhave := dcongr_arg (fun x => (normalizeIso x h).hom) (normalizeAux_congr p (Quot.mk _ \u03b7'))\n[GOAL]\ncase mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\np : Path a a\u271d\nthis :\n (normalizeIso (normalizeAux p f\u271d) h).hom =\n eqToHom\n (_ :\n (\u2191(preinclusion B)).map { as := normalizeAux p f\u271d } \u226b h =\n (\u2191(preinclusion B)).map { as := normalizeAux p g\u271d } \u226b h) \u226b\n (normalizeIso (normalizeAux p g\u271d) h).hom \u226b\n eqToHom\n (_ :\n (\u2191(preinclusion B)).map { as := normalizeAux (normalizeAux p g\u271d) h } =\n (\u2191(preinclusion B)).map { as := normalizeAux (normalizeAux p f\u271d) h })\n\u22a2 (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n ((normalizeIso p f\u271d).hom \u25b7 h \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d })) \u25b7\n h) \u226b\n (normalizeIso (normalizeAux p g\u271d) h).hom =\n ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n (normalizeIso p f\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p f\u271d) h).hom) \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux (normalizeAux p f\u271d) h } = { as := normalizeAux (normalizeAux p g\u271d) h }))\n[PROOFSTEP]\ndsimp at this \n[GOAL]\ncase mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n \u2200 (p : Path a a\u271d),\n (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\np : Path a a\u271d\nthis :\n (normalizeIso (normalizeAux p f\u271d) h).hom =\n eqToHom\n (_ :\n (\u2191(preinclusion B)).map { as := normalizeAux p f\u271d } \u226b h =\n (\u2191(preinclusion B)).map { as := normalizeAux p g\u271d } \u226b h) \u226b\n (normalizeIso (normalizeAux p g\u271d) h).hom \u226b\n eqToHom\n (_ :\n (\u2191(preinclusion B)).map { as := normalizeAux (normalizeAux p g\u271d) h } =\n (\u2191(preinclusion B)).map { as := normalizeAux (normalizeAux p f\u271d) h })\n\u22a2 (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n ((normalizeIso p f\u271d).hom \u25b7 h \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d })) \u25b7\n h) \u226b\n (normalizeIso (normalizeAux p g\u271d) h).hom =\n ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n (normalizeIso p f\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p f\u271d) h).hom) \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux (normalizeAux p f\u271d) h } = { as := normalizeAux (normalizeAux p g\u271d) h }))\n[PROOFSTEP]\nsimp [this]\n[GOAL]\ncase mk.associator\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d d\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d : b\u271d \u27f6 c\u271d\nh\u271d : c\u271d \u27f6 d\u271d\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.associator f\u271d g\u271d h\u271d) \u226b (normalizeIso p (f\u271d \u226b g\u271d \u226b h\u271d)).hom =\n (normalizeIso p ((f\u271d \u226b g\u271d) \u226b h\u271d)).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux p ((f\u271d \u226b g\u271d) \u226b h\u271d) } = { as := normalizeAux p (f\u271d \u226b g\u271d \u226b h\u271d) }))\n[PROOFSTEP]\n\n| _ => simp\n[GOAL]\ncase mk.associator\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d d\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d : b\u271d \u27f6 c\u271d\nh\u271d : c\u271d \u27f6 d\u271d\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.associator f\u271d g\u271d h\u271d) \u226b (normalizeIso p (f\u271d \u226b g\u271d \u226b h\u271d)).hom =\n (normalizeIso p ((f\u271d \u226b g\u271d) \u226b h\u271d)).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux p ((f\u271d \u226b g\u271d) \u226b h\u271d) } = { as := normalizeAux p (f\u271d \u226b g\u271d \u226b h\u271d) }))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.associator_inv\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d d\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d : b\u271d \u27f6 c\u271d\nh\u271d : c\u271d \u27f6 d\u271d\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.associator_inv f\u271d g\u271d h\u271d) \u226b\n (normalizeIso p ((f\u271d \u226b g\u271d) \u226b h\u271d)).hom =\n (normalizeIso p (f\u271d \u226b g\u271d \u226b h\u271d)).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux p (f\u271d \u226b g\u271d \u226b h\u271d) } = { as := normalizeAux p ((f\u271d \u226b g\u271d) \u226b h\u271d) }))\n[PROOFSTEP]\n\n| _ => simp\n[GOAL]\ncase mk.associator_inv\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d d\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d : b\u271d \u27f6 c\u271d\nh\u271d : c\u271d \u27f6 d\u271d\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.associator_inv f\u271d g\u271d h\u271d) \u226b\n (normalizeIso p ((f\u271d \u226b g\u271d) \u226b h\u271d)).hom =\n (normalizeIso p (f\u271d \u226b g\u271d \u226b h\u271d)).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux p (f\u271d \u226b g\u271d \u226b h\u271d) } = { as := normalizeAux p ((f\u271d \u226b g\u271d) \u226b h\u271d) }))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.right_unitor\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.right_unitor f\u271d) \u226b (normalizeIso p f\u271d).hom =\n (normalizeIso p (f\u271d \u226b \ud835\udfd9 b\u271d)).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux p (f\u271d \u226b \ud835\udfd9 b\u271d) } = { as := normalizeAux p f\u271d }))\n[PROOFSTEP]\n\n| _ => simp\n[GOAL]\ncase mk.right_unitor\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.right_unitor f\u271d) \u226b (normalizeIso p f\u271d).hom =\n (normalizeIso p (f\u271d \u226b \ud835\udfd9 b\u271d)).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux p (f\u271d \u226b \ud835\udfd9 b\u271d) } = { as := normalizeAux p f\u271d }))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.right_unitor_inv\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.right_unitor_inv f\u271d) \u226b (normalizeIso p (f\u271d \u226b \ud835\udfd9 b\u271d)).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p (f\u271d \u226b \ud835\udfd9 b\u271d) }))\n[PROOFSTEP]\n\n| _ => simp\n[GOAL]\ncase mk.right_unitor_inv\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.right_unitor_inv f\u271d) \u226b (normalizeIso p (f\u271d \u226b \ud835\udfd9 b\u271d)).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p (f\u271d \u226b \ud835\udfd9 b\u271d) }))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.left_unitor\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.left_unitor f\u271d) \u226b (normalizeIso p f\u271d).hom =\n (normalizeIso p (\ud835\udfd9 a\u271d \u226b f\u271d)).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux p (\ud835\udfd9 a\u271d \u226b f\u271d) } = { as := normalizeAux p f\u271d }))\n[PROOFSTEP]\n\n| _ => simp\n[GOAL]\ncase mk.left_unitor\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.left_unitor f\u271d) \u226b (normalizeIso p f\u271d).hom =\n (normalizeIso p (\ud835\udfd9 a\u271d \u226b f\u271d)).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux p (\ud835\udfd9 a\u271d \u226b f\u271d) } = { as := normalizeAux p f\u271d }))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.left_unitor_inv\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.left_unitor_inv f\u271d) \u226b (normalizeIso p (\ud835\udfd9 a\u271d \u226b f\u271d)).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p (\ud835\udfd9 a\u271d \u226b f\u271d) }))\n[PROOFSTEP]\n\n| _ => simp\n[GOAL]\ncase mk.left_unitor_inv\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.left_unitor_inv f\u271d) \u226b (normalizeIso p (\ud835\udfd9 a\u271d \u226b f\u271d)).hom =\n (normalizeIso p f\u271d).hom \u226b\n PrelaxFunctor.map\u2082 (preinclusion B)\n (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p (\ud835\udfd9 a\u271d \u226b f\u271d) }))\n[PROOFSTEP]\nsimp\n[GOAL]\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf : Hom a b\ng : Hom b c\n\u22a2 normalizeAux nil (Hom.comp f g) = comp (normalizeAux nil f) (normalizeAux nil g)\n[PROOFSTEP]\ninduction g generalizing a with\n| id => rfl\n| of => rfl\n| comp g _ ihf ihg => erw [ihg (f.comp g), ihf f, ihg g, comp_assoc]\n[GOAL]\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf : Hom a b\ng : Hom b c\n\u22a2 normalizeAux nil (Hom.comp f g) = comp (normalizeAux nil f) (normalizeAux nil g)\n[PROOFSTEP]\ninduction g generalizing a with\n| id => rfl\n| of => rfl\n| comp g _ ihf ihg => erw [ihg (f.comp g), ihf f, ihg g, comp_assoc]\n[GOAL]\ncase id\nB : Type u\ninst\u271d : Quiver B\nb c a\u271d a : B\nf : Hom a a\u271d\n\u22a2 normalizeAux nil (Hom.comp f (Hom.id a\u271d)) = comp (normalizeAux nil f) (normalizeAux nil (Hom.id a\u271d))\n[PROOFSTEP]\n\n| id => rfl\n[GOAL]\ncase id\nB : Type u\ninst\u271d : Quiver B\nb c a\u271d a : B\nf : Hom a a\u271d\n\u22a2 normalizeAux nil (Hom.comp f (Hom.id a\u271d)) = comp (normalizeAux nil f) (normalizeAux nil (Hom.id a\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase of\nB : Type u\ninst\u271d : Quiver B\nb c a\u271d b\u271d : B\nf\u271d : a\u271d \u27f6 b\u271d\na : B\nf : Hom a a\u271d\n\u22a2 normalizeAux nil (Hom.comp f (Hom.of f\u271d)) = comp (normalizeAux nil f) (normalizeAux nil (Hom.of f\u271d))\n[PROOFSTEP]\n\n| of => rfl\n[GOAL]\ncase of\nB : Type u\ninst\u271d : Quiver B\nb c a\u271d b\u271d : B\nf\u271d : a\u271d \u27f6 b\u271d\na : B\nf : Hom a a\u271d\n\u22a2 normalizeAux nil (Hom.comp f (Hom.of f\u271d)) = comp (normalizeAux nil f) (normalizeAux nil (Hom.of f\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase comp\nB : Type u\ninst\u271d : Quiver B\nb c a\u271d b\u271d c\u271d : B\ng : Hom a\u271d b\u271d\ng\u271d : Hom b\u271d c\u271d\nihf : \u2200 {a : B} (f : Hom a a\u271d), normalizeAux nil (Hom.comp f g) = comp (normalizeAux nil f) (normalizeAux nil g)\nihg : \u2200 {a : B} (f : Hom a b\u271d), normalizeAux nil (Hom.comp f g\u271d) = comp (normalizeAux nil f) (normalizeAux nil g\u271d)\na : B\nf : Hom a a\u271d\n\u22a2 normalizeAux nil (Hom.comp f (Hom.comp g g\u271d)) = comp (normalizeAux nil f) (normalizeAux nil (Hom.comp g g\u271d))\n[PROOFSTEP]\n\n| comp g _ ihf ihg => erw [ihg (f.comp g), ihf f, ihg g, comp_assoc]\n[GOAL]\ncase comp\nB : Type u\ninst\u271d : Quiver B\nb c a\u271d b\u271d c\u271d : B\ng : Hom a\u271d b\u271d\ng\u271d : Hom b\u271d c\u271d\nihf : \u2200 {a : B} (f : Hom a a\u271d), normalizeAux nil (Hom.comp f g) = comp (normalizeAux nil f) (normalizeAux nil g)\nihg : \u2200 {a : B} (f : Hom a b\u271d), normalizeAux nil (Hom.comp f g\u271d) = comp (normalizeAux nil f) (normalizeAux nil g\u271d)\na : B\nf : Hom a a\u271d\n\u22a2 normalizeAux nil (Hom.comp f (Hom.comp g g\u271d)) = comp (normalizeAux nil f) (normalizeAux nil (Hom.comp g g\u271d))\n[PROOFSTEP]\nerw [ihg (f.comp g), ihf f, ihg g, comp_assoc]\n[GOAL]\nB : Type u\ninst\u271d : Quiver B\na b : FreeBicategory B\n\u22a2 \u2200 {X Y : a \u27f6 b} (f : X \u27f6 Y),\n (\ud835\udfed (a \u27f6 b)).map f \u226b ((fun f => (\u03bb_ f).symm \u226a\u226b normalizeIso nil f) Y).hom =\n ((fun f => (\u03bb_ f).symm \u226a\u226b normalizeIso nil f) X).hom \u226b\n (Pseudofunctor.mapFunctor (normalize B) a b \u22d9 inclusionPath a b).map f\n[PROOFSTEP]\nintro f g \u03b7\n[GOAL]\nB : Type u\ninst\u271d : Quiver B\na b : FreeBicategory B\nf g : a \u27f6 b\n\u03b7 : f \u27f6 g\n\u22a2 (\ud835\udfed (a \u27f6 b)).map \u03b7 \u226b ((fun f => (\u03bb_ f).symm \u226a\u226b normalizeIso nil f) g).hom =\n ((fun f => (\u03bb_ f).symm \u226a\u226b normalizeIso nil f) f).hom \u226b\n (Pseudofunctor.mapFunctor (normalize B) a b \u22d9 inclusionPath a b).map \u03b7\n[PROOFSTEP]\nerw [leftUnitor_inv_naturality_assoc, assoc]\n[GOAL]\nB : Type u\ninst\u271d : Quiver B\na b : FreeBicategory B\nf g : a \u27f6 b\n\u03b7 : f \u27f6 g\n\u22a2 (\u03bb_ ((\ud835\udfed (a \u27f6 b)).obj f)).inv \u226b \ud835\udfd9 a \u25c1 (\ud835\udfed (a \u27f6 b)).map \u03b7 \u226b (normalizeIso nil g).hom =\n (\u03bb_ f).symm.hom \u226b (normalizeIso nil f).hom \u226b (Pseudofunctor.mapFunctor (normalize B) a b \u22d9 inclusionPath a b).map \u03b7\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nB : Type u\ninst\u271d : Quiver B\na b : FreeBicategory B\nf g : a \u27f6 b\n\u03b7 : f \u27f6 g\n\u22a2 \ud835\udfd9 a \u25c1 (\ud835\udfed (a \u27f6 b)).map \u03b7 \u226b (normalizeIso nil g).hom =\n (normalizeIso nil f).hom \u226b (Pseudofunctor.mapFunctor (normalize B) a b \u22d9 inclusionPath a b).map \u03b7\n[PROOFSTEP]\nexact normalize_naturality nil \u03b7\n[GOAL]\nB : Type u\ninst\u271d : Quiver B\na b : B\nf : Discrete (Path a b)\n\u22a2 (inclusionPath a b \u22d9 Pseudofunctor.mapFunctor (normalize B) a b).obj f = (\ud835\udfed (Discrete (Path a b))).obj f\n[PROOFSTEP]\ninduction' f with f\n[GOAL]\ncase mk\nB : Type u\ninst\u271d : Quiver B\na b : B\nf : Path a b\n\u22a2 (inclusionPath a b \u22d9 Pseudofunctor.mapFunctor (normalize B) a b).obj { as := f } =\n (\ud835\udfed (Discrete (Path a b))).obj { as := f }\n[PROOFSTEP]\ninduction' f with _ _ _ _ ih\n[GOAL]\ncase mk.nil\nB : Type u\ninst\u271d : Quiver B\na b : B\n\u22a2 (inclusionPath a a \u22d9 Pseudofunctor.mapFunctor (normalize B) a a).obj { as := nil } =\n (\ud835\udfed (Discrete (Path a a))).obj { as := nil }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.cons\nB : Type u\ninst\u271d : Quiver B\na b b\u271d c\u271d : B\na\u271d\u00b9 : Path a b\u271d\na\u271d : b\u271d \u27f6 c\u271d\nih :\n (inclusionPath a b\u271d \u22d9 Pseudofunctor.mapFunctor (normalize B) a b\u271d).obj { as := a\u271d\u00b9 } =\n (\ud835\udfed (Discrete (Path a b\u271d))).obj { as := a\u271d\u00b9 }\n\u22a2 (inclusionPath a c\u271d \u22d9 Pseudofunctor.mapFunctor (normalize B) a c\u271d).obj { as := cons a\u271d\u00b9 a\u271d } =\n (\ud835\udfed (Discrete (Path a c\u271d))).obj { as := cons a\u271d\u00b9 a\u271d }\n[PROOFSTEP]\next1\n[GOAL]\ncase mk.cons.as\nB : Type u\ninst\u271d : Quiver B\na b b\u271d c\u271d : B\na\u271d\u00b9 : Path a b\u271d\na\u271d : b\u271d \u27f6 c\u271d\nih :\n (inclusionPath a b\u271d \u22d9 Pseudofunctor.mapFunctor (normalize B) a b\u271d).obj { as := a\u271d\u00b9 } =\n (\ud835\udfed (Discrete (Path a b\u271d))).obj { as := a\u271d\u00b9 }\n\u22a2 ((inclusionPath a c\u271d \u22d9 Pseudofunctor.mapFunctor (normalize B) a c\u271d).obj { as := cons a\u271d\u00b9 a\u271d }).as =\n ((\ud835\udfed (Discrete (Path a c\u271d))).obj { as := cons a\u271d\u00b9 a\u271d }).as\n[PROOFSTEP]\ninjection ih with ih\n[GOAL]\ncase mk.cons.as\nB : Type u\ninst\u271d : Quiver B\na b b\u271d c\u271d : B\na\u271d\u00b9 : Path a b\u271d\na\u271d : b\u271d \u27f6 c\u271d\nih : normalizeAux nil ((inclusionPath a b\u271d).obj { as := a\u271d\u00b9 }) = a\u271d\u00b9\n\u22a2 ((inclusionPath a c\u271d \u22d9 Pseudofunctor.mapFunctor (normalize B) a c\u271d).obj { as := cons a\u271d\u00b9 a\u271d }).as =\n ((\ud835\udfed (Discrete (Path a c\u271d))).obj { as := cons a\u271d\u00b9 a\u271d }).as\n[PROOFSTEP]\nconv =>\n rhs\n rw [\u2190 ih]\n[GOAL]\nB : Type u\ninst\u271d : Quiver B\na b b\u271d c\u271d : B\na\u271d\u00b9 : Path a b\u271d\na\u271d : b\u271d \u27f6 c\u271d\nih : normalizeAux nil ((inclusionPath a b\u271d).obj { as := a\u271d\u00b9 }) = a\u271d\u00b9\n| ((inclusionPath a c\u271d \u22d9 Pseudofunctor.mapFunctor (normalize B) a c\u271d).obj { as := cons a\u271d\u00b9 a\u271d }).as =\n ((\ud835\udfed (Discrete (Path a c\u271d))).obj { as := cons a\u271d\u00b9 a\u271d }).as\n[PROOFSTEP]\n rhs\n rw [\u2190 ih]\n[GOAL]\nB : Type u\ninst\u271d : Quiver B\na b b\u271d c\u271d : B\na\u271d\u00b9 : Path a b\u271d\na\u271d : b\u271d \u27f6 c\u271d\nih : normalizeAux nil ((inclusionPath a b\u271d).obj { as := a\u271d\u00b9 }) = a\u271d\u00b9\n| ((inclusionPath a c\u271d \u22d9 Pseudofunctor.mapFunctor (normalize B) a c\u271d).obj { as := cons a\u271d\u00b9 a\u271d }).as =\n ((\ud835\udfed (Discrete (Path a c\u271d))).obj { as := cons a\u271d\u00b9 a\u271d }).as\n[PROOFSTEP]\n rhs\n rw [\u2190 ih]\n[GOAL]\nB : Type u\ninst\u271d : Quiver B\na b b\u271d c\u271d : B\na\u271d\u00b9 : Path a b\u271d\na\u271d : b\u271d \u27f6 c\u271d\nih : normalizeAux nil ((inclusionPath a b\u271d).obj { as := a\u271d\u00b9 }) = a\u271d\u00b9\n| ((inclusionPath a c\u271d \u22d9 Pseudofunctor.mapFunctor (normalize B) a c\u271d).obj { as := cons a\u271d\u00b9 a\u271d }).as =\n ((\ud835\udfed (Discrete (Path a c\u271d))).obj { as := cons a\u271d\u00b9 a\u271d }).as\n[PROOFSTEP]\nrhs\n[GOAL]\nB : Type u\ninst\u271d : Quiver B\na b b\u271d c\u271d : B\na\u271d\u00b9 : Path a b\u271d\na\u271d : b\u271d \u27f6 c\u271d\nih : normalizeAux nil ((inclusionPath a b\u271d).obj { as := a\u271d\u00b9 }) = a\u271d\u00b9\n| ((\ud835\udfed (Discrete (Path a c\u271d))).obj { as := cons a\u271d\u00b9 a\u271d }).as\n[PROOFSTEP]\nrw [\u2190 ih]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Bicategory.Coherence", "llama_tokens": 27435, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.4921881357207956, "lm_q2_score": 0.018264278819436312, "lm_q1q2_score": 0.008989461342423172}} {"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.105, u_1} D\n\u22a2 CompleteLattice (MorphismProperty C)\n[PROOFSTEP]\ndsimp only [MorphismProperty]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.105, u_1} D\n\u22a2 CompleteLattice (\u2983X Y : C\u2984 \u2192 (X \u27f6 Y) \u2192 Prop)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.6347, u_1} D\nX Y : C\nf : X \u27f6 Y\n\u22a2 \u22a4 f\n[PROOFSTEP]\nsimp only [top_eq]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.10321, u_1} D\nP : MorphismProperty C\nX\u271d Y\u271d Z\u271d : C\ne : X\u271d \u2245 Y\u271d\nf : Y\u271d \u27f6 Z\u271d\nx\u271d : isoClosure P f\nw\u271d\u00b9 w\u271d : C\nf' : w\u271d\u00b9 \u27f6 w\u271d\nhf' : P f'\niso : Arrow.mk f' \u2245 Arrow.mk f\n\u22a2 (asIso iso.hom.left \u226a\u226b e.symm).hom \u226b (Arrow.mk (e.hom \u226b f)).hom = (Arrow.mk f').hom \u226b (asIso iso.hom.right).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.10321, u_1} D\nP : MorphismProperty C\nX\u271d Y\u271d Z\u271d : C\ne : Y\u271d \u2245 Z\u271d\nf : X\u271d \u27f6 Y\u271d\nx\u271d : isoClosure P f\nw\u271d\u00b9 w\u271d : C\nf' : w\u271d\u00b9 \u27f6 w\u271d\nhf' : P f'\niso : Arrow.mk f' \u2245 Arrow.mk f\n\u22a2 (asIso iso.hom.left).hom \u226b (Arrow.mk (f \u226b e.hom)).hom = (Arrow.mk f').hom \u226b (asIso iso.hom.right \u226a\u226b e).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.20276, u_1} D\nP : MorphismProperty C\nhP : RespectsIso P\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d : IsIso f\nh : P (f \u226b g)\n\u22a2 P g\n[PROOFSTEP]\nsimpa using hP.1 (asIso f).symm (f \u226b g) h\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.21442, u_1} D\nP : MorphismProperty C\nhP : RespectsIso P\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d : IsIso g\nh : P (f \u226b g)\n\u22a2 P f\n[PROOFSTEP]\nsimpa using hP.2 (asIso g).symm (f \u226b g) h\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.22704, u_1} D\nP : MorphismProperty C\nhP : RespectsIso P\nf g : Arrow C\ne : f \u2245 g\n\u22a2 P f.hom \u2194 P g.hom\n[PROOFSTEP]\nrw [\u2190 Arrow.inv_left_hom_right e.hom, hP.cancel_left_isIso, hP.cancel_right_isIso]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.23989, u_1} D\nP : MorphismProperty C\nhP : \u2200 (f g : Arrow C), (f \u2245 g) \u2192 P f.hom \u2192 P g.hom\n\u22a2 RespectsIso P\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.23989, u_1} D\nP : MorphismProperty C\nhP : \u2200 (f g : Arrow C), (f \u2245 g) \u2192 P f.hom \u2192 P g.hom\n\u22a2 \u2200 {X Y Z : C} (e : X \u2245 Y) (f : Y \u27f6 Z), P f \u2192 P (e.hom \u226b f)\n[PROOFSTEP]\nintro X Y Z e f hf\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.23989, u_1} D\nP : MorphismProperty C\nhP : \u2200 (f g : Arrow C), (f \u2245 g) \u2192 P f.hom \u2192 P g.hom\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\nhf : P f\n\u22a2 P (e.hom \u226b f)\n[PROOFSTEP]\nrefine' hP (Arrow.mk f) (Arrow.mk (e.hom \u226b f)) (Arrow.isoMk e.symm (Iso.refl _) _) hf\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.23989, u_1} D\nP : MorphismProperty C\nhP : \u2200 (f g : Arrow C), (f \u2245 g) \u2192 P f.hom \u2192 P g.hom\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\nhf : P f\n\u22a2 e.symm.hom \u226b (Arrow.mk (e.hom \u226b f)).hom = (Arrow.mk f).hom \u226b (Iso.refl (Arrow.mk f).right).hom\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.23989, u_1} D\nP : MorphismProperty C\nhP : \u2200 (f g : Arrow C), (f \u2245 g) \u2192 P f.hom \u2192 P g.hom\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\nhf : P f\n\u22a2 e.inv \u226b e.hom \u226b f = f \u226b \ud835\udfd9 Z\n[PROOFSTEP]\nsimp only [Iso.inv_hom_id_assoc, Category.comp_id]\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.23989, u_1} D\nP : MorphismProperty C\nhP : \u2200 (f g : Arrow C), (f \u2245 g) \u2192 P f.hom \u2192 P g.hom\n\u22a2 \u2200 {X Y Z : C} (e : Y \u2245 Z) (f : X \u27f6 Y), P f \u2192 P (f \u226b e.hom)\n[PROOFSTEP]\nintro X Y Z e f hf\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.23989, u_1} D\nP : MorphismProperty C\nhP : \u2200 (f g : Arrow C), (f \u2245 g) \u2192 P f.hom \u2192 P g.hom\nX Y Z : C\ne : Y \u2245 Z\nf : X \u27f6 Y\nhf : P f\n\u22a2 P (f \u226b e.hom)\n[PROOFSTEP]\nrefine' hP (Arrow.mk f) (Arrow.mk (f \u226b e.hom)) (Arrow.isoMk (Iso.refl _) e _) hf\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.23989, u_1} D\nP : MorphismProperty C\nhP : \u2200 (f g : Arrow C), (f \u2245 g) \u2192 P f.hom \u2192 P g.hom\nX Y Z : C\ne : Y \u2245 Z\nf : X \u27f6 Y\nhf : P f\n\u22a2 (Iso.refl (Arrow.mk f).left).hom \u226b (Arrow.mk (f \u226b e.hom)).hom = (Arrow.mk f).hom \u226b e.hom\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.23989, u_1} D\nP : MorphismProperty C\nhP : \u2200 (f g : Arrow C), (f \u2245 g) \u2192 P f.hom \u2192 P g.hom\nX Y Z : C\ne : Y \u2245 Z\nf : X \u27f6 Y\nhf : P f\n\u22a2 \ud835\udfd9 X \u226b f \u226b e.hom = f \u226b e.hom\n[PROOFSTEP]\nsimp only [Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.26090, u_1} D\nP : MorphismProperty C\ninst\u271d : HasPullbacks C\nhP\u2081 : RespectsIso P\nhP\u2082 : \u2200 (X Y S : C) (f : X \u27f6 S) (g : Y \u27f6 S), P g \u2192 P pullback.fst\nX Y Y' S : C\nf : X \u27f6 S\ng : Y \u27f6 S\nf' : Y' \u27f6 Y\ng' : Y' \u27f6 X\nsq : IsPullback f' g' g f\nhg : P g\n\u22a2 P g'\n[PROOFSTEP]\nlet e := sq.flip.isoPullback\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.26090, u_1} D\nP : MorphismProperty C\ninst\u271d : HasPullbacks C\nhP\u2081 : RespectsIso P\nhP\u2082 : \u2200 (X Y S : C) (f : X \u27f6 S) (g : Y \u27f6 S), P g \u2192 P pullback.fst\nX Y Y' S : C\nf : X \u27f6 S\ng : Y \u27f6 S\nf' : Y' \u27f6 Y\ng' : Y' \u27f6 X\nsq : IsPullback f' g' g f\nhg : P g\ne : Y' \u2245 pullback f g := IsPullback.isoPullback (_ : IsPullback g' f' f g)\n\u22a2 P g'\n[PROOFSTEP]\nrw [\u2190 hP\u2081.cancel_left_isIso e.inv, sq.flip.isoPullback_inv_fst]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.26090, u_1} D\nP : MorphismProperty C\ninst\u271d : HasPullbacks C\nhP\u2081 : RespectsIso P\nhP\u2082 : \u2200 (X Y S : C) (f : X \u27f6 S) (g : Y \u27f6 S), P g \u2192 P pullback.fst\nX Y Y' S : C\nf : X \u27f6 S\ng : Y \u27f6 S\nf' : Y' \u27f6 Y\ng' : Y' \u27f6 X\nsq : IsPullback f' g' g f\nhg : P g\ne : Y' \u2245 pullback f g := IsPullback.isoPullback (_ : IsPullback g' f' f g)\n\u22a2 P pullback.fst\n[PROOFSTEP]\nexact hP\u2082 _ _ _ f g hg\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.27180, u_1} D\nP : MorphismProperty C\nhP : StableUnderBaseChange P\n\u22a2 RespectsIso P\n[PROOFSTEP]\napply RespectsIso.of_respects_arrow_iso\n[GOAL]\ncase hP\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.27180, u_1} D\nP : MorphismProperty C\nhP : StableUnderBaseChange P\n\u22a2 \u2200 (f g : Arrow C), (f \u2245 g) \u2192 P f.hom \u2192 P g.hom\n[PROOFSTEP]\nintro f g e\n[GOAL]\ncase hP\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.27180, u_1} D\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nf g : Arrow C\ne : f \u2245 g\n\u22a2 P f.hom \u2192 P g.hom\n[PROOFSTEP]\nexact hP (IsPullback.of_horiz_isIso (CommSq.mk e.inv.w))\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.30641, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nS S' : C\nf : S' \u27f6 S\nX Y : Over S\ng : X \u27f6 Y\nH : P g.left\n\u22a2 P ((baseChange f).map g).left\n[PROOFSTEP]\nlet e := pullbackRightPullbackFstIso Y.hom f g.left \u226a\u226b pullback.congrHom (g.w.trans (Category.comp_id _)) rfl\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.30641, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nS S' : C\nf : S' \u27f6 S\nX Y : Over S\ng : X \u27f6 Y\nH : P g.left\ne : pullback g.left pullback.fst \u2245 pullback X.hom f :=\n pullbackRightPullbackFstIso Y.hom f g.left \u226a\u226b pullback.congrHom (_ : (\ud835\udfed C).map g.left \u226b Y.hom = X.hom) (_ : f = f)\n\u22a2 P ((baseChange f).map g).left\n[PROOFSTEP]\nhave : e.inv \u226b pullback.snd = ((baseChange f).map g).left := by ext <;> dsimp <;> simp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.30641, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nS S' : C\nf : S' \u27f6 S\nX Y : Over S\ng : X \u27f6 Y\nH : P g.left\ne : pullback g.left pullback.fst \u2245 pullback X.hom f :=\n pullbackRightPullbackFstIso Y.hom f g.left \u226a\u226b pullback.congrHom (_ : (\ud835\udfed C).map g.left \u226b Y.hom = X.hom) (_ : f = f)\n\u22a2 e.inv \u226b pullback.snd = ((baseChange f).map g).left\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.30641, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nS S' : C\nf : S' \u27f6 S\nX Y : Over S\ng : X \u27f6 Y\nH : P g.left\ne : pullback g.left pullback.fst \u2245 pullback X.hom f :=\n pullbackRightPullbackFstIso Y.hom f g.left \u226a\u226b pullback.congrHom (_ : (\ud835\udfed C).map g.left \u226b Y.hom = X.hom) (_ : f = f)\n\u22a2 (e.inv \u226b pullback.snd) \u226b pullback.fst = ((baseChange f).map g).left \u226b pullback.fst\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.30641, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nS S' : C\nf : S' \u27f6 S\nX Y : Over S\ng : X \u27f6 Y\nH : P g.left\ne : pullback g.left pullback.fst \u2245 pullback X.hom f :=\n pullbackRightPullbackFstIso Y.hom f g.left \u226a\u226b pullback.congrHom (_ : (\ud835\udfed C).map g.left \u226b Y.hom = X.hom) (_ : f = f)\n\u22a2 (e.inv \u226b pullback.snd) \u226b pullback.snd = ((baseChange f).map g).left \u226b pullback.snd\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.30641, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nS S' : C\nf : S' \u27f6 S\nX Y : Over S\ng : X \u27f6 Y\nH : P g.left\ne : pullback g.left pullback.fst \u2245 pullback X.hom f :=\n pullbackRightPullbackFstIso Y.hom f g.left \u226a\u226b pullback.congrHom (_ : (\ud835\udfed C).map g.left \u226b Y.hom = X.hom) (_ : f = f)\n\u22a2 (((pullback.congrHom (_ : g.left \u226b Y.hom = X.hom) (_ : f = f)).inv \u226b\n (pullbackRightPullbackFstIso Y.hom f g.left).inv) \u226b\n pullback.snd) \u226b\n pullback.fst =\n pullback.map X.hom f Y.hom f g.left (\ud835\udfd9 S') (\ud835\udfd9 S) (_ : X.hom \u226b \ud835\udfd9 S = g.left \u226b Y.hom) (_ : f \u226b \ud835\udfd9 S = \ud835\udfd9 S' \u226b f) \u226b\n pullback.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.30641, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nS S' : C\nf : S' \u27f6 S\nX Y : Over S\ng : X \u27f6 Y\nH : P g.left\ne : pullback g.left pullback.fst \u2245 pullback X.hom f :=\n pullbackRightPullbackFstIso Y.hom f g.left \u226a\u226b pullback.congrHom (_ : (\ud835\udfed C).map g.left \u226b Y.hom = X.hom) (_ : f = f)\n\u22a2 (((pullback.congrHom (_ : g.left \u226b Y.hom = X.hom) (_ : f = f)).inv \u226b\n (pullbackRightPullbackFstIso Y.hom f g.left).inv) \u226b\n pullback.snd) \u226b\n pullback.snd =\n pullback.map X.hom f Y.hom f g.left (\ud835\udfd9 S') (\ud835\udfd9 S) (_ : X.hom \u226b \ud835\udfd9 S = g.left \u226b Y.hom) (_ : f \u226b \ud835\udfd9 S = \ud835\udfd9 S' \u226b f) \u226b\n pullback.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.30641, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nS S' : C\nf : S' \u27f6 S\nX Y : Over S\ng : X \u27f6 Y\nH : P g.left\ne : pullback g.left pullback.fst \u2245 pullback X.hom f :=\n pullbackRightPullbackFstIso Y.hom f g.left \u226a\u226b pullback.congrHom (_ : (\ud835\udfed C).map g.left \u226b Y.hom = X.hom) (_ : f = f)\nthis : e.inv \u226b pullback.snd = ((baseChange f).map g).left\n\u22a2 P ((baseChange f).map g).left\n[PROOFSTEP]\nrw [\u2190 this, hP.respectsIso.cancel_left_isIso]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.30641, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nS S' : C\nf : S' \u27f6 S\nX Y : Over S\ng : X \u27f6 Y\nH : P g.left\ne : pullback g.left pullback.fst \u2245 pullback X.hom f :=\n pullbackRightPullbackFstIso Y.hom f g.left \u226a\u226b pullback.congrHom (_ : (\ud835\udfed C).map g.left \u226b Y.hom = X.hom) (_ : f = f)\nthis : e.inv \u226b pullback.snd = ((baseChange f).map g).left\n\u22a2 P pullback.snd\n[PROOFSTEP]\nexact hP.snd _ _ H\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.39138, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : StableUnderComposition P\nS X X' Y Y' : C\nf : X \u27f6 S\ng : Y \u27f6 S\nf' : X' \u27f6 S\ng' : Y' \u27f6 S\ni\u2081 : X \u27f6 X'\ni\u2082 : Y \u27f6 Y'\nh\u2081 : P i\u2081\nh\u2082 : P i\u2082\ne\u2081 : f = i\u2081 \u226b f'\ne\u2082 : g = i\u2082 \u226b g'\n\u22a2 P (pullback.map f g f' g' i\u2081 i\u2082 (\ud835\udfd9 S) (_ : f \u226b \ud835\udfd9 S = i\u2081 \u226b f') (_ : g \u226b \ud835\udfd9 S = i\u2082 \u226b g'))\n[PROOFSTEP]\nhave :\n pullback.map f g f' g' i\u2081 i\u2082 (\ud835\udfd9 _) ((Category.comp_id _).trans e\u2081) ((Category.comp_id _).trans e\u2082) =\n ((pullbackSymmetry _ _).hom \u226b ((baseChange _).map (Over.homMk _ e\u2082.symm : Over.mk g \u27f6 Over.mk g')).left) \u226b\n (pullbackSymmetry _ _).hom \u226b ((baseChange g').map (Over.homMk _ e\u2081.symm : Over.mk f \u27f6 Over.mk f')).left :=\n by ext <;> dsimp <;> simp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.39138, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : StableUnderComposition P\nS X X' Y Y' : C\nf : X \u27f6 S\ng : Y \u27f6 S\nf' : X' \u27f6 S\ng' : Y' \u27f6 S\ni\u2081 : X \u27f6 X'\ni\u2082 : Y \u27f6 Y'\nh\u2081 : P i\u2081\nh\u2082 : P i\u2082\ne\u2081 : f = i\u2081 \u226b f'\ne\u2082 : g = i\u2082 \u226b g'\n\u22a2 pullback.map f g f' g' i\u2081 i\u2082 (\ud835\udfd9 S) (_ : f \u226b \ud835\udfd9 S = i\u2081 \u226b f') (_ : g \u226b \ud835\udfd9 S = i\u2082 \u226b g') =\n ((pullbackSymmetry (Over.mk f).hom (Over.mk g).hom).hom \u226b ((baseChange (Over.mk f).hom).map (Over.homMk i\u2082)).left) \u226b\n (pullbackSymmetry (Over.mk g').hom (Over.mk f).hom).hom \u226b ((baseChange g').map (Over.homMk i\u2081)).left\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.39138, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : StableUnderComposition P\nS X X' Y Y' : C\nf : X \u27f6 S\ng : Y \u27f6 S\nf' : X' \u27f6 S\ng' : Y' \u27f6 S\ni\u2081 : X \u27f6 X'\ni\u2082 : Y \u27f6 Y'\nh\u2081 : P i\u2081\nh\u2082 : P i\u2082\ne\u2081 : f = i\u2081 \u226b f'\ne\u2082 : g = i\u2082 \u226b g'\n\u22a2 pullback.map f g f' g' i\u2081 i\u2082 (\ud835\udfd9 S) (_ : f \u226b \ud835\udfd9 S = i\u2081 \u226b f') (_ : g \u226b \ud835\udfd9 S = i\u2082 \u226b g') \u226b pullback.fst =\n (((pullbackSymmetry (Over.mk f).hom (Over.mk g).hom).hom \u226b\n ((baseChange (Over.mk f).hom).map (Over.homMk i\u2082)).left) \u226b\n (pullbackSymmetry (Over.mk g').hom (Over.mk f).hom).hom \u226b ((baseChange g').map (Over.homMk i\u2081)).left) \u226b\n pullback.fst\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.39138, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : StableUnderComposition P\nS X X' Y Y' : C\nf : X \u27f6 S\ng : Y \u27f6 S\nf' : X' \u27f6 S\ng' : Y' \u27f6 S\ni\u2081 : X \u27f6 X'\ni\u2082 : Y \u27f6 Y'\nh\u2081 : P i\u2081\nh\u2082 : P i\u2082\ne\u2081 : f = i\u2081 \u226b f'\ne\u2082 : g = i\u2082 \u226b g'\n\u22a2 pullback.map f g f' g' i\u2081 i\u2082 (\ud835\udfd9 S) (_ : f \u226b \ud835\udfd9 S = i\u2081 \u226b f') (_ : g \u226b \ud835\udfd9 S = i\u2082 \u226b g') \u226b pullback.snd =\n (((pullbackSymmetry (Over.mk f).hom (Over.mk g).hom).hom \u226b\n ((baseChange (Over.mk f).hom).map (Over.homMk i\u2082)).left) \u226b\n (pullbackSymmetry (Over.mk g').hom (Over.mk f).hom).hom \u226b ((baseChange g').map (Over.homMk i\u2081)).left) \u226b\n pullback.snd\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2080\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.39138, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : StableUnderComposition P\nS X X' Y Y' : C\nf : X \u27f6 S\ng : Y \u27f6 S\nf' : X' \u27f6 S\ng' : Y' \u27f6 S\ni\u2081 : X \u27f6 X'\ni\u2082 : Y \u27f6 Y'\nh\u2081 : P i\u2081\nh\u2082 : P i\u2082\ne\u2081 : f = i\u2081 \u226b f'\ne\u2082 : g = i\u2082 \u226b g'\n\u22a2 pullback.map f g f' g' i\u2081 i\u2082 (\ud835\udfd9 S) (_ : f \u226b \ud835\udfd9 S = i\u2081 \u226b f') (_ : g \u226b \ud835\udfd9 S = i\u2082 \u226b g') \u226b pullback.fst =\n (((pullbackSymmetry f g).hom \u226b\n pullback.map g f g' f i\u2082 (\ud835\udfd9 X) (\ud835\udfd9 S) (_ : (Over.mk g).hom \u226b \ud835\udfd9 S = (Over.homMk i\u2082).left \u226b (Over.mk g').hom)\n (_ : f \u226b \ud835\udfd9 S = \ud835\udfd9 X \u226b f)) \u226b\n (pullbackSymmetry g' f).hom \u226b\n pullback.map f g' f' g' i\u2081 (\ud835\udfd9 Y') (\ud835\udfd9 S) (_ : (Over.mk f).hom \u226b \ud835\udfd9 S = (Over.homMk i\u2081).left \u226b (Over.mk f').hom)\n (_ : g' \u226b \ud835\udfd9 S = \ud835\udfd9 Y' \u226b g')) \u226b\n pullback.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2081\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.39138, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : StableUnderComposition P\nS X X' Y Y' : C\nf : X \u27f6 S\ng : Y \u27f6 S\nf' : X' \u27f6 S\ng' : Y' \u27f6 S\ni\u2081 : X \u27f6 X'\ni\u2082 : Y \u27f6 Y'\nh\u2081 : P i\u2081\nh\u2082 : P i\u2082\ne\u2081 : f = i\u2081 \u226b f'\ne\u2082 : g = i\u2082 \u226b g'\n\u22a2 pullback.map f g f' g' i\u2081 i\u2082 (\ud835\udfd9 S) (_ : f \u226b \ud835\udfd9 S = i\u2081 \u226b f') (_ : g \u226b \ud835\udfd9 S = i\u2082 \u226b g') \u226b pullback.snd =\n (((pullbackSymmetry f g).hom \u226b\n pullback.map g f g' f i\u2082 (\ud835\udfd9 X) (\ud835\udfd9 S) (_ : (Over.mk g).hom \u226b \ud835\udfd9 S = (Over.homMk i\u2082).left \u226b (Over.mk g').hom)\n (_ : f \u226b \ud835\udfd9 S = \ud835\udfd9 X \u226b f)) \u226b\n (pullbackSymmetry g' f).hom \u226b\n pullback.map f g' f' g' i\u2081 (\ud835\udfd9 Y') (\ud835\udfd9 S) (_ : (Over.mk f).hom \u226b \ud835\udfd9 S = (Over.homMk i\u2081).left \u226b (Over.mk f').hom)\n (_ : g' \u226b \ud835\udfd9 S = \ud835\udfd9 Y' \u226b g')) \u226b\n pullback.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.39138, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : StableUnderComposition P\nS X X' Y Y' : C\nf : X \u27f6 S\ng : Y \u27f6 S\nf' : X' \u27f6 S\ng' : Y' \u27f6 S\ni\u2081 : X \u27f6 X'\ni\u2082 : Y \u27f6 Y'\nh\u2081 : P i\u2081\nh\u2082 : P i\u2082\ne\u2081 : f = i\u2081 \u226b f'\ne\u2082 : g = i\u2082 \u226b g'\nthis :\n pullback.map f g f' g' i\u2081 i\u2082 (\ud835\udfd9 S) (_ : f \u226b \ud835\udfd9 S = i\u2081 \u226b f') (_ : g \u226b \ud835\udfd9 S = i\u2082 \u226b g') =\n ((pullbackSymmetry (Over.mk f).hom (Over.mk g).hom).hom \u226b ((baseChange (Over.mk f).hom).map (Over.homMk i\u2082)).left) \u226b\n (pullbackSymmetry (Over.mk g').hom (Over.mk f).hom).hom \u226b ((baseChange g').map (Over.homMk i\u2081)).left\n\u22a2 P (pullback.map f g f' g' i\u2081 i\u2082 (\ud835\udfd9 S) (_ : f \u226b \ud835\udfd9 S = i\u2081 \u226b f') (_ : g \u226b \ud835\udfd9 S = i\u2082 \u226b g'))\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.39138, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : StableUnderComposition P\nS X X' Y Y' : C\nf : X \u27f6 S\ng : Y \u27f6 S\nf' : X' \u27f6 S\ng' : Y' \u27f6 S\ni\u2081 : X \u27f6 X'\ni\u2082 : Y \u27f6 Y'\nh\u2081 : P i\u2081\nh\u2082 : P i\u2082\ne\u2081 : f = i\u2081 \u226b f'\ne\u2082 : g = i\u2082 \u226b g'\nthis :\n pullback.map f g f' g' i\u2081 i\u2082 (\ud835\udfd9 S) (_ : f \u226b \ud835\udfd9 S = i\u2081 \u226b f') (_ : g \u226b \ud835\udfd9 S = i\u2082 \u226b g') =\n ((pullbackSymmetry (Over.mk f).hom (Over.mk g).hom).hom \u226b ((baseChange (Over.mk f).hom).map (Over.homMk i\u2082)).left) \u226b\n (pullbackSymmetry (Over.mk g').hom (Over.mk f).hom).hom \u226b ((baseChange g').map (Over.homMk i\u2081)).left\n\u22a2 P\n (((pullbackSymmetry (Over.mk f).hom (Over.mk g).hom).hom \u226b\n ((baseChange (Over.mk f).hom).map (Over.homMk i\u2082)).left) \u226b\n (pullbackSymmetry (Over.mk g').hom (Over.mk f).hom).hom \u226b ((baseChange g').map (Over.homMk i\u2081)).left)\n[PROOFSTEP]\napply hP'\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.39138, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : StableUnderComposition P\nS X X' Y Y' : C\nf : X \u27f6 S\ng : Y \u27f6 S\nf' : X' \u27f6 S\ng' : Y' \u27f6 S\ni\u2081 : X \u27f6 X'\ni\u2082 : Y \u27f6 Y'\nh\u2081 : P i\u2081\nh\u2082 : P i\u2082\ne\u2081 : f = i\u2081 \u226b f'\ne\u2082 : g = i\u2082 \u226b g'\nthis :\n pullback.map f g f' g' i\u2081 i\u2082 (\ud835\udfd9 S) (_ : f \u226b \ud835\udfd9 S = i\u2081 \u226b f') (_ : g \u226b \ud835\udfd9 S = i\u2082 \u226b g') =\n ((pullbackSymmetry (Over.mk f).hom (Over.mk g).hom).hom \u226b ((baseChange (Over.mk f).hom).map (Over.homMk i\u2082)).left) \u226b\n (pullbackSymmetry (Over.mk g').hom (Over.mk f).hom).hom \u226b ((baseChange g').map (Over.homMk i\u2081)).left\n\u22a2 P ((pullbackSymmetry (Over.mk f).hom (Over.mk g).hom).hom \u226b ((baseChange (Over.mk f).hom).map (Over.homMk i\u2082)).left)\n[PROOFSTEP]\nrw [hP.respectsIso.cancel_left_isIso]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.39138, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : StableUnderComposition P\nS X X' Y Y' : C\nf : X \u27f6 S\ng : Y \u27f6 S\nf' : X' \u27f6 S\ng' : Y' \u27f6 S\ni\u2081 : X \u27f6 X'\ni\u2082 : Y \u27f6 Y'\nh\u2081 : P i\u2081\nh\u2082 : P i\u2082\ne\u2081 : f = i\u2081 \u226b f'\ne\u2082 : g = i\u2082 \u226b g'\nthis :\n pullback.map f g f' g' i\u2081 i\u2082 (\ud835\udfd9 S) (_ : f \u226b \ud835\udfd9 S = i\u2081 \u226b f') (_ : g \u226b \ud835\udfd9 S = i\u2082 \u226b g') =\n ((pullbackSymmetry (Over.mk f).hom (Over.mk g).hom).hom \u226b ((baseChange (Over.mk f).hom).map (Over.homMk i\u2082)).left) \u226b\n (pullbackSymmetry (Over.mk g').hom (Over.mk f).hom).hom \u226b ((baseChange g').map (Over.homMk i\u2081)).left\n\u22a2 P ((pullbackSymmetry (Over.mk g').hom (Over.mk f).hom).hom \u226b ((baseChange g').map (Over.homMk i\u2081)).left)\n[PROOFSTEP]\nrw [hP.respectsIso.cancel_left_isIso]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.39138, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : StableUnderComposition P\nS X X' Y Y' : C\nf : X \u27f6 S\ng : Y \u27f6 S\nf' : X' \u27f6 S\ng' : Y' \u27f6 S\ni\u2081 : X \u27f6 X'\ni\u2082 : Y \u27f6 Y'\nh\u2081 : P i\u2081\nh\u2082 : P i\u2082\ne\u2081 : f = i\u2081 \u226b f'\ne\u2082 : g = i\u2082 \u226b g'\nthis :\n pullback.map f g f' g' i\u2081 i\u2082 (\ud835\udfd9 S) (_ : f \u226b \ud835\udfd9 S = i\u2081 \u226b f') (_ : g \u226b \ud835\udfd9 S = i\u2082 \u226b g') =\n ((pullbackSymmetry (Over.mk f).hom (Over.mk g).hom).hom \u226b ((baseChange (Over.mk f).hom).map (Over.homMk i\u2082)).left) \u226b\n (pullbackSymmetry (Over.mk g').hom (Over.mk f).hom).hom \u226b ((baseChange g').map (Over.homMk i\u2081)).left\n\u22a2 P ((baseChange (Over.mk f).hom).map (Over.homMk i\u2082)).left\ncase a\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.39138, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : StableUnderComposition P\nS X X' Y Y' : C\nf : X \u27f6 S\ng : Y \u27f6 S\nf' : X' \u27f6 S\ng' : Y' \u27f6 S\ni\u2081 : X \u27f6 X'\ni\u2082 : Y \u27f6 Y'\nh\u2081 : P i\u2081\nh\u2082 : P i\u2082\ne\u2081 : f = i\u2081 \u226b f'\ne\u2082 : g = i\u2082 \u226b g'\nthis :\n pullback.map f g f' g' i\u2081 i\u2082 (\ud835\udfd9 S) (_ : f \u226b \ud835\udfd9 S = i\u2081 \u226b f') (_ : g \u226b \ud835\udfd9 S = i\u2082 \u226b g') =\n ((pullbackSymmetry (Over.mk f).hom (Over.mk g).hom).hom \u226b ((baseChange (Over.mk f).hom).map (Over.homMk i\u2082)).left) \u226b\n (pullbackSymmetry (Over.mk g').hom (Over.mk f).hom).hom \u226b ((baseChange g').map (Over.homMk i\u2081)).left\n\u22a2 P ((baseChange g').map (Over.homMk i\u2081)).left\n[PROOFSTEP]\nexacts [hP.baseChange_map _ (Over.homMk _ e\u2082.symm : Over.mk g \u27f6 Over.mk g') h\u2082,\n hP.baseChange_map _ (Over.homMk _ e\u2081.symm : Over.mk f \u27f6 Over.mk f') h\u2081]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.47372, u_1} D\nP : MorphismProperty C\ninst\u271d : HasPushouts C\nhP\u2081 : RespectsIso P\nhP\u2082 : \u2200 (A B A' : C) (f : A \u27f6 A') (g : A \u27f6 B), P f \u2192 P pushout.inr\nA A' B B' : C\nf : A \u27f6 A'\ng : A \u27f6 B\nf' : B \u27f6 B'\ng' : A' \u27f6 B'\nsq : IsPushout g f f' g'\nhf : P f\n\u22a2 P f'\n[PROOFSTEP]\nlet e := sq.flip.isoPushout\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.47372, u_1} D\nP : MorphismProperty C\ninst\u271d : HasPushouts C\nhP\u2081 : RespectsIso P\nhP\u2082 : \u2200 (A B A' : C) (f : A \u27f6 A') (g : A \u27f6 B), P f \u2192 P pushout.inr\nA A' B B' : C\nf : A \u27f6 A'\ng : A \u27f6 B\nf' : B \u27f6 B'\ng' : A' \u27f6 B'\nsq : IsPushout g f f' g'\nhf : P f\ne : B' \u2245 pushout f g := IsPushout.isoPushout (_ : IsPushout f g g' f')\n\u22a2 P f'\n[PROOFSTEP]\nrw [\u2190 hP\u2081.cancel_right_isIso _ e.hom, sq.flip.inr_isoPushout_hom]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.47372, u_1} D\nP : MorphismProperty C\ninst\u271d : HasPushouts C\nhP\u2081 : RespectsIso P\nhP\u2082 : \u2200 (A B A' : C) (f : A \u27f6 A') (g : A \u27f6 B), P f \u2192 P pushout.inr\nA A' B B' : C\nf : A \u27f6 A'\ng : A \u27f6 B\nf' : B \u27f6 B'\ng' : A' \u27f6 B'\nsq : IsPushout g f f' g'\nhf : P f\ne : B' \u2245 pushout f g := IsPushout.isoPushout (_ : IsPushout f g g' f')\n\u22a2 P pushout.inr\n[PROOFSTEP]\nexact hP\u2082 _ _ _ f g hf\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b3 : Category.{?u.53343, u_1} D\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\ninst\u271d\u00b2 : Category.{u_5, u_2} C\u2081\ninst\u271d\u00b9 : Category.{u_6, u_3} C\u2082\ninst\u271d : Category.{u_7, u_4} C\u2083\nW : MorphismProperty C\u2081\nF : C\u2081 \u2964 C\u2082\nhF : IsInvertedBy W F\nG : C\u2082 \u2964 C\u2083\nX Y : C\u2081\nf : X \u27f6 Y\nhf : W f\n\u22a2 IsIso ((F \u22d9 G).map f)\n[PROOFSTEP]\nhaveI := hF f hf\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b3 : Category.{?u.53343, u_1} D\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\ninst\u271d\u00b2 : Category.{u_5, u_2} C\u2081\ninst\u271d\u00b9 : Category.{u_6, u_3} C\u2082\ninst\u271d : Category.{u_7, u_4} C\u2083\nW : MorphismProperty C\u2081\nF : C\u2081 \u2964 C\u2082\nhF : IsInvertedBy W F\nG : C\u2082 \u2964 C\u2083\nX Y : C\u2081\nf : X \u27f6 Y\nhf : W f\nthis : IsIso (F.map f)\n\u22a2 IsIso ((F \u22d9 G).map f)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b3 : Category.{?u.53343, u_1} D\nC\u2081 : Type u_2\nC\u2082 : Type u_3\nC\u2083 : Type u_4\ninst\u271d\u00b2 : Category.{u_5, u_2} C\u2081\ninst\u271d\u00b9 : Category.{u_6, u_3} C\u2082\ninst\u271d : Category.{u_7, u_4} C\u2083\nW : MorphismProperty C\u2081\nF : C\u2081 \u2964 C\u2082\nhF : IsInvertedBy W F\nG : C\u2082 \u2964 C\u2083\nX Y : C\u2081\nf : X \u27f6 Y\nhf : W f\nthis : IsIso (F.map f)\n\u22a2 IsIso (G.map (F.map f))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nL : C \u2964 D\nh : IsInvertedBy W L\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nhf : MorphismProperty.op W f\n\u22a2 IsIso (L.op.map f)\n[PROOFSTEP]\nhaveI := h f.unop hf\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nL : C \u2964 D\nh : IsInvertedBy W L\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nhf : MorphismProperty.op W f\nthis : IsIso (L.map f.unop)\n\u22a2 IsIso (L.op.map f)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nL : C \u2964 D\nh : IsInvertedBy W L\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nhf : MorphismProperty.op W f\nthis : IsIso (L.map f.unop)\n\u22a2 IsIso (L.map f.unop).op\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nL : C\u1d52\u1d56 \u2964 D\nh : IsInvertedBy (MorphismProperty.op W) L\nX Y : C\nf : X \u27f6 Y\nhf : W f\n\u22a2 IsIso (L.rightOp.map f)\n[PROOFSTEP]\nhaveI := h f.op hf\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nL : C\u1d52\u1d56 \u2964 D\nh : IsInvertedBy (MorphismProperty.op W) L\nX Y : C\nf : X \u27f6 Y\nhf : W f\nthis : IsIso (L.map f.op)\n\u22a2 IsIso (L.rightOp.map f)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nL : C\u1d52\u1d56 \u2964 D\nh : IsInvertedBy (MorphismProperty.op W) L\nX Y : C\nf : X \u27f6 Y\nhf : W f\nthis : IsIso (L.map f.op)\n\u22a2 IsIso (L.map f.op).op\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nL : C \u2964 D\u1d52\u1d56\nh : IsInvertedBy W L\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nhf : MorphismProperty.op W f\n\u22a2 IsIso (L.leftOp.map f)\n[PROOFSTEP]\nhaveI := h f.unop hf\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nL : C \u2964 D\u1d52\u1d56\nh : IsInvertedBy W L\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nhf : MorphismProperty.op W f\nthis : IsIso (L.map f.unop)\n\u22a2 IsIso (L.leftOp.map f)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nL : C \u2964 D\u1d52\u1d56\nh : IsInvertedBy W L\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\nhf : MorphismProperty.op W f\nthis : IsIso (L.map f.unop)\n\u22a2 IsIso (L.map f.unop).unop\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nL : C\u1d52\u1d56 \u2964 D\u1d52\u1d56\nh : IsInvertedBy (MorphismProperty.op W) L\nX Y : C\nf : X \u27f6 Y\nhf : W f\n\u22a2 IsIso ((Functor.unop L).map f)\n[PROOFSTEP]\nhaveI := h f.op hf\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nL : C\u1d52\u1d56 \u2964 D\u1d52\u1d56\nh : IsInvertedBy (MorphismProperty.op W) L\nX Y : C\nf : X \u27f6 Y\nhf : W f\nthis : IsIso (L.map f.op)\n\u22a2 IsIso ((Functor.unop L).map f)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nL : C\u1d52\u1d56 \u2964 D\u1d52\u1d56\nh : IsInvertedBy (MorphismProperty.op W) L\nX Y : C\nf : X \u27f6 Y\nhf : W f\nthis : IsIso (L.map f.op)\n\u22a2 IsIso (L.map f.op).unop\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u2081 F\u2082 : C \u2964 D\napp : (X : C) \u2192 F\u2081.obj X \u27f6 F\u2082.obj X\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : naturalityProperty app f\nhg : naturalityProperty app g\n\u22a2 naturalityProperty app (f \u226b g)\n[PROOFSTEP]\nsimp only [naturalityProperty] at hf hg \u22a2\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u2081 F\u2082 : C \u2964 D\napp : (X : C) \u2192 F\u2081.obj X \u27f6 F\u2082.obj X\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : F\u2081.map f \u226b app Y = app X \u226b F\u2082.map f\nhg : F\u2081.map g \u226b app Z = app Y \u226b F\u2082.map g\n\u22a2 F\u2081.map (f \u226b g) \u226b app Z = app X \u226b F\u2082.map (f \u226b g)\n[PROOFSTEP]\nsimp only [Functor.map_comp, Category.assoc, hg]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u2081 F\u2082 : C \u2964 D\napp : (X : C) \u2192 F\u2081.obj X \u27f6 F\u2082.obj X\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : F\u2081.map f \u226b app Y = app X \u226b F\u2082.map f\nhg : F\u2081.map g \u226b app Z = app Y \u226b F\u2082.map g\n\u22a2 F\u2081.map f \u226b app Y \u226b F\u2082.map g = app X \u226b F\u2082.map f \u226b F\u2082.map g\n[PROOFSTEP]\nslice_lhs 1 2 => rw [hf]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u2081 F\u2082 : C \u2964 D\napp : (X : C) \u2192 F\u2081.obj X \u27f6 F\u2082.obj X\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : F\u2081.map f \u226b app Y = app X \u226b F\u2082.map f\nhg : F\u2081.map g \u226b app Z = app Y \u226b F\u2082.map g\n| F\u2081.map f \u226b app Y\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u2081 F\u2082 : C \u2964 D\napp : (X : C) \u2192 F\u2081.obj X \u27f6 F\u2082.obj X\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : F\u2081.map f \u226b app Y = app X \u226b F\u2082.map f\nhg : F\u2081.map g \u226b app Z = app Y \u226b F\u2082.map g\n| F\u2082.map g\n[PROOFSTEP]\nrw [hf]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u2081 F\u2082 : C \u2964 D\napp : (X : C) \u2192 F\u2081.obj X \u27f6 F\u2082.obj X\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : F\u2081.map f \u226b app Y = app X \u226b F\u2082.map f\nhg : F\u2081.map g \u226b app Z = app Y \u226b F\u2082.map g\n| F\u2081.map f \u226b app Y\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u2081 F\u2082 : C \u2964 D\napp : (X : C) \u2192 F\u2081.obj X \u27f6 F\u2082.obj X\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : F\u2081.map f \u226b app Y = app X \u226b F\u2082.map f\nhg : F\u2081.map g \u226b app Z = app Y \u226b F\u2082.map g\n| F\u2082.map g\n[PROOFSTEP]\nrw [hf]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u2081 F\u2082 : C \u2964 D\napp : (X : C) \u2192 F\u2081.obj X \u27f6 F\u2082.obj X\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : F\u2081.map f \u226b app Y = app X \u226b F\u2082.map f\nhg : F\u2081.map g \u226b app Z = app Y \u226b F\u2082.map g\n| F\u2081.map f \u226b app Y\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u2081 F\u2082 : C \u2964 D\napp : (X : C) \u2192 F\u2081.obj X \u27f6 F\u2082.obj X\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : F\u2081.map f \u226b app Y = app X \u226b F\u2082.map f\nhg : F\u2081.map g \u226b app Z = app Y \u226b F\u2082.map g\n| F\u2082.map g\n[PROOFSTEP]\nrw [hf]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u2081 F\u2082 : C \u2964 D\napp : (X : C) \u2192 F\u2081.obj X \u27f6 F\u2082.obj X\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : F\u2081.map f \u226b app Y = app X \u226b F\u2082.map f\nhg : F\u2081.map g \u226b app Z = app Y \u226b F\u2082.map g\n\u22a2 (app X \u226b F\u2082.map f) \u226b F\u2082.map g = app X \u226b F\u2082.map f \u226b F\u2082.map g\n[PROOFSTEP]\nrw [Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u2081 F\u2082 : C \u2964 D\napp : (X : C) \u2192 F\u2081.obj X \u27f6 F\u2082.obj X\nX Y : C\ne : X \u2245 Y\nhe : naturalityProperty app e.hom\n\u22a2 naturalityProperty app e.inv\n[PROOFSTEP]\nsimp only [naturalityProperty] at he \u22a2\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u2081 F\u2082 : C \u2964 D\napp : (X : C) \u2192 F\u2081.obj X \u27f6 F\u2082.obj X\nX Y : C\ne : X \u2245 Y\nhe : F\u2081.map e.hom \u226b app Y = app X \u226b F\u2082.map e.hom\n\u22a2 F\u2081.map e.inv \u226b app X = app Y \u226b F\u2082.map e.inv\n[PROOFSTEP]\nrw [\u2190 cancel_epi (F\u2081.map e.hom)]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u2081 F\u2082 : C \u2964 D\napp : (X : C) \u2192 F\u2081.obj X \u27f6 F\u2082.obj X\nX Y : C\ne : X \u2245 Y\nhe : F\u2081.map e.hom \u226b app Y = app X \u226b F\u2082.map e.hom\n\u22a2 F\u2081.map e.hom \u226b F\u2081.map e.inv \u226b app X = F\u2081.map e.hom \u226b app Y \u226b F\u2082.map e.inv\n[PROOFSTEP]\nslice_rhs 1 2 => rw [he]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u2081 F\u2082 : C \u2964 D\napp : (X : C) \u2192 F\u2081.obj X \u27f6 F\u2082.obj X\nX Y : C\ne : X \u2245 Y\nhe : F\u2081.map e.hom \u226b app Y = app X \u226b F\u2082.map e.hom\n| F\u2081.map e.hom \u226b app Y\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u2081 F\u2082 : C \u2964 D\napp : (X : C) \u2192 F\u2081.obj X \u27f6 F\u2082.obj X\nX Y : C\ne : X \u2245 Y\nhe : F\u2081.map e.hom \u226b app Y = app X \u226b F\u2082.map e.hom\n| F\u2082.map e.inv\n[PROOFSTEP]\nrw [he]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u2081 F\u2082 : C \u2964 D\napp : (X : C) \u2192 F\u2081.obj X \u27f6 F\u2082.obj X\nX Y : C\ne : X \u2245 Y\nhe : F\u2081.map e.hom \u226b app Y = app X \u226b F\u2082.map e.hom\n| F\u2081.map e.hom \u226b app Y\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u2081 F\u2082 : C \u2964 D\napp : (X : C) \u2192 F\u2081.obj X \u27f6 F\u2082.obj X\nX Y : C\ne : X \u2245 Y\nhe : F\u2081.map e.hom \u226b app Y = app X \u226b F\u2082.map e.hom\n| F\u2082.map e.inv\n[PROOFSTEP]\nrw [he]\n[GOAL]\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u2081 F\u2082 : C \u2964 D\napp : (X : C) \u2192 F\u2081.obj X \u27f6 F\u2082.obj X\nX Y : C\ne : X \u2245 Y\nhe : F\u2081.map e.hom \u226b app Y = app X \u226b F\u2082.map e.hom\n| F\u2081.map e.hom \u226b app Y\ncase a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u2081 F\u2082 : C \u2964 D\napp : (X : C) \u2192 F\u2081.obj X \u27f6 F\u2082.obj X\nX Y : C\ne : X \u2245 Y\nhe : F\u2081.map e.hom \u226b app Y = app X \u226b F\u2082.map e.hom\n| F\u2082.map e.inv\n[PROOFSTEP]\nrw [he]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nF\u2081 F\u2082 : C \u2964 D\napp : (X : C) \u2192 F\u2081.obj X \u27f6 F\u2082.obj X\nX Y : C\ne : X \u2245 Y\nhe : F\u2081.map e.hom \u226b app Y = app X \u226b F\u2082.map e.hom\n\u22a2 F\u2081.map e.hom \u226b F\u2081.map e.inv \u226b app X = (app X \u226b F\u2082.map e.hom) \u226b F\u2082.map e.inv\n[PROOFSTEP]\nsimp only [Category.assoc, \u2190 F\u2081.map_comp_assoc, \u2190 F\u2082.map_comp, e.hom_inv_id, Functor.map_id, Category.id_comp,\n Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty D\nh : RespectsIso P\nF : C \u2964 D\n\u22a2 RespectsIso (MorphismProperty.inverseImage P F)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty D\nh : RespectsIso P\nF : C \u2964 D\n\u22a2 \u2200 {X Y Z : C} (e : X \u2245 Y) (f : Y \u27f6 Z),\n MorphismProperty.inverseImage P F f \u2192 MorphismProperty.inverseImage P F (e.hom \u226b f)\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty D\nh : RespectsIso P\nF : C \u2964 D\n\u22a2 \u2200 {X Y Z : C} (e : Y \u2245 Z) (f : X \u27f6 Y),\n MorphismProperty.inverseImage P F f \u2192 MorphismProperty.inverseImage P F (f \u226b e.hom)\n[PROOFSTEP]\nall_goals\n intro X Y Z e f hf\n dsimp [MorphismProperty.inverseImage]\n rw [F.map_comp]\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty D\nh : RespectsIso P\nF : C \u2964 D\n\u22a2 \u2200 {X Y Z : C} (e : X \u2245 Y) (f : Y \u27f6 Z),\n MorphismProperty.inverseImage P F f \u2192 MorphismProperty.inverseImage P F (e.hom \u226b f)\n[PROOFSTEP]\nintro X Y Z e f hf\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty D\nh : RespectsIso P\nF : C \u2964 D\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\nhf : MorphismProperty.inverseImage P F f\n\u22a2 MorphismProperty.inverseImage P F (e.hom \u226b f)\n[PROOFSTEP]\ndsimp [MorphismProperty.inverseImage]\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty D\nh : RespectsIso P\nF : C \u2964 D\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\nhf : MorphismProperty.inverseImage P F f\n\u22a2 P (F.map (e.hom \u226b f))\n[PROOFSTEP]\nrw [F.map_comp]\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty D\nh : RespectsIso P\nF : C \u2964 D\n\u22a2 \u2200 {X Y Z : C} (e : Y \u2245 Z) (f : X \u27f6 Y),\n MorphismProperty.inverseImage P F f \u2192 MorphismProperty.inverseImage P F (f \u226b e.hom)\n[PROOFSTEP]\nintro X Y Z e f hf\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty D\nh : RespectsIso P\nF : C \u2964 D\nX Y Z : C\ne : Y \u2245 Z\nf : X \u27f6 Y\nhf : MorphismProperty.inverseImage P F f\n\u22a2 MorphismProperty.inverseImage P F (f \u226b e.hom)\n[PROOFSTEP]\ndsimp [MorphismProperty.inverseImage]\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty D\nh : RespectsIso P\nF : C \u2964 D\nX Y Z : C\ne : Y \u2245 Z\nf : X \u27f6 Y\nhf : MorphismProperty.inverseImage P F f\n\u22a2 P (F.map (f \u226b e.hom))\n[PROOFSTEP]\nrw [F.map_comp]\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty D\nh : RespectsIso P\nF : C \u2964 D\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\nhf : MorphismProperty.inverseImage P F f\n\u22a2 P (F.map e.hom \u226b F.map f)\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty D\nh : RespectsIso P\nF : C \u2964 D\nX Y Z : C\ne : Y \u2245 Z\nf : X \u27f6 Y\nhf : MorphismProperty.inverseImage P F f\n\u22a2 P (F.map f \u226b F.map e.hom)\n[PROOFSTEP]\nexacts [h.1 (F.mapIso e) (F.map f) hf, h.2 (F.mapIso e) (F.map f) hf]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty D\nh : StableUnderComposition P\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : MorphismProperty.inverseImage P F f\nhg : MorphismProperty.inverseImage P F g\n\u22a2 MorphismProperty.inverseImage P F (f \u226b g)\n[PROOFSTEP]\nsimpa only [\u2190 F.map_comp] using h (F.map f) (F.map g) hf hg\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.65654, u_1} D\nX Y : C\nf : X \u27f6 Y\n\u22a2 isomorphisms C f \u2194 IsIso f\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.65803, u_1} D\nX Y : C\nf : X \u27f6 Y\n\u22a2 monomorphisms C f \u2194 Mono f\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.65952, u_1} D\nX Y : C\nf : X \u27f6 Y\n\u22a2 epimorphisms C f \u2194 Epi f\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.66779, u_1} D\n\u22a2 RespectsIso (MorphismProperty.monomorphisms C)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.66779, u_1} D\n\u22a2 \u2200 {X Y Z : C} (e : X \u2245 Y) (f : Y \u27f6 Z),\n MorphismProperty.monomorphisms C f \u2192 MorphismProperty.monomorphisms C (e.hom \u226b f)\n[PROOFSTEP]\nintro X Y Z e f\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.66779, u_1} D\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\n\u22a2 MorphismProperty.monomorphisms C f \u2192 MorphismProperty.monomorphisms C (e.hom \u226b f)\n[PROOFSTEP]\nsimp only [monomorphisms.iff]\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.66779, u_1} D\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\n\u22a2 Mono f \u2192 Mono (e.hom \u226b f)\n[PROOFSTEP]\nintro\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.66779, u_1} D\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\na\u271d : Mono f\n\u22a2 Mono (e.hom \u226b f)\n[PROOFSTEP]\napply mono_comp\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.66779, u_1} D\n\u22a2 \u2200 {X Y Z : C} (e : Y \u2245 Z) (f : X \u27f6 Y),\n MorphismProperty.monomorphisms C f \u2192 MorphismProperty.monomorphisms C (f \u226b e.hom)\n[PROOFSTEP]\nintro X Y Z e f\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.66779, u_1} D\nX Y Z : C\ne : Y \u2245 Z\nf : X \u27f6 Y\n\u22a2 MorphismProperty.monomorphisms C f \u2192 MorphismProperty.monomorphisms C (f \u226b e.hom)\n[PROOFSTEP]\nsimp only [monomorphisms.iff]\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.66779, u_1} D\nX Y Z : C\ne : Y \u2245 Z\nf : X \u27f6 Y\n\u22a2 Mono f \u2192 Mono (f \u226b e.hom)\n[PROOFSTEP]\nintro\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.66779, u_1} D\nX Y Z : C\ne : Y \u2245 Z\nf : X \u27f6 Y\na\u271d : Mono f\n\u22a2 Mono (f \u226b e.hom)\n[PROOFSTEP]\napply mono_comp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.67468, u_1} D\n\u22a2 RespectsIso (MorphismProperty.epimorphisms C)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.67468, u_1} D\n\u22a2 \u2200 {X Y Z : C} (e : X \u2245 Y) (f : Y \u27f6 Z), MorphismProperty.epimorphisms C f \u2192 MorphismProperty.epimorphisms C (e.hom \u226b f)\n[PROOFSTEP]\nintro X Y Z e f\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.67468, u_1} D\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\n\u22a2 MorphismProperty.epimorphisms C f \u2192 MorphismProperty.epimorphisms C (e.hom \u226b f)\n[PROOFSTEP]\nsimp only [epimorphisms.iff]\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.67468, u_1} D\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\n\u22a2 Epi f \u2192 Epi (e.hom \u226b f)\n[PROOFSTEP]\nintro\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.67468, u_1} D\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\na\u271d : Epi f\n\u22a2 Epi (e.hom \u226b f)\n[PROOFSTEP]\napply epi_comp\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.67468, u_1} D\n\u22a2 \u2200 {X Y Z : C} (e : Y \u2245 Z) (f : X \u27f6 Y), MorphismProperty.epimorphisms C f \u2192 MorphismProperty.epimorphisms C (f \u226b e.hom)\n[PROOFSTEP]\nintro X Y Z e f\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.67468, u_1} D\nX Y Z : C\ne : Y \u2245 Z\nf : X \u27f6 Y\n\u22a2 MorphismProperty.epimorphisms C f \u2192 MorphismProperty.epimorphisms C (f \u226b e.hom)\n[PROOFSTEP]\nsimp only [epimorphisms.iff]\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.67468, u_1} D\nX Y Z : C\ne : Y \u2245 Z\nf : X \u27f6 Y\n\u22a2 Epi f \u2192 Epi (f \u226b e.hom)\n[PROOFSTEP]\nintro\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.67468, u_1} D\nX Y Z : C\ne : Y \u2245 Z\nf : X \u27f6 Y\na\u271d : Epi f\n\u22a2 Epi (f \u226b e.hom)\n[PROOFSTEP]\napply epi_comp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.68149, u_1} D\n\u22a2 RespectsIso (MorphismProperty.isomorphisms C)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.68149, u_1} D\n\u22a2 \u2200 {X Y Z : C} (e : X \u2245 Y) (f : Y \u27f6 Z), MorphismProperty.isomorphisms C f \u2192 MorphismProperty.isomorphisms C (e.hom \u226b f)\n[PROOFSTEP]\nintro X Y Z e f\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.68149, u_1} D\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\n\u22a2 MorphismProperty.isomorphisms C f \u2192 MorphismProperty.isomorphisms C (e.hom \u226b f)\n[PROOFSTEP]\nsimp only [isomorphisms.iff]\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.68149, u_1} D\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\n\u22a2 IsIso f \u2192 IsIso (e.hom \u226b f)\n[PROOFSTEP]\nintro\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.68149, u_1} D\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\na\u271d : IsIso f\n\u22a2 IsIso (e.hom \u226b f)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.68149, u_1} D\n\u22a2 \u2200 {X Y Z : C} (e : Y \u2245 Z) (f : X \u27f6 Y), MorphismProperty.isomorphisms C f \u2192 MorphismProperty.isomorphisms C (f \u226b e.hom)\n[PROOFSTEP]\nintro X Y Z e f\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.68149, u_1} D\nX Y Z : C\ne : Y \u2245 Z\nf : X \u27f6 Y\n\u22a2 MorphismProperty.isomorphisms C f \u2192 MorphismProperty.isomorphisms C (f \u226b e.hom)\n[PROOFSTEP]\nsimp only [isomorphisms.iff]\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.68149, u_1} D\nX Y Z : C\ne : Y \u2245 Z\nf : X \u27f6 Y\n\u22a2 IsIso f \u2192 IsIso (f \u226b e.hom)\n[PROOFSTEP]\nintro\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.68149, u_1} D\nX Y Z : C\ne : Y \u2245 Z\nf : X \u27f6 Y\na\u271d : IsIso f\n\u22a2 IsIso (f \u226b e.hom)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.68670, u_1} D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : MorphismProperty.isomorphisms C f\nhg : MorphismProperty.isomorphisms C g\n\u22a2 MorphismProperty.isomorphisms C (f \u226b g)\n[PROOFSTEP]\nrw [isomorphisms.iff] at hf hg \u22a2\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.68670, u_1} D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : IsIso f\nhg : IsIso g\n\u22a2 IsIso (f \u226b g)\n[PROOFSTEP]\nhaveI := hf\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.68670, u_1} D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : IsIso f\nhg : IsIso g\nthis : IsIso f\n\u22a2 IsIso (f \u226b g)\n[PROOFSTEP]\nhaveI := hg\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.68670, u_1} D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : IsIso f\nhg : IsIso g\nthis\u271d : IsIso f\nthis : IsIso g\n\u22a2 IsIso (f \u226b g)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.69618, u_1} D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : MorphismProperty.monomorphisms C f\nhg : MorphismProperty.monomorphisms C g\n\u22a2 MorphismProperty.monomorphisms C (f \u226b g)\n[PROOFSTEP]\nrw [monomorphisms.iff] at hf hg \u22a2\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.69618, u_1} D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : Mono f\nhg : Mono g\n\u22a2 Mono (f \u226b g)\n[PROOFSTEP]\nhaveI := hf\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.69618, u_1} D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : Mono f\nhg : Mono g\nthis : Mono f\n\u22a2 Mono (f \u226b g)\n[PROOFSTEP]\nhaveI := hg\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.69618, u_1} D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : Mono f\nhg : Mono g\nthis\u271d : Mono f\nthis : Mono g\n\u22a2 Mono (f \u226b g)\n[PROOFSTEP]\napply mono_comp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.70586, u_1} D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : MorphismProperty.epimorphisms C f\nhg : MorphismProperty.epimorphisms C g\n\u22a2 MorphismProperty.epimorphisms C (f \u226b g)\n[PROOFSTEP]\nrw [epimorphisms.iff] at hf hg \u22a2\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.70586, u_1} D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : Epi f\nhg : Epi g\n\u22a2 Epi (f \u226b g)\n[PROOFSTEP]\nhaveI := hf\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.70586, u_1} D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : Epi f\nhg : Epi g\nthis : Epi f\n\u22a2 Epi (f \u226b g)\n[PROOFSTEP]\nhaveI := hg\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.70586, u_1} D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : Epi f\nhg : Epi g\nthis\u271d : Epi f\nthis : Epi g\n\u22a2 Epi (f \u226b g)\n[PROOFSTEP]\napply epi_comp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nF\u2081 F\u2082 : FunctorsInverting W D\nh : F\u2081.obj = F\u2082.obj\n\u22a2 F\u2081 = F\u2082\n[PROOFSTEP]\ncases F\u2081\n[GOAL]\ncase mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nF\u2082 : FunctorsInverting W D\nobj\u271d : C \u2964 D\nproperty\u271d : IsInvertedBy W obj\u271d\nh : { obj := obj\u271d, property := property\u271d }.obj = F\u2082.obj\n\u22a2 { obj := obj\u271d, property := property\u271d } = F\u2082\n[PROOFSTEP]\ncases F\u2082\n[GOAL]\ncase mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nobj\u271d\u00b9 : C \u2964 D\nproperty\u271d\u00b9 : IsInvertedBy W obj\u271d\u00b9\nobj\u271d : C \u2964 D\nproperty\u271d : IsInvertedBy W obj\u271d\nh : { obj := obj\u271d\u00b9, property := property\u271d\u00b9 }.obj = { obj := obj\u271d, property := property\u271d }.obj\n\u22a2 { obj := obj\u271d\u00b9, property := property\u271d\u00b9 } = { obj := obj\u271d, property := property\u271d }\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nobj\u271d : C \u2964 D\nproperty\u271d\u00b9 : IsInvertedBy W obj\u271d\nproperty\u271d : IsInvertedBy W { obj := obj\u271d, property := property\u271d\u00b9 }.obj\n\u22a2 { obj := obj\u271d, property := property\u271d\u00b9 } =\n { obj := { obj := obj\u271d, property := property\u271d\u00b9 }.obj, property := property\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nF\u2081 F\u2082 : C \u2964 D\ne : F\u2081 \u2245 F\u2082\n\u22a2 IsInvertedBy W F\u2081 \u2194 IsInvertedBy W F\u2082\n[PROOFSTEP]\nsuffices \u2200 (X Y : C) (f : X \u27f6 Y), IsIso (F\u2081.map f) \u2194 IsIso (F\u2082.map f)\n by\n constructor\n exact fun h X Y f hf => by\n rw [\u2190 this]\n exact h f hf\n exact fun h X Y f hf => by\n rw [this]\n exact h f hf\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nF\u2081 F\u2082 : C \u2964 D\ne : F\u2081 \u2245 F\u2082\nthis : \u2200 (X Y : C) (f : X \u27f6 Y), IsIso (F\u2081.map f) \u2194 IsIso (F\u2082.map f)\n\u22a2 IsInvertedBy W F\u2081 \u2194 IsInvertedBy W F\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nF\u2081 F\u2082 : C \u2964 D\ne : F\u2081 \u2245 F\u2082\nthis : \u2200 (X Y : C) (f : X \u27f6 Y), IsIso (F\u2081.map f) \u2194 IsIso (F\u2082.map f)\n\u22a2 IsInvertedBy W F\u2081 \u2192 IsInvertedBy W F\u2082\ncase mpr\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nF\u2081 F\u2082 : C \u2964 D\ne : F\u2081 \u2245 F\u2082\nthis : \u2200 (X Y : C) (f : X \u27f6 Y), IsIso (F\u2081.map f) \u2194 IsIso (F\u2082.map f)\n\u22a2 IsInvertedBy W F\u2082 \u2192 IsInvertedBy W F\u2081\n[PROOFSTEP]\nexact fun h X Y f hf => by\n rw [\u2190 this]\n exact h f hf\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nF\u2081 F\u2082 : C \u2964 D\ne : F\u2081 \u2245 F\u2082\nthis : \u2200 (X Y : C) (f : X \u27f6 Y), IsIso (F\u2081.map f) \u2194 IsIso (F\u2082.map f)\nh : IsInvertedBy W F\u2081\nX Y : C\nf : X \u27f6 Y\nhf : W f\n\u22a2 IsIso (F\u2082.map f)\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nF\u2081 F\u2082 : C \u2964 D\ne : F\u2081 \u2245 F\u2082\nthis : \u2200 (X Y : C) (f : X \u27f6 Y), IsIso (F\u2081.map f) \u2194 IsIso (F\u2082.map f)\nh : IsInvertedBy W F\u2081\nX Y : C\nf : X \u27f6 Y\nhf : W f\n\u22a2 IsIso (F\u2081.map f)\n[PROOFSTEP]\nexact h f hf\n[GOAL]\ncase mpr\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nF\u2081 F\u2082 : C \u2964 D\ne : F\u2081 \u2245 F\u2082\nthis : \u2200 (X Y : C) (f : X \u27f6 Y), IsIso (F\u2081.map f) \u2194 IsIso (F\u2082.map f)\n\u22a2 IsInvertedBy W F\u2082 \u2192 IsInvertedBy W F\u2081\n[PROOFSTEP]\nexact fun h X Y f hf => by\n rw [this]\n exact h f hf\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nF\u2081 F\u2082 : C \u2964 D\ne : F\u2081 \u2245 F\u2082\nthis : \u2200 (X Y : C) (f : X \u27f6 Y), IsIso (F\u2081.map f) \u2194 IsIso (F\u2082.map f)\nh : IsInvertedBy W F\u2082\nX Y : C\nf : X \u27f6 Y\nhf : W f\n\u22a2 IsIso (F\u2081.map f)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nF\u2081 F\u2082 : C \u2964 D\ne : F\u2081 \u2245 F\u2082\nthis : \u2200 (X Y : C) (f : X \u27f6 Y), IsIso (F\u2081.map f) \u2194 IsIso (F\u2082.map f)\nh : IsInvertedBy W F\u2082\nX Y : C\nf : X \u27f6 Y\nhf : W f\n\u22a2 IsIso (F\u2082.map f)\n[PROOFSTEP]\nexact h f hf\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nF\u2081 F\u2082 : C \u2964 D\ne : F\u2081 \u2245 F\u2082\n\u22a2 \u2200 (X Y : C) (f : X \u27f6 Y), IsIso (F\u2081.map f) \u2194 IsIso (F\u2082.map f)\n[PROOFSTEP]\nintro X Y f\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nF\u2081 F\u2082 : C \u2964 D\ne : F\u2081 \u2245 F\u2082\nX Y : C\nf : X \u27f6 Y\n\u22a2 IsIso (F\u2081.map f) \u2194 IsIso (F\u2082.map f)\n[PROOFSTEP]\nexact (RespectsIso.isomorphisms D).arrow_mk_iso_iff (Arrow.isoMk (e.app X) (e.app Y) (by simp))\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nW : MorphismProperty C\nF\u2081 F\u2082 : C \u2964 D\ne : F\u2081 \u2245 F\u2082\nX Y : C\nf : X \u27f6 Y\n\u22a2 (e.app X).hom \u226b (Arrow.mk (F\u2082.map f)).hom = (Arrow.mk (F\u2081.map f)).hom \u226b (e.app Y).hom\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.75953, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\n\u22a2 RespectsIso (MorphismProperty.diagonal P)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.75953, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\n\u22a2 \u2200 {X Y Z : C} (e : X \u2245 Y) (f : Y \u27f6 Z), MorphismProperty.diagonal P f \u2192 MorphismProperty.diagonal P (e.hom \u226b f)\n[PROOFSTEP]\nintrov H\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.75953, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\nH : MorphismProperty.diagonal P f\n\u22a2 MorphismProperty.diagonal P (e.hom \u226b f)\n[PROOFSTEP]\nrwa [diagonal_iff, pullback.diagonal_comp, hP.cancel_left_isIso, hP.cancel_left_isIso, \u2190\n hP.cancel_right_isIso _ (pullback.map (e.hom \u226b f) (e.hom \u226b f) f f e.hom e.hom (\ud835\udfd9 Z) (by simp) (by simp)), \u2190\n pullback.condition, hP.cancel_left_isIso]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.75953, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\nH : MorphismProperty.diagonal P f\n\u22a2 (e.hom \u226b f) \u226b \ud835\udfd9 Z = e.hom \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.75953, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\nH : MorphismProperty.diagonal P f\n\u22a2 (e.hom \u226b f) \u226b \ud835\udfd9 Z = e.hom \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.75953, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\n\u22a2 \u2200 {X Y Z : C} (e : Y \u2245 Z) (f : X \u27f6 Y), MorphismProperty.diagonal P f \u2192 MorphismProperty.diagonal P (f \u226b e.hom)\n[PROOFSTEP]\nintrov H\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.75953, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\nX Y Z : C\ne : Y \u2245 Z\nf : X \u27f6 Y\nH : MorphismProperty.diagonal P f\n\u22a2 MorphismProperty.diagonal P (f \u226b e.hom)\n[PROOFSTEP]\ndelta diagonal\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.75953, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : RespectsIso P\nX Y Z : C\ne : Y \u2245 Z\nf : X \u27f6 Y\nH : MorphismProperty.diagonal P f\n\u22a2 P (pullback.diagonal (f \u226b e.hom))\n[PROOFSTEP]\nrwa [pullback.diagonal_comp, hP.cancel_right_isIso]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.83850, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nhP' : RespectsIso P\nhP'' : StableUnderBaseChange P\n\u22a2 StableUnderComposition (MorphismProperty.diagonal P)\n[PROOFSTEP]\nintrov X h\u2081 h\u2082\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.83850, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nhP' : RespectsIso P\nhP'' : StableUnderBaseChange P\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nh\u2081 : MorphismProperty.diagonal P f\nh\u2082 : MorphismProperty.diagonal P g\n\u22a2 MorphismProperty.diagonal P (f \u226b g)\n[PROOFSTEP]\nrw [diagonal_iff, pullback.diagonal_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.83850, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nhP' : RespectsIso P\nhP'' : StableUnderBaseChange P\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nh\u2081 : MorphismProperty.diagonal P f\nh\u2082 : MorphismProperty.diagonal P g\n\u22a2 P (pullback.diagonal f \u226b (pullbackDiagonalMapIdIso f f g).inv \u226b pullback.snd)\n[PROOFSTEP]\nexact hP _ _ h\u2081 (by simpa [hP'.cancel_left_isIso] using hP''.snd _ _ h\u2082)\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.83850, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nhP' : RespectsIso P\nhP'' : StableUnderBaseChange P\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nh\u2081 : MorphismProperty.diagonal P f\nh\u2082 : MorphismProperty.diagonal P g\n\u22a2 P ((pullbackDiagonalMapIdIso f f g).inv \u226b pullback.snd)\n[PROOFSTEP]\nsimpa [hP'.cancel_left_isIso] using hP''.snd _ _ h\u2082\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.86028, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\n\u22a2 \u2200 (X Y S : C) (f : X \u27f6 S) (g : Y \u27f6 S), MorphismProperty.diagonal P g \u2192 MorphismProperty.diagonal P pullback.fst\n[PROOFSTEP]\nintrov h\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.86028, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nX Y S : C\nf : X \u27f6 S\ng : Y \u27f6 S\nh : MorphismProperty.diagonal P g\n\u22a2 MorphismProperty.diagonal P pullback.fst\n[PROOFSTEP]\nrw [diagonal_iff, diagonal_pullback_fst, hP'.cancel_left_isIso, hP'.cancel_right_isIso]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.86028, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nX Y S : C\nf : X \u27f6 S\ng : Y \u27f6 S\nh : MorphismProperty.diagonal P g\n\u22a2 P ((baseChange f).map (Over.homMk (pullback.diagonal g))).left\n[PROOFSTEP]\nexact hP.baseChange_map f _ (by simpa)\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.86028, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nX Y S : C\nf : X \u27f6 S\ng : Y \u27f6 S\nh : MorphismProperty.diagonal P g\n\u22a2 P (Over.homMk (pullback.diagonal g)).left\n[PROOFSTEP]\nsimpa\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.88038, u_1} D\nP : MorphismProperty C\n\u22a2 RespectsIso (universally P)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.88038, u_1} D\nP : MorphismProperty C\n\u22a2 \u2200 {X Y Z : C} (e : X \u2245 Y) (f : Y \u27f6 Z), universally P f \u2192 universally P (e.hom \u226b f)\n[PROOFSTEP]\nintro X Y Z e f hf X' Z' i\u2081 i\u2082 f' H\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.88038, u_1} D\nP : MorphismProperty C\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\nhf : universally P f\nX' Z' : C\ni\u2081 : X' \u27f6 X\ni\u2082 : Z' \u27f6 Z\nf' : X' \u27f6 Z'\nH : IsPullback f' i\u2081 i\u2082 (e.hom \u226b f)\n\u22a2 P f'\n[PROOFSTEP]\nhave : IsPullback (\ud835\udfd9 _) (i\u2081 \u226b e.hom) i\u2081 e.inv :=\n IsPullback.of_horiz_isIso \u27e8by rw [Category.id_comp, Category.assoc, e.hom_inv_id, Category.comp_id]\u27e9\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.88038, u_1} D\nP : MorphismProperty C\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\nhf : universally P f\nX' Z' : C\ni\u2081 : X' \u27f6 X\ni\u2082 : Z' \u27f6 Z\nf' : X' \u27f6 Z'\nH : IsPullback f' i\u2081 i\u2082 (e.hom \u226b f)\n\u22a2 \ud835\udfd9 X' \u226b i\u2081 = (i\u2081 \u226b e.hom) \u226b e.inv\n[PROOFSTEP]\nrw [Category.id_comp, Category.assoc, e.hom_inv_id, Category.comp_id]\n[GOAL]\ncase left\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.88038, u_1} D\nP : MorphismProperty C\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\nhf : universally P f\nX' Z' : C\ni\u2081 : X' \u27f6 X\ni\u2082 : Z' \u27f6 Z\nf' : X' \u27f6 Z'\nH : IsPullback f' i\u2081 i\u2082 (e.hom \u226b f)\nthis : IsPullback (\ud835\udfd9 X') (i\u2081 \u226b e.hom) i\u2081 e.inv\n\u22a2 P f'\n[PROOFSTEP]\nexact hf _ _ _ (by simpa only [Iso.inv_hom_id_assoc, Category.id_comp] using this.paste_horiz H)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.88038, u_1} D\nP : MorphismProperty C\nX Y Z : C\ne : X \u2245 Y\nf : Y \u27f6 Z\nhf : universally P f\nX' Z' : C\ni\u2081 : X' \u27f6 X\ni\u2082 : Z' \u27f6 Z\nf' : X' \u27f6 Z'\nH : IsPullback f' i\u2081 i\u2082 (e.hom \u226b f)\nthis : IsPullback (\ud835\udfd9 X') (i\u2081 \u226b e.hom) i\u2081 e.inv\n\u22a2 IsPullback f' ?m.90182 ?m.90183 f\n[PROOFSTEP]\nsimpa only [Iso.inv_hom_id_assoc, Category.id_comp] using this.paste_horiz H\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.88038, u_1} D\nP : MorphismProperty C\n\u22a2 \u2200 {X Y Z : C} (e : Y \u2245 Z) (f : X \u27f6 Y), universally P f \u2192 universally P (f \u226b e.hom)\n[PROOFSTEP]\nintro X Y Z e f hf X' Z' i\u2081 i\u2082 f' H\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.88038, u_1} D\nP : MorphismProperty C\nX Y Z : C\ne : Y \u2245 Z\nf : X \u27f6 Y\nhf : universally P f\nX' Z' : C\ni\u2081 : X' \u27f6 X\ni\u2082 : Z' \u27f6 Z\nf' : X' \u27f6 Z'\nH : IsPullback f' i\u2081 i\u2082 (f \u226b e.hom)\n\u22a2 P f'\n[PROOFSTEP]\nhave : IsPullback (\ud835\udfd9 _) i\u2082 (i\u2082 \u226b e.inv) e.inv := IsPullback.of_horiz_isIso \u27e8Category.id_comp _\u27e9\n[GOAL]\ncase right\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.88038, u_1} D\nP : MorphismProperty C\nX Y Z : C\ne : Y \u2245 Z\nf : X \u27f6 Y\nhf : universally P f\nX' Z' : C\ni\u2081 : X' \u27f6 X\ni\u2082 : Z' \u27f6 Z\nf' : X' \u27f6 Z'\nH : IsPullback f' i\u2081 i\u2082 (f \u226b e.hom)\nthis : IsPullback (\ud835\udfd9 Z') i\u2082 (i\u2082 \u226b e.inv) e.inv\n\u22a2 P f'\n[PROOFSTEP]\nexact\n hf _ _ _ (by simpa only [Category.assoc, Iso.hom_inv_id, Category.comp_id, Category.comp_id] using H.paste_horiz this)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.88038, u_1} D\nP : MorphismProperty C\nX Y Z : C\ne : Y \u2245 Z\nf : X \u27f6 Y\nhf : universally P f\nX' Z' : C\ni\u2081 : X' \u27f6 X\ni\u2082 : Z' \u27f6 Z\nf' : X' \u27f6 Z'\nH : IsPullback f' i\u2081 i\u2082 (f \u226b e.hom)\nthis : IsPullback (\ud835\udfd9 Z') i\u2082 (i\u2082 \u226b e.inv) e.inv\n\u22a2 IsPullback f' ?m.91717 ?m.91718 f\n[PROOFSTEP]\nsimpa only [Category.assoc, Iso.hom_inv_id, Category.comp_id, Category.comp_id] using H.paste_horiz this\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.92183, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\n\u22a2 StableUnderComposition (MorphismProperty.universally P)\n[PROOFSTEP]\nintro X Y Z f g hf hg X' Z' i\u2081 i\u2082 f' H\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.92183, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : MorphismProperty.universally P f\nhg : MorphismProperty.universally P g\nX' Z' : C\ni\u2081 : X' \u27f6 X\ni\u2082 : Z' \u27f6 Z\nf' : X' \u27f6 Z'\nH : IsPullback f' i\u2081 i\u2082 (f \u226b g)\n\u22a2 P f'\n[PROOFSTEP]\nhave := pullback.lift_fst _ _ (H.w.trans (Category.assoc _ _ _).symm)\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.92183, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : MorphismProperty.universally P f\nhg : MorphismProperty.universally P g\nX' Z' : C\ni\u2081 : X' \u27f6 X\ni\u2082 : Z' \u27f6 Z\nf' : X' \u27f6 Z'\nH : IsPullback f' i\u2081 i\u2082 (f \u226b g)\nthis : pullback.lift f' (i\u2081 \u226b f) (_ : f' \u226b i\u2082 = (i\u2081 \u226b f) \u226b g) \u226b pullback.fst = f'\n\u22a2 P f'\n[PROOFSTEP]\nrw [\u2190 this] at H \u22a2\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.92183, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : MorphismProperty.universally P f\nhg : MorphismProperty.universally P g\nX' Z' : C\ni\u2081 : X' \u27f6 X\ni\u2082 : Z' \u27f6 Z\nf' : X' \u27f6 Z'\nH\u271d : IsPullback f' i\u2081 i\u2082 (f \u226b g)\nH : IsPullback (pullback.lift f' (i\u2081 \u226b f) (_ : f' \u226b i\u2082 = (i\u2081 \u226b f) \u226b g) \u226b pullback.fst) i\u2081 i\u2082 (f \u226b g)\nthis : pullback.lift f' (i\u2081 \u226b f) (_ : f' \u226b i\u2082 = (i\u2081 \u226b f) \u226b g) \u226b pullback.fst = f'\n\u22a2 P (pullback.lift f' (i\u2081 \u226b f) (_ : f' \u226b i\u2082 = (i\u2081 \u226b f) \u226b g) \u226b pullback.fst)\n[PROOFSTEP]\napply hP _ _ _ (hg _ _ _ <| IsPullback.of_hasPullback _ _)\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.92183, u_1} D\ninst\u271d : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : MorphismProperty.universally P f\nhg : MorphismProperty.universally P g\nX' Z' : C\ni\u2081 : X' \u27f6 X\ni\u2082 : Z' \u27f6 Z\nf' : X' \u27f6 Z'\nH\u271d : IsPullback f' i\u2081 i\u2082 (f \u226b g)\nH : IsPullback (pullback.lift f' (i\u2081 \u226b f) (_ : f' \u226b i\u2082 = (i\u2081 \u226b f) \u226b g) \u226b pullback.fst) i\u2081 i\u2082 (f \u226b g)\nthis : pullback.lift f' (i\u2081 \u226b f) (_ : f' \u226b i\u2082 = (i\u2081 \u226b f) \u226b g) \u226b pullback.fst = f'\n\u22a2 P (pullback.lift f' (i\u2081 \u226b f) (_ : f' \u226b i\u2082 = (i\u2081 \u226b f) \u226b g))\n[PROOFSTEP]\nexact hf _ _ _ (H.of_right (pullback.lift_snd _ _ _) (IsPullback.of_hasPullback i\u2082 g))\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.93105, u_1} D\nP : MorphismProperty C\n\u22a2 universally P \u2264 P\n[PROOFSTEP]\nintro X Y f hf\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.93105, u_1} D\nP : MorphismProperty C\nX Y : C\nf : X \u27f6 Y\nhf : universally P f\n\u22a2 P f\n[PROOFSTEP]\nexact hf (\ud835\udfd9 _) (\ud835\udfd9 _) _ (IsPullback.of_vert_isIso \u27e8by rw [Category.comp_id, Category.id_comp]\u27e9)\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{?u.93105, u_1} D\nP : MorphismProperty C\nX Y : C\nf : X \u27f6 Y\nhf : universally P f\n\u22a2 f \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f\n[PROOFSTEP]\nrw [Category.comp_id, Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.95780, u_1} D\ninst\u271d : ConcreteCategory C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : MorphismProperty.injective C f\nhg : MorphismProperty.injective C g\n\u22a2 MorphismProperty.injective C (f \u226b g)\n[PROOFSTEP]\ndelta MorphismProperty.injective\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.95780, u_1} D\ninst\u271d : ConcreteCategory C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : MorphismProperty.injective C f\nhg : MorphismProperty.injective C g\n\u22a2 Injective \u2191(f \u226b g)\n[PROOFSTEP]\nrw [coe_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.95780, u_1} D\ninst\u271d : ConcreteCategory C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : MorphismProperty.injective C f\nhg : MorphismProperty.injective C g\n\u22a2 Injective (\u2191g \u2218 \u2191f)\n[PROOFSTEP]\nexact hg.comp hf\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.96246, u_1} D\ninst\u271d : ConcreteCategory C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : MorphismProperty.surjective C f\nhg : MorphismProperty.surjective C g\n\u22a2 MorphismProperty.surjective C (f \u226b g)\n[PROOFSTEP]\ndelta MorphismProperty.surjective\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.96246, u_1} D\ninst\u271d : ConcreteCategory C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : MorphismProperty.surjective C f\nhg : MorphismProperty.surjective C g\n\u22a2 Surjective \u2191(f \u226b g)\n[PROOFSTEP]\nrw [coe_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.96246, u_1} D\ninst\u271d : ConcreteCategory C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : MorphismProperty.surjective C f\nhg : MorphismProperty.surjective C g\n\u22a2 Surjective (\u2191g \u2218 \u2191f)\n[PROOFSTEP]\nexact hg.comp hf\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.96710, u_1} D\ninst\u271d : ConcreteCategory C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : MorphismProperty.bijective C f\nhg : MorphismProperty.bijective C g\n\u22a2 MorphismProperty.bijective C (f \u226b g)\n[PROOFSTEP]\ndelta MorphismProperty.bijective\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.96710, u_1} D\ninst\u271d : ConcreteCategory C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : MorphismProperty.bijective C f\nhg : MorphismProperty.bijective C g\n\u22a2 Bijective \u2191(f \u226b g)\n[PROOFSTEP]\nrw [coe_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{?u.96710, u_1} D\ninst\u271d : ConcreteCategory C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : MorphismProperty.bijective C f\nhg : MorphismProperty.bijective C g\n\u22a2 Bijective (\u2191g \u2218 \u2191f)\n[PROOFSTEP]\nexact hg.comp hf\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.MorphismProperty", "llama_tokens": 34957, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.39981165504266236, "lm_q2_score": 0.02228618614801742, "lm_q1q2_score": 0.0089102769684277}} {"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX : C\n\u22a2 PartialOrder (Subobject X)\n[PROOFSTEP]\ndsimp only [Subobject]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX : C\n\u22a2 PartialOrder (ThinSkeleton (MonoOver X))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX : C\np : Subobject X \u2192 Prop\nh : \u2200 \u2983A : C\u2984 (f : A \u27f6 X) [inst : Mono f], p (mk f)\nP : Subobject X\n\u22a2 p P\n[PROOFSTEP]\napply Quotient.inductionOn'\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX : C\np : Subobject X \u2192 Prop\nh : \u2200 \u2983A : C\u2984 (f : A \u27f6 X) [inst : Mono f], p (mk f)\nP : Subobject X\n\u22a2 \u2200 (a : MonoOver X), p (Quotient.mk'' a)\n[PROOFSTEP]\nintro a\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX : C\np : Subobject X \u2192 Prop\nh : \u2200 \u2983A : C\u2984 (f : A \u27f6 X) [inst : Mono f], p (mk f)\nP : Subobject X\na : MonoOver X\n\u22a2 p (Quotient.mk'' a)\n[PROOFSTEP]\nexact h a.arrow\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX : C\np : Subobject X \u2192 Subobject X \u2192 Prop\nh : \u2200 \u2983A B : C\u2984 (f : A \u27f6 X) (g : B \u27f6 X) [inst : Mono f] [inst_1 : Mono g], p (mk f) (mk g)\nP Q : Subobject X\n\u22a2 p P Q\n[PROOFSTEP]\napply Quotient.inductionOn\u2082'\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX : C\np : Subobject X \u2192 Subobject X \u2192 Prop\nh : \u2200 \u2983A B : C\u2984 (f : A \u27f6 X) (g : B \u27f6 X) [inst : Mono f] [inst_1 : Mono g], p (mk f) (mk g)\nP Q : Subobject X\n\u22a2 \u2200 (a\u2081 a\u2082 : MonoOver X), p (Quotient.mk'' a\u2081) (Quotient.mk'' a\u2082)\n[PROOFSTEP]\nintro a b\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX : C\np : Subobject X \u2192 Subobject X \u2192 Prop\nh : \u2200 \u2983A B : C\u2984 (f : A \u27f6 X) (g : B \u27f6 X) [inst : Mono f] [inst_1 : Mono g], p (mk f) (mk g)\nP Q : Subobject X\na b : MonoOver X\n\u22a2 p (Quotient.mk'' a) (Quotient.mk'' b)\n[PROOFSTEP]\nexact h a.arrow b.arrow\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nA : C\nX Y : Subobject A\nh : X = Y\n\u22a2 eqToHom (_ : underlying.obj X = underlying.obj Y) \u226b arrow Y = arrow X\n[PROOFSTEP]\ninduction h\n[GOAL]\ncase refl\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nA : C\nX Y : Subobject A\n\u22a2 eqToHom (_ : underlying.obj X = underlying.obj X) \u226b arrow X = arrow X\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nP : Subobject X\nQ : MonoOver X\n\u22a2 mk (arrow (Quotient.mk'' Q)) = Quotient.mk'' Q\n[PROOFSTEP]\nobtain \u27e8e\u27e9 := @Quotient.mk_out' _ (isIsomorphicSetoid _) Q\n[GOAL]\ncase intro\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nP : Subobject X\nQ : MonoOver X\ne : Quotient.out' (Quotient.mk'' Q) \u2245 Q\n\u22a2 mk (arrow (Quotient.mk'' Q)) = Quotient.mk'' Q\n[PROOFSTEP]\nexact Quotient.sound' \u27e8MonoOver.isoMk (Iso.refl _) \u226a\u226b e\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nB : C\nX Y : Subobject B\nf : underlying.obj X \u27f6 underlying.obj Y\nw : f \u226b arrow Y = arrow X\n\u22a2 X \u2264 Y\n[PROOFSTEP]\nconvert mk_le_mk_of_comm _ w\n[GOAL]\ncase h.e'_3\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nB : C\nX Y : Subobject B\nf : underlying.obj X \u27f6 underlying.obj Y\nw : f \u226b arrow Y = arrow X\n\u22a2 X = mk (arrow X)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_4\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nB : C\nX Y : Subobject B\nf : underlying.obj X \u27f6 underlying.obj Y\nw : f \u226b arrow Y = arrow X\n\u22a2 Y = mk (arrow Y)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A : C\nX : Subobject B\nf : A \u27f6 B\ninst\u271d : Mono f\ng : underlying.obj X \u27f6 A\nw : g \u226b f = arrow X\n\u22a2 (g \u226b (underlyingIso f).inv) \u226b arrow (mk f) = arrow X\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A : C\nX : Subobject B\nf : A \u27f6 B\ninst\u271d : Mono f\ng : A \u27f6 underlying.obj X\nw : g \u226b arrow X = f\n\u22a2 ((underlyingIso f).hom \u226b g) \u226b arrow X = arrow (mk f)\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A : C\nX : Subobject B\nf : A \u27f6 B\ninst\u271d : Mono f\ni : underlying.obj X \u2245 A\nw : i.hom \u226b f = arrow X\n\u22a2 (i \u226a\u226b (underlyingIso f).symm).hom \u226b arrow (mk f) = arrow X\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A : C\nX : Subobject B\nf : A \u27f6 B\ninst\u271d : Mono f\ni : A \u2245 underlying.obj X\nw : i.hom \u226b arrow X = f\n\u22a2 i.symm.hom \u226b f = arrow X\n[PROOFSTEP]\nrw [Iso.symm_hom, Iso.inv_comp_eq, w]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nB A\u2081 A\u2082 : C\nf : A\u2081 \u27f6 B\ng : A\u2082 \u27f6 B\ninst\u271d\u00b9 : Mono f\ninst\u271d : Mono g\ni : A\u2081 \u2245 A\u2082\nw : i.hom \u226b g = f\n\u22a2 (underlyingIso f \u226a\u226b i).hom \u226b g = arrow (mk f)\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nB : C\nX Y : Subobject B\nh : X \u2264 Y\n\u22a2 Mono (ofLE X Y h)\n[PROOFSTEP]\nfconstructor\n[GOAL]\ncase right_cancellation\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nB : C\nX Y : Subobject B\nh : X \u2264 Y\n\u22a2 \u2200 {Z : C} (g h_1 : Z \u27f6 underlying.obj X), g \u226b ofLE X Y h = h_1 \u226b ofLE X Y h \u2192 g = h_1\n[PROOFSTEP]\nintro Z f g w\n[GOAL]\ncase right_cancellation\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z\u271d : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nB : C\nX Y : Subobject B\nh : X \u2264 Y\nZ : C\nf g : Z \u27f6 underlying.obj X\nw : f \u226b ofLE X Y h = g \u226b ofLE X Y h\n\u22a2 f = g\n[PROOFSTEP]\nreplace w := w =\u226b Y.arrow\n[GOAL]\ncase right_cancellation\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z\u271d : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nB : C\nX Y : Subobject B\nh : X \u2264 Y\nZ : C\nf g : Z \u27f6 underlying.obj X\nw : (f \u226b ofLE X Y h) \u226b arrow Y = (g \u226b ofLE X Y h) \u226b arrow Y\n\u22a2 f = g\n[PROOFSTEP]\next\n[GOAL]\ncase right_cancellation.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z\u271d : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nB : C\nX Y : Subobject B\nh : X \u2264 Y\nZ : C\nf g : Z \u27f6 underlying.obj X\nw : (f \u226b ofLE X Y h) \u226b arrow Y = (g \u226b ofLE X Y h) \u226b arrow Y\n\u22a2 f \u226b arrow X = g \u226b arrow X\n[PROOFSTEP]\nsimpa using w\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nB A\u2081 A\u2082 : C\nf\u2081 : A\u2081 \u27f6 B\nf\u2082 : A\u2082 \u27f6 B\ninst\u271d\u00b9 : Mono f\u2081\ninst\u271d : Mono f\u2082\ng : A\u2081 \u27f6 A\u2082\nw : g \u226b f\u2082 = f\u2081\n\u22a2 ofLE (mk f\u2081) (mk f\u2082) (_ : mk f\u2081 \u2264 mk f\u2082) = (underlyingIso f\u2081).hom \u226b g \u226b (underlyingIso f\u2082).inv\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nB A\u2081 A\u2082 : C\nf\u2081 : A\u2081 \u27f6 B\nf\u2082 : A\u2082 \u27f6 B\ninst\u271d\u00b9 : Mono f\u2081\ninst\u271d : Mono f\u2082\ng : A\u2081 \u27f6 A\u2082\nw : g \u226b f\u2082 = f\u2081\n\u22a2 ofLE (mk f\u2081) (mk f\u2082) (_ : mk f\u2081 \u2264 mk f\u2082) \u226b arrow (mk f\u2082) =\n ((underlyingIso f\u2081).hom \u226b g \u226b (underlyingIso f\u2082).inv) \u226b arrow (mk f\u2082)\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A : C\nX : Subobject B\nf : A \u27f6 B\ninst\u271d : Mono f\nh : X \u2264 mk f\n\u22a2 Mono (ofLEMk X f h)\n[PROOFSTEP]\ndsimp only [ofLEMk]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A : C\nX : Subobject B\nf : A \u27f6 B\ninst\u271d : Mono f\nh : X \u2264 mk f\n\u22a2 Mono (ofLE X (mk f) h \u226b (underlyingIso f).hom)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A : C\nX : Subobject B\nf : A \u27f6 B\ninst\u271d : Mono f\nh : X \u2264 mk f\n\u22a2 ofLEMk X f h \u226b f = arrow X\n[PROOFSTEP]\nsimp [ofLEMk]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A : C\nf : A \u27f6 B\ninst\u271d : Mono f\nX : Subobject B\nh : mk f \u2264 X\n\u22a2 Mono (ofMkLE f X h)\n[PROOFSTEP]\ndsimp only [ofMkLE]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A : C\nf : A \u27f6 B\ninst\u271d : Mono f\nX : Subobject B\nh : mk f \u2264 X\n\u22a2 Mono ((underlyingIso f).inv \u226b ofLE (mk f) X h)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A : C\nf : A \u27f6 B\ninst\u271d : Mono f\nX : Subobject B\nh : mk f \u2264 X\n\u22a2 ofMkLE f X h \u226b arrow X = f\n[PROOFSTEP]\nsimp [ofMkLE]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nB A\u2081 A\u2082 : C\nf : A\u2081 \u27f6 B\ng : A\u2082 \u27f6 B\ninst\u271d\u00b9 : Mono f\ninst\u271d : Mono g\nh : mk f \u2264 mk g\n\u22a2 Mono (ofMkLEMk f g h)\n[PROOFSTEP]\ndsimp only [ofMkLEMk]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nB A\u2081 A\u2082 : C\nf : A\u2081 \u27f6 B\ng : A\u2082 \u27f6 B\ninst\u271d\u00b9 : Mono f\ninst\u271d : Mono g\nh : mk f \u2264 mk g\n\u22a2 Mono ((underlyingIso f).inv \u226b ofLE (mk f) (mk g) h \u226b (underlyingIso g).hom)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nB A\u2081 A\u2082 : C\nf : A\u2081 \u27f6 B\ng : A\u2082 \u27f6 B\ninst\u271d\u00b9 : Mono f\ninst\u271d : Mono g\nh : mk f \u2264 mk g\n\u22a2 ofMkLEMk f g h \u226b g = f\n[PROOFSTEP]\nsimp [ofMkLEMk]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z\u271d : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nB : C\nX Y Z : Subobject B\nh\u2081 : X \u2264 Y\nh\u2082 : Y \u2264 Z\n\u22a2 ofLE X Y h\u2081 \u226b ofLE Y Z h\u2082 = ofLE X Z (_ : X \u2264 Z)\n[PROOFSTEP]\nsimp only [ofLE, \u2190 Functor.map_comp underlying]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z\u271d : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nB : C\nX Y Z : Subobject B\nh\u2081 : X \u2264 Y\nh\u2082 : Y \u2264 Z\n\u22a2 underlying.map (LE.le.hom h\u2081 \u226b LE.le.hom h\u2082) = underlying.map (LE.le.hom (_ : X \u2264 Z))\n[PROOFSTEP]\ncongr 1\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A : C\nX Y : Subobject B\nf : A \u27f6 B\ninst\u271d : Mono f\nh\u2081 : X \u2264 Y\nh\u2082 : Y \u2264 mk f\n\u22a2 ofLE X Y h\u2081 \u226b ofLEMk Y f h\u2082 = ofLEMk X f (_ : X \u2264 mk f)\n[PROOFSTEP]\nsimp only [ofMkLE, ofLEMk, ofLE, \u2190 Functor.map_comp_assoc underlying]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A : C\nX Y : Subobject B\nf : A \u27f6 B\ninst\u271d : Mono f\nh\u2081 : X \u2264 Y\nh\u2082 : Y \u2264 mk f\n\u22a2 underlying.map (LE.le.hom h\u2081 \u226b LE.le.hom h\u2082) \u226b (underlyingIso f).hom =\n underlying.map (LE.le.hom (_ : X \u2264 mk f)) \u226b (underlyingIso f).hom\n[PROOFSTEP]\ncongr 1\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A : C\nX : Subobject B\nf : A \u27f6 B\ninst\u271d : Mono f\nY : Subobject B\nh\u2081 : X \u2264 mk f\nh\u2082 : mk f \u2264 Y\n\u22a2 ofLEMk X f h\u2081 \u226b ofMkLE f Y h\u2082 = ofLE X Y (_ : X \u2264 Y)\n[PROOFSTEP]\nsimp only [ofMkLE, ofLEMk, ofLE, \u2190 Functor.map_comp underlying, assoc, Iso.hom_inv_id_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A : C\nX : Subobject B\nf : A \u27f6 B\ninst\u271d : Mono f\nY : Subobject B\nh\u2081 : X \u2264 mk f\nh\u2082 : mk f \u2264 Y\n\u22a2 underlying.map (LE.le.hom h\u2081 \u226b LE.le.hom h\u2082) = underlying.map (LE.le.hom (_ : X \u2264 Y))\n[PROOFSTEP]\ncongr 1\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nB A\u2081 A\u2082 : C\nX : Subobject B\nf : A\u2081 \u27f6 B\ninst\u271d\u00b9 : Mono f\ng : A\u2082 \u27f6 B\ninst\u271d : Mono g\nh\u2081 : X \u2264 mk f\nh\u2082 : mk f \u2264 mk g\n\u22a2 ofLEMk X f h\u2081 \u226b ofMkLEMk f g h\u2082 = ofLEMk X g (_ : X \u2264 mk g)\n[PROOFSTEP]\nsimp only [ofMkLE, ofLEMk, ofLE, ofMkLEMk, \u2190 Functor.map_comp_assoc underlying, assoc, Iso.hom_inv_id_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nB A\u2081 A\u2082 : C\nX : Subobject B\nf : A\u2081 \u27f6 B\ninst\u271d\u00b9 : Mono f\ng : A\u2082 \u27f6 B\ninst\u271d : Mono g\nh\u2081 : X \u2264 mk f\nh\u2082 : mk f \u2264 mk g\n\u22a2 underlying.map (LE.le.hom h\u2081 \u226b LE.le.hom h\u2082) \u226b (underlyingIso g).hom =\n underlying.map (LE.le.hom (_ : X \u2264 mk g)) \u226b (underlyingIso g).hom\n[PROOFSTEP]\ncongr 1\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A\u2081 : C\nf : A\u2081 \u27f6 B\ninst\u271d : Mono f\nX Y : Subobject B\nh\u2081 : mk f \u2264 X\nh\u2082 : X \u2264 Y\n\u22a2 ofMkLE f X h\u2081 \u226b ofLE X Y h\u2082 = ofMkLE f Y (_ : mk f \u2264 Y)\n[PROOFSTEP]\nsimp only [ofMkLE, ofLEMk, ofLE, ofMkLEMk, \u2190 Functor.map_comp underlying, assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A\u2081 : C\nf : A\u2081 \u27f6 B\ninst\u271d : Mono f\nX Y : Subobject B\nh\u2081 : mk f \u2264 X\nh\u2082 : X \u2264 Y\n\u22a2 (underlyingIso f).inv \u226b underlying.map (LE.le.hom h\u2081 \u226b LE.le.hom h\u2082) =\n (underlyingIso f).inv \u226b underlying.map (LE.le.hom (_ : mk f \u2264 Y))\n[PROOFSTEP]\ncongr 1\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nB A\u2081 A\u2082 : C\nf : A\u2081 \u27f6 B\ninst\u271d\u00b9 : Mono f\nX : Subobject B\ng : A\u2082 \u27f6 B\ninst\u271d : Mono g\nh\u2081 : mk f \u2264 X\nh\u2082 : X \u2264 mk g\n\u22a2 ofMkLE f X h\u2081 \u226b ofLEMk X g h\u2082 = ofMkLEMk f g (_ : mk f \u2264 mk g)\n[PROOFSTEP]\nsimp only [ofMkLE, ofLEMk, ofLE, ofMkLEMk, \u2190 Functor.map_comp_assoc underlying, assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nB A\u2081 A\u2082 : C\nf : A\u2081 \u27f6 B\ninst\u271d\u00b9 : Mono f\nX : Subobject B\ng : A\u2082 \u27f6 B\ninst\u271d : Mono g\nh\u2081 : mk f \u2264 X\nh\u2082 : X \u2264 mk g\n\u22a2 (underlyingIso f).inv \u226b underlying.map (LE.le.hom h\u2081 \u226b LE.le.hom h\u2082) \u226b (underlyingIso g).hom =\n (underlyingIso f).inv \u226b underlying.map (LE.le.hom (_ : mk f \u2264 mk g)) \u226b (underlyingIso g).hom\n[PROOFSTEP]\ncongr 1\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nB A\u2081 A\u2082 : C\nf : A\u2081 \u27f6 B\ninst\u271d\u00b9 : Mono f\ng : A\u2082 \u27f6 B\ninst\u271d : Mono g\nX : Subobject B\nh\u2081 : mk f \u2264 mk g\nh\u2082 : mk g \u2264 X\n\u22a2 ofMkLEMk f g h\u2081 \u226b ofMkLE g X h\u2082 = ofMkLE f X (_ : mk f \u2264 X)\n[PROOFSTEP]\nsimp only [ofMkLE, ofLEMk, ofLE, ofMkLEMk, \u2190 Functor.map_comp underlying, assoc, Iso.hom_inv_id_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nB A\u2081 A\u2082 : C\nf : A\u2081 \u27f6 B\ninst\u271d\u00b9 : Mono f\ng : A\u2082 \u27f6 B\ninst\u271d : Mono g\nX : Subobject B\nh\u2081 : mk f \u2264 mk g\nh\u2082 : mk g \u2264 X\n\u22a2 (underlyingIso f).inv \u226b underlying.map (LE.le.hom h\u2081 \u226b LE.le.hom h\u2082) =\n (underlyingIso f).inv \u226b underlying.map (LE.le.hom (_ : mk f \u2264 X))\n[PROOFSTEP]\ncongr 1\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nB A\u2081 A\u2082 A\u2083 : C\nf : A\u2081 \u27f6 B\ninst\u271d\u00b2 : Mono f\ng : A\u2082 \u27f6 B\ninst\u271d\u00b9 : Mono g\nh : A\u2083 \u27f6 B\ninst\u271d : Mono h\nh\u2081 : mk f \u2264 mk g\nh\u2082 : mk g \u2264 mk h\n\u22a2 ofMkLEMk f g h\u2081 \u226b ofMkLEMk g h h\u2082 = ofMkLEMk f h (_ : mk f \u2264 mk h)\n[PROOFSTEP]\nsimp only [ofMkLE, ofLEMk, ofLE, ofMkLEMk, \u2190 Functor.map_comp_assoc underlying, assoc, Iso.hom_inv_id_assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nB A\u2081 A\u2082 A\u2083 : C\nf : A\u2081 \u27f6 B\ninst\u271d\u00b2 : Mono f\ng : A\u2082 \u27f6 B\ninst\u271d\u00b9 : Mono g\nh : A\u2083 \u27f6 B\ninst\u271d : Mono h\nh\u2081 : mk f \u2264 mk g\nh\u2082 : mk g \u2264 mk h\n\u22a2 (underlyingIso f).inv \u226b underlying.map (LE.le.hom h\u2081 \u226b LE.le.hom h\u2082) \u226b (underlyingIso h).hom =\n (underlyingIso f).inv \u226b underlying.map (LE.le.hom (_ : mk f \u2264 mk h)) \u226b (underlyingIso h).hom\n[PROOFSTEP]\ncongr 1\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nB : C\nX : Subobject B\n\u22a2 ofLE X X (_ : X \u2264 X) = \ud835\udfd9 (underlying.obj X)\n[PROOFSTEP]\napply (cancel_mono X.arrow).mp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nB : C\nX : Subobject B\n\u22a2 ofLE X X (_ : X \u2264 X) \u226b arrow X = \ud835\udfd9 (underlying.obj X) \u226b arrow X\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A\u2081 : C\nf : A\u2081 \u27f6 B\ninst\u271d : Mono f\n\u22a2 ofMkLEMk f f (_ : mk f \u2264 mk f) = \ud835\udfd9 A\u2081\n[PROOFSTEP]\napply (cancel_mono f).mp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A\u2081 : C\nf : A\u2081 \u27f6 B\ninst\u271d : Mono f\n\u22a2 ofMkLEMk f f (_ : mk f \u2264 mk f) \u226b f = \ud835\udfd9 A\u2081 \u226b f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nA : C\nB : D\ne : MonoOver A \u224c MonoOver B\n\u22a2 \ud835\udfed (Subobject A) \u2245 lower e.functor \u22d9 lower e.inverse\n[PROOFSTEP]\napply eqToIso\n[GOAL]\ncase p\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nA : C\nB : D\ne : MonoOver A \u224c MonoOver B\n\u22a2 \ud835\udfed (Subobject A) = lower e.functor \u22d9 lower e.inverse\n[PROOFSTEP]\nconvert ThinSkeleton.map_iso_eq e.unitIso\n[GOAL]\ncase h.e'_2\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nA : C\nB : D\ne : MonoOver A \u224c MonoOver B\n\u22a2 \ud835\udfed (Subobject A) = ThinSkeleton.map (\ud835\udfed (MonoOver A))\n[PROOFSTEP]\nexact ThinSkeleton.map_id_eq.symm\n[GOAL]\ncase h.e'_3\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nA : C\nB : D\ne : MonoOver A \u224c MonoOver B\n\u22a2 lower e.functor \u22d9 lower e.inverse = ThinSkeleton.map (e.functor \u22d9 e.inverse)\n[PROOFSTEP]\nexact (ThinSkeleton.map_comp_eq _ _).symm\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nA : C\nB : D\ne : MonoOver A \u224c MonoOver B\n\u22a2 lower e.inverse \u22d9 lower e.functor \u2245 \ud835\udfed (Subobject B)\n[PROOFSTEP]\napply eqToIso\n[GOAL]\ncase p\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nA : C\nB : D\ne : MonoOver A \u224c MonoOver B\n\u22a2 lower e.inverse \u22d9 lower e.functor = \ud835\udfed (Subobject B)\n[PROOFSTEP]\nconvert ThinSkeleton.map_iso_eq e.counitIso\n[GOAL]\ncase h.e'_2\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nA : C\nB : D\ne : MonoOver A \u224c MonoOver B\n\u22a2 lower e.inverse \u22d9 lower e.functor = ThinSkeleton.map (e.inverse \u22d9 e.functor)\n[PROOFSTEP]\nexact (ThinSkeleton.map_comp_eq _ _).symm\n[GOAL]\ncase h.e'_3\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nA : C\nB : D\ne : MonoOver A \u224c MonoOver B\n\u22a2 \ud835\udfed (Subobject B) = ThinSkeleton.map (\ud835\udfed (MonoOver B))\n[PROOFSTEP]\nexact ThinSkeleton.map_id_eq.symm\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasPullbacks C\nx : Subobject X\n\u22a2 (pullback (\ud835\udfd9 X)).obj x = x\n[PROOFSTEP]\ninduction' x using Quotient.inductionOn' with f\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasPullbacks C\nf : MonoOver X\n\u22a2 (pullback (\ud835\udfd9 X)).obj (Quotient.mk'' f) = Quotient.mk'' f\n[PROOFSTEP]\nexact Quotient.sound \u27e8MonoOver.pullbackId.app f\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasPullbacks C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nx : Subobject Z\n\u22a2 (pullback (f \u226b g)).obj x = (pullback f).obj ((pullback g).obj x)\n[PROOFSTEP]\ninduction' x using Quotient.inductionOn' with t\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasPullbacks C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nt : MonoOver Z\n\u22a2 (pullback (f \u226b g)).obj (Quotient.mk'' t) = (pullback f).obj ((pullback g).obj (Quotient.mk'' t))\n[PROOFSTEP]\nexact Quotient.sound \u27e8(MonoOver.pullbackComp _ _).app t\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nx : Subobject X\n\u22a2 (map (\ud835\udfd9 X)).obj x = x\n[PROOFSTEP]\ninduction' x using Quotient.inductionOn' with f\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nf : MonoOver X\n\u22a2 (map (\ud835\udfd9 X)).obj (Quotient.mk'' f) = Quotient.mk'' f\n[PROOFSTEP]\nexact Quotient.sound \u27e8MonoOver.mapId.app f\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : Mono f\ninst\u271d : Mono g\nx : Subobject X\n\u22a2 (map (f \u226b g)).obj x = (map g).obj ((map f).obj x)\n[PROOFSTEP]\ninduction' x using Quotient.inductionOn' with t\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b9 : Mono f\ninst\u271d : Mono g\nt : MonoOver X\n\u22a2 (map (f \u226b g)).obj (Quotient.mk'' t) = (map g).obj ((map f).obj (Quotient.mk'' t))\n[PROOFSTEP]\nexact Quotient.sound \u27e8(MonoOver.mapComp _ _).app t\u27e9\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : X \u2245 Y\ng : Subobject X\n\u22a2 (map e.inv).obj ((map e.hom).obj g) = g\n[PROOFSTEP]\nsimp_rw [\u2190 map_comp, e.hom_inv_id, map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : X \u2245 Y\ng : Subobject Y\n\u22a2 (map e.hom).obj ((map e.inv).obj g) = g\n[PROOFSTEP]\nsimp_rw [\u2190 map_comp, e.inv_hom_id, map_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : X \u2245 Y\nA B : Subobject X\n\u22a2 \u2191{ toFun := (map e.hom).toPrefunctor.obj, invFun := (map e.inv).toPrefunctor.obj,\n left_inv := (_ : \u2200 (g : Subobject X), (map e.inv).obj ((map e.hom).obj g) = g),\n right_inv := (_ : \u2200 (g : Subobject Y), (map e.hom).obj ((map e.inv).obj g) = g) }\n A \u2264\n \u2191{ toFun := (map e.hom).toPrefunctor.obj, invFun := (map e.inv).toPrefunctor.obj,\n left_inv := (_ : \u2200 (g : Subobject X), (map e.inv).obj ((map e.hom).obj g) = g),\n right_inv := (_ : \u2200 (g : Subobject Y), (map e.hom).obj ((map e.inv).obj g) = g) }\n B \u2194\n A \u2264 B\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : X \u2245 Y\nA B : Subobject X\n\u22a2 (map e.hom).obj A \u2264 (map e.hom).obj B \u2194 A \u2264 B\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : X \u2245 Y\nA B : Subobject X\n\u22a2 (map e.hom).obj A \u2264 (map e.hom).obj B \u2192 A \u2264 B\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : X \u2245 Y\nA B : Subobject X\nh : (map e.hom).obj A \u2264 (map e.hom).obj B\n\u22a2 A \u2264 B\n[PROOFSTEP]\napply_fun (map e.inv).obj at h \n[GOAL]\ncase mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : X \u2245 Y\nA B : Subobject X\nh : (map e.inv).obj ((map e.hom).obj A) \u2264 (map e.inv).obj ((map e.hom).obj B)\n\u22a2 A \u2264 B\n[PROOFSTEP]\nsimpa only [\u2190 map_comp, e.hom_inv_id, map_id] using h\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : X \u2245 Y\nA B : Subobject X\nh : (map e.hom).obj A \u2264 (map e.hom).obj B\n\u22a2 Monotone (map e.inv).toPrefunctor.obj\n[PROOFSTEP]\napply Functor.monotone\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : X \u2245 Y\nA B : Subobject X\n\u22a2 A \u2264 B \u2192 (map e.hom).obj A \u2264 (map e.hom).obj B\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : X \u2245 Y\nA B : Subobject X\nh : A \u2264 B\n\u22a2 (map e.hom).obj A \u2264 (map e.hom).obj B\n[PROOFSTEP]\napply_fun (map e.hom).obj at h \n[GOAL]\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : X \u2245 Y\nA B : Subobject X\nh : (map e.hom).obj A \u2264 (map e.hom).obj B\n\u22a2 (map e.hom).obj A \u2264 (map e.hom).obj B\n[PROOFSTEP]\nexact h\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\ne : X \u2245 Y\nA B : Subobject X\nh : A \u2264 B\n\u22a2 Monotone (map e.hom).toPrefunctor.obj\n[PROOFSTEP]\napply Functor.monotone\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : HasPullbacks C\nf : X \u27f6 Y\ninst\u271d : Mono f\ng : Subobject X\n\u22a2 (pullback f).obj ((map f).obj g) = g\n[PROOFSTEP]\nrevert g\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : HasPullbacks C\nf : X \u27f6 Y\ninst\u271d : Mono f\n\u22a2 \u2200 (g : Subobject X), (pullback f).obj ((map f).obj g) = g\n[PROOFSTEP]\nexact Quotient.ind (fun g' => Quotient.sound \u27e8(MonoOver.pullbackMapSelf f).app _\u27e9)\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : HasPullbacks C\nX Y Z W : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : Y \u27f6 W\nk : Z \u27f6 W\ninst\u271d\u00b9 : Mono h\ninst\u271d : Mono g\ncomm : f \u226b h = g \u226b k\nt : IsLimit (PullbackCone.mk f g comm)\np : Subobject Y\n\u22a2 (map g).obj ((pullback f).obj p) = (pullback k).obj ((map h).obj p)\n[PROOFSTEP]\nrevert p\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : HasPullbacks C\nX Y Z W : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : Y \u27f6 W\nk : Z \u27f6 W\ninst\u271d\u00b9 : Mono h\ninst\u271d : Mono g\ncomm : f \u226b h = g \u226b k\nt : IsLimit (PullbackCone.mk f g comm)\n\u22a2 \u2200 (p : Subobject Y), (map g).obj ((pullback f).obj p) = (pullback k).obj ((map h).obj p)\n[PROOFSTEP]\napply Quotient.ind'\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : HasPullbacks C\nX Y Z W : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : Y \u27f6 W\nk : Z \u27f6 W\ninst\u271d\u00b9 : Mono h\ninst\u271d : Mono g\ncomm : f \u226b h = g \u226b k\nt : IsLimit (PullbackCone.mk f g comm)\n\u22a2 \u2200 (a : MonoOver Y),\n (map g).obj ((pullback f).obj (Quotient.mk'' a)) = (pullback k).obj ((map h).obj (Quotient.mk'' a))\n[PROOFSTEP]\nintro a\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : HasPullbacks C\nX Y Z W : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : Y \u27f6 W\nk : Z \u27f6 W\ninst\u271d\u00b9 : Mono h\ninst\u271d : Mono g\ncomm : f \u226b h = g \u226b k\nt : IsLimit (PullbackCone.mk f g comm)\na : MonoOver Y\n\u22a2 (map g).obj ((pullback f).obj (Quotient.mk'' a)) = (pullback k).obj ((map h).obj (Quotient.mk'' a))\n[PROOFSTEP]\napply Quotient.sound\n[GOAL]\ncase h.a\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : HasPullbacks C\nX Y Z W : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : Y \u27f6 W\nk : Z \u27f6 W\ninst\u271d\u00b9 : Mono h\ninst\u271d : Mono g\ncomm : f \u226b h = g \u226b k\nt : IsLimit (PullbackCone.mk f g comm)\na : MonoOver Y\n\u22a2 (MonoOver.map g).obj ((MonoOver.pullback f).obj a) \u2248 (MonoOver.pullback k).obj ((MonoOver.map h).obj a)\n[PROOFSTEP]\napply ThinSkeleton.equiv_of_both_ways\n[GOAL]\ncase h.a.f\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : HasPullbacks C\nX Y Z W : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : Y \u27f6 W\nk : Z \u27f6 W\ninst\u271d\u00b9 : Mono h\ninst\u271d : Mono g\ncomm : f \u226b h = g \u226b k\nt : IsLimit (PullbackCone.mk f g comm)\na : MonoOver Y\n\u22a2 (MonoOver.map g).obj ((MonoOver.pullback f).obj a) \u27f6 (MonoOver.pullback k).obj ((MonoOver.map h).obj a)\n[PROOFSTEP]\nrefine' MonoOver.homMk (pullback.lift pullback.fst _ _) (pullback.lift_snd _ _ _)\n[GOAL]\ncase h.a.f\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : HasPullbacks C\nX Y Z W : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : Y \u27f6 W\nk : Z \u27f6 W\ninst\u271d\u00b9 : Mono h\ninst\u271d : Mono g\ncomm : f \u226b h = g \u226b k\nt : IsLimit (PullbackCone.mk f g comm)\na : MonoOver Y\n\u22a2 pullback.fst \u226b ((MonoOver.forget W).obj ((MonoOver.map h).obj a)).hom =\n MonoOver.arrow ((MonoOver.map g).obj ((MonoOver.pullback f).obj a)) \u226b k\n[PROOFSTEP]\nchange _ \u226b a.arrow \u226b h = (pullback.snd \u226b g) \u226b _\n[GOAL]\ncase h.a.f\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : HasPullbacks C\nX Y Z W : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : Y \u27f6 W\nk : Z \u27f6 W\ninst\u271d\u00b9 : Mono h\ninst\u271d : Mono g\ncomm : f \u226b h = g \u226b k\nt : IsLimit (PullbackCone.mk f g comm)\na : MonoOver Y\n\u22a2 pullback.fst \u226b MonoOver.arrow a \u226b h = (pullback.snd \u226b g) \u226b k\n[PROOFSTEP]\nrw [assoc, \u2190 comm, pullback.condition_assoc]\n[GOAL]\ncase h.a.g\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : HasPullbacks C\nX Y Z W : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : Y \u27f6 W\nk : Z \u27f6 W\ninst\u271d\u00b9 : Mono h\ninst\u271d : Mono g\ncomm : f \u226b h = g \u226b k\nt : IsLimit (PullbackCone.mk f g comm)\na : MonoOver Y\n\u22a2 (MonoOver.pullback k).obj ((MonoOver.map h).obj a) \u27f6 (MonoOver.map g).obj ((MonoOver.pullback f).obj a)\n[PROOFSTEP]\nrefine'\n MonoOver.homMk\n (pullback.lift pullback.fst (PullbackCone.IsLimit.lift t (pullback.fst \u226b a.arrow) pullback.snd _)\n (PullbackCone.IsLimit.lift_fst _ _ _ _).symm)\n _\n[GOAL]\ncase h.a.g.refine'_1\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : HasPullbacks C\nX Y Z W : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : Y \u27f6 W\nk : Z \u27f6 W\ninst\u271d\u00b9 : Mono h\ninst\u271d : Mono g\ncomm : f \u226b h = g \u226b k\nt : IsLimit (PullbackCone.mk f g comm)\na : MonoOver Y\n\u22a2 (pullback.fst \u226b ((MonoOver.forget Y).obj a).hom) \u226b h = pullback.snd \u226b k\n[PROOFSTEP]\nrw [\u2190 pullback.condition, assoc]\n[GOAL]\ncase h.a.g.refine'_1\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : HasPullbacks C\nX Y Z W : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : Y \u27f6 W\nk : Z \u27f6 W\ninst\u271d\u00b9 : Mono h\ninst\u271d : Mono g\ncomm : f \u226b h = g \u226b k\nt : IsLimit (PullbackCone.mk f g comm)\na : MonoOver Y\n\u22a2 pullback.fst \u226b ((MonoOver.forget Y).obj a).hom \u226b h =\n pullback.fst \u226b ((MonoOver.forget W).obj ((MonoOver.map h).obj a)).hom\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.a.g.refine'_2\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : HasPullbacks C\nX Y Z W : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : Y \u27f6 W\nk : Z \u27f6 W\ninst\u271d\u00b9 : Mono h\ninst\u271d : Mono g\ncomm : f \u226b h = g \u226b k\nt : IsLimit (PullbackCone.mk f g comm)\na : MonoOver Y\n\u22a2 pullback.lift pullback.fst\n (PullbackCone.IsLimit.lift t (pullback.fst \u226b MonoOver.arrow a) pullback.snd\n (_ : (pullback.fst \u226b ((MonoOver.forget Y).obj a).hom) \u226b h = pullback.snd \u226b k))\n (_ :\n pullback.fst \u226b ((MonoOver.forget Y).obj a).hom =\n PullbackCone.IsLimit.lift t (pullback.fst \u226b ((MonoOver.forget Y).obj a).hom) pullback.snd\n (_ : (pullback.fst \u226b ((MonoOver.forget Y).obj a).hom) \u226b h = pullback.snd \u226b k) \u226b\n PullbackCone.fst (PullbackCone.mk f g comm)) \u226b\n MonoOver.arrow ((MonoOver.map g).obj ((MonoOver.pullback f).obj a)) =\n MonoOver.arrow ((MonoOver.pullback k).obj ((MonoOver.map h).obj a))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.a.g.refine'_2\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : HasPullbacks C\nX Y Z W : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : Y \u27f6 W\nk : Z \u27f6 W\ninst\u271d\u00b9 : Mono h\ninst\u271d : Mono g\ncomm : f \u226b h = g \u226b k\nt : IsLimit (PullbackCone.mk f g comm)\na : MonoOver Y\n\u22a2 pullback.lift pullback.fst\n (PullbackCone.IsLimit.lift t (pullback.fst \u226b MonoOver.arrow a) pullback.snd\n (_ : (pullback.fst \u226b MonoOver.arrow a) \u226b h = pullback.snd \u226b k))\n (_ :\n pullback.fst \u226b MonoOver.arrow a =\n PullbackCone.IsLimit.lift t (pullback.fst \u226b MonoOver.arrow a) pullback.snd\n (_ : (pullback.fst \u226b MonoOver.arrow a) \u226b h = pullback.snd \u226b k) \u226b\n f) \u226b\n pullback.snd \u226b g =\n pullback.snd\n[PROOFSTEP]\nrw [pullback.lift_snd_assoc]\n[GOAL]\ncase h.a.g.refine'_2\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z\u271d : C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : HasPullbacks C\nX Y Z W : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nh : Y \u27f6 W\nk : Z \u27f6 W\ninst\u271d\u00b9 : Mono h\ninst\u271d : Mono g\ncomm : f \u226b h = g \u226b k\nt : IsLimit (PullbackCone.mk f g comm)\na : MonoOver Y\n\u22a2 PullbackCone.IsLimit.lift t (pullback.fst \u226b MonoOver.arrow a) pullback.snd\n (_ : (pullback.fst \u226b MonoOver.arrow a) \u226b h = pullback.snd \u226b k) \u226b\n g =\n pullback.snd\n[PROOFSTEP]\napply PullbackCone.IsLimit.lift_snd\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Subobject.Basic", "llama_tokens": 16630, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.40733340004593016, "lm_q2_score": 0.02002344242249784, "lm_q1q2_score": 0.008156216882579961}} {"text": "[GOAL]\nF : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\ninst\u271d : LawfulApplicative F\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng : \u03c3 \u2192 \u03b2\nx : F \u03b1\ny : F \u03c3\n\u22a2 (Seq.seq (f <$> x) fun x => g <$> y) = Seq.seq ((flip (fun x x_1 => x \u2218 x_1) g \u2218 f) <$> x) fun x => y\n[PROOFSTEP]\nsimp [flip, functor_norm]\n[GOAL]\nF : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\ninst\u271d : LawfulApplicative F\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nf : \u03b1 \u2192 \u03b2\n\u22a2 (fun x x_1 => Seq.seq x fun x => x_1) (pure f) = (fun x x_1 => x <$> x_1) f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\ninst\u271d : LawfulApplicative F\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nf : \u03b1 \u2192 \u03b2\nx\u271d : F \u03b1\n\u22a2 (fun x x_1 => Seq.seq x fun x => x_1) (pure f) x\u271d = (fun x x_1 => x <$> x_1) f x\u271d\n[PROOFSTEP]\nsimp [functor_norm]\n[GOAL]\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\np2 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns2 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL1 : LawfulApplicative F\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\n\u22a2 mk = mk\n[PROOFSTEP]\nobtain rfl : @p1 = @p2 := by\n funext \u03b1 x\n apply H1\n[GOAL]\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\np2 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns2 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL1 : LawfulApplicative F\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\n\u22a2 p1 = p2\n[PROOFSTEP]\nfunext \u03b1 x\n[GOAL]\ncase h.h\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\np2 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns2 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL1 : LawfulApplicative F\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\n\u03b1 : Type u\nx : \u03b1\n\u22a2 p1 x = p2 x\n[PROOFSTEP]\napply H1\n[GOAL]\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\ns2 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL1 : LawfulApplicative F\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\n\u22a2 mk = mk\n[PROOFSTEP]\nobtain rfl : @s1 = @s2 := by\n funext \u03b1 \u03b2 f x\n exact H2 f (x Unit.unit)\n[GOAL]\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\ns2 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL1 : LawfulApplicative F\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\n\u22a2 s1 = s2\n[PROOFSTEP]\nfunext \u03b1 \u03b2 f x\n[GOAL]\ncase h.h.h.h\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2\u271d \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\ns2 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL1 : LawfulApplicative F\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\n\u03b1 \u03b2 : Type u\nf : F (\u03b1 \u2192 \u03b2)\nx : Unit \u2192 F \u03b1\n\u22a2 s1 f x = s2 f x\n[PROOFSTEP]\nexact H2 f (x Unit.unit)\n[GOAL]\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL1 : LawfulApplicative F\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\n\u22a2 mk = mk\n[PROOFSTEP]\nobtain \u27e8seqLeft_eq1, seqRight_eq1, pure_seq1, -\u27e9 := L1\n[GOAL]\ncase mk\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u22a2 mk = mk\n[PROOFSTEP]\nobtain \u27e8seqLeft_eq2, seqRight_eq2, pure_seq2, -\u27e9 := L2\n[GOAL]\ncase mk.mk\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u22a2 mk = mk\n[PROOFSTEP]\nobtain rfl : F1 = F2 := by\n apply Functor.ext\n intros\n exact (pure_seq1 _ _).symm.trans (pure_seq2 _ _)\n[GOAL]\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u22a2 F1 = F2\n[PROOFSTEP]\napply Functor.ext\n[GOAL]\ncase a\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u22a2 \u2200 (\u03b1 \u03b2 : Type u) (f : \u03b1 \u2192 \u03b2) (x : F \u03b1), f <$> x = f <$> x\n[PROOFSTEP]\nintros\n[GOAL]\ncase a\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u03b1\u271d \u03b2\u271d : Type u\nf\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : F \u03b1\u271d\n\u22a2 f\u271d <$> x\u271d = f\u271d <$> x\u271d\n[PROOFSTEP]\nexact (pure_seq1 _ _).symm.trans (pure_seq2 _ _)\n[GOAL]\ncase mk.mk\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u22a2 mk = mk\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_toSeqLeft.e_seqLeft\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u22a2 sl1 = sl2\n[PROOFSTEP]\nfunext \u03b1 \u03b2 x y\n[GOAL]\ncase mk.mk.e_toSeqRight.e_seqRight\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u22a2 sr1 = sr2\n[PROOFSTEP]\nfunext \u03b1 \u03b2 x y\n[GOAL]\ncase mk.mk.e_toSeqLeft.e_seqLeft.h.h.h.h\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2\u271d \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u03b1 \u03b2 : Type u\nx : F \u03b1\ny : Unit \u2192 F \u03b2\n\u22a2 sl1 x y = sl2 x y\n[PROOFSTEP]\nexact (seqLeft_eq1 _ (y Unit.unit)).trans (seqLeft_eq2 _ _).symm\n[GOAL]\ncase mk.mk.e_toSeqRight.e_seqRight.h.h.h.h\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2\u271d \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u03b1 \u03b2 : Type u\nx : F \u03b1\ny : Unit \u2192 F \u03b2\n\u22a2 sr1 x y = sr2 x y\n[PROOFSTEP]\nexact (seqRight_eq1 _ (y Unit.unit)).trans (seqRight_eq2 _ (y Unit.unit)).symm\n[GOAL]\n\u22a2 CommApplicative Id\n[PROOFSTEP]\nrefine' { .. }\n[GOAL]\ncase refine'_1\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.6279}, Functor.mapConst = Functor.map \u2218 Function.const \u03b2\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\n\u22a2 \u2200 {\u03b1 : Type ?u.6279} (x : Id \u03b1), id <$> x = x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_3\n\u22a2 \u2200 {\u03b1 \u03b2 \u03b3 : Type ?u.6279} (g : \u03b1 \u2192 \u03b2) (h : \u03b2 \u2192 \u03b3) (x : Id \u03b1), (h \u2218 g) <$> x = h <$> g <$> x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_4\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.6279} (x : Id \u03b1) (y : Id \u03b2),\n (SeqLeft.seqLeft x fun x => y) = Seq.seq (Function.const \u03b2 <$> x) fun x => y\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_5\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.6279} (x : Id \u03b1) (y : Id \u03b2),\n (SeqRight.seqRight x fun x => y) = Seq.seq (Function.const \u03b1 id <$> x) fun x => y\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_6\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.6279} (g : \u03b1 \u2192 \u03b2) (x : Id \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_7\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.6279} (g : \u03b1 \u2192 \u03b2) (x : \u03b1), g <$> pure x = pure (g x)\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_8\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.6279} (g : Id (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_9\n\u22a2 \u2200 {\u03b1 \u03b2 \u03b3 : Type ?u.6279} (x : Id \u03b1) (g : Id (\u03b1 \u2192 \u03b2)) (h : Id (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (Function.comp <$> h) fun x => g) fun x_1 => x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_10\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.6279} (a : Id \u03b1) (b : Id \u03b2),\n (Seq.seq (Prod.mk <$> a) fun x => b) = Seq.seq ((fun b a => (a, b)) <$> b) fun x => a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\n\u03b1\u271d \u03b2\u271d : Type ?u.6279\n\u22a2 Functor.mapConst = Functor.map \u2218 Function.const \u03b2\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\n\u03b1\u271d : Type ?u.6279\nx\u271d : Id \u03b1\u271d\n\u22a2 id <$> x\u271d = x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_3\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.6279\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nh\u271d : \u03b2\u271d \u2192 \u03b3\u271d\nx\u271d : Id \u03b1\u271d\n\u22a2 (h\u271d \u2218 g\u271d) <$> x\u271d = h\u271d <$> g\u271d <$> x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_4\n\u03b1\u271d \u03b2\u271d : Type ?u.6279\nx\u271d : Id \u03b1\u271d\ny\u271d : Id \u03b2\u271d\n\u22a2 (SeqLeft.seqLeft x\u271d fun x => y\u271d) = Seq.seq (Function.const \u03b2\u271d <$> x\u271d) fun x => y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_5\n\u03b1\u271d \u03b2\u271d : Type ?u.6279\nx\u271d : Id \u03b1\u271d\ny\u271d : Id \u03b2\u271d\n\u22a2 (SeqRight.seqRight x\u271d fun x => y\u271d) = Seq.seq (Function.const \u03b1\u271d id <$> x\u271d) fun x => y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_6\n\u03b1\u271d \u03b2\u271d : Type ?u.6279\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : Id \u03b1\u271d\n\u22a2 (Seq.seq (pure g\u271d) fun x => x\u271d) = g\u271d <$> x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_7\n\u03b1\u271d \u03b2\u271d : Type ?u.6279\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : \u03b1\u271d\n\u22a2 g\u271d <$> pure x\u271d = pure (g\u271d x\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_8\n\u03b1\u271d \u03b2\u271d : Type ?u.6279\ng\u271d : Id (\u03b1\u271d \u2192 \u03b2\u271d)\nx\u271d : \u03b1\u271d\n\u22a2 (Seq.seq g\u271d fun x => pure x\u271d) = (fun h => h x\u271d) <$> g\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_9\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.6279\nx\u271d : Id \u03b1\u271d\ng\u271d : Id (\u03b1\u271d \u2192 \u03b2\u271d)\nh\u271d : Id (\u03b2\u271d \u2192 \u03b3\u271d)\n\u22a2 (Seq.seq h\u271d fun x => Seq.seq g\u271d fun x => x\u271d) = Seq.seq (Seq.seq (Function.comp <$> h\u271d) fun x => g\u271d) fun x => x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_10\n\u03b1\u271d \u03b2\u271d : Type ?u.6279\na\u271d : Id \u03b1\u271d\nb\u271d : Id \u03b2\u271d\n\u22a2 (Seq.seq (Prod.mk <$> a\u271d) fun x => b\u271d) = Seq.seq ((fun b a => (a, b)) <$> b\u271d) fun x => a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u \u2192 Type w\nG : Type v \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type v\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 run (f <$> pure x) = run (pure (f x))\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u \u2192 Type w\nG : Type v \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type v\nf : Comp F G (\u03b1 \u2192 \u03b2)\nx : \u03b1\n\u22a2 run (Seq.seq f fun x_1 => pure x) = run ((fun g => g x) <$> f)\n[PROOFSTEP]\nsimp [(\u00b7 \u2218 \u00b7), functor_norm]\n[GOAL]\nF : Type u \u2192 Type w\nG : Type v \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type v\nx : Comp F G \u03b1\nf : Comp F G (\u03b1 \u2192 \u03b2)\ng : Comp F G (\u03b2 \u2192 \u03b3)\n\u22a2 run (Seq.seq g fun x_1 => Seq.seq f fun x_2 => x) =\n run (Seq.seq (Seq.seq (Function.comp <$> g) fun x => f) fun x_1 => x)\n[PROOFSTEP]\nsimp [(\u00b7 \u2218 \u00b7), functor_norm]\n[GOAL]\nF : Type u \u2192 Type w\nG : Type v \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type v\nf : \u03b1 \u2192 \u03b2\nx : Comp F G \u03b1\n\u22a2 run (Seq.seq (pure f) fun x_1 => x) = run (f <$> x)\n[PROOFSTEP]\nsimp [Applicative.pure_seq_eq_map', functor_norm]\n[GOAL]\nF : Type u \u2192 Type w\nG : Type v \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type v\n\u22a2 \u2200 {\u03b1 \u03b2 : Type v} (x : Comp F G \u03b1) (y : Comp F G \u03b2),\n (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\n[PROOFSTEP]\nintros\n[GOAL]\nF : Type u \u2192 Type w\nG : Type v \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 \u03b1\u271d \u03b2\u271d : Type v\nx\u271d : Comp F G \u03b1\u271d\ny\u271d : Comp F G \u03b2\u271d\n\u22a2 (SeqLeft.seqLeft x\u271d fun x => y\u271d) = Seq.seq (const \u03b2\u271d <$> x\u271d) fun x => y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u \u2192 Type w\nG : Type v \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type v\n\u22a2 \u2200 {\u03b1 \u03b2 : Type v} (x : Comp F G \u03b1) (y : Comp F G \u03b2),\n (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\n[PROOFSTEP]\nintros\n[GOAL]\nF : Type u \u2192 Type w\nG : Type v \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 \u03b1\u271d \u03b2\u271d : Type v\nx\u271d : Comp F G \u03b1\u271d\ny\u271d : Comp F G \u03b2\u271d\n\u22a2 (SeqRight.seqRight x\u271d fun x => y\u271d) = Seq.seq (const \u03b1\u271d id <$> x\u271d) fun x => y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nF\u271d : Type u \u2192 Type w\nG : Type v \u2192 Type u\ninst\u271d\u2074 : Applicative F\u271d\ninst\u271d\u00b3 : Applicative G\ninst\u271d\u00b2 : LawfulApplicative F\u271d\ninst\u271d\u00b9 : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type v\nF : Type u_1 \u2192 Type u_2\nAF : Applicative F\ninst\u271d : LawfulApplicative F\n\u03b1\u271d \u03b2\u271d : Type u_1\nf : F (\u03b1\u271d \u2192 \u03b2\u271d)\nx : F \u03b1\u271d\n\u22a2 (Seq.seq (id <$> f) fun x_1 => x) = Seq.seq f fun x_1 => x\n[PROOFSTEP]\nrw [id_map]\n[GOAL]\nF : Type u \u2192 Type w\nG : Type v \u2192 Type u\ninst\u271d\u2077 : Applicative F\ninst\u271d\u2076 : Applicative G\ninst\u271d\u2075 : LawfulApplicative F\ninst\u271d\u2074 : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type v\nf : Type u \u2192 Type w\ng : Type v \u2192 Type u\ninst\u271d\u00b3 : Applicative f\ninst\u271d\u00b2 : Applicative g\ninst\u271d\u00b9 : CommApplicative f\ninst\u271d : CommApplicative g\n\u22a2 CommApplicative (Comp f g)\n[PROOFSTEP]\nrefine' { @instLawfulApplicativeComp f g _ _ _ _ with .. }\n[GOAL]\nF : Type u \u2192 Type w\nG : Type v \u2192 Type u\ninst\u271d\u2077 : Applicative F\ninst\u271d\u2076 : Applicative G\ninst\u271d\u2075 : LawfulApplicative F\ninst\u271d\u2074 : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type v\nf : Type u \u2192 Type w\ng : Type v \u2192 Type u\ninst\u271d\u00b3 : Applicative f\ninst\u271d\u00b2 : Applicative g\ninst\u271d\u00b9 : CommApplicative f\ninst\u271d : CommApplicative g\nsrc\u271d : LawfulApplicative (Comp f g) := instLawfulApplicativeComp\n\u22a2 \u2200 {\u03b1 \u03b2 : Type v} (a : Comp f g \u03b1) (b : Comp f g \u03b2),\n (Seq.seq (Prod.mk <$> a) fun x => b) = Seq.seq ((fun b a => (a, b)) <$> b) fun x => a\n[PROOFSTEP]\nintros\n[GOAL]\nF : Type u \u2192 Type w\nG : Type v \u2192 Type u\ninst\u271d\u2077 : Applicative F\ninst\u271d\u2076 : Applicative G\ninst\u271d\u2075 : LawfulApplicative F\ninst\u271d\u2074 : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type v\nf : Type u \u2192 Type w\ng : Type v \u2192 Type u\ninst\u271d\u00b3 : Applicative f\ninst\u271d\u00b2 : Applicative g\ninst\u271d\u00b9 : CommApplicative f\ninst\u271d : CommApplicative g\nsrc\u271d : LawfulApplicative (Comp f g) := instLawfulApplicativeComp\n\u03b1\u271d \u03b2\u271d : Type v\na\u271d : Comp f g \u03b1\u271d\nb\u271d : Comp f g \u03b2\u271d\n\u22a2 (Seq.seq (Prod.mk <$> a\u271d) fun x => b\u271d) = Seq.seq ((fun b a => (a, b)) <$> b\u271d) fun x => a\u271d\n[PROOFSTEP]\nsimp! [map, Seq.seq, functor_norm]\n[GOAL]\nF : Type u \u2192 Type w\nG : Type v \u2192 Type u\ninst\u271d\u2077 : Applicative F\ninst\u271d\u2076 : Applicative G\ninst\u271d\u2075 : LawfulApplicative F\ninst\u271d\u2074 : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type v\nf : Type u \u2192 Type w\ng : Type v \u2192 Type u\ninst\u271d\u00b3 : Applicative f\ninst\u271d\u00b2 : Applicative g\ninst\u271d\u00b9 : CommApplicative f\ninst\u271d : CommApplicative g\nsrc\u271d : LawfulApplicative (Comp f g) := instLawfulApplicativeComp\n\u03b1\u271d \u03b2\u271d : Type v\na\u271d : Comp f g \u03b1\u271d\nb\u271d : Comp f g \u03b2\u271d\n\u22a2 mk (Seq.seq ((fun x x_1 => Seq.seq x fun x => x_1) <$> mk ((fun x => Prod.mk <$> x) <$> a\u271d)) fun x => b\u271d) =\n mk\n (Seq.seq ((fun x x_1 => Seq.seq x fun x => x_1) <$> mk ((fun x => (fun b a => (a, b)) <$> x) <$> b\u271d)) fun x => a\u271d)\n[PROOFSTEP]\nrw [commutative_map]\n[GOAL]\nF : Type u \u2192 Type w\nG : Type v \u2192 Type u\ninst\u271d\u2077 : Applicative F\ninst\u271d\u2076 : Applicative G\ninst\u271d\u2075 : LawfulApplicative F\ninst\u271d\u2074 : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type v\nf : Type u \u2192 Type w\ng : Type v \u2192 Type u\ninst\u271d\u00b3 : Applicative f\ninst\u271d\u00b2 : Applicative g\ninst\u271d\u00b9 : CommApplicative f\ninst\u271d : CommApplicative g\nsrc\u271d : LawfulApplicative (Comp f g) := instLawfulApplicativeComp\n\u03b1\u271d \u03b2\u271d : Type v\na\u271d : Comp f g \u03b1\u271d\nb\u271d : Comp f g \u03b2\u271d\n\u22a2 mk (Seq.seq ((flip fun x x_1 => Seq.seq x fun x => x_1) <$> b\u271d) fun x => mk ((fun x => Prod.mk <$> x) <$> a\u271d)) =\n mk\n (Seq.seq ((fun x x_1 => Seq.seq x fun x => x_1) <$> mk ((fun x => (fun b a => (a, b)) <$> x) <$> b\u271d)) fun x => a\u271d)\n[PROOFSTEP]\nsimp [Comp.mk, flip, (\u00b7 \u2218 \u00b7), functor_norm]\n[GOAL]\nF : Type u \u2192 Type w\nG : Type v \u2192 Type u\ninst\u271d\u2077 : Applicative F\ninst\u271d\u2076 : Applicative G\ninst\u271d\u2075 : LawfulApplicative F\ninst\u271d\u2074 : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type v\nf : Type u \u2192 Type w\ng : Type v \u2192 Type u\ninst\u271d\u00b3 : Applicative f\ninst\u271d\u00b2 : Applicative g\ninst\u271d\u00b9 : CommApplicative f\ninst\u271d : CommApplicative g\nsrc\u271d : LawfulApplicative (Comp f g) := instLawfulApplicativeComp\n\u03b1\u271d \u03b2\u271d : Type v\na\u271d : Comp f g \u03b1\u271d\nb\u271d : Comp f g \u03b2\u271d\n\u22a2 (Seq.seq ((fun x x_1 => Seq.seq (Prod.mk <$> x_1) fun x_2 => x) <$> b\u271d) fun x => a\u271d) =\n Seq.seq ((fun x x_1 => Seq.seq ((fun b a => (a, b)) <$> x) fun x => x_1) <$> b\u271d) fun x => a\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_a\nF : Type u \u2192 Type w\nG : Type v \u2192 Type u\ninst\u271d\u2077 : Applicative F\ninst\u271d\u2076 : Applicative G\ninst\u271d\u2075 : LawfulApplicative F\ninst\u271d\u2074 : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type v\nf : Type u \u2192 Type w\ng : Type v \u2192 Type u\ninst\u271d\u00b3 : Applicative f\ninst\u271d\u00b2 : Applicative g\ninst\u271d\u00b9 : CommApplicative f\ninst\u271d : CommApplicative g\nsrc\u271d : LawfulApplicative (Comp f g) := instLawfulApplicativeComp\n\u03b1\u271d \u03b2\u271d : Type v\na\u271d : Comp f g \u03b1\u271d\nb\u271d : Comp f g \u03b2\u271d\n\u22a2 (fun x x_1 => Seq.seq (Prod.mk <$> x_1) fun x_2 => x) = fun x x_1 => Seq.seq ((fun b a => (a, b)) <$> x) fun x => x_1\n[PROOFSTEP]\nfunext x y\n[GOAL]\ncase e_a.e_a.h.h\nF : Type u \u2192 Type w\nG : Type v \u2192 Type u\ninst\u271d\u2077 : Applicative F\ninst\u271d\u2076 : Applicative G\ninst\u271d\u2075 : LawfulApplicative F\ninst\u271d\u2074 : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type v\nf : Type u \u2192 Type w\ng : Type v \u2192 Type u\ninst\u271d\u00b3 : Applicative f\ninst\u271d\u00b2 : Applicative g\ninst\u271d\u00b9 : CommApplicative f\ninst\u271d : CommApplicative g\nsrc\u271d : LawfulApplicative (Comp f g) := instLawfulApplicativeComp\n\u03b1\u271d \u03b2\u271d : Type v\na\u271d : Comp f g \u03b1\u271d\nb\u271d : Comp f g \u03b2\u271d\nx : g \u03b2\u271d\ny : g \u03b1\u271d\n\u22a2 (Seq.seq (Prod.mk <$> y) fun x_1 => x) = Seq.seq ((fun b a => (a, b)) <$> x) fun x => y\n[PROOFSTEP]\nrw [commutative_map]\n[GOAL]\ncase e_a.e_a.h.h\nF : Type u \u2192 Type w\nG : Type v \u2192 Type u\ninst\u271d\u2077 : Applicative F\ninst\u271d\u2076 : Applicative G\ninst\u271d\u2075 : LawfulApplicative F\ninst\u271d\u2074 : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type v\nf : Type u \u2192 Type w\ng : Type v \u2192 Type u\ninst\u271d\u00b3 : Applicative f\ninst\u271d\u00b2 : Applicative g\ninst\u271d\u00b9 : CommApplicative f\ninst\u271d : CommApplicative g\nsrc\u271d : LawfulApplicative (Comp f g) := instLawfulApplicativeComp\n\u03b1\u271d \u03b2\u271d : Type v\na\u271d : Comp f g \u03b1\u271d\nb\u271d : Comp f g \u03b2\u271d\nx : g \u03b2\u271d\ny : g \u03b1\u271d\n\u22a2 (Seq.seq (flip Prod.mk <$> x) fun x => y) = Seq.seq ((fun b a => (a, b)) <$> x) fun x => y\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u22a2 LawfulApplicative (Const \u03b1)\n[PROOFSTEP]\nrefine' { .. }\n[GOAL]\ncase refine'_1\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u22a2 \u2200 {\u03b1_1 \u03b2 : Type ?u.61710}, mapConst = map \u2218 Function.const \u03b2\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u22a2 \u2200 {\u03b1_1 : Type ?u.61710} (x : Const \u03b1 \u03b1_1), id <$> x = x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_3\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u22a2 \u2200 {\u03b1_1 \u03b2 \u03b3 : Type ?u.61710} (g : \u03b1_1 \u2192 \u03b2) (h : \u03b2 \u2192 \u03b3) (x : Const \u03b1 \u03b1_1), (h \u2218 g) <$> x = h <$> g <$> x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_4\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u22a2 \u2200 {\u03b1_1 \u03b2 : Type ?u.61710} (x : Const \u03b1 \u03b1_1) (y : Const \u03b1 \u03b2),\n (SeqLeft.seqLeft x fun x => y) = Seq.seq (Function.const \u03b2 <$> x) fun x => y\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_5\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u22a2 \u2200 {\u03b1_1 \u03b2 : Type ?u.61710} (x : Const \u03b1 \u03b1_1) (y : Const \u03b1 \u03b2),\n (SeqRight.seqRight x fun x => y) = Seq.seq (Function.const \u03b1_1 id <$> x) fun x => y\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_6\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u22a2 \u2200 {\u03b1_1 \u03b2 : Type ?u.61710} (g : \u03b1_1 \u2192 \u03b2) (x : Const \u03b1 \u03b1_1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_7\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u22a2 \u2200 {\u03b1_1 \u03b2 : Type ?u.61710} (g : \u03b1_1 \u2192 \u03b2) (x : \u03b1_1), g <$> pure x = pure (g x)\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_8\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u22a2 \u2200 {\u03b1_1 \u03b2 : Type ?u.61710} (g : Const \u03b1 (\u03b1_1 \u2192 \u03b2)) (x : \u03b1_1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_9\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u22a2 \u2200 {\u03b1_1 \u03b2 \u03b3 : Type ?u.61710} (x : Const \u03b1 \u03b1_1) (g : Const \u03b1 (\u03b1_1 \u2192 \u03b2)) (h : Const \u03b1 (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (Function.comp <$> h) fun x => g) fun x_1 => x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.61710\n\u22a2 mapConst = map \u2218 Function.const \u03b2\u271d\n[PROOFSTEP]\nsimp [mul_assoc, (\u00b7 <$> \u00b7), Seq.seq, pure]\n[GOAL]\ncase refine'_2\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u03b1\u271d : Type ?u.61710\nx\u271d : Const \u03b1 \u03b1\u271d\n\u22a2 id <$> x\u271d = x\u271d\n[PROOFSTEP]\nsimp [mul_assoc, (\u00b7 <$> \u00b7), Seq.seq, pure]\n[GOAL]\ncase refine'_3\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.61710\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nh\u271d : \u03b2\u271d \u2192 \u03b3\u271d\nx\u271d : Const \u03b1 \u03b1\u271d\n\u22a2 (h\u271d \u2218 g\u271d) <$> x\u271d = h\u271d <$> g\u271d <$> x\u271d\n[PROOFSTEP]\nsimp [mul_assoc, (\u00b7 <$> \u00b7), Seq.seq, pure]\n[GOAL]\ncase refine'_4\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.61710\nx\u271d : Const \u03b1 \u03b1\u271d\ny\u271d : Const \u03b1 \u03b2\u271d\n\u22a2 (SeqLeft.seqLeft x\u271d fun x => y\u271d) = Seq.seq (Function.const \u03b2\u271d <$> x\u271d) fun x => y\u271d\n[PROOFSTEP]\nsimp [mul_assoc, (\u00b7 <$> \u00b7), Seq.seq, pure]\n[GOAL]\ncase refine'_5\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.61710\nx\u271d : Const \u03b1 \u03b1\u271d\ny\u271d : Const \u03b1 \u03b2\u271d\n\u22a2 (SeqRight.seqRight x\u271d fun x => y\u271d) = Seq.seq (Function.const \u03b1\u271d id <$> x\u271d) fun x => y\u271d\n[PROOFSTEP]\nsimp [mul_assoc, (\u00b7 <$> \u00b7), Seq.seq, pure]\n[GOAL]\ncase refine'_6\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.61710\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : Const \u03b1 \u03b1\u271d\n\u22a2 (Seq.seq (pure g\u271d) fun x => x\u271d) = g\u271d <$> x\u271d\n[PROOFSTEP]\nsimp [mul_assoc, (\u00b7 <$> \u00b7), Seq.seq, pure]\n[GOAL]\ncase refine'_7\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.61710\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : \u03b1\u271d\n\u22a2 g\u271d <$> pure x\u271d = pure (g\u271d x\u271d)\n[PROOFSTEP]\nsimp [mul_assoc, (\u00b7 <$> \u00b7), Seq.seq, pure]\n[GOAL]\ncase refine'_8\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.61710\ng\u271d : Const \u03b1 (\u03b1\u271d \u2192 \u03b2\u271d)\nx\u271d : \u03b1\u271d\n\u22a2 (Seq.seq g\u271d fun x => pure x\u271d) = (fun h => h x\u271d) <$> g\u271d\n[PROOFSTEP]\nsimp [mul_assoc, (\u00b7 <$> \u00b7), Seq.seq, pure]\n[GOAL]\ncase refine'_9\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.61710\nx\u271d : Const \u03b1 \u03b1\u271d\ng\u271d : Const \u03b1 (\u03b1\u271d \u2192 \u03b2\u271d)\nh\u271d : Const \u03b1 (\u03b2\u271d \u2192 \u03b3\u271d)\n\u22a2 (Seq.seq h\u271d fun x => Seq.seq g\u271d fun x => x\u271d) = Seq.seq (Seq.seq (Function.comp <$> h\u271d) fun x => g\u271d) fun x => x\u271d\n[PROOFSTEP]\nsimp [mul_assoc, (\u00b7 <$> \u00b7), Seq.seq, pure]\n[GOAL]\ncase refine'_1\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.61710\n\u22a2 mapConst = Const.map \u2218 Function.const \u03b2\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u03b1\u271d : Type ?u.61710\nx\u271d : Const \u03b1 \u03b1\u271d\n\u22a2 Const.map id x\u271d = x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_3\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.61710\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nh\u271d : \u03b2\u271d \u2192 \u03b3\u271d\nx\u271d : Const \u03b1 \u03b1\u271d\n\u22a2 Const.map (h\u271d \u2218 g\u271d) x\u271d = Const.map h\u271d (Const.map g\u271d x\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_4\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.61710\nx\u271d : Const \u03b1 \u03b1\u271d\ny\u271d : Const \u03b1 \u03b2\u271d\n\u22a2 (SeqLeft.seqLeft x\u271d fun x => y\u271d) = Const.map (Function.const \u03b2\u271d) x\u271d * y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_5\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.61710\nx\u271d : Const \u03b1 \u03b1\u271d\ny\u271d : Const \u03b1 \u03b2\u271d\n\u22a2 (SeqRight.seqRight x\u271d fun x => y\u271d) = Const.map (Function.const \u03b1\u271d id) x\u271d * y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_6\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.61710\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : Const \u03b1 \u03b1\u271d\n\u22a2 x\u271d = Const.map g\u271d x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_7\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.61710\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : \u03b1\u271d\n\u22a2 Const.map g\u271d 1 = 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_8\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.61710\ng\u271d : Const \u03b1 (\u03b1\u271d \u2192 \u03b2\u271d)\nx\u271d : \u03b1\u271d\n\u22a2 g\u271d = Const.map (fun h => h x\u271d) g\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_9\n\u03b1 : Type ?u.61704\ninst\u271d : Monoid \u03b1\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.61710\nx\u271d : Const \u03b1 \u03b1\u271d\ng\u271d : Const \u03b1 (\u03b1\u271d \u2192 \u03b2\u271d)\nh\u271d : Const \u03b1 (\u03b2\u271d \u2192 \u03b3\u271d)\n\u22a2 h\u271d * (g\u271d * x\u271d) = Const.map Function.comp h\u271d * (g\u271d * x\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u22a2 LawfulApplicative (AddConst \u03b1)\n[PROOFSTEP]\nrefine' { .. }\n[GOAL]\ncase refine'_1\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u22a2 \u2200 {\u03b1_1 \u03b2 : Type ?u.64153}, mapConst = map \u2218 Function.const \u03b2\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u22a2 \u2200 {\u03b1_1 : Type ?u.64153} (x : AddConst \u03b1 \u03b1_1), id <$> x = x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_3\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u22a2 \u2200 {\u03b1_1 \u03b2 \u03b3 : Type ?u.64153} (g : \u03b1_1 \u2192 \u03b2) (h : \u03b2 \u2192 \u03b3) (x : AddConst \u03b1 \u03b1_1), (h \u2218 g) <$> x = h <$> g <$> x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_4\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u22a2 \u2200 {\u03b1_1 \u03b2 : Type ?u.64153} (x : AddConst \u03b1 \u03b1_1) (y : AddConst \u03b1 \u03b2),\n (SeqLeft.seqLeft x fun x => y) = Seq.seq (Function.const \u03b2 <$> x) fun x => y\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_5\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u22a2 \u2200 {\u03b1_1 \u03b2 : Type ?u.64153} (x : AddConst \u03b1 \u03b1_1) (y : AddConst \u03b1 \u03b2),\n (SeqRight.seqRight x fun x => y) = Seq.seq (Function.const \u03b1_1 id <$> x) fun x => y\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_6\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u22a2 \u2200 {\u03b1_1 \u03b2 : Type ?u.64153} (g : \u03b1_1 \u2192 \u03b2) (x : AddConst \u03b1 \u03b1_1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_7\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u22a2 \u2200 {\u03b1_1 \u03b2 : Type ?u.64153} (g : \u03b1_1 \u2192 \u03b2) (x : \u03b1_1), g <$> pure x = pure (g x)\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_8\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u22a2 \u2200 {\u03b1_1 \u03b2 : Type ?u.64153} (g : AddConst \u03b1 (\u03b1_1 \u2192 \u03b2)) (x : \u03b1_1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_9\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u22a2 \u2200 {\u03b1_1 \u03b2 \u03b3 : Type ?u.64153} (x : AddConst \u03b1 \u03b1_1) (g : AddConst \u03b1 (\u03b1_1 \u2192 \u03b2)) (h : AddConst \u03b1 (\u03b2 \u2192 \u03b3)),\n (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (Function.comp <$> h) fun x => g) fun x_1 => x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.64153\n\u22a2 mapConst = map \u2218 Function.const \u03b2\u271d\n[PROOFSTEP]\nsimp [add_assoc, (\u00b7 <$> \u00b7), Seq.seq, pure]\n[GOAL]\ncase refine'_2\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u03b1\u271d : Type ?u.64153\nx\u271d : AddConst \u03b1 \u03b1\u271d\n\u22a2 id <$> x\u271d = x\u271d\n[PROOFSTEP]\nsimp [add_assoc, (\u00b7 <$> \u00b7), Seq.seq, pure]\n[GOAL]\ncase refine'_3\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.64153\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nh\u271d : \u03b2\u271d \u2192 \u03b3\u271d\nx\u271d : AddConst \u03b1 \u03b1\u271d\n\u22a2 (h\u271d \u2218 g\u271d) <$> x\u271d = h\u271d <$> g\u271d <$> x\u271d\n[PROOFSTEP]\nsimp [add_assoc, (\u00b7 <$> \u00b7), Seq.seq, pure]\n[GOAL]\ncase refine'_4\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.64153\nx\u271d : AddConst \u03b1 \u03b1\u271d\ny\u271d : AddConst \u03b1 \u03b2\u271d\n\u22a2 (SeqLeft.seqLeft x\u271d fun x => y\u271d) = Seq.seq (Function.const \u03b2\u271d <$> x\u271d) fun x => y\u271d\n[PROOFSTEP]\nsimp [add_assoc, (\u00b7 <$> \u00b7), Seq.seq, pure]\n[GOAL]\ncase refine'_5\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.64153\nx\u271d : AddConst \u03b1 \u03b1\u271d\ny\u271d : AddConst \u03b1 \u03b2\u271d\n\u22a2 (SeqRight.seqRight x\u271d fun x => y\u271d) = Seq.seq (Function.const \u03b1\u271d id <$> x\u271d) fun x => y\u271d\n[PROOFSTEP]\nsimp [add_assoc, (\u00b7 <$> \u00b7), Seq.seq, pure]\n[GOAL]\ncase refine'_6\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.64153\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : AddConst \u03b1 \u03b1\u271d\n\u22a2 (Seq.seq (pure g\u271d) fun x => x\u271d) = g\u271d <$> x\u271d\n[PROOFSTEP]\nsimp [add_assoc, (\u00b7 <$> \u00b7), Seq.seq, pure]\n[GOAL]\ncase refine'_7\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.64153\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : \u03b1\u271d\n\u22a2 g\u271d <$> pure x\u271d = pure (g\u271d x\u271d)\n[PROOFSTEP]\nsimp [add_assoc, (\u00b7 <$> \u00b7), Seq.seq, pure]\n[GOAL]\ncase refine'_8\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.64153\ng\u271d : AddConst \u03b1 (\u03b1\u271d \u2192 \u03b2\u271d)\nx\u271d : \u03b1\u271d\n\u22a2 (Seq.seq g\u271d fun x => pure x\u271d) = (fun h => h x\u271d) <$> g\u271d\n[PROOFSTEP]\nsimp [add_assoc, (\u00b7 <$> \u00b7), Seq.seq, pure]\n[GOAL]\ncase refine'_9\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.64153\nx\u271d : AddConst \u03b1 \u03b1\u271d\ng\u271d : AddConst \u03b1 (\u03b1\u271d \u2192 \u03b2\u271d)\nh\u271d : AddConst \u03b1 (\u03b2\u271d \u2192 \u03b3\u271d)\n\u22a2 (Seq.seq h\u271d fun x => Seq.seq g\u271d fun x => x\u271d) = Seq.seq (Seq.seq (Function.comp <$> h\u271d) fun x => g\u271d) fun x => x\u271d\n[PROOFSTEP]\nsimp [add_assoc, (\u00b7 <$> \u00b7), Seq.seq, pure]\n[GOAL]\ncase refine'_1\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.64153\n\u22a2 mapConst = Const.map \u2218 Function.const \u03b2\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u03b1\u271d : Type ?u.64153\nx\u271d : AddConst \u03b1 \u03b1\u271d\n\u22a2 Const.map id x\u271d = x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_3\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.64153\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nh\u271d : \u03b2\u271d \u2192 \u03b3\u271d\nx\u271d : AddConst \u03b1 \u03b1\u271d\n\u22a2 Const.map (h\u271d \u2218 g\u271d) x\u271d = Const.map h\u271d (Const.map g\u271d x\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_4\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.64153\nx\u271d : AddConst \u03b1 \u03b1\u271d\ny\u271d : AddConst \u03b1 \u03b2\u271d\n\u22a2 (SeqLeft.seqLeft x\u271d fun x => y\u271d) = Const.map (Function.const \u03b2\u271d) x\u271d + y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_5\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.64153\nx\u271d : AddConst \u03b1 \u03b1\u271d\ny\u271d : AddConst \u03b1 \u03b2\u271d\n\u22a2 (SeqRight.seqRight x\u271d fun x => y\u271d) = Const.map (Function.const \u03b1\u271d id) x\u271d + y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_6\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.64153\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : AddConst \u03b1 \u03b1\u271d\n\u22a2 x\u271d = Const.map g\u271d x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_7\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.64153\ng\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : \u03b1\u271d\n\u22a2 Const.map g\u271d 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_8\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u03b1\u271d \u03b2\u271d : Type ?u.64153\ng\u271d : AddConst \u03b1 (\u03b1\u271d \u2192 \u03b2\u271d)\nx\u271d : \u03b1\u271d\n\u22a2 g\u271d = Const.map (fun h => h x\u271d) g\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_9\n\u03b1 : Type ?u.64148\ninst\u271d : AddMonoid \u03b1\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.64153\nx\u271d : AddConst \u03b1 \u03b1\u271d\ng\u271d : AddConst \u03b1 (\u03b1\u271d \u2192 \u03b2\u271d)\nh\u271d : AddConst \u03b1 (\u03b2\u271d \u2192 \u03b3\u271d)\n\u22a2 h\u271d + (g\u271d + x\u271d) = Const.map Function.comp h\u271d + (g\u271d + x\u271d)\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Control.Applicative", "llama_tokens": 26015, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.4035668537353745, "lm_q2_score": 0.019419347508538933, "lm_q1q2_score": 0.007837004975614941}} {"text": "[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\nX : Type u\ninst\u271d : MulAction M X\n\u22a2 Category.{?u.1482, u} (ActionCategory M X)\n[PROOFSTEP]\ndsimp only [ActionCategory]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\nX : Type u\ninst\u271d : MulAction M X\n\u22a2 Category.{?u.1482, u} (Functor.Elements (actionAsFunctor M X))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\nX : Type u\ninst\u271d : MulAction M X\nx : ActionCategory M X\n\u22a2 { fst := (), snd := ActionCategory.back x } = x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\nX : Type u\ninst\u271d : MulAction M X\nfst\u271d : SingleObj M\nsnd\u271d : (actionAsFunctor M X).obj fst\u271d\n\u22a2 { fst := (), snd := ActionCategory.back { fst := fst\u271d, snd := snd\u271d } } = { fst := fst\u271d, snd := snd\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : Monoid M\nX : Type u\ninst\u271d\u00b2 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G X\nP : \u2983a b : ActionCategory G X\u2984 \u2192 (a \u27f6 b) \u2192 Sort u_3\nhyp : (t : X) \u2192 (g : G) \u2192 P (homOfPair t g)\na b : ActionCategory G X\nf : a \u27f6 b\n\u22a2 P f\n[PROOFSTEP]\nrefine' cast _ (hyp b.back f.val)\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : Monoid M\nX : Type u\ninst\u271d\u00b2 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G X\nP : \u2983a b : ActionCategory G X\u2984 \u2192 (a \u27f6 b) \u2192 Sort u_3\nhyp : (t : X) \u2192 (g : G) \u2192 P (homOfPair t g)\na b : ActionCategory G X\nf : a \u27f6 b\n\u22a2 P (homOfPair (ActionCategory.back b) \u2191f) = P f\n[PROOFSTEP]\nrcases a with \u27e8\u27e8\u27e9, a : X\u27e9\n[GOAL]\ncase mk.unit\nM : Type u_1\ninst\u271d\u00b3 : Monoid M\nX : Type u\ninst\u271d\u00b2 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G X\nP : \u2983a b : ActionCategory G X\u2984 \u2192 (a \u27f6 b) \u2192 Sort u_3\nhyp : (t : X) \u2192 (g : G) \u2192 P (homOfPair t g)\nb : ActionCategory G X\na : X\nf : { fst := PUnit.unit, snd := a } \u27f6 b\n\u22a2 P (homOfPair (ActionCategory.back b) \u2191f) = P f\n[PROOFSTEP]\nrcases b with \u27e8\u27e8\u27e9, b : X\u27e9\n[GOAL]\ncase mk.unit.mk.unit\nM : Type u_1\ninst\u271d\u00b3 : Monoid M\nX : Type u\ninst\u271d\u00b2 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G X\nP : \u2983a b : ActionCategory G X\u2984 \u2192 (a \u27f6 b) \u2192 Sort u_3\nhyp : (t : X) \u2192 (g : G) \u2192 P (homOfPair t g)\na b : X\nf : { fst := PUnit.unit, snd := a } \u27f6 { fst := PUnit.unit, snd := b }\n\u22a2 P (homOfPair (ActionCategory.back { fst := PUnit.unit, snd := b }) \u2191f) = P f\n[PROOFSTEP]\nrcases f with \u27e8g : G, h : g \u2022 a = b\u27e9\n[GOAL]\ncase mk.unit.mk.unit.mk\nM : Type u_1\ninst\u271d\u00b3 : Monoid M\nX : Type u\ninst\u271d\u00b2 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G X\nP : \u2983a b : ActionCategory G X\u2984 \u2192 (a \u27f6 b) \u2192 Sort u_3\nhyp : (t : X) \u2192 (g : G) \u2192 P (homOfPair t g)\na b : X\ng : G\nh : g \u2022 a = b\n\u22a2 P (homOfPair (ActionCategory.back { fst := PUnit.unit, snd := b }) \u2191{ val := g, property := h }) =\n P { val := g, property := h }\n[PROOFSTEP]\ncases inv_smul_eq_iff.mpr h.symm\n[GOAL]\ncase mk.unit.mk.unit.mk.refl\nM : Type u_1\ninst\u271d\u00b3 : Monoid M\nX : Type u\ninst\u271d\u00b2 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G X\nP : \u2983a b : ActionCategory G X\u2984 \u2192 (a \u27f6 b) \u2192 Sort u_3\nhyp : (t : X) \u2192 (g : G) \u2192 P (homOfPair t g)\nb : X\ng : G\nh : g \u2022 g\u207b\u00b9 \u2022 b = b\n\u22a2 P (homOfPair (ActionCategory.back { fst := PUnit.unit, snd := b }) \u2191{ val := g, property := h }) =\n P { val := g, property := h }\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : Monoid M\nX : Type u\ninst\u271d\u00b2 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G X\na' b' : ActionCategory G X\nf : a' \u27f6 b'\na b : X\ng : G\nha : a' = { fst := (), snd := a }\nhb : b' = { fst := (), snd := b }\nhg : a = g\u207b\u00b9 \u2022 b\n\u22a2 a' = { fst := (), snd := g\u207b\u00b9 \u2022 b }\n[PROOFSTEP]\nrw [ha, hg]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : Monoid M\nX : Type u\ninst\u271d\u00b2 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G X\na' b' : ActionCategory G X\nf : a' \u27f6 b'\na b : X\ng : G\nha : a' = { fst := (), snd := a }\nhb : b' = { fst := (), snd := b }\nhg : a = g\u207b\u00b9 \u2022 b\n\u22a2 { fst := (), snd := b } = b'\n[PROOFSTEP]\nrw [hb]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : Monoid M\nX : Type u\ninst\u271d\u00b2 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G X\na' b' : ActionCategory G X\nf : a' \u27f6 b'\n\u22a2 \u2203 a b g ha hb hg,\n f = eqToHom (_ : a' = { fst := (), snd := g\u207b\u00b9 \u2022 b }) \u226b homOfPair b g \u226b eqToHom (_ : { fst := (), snd := b } = b')\n[PROOFSTEP]\nrevert a' b' f\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : Monoid M\nX : Type u\ninst\u271d\u00b2 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G X\n\u22a2 \u2200 \u2983a' b' : ActionCategory G X\u2984 (f : a' \u27f6 b'),\n \u2203 a b g ha hb hg,\n f = eqToHom (_ : a' = { fst := (), snd := g\u207b\u00b9 \u2022 b }) \u226b homOfPair b g \u226b eqToHom (_ : { fst := (), snd := b } = b')\n[PROOFSTEP]\nexact ActionCategory.cases (fun t g => \u27e8g\u207b\u00b9 \u2022 t, t, g, rfl, rfl, rfl, by simp\u27e9)\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : Monoid M\nX : Type u\ninst\u271d\u00b2 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b9 : Group G\ninst\u271d : MulAction G X\nt : X\ng : G\n\u22a2 homOfPair t g =\n eqToHom (_ : { fst := (), snd := g\u207b\u00b9 \u2022 t } = { fst := (), snd := g\u207b\u00b9 \u2022 t }) \u226b\n homOfPair t g \u226b eqToHom (_ : { fst := (), snd := t } = { fst := (), snd := t })\n[PROOFSTEP]\nsimp\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : ActionCategory G X \u2964 SingleObj H\n\u22a2 \u2200 {a b : ActionCategory G X} {f : a \u27f6 b}, F.map f = F.map (homOfPair (ActionCategory.back b) \u2191f)\n[PROOFSTEP]\napply ActionCategory.cases\n[GOAL]\ncase hyp\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : ActionCategory G X \u2964 SingleObj H\n\u22a2 \u2200 (t : X) (g : G),\n F.map (homOfPair t g) = F.map (homOfPair (ActionCategory.back { fst := (), snd := t }) \u2191(homOfPair t g))\n[PROOFSTEP]\nintros\n[GOAL]\ncase hyp\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : ActionCategory G X \u2964 SingleObj H\nt\u271d : X\ng\u271d : G\n\u22a2 F.map (homOfPair t\u271d g\u271d) = F.map (homOfPair (ActionCategory.back { fst := (), snd := t\u271d }) \u2191(homOfPair t\u271d g\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : ActionCategory G X \u2964 SingleObj H\nF_map_eq : \u2200 {a b : ActionCategory G X} {f : a \u27f6 b}, F.map f = F.map (homOfPair (ActionCategory.back b) \u2191f)\n\u22a2 (fun g => { left := fun b => F.map (homOfPair b g), right := g }) 1 = 1\n[PROOFSTEP]\ncongr\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : ActionCategory G X \u2964 SingleObj H\nF_map_eq : \u2200 {a b : ActionCategory G X} {f : a \u27f6 b}, F.map f = F.map (homOfPair (ActionCategory.back b) \u2191f)\n\u22a2 (fun g => { left := fun b => F.map (homOfPair b g), right := g }) 1 = 1\n[PROOFSTEP]\ndsimp\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : ActionCategory G X \u2964 SingleObj H\nF_map_eq : \u2200 {a b : ActionCategory G X} {f : a \u27f6 b}, F.map f = F.map (homOfPair (ActionCategory.back b) \u2191f)\n\u22a2 { left := fun b => F.map (homOfPair b 1), right := 1 } = 1\n[PROOFSTEP]\next1\n[GOAL]\ncase left\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : ActionCategory G X \u2964 SingleObj H\nF_map_eq : \u2200 {a b : ActionCategory G X} {f : a \u27f6 b}, F.map f = F.map (homOfPair (ActionCategory.back b) \u2191f)\n\u22a2 { left := fun b => F.map (homOfPair b 1), right := 1 }.left = 1.left\ncase right\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : ActionCategory G X \u2964 SingleObj H\nF_map_eq : \u2200 {a b : ActionCategory G X} {f : a \u27f6 b}, F.map f = F.map (homOfPair (ActionCategory.back b) \u2191f)\n\u22a2 { left := fun b => F.map (homOfPair b 1), right := 1 }.right = 1.right\n[PROOFSTEP]\next b\n[GOAL]\ncase left.h\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : ActionCategory G X \u2964 SingleObj H\nF_map_eq : \u2200 {a b : ActionCategory G X} {f : a \u27f6 b}, F.map f = F.map (homOfPair (ActionCategory.back b) \u2191f)\nb : X\n\u22a2 left { left := fun b => F.map (homOfPair b 1), right := 1 } b = left 1 b\ncase right\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : ActionCategory G X \u2964 SingleObj H\nF_map_eq : \u2200 {a b : ActionCategory G X} {f : a \u27f6 b}, F.map f = F.map (homOfPair (ActionCategory.back b) \u2191f)\n\u22a2 { left := fun b => F.map (homOfPair b 1), right := 1 }.right = 1.right\n[PROOFSTEP]\nexact F_map_eq.symm.trans (F.map_id b)\n[GOAL]\ncase right\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : ActionCategory G X \u2964 SingleObj H\nF_map_eq : \u2200 {a b : ActionCategory G X} {f : a \u27f6 b}, F.map f = F.map (homOfPair (ActionCategory.back b) \u2191f)\n\u22a2 { left := fun b => F.map (homOfPair b 1), right := 1 }.right = 1.right\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : ActionCategory G X \u2964 SingleObj H\nF_map_eq : \u2200 {a b : ActionCategory G X} {f : a \u27f6 b}, F.map f = F.map (homOfPair (ActionCategory.back b) \u2191f)\n\u22a2 \u2200 (x y : G),\n OneHom.toFun\n { toFun := fun g => { left := fun b => F.map (homOfPair b g), right := g },\n map_one' := (_ : (fun g => { left := fun b => F.map (homOfPair b g), right := g }) 1 = 1) }\n (x * y) =\n OneHom.toFun\n { toFun := fun g => { left := fun b => F.map (homOfPair b g), right := g },\n map_one' := (_ : (fun g => { left := fun b => F.map (homOfPair b g), right := g }) 1 = 1) }\n x *\n OneHom.toFun\n { toFun := fun g => { left := fun b => F.map (homOfPair b g), right := g },\n map_one' := (_ : (fun g => { left := fun b => F.map (homOfPair b g), right := g }) 1 = 1) }\n y\n[PROOFSTEP]\nintro g h\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : ActionCategory G X \u2964 SingleObj H\nF_map_eq : \u2200 {a b : ActionCategory G X} {f : a \u27f6 b}, F.map f = F.map (homOfPair (ActionCategory.back b) \u2191f)\ng h : G\n\u22a2 OneHom.toFun\n { toFun := fun g => { left := fun b => F.map (homOfPair b g), right := g },\n map_one' := (_ : (fun g => { left := fun b => F.map (homOfPair b g), right := g }) 1 = 1) }\n (g * h) =\n OneHom.toFun\n { toFun := fun g => { left := fun b => F.map (homOfPair b g), right := g },\n map_one' := (_ : (fun g => { left := fun b => F.map (homOfPair b g), right := g }) 1 = 1) }\n g *\n OneHom.toFun\n { toFun := fun g => { left := fun b => F.map (homOfPair b g), right := g },\n map_one' := (_ : (fun g => { left := fun b => F.map (homOfPair b g), right := g }) 1 = 1) }\n h\n[PROOFSTEP]\ncongr\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : ActionCategory G X \u2964 SingleObj H\nF_map_eq : \u2200 {a b : ActionCategory G X} {f : a \u27f6 b}, F.map f = F.map (homOfPair (ActionCategory.back b) \u2191f)\ng h : G\n\u22a2 OneHom.toFun\n { toFun := fun g => { left := fun b => F.map (homOfPair b g), right := g },\n map_one' := (_ : (fun g => { left := fun b => F.map (homOfPair b g), right := g }) 1 = 1) }\n (g * h) =\n OneHom.toFun\n { toFun := fun g => { left := fun b => F.map (homOfPair b g), right := g },\n map_one' := (_ : (fun g => { left := fun b => F.map (homOfPair b g), right := g }) 1 = 1) }\n g *\n OneHom.toFun\n { toFun := fun g => { left := fun b => F.map (homOfPair b g), right := g },\n map_one' := (_ : (fun g => { left := fun b => F.map (homOfPair b g), right := g }) 1 = 1) }\n h\n[PROOFSTEP]\next b\n[GOAL]\ncase left.h\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : ActionCategory G X \u2964 SingleObj H\nF_map_eq : \u2200 {a b : ActionCategory G X} {f : a \u27f6 b}, F.map f = F.map (homOfPair (ActionCategory.back b) \u2191f)\ng h : G\nb : X\n\u22a2 left\n (OneHom.toFun\n { toFun := fun g => { left := fun b => F.map (homOfPair b g), right := g },\n map_one' := (_ : (fun g => { left := fun b => F.map (homOfPair b g), right := g }) 1 = 1) }\n (g * h))\n b =\n left\n (OneHom.toFun\n { toFun := fun g => { left := fun b => F.map (homOfPair b g), right := g },\n map_one' := (_ : (fun g => { left := fun b => F.map (homOfPair b g), right := g }) 1 = 1) }\n g *\n OneHom.toFun\n { toFun := fun g => { left := fun b => F.map (homOfPair b g), right := g },\n map_one' := (_ : (fun g => { left := fun b => F.map (homOfPair b g), right := g }) 1 = 1) }\n h)\n b\ncase right\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : ActionCategory G X \u2964 SingleObj H\nF_map_eq : \u2200 {a b : ActionCategory G X} {f : a \u27f6 b}, F.map f = F.map (homOfPair (ActionCategory.back b) \u2191f)\ng h : G\n\u22a2 (OneHom.toFun\n { toFun := fun g => { left := fun b => F.map (homOfPair b g), right := g },\n map_one' := (_ : (fun g => { left := fun b => F.map (homOfPair b g), right := g }) 1 = 1) }\n (g * h)).right =\n (OneHom.toFun\n { toFun := fun g => { left := fun b => F.map (homOfPair b g), right := g },\n map_one' := (_ : (fun g => { left := fun b => F.map (homOfPair b g), right := g }) 1 = 1) }\n g *\n OneHom.toFun\n { toFun := fun g => { left := fun b => F.map (homOfPair b g), right := g },\n map_one' := (_ : (fun g => { left := fun b => F.map (homOfPair b g), right := g }) 1 = 1) }\n h).right\n[PROOFSTEP]\nexact F_map_eq.symm.trans (F.map_comp (homOfPair (g\u207b\u00b9 \u2022 b) h) (homOfPair b g))\n[GOAL]\ncase right\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : ActionCategory G X \u2964 SingleObj H\nF_map_eq : \u2200 {a b : ActionCategory G X} {f : a \u27f6 b}, F.map f = F.map (homOfPair (ActionCategory.back b) \u2191f)\ng h : G\n\u22a2 (OneHom.toFun\n { toFun := fun g => { left := fun b => F.map (homOfPair b g), right := g },\n map_one' := (_ : (fun g => { left := fun b => F.map (homOfPair b g), right := g }) 1 = 1) }\n (g * h)).right =\n (OneHom.toFun\n { toFun := fun g => { left := fun b => F.map (homOfPair b g), right := g },\n map_one' := (_ : (fun g => { left := fun b => F.map (homOfPair b g), right := g }) 1 = 1) }\n g *\n OneHom.toFun\n { toFun := fun g => { left := fun b => F.map (homOfPair b g), right := g },\n map_one' := (_ : (fun g => { left := fun b => F.map (homOfPair b g), right := g }) 1 = 1) }\n h).right\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx\u271d : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : G \u2192* (X \u2192 H) \u22ca[mulAutArrow] G\nsane : \u2200 (g : G), (\u2191F g).right = g\nx : ActionCategory G X\n\u22a2 { obj := fun x => (), map := fun {x b} f => left (\u2191F \u2191f) (ActionCategory.back b) }.map (\ud835\udfd9 x) =\n \ud835\udfd9 ({ obj := fun x => (), map := fun {x b} f => left (\u2191F \u2191f) (ActionCategory.back b) }.obj x)\n[PROOFSTEP]\ndsimp\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx\u271d : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : G \u2192* (X \u2192 H) \u22ca[mulAutArrow] G\nsane : \u2200 (g : G), (\u2191F g).right = g\nx : ActionCategory G X\n\u22a2 left (\u2191F 1) (ActionCategory.back x) = \ud835\udfd9 ()\n[PROOFSTEP]\nrw [F.map_one]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx\u271d : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : G \u2192* (X \u2192 H) \u22ca[mulAutArrow] G\nsane : \u2200 (g : G), (\u2191F g).right = g\nx : ActionCategory G X\n\u22a2 left 1 (ActionCategory.back x) = \ud835\udfd9 ()\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : G \u2192* (X \u2192 H) \u22ca[mulAutArrow] G\nsane : \u2200 (g : G), (\u2191F g).right = g\nX\u271d Y\u271d Z\u271d : ActionCategory G X\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun x => (), map := fun {x b} f => left (\u2191F \u2191f) (ActionCategory.back b) }.map (f \u226b g) =\n { obj := fun x => (), map := fun {x b} f => left (\u2191F \u2191f) (ActionCategory.back b) }.map f \u226b\n { obj := fun x => (), map := fun {x b} f => left (\u2191F \u2191f) (ActionCategory.back b) }.map g\n[PROOFSTEP]\nobtain \u27e8_, z, \u03b3\u2081, rfl, rfl, rfl, rfl\u27e9 := ActionCategory.cases' g\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : G \u2192* (X \u2192 H) \u22ca[mulAutArrow] G\nsane : \u2200 (g : G), (\u2191F g).right = g\nX\u271d : ActionCategory G X\nz : X\n\u03b3\u2081 : G\nf : X\u271d \u27f6 { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z }\n\u22a2 { obj := fun x => (), map := fun {x b} f => left (\u2191F \u2191f) (ActionCategory.back b) }.map\n (f \u226b\n eqToHom (_ : { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z } = { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z }) \u226b\n homOfPair z \u03b3\u2081 \u226b eqToHom (_ : { fst := (), snd := z } = { fst := (), snd := z })) =\n { obj := fun x => (), map := fun {x b} f => left (\u2191F \u2191f) (ActionCategory.back b) }.map f \u226b\n { obj := fun x => (), map := fun {x b} f => left (\u2191F \u2191f) (ActionCategory.back b) }.map\n (eqToHom (_ : { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z } = { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z }) \u226b\n homOfPair z \u03b3\u2081 \u226b eqToHom (_ : { fst := (), snd := z } = { fst := (), snd := z }))\n[PROOFSTEP]\nobtain \u27e8_, y, \u03b3\u2082, rfl, hy, rfl, rfl\u27e9 := ActionCategory.cases' f\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : G \u2192* (X \u2192 H) \u22ca[mulAutArrow] G\nsane : \u2200 (g : G), (\u2191F g).right = g\nz : X\n\u03b3\u2081 : G\ny : X\n\u03b3\u2082 : G\nhy : { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z } = { fst := (), snd := y }\n\u22a2 { obj := fun x => (), map := fun {x b} f => left (\u2191F \u2191f) (ActionCategory.back b) }.map\n ((eqToHom (_ : { fst := (), snd := \u03b3\u2082\u207b\u00b9 \u2022 y } = { fst := (), snd := \u03b3\u2082\u207b\u00b9 \u2022 y }) \u226b\n homOfPair y \u03b3\u2082 \u226b eqToHom (_ : { fst := (), snd := y } = { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z })) \u226b\n eqToHom (_ : { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z } = { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z }) \u226b\n homOfPair z \u03b3\u2081 \u226b eqToHom (_ : { fst := (), snd := z } = { fst := (), snd := z })) =\n { obj := fun x => (), map := fun {x b} f => left (\u2191F \u2191f) (ActionCategory.back b) }.map\n (eqToHom (_ : { fst := (), snd := \u03b3\u2082\u207b\u00b9 \u2022 y } = { fst := (), snd := \u03b3\u2082\u207b\u00b9 \u2022 y }) \u226b\n homOfPair y \u03b3\u2082 \u226b eqToHom (_ : { fst := (), snd := y } = { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z })) \u226b\n { obj := fun x => (), map := fun {x b} f => left (\u2191F \u2191f) (ActionCategory.back b) }.map\n (eqToHom (_ : { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z } = { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z }) \u226b\n homOfPair z \u03b3\u2081 \u226b eqToHom (_ : { fst := (), snd := z } = { fst := (), snd := z }))\n[PROOFSTEP]\nobtain rfl : y = \u03b3\u2081\u207b\u00b9 \u2022 z := congr_arg Sigma.snd hy.symm\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : G \u2192* (X \u2192 H) \u22ca[mulAutArrow] G\nsane : \u2200 (g : G), (\u2191F g).right = g\nz : X\n\u03b3\u2081 \u03b3\u2082 : G\nhy : { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z } = { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z }\n\u22a2 { obj := fun x => (), map := fun {x b} f => left (\u2191F \u2191f) (ActionCategory.back b) }.map\n ((eqToHom (_ : { fst := (), snd := \u03b3\u2082\u207b\u00b9 \u2022 \u03b3\u2081\u207b\u00b9 \u2022 z } = { fst := (), snd := \u03b3\u2082\u207b\u00b9 \u2022 \u03b3\u2081\u207b\u00b9 \u2022 z }) \u226b\n homOfPair (\u03b3\u2081\u207b\u00b9 \u2022 z) \u03b3\u2082 \u226b eqToHom (_ : { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z } = { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z })) \u226b\n eqToHom (_ : { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z } = { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z }) \u226b\n homOfPair z \u03b3\u2081 \u226b eqToHom (_ : { fst := (), snd := z } = { fst := (), snd := z })) =\n { obj := fun x => (), map := fun {x b} f => left (\u2191F \u2191f) (ActionCategory.back b) }.map\n (eqToHom (_ : { fst := (), snd := \u03b3\u2082\u207b\u00b9 \u2022 \u03b3\u2081\u207b\u00b9 \u2022 z } = { fst := (), snd := \u03b3\u2082\u207b\u00b9 \u2022 \u03b3\u2081\u207b\u00b9 \u2022 z }) \u226b\n homOfPair (\u03b3\u2081\u207b\u00b9 \u2022 z) \u03b3\u2082 \u226b eqToHom (_ : { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z } = { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z })) \u226b\n { obj := fun x => (), map := fun {x b} f => left (\u2191F \u2191f) (ActionCategory.back b) }.map\n (eqToHom (_ : { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z } = { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z }) \u226b\n homOfPair z \u03b3\u2081 \u226b eqToHom (_ : { fst := (), snd := z } = { fst := (), snd := z }))\n[PROOFSTEP]\nsimp [sane]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nX : Type u\ninst\u271d\u00b3 : MulAction M X\nx : X\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\nH : Type u_3\ninst\u271d : Group H\nF : G \u2192* (X \u2192 H) \u22ca[mulAutArrow] G\nsane : \u2200 (g : G), (\u2191F g).right = g\nz : X\n\u03b3\u2081 \u03b3\u2082 : G\nhy : { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z } = { fst := (), snd := \u03b3\u2081\u207b\u00b9 \u2022 z }\n\u22a2 left (\u2191F \u03b3\u2081) z * (\u03b3\u2081 \u2022 (\u2191F \u03b3\u2082).left) z = left (\u2191F \u03b3\u2082) (\u03b3\u2081\u207b\u00b9 \u2022 z) \u226b left (\u2191F \u03b3\u2081) z\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Action", "llama_tokens": 10049, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.40356685373537454, "lm_q2_score": 0.018833128238333205, "lm_q1q2_score": 0.007600426309138968}} {"text": "[GOAL]\nX : Compactum\nx : X.A\n\u22a2 str X (incl X x) = x\n[PROOFSTEP]\nchange ((\u03b2).\u03b7.app _ \u226b X.a) _ = _\n[GOAL]\nX : Compactum\nx : X.A\n\u22a2 (NatTrans.app (Monad.\u03b7 \u03b2) X.A \u226b X.a) x = x\n[PROOFSTEP]\nrw [Monad.Algebra.unit]\n[GOAL]\nX : Compactum\nx : X.A\n\u22a2 \ud835\udfd9 X.A x = x\n[PROOFSTEP]\nrfl\n[GOAL]\nX Y : Compactum\nf : X \u27f6 Y\nxs : Ultrafilter X.A\n\u22a2 Monad.Algebra.Hom.f f (str X xs) = str Y (Ultrafilter.map f.f xs)\n[PROOFSTEP]\nchange (X.a \u226b f.f) _ = _\n[GOAL]\nX Y : Compactum\nf : X \u27f6 Y\nxs : Ultrafilter X.A\n\u22a2 (X.a \u226b f.f) xs = str Y (Ultrafilter.map f.f xs)\n[PROOFSTEP]\nrw [\u2190 f.h]\n[GOAL]\nX Y : Compactum\nf : X \u27f6 Y\nxs : Ultrafilter X.A\n\u22a2 (\u03b2.toFunctor.map f.f \u226b Y.a) xs = str Y (Ultrafilter.map f.f xs)\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Compactum\nuux : Ultrafilter (Ultrafilter X.A)\n\u22a2 str X (join X uux) = str X (Ultrafilter.map (str X) uux)\n[PROOFSTEP]\nchange ((\u03b2).\u03bc.app _ \u226b X.a) _ = _\n[GOAL]\nX : Compactum\nuux : Ultrafilter (Ultrafilter X.A)\n\u22a2 (NatTrans.app (Monad.\u03bc \u03b2) X.A \u226b X.a) uux = str X (Ultrafilter.map (str X) uux)\n[PROOFSTEP]\nrw [Monad.Algebra.assoc]\n[GOAL]\nX : Compactum\nuux : Ultrafilter (Ultrafilter X.A)\n\u22a2 (\u03b2.toFunctor.map X.a \u226b X.a) uux = str X (Ultrafilter.map (str X) uux)\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Compactum\nS : Set X.A\n\u22a2 IsClosed S \u2194 \u2200 (F : Ultrafilter X.A), S \u2208 F \u2192 str X F \u2208 S\n[PROOFSTEP]\nrw [\u2190 isOpen_compl_iff]\n[GOAL]\nX : Compactum\nS : Set X.A\n\u22a2 IsOpen S\u1d9c \u2194 \u2200 (F : Ultrafilter X.A), S \u2208 F \u2192 str X F \u2208 S\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nX : Compactum\nS : Set X.A\n\u22a2 IsOpen S\u1d9c \u2192 \u2200 (F : Ultrafilter X.A), S \u2208 F \u2192 str X F \u2208 S\n[PROOFSTEP]\nintro cond F h\n[GOAL]\ncase mp\nX : Compactum\nS : Set X.A\ncond : IsOpen S\u1d9c\nF : Ultrafilter X.A\nh : S \u2208 F\n\u22a2 str X F \u2208 S\n[PROOFSTEP]\nby_contra c\n[GOAL]\ncase mp\nX : Compactum\nS : Set X.A\ncond : IsOpen S\u1d9c\nF : Ultrafilter X.A\nh : S \u2208 F\nc : \u00acstr X F \u2208 S\n\u22a2 False\n[PROOFSTEP]\nspecialize cond F c\n[GOAL]\ncase mp\nX : Compactum\nS : Set X.A\nF : Ultrafilter X.A\nh : S \u2208 F\nc : \u00acstr X F \u2208 S\ncond : S\u1d9c \u2208 F\n\u22a2 False\n[PROOFSTEP]\nrw [compl_mem_iff_not_mem] at cond \n[GOAL]\ncase mp\nX : Compactum\nS : Set X.A\nF : Ultrafilter X.A\nh : S \u2208 F\nc : \u00acstr X F \u2208 S\ncond : \u00acS \u2208 F\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase mpr\nX : Compactum\nS : Set X.A\n\u22a2 (\u2200 (F : Ultrafilter X.A), S \u2208 F \u2192 str X F \u2208 S) \u2192 IsOpen S\u1d9c\n[PROOFSTEP]\nintro h1 F h2\n[GOAL]\ncase mpr\nX : Compactum\nS : Set X.A\nh1 : \u2200 (F : Ultrafilter X.A), S \u2208 F \u2192 str X F \u2208 S\nF : Ultrafilter X.A\nh2 : str X F \u2208 S\u1d9c\n\u22a2 S\u1d9c \u2208 F\n[PROOFSTEP]\nspecialize h1 F\n[GOAL]\ncase mpr\nX : Compactum\nS : Set X.A\nF : Ultrafilter X.A\nh2 : str X F \u2208 S\u1d9c\nh1 : S \u2208 F \u2192 str X F \u2208 S\n\u22a2 S\u1d9c \u2208 F\n[PROOFSTEP]\ncases' F.mem_or_compl_mem S with h h\n[GOAL]\ncase mpr.inl\nX : Compactum\nS : Set X.A\nF : Ultrafilter X.A\nh2 : str X F \u2208 S\u1d9c\nh1 : S \u2208 F \u2192 str X F \u2208 S\nh : S \u2208 F\n\u22a2 S\u1d9c \u2208 F\ncase mpr.inr\nX : Compactum\nS : Set X.A\nF : Ultrafilter X.A\nh2 : str X F \u2208 S\u1d9c\nh1 : S \u2208 F \u2192 str X F \u2208 S\nh : S\u1d9c \u2208 F\n\u22a2 S\u1d9c \u2208 F\n[PROOFSTEP]\nexacts [absurd (h1 h) h2, h]\n[GOAL]\nX : Compactum\n\u22a2 CompactSpace X.A\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase isCompact_univ\nX : Compactum\n\u22a2 IsCompact Set.univ\n[PROOFSTEP]\nrw [isCompact_iff_ultrafilter_le_nhds]\n[GOAL]\ncase isCompact_univ\nX : Compactum\n\u22a2 \u2200 (f : Ultrafilter X.A), \u2191f \u2264 \ud835\udcdf Set.univ \u2192 \u2203 a, a \u2208 Set.univ \u2227 \u2191f \u2264 \ud835\udcdd a\n[PROOFSTEP]\nintro F _\n[GOAL]\ncase isCompact_univ\nX : Compactum\nF : Ultrafilter X.A\na\u271d : \u2191F \u2264 \ud835\udcdf Set.univ\n\u22a2 \u2203 a, a \u2208 Set.univ \u2227 \u2191F \u2264 \ud835\udcdd a\n[PROOFSTEP]\nrefine' \u27e8X.str F, by tauto, _\u27e9\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\na\u271d : \u2191F \u2264 \ud835\udcdf Set.univ\n\u22a2 str X F \u2208 Set.univ\n[PROOFSTEP]\ntauto\n[GOAL]\ncase isCompact_univ\nX : Compactum\nF : Ultrafilter X.A\na\u271d : \u2191F \u2264 \ud835\udcdf Set.univ\n\u22a2 \u2191F \u2264 \ud835\udcdd (str X F)\n[PROOFSTEP]\nrw [le_nhds_iff]\n[GOAL]\ncase isCompact_univ\nX : Compactum\nF : Ultrafilter X.A\na\u271d : \u2191F \u2264 \ud835\udcdf Set.univ\n\u22a2 \u2200 (s : Set X.A), str X F \u2208 s \u2192 IsOpen s \u2192 s \u2208 \u2191F\n[PROOFSTEP]\nintro S h1 h2\n[GOAL]\ncase isCompact_univ\nX : Compactum\nF : Ultrafilter X.A\na\u271d : \u2191F \u2264 \ud835\udcdf Set.univ\nS : Set X.A\nh1 : str X F \u2208 S\nh2 : IsOpen S\n\u22a2 S \u2208 \u2191F\n[PROOFSTEP]\nexact h2 F h1\n[GOAL]\nX : Compactum\nA B : Set X.A\n\u22a2 Compactum.basic (A \u2229 B) = Compactum.basic A \u2229 Compactum.basic B\n[PROOFSTEP]\next G\n[GOAL]\ncase h\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\n\u22a2 G \u2208 Compactum.basic (A \u2229 B) \u2194 G \u2208 Compactum.basic A \u2229 Compactum.basic B\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\n\u22a2 G \u2208 Compactum.basic (A \u2229 B) \u2192 G \u2208 Compactum.basic A \u2229 Compactum.basic B\n[PROOFSTEP]\nintro hG\n[GOAL]\ncase h.mp\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\nhG : G \u2208 Compactum.basic (A \u2229 B)\n\u22a2 G \u2208 Compactum.basic A \u2229 Compactum.basic B\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp.left\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\nhG : G \u2208 Compactum.basic (A \u2229 B)\n\u22a2 G \u2208 Compactum.basic A\n[PROOFSTEP]\nfilter_upwards [hG] with _\n[GOAL]\ncase h.mp.right\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\nhG : G \u2208 Compactum.basic (A \u2229 B)\n\u22a2 G \u2208 Compactum.basic B\n[PROOFSTEP]\nfilter_upwards [hG] with _\n[GOAL]\ncase h\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\nhG : G \u2208 Compactum.basic (A \u2229 B)\na\u271d : X.A\n\u22a2 a\u271d \u2208 A \u2229 B \u2192 a\u271d \u2208 A\ncase h X : Compactum A B : Set X.A G : Ultrafilter X.A hG : G \u2208 Compactum.basic (A \u2229 B) a\u271d : X.A \u22a2 a\u271d \u2208 A \u2229 B \u2192 a\u271d \u2208 B\n[PROOFSTEP]\nexacts [And.left, And.right]\n[GOAL]\ncase h.mpr\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\n\u22a2 G \u2208 Compactum.basic A \u2229 Compactum.basic B \u2192 G \u2208 Compactum.basic (A \u2229 B)\n[PROOFSTEP]\nrintro \u27e8h1, h2\u27e9\n[GOAL]\ncase h.mpr.intro\nX : Compactum\nA B : Set X.A\nG : Ultrafilter X.A\nh1 : G \u2208 Compactum.basic A\nh2 : G \u2208 Compactum.basic B\n\u22a2 G \u2208 Compactum.basic (A \u2229 B)\n[PROOFSTEP]\nexact inter_mem h1 h2\n[GOAL]\nX : Compactum\nA : Set X.A\na : X.A\nha : a \u2208 A\n\u22a2 str X (incl X a) = a\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Compactum\nA : Set X.A\n\u22a2 Compactum.cl (Compactum.cl A) \u2286 Compactum.cl A\n[PROOFSTEP]\nrintro _\n \u27e8F, hF, rfl\u27e9\n -- Notation to be used in this proof.\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\n\u22a2 str X F \u2208 Compactum.cl A\n[PROOFSTEP]\nlet fsu := Finset (Set (Ultrafilter X))\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\n\u22a2 str X F \u2208 Compactum.cl A\n[PROOFSTEP]\nlet ssu := Set (Set (Ultrafilter X))\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u22a2 str X F \u2208 Compactum.cl A\n[PROOFSTEP]\nlet \u03b9 : fsu \u2192 ssu := fun x \u21a6 \u2191x\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\n\u22a2 str X F \u2208 Compactum.cl A\n[PROOFSTEP]\nlet C0 : ssu := {Z | \u2203 B \u2208 F, X.str \u207b\u00b9' B = Z}\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\n\u22a2 str X F \u2208 Compactum.cl A\n[PROOFSTEP]\nlet AA := {G : Ultrafilter X | A \u2208 G}\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\n\u22a2 str X F \u2208 Compactum.cl A\n[PROOFSTEP]\nlet C1 := insert AA C0\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\n\u22a2 str X F \u2208 Compactum.cl A\n[PROOFSTEP]\nlet C2 := finiteInterClosure C1\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\n\u22a2 str X F \u2208 Compactum.cl A\n[PROOFSTEP]\nhave claim1 : \u2200 (B) (_ : B \u2208 C0) (C) (_ : C \u2208 C0), B \u2229 C \u2208 C0 :=\n by\n rintro B \u27e8Q, hQ, rfl\u27e9 C \u27e8R, hR, rfl\u27e9\n use Q \u2229 R\n simp only [and_true_iff, eq_self_iff_true, Set.preimage_inter]\n exact inter_sets _ hQ hR\n[GOAL]\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\n\u22a2 \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\n[PROOFSTEP]\nrintro B \u27e8Q, hQ, rfl\u27e9 C \u27e8R, hR, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nQ : Set X.A\nhQ : Q \u2208 F\nR : Set X.A\nhR : R \u2208 F\n\u22a2 str X \u207b\u00b9' Q \u2229 str X \u207b\u00b9' R \u2208 C0\n[PROOFSTEP]\nuse Q \u2229 R\n[GOAL]\ncase h\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nQ : Set X.A\nhQ : Q \u2208 F\nR : Set X.A\nhR : R \u2208 F\n\u22a2 Q \u2229 R \u2208 F \u2227 str X \u207b\u00b9' (Q \u2229 R) = str X \u207b\u00b9' Q \u2229 str X \u207b\u00b9' R\n[PROOFSTEP]\nsimp only [and_true_iff, eq_self_iff_true, Set.preimage_inter]\n[GOAL]\ncase h\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nQ : Set X.A\nhQ : Q \u2208 F\nR : Set X.A\nhR : R \u2208 F\n\u22a2 Q \u2229 R \u2208 F\n[PROOFSTEP]\nexact inter_sets _ hQ hR\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\n\u22a2 str X F \u2208 Compactum.cl A\n[PROOFSTEP]\nhave claim2 : \u2200 B \u2208 C0, Set.Nonempty B := by\n rintro B \u27e8Q, hQ, rfl\u27e9\n obtain \u27e8q\u27e9 := Filter.nonempty_of_mem hQ\n use X.incl q\n simpa\n -- The intersection of AA with every set in C0 is nonempty.\n[GOAL]\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\n\u22a2 \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\n[PROOFSTEP]\nrintro B \u27e8Q, hQ, rfl\u27e9\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nQ : Set X.A\nhQ : Q \u2208 F\n\u22a2 Set.Nonempty (str X \u207b\u00b9' Q)\n[PROOFSTEP]\nobtain \u27e8q\u27e9 := Filter.nonempty_of_mem hQ\n[GOAL]\ncase intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nQ : Set X.A\nhQ : Q \u2208 F\nq : X.A\nh\u271d : q \u2208 Q\n\u22a2 Set.Nonempty (str X \u207b\u00b9' Q)\n[PROOFSTEP]\nuse X.incl q\n[GOAL]\ncase h\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nQ : Set X.A\nhQ : Q \u2208 F\nq : X.A\nh\u271d : q \u2208 Q\n\u22a2 incl X q \u2208 str X \u207b\u00b9' Q\n[PROOFSTEP]\nsimpa\n -- The intersection of AA with every set in C0 is nonempty.\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\n\u22a2 str X F \u2208 Compactum.cl A\n[PROOFSTEP]\nhave claim3 : \u2200 B \u2208 C0, (AA \u2229 B).Nonempty := by\n rintro B \u27e8Q, hQ, rfl\u27e9\n have : (Q \u2229 cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF)\n rcases this with \u27e8q, hq1, P, hq2, hq3\u27e9\n refine' \u27e8P, hq2, _\u27e9\n rw [\u2190 hq3] at hq1 \n simpa\n -- Suffices to show that the intersection of any finite subcollection of C1 is nonempty.\n[GOAL]\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\n\u22a2 \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\n[PROOFSTEP]\nrintro B \u27e8Q, hQ, rfl\u27e9\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nQ : Set X.A\nhQ : Q \u2208 F\n\u22a2 Set.Nonempty (AA \u2229 str X \u207b\u00b9' Q)\n[PROOFSTEP]\nhave : (Q \u2229 cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF)\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nQ : Set X.A\nhQ : Q \u2208 F\nthis : Set.Nonempty (Q \u2229 Compactum.cl A)\n\u22a2 Set.Nonempty (AA \u2229 str X \u207b\u00b9' Q)\n[PROOFSTEP]\nrcases this with \u27e8q, hq1, P, hq2, hq3\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nQ : Set X.A\nhQ : Q \u2208 F\nq : X.A\nhq1 : q \u2208 Q\nP : Ultrafilter X.A\nhq2 : P \u2208 Compactum.basic A\nhq3 : str X P = q\n\u22a2 Set.Nonempty (AA \u2229 str X \u207b\u00b9' Q)\n[PROOFSTEP]\nrefine' \u27e8P, hq2, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nQ : Set X.A\nhQ : Q \u2208 F\nq : X.A\nhq1 : q \u2208 Q\nP : Ultrafilter X.A\nhq2 : P \u2208 Compactum.basic A\nhq3 : str X P = q\n\u22a2 P \u2208 str X \u207b\u00b9' Q\n[PROOFSTEP]\nrw [\u2190 hq3] at hq1 \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nQ : Set X.A\nhQ : Q \u2208 F\nq : X.A\nP : Ultrafilter X.A\nhq1 : str X P \u2208 Q\nhq2 : P \u2208 Compactum.basic A\nhq3 : str X P = q\n\u22a2 P \u2208 str X \u207b\u00b9' Q\n[PROOFSTEP]\nsimpa\n -- Suffices to show that the intersection of any finite subcollection of C1 is nonempty.\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\n\u22a2 str X F \u2208 Compactum.cl A\n[PROOFSTEP]\nsuffices \u2200 T : fsu, \u03b9 T \u2286 C1 \u2192 (\u22c2\u2080 \u03b9 T).Nonempty\n by\n obtain \u27e8G, h1\u27e9 := exists_ultrafilter_of_finite_inter_nonempty _ this\n use X.join G\n have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr \u27e8S, hS, rfl\u27e9)\n rw [join_distrib, this]\n exact\n \u27e8h1 (Or.inl rfl), rfl\u27e9\n -- C2 is closed under finite intersections (by construction!).\n[GOAL]\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nthis : \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)\n\u22a2 str X F \u2208 Compactum.cl A\n[PROOFSTEP]\nobtain \u27e8G, h1\u27e9 := exists_ultrafilter_of_finite_inter_nonempty _ this\n[GOAL]\ncase intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nthis : \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)\nG : Ultrafilter (Ultrafilter X.A)\nh1 : C1 \u2286 G.sets\n\u22a2 str X F \u2208 Compactum.cl A\n[PROOFSTEP]\nuse X.join G\n[GOAL]\ncase h\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nthis : \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)\nG : Ultrafilter (Ultrafilter X.A)\nh1 : C1 \u2286 G.sets\n\u22a2 join X G \u2208 Compactum.basic A \u2227 str X (join X G) = str X F\n[PROOFSTEP]\nhave : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr \u27e8S, hS, rfl\u27e9)\n[GOAL]\ncase h\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nthis\u271d : \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)\nG : Ultrafilter (Ultrafilter X.A)\nh1 : C1 \u2286 G.sets\nthis : Ultrafilter.map (str X) G = F\n\u22a2 join X G \u2208 Compactum.basic A \u2227 str X (join X G) = str X F\n[PROOFSTEP]\nrw [join_distrib, this]\n[GOAL]\ncase h\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nthis\u271d : \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)\nG : Ultrafilter (Ultrafilter X.A)\nh1 : C1 \u2286 G.sets\nthis : Ultrafilter.map (str X) G = F\n\u22a2 join X G \u2208 Compactum.basic A \u2227 str X F = str X F\n[PROOFSTEP]\nexact\n \u27e8h1 (Or.inl rfl), rfl\u27e9\n -- C2 is closed under finite intersections (by construction!).\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\n\u22a2 \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)\n[PROOFSTEP]\nhave claim4 := finiteInterClosure_finiteInter C1\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\n\u22a2 \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)\n[PROOFSTEP]\nhave claim5 : FiniteInter C0 :=\n \u27e8\u27e8_, univ_mem, Set.preimage_univ\u27e9, claim1\u27e9\n -- Every element of C2 is nonempty.\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\n\u22a2 \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)\n[PROOFSTEP]\nhave claim6 : \u2200 P \u2208 C2, (P : Set (Ultrafilter X)).Nonempty :=\n by\n suffices \u2200 P \u2208 C2, P \u2208 C0 \u2228 \u2203 Q \u2208 C0, P = AA \u2229 Q by\n intro P hP\n cases' this P hP with h h\n \u00b7 exact claim2 _ h\n \u00b7 rcases h with \u27e8Q, hQ, rfl\u27e9\n exact claim3 _ hQ\n intro P hP\n exact claim5.finiteInterClosure_insert _ hP\n[GOAL]\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\n\u22a2 \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 Set.Nonempty P\n[PROOFSTEP]\nsuffices \u2200 P \u2208 C2, P \u2208 C0 \u2228 \u2203 Q \u2208 C0, P = AA \u2229 Q by\n intro P hP\n cases' this P hP with h h\n \u00b7 exact claim2 _ h\n \u00b7 rcases h with \u27e8Q, hQ, rfl\u27e9\n exact claim3 _ hQ\n[GOAL]\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nthis : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 P \u2208 C0 \u2228 \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q\n\u22a2 \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 Set.Nonempty P\n[PROOFSTEP]\nintro P hP\n[GOAL]\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nthis : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 P \u2208 C0 \u2228 \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q\nP : Set (Ultrafilter X.A)\nhP : P \u2208 C2\n\u22a2 Set.Nonempty P\n[PROOFSTEP]\ncases' this P hP with h h\n[GOAL]\ncase inl\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nthis : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 P \u2208 C0 \u2228 \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q\nP : Set (Ultrafilter X.A)\nhP : P \u2208 C2\nh : P \u2208 C0\n\u22a2 Set.Nonempty P\n[PROOFSTEP]\nexact claim2 _ h\n[GOAL]\ncase inr\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nthis : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 P \u2208 C0 \u2228 \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q\nP : Set (Ultrafilter X.A)\nhP : P \u2208 C2\nh : \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q\n\u22a2 Set.Nonempty P\n[PROOFSTEP]\nrcases h with \u27e8Q, hQ, rfl\u27e9\n[GOAL]\ncase inr.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nthis : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 P \u2208 C0 \u2228 \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q\nQ : Set (Ultrafilter X.A)\nhQ : Q \u2208 C0\nhP : AA \u2229 Q \u2208 C2\n\u22a2 Set.Nonempty (AA \u2229 Q)\n[PROOFSTEP]\nexact claim3 _ hQ\n[GOAL]\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\n\u22a2 \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 P \u2208 C0 \u2228 \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q\n[PROOFSTEP]\nintro P hP\n[GOAL]\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nP : Set (Ultrafilter X.A)\nhP : P \u2208 C2\n\u22a2 P \u2208 C0 \u2228 \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q\n[PROOFSTEP]\nexact claim5.finiteInterClosure_insert _ hP\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nclaim6 : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 Set.Nonempty P\n\u22a2 \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)\n[PROOFSTEP]\nintro T hT\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nclaim6 : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 Set.Nonempty P\nT : fsu\nhT : \u03b9 T \u2286 C1\n\u22a2 Set.Nonempty (\u22c2\u2080 \u03b9 T)\n[PROOFSTEP]\nsuffices \u22c2\u2080 \u03b9 T \u2208 C2 by exact claim6 _ this\n[GOAL]\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nclaim6 : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 Set.Nonempty P\nT : fsu\nhT : \u03b9 T \u2286 C1\nthis : \u22c2\u2080 \u03b9 T \u2208 C2\n\u22a2 Set.Nonempty (\u22c2\u2080 \u03b9 T)\n[PROOFSTEP]\nexact claim6 _ this\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nclaim6 : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 Set.Nonempty P\nT : fsu\nhT : \u03b9 T \u2286 C1\n\u22a2 \u22c2\u2080 \u03b9 T \u2208 C2\n[PROOFSTEP]\napply claim4.finiteInter_mem T\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nclaim6 : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 Set.Nonempty P\nT : fsu\nhT : \u03b9 T \u2286 C1\n\u22a2 \u2191T \u2286 finiteInterClosure C1\n[PROOFSTEP]\nintro t ht\n[GOAL]\ncase intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nclaim6 : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 Set.Nonempty P\nT : fsu\nhT : \u03b9 T \u2286 C1\nt : Set (Ultrafilter X.A)\nht : t \u2208 \u2191T\n\u22a2 t \u2208 finiteInterClosure C1\n[PROOFSTEP]\nrefine' finiteInterClosure.basic (@hT t ht)\n[GOAL]\nX : Compactum\nA : Set X.A\n\u22a2 IsClosed (Compactum.cl A)\n[PROOFSTEP]\nrw [isClosed_iff]\n[GOAL]\nX : Compactum\nA : Set X.A\n\u22a2 \u2200 (F : Ultrafilter X.A), Compactum.cl A \u2208 F \u2192 str X F \u2208 Compactum.cl A\n[PROOFSTEP]\nintro F hF\n[GOAL]\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : Compactum.cl A \u2208 F\n\u22a2 str X F \u2208 Compactum.cl A\n[PROOFSTEP]\nexact cl_cl _ \u27e8F, hF, rfl\u27e9\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\n\u22a2 \u2191F \u2264 \ud835\udcdd x \u2192 str X F = x\n[PROOFSTEP]\nlet fsu := Finset (Set (Ultrafilter X))\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\n\u22a2 \u2191F \u2264 \ud835\udcdd x \u2192 str X F = x\n[PROOFSTEP]\nlet ssu := Set (Set (Ultrafilter X))\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u22a2 \u2191F \u2264 \ud835\udcdd x \u2192 str X F = x\n[PROOFSTEP]\nlet \u03b9 : fsu \u2192 ssu := fun x \u21a6 \u2191x\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\n\u22a2 \u2191F \u2264 \ud835\udcdd x \u2192 str X F = x\n[PROOFSTEP]\nlet T0 : ssu := {S | \u2203 A \u2208 F, S = basic A}\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\n\u22a2 \u2191F \u2264 \ud835\udcdd x \u2192 str X F = x\n[PROOFSTEP]\nlet AA := X.str \u207b\u00b9' { x }\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\n\u22a2 \u2191F \u2264 \ud835\udcdd x \u2192 str X F = x\n[PROOFSTEP]\nlet T1 := insert AA T0\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\n\u22a2 \u2191F \u2264 \ud835\udcdd x \u2192 str X F = x\n[PROOFSTEP]\nlet T2 := finiteInterClosure T1\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\n\u22a2 \u2191F \u2264 \ud835\udcdd x \u2192 str X F = x\n[PROOFSTEP]\nintro cond\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\n\u22a2 str X F = x\n[PROOFSTEP]\nhave claim1 : \u2200 A : Set X, IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A :=\n by\n intro A hA h\n by_contra H\n rw [le_nhds_iff] at cond \n specialize cond A\u1d9c H hA.isOpen_compl\n rw [Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_not_mem] at cond \n contradiction\n -- If A \u2208 F, then x \u2208 cl A.\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\n\u22a2 \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\n[PROOFSTEP]\nintro A hA h\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nA : Set X.A\nhA : IsClosed A\nh : A \u2208 F\n\u22a2 x \u2208 A\n[PROOFSTEP]\nby_contra H\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nA : Set X.A\nhA : IsClosed A\nh : A \u2208 F\nH : \u00acx \u2208 A\n\u22a2 False\n[PROOFSTEP]\nrw [le_nhds_iff] at cond \n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2200 (s : Set X.A), x \u2208 s \u2192 IsOpen s \u2192 s \u2208 \u2191F\nA : Set X.A\nhA : IsClosed A\nh : A \u2208 F\nH : \u00acx \u2208 A\n\u22a2 False\n[PROOFSTEP]\nspecialize cond A\u1d9c H hA.isOpen_compl\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\nA : Set X.A\nhA : IsClosed A\nh : A \u2208 F\nH : \u00acx \u2208 A\ncond : A\u1d9c \u2208 \u2191F\n\u22a2 False\n[PROOFSTEP]\nrw [Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_not_mem] at cond \n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\nA : Set X.A\nhA : IsClosed A\nh : A \u2208 F\nH : \u00acx \u2208 A\ncond : \u00acA \u2208 F\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n -- If A \u2208 F, then x \u2208 cl A.\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\n\u22a2 str X F = x\n[PROOFSTEP]\nhave claim2 : \u2200 A : Set X, A \u2208 F \u2192 x \u2208 cl A := by\n intro A hA\n exact\n claim1 (cl A) (isClosed_cl A)\n (mem_of_superset hA (subset_cl A))\n -- T0 is closed under intersections.\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\n\u22a2 \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\n[PROOFSTEP]\nintro A hA\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nA : Set X.A\nhA : A \u2208 F\n\u22a2 x \u2208 Compactum.cl A\n[PROOFSTEP]\nexact\n claim1 (cl A) (isClosed_cl A)\n (mem_of_superset hA (subset_cl A))\n -- T0 is closed under intersections.\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\n\u22a2 str X F = x\n[PROOFSTEP]\nhave claim3 : \u2200 (S1) (_ : S1 \u2208 T0) (S2) (_ : S2 \u2208 T0), S1 \u2229 S2 \u2208 T0 :=\n by\n rintro S1 \u27e8S1, hS1, rfl\u27e9 S2 \u27e8S2, hS2, rfl\u27e9\n exact\n \u27e8S1 \u2229 S2, inter_mem hS1 hS2, by simp [basic_inter]\u27e9\n -- For every S \u2208 T0, the intersection AA \u2229 S is nonempty.\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\n\u22a2 \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\n[PROOFSTEP]\nrintro S1 \u27e8S1, hS1, rfl\u27e9 S2 \u27e8S2, hS2, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nS1 : Set X.A\nhS1 : S1 \u2208 F\nS2 : Set X.A\nhS2 : S2 \u2208 F\n\u22a2 Compactum.basic S1 \u2229 Compactum.basic S2 \u2208 T0\n[PROOFSTEP]\nexact\n \u27e8S1 \u2229 S2, inter_mem hS1 hS2, by simp [basic_inter]\u27e9\n -- For every S \u2208 T0, the intersection AA \u2229 S is nonempty.\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nS1 : Set X.A\nhS1 : S1 \u2208 F\nS2 : Set X.A\nhS2 : S2 \u2208 F\n\u22a2 Compactum.basic S1 \u2229 Compactum.basic S2 = Compactum.basic (S1 \u2229 S2)\n[PROOFSTEP]\nsimp [basic_inter]\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\n\u22a2 str X F = x\n[PROOFSTEP]\nhave claim4 : \u2200 S \u2208 T0, (AA \u2229 S).Nonempty := by\n rintro S \u27e8S, hS, rfl\u27e9\n rcases claim2 _ hS with \u27e8G, hG, hG2\u27e9\n exact\n \u27e8G, hG2, hG\u27e9\n -- Every element of T0 is nonempty.\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\n\u22a2 \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\n[PROOFSTEP]\nrintro S \u27e8S, hS, rfl\u27e9\n[GOAL]\ncase intro.intro\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nS : Set X.A\nhS : S \u2208 F\n\u22a2 Set.Nonempty (AA \u2229 Compactum.basic S)\n[PROOFSTEP]\nrcases claim2 _ hS with \u27e8G, hG, hG2\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nS : Set X.A\nhS : S \u2208 F\nG : Ultrafilter X.A\nhG : G \u2208 Compactum.basic S\nhG2 : str X G = x\n\u22a2 Set.Nonempty (AA \u2229 Compactum.basic S)\n[PROOFSTEP]\nexact\n \u27e8G, hG2, hG\u27e9\n -- Every element of T0 is nonempty.\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\n\u22a2 str X F = x\n[PROOFSTEP]\nhave claim5 : \u2200 S \u2208 T0, Set.Nonempty S := by\n rintro S \u27e8S, hS, rfl\u27e9\n exact\n \u27e8F, hS\u27e9\n -- Every element of T2 is nonempty.\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\n\u22a2 \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\n[PROOFSTEP]\nrintro S \u27e8S, hS, rfl\u27e9\n[GOAL]\ncase intro.intro\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nS : Set X.A\nhS : S \u2208 F\n\u22a2 Set.Nonempty (Compactum.basic S)\n[PROOFSTEP]\nexact\n \u27e8F, hS\u27e9\n -- Every element of T2 is nonempty.\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\n\u22a2 str X F = x\n[PROOFSTEP]\nhave claim6 : \u2200 S \u2208 T2, Set.Nonempty S :=\n by\n suffices \u2200 S \u2208 T2, S \u2208 T0 \u2228 \u2203 Q \u2208 T0, S = AA \u2229 Q by\n intro S hS\n cases' this _ hS with h h\n \u00b7 exact claim5 S h\n \u00b7 rcases h with \u27e8Q, hQ, rfl\u27e9\n exact claim4 Q hQ\n intro S hS\n apply finiteInterClosure_insert\n \u00b7 constructor\n \u00b7 use Set.univ\n refine' \u27e8Filter.univ_sets _, _\u27e9\n ext\n refine' \u27e8_, by tauto\u27e9\n \u00b7 intro\n apply Filter.univ_sets\n \u00b7 exact claim3\n \u00b7 exact hS\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\n\u22a2 \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 Set.Nonempty S\n[PROOFSTEP]\nsuffices \u2200 S \u2208 T2, S \u2208 T0 \u2228 \u2203 Q \u2208 T0, S = AA \u2229 Q by\n intro S hS\n cases' this _ hS with h h\n \u00b7 exact claim5 S h\n \u00b7 rcases h with \u27e8Q, hQ, rfl\u27e9\n exact claim4 Q hQ\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nthis : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 S \u2208 T0 \u2228 \u2203 Q, Q \u2208 T0 \u2227 S = AA \u2229 Q\n\u22a2 \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 Set.Nonempty S\n[PROOFSTEP]\nintro S hS\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nthis : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 S \u2208 T0 \u2228 \u2203 Q, Q \u2208 T0 \u2227 S = AA \u2229 Q\nS : Set (Ultrafilter X.A)\nhS : S \u2208 T2\n\u22a2 Set.Nonempty S\n[PROOFSTEP]\ncases' this _ hS with h h\n[GOAL]\ncase inl\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nthis : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 S \u2208 T0 \u2228 \u2203 Q, Q \u2208 T0 \u2227 S = AA \u2229 Q\nS : Set (Ultrafilter X.A)\nhS : S \u2208 T2\nh : S \u2208 T0\n\u22a2 Set.Nonempty S\n[PROOFSTEP]\nexact claim5 S h\n[GOAL]\ncase inr\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nthis : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 S \u2208 T0 \u2228 \u2203 Q, Q \u2208 T0 \u2227 S = AA \u2229 Q\nS : Set (Ultrafilter X.A)\nhS : S \u2208 T2\nh : \u2203 Q, Q \u2208 T0 \u2227 S = AA \u2229 Q\n\u22a2 Set.Nonempty S\n[PROOFSTEP]\nrcases h with \u27e8Q, hQ, rfl\u27e9\n[GOAL]\ncase inr.intro.intro\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nthis : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 S \u2208 T0 \u2228 \u2203 Q, Q \u2208 T0 \u2227 S = AA \u2229 Q\nQ : Set (Ultrafilter X.A)\nhQ : Q \u2208 T0\nhS : AA \u2229 Q \u2208 T2\n\u22a2 Set.Nonempty (AA \u2229 Q)\n[PROOFSTEP]\nexact claim4 Q hQ\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\n\u22a2 \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 S \u2208 T0 \u2228 \u2203 Q, Q \u2208 T0 \u2227 S = AA \u2229 Q\n[PROOFSTEP]\nintro S hS\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nS : Set (Ultrafilter X.A)\nhS : S \u2208 T2\n\u22a2 S \u2208 T0 \u2228 \u2203 Q, Q \u2208 T0 \u2227 S = AA \u2229 Q\n[PROOFSTEP]\napply finiteInterClosure_insert\n[GOAL]\ncase cond\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nS : Set (Ultrafilter X.A)\nhS : S \u2208 T2\n\u22a2 FiniteInter T0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cond.univ_mem\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nS : Set (Ultrafilter X.A)\nhS : S \u2208 T2\n\u22a2 Set.univ \u2208 T0\n[PROOFSTEP]\nuse Set.univ\n[GOAL]\ncase h\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nS : Set (Ultrafilter X.A)\nhS : S \u2208 T2\n\u22a2 Set.univ \u2208 F \u2227 Set.univ = Compactum.basic Set.univ\n[PROOFSTEP]\nrefine' \u27e8Filter.univ_sets _, _\u27e9\n[GOAL]\ncase h\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nS : Set (Ultrafilter X.A)\nhS : S \u2208 T2\n\u22a2 Set.univ = Compactum.basic Set.univ\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nS : Set (Ultrafilter X.A)\nhS : S \u2208 T2\nx\u271d : Ultrafilter X.A\n\u22a2 x\u271d \u2208 Set.univ \u2194 x\u271d \u2208 Compactum.basic Set.univ\n[PROOFSTEP]\nrefine' \u27e8_, by tauto\u27e9\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nS : Set (Ultrafilter X.A)\nhS : S \u2208 T2\nx\u271d : Ultrafilter X.A\n\u22a2 x\u271d \u2208 Compactum.basic Set.univ \u2192 x\u271d \u2208 Set.univ\n[PROOFSTEP]\ntauto\n[GOAL]\ncase h.h\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nS : Set (Ultrafilter X.A)\nhS : S \u2208 T2\nx\u271d : Ultrafilter X.A\n\u22a2 x\u271d \u2208 Set.univ \u2192 x\u271d \u2208 Compactum.basic Set.univ\n[PROOFSTEP]\nintro\n[GOAL]\ncase h.h\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nS : Set (Ultrafilter X.A)\nhS : S \u2208 T2\nx\u271d : Ultrafilter X.A\na\u271d : x\u271d \u2208 Set.univ\n\u22a2 x\u271d \u2208 Compactum.basic Set.univ\n[PROOFSTEP]\napply Filter.univ_sets\n[GOAL]\ncase cond.inter_mem\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nS : Set (Ultrafilter X.A)\nhS : S \u2208 T2\n\u22a2 \u2200 \u2983s : Set (Ultrafilter X.A)\u2984, s \u2208 T0 \u2192 \u2200 \u2983t : Set (Ultrafilter X.A)\u2984, t \u2208 T0 \u2192 s \u2229 t \u2208 T0\n[PROOFSTEP]\nexact claim3\n[GOAL]\ncase H\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nS : Set (Ultrafilter X.A)\nhS : S \u2208 T2\n\u22a2 S \u2208 finiteInterClosure (insert AA T0)\n[PROOFSTEP]\nexact hS\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nclaim6 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 Set.Nonempty S\n\u22a2 str X F = x\n[PROOFSTEP]\nsuffices \u2200 F : fsu, \u2191F \u2286 T1 \u2192 (\u22c2\u2080 \u03b9 F).Nonempty\n by\n obtain \u27e8G, h1\u27e9 := Ultrafilter.exists_ultrafilter_of_finite_inter_nonempty _ this\n have c1 : X.join G = F := Ultrafilter.coe_le_coe.1 fun P hP => h1 (Or.inr \u27e8P, hP, rfl\u27e9)\n have c2 : G.map X.str = X.incl x :=\n by\n refine' Ultrafilter.coe_le_coe.1 fun P hP => _\n apply mem_of_superset (h1 (Or.inl rfl))\n rintro x \u27e8rfl\u27e9\n exact hP\n simp [\u2190 c1, c2]\n -- Finish...\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nclaim6 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 Set.Nonempty S\nthis : \u2200 (F : fsu), \u2191F \u2286 T1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 F)\n\u22a2 str X F = x\n[PROOFSTEP]\nobtain \u27e8G, h1\u27e9 := Ultrafilter.exists_ultrafilter_of_finite_inter_nonempty _ this\n[GOAL]\ncase intro\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nclaim6 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 Set.Nonempty S\nthis : \u2200 (F : fsu), \u2191F \u2286 T1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 F)\nG : Ultrafilter (Ultrafilter X.A)\nh1 : T1 \u2286 G.sets\n\u22a2 str X F = x\n[PROOFSTEP]\nhave c1 : X.join G = F := Ultrafilter.coe_le_coe.1 fun P hP => h1 (Or.inr \u27e8P, hP, rfl\u27e9)\n[GOAL]\ncase intro\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nclaim6 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 Set.Nonempty S\nthis : \u2200 (F : fsu), \u2191F \u2286 T1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 F)\nG : Ultrafilter (Ultrafilter X.A)\nh1 : T1 \u2286 G.sets\nc1 : join X G = F\n\u22a2 str X F = x\n[PROOFSTEP]\nhave c2 : G.map X.str = X.incl x := by\n refine' Ultrafilter.coe_le_coe.1 fun P hP => _\n apply mem_of_superset (h1 (Or.inl rfl))\n rintro x \u27e8rfl\u27e9\n exact hP\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nclaim6 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 Set.Nonempty S\nthis : \u2200 (F : fsu), \u2191F \u2286 T1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 F)\nG : Ultrafilter (Ultrafilter X.A)\nh1 : T1 \u2286 G.sets\nc1 : join X G = F\n\u22a2 Ultrafilter.map (str X) G = incl X x\n[PROOFSTEP]\nrefine' Ultrafilter.coe_le_coe.1 fun P hP => _\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nclaim6 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 Set.Nonempty S\nthis : \u2200 (F : fsu), \u2191F \u2286 T1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 F)\nG : Ultrafilter (Ultrafilter X.A)\nh1 : T1 \u2286 G.sets\nc1 : join X G = F\nP : Set X.A\nhP : P \u2208 \u2191(incl X x)\n\u22a2 P \u2208 \u2191(Ultrafilter.map (str X) G)\n[PROOFSTEP]\napply mem_of_superset (h1 (Or.inl rfl))\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nclaim6 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 Set.Nonempty S\nthis : \u2200 (F : fsu), \u2191F \u2286 T1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 F)\nG : Ultrafilter (Ultrafilter X.A)\nh1 : T1 \u2286 G.sets\nc1 : join X G = F\nP : Set X.A\nhP : P \u2208 \u2191(incl X x)\n\u22a2 AA \u2286 str X \u207b\u00b9' P\n[PROOFSTEP]\nrintro x \u27e8rfl\u27e9\n[GOAL]\ncase refl\nX : Compactum\nF : Ultrafilter X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nG : Ultrafilter (Ultrafilter X.A)\nc1 : join X G = F\nP : Set X.A\nx : Ultrafilter X.A\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {str X x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd (str X x)\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 str X x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 str X x \u2208 Compactum.cl A\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim6 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 Set.Nonempty S\nthis : \u2200 (F : fsu), \u2191F \u2286 T1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 F)\nh1 : T1 \u2286 G.sets\nhP : P \u2208 \u2191(incl X (str X x))\n\u22a2 x \u2208 str X \u207b\u00b9' P\n[PROOFSTEP]\nexact hP\n[GOAL]\ncase intro\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nclaim6 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 Set.Nonempty S\nthis : \u2200 (F : fsu), \u2191F \u2286 T1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 F)\nG : Ultrafilter (Ultrafilter X.A)\nh1 : T1 \u2286 G.sets\nc1 : join X G = F\nc2 : Ultrafilter.map (str X) G = incl X x\n\u22a2 str X F = x\n[PROOFSTEP]\nsimp [\u2190 c1, c2]\n -- Finish...\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nclaim6 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 Set.Nonempty S\n\u22a2 \u2200 (F : fsu), \u2191F \u2286 T1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 F)\n[PROOFSTEP]\nintro T hT\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nclaim6 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 Set.Nonempty S\nT : fsu\nhT : \u2191T \u2286 T1\n\u22a2 Set.Nonempty (\u22c2\u2080 \u03b9 T)\n[PROOFSTEP]\nrefine' claim6 _ (finiteInter_mem (.finiteInterClosure_finiteInter _) _ _)\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nclaim6 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 Set.Nonempty S\nT : fsu\nhT : \u2191T \u2286 T1\n\u22a2 \u2191T \u2286 finiteInterClosure T1\n[PROOFSTEP]\nintro t ht\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nT0 : ssu := {S | \u2203 A, A \u2208 F \u2227 S = Compactum.basic A}\nAA : Set (Ultrafilter X.A) := str X \u207b\u00b9' {x}\nT1 : ssu := insert AA T0\nT2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1\ncond : \u2191F \u2264 \ud835\udcdd x\nclaim1 : \u2200 (A : Set X.A), IsClosed A \u2192 A \u2208 F \u2192 x \u2208 A\nclaim2 : \u2200 (A : Set X.A), A \u2208 F \u2192 x \u2208 Compactum.cl A\nclaim3 : \u2200 (S1 : Set (Ultrafilter X.A)), S1 \u2208 T0 \u2192 \u2200 (S2 : Set (Ultrafilter X.A)), S2 \u2208 T0 \u2192 S1 \u2229 S2 \u2208 T0\nclaim4 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty (AA \u2229 S)\nclaim5 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T0 \u2192 Set.Nonempty S\nclaim6 : \u2200 (S : Set (Ultrafilter X.A)), S \u2208 T2 \u2192 Set.Nonempty S\nT : fsu\nhT : \u2191T \u2286 T1\nt : Set (Ultrafilter X.A)\nht : t \u2208 \u2191T\n\u22a2 t \u2208 finiteInterClosure T1\n[PROOFSTEP]\nexact finiteInterClosure.basic (@hT t ht)\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\nx : X.A\nh : str X F = x\ns : Set X.A\nhx : x \u2208 s\nhs : IsOpen s\n\u22a2 str X F \u2208 s\n[PROOFSTEP]\nrwa [h]\n[GOAL]\nX : Compactum\n\u22a2 T2Space X.A\n[PROOFSTEP]\nrw [t2_iff_ultrafilter]\n[GOAL]\nX : Compactum\n\u22a2 \u2200 {x y : X.A} (f : Ultrafilter X.A), \u2191f \u2264 \ud835\udcdd x \u2192 \u2191f \u2264 \ud835\udcdd y \u2192 x = y\n[PROOFSTEP]\nintro _ _ F hx hy\n[GOAL]\nX : Compactum\nx\u271d y\u271d : X.A\nF : Ultrafilter X.A\nhx : \u2191F \u2264 \ud835\udcdd x\u271d\nhy : \u2191F \u2264 \ud835\udcdd y\u271d\n\u22a2 x\u271d = y\u271d\n[PROOFSTEP]\nrw [\u2190 str_eq_of_le_nhds _ _ hx, \u2190 str_eq_of_le_nhds _ _ hy]\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\n\u22a2 Ultrafilter.lim F = str X F\n[PROOFSTEP]\nrw [Ultrafilter.lim_eq_iff_le_nhds, le_nhds_iff]\n[GOAL]\nX : Compactum\nF : Ultrafilter X.A\n\u22a2 \u2200 (s : Set X.A), str X F \u2208 s \u2192 IsOpen s \u2192 s \u2208 \u2191F\n[PROOFSTEP]\ntauto\n[GOAL]\nX : Compactum\nA : Set X.A\n\u22a2 Compactum.cl A = closure A\n[PROOFSTEP]\next\n[GOAL]\ncase h\nX : Compactum\nA : Set X.A\nx\u271d : X.A\n\u22a2 x\u271d \u2208 Compactum.cl A \u2194 x\u271d \u2208 closure A\n[PROOFSTEP]\nrw [mem_closure_iff_ultrafilter]\n[GOAL]\ncase h\nX : Compactum\nA : Set X.A\nx\u271d : X.A\n\u22a2 x\u271d \u2208 Compactum.cl A \u2194 \u2203 u, A \u2208 u \u2227 \u2191u \u2264 \ud835\udcdd x\u271d\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nX : Compactum\nA : Set X.A\nx\u271d : X.A\n\u22a2 x\u271d \u2208 Compactum.cl A \u2192 \u2203 u, A \u2208 u \u2227 \u2191u \u2264 \ud835\udcdd x\u271d\n[PROOFSTEP]\nrintro \u27e8F, h1, h2\u27e9\n[GOAL]\ncase h.mp.intro.intro\nX : Compactum\nA : Set X.A\nx\u271d : X.A\nF : Ultrafilter X.A\nh1 : F \u2208 Compactum.basic A\nh2 : str X F = x\u271d\n\u22a2 \u2203 u, A \u2208 u \u2227 \u2191u \u2264 \ud835\udcdd x\u271d\n[PROOFSTEP]\nexact \u27e8F, h1, le_nhds_of_str_eq _ _ h2\u27e9\n[GOAL]\ncase h.mpr\nX : Compactum\nA : Set X.A\nx\u271d : X.A\n\u22a2 (\u2203 u, A \u2208 u \u2227 \u2191u \u2264 \ud835\udcdd x\u271d) \u2192 x\u271d \u2208 Compactum.cl A\n[PROOFSTEP]\nrintro \u27e8F, h1, h2\u27e9\n[GOAL]\ncase h.mpr.intro.intro\nX : Compactum\nA : Set X.A\nx\u271d : X.A\nF : Ultrafilter X.A\nh1 : A \u2208 F\nh2 : \u2191F \u2264 \ud835\udcdd x\u271d\n\u22a2 x\u271d \u2208 Compactum.cl A\n[PROOFSTEP]\nexact \u27e8F, h1, str_eq_of_le_nhds _ _ h2\u27e9\n[GOAL]\nX Y : Compactum\nf : X \u27f6 Y\n\u22a2 Continuous f.f\n[PROOFSTEP]\nrw [continuous_iff_ultrafilter]\n[GOAL]\nX Y : Compactum\nf : X \u27f6 Y\n\u22a2 \u2200 (x : X.A) (g : Ultrafilter X.A), \u2191g \u2264 \ud835\udcdd x \u2192 Tendsto f.f (\u2191g) (\ud835\udcdd (Monad.Algebra.Hom.f f x))\n[PROOFSTEP]\nintro x g h\n[GOAL]\nX Y : Compactum\nf : X \u27f6 Y\nx : X.A\ng : Ultrafilter X.A\nh : \u2191g \u2264 \ud835\udcdd x\n\u22a2 Tendsto f.f (\u2191g) (\ud835\udcdd (Monad.Algebra.Hom.f f x))\n[PROOFSTEP]\nrw [Tendsto, \u2190 coe_map]\n[GOAL]\nX Y : Compactum\nf : X \u27f6 Y\nx : X.A\ng : Ultrafilter X.A\nh : \u2191g \u2264 \ud835\udcdd x\n\u22a2 \u2191(Ultrafilter.map f.f g) \u2264 \ud835\udcdd (Monad.Algebra.Hom.f f x)\n[PROOFSTEP]\napply le_nhds_of_str_eq\n[GOAL]\ncase a\nX Y : Compactum\nf : X \u27f6 Y\nx : X.A\ng : Ultrafilter X.A\nh : \u2191g \u2264 \ud835\udcdd x\n\u22a2 str Y (Ultrafilter.map f.f g) = Monad.Algebra.Hom.f f x\n[PROOFSTEP]\nrw [\u2190 str_hom_commute, str_eq_of_le_nhds _ x _]\n[GOAL]\nX Y : Compactum\nf : X \u27f6 Y\nx : X.A\ng : Ultrafilter X.A\nh : \u2191g \u2264 \ud835\udcdd x\n\u22a2 \u2191g \u2264 \ud835\udcdd x\n[PROOFSTEP]\napply h\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\n\u22a2 NatTrans.app (Monad.\u03b7 \u03b2) X \u226b Ultrafilter.lim = \ud835\udfd9 X\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nx : (\ud835\udfed (Type u_1)).obj X\n\u22a2 (NatTrans.app (Monad.\u03b7 \u03b2) X \u226b Ultrafilter.lim) x = \ud835\udfd9 X x\n[PROOFSTEP]\nexact lim_eq (pure_le_nhds _)\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\n\u22a2 NatTrans.app (Monad.\u03bc \u03b2) X \u226b Ultrafilter.lim = \u03b2.toFunctor.map Ultrafilter.lim \u226b Ultrafilter.lim\n[PROOFSTEP]\next FF\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nFF : (\u03b2.toFunctor \u22d9 \u03b2.toFunctor).obj X\n\u22a2 (NatTrans.app (Monad.\u03bc \u03b2) X \u226b Ultrafilter.lim) FF = (\u03b2.toFunctor.map Ultrafilter.lim \u226b Ultrafilter.lim) FF\n[PROOFSTEP]\nchange Ultrafilter (Ultrafilter X) at FF \n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nFF : Ultrafilter (Ultrafilter X)\n\u22a2 (NatTrans.app (Monad.\u03bc \u03b2) X \u226b Ultrafilter.lim) FF = (\u03b2.toFunctor.map Ultrafilter.lim \u226b Ultrafilter.lim) FF\n[PROOFSTEP]\nset x := (Ultrafilter.map Ultrafilter.lim FF).lim with c1\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nFF : Ultrafilter (Ultrafilter X)\nx : X := Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc1 : x = Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\n\u22a2 (NatTrans.app (Monad.\u03bc \u03b2) X \u226b Ultrafilter.lim) FF = (\u03b2.toFunctor.map Ultrafilter.lim \u226b Ultrafilter.lim) FF\n[PROOFSTEP]\nhave c2 : \u2200 (U : Set X) (F : Ultrafilter X), F.lim \u2208 U \u2192 IsOpen U \u2192 U \u2208 F :=\n by\n intro U F h1 hU\n exact isOpen_iff_ultrafilter.mp hU _ h1 _ (Ultrafilter.le_nhds_lim _)\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nFF : Ultrafilter (Ultrafilter X)\nx : X := Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc1 : x = Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\n\u22a2 \u2200 (U : Set X) (F : Ultrafilter X), Ultrafilter.lim F \u2208 U \u2192 IsOpen U \u2192 U \u2208 F\n[PROOFSTEP]\nintro U F h1 hU\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nFF : Ultrafilter (Ultrafilter X)\nx : X := Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc1 : x = Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nU : Set X\nF : Ultrafilter X\nh1 : Ultrafilter.lim F \u2208 U\nhU : IsOpen U\n\u22a2 U \u2208 F\n[PROOFSTEP]\nexact isOpen_iff_ultrafilter.mp hU _ h1 _ (Ultrafilter.le_nhds_lim _)\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nFF : Ultrafilter (Ultrafilter X)\nx : X := Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc1 : x = Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc2 : \u2200 (U : Set X) (F : Ultrafilter X), Ultrafilter.lim F \u2208 U \u2192 IsOpen U \u2192 U \u2208 F\n\u22a2 (NatTrans.app (Monad.\u03bc \u03b2) X \u226b Ultrafilter.lim) FF = (\u03b2.toFunctor.map Ultrafilter.lim \u226b Ultrafilter.lim) FF\n[PROOFSTEP]\nhave c3 : \u2191(Ultrafilter.map Ultrafilter.lim FF) \u2264 \ud835\udcdd x :=\n by\n rw [le_nhds_iff]\n intro U hx hU\n exact mem_coe.2 (c2 _ _ (by rwa [\u2190 c1]) hU)\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nFF : Ultrafilter (Ultrafilter X)\nx : X := Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc1 : x = Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc2 : \u2200 (U : Set X) (F : Ultrafilter X), Ultrafilter.lim F \u2208 U \u2192 IsOpen U \u2192 U \u2208 F\n\u22a2 \u2191(Ultrafilter.map Ultrafilter.lim FF) \u2264 \ud835\udcdd x\n[PROOFSTEP]\nrw [le_nhds_iff]\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nFF : Ultrafilter (Ultrafilter X)\nx : X := Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc1 : x = Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc2 : \u2200 (U : Set X) (F : Ultrafilter X), Ultrafilter.lim F \u2208 U \u2192 IsOpen U \u2192 U \u2208 F\n\u22a2 \u2200 (s : Set X), x \u2208 s \u2192 IsOpen s \u2192 s \u2208 \u2191(Ultrafilter.map Ultrafilter.lim FF)\n[PROOFSTEP]\nintro U hx hU\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nFF : Ultrafilter (Ultrafilter X)\nx : X := Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc1 : x = Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc2 : \u2200 (U : Set X) (F : Ultrafilter X), Ultrafilter.lim F \u2208 U \u2192 IsOpen U \u2192 U \u2208 F\nU : Set X\nhx : x \u2208 U\nhU : IsOpen U\n\u22a2 U \u2208 \u2191(Ultrafilter.map Ultrafilter.lim FF)\n[PROOFSTEP]\nexact mem_coe.2 (c2 _ _ (by rwa [\u2190 c1]) hU)\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nFF : Ultrafilter (Ultrafilter X)\nx : X := Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc1 : x = Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc2 : \u2200 (U : Set X) (F : Ultrafilter X), Ultrafilter.lim F \u2208 U \u2192 IsOpen U \u2192 U \u2208 F\nU : Set X\nhx : x \u2208 U\nhU : IsOpen U\n\u22a2 Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF) \u2208 U\n[PROOFSTEP]\nrwa [\u2190 c1]\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nFF : Ultrafilter (Ultrafilter X)\nx : X := Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc1 : x = Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc2 : \u2200 (U : Set X) (F : Ultrafilter X), Ultrafilter.lim F \u2208 U \u2192 IsOpen U \u2192 U \u2208 F\nc3 : \u2191(Ultrafilter.map Ultrafilter.lim FF) \u2264 \ud835\udcdd x\n\u22a2 (NatTrans.app (Monad.\u03bc \u03b2) X \u226b Ultrafilter.lim) FF = (\u03b2.toFunctor.map Ultrafilter.lim \u226b Ultrafilter.lim) FF\n[PROOFSTEP]\nhave c4 : \u2200 U : Set X, x \u2208 U \u2192 IsOpen U \u2192 {G : Ultrafilter X | U \u2208 G} \u2208 FF :=\n by\n intro U hx hU\n suffices Ultrafilter.lim \u207b\u00b9' U \u2208 FF by\n apply mem_of_superset this\n intro P hP\n exact c2 U P hP hU\n exact @c3 U (IsOpen.mem_nhds hU hx)\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nFF : Ultrafilter (Ultrafilter X)\nx : X := Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc1 : x = Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc2 : \u2200 (U : Set X) (F : Ultrafilter X), Ultrafilter.lim F \u2208 U \u2192 IsOpen U \u2192 U \u2208 F\nc3 : \u2191(Ultrafilter.map Ultrafilter.lim FF) \u2264 \ud835\udcdd x\n\u22a2 \u2200 (U : Set X), x \u2208 U \u2192 IsOpen U \u2192 {G | U \u2208 G} \u2208 FF\n[PROOFSTEP]\nintro U hx hU\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nFF : Ultrafilter (Ultrafilter X)\nx : X := Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc1 : x = Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc2 : \u2200 (U : Set X) (F : Ultrafilter X), Ultrafilter.lim F \u2208 U \u2192 IsOpen U \u2192 U \u2208 F\nc3 : \u2191(Ultrafilter.map Ultrafilter.lim FF) \u2264 \ud835\udcdd x\nU : Set X\nhx : x \u2208 U\nhU : IsOpen U\n\u22a2 {G | U \u2208 G} \u2208 FF\n[PROOFSTEP]\nsuffices Ultrafilter.lim \u207b\u00b9' U \u2208 FF by\n apply mem_of_superset this\n intro P hP\n exact c2 U P hP hU\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nFF : Ultrafilter (Ultrafilter X)\nx : X := Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc1 : x = Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc2 : \u2200 (U : Set X) (F : Ultrafilter X), Ultrafilter.lim F \u2208 U \u2192 IsOpen U \u2192 U \u2208 F\nc3 : \u2191(Ultrafilter.map Ultrafilter.lim FF) \u2264 \ud835\udcdd x\nU : Set X\nhx : x \u2208 U\nhU : IsOpen U\nthis : Ultrafilter.lim \u207b\u00b9' U \u2208 FF\n\u22a2 {G | U \u2208 G} \u2208 FF\n[PROOFSTEP]\napply mem_of_superset this\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nFF : Ultrafilter (Ultrafilter X)\nx : X := Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc1 : x = Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc2 : \u2200 (U : Set X) (F : Ultrafilter X), Ultrafilter.lim F \u2208 U \u2192 IsOpen U \u2192 U \u2208 F\nc3 : \u2191(Ultrafilter.map Ultrafilter.lim FF) \u2264 \ud835\udcdd x\nU : Set X\nhx : x \u2208 U\nhU : IsOpen U\nthis : Ultrafilter.lim \u207b\u00b9' U \u2208 FF\n\u22a2 Ultrafilter.lim \u207b\u00b9' U \u2286 {G | U \u2208 G}\n[PROOFSTEP]\nintro P hP\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nFF : Ultrafilter (Ultrafilter X)\nx : X := Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc1 : x = Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc2 : \u2200 (U : Set X) (F : Ultrafilter X), Ultrafilter.lim F \u2208 U \u2192 IsOpen U \u2192 U \u2208 F\nc3 : \u2191(Ultrafilter.map Ultrafilter.lim FF) \u2264 \ud835\udcdd x\nU : Set X\nhx : x \u2208 U\nhU : IsOpen U\nthis : Ultrafilter.lim \u207b\u00b9' U \u2208 FF\nP : Ultrafilter X\nhP : P \u2208 Ultrafilter.lim \u207b\u00b9' U\n\u22a2 P \u2208 {G | U \u2208 G}\n[PROOFSTEP]\nexact c2 U P hP hU\n[GOAL]\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nFF : Ultrafilter (Ultrafilter X)\nx : X := Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc1 : x = Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc2 : \u2200 (U : Set X) (F : Ultrafilter X), Ultrafilter.lim F \u2208 U \u2192 IsOpen U \u2192 U \u2208 F\nc3 : \u2191(Ultrafilter.map Ultrafilter.lim FF) \u2264 \ud835\udcdd x\nU : Set X\nhx : x \u2208 U\nhU : IsOpen U\n\u22a2 Ultrafilter.lim \u207b\u00b9' U \u2208 FF\n[PROOFSTEP]\nexact @c3 U (IsOpen.mem_nhds hU hx)\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nFF : Ultrafilter (Ultrafilter X)\nx : X := Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc1 : x = Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc2 : \u2200 (U : Set X) (F : Ultrafilter X), Ultrafilter.lim F \u2208 U \u2192 IsOpen U \u2192 U \u2208 F\nc3 : \u2191(Ultrafilter.map Ultrafilter.lim FF) \u2264 \ud835\udcdd x\nc4 : \u2200 (U : Set X), x \u2208 U \u2192 IsOpen U \u2192 {G | U \u2208 G} \u2208 FF\n\u22a2 (NatTrans.app (Monad.\u03bc \u03b2) X \u226b Ultrafilter.lim) FF = (\u03b2.toFunctor.map Ultrafilter.lim \u226b Ultrafilter.lim) FF\n[PROOFSTEP]\napply lim_eq\n[GOAL]\ncase h.h\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nFF : Ultrafilter (Ultrafilter X)\nx : X := Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc1 : x = Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc2 : \u2200 (U : Set X) (F : Ultrafilter X), Ultrafilter.lim F \u2208 U \u2192 IsOpen U \u2192 U \u2208 F\nc3 : \u2191(Ultrafilter.map Ultrafilter.lim FF) \u2264 \ud835\udcdd x\nc4 : \u2200 (U : Set X), x \u2208 U \u2192 IsOpen U \u2192 {G | U \u2208 G} \u2208 FF\n\u22a2 \u2191(NatTrans.app (Monad.\u03bc \u03b2) X FF) \u2264 \ud835\udcdd ((\u03b2.toFunctor.map Ultrafilter.lim \u226b Ultrafilter.lim) FF)\n[PROOFSTEP]\nrw [le_nhds_iff]\n[GOAL]\ncase h.h\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\nFF : Ultrafilter (Ultrafilter X)\nx : X := Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc1 : x = Ultrafilter.lim (Ultrafilter.map Ultrafilter.lim FF)\nc2 : \u2200 (U : Set X) (F : Ultrafilter X), Ultrafilter.lim F \u2208 U \u2192 IsOpen U \u2192 U \u2208 F\nc3 : \u2191(Ultrafilter.map Ultrafilter.lim FF) \u2264 \ud835\udcdd x\nc4 : \u2200 (U : Set X), x \u2208 U \u2192 IsOpen U \u2192 {G | U \u2208 G} \u2208 FF\n\u22a2 \u2200 (s : Set X),\n (\u03b2.toFunctor.map Ultrafilter.lim \u226b Ultrafilter.lim) FF \u2208 s \u2192 IsOpen s \u2192 s \u2208 \u2191(NatTrans.app (Monad.\u03bc \u03b2) X FF)\n[PROOFSTEP]\nexact c4\n[GOAL]\nX Y : Compactum\nf : X.A \u2192 Y.A\ncont : Continuous f\n\u22a2 \u03b2.toFunctor.map f \u226b Y.a = X.a \u226b f\n[PROOFSTEP]\nrw [continuous_iff_ultrafilter] at cont \n[GOAL]\nX Y : Compactum\nf : X.A \u2192 Y.A\ncont : \u2200 (x : X.A) (g : Ultrafilter X.A), \u2191g \u2264 \ud835\udcdd x \u2192 Tendsto f (\u2191g) (\ud835\udcdd (f x))\n\u22a2 \u03b2.toFunctor.map f \u226b Y.a = X.a \u226b f\n[PROOFSTEP]\next (F : Ultrafilter X)\n[GOAL]\ncase h\nX Y : Compactum\nf : X.A \u2192 Y.A\ncont : \u2200 (x : X.A) (g : Ultrafilter X.A), \u2191g \u2264 \ud835\udcdd x \u2192 Tendsto f (\u2191g) (\ud835\udcdd (f x))\nF : Ultrafilter X.A\n\u22a2 (\u03b2.toFunctor.map f \u226b Y.a) F = (X.a \u226b f) F\n[PROOFSTEP]\nspecialize cont (X.str F) F (le_nhds_of_str_eq F (X.str F) rfl)\n[GOAL]\ncase h\nX Y : Compactum\nf : X.A \u2192 Y.A\nF : Ultrafilter X.A\ncont : Tendsto f (\u2191F) (\ud835\udcdd (f (str X F)))\n\u22a2 (\u03b2.toFunctor.map f \u226b Y.a) F = (X.a \u226b f) F\n[PROOFSTEP]\nsimp only [types_comp_apply, ofTypeFunctor_map]\n[GOAL]\ncase h\nX Y : Compactum\nf : X.A \u2192 Y.A\nF : Ultrafilter X.A\ncont : Tendsto f (\u2191F) (\ud835\udcdd (f (str X F)))\n\u22a2 Monad.Algebra.a Y (\u03b2.toFunctor.map f F) = f (Monad.Algebra.a X F)\n[PROOFSTEP]\nexact str_eq_of_le_nhds (Ultrafilter.map f F) _ cont\n[GOAL]\n\u22a2 \u2200 {X Y : Compactum}, Function.Injective compactumToCompHaus.map\n[PROOFSTEP]\nintro _ _ _ _ h\n[GOAL]\nX\u271d Y\u271d : Compactum\na\u2081\u271d a\u2082\u271d : X\u271d \u27f6 Y\u271d\nh : compactumToCompHaus.map a\u2081\u271d = compactumToCompHaus.map a\u2082\u271d\n\u22a2 a\u2081\u271d = a\u2082\u271d\n[PROOFSTEP]\napply Monad.Algebra.Hom.ext\n[GOAL]\ncase f\nX\u271d Y\u271d : Compactum\na\u2081\u271d a\u2082\u271d : X\u271d \u27f6 Y\u271d\nh : compactumToCompHaus.map a\u2081\u271d = compactumToCompHaus.map a\u2082\u271d\n\u22a2 a\u2081\u271d.f = a\u2082\u271d.f\n[PROOFSTEP]\napply congrArg (fun f => f.toFun) h\n[GOAL]\nD : CompHaus\nx\u271d : Set \u2191D.toTop\nh : IsOpen x\u271d\n\u22a2 IsOpen (id \u207b\u00b9' x\u271d)\n[PROOFSTEP]\nrw [isOpen_iff_ultrafilter'] at h \n[GOAL]\nD : CompHaus\nx\u271d : Set \u2191D.toTop\nh : \u2200 (F : Ultrafilter \u2191D.toTop), Ultrafilter.lim F \u2208 x\u271d \u2192 x\u271d \u2208 \u2191F\n\u22a2 IsOpen (id \u207b\u00b9' x\u271d)\n[PROOFSTEP]\nexact h\n[GOAL]\nD : CompHaus\nx\u271d : Set \u2191(compactumToCompHaus.obj (Compactum.ofTopologicalSpace \u2191D.toTop)).toTop\nh1 : IsOpen x\u271d\n\u22a2 IsOpen (id \u207b\u00b9' x\u271d)\n[PROOFSTEP]\nrw [isOpen_iff_ultrafilter']\n[GOAL]\nD : CompHaus\nx\u271d : Set \u2191(compactumToCompHaus.obj (Compactum.ofTopologicalSpace \u2191D.toTop)).toTop\nh1 : IsOpen x\u271d\n\u22a2 \u2200 (F : Ultrafilter \u2191D.toTop), Ultrafilter.lim F \u2208 id \u207b\u00b9' x\u271d \u2192 id \u207b\u00b9' x\u271d \u2208 \u2191F\n[PROOFSTEP]\nintro _ h2\n[GOAL]\nD : CompHaus\nx\u271d : Set \u2191(compactumToCompHaus.obj (Compactum.ofTopologicalSpace \u2191D.toTop)).toTop\nh1 : IsOpen x\u271d\nF\u271d : Ultrafilter \u2191D.toTop\nh2 : Ultrafilter.lim F\u271d \u2208 id \u207b\u00b9' x\u271d\n\u22a2 id \u207b\u00b9' x\u271d \u2208 \u2191F\u271d\n[PROOFSTEP]\nexact h1 _ h2\n[GOAL]\n\u22a2 IsEquivalence compactumToCompHaus\n[PROOFSTEP]\nhave := compactumToCompHaus.full\n[GOAL]\nthis : Full compactumToCompHaus\n\u22a2 IsEquivalence compactumToCompHaus\n[PROOFSTEP]\nhave := compactumToCompHaus.faithful\n[GOAL]\nthis\u271d : Full compactumToCompHaus\nthis : Faithful compactumToCompHaus\n\u22a2 IsEquivalence compactumToCompHaus\n[PROOFSTEP]\nhave := compactumToCompHaus.essSurj\n[GOAL]\nthis\u271d\u00b9 : Full compactumToCompHaus\nthis\u271d : Faithful compactumToCompHaus\nthis : EssSurj compactumToCompHaus\n\u22a2 IsEquivalence compactumToCompHaus\n[PROOFSTEP]\napply Equivalence.ofFullyFaithfullyEssSurj _\n[GOAL]\n\u22a2 CreatesLimits (forget CompHaus)\n[PROOFSTEP]\nlet e : forget CompHaus \u2245 compactumToCompHaus.inv \u22d9 Compactum.forget :=\n (((forget CompHaus).leftUnitor.symm \u226a\u226b\n isoWhiskerRight compactumToCompHaus.asEquivalence.symm.unitIso (forget CompHaus)) \u226a\u226b\n compactumToCompHaus.inv.associator compactumToCompHaus (forget CompHaus)) \u226a\u226b\n isoWhiskerLeft _ compactumToCompHausCompForget\n[GOAL]\ne : forget CompHaus \u2245 Functor.inv compactumToCompHaus \u22d9 Compactum.forget :=\n (((Functor.leftUnitor (forget CompHaus)).symm \u226a\u226b\n isoWhiskerRight (CategoryTheory.Equivalence.symm (Functor.asEquivalence compactumToCompHaus)).unitIso\n (forget CompHaus)) \u226a\u226b\n Functor.associator (Functor.inv compactumToCompHaus) compactumToCompHaus (forget CompHaus)) \u226a\u226b\n isoWhiskerLeft (Functor.inv compactumToCompHaus) compactumToCompHausCompForget\n\u22a2 CreatesLimits (forget CompHaus)\n[PROOFSTEP]\nexact createsLimitsOfNatIso e.symm\n[GOAL]\n\u22a2 CreatesLimits (forget Profinite)\n[PROOFSTEP]\nchange CreatesLimits (profiniteToCompHaus \u22d9 forget _)\n[GOAL]\n\u22a2 CreatesLimits (profiniteToCompHaus \u22d9 forget CompHaus)\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.Topology.Category.Compactum", "llama_tokens": 51519, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.43398146480389854, "lm_q2_score": 0.016914911439287056, "lm_q1q2_score": 0.007340758043450016}} {"text": "[GOAL]\nX : TopCat\nT : \u2191X \u2192 Type v\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 { obj := fun U => (x : { x // x \u2208 U.unop }) \u2192 T \u2191x,\n map := fun {U V} i g x => g ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U.unop) }) x) }.map\n (\ud835\udfd9 U) =\n \ud835\udfd9\n ({ obj := fun U => (x : { x // x \u2208 U.unop }) \u2192 T \u2191x,\n map := fun {U V} i g x => g ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U.unop) }) x) }.obj\n U)\n[PROOFSTEP]\next g\n[GOAL]\ncase h\nX : TopCat\nT : \u2191X \u2192 Type v\nU : (Opens \u2191X)\u1d52\u1d56\ng :\n { obj := fun U => (x : { x // x \u2208 U.unop }) \u2192 T \u2191x,\n map := fun {U V} i g x => g ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U.unop) }) x) }.obj\n U\n\u22a2 { obj := fun U => (x : { x // x \u2208 U.unop }) \u2192 T \u2191x,\n map := fun {U V} i g x => g ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U.unop) }) x) }.map\n (\ud835\udfd9 U) g =\n \ud835\udfd9\n ({ obj := fun U => (x : { x // x \u2208 U.unop }) \u2192 T \u2191x,\n map := fun {U V} i g x => g ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U.unop) }) x) }.obj\n U)\n g\n[PROOFSTEP]\nrfl\n[GOAL]\nX : TopCat\nT : Type v\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 { obj := fun U => { x // x \u2208 U.unop } \u2192 T,\n map := fun {U V} i g => g \u2218 fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U.unop) } }.map\n (\ud835\udfd9 U) =\n \ud835\udfd9\n ({ obj := fun U => { x // x \u2208 U.unop } \u2192 T,\n map := fun {U V} i g => g \u2218 fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U.unop) } }.obj\n U)\n[PROOFSTEP]\next g\n[GOAL]\ncase h\nX : TopCat\nT : Type v\nU : (Opens \u2191X)\u1d52\u1d56\ng :\n { obj := fun U => { x // x \u2208 U.unop } \u2192 T,\n map := fun {U V} i g => g \u2218 fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U.unop) } }.obj\n U\n\u22a2 { obj := fun U => { x // x \u2208 U.unop } \u2192 T,\n map := fun {U V} i g => g \u2218 fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U.unop) } }.map\n (\ud835\udfd9 U) g =\n \ud835\udfd9\n ({ obj := fun U => { x // x \u2208 U.unop } \u2192 T,\n map := fun {U V} i g => g \u2218 fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U.unop) } }.obj\n U)\n g\n[PROOFSTEP]\nrfl\n[GOAL]\nX\u271d : TopCat\nX : TopCat\u1d52\u1d56\nR : TopCommRingCat\n\u22a2 CommRing C(\u2191X.unop, \u2191((forget\u2082 TopCommRingCat TopCat).obj R))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX\u271d : TopCat\nX Y : TopCat\u1d52\u1d56\nf : X \u27f6 Y\nR : TopCommRingCat\n\u22a2 \u2200 (x y : \u2191(continuousFunctions X R)),\n OneHom.toFun { toFun := fun g => f.unop \u226b g, map_one' := (_ : (fun g => f.unop \u226b g) 1 = (fun g => f.unop \u226b g) 1) }\n (x * y) =\n OneHom.toFun { toFun := fun g => f.unop \u226b g, map_one' := (_ : (fun g => f.unop \u226b g) 1 = (fun g => f.unop \u226b g) 1) }\n x *\n OneHom.toFun\n { toFun := fun g => f.unop \u226b g, map_one' := (_ : (fun g => f.unop \u226b g) 1 = (fun g => f.unop \u226b g) 1) } y\n[PROOFSTEP]\naesop_cat\n[GOAL]\nX\u271d : TopCat\nX Y : TopCat\u1d52\u1d56\nf : X \u27f6 Y\nR : TopCommRingCat\n\u22a2 \u2200 (x y : \u2191(continuousFunctions X R)),\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := fun g => f.unop \u226b g, map_one' := (_ : (fun g => f.unop \u226b g) 1 = (fun g => f.unop \u226b g) 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191(continuousFunctions X R)),\n OneHom.toFun\n { toFun := fun g => f.unop \u226b g,\n map_one' := (_ : (fun g => f.unop \u226b g) 1 = (fun g => f.unop \u226b g) 1) }\n (x * y) =\n OneHom.toFun\n { toFun := fun g => f.unop \u226b g,\n map_one' := (_ : (fun g => f.unop \u226b g) 1 = (fun g => f.unop \u226b g) 1) }\n x *\n OneHom.toFun\n { toFun := fun g => f.unop \u226b g,\n map_one' := (_ : (fun g => f.unop \u226b g) 1 = (fun g => f.unop \u226b g) 1) }\n y) })\n (x + y) =\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := fun g => f.unop \u226b g, map_one' := (_ : (fun g => f.unop \u226b g) 1 = (fun g => f.unop \u226b g) 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191(continuousFunctions X R)),\n OneHom.toFun\n { toFun := fun g => f.unop \u226b g,\n map_one' := (_ : (fun g => f.unop \u226b g) 1 = (fun g => f.unop \u226b g) 1) }\n (x * y) =\n OneHom.toFun\n { toFun := fun g => f.unop \u226b g,\n map_one' := (_ : (fun g => f.unop \u226b g) 1 = (fun g => f.unop \u226b g) 1) }\n x *\n OneHom.toFun\n { toFun := fun g => f.unop \u226b g,\n map_one' := (_ : (fun g => f.unop \u226b g) 1 = (fun g => f.unop \u226b g) 1) }\n y) })\n x +\n OneHom.toFun\n (\u2191{\n toOneHom :=\n { toFun := fun g => f.unop \u226b g, map_one' := (_ : (fun g => f.unop \u226b g) 1 = (fun g => f.unop \u226b g) 1) },\n map_mul' :=\n (_ :\n \u2200 (x y : \u2191(continuousFunctions X R)),\n OneHom.toFun\n { toFun := fun g => f.unop \u226b g,\n map_one' := (_ : (fun g => f.unop \u226b g) 1 = (fun g => f.unop \u226b g) 1) }\n (x * y) =\n OneHom.toFun\n { toFun := fun g => f.unop \u226b g,\n map_one' := (_ : (fun g => f.unop \u226b g) 1 = (fun g => f.unop \u226b g) 1) }\n x *\n OneHom.toFun\n { toFun := fun g => f.unop \u226b g,\n map_one' := (_ : (fun g => f.unop \u226b g) 1 = (fun g => f.unop \u226b g) 1) }\n y) })\n y\n[PROOFSTEP]\naesop_cat\n[GOAL]\nX\u271d : TopCat\nR : TopCommRingCat\nX : TopCat\u1d52\u1d56\n\u22a2 { obj := fun X => continuousFunctions X R, map := fun {X Y} f => continuousFunctions.pullback f R }.map (\ud835\udfd9 X) =\n \ud835\udfd9 ({ obj := fun X => continuousFunctions X R, map := fun {X Y} f => continuousFunctions.pullback f R }.obj X)\n[PROOFSTEP]\next\n[GOAL]\ncase w\nX\u271d : TopCat\nR : TopCommRingCat\nX : TopCat\u1d52\u1d56\nx\u271d :\n (forget CommRingCat).obj\n ({ obj := fun X => continuousFunctions X R, map := fun {X Y} f => continuousFunctions.pullback f R }.obj X)\n\u22a2 \u2191({ obj := fun X => continuousFunctions X R, map := fun {X Y} f => continuousFunctions.pullback f R }.map (\ud835\udfd9 X)) x\u271d =\n \u2191(\ud835\udfd9 ({ obj := fun X => continuousFunctions X R, map := fun {X Y} f => continuousFunctions.pullback f R }.obj X)) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nX\u271d : TopCat\nX : TopCommRingCat\n\u22a2 {\n obj := fun R =>\n CategoryTheory.Functor.mk\n { obj := fun X => continuousFunctions X R, map := fun {X Y} f => continuousFunctions.pullback f R },\n map := fun {R S} \u03c6 => NatTrans.mk fun X => continuousFunctions.map X \u03c6 }.map\n (\ud835\udfd9 X) =\n \ud835\udfd9\n ({\n obj := fun R =>\n CategoryTheory.Functor.mk\n { obj := fun X => continuousFunctions X R, map := fun {X Y} f => continuousFunctions.pullback f R },\n map := fun {R S} \u03c6 => NatTrans.mk fun X => continuousFunctions.map X \u03c6 }.obj\n X)\n[PROOFSTEP]\next\n[GOAL]\ncase w.h.w\nX\u271d : TopCat\nX : TopCommRingCat\nx\u271d\u00b9 : TopCat\u1d52\u1d56\nx\u271d :\n (forget CommRingCat).obj\n (({\n obj := fun R =>\n CategoryTheory.Functor.mk\n { obj := fun X => continuousFunctions X R, map := fun {X Y} f => continuousFunctions.pullback f R },\n map := fun {R S} \u03c6 => NatTrans.mk fun X => continuousFunctions.map X \u03c6 }.obj\n X).obj\n x\u271d\u00b9)\n\u22a2 \u2191(NatTrans.app\n ({\n obj := fun R =>\n CategoryTheory.Functor.mk\n { obj := fun X => continuousFunctions X R, map := fun {X Y} f => continuousFunctions.pullback f R },\n map := fun {R S} \u03c6 => NatTrans.mk fun X => continuousFunctions.map X \u03c6 }.map\n (\ud835\udfd9 X))\n x\u271d\u00b9)\n x\u271d =\n \u2191(NatTrans.app\n (\ud835\udfd9\n ({\n obj := fun R =>\n CategoryTheory.Functor.mk\n { obj := fun X => continuousFunctions X R,\n map := fun {X Y} f => continuousFunctions.pullback f R },\n map := fun {R S} \u03c6 => NatTrans.mk fun X => continuousFunctions.map X \u03c6 }.obj\n X))\n x\u271d\u00b9)\n x\u271d\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Topology.Sheaves.PresheafOfFunctions", "llama_tokens": 3317, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.4687906414693485, "lm_q2_score": 0.014957083739073663, "lm_q1q2_score": 0.007011740880551104}} {"text": "[GOAL]\nR : CommRingCat\u1d52\u1d56\n\u22a2 { obj := fun R => topObj R.unop, map := fun {R S} f => topMap f.unop }.map (\ud835\udfd9 R) =\n \ud835\udfd9 ({ obj := fun R => topObj R.unop, map := fun {R S} f => topMap f.unop }.obj R)\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : CommRingCat\u1d52\u1d56\n\u22a2 topMap (\ud835\udfd9 R.unop) = \ud835\udfd9 (topObj R.unop)\n[PROOFSTEP]\nrw [Spec.topMap_id]\n[GOAL]\nR S T : CommRingCat\u1d52\u1d56\nf : R \u27f6 S\ng : S \u27f6 T\n\u22a2 { obj := fun R => topObj R.unop, map := fun {R S} f => topMap f.unop }.map (f \u226b g) =\n { obj := fun R => topObj R.unop, map := fun {R S} f => topMap f.unop }.map f \u226b\n { obj := fun R => topObj R.unop, map := fun {R S} f => topMap f.unop }.map g\n[PROOFSTEP]\ndsimp\n[GOAL]\nR S T : CommRingCat\u1d52\u1d56\nf : R \u27f6 S\ng : S \u27f6 T\n\u22a2 topMap (g.unop \u226b f.unop) = topMap f.unop \u226b topMap g.unop\n[PROOFSTEP]\nrw [Spec.topMap_comp]\n[GOAL]\nR : CommRingCat\n\u22a2 (sheafedSpaceMap (\ud835\udfd9 R)).c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (TopologicalSpace.Opens.map (sheafedSpaceMap (\ud835\udfd9 R)).base).op =\n (TopologicalSpace.Opens.map (\ud835\udfd9 (sheafedSpaceObj R)).base).op))\n (sheafedSpaceObj R).toPresheafedSpace.presheaf =\n (\ud835\udfd9 (sheafedSpaceObj R)).c\n[PROOFSTEP]\next U\n[GOAL]\ncase w.w\nR : CommRingCat\nU : TopologicalSpace.Opens \u2191\u2191(sheafedSpaceObj R).toPresheafedSpace\nx\u271d : (forget CommRingCat).obj ((sheafedSpaceObj R).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 \u2191(NatTrans.app\n ((sheafedSpaceMap (\ud835\udfd9 R)).c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (TopologicalSpace.Opens.map (sheafedSpaceMap (\ud835\udfd9 R)).base).op =\n (TopologicalSpace.Opens.map (\ud835\udfd9 (sheafedSpaceObj R)).base).op))\n (sheafedSpaceObj R).toPresheafedSpace.presheaf)\n (op U))\n x\u271d =\n \u2191(NatTrans.app (\ud835\udfd9 (sheafedSpaceObj R)).c (op U)) x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.w\nR : CommRingCat\nU : TopologicalSpace.Opens \u2191\u2191(sheafedSpaceObj R).toPresheafedSpace\nx\u271d : (forget CommRingCat).obj ((sheafedSpaceObj R).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 \u2191(NatTrans.app\n ((sheafedSpaceMap (\ud835\udfd9 R)).c \u226b\n whiskerRight (\ud835\udfd9 (TopologicalSpace.Opens.map (topMap (\ud835\udfd9 R))).op) (structureSheaf \u2191R).val)\n (op U))\n x\u271d =\n \u2191(NatTrans.app (\ud835\udfd9 (sheafedSpaceObj R)).c (op U)) x\u271d\n[PROOFSTEP]\nerw [NatTrans.comp_app, sheafedSpaceMap_c_app, PresheafedSpace.id_c_app, comap_id]\n[GOAL]\ncase w.w\nR : CommRingCat\nU : TopologicalSpace.Opens \u2191\u2191(sheafedSpaceObj R).toPresheafedSpace\nx\u271d : (forget CommRingCat).obj ((sheafedSpaceObj R).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 \u2191(eqToHom\n (_ :\n (structureSheaf \u2191R).val.obj (op (op U).unop) =\n (structureSheaf \u2191R).val.obj (op ((TopologicalSpace.Opens.map (topMap (\ud835\udfd9 R))).obj (op U).unop))) \u226b\n NatTrans.app (whiskerRight (\ud835\udfd9 (TopologicalSpace.Opens.map (topMap (\ud835\udfd9 R))).op) (structureSheaf \u2191R).val) (op U))\n x\u271d =\n \u2191((sheafedSpaceObj R).toPresheafedSpace.presheaf.map (\ud835\udfd9 (op U))) x\u271d\ncase w.w.hUV\nR : CommRingCat\nU : TopologicalSpace.Opens \u2191\u2191(sheafedSpaceObj R).toPresheafedSpace\nx\u271d : (forget CommRingCat).obj ((sheafedSpaceObj R).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 (op U).unop = (TopologicalSpace.Opens.map (topMap (\ud835\udfd9 R))).obj (op U).unop\n[PROOFSTEP]\nswap\n[GOAL]\ncase w.w.hUV\nR : CommRingCat\nU : TopologicalSpace.Opens \u2191\u2191(sheafedSpaceObj R).toPresheafedSpace\nx\u271d : (forget CommRingCat).obj ((sheafedSpaceObj R).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 (op U).unop = (TopologicalSpace.Opens.map (topMap (\ud835\udfd9 R))).obj (op U).unop\n[PROOFSTEP]\nrw [Spec.topMap_id, TopologicalSpace.Opens.map_id_obj_unop]\n[GOAL]\ncase w.w\nR : CommRingCat\nU : TopologicalSpace.Opens \u2191\u2191(sheafedSpaceObj R).toPresheafedSpace\nx\u271d : (forget CommRingCat).obj ((sheafedSpaceObj R).toPresheafedSpace.presheaf.obj (op U))\n\u22a2 \u2191(eqToHom\n (_ :\n (structureSheaf \u2191R).val.obj (op (op U).unop) =\n (structureSheaf \u2191R).val.obj (op ((TopologicalSpace.Opens.map (topMap (\ud835\udfd9 R))).obj (op U).unop))) \u226b\n NatTrans.app (whiskerRight (\ud835\udfd9 (TopologicalSpace.Opens.map (topMap (\ud835\udfd9 R))).op) (structureSheaf \u2191R).val) (op U))\n x\u271d =\n \u2191((sheafedSpaceObj R).toPresheafedSpace.presheaf.map (\ud835\udfd9 (op U))) x\u271d\n[PROOFSTEP]\nsimp [eqToHom_map]\n[GOAL]\nR S T : CommRingCat\nf : R \u27f6 S\ng : S \u27f6 T\n\u22a2 (sheafedSpaceMap (f \u226b g)).c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (TopologicalSpace.Opens.map (sheafedSpaceMap (f \u226b g)).base).op =\n (TopologicalSpace.Opens.map (sheafedSpaceMap g \u226b sheafedSpaceMap f).base).op))\n (sheafedSpaceObj T).toPresheafedSpace.presheaf =\n (sheafedSpaceMap g \u226b sheafedSpaceMap f).c\n[PROOFSTEP]\next\n -- Porting note : was one liner\n -- `dsimp, rw category_theory.functor.map_id, rw category.comp_id, erw comap_comp f g, refl`\n[GOAL]\ncase w.w\nR S T : CommRingCat\nf : R \u27f6 S\ng : S \u27f6 T\nU\u271d : TopologicalSpace.Opens \u2191\u2191(sheafedSpaceObj R).toPresheafedSpace\nx\u271d : (forget CommRingCat).obj ((sheafedSpaceObj R).toPresheafedSpace.presheaf.obj (op U\u271d))\n\u22a2 \u2191(NatTrans.app\n ((sheafedSpaceMap (f \u226b g)).c \u226b\n whiskerRight\n (eqToHom\n (_ :\n (TopologicalSpace.Opens.map (sheafedSpaceMap (f \u226b g)).base).op =\n (TopologicalSpace.Opens.map (sheafedSpaceMap g \u226b sheafedSpaceMap f).base).op))\n (sheafedSpaceObj T).toPresheafedSpace.presheaf)\n (op U\u271d))\n x\u271d =\n \u2191(NatTrans.app (sheafedSpaceMap g \u226b sheafedSpaceMap f).c (op U\u271d)) x\u271d\n[PROOFSTEP]\nrw [NatTrans.comp_app, sheafedSpaceMap_c_app, whiskerRight_app, eqToHom_refl]\n[GOAL]\ncase w.w\nR S T : CommRingCat\nf : R \u27f6 S\ng : S \u27f6 T\nU\u271d : TopologicalSpace.Opens \u2191\u2191(sheafedSpaceObj R).toPresheafedSpace\nx\u271d : (forget CommRingCat).obj ((sheafedSpaceObj R).toPresheafedSpace.presheaf.obj (op U\u271d))\n\u22a2 \u2191(comap (f \u226b g) (op U\u271d).unop ((TopologicalSpace.Opens.map (topMap (f \u226b g))).obj (op U\u271d).unop)\n (_ :\n \u2200 (x : \u2191(PrimeSpectrum.Top \u2191T)),\n x \u2208 ((TopologicalSpace.Opens.map (topMap (f \u226b g))).obj (op U\u271d).unop).carrier \u2192\n x \u2208 ((TopologicalSpace.Opens.map (topMap (f \u226b g))).obj (op U\u271d).unop).carrier) \u226b\n (sheafedSpaceObj T).toPresheafedSpace.presheaf.map\n (NatTrans.app (\ud835\udfd9 (TopologicalSpace.Opens.map (sheafedSpaceMap (f \u226b g)).base).op) (op U\u271d)))\n x\u271d =\n \u2191(NatTrans.app (sheafedSpaceMap g \u226b sheafedSpaceMap f).c (op U\u271d)) x\u271d\n[PROOFSTEP]\nerw [(sheafedSpaceObj T).presheaf.map_id, Category.comp_id, comap_comp]\n[GOAL]\ncase w.w\nR S T : CommRingCat\nf : R \u27f6 S\ng : S \u27f6 T\nU\u271d : TopologicalSpace.Opens \u2191\u2191(sheafedSpaceObj R).toPresheafedSpace\nx\u271d : (forget CommRingCat).obj ((sheafedSpaceObj R).toPresheafedSpace.presheaf.obj (op U\u271d))\n\u22a2 \u2191(RingHom.comp (comap g ?w.w.V ((TopologicalSpace.Opens.map (topMap (f \u226b g))).obj (op U\u271d).unop) ?w.w.hVW)\n (comap f (op U\u271d).unop ?w.w.V ?w.w.hUV))\n x\u271d =\n \u2191(NatTrans.app (sheafedSpaceMap g \u226b sheafedSpaceMap f).c (op U\u271d)) x\u271d\ncase w.w.V\nR S T : CommRingCat\nf : R \u27f6 S\ng : S \u27f6 T\nU\u271d : TopologicalSpace.Opens \u2191\u2191(sheafedSpaceObj R).toPresheafedSpace\nx\u271d : (forget CommRingCat).obj ((sheafedSpaceObj R).toPresheafedSpace.presheaf.obj (op U\u271d))\n\u22a2 TopologicalSpace.Opens \u2191(PrimeSpectrum.Top \u2191S)\ncase w.w.hUV\nR S T : CommRingCat\nf : R \u27f6 S\ng : S \u27f6 T\nU\u271d : TopologicalSpace.Opens \u2191\u2191(sheafedSpaceObj R).toPresheafedSpace\nx\u271d : (forget CommRingCat).obj ((sheafedSpaceObj R).toPresheafedSpace.presheaf.obj (op U\u271d))\n\u22a2 \u2200 (p : \u2191(PrimeSpectrum.Top \u2191S)), p \u2208 ?w.w.V \u2192 \u2191(PrimeSpectrum.comap f) p \u2208 (op U\u271d).unop\ncase w.w.hVW\nR S T : CommRingCat\nf : R \u27f6 S\ng : S \u27f6 T\nU\u271d : TopologicalSpace.Opens \u2191\u2191(sheafedSpaceObj R).toPresheafedSpace\nx\u271d : (forget CommRingCat).obj ((sheafedSpaceObj R).toPresheafedSpace.presheaf.obj (op U\u271d))\n\u22a2 \u2200 (p : \u2191(PrimeSpectrum.Top \u2191T)),\n p \u2208 (TopologicalSpace.Opens.map (topMap (f \u226b g))).obj (op U\u271d).unop \u2192 \u2191(PrimeSpectrum.comap g) p \u2208 ?w.w.V\n[PROOFSTEP]\nrfl\n[GOAL]\nR : CommRingCat\u1d52\u1d56\n\u22a2 { obj := fun R => sheafedSpaceObj R.unop, map := fun {X Y} f => sheafedSpaceMap f.unop }.map (\ud835\udfd9 R) =\n \ud835\udfd9 ({ obj := fun R => sheafedSpaceObj R.unop, map := fun {X Y} f => sheafedSpaceMap f.unop }.obj R)\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : CommRingCat\u1d52\u1d56\n\u22a2 sheafedSpaceMap (\ud835\udfd9 R.unop) = \ud835\udfd9 (sheafedSpaceObj R.unop)\n[PROOFSTEP]\nrw [Spec.sheafedSpaceMap_id]\n[GOAL]\nX\u271d Y\u271d Z\u271d : CommRingCat\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun R => sheafedSpaceObj R.unop, map := fun {X Y} f => sheafedSpaceMap f.unop }.map (f \u226b g) =\n { obj := fun R => sheafedSpaceObj R.unop, map := fun {X Y} f => sheafedSpaceMap f.unop }.map f \u226b\n { obj := fun R => sheafedSpaceObj R.unop, map := fun {X Y} f => sheafedSpaceMap f.unop }.map g\n[PROOFSTEP]\ndsimp\n[GOAL]\nX\u271d Y\u271d Z\u271d : CommRingCat\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 sheafedSpaceMap (g.unop \u226b f.unop) = sheafedSpaceMap f.unop \u226b sheafedSpaceMap g.unop\n[PROOFSTEP]\nrw [Spec.sheafedSpaceMap_comp]\n[GOAL]\nX : RingedSpace\nR : CommRingCat\n\u03b1 \u03b2 : X \u27f6 sheafedSpaceObj R\nw : \u03b1.base = \u03b2.base\nr : \u2191R\nU : TopologicalSpace.Opens (PrimeSpectrum \u2191R) := PrimeSpectrum.basicOpen r\n\u22a2 (TopologicalSpace.Opens.map \u03b1.base).op.obj (op U) = (TopologicalSpace.Opens.map \u03b2.base).op.obj (op U)\n[PROOFSTEP]\nrw [w]\n[GOAL]\nX : RingedSpace\nR : CommRingCat\n\u03b1 \u03b2 : X \u27f6 sheafedSpaceObj R\nw : \u03b1.base = \u03b2.base\nh :\n \u2200 (r : \u2191R),\n let U := PrimeSpectrum.basicOpen r;\n (toOpen (\u2191R) U \u226b NatTrans.app \u03b1.c (op U)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (TopologicalSpace.Opens.map \u03b1.base).op.obj (op U) = (TopologicalSpace.Opens.map \u03b2.base).op.obj (op U))) =\n toOpen (\u2191R) U \u226b NatTrans.app \u03b2.c (op U)\n\u22a2 \u03b1 = \u03b2\n[PROOFSTEP]\next : 1\n[GOAL]\ncase w\nX : RingedSpace\nR : CommRingCat\n\u03b1 \u03b2 : X \u27f6 sheafedSpaceObj R\nw : \u03b1.base = \u03b2.base\nh :\n \u2200 (r : \u2191R),\n let U := PrimeSpectrum.basicOpen r;\n (toOpen (\u2191R) U \u226b NatTrans.app \u03b1.c (op U)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (TopologicalSpace.Opens.map \u03b1.base).op.obj (op U) = (TopologicalSpace.Opens.map \u03b2.base).op.obj (op U))) =\n toOpen (\u2191R) U \u226b NatTrans.app \u03b2.c (op U)\n\u22a2 \u03b1.base = \u03b2.base\n[PROOFSTEP]\nexact w\n[GOAL]\ncase h\nX : RingedSpace\nR : CommRingCat\n\u03b1 \u03b2 : X \u27f6 sheafedSpaceObj R\nw : \u03b1.base = \u03b2.base\nh :\n \u2200 (r : \u2191R),\n let U := PrimeSpectrum.basicOpen r;\n (toOpen (\u2191R) U \u226b NatTrans.app \u03b1.c (op U)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (TopologicalSpace.Opens.map \u03b1.base).op.obj (op U) = (TopologicalSpace.Opens.map \u03b2.base).op.obj (op U))) =\n toOpen (\u2191R) U \u226b NatTrans.app \u03b2.c (op U)\n\u22a2 \u03b1.c \u226b\n whiskerRight (eqToHom (_ : (TopologicalSpace.Opens.map \u03b1.base).op = (TopologicalSpace.Opens.map \u03b2.base).op))\n X.presheaf =\n \u03b2.c\n[PROOFSTEP]\napply ((TopCat.Sheaf.pushforward \u03b2.base).obj X.sheaf).hom_ext _ PrimeSpectrum.isBasis_basic_opens\n[GOAL]\ncase h\nX : RingedSpace\nR : CommRingCat\n\u03b1 \u03b2 : X \u27f6 sheafedSpaceObj R\nw : \u03b1.base = \u03b2.base\nh :\n \u2200 (r : \u2191R),\n let U := PrimeSpectrum.basicOpen r;\n (toOpen (\u2191R) U \u226b NatTrans.app \u03b1.c (op U)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (TopologicalSpace.Opens.map \u03b1.base).op.obj (op U) = (TopologicalSpace.Opens.map \u03b2.base).op.obj (op U))) =\n toOpen (\u2191R) U \u226b NatTrans.app \u03b2.c (op U)\n\u22a2 \u2200 (i : \u2191R),\n NatTrans.app\n (\u03b1.c \u226b\n whiskerRight (eqToHom (_ : (TopologicalSpace.Opens.map \u03b1.base).op = (TopologicalSpace.Opens.map \u03b2.base).op))\n X.presheaf)\n (op (PrimeSpectrum.basicOpen i)) =\n NatTrans.app \u03b2.c (op (PrimeSpectrum.basicOpen i))\n[PROOFSTEP]\nintro r\n[GOAL]\ncase h\nX : RingedSpace\nR : CommRingCat\n\u03b1 \u03b2 : X \u27f6 sheafedSpaceObj R\nw : \u03b1.base = \u03b2.base\nh :\n \u2200 (r : \u2191R),\n let U := PrimeSpectrum.basicOpen r;\n (toOpen (\u2191R) U \u226b NatTrans.app \u03b1.c (op U)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (TopologicalSpace.Opens.map \u03b1.base).op.obj (op U) = (TopologicalSpace.Opens.map \u03b2.base).op.obj (op U))) =\n toOpen (\u2191R) U \u226b NatTrans.app \u03b2.c (op U)\nr : \u2191R\n\u22a2 NatTrans.app\n (\u03b1.c \u226b\n whiskerRight (eqToHom (_ : (TopologicalSpace.Opens.map \u03b1.base).op = (TopologicalSpace.Opens.map \u03b2.base).op))\n X.presheaf)\n (op (PrimeSpectrum.basicOpen r)) =\n NatTrans.app \u03b2.c (op (PrimeSpectrum.basicOpen r))\n[PROOFSTEP]\napply\n (StructureSheaf.to_basicOpen_epi R r).1\n -- Porting note : was a one-liner `simpa using h r`\n[GOAL]\ncase h.a\nX : RingedSpace\nR : CommRingCat\n\u03b1 \u03b2 : X \u27f6 sheafedSpaceObj R\nw : \u03b1.base = \u03b2.base\nh :\n \u2200 (r : \u2191R),\n let U := PrimeSpectrum.basicOpen r;\n (toOpen (\u2191R) U \u226b NatTrans.app \u03b1.c (op U)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (TopologicalSpace.Opens.map \u03b1.base).op.obj (op U) = (TopologicalSpace.Opens.map \u03b2.base).op.obj (op U))) =\n toOpen (\u2191R) U \u226b NatTrans.app \u03b2.c (op U)\nr : \u2191R\n\u22a2 toOpen (\u2191R) (PrimeSpectrum.basicOpen r) \u226b\n NatTrans.app\n (\u03b1.c \u226b\n whiskerRight (eqToHom (_ : (TopologicalSpace.Opens.map \u03b1.base).op = (TopologicalSpace.Opens.map \u03b2.base).op))\n X.presheaf)\n (op (PrimeSpectrum.basicOpen r)) =\n toOpen (\u2191R) (PrimeSpectrum.basicOpen r) \u226b NatTrans.app \u03b2.c (op (PrimeSpectrum.basicOpen r))\n[PROOFSTEP]\nspecialize h r\n[GOAL]\ncase h.a\nX : RingedSpace\nR : CommRingCat\n\u03b1 \u03b2 : X \u27f6 sheafedSpaceObj R\nw : \u03b1.base = \u03b2.base\nr : \u2191R\nh :\n let U := PrimeSpectrum.basicOpen r;\n (toOpen (\u2191R) U \u226b NatTrans.app \u03b1.c (op U)) \u226b\n X.presheaf.map\n (eqToHom\n (_ : (TopologicalSpace.Opens.map \u03b1.base).op.obj (op U) = (TopologicalSpace.Opens.map \u03b2.base).op.obj (op U))) =\n toOpen (\u2191R) U \u226b NatTrans.app \u03b2.c (op U)\n\u22a2 toOpen (\u2191R) (PrimeSpectrum.basicOpen r) \u226b\n NatTrans.app\n (\u03b1.c \u226b\n whiskerRight (eqToHom (_ : (TopologicalSpace.Opens.map \u03b1.base).op = (TopologicalSpace.Opens.map \u03b2.base).op))\n X.presheaf)\n (op (PrimeSpectrum.basicOpen r)) =\n toOpen (\u2191R) (PrimeSpectrum.basicOpen r) \u226b NatTrans.app \u03b2.c (op (PrimeSpectrum.basicOpen r))\n[PROOFSTEP]\nsimp only [sheafedSpaceObj_carrier, Functor.op_obj, unop_op, TopCat.Presheaf.pushforwardObj_obj,\n sheafedSpaceObj_presheaf, Category.assoc] at h \n[GOAL]\ncase h.a\nX : RingedSpace\nR : CommRingCat\n\u03b1 \u03b2 : X \u27f6 sheafedSpaceObj R\nw : \u03b1.base = \u03b2.base\nr : \u2191R\nh :\n toOpen (\u2191R) (PrimeSpectrum.basicOpen r) \u226b\n NatTrans.app \u03b1.c (op (PrimeSpectrum.basicOpen r)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (TopologicalSpace.Opens.map \u03b1.base).op.obj (op (PrimeSpectrum.basicOpen r)) =\n (TopologicalSpace.Opens.map \u03b2.base).op.obj (op (PrimeSpectrum.basicOpen r)))) =\n toOpen (\u2191R) (PrimeSpectrum.basicOpen r) \u226b NatTrans.app \u03b2.c (op (PrimeSpectrum.basicOpen r))\n\u22a2 toOpen (\u2191R) (PrimeSpectrum.basicOpen r) \u226b\n NatTrans.app\n (\u03b1.c \u226b\n whiskerRight (eqToHom (_ : (TopologicalSpace.Opens.map \u03b1.base).op = (TopologicalSpace.Opens.map \u03b2.base).op))\n X.presheaf)\n (op (PrimeSpectrum.basicOpen r)) =\n toOpen (\u2191R) (PrimeSpectrum.basicOpen r) \u226b NatTrans.app \u03b2.c (op (PrimeSpectrum.basicOpen r))\n[PROOFSTEP]\nrw [NatTrans.comp_app, \u2190 h]\n[GOAL]\ncase h.a\nX : RingedSpace\nR : CommRingCat\n\u03b1 \u03b2 : X \u27f6 sheafedSpaceObj R\nw : \u03b1.base = \u03b2.base\nr : \u2191R\nh :\n toOpen (\u2191R) (PrimeSpectrum.basicOpen r) \u226b\n NatTrans.app \u03b1.c (op (PrimeSpectrum.basicOpen r)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (TopologicalSpace.Opens.map \u03b1.base).op.obj (op (PrimeSpectrum.basicOpen r)) =\n (TopologicalSpace.Opens.map \u03b2.base).op.obj (op (PrimeSpectrum.basicOpen r)))) =\n toOpen (\u2191R) (PrimeSpectrum.basicOpen r) \u226b NatTrans.app \u03b2.c (op (PrimeSpectrum.basicOpen r))\n\u22a2 toOpen (\u2191R) (PrimeSpectrum.basicOpen r) \u226b\n NatTrans.app \u03b1.c (op (PrimeSpectrum.basicOpen r)) \u226b\n NatTrans.app\n (whiskerRight (eqToHom (_ : (TopologicalSpace.Opens.map \u03b1.base).op = (TopologicalSpace.Opens.map \u03b2.base).op))\n X.presheaf)\n (op (PrimeSpectrum.basicOpen r)) =\n toOpen (\u2191R) (PrimeSpectrum.basicOpen r) \u226b\n NatTrans.app \u03b1.c (op (PrimeSpectrum.basicOpen r)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (TopologicalSpace.Opens.map \u03b1.base).op.obj (op (PrimeSpectrum.basicOpen r)) =\n (TopologicalSpace.Opens.map \u03b2.base).op.obj (op (PrimeSpectrum.basicOpen r))))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.a.e_a.e_a\nX : RingedSpace\nR : CommRingCat\n\u03b1 \u03b2 : X \u27f6 sheafedSpaceObj R\nw : \u03b1.base = \u03b2.base\nr : \u2191R\nh :\n toOpen (\u2191R) (PrimeSpectrum.basicOpen r) \u226b\n NatTrans.app \u03b1.c (op (PrimeSpectrum.basicOpen r)) \u226b\n X.presheaf.map\n (eqToHom\n (_ :\n (TopologicalSpace.Opens.map \u03b1.base).op.obj (op (PrimeSpectrum.basicOpen r)) =\n (TopologicalSpace.Opens.map \u03b2.base).op.obj (op (PrimeSpectrum.basicOpen r)))) =\n toOpen (\u2191R) (PrimeSpectrum.basicOpen r) \u226b NatTrans.app \u03b2.c (op (PrimeSpectrum.basicOpen r))\n\u22a2 NatTrans.app\n (whiskerRight (eqToHom (_ : (TopologicalSpace.Opens.map \u03b1.base).op = (TopologicalSpace.Opens.map \u03b2.base).op))\n X.presheaf)\n (op (PrimeSpectrum.basicOpen r)) =\n X.presheaf.map\n (eqToHom\n (_ :\n (TopologicalSpace.Opens.map \u03b1.base).op.obj (op (PrimeSpectrum.basicOpen r)) =\n (TopologicalSpace.Opens.map \u03b2.base).op.obj (op (PrimeSpectrum.basicOpen r))))\n[PROOFSTEP]\nsimp\n[GOAL]\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191S\n\u22a2 toStalk (\u2191R) (\u2191(PrimeSpectrum.comap f) p) \u226b PresheafedSpace.stalkMap (Spec.sheafedSpaceMap f) p = f \u226b toStalk (\u2191S) p\n[PROOFSTEP]\nerw [\u2190 toOpen_germ S \u22a4 \u27e8p, trivial\u27e9, \u2190 toOpen_germ R \u22a4 \u27e8PrimeSpectrum.comap f p, trivial\u27e9, Category.assoc,\n PresheafedSpace.stalkMap_germ (Spec.sheafedSpaceMap f) \u22a4 \u27e8p, trivial\u27e9, Spec.sheafedSpaceMap_c_app,\n toOpen_comp_comap_assoc]\n[GOAL]\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191S\n\u22a2 CommRingCat.ofHom f \u226b\n toOpen (\u2191S) (\u2191(TopologicalSpace.Opens.comap (PrimeSpectrum.comap f)) \u22a4) \u226b\n TopCat.Presheaf.germ (Spec.sheafedSpaceObj S).toPresheafedSpace.presheaf { val := p, property := trivial } =\n f \u226b toOpen \u2191S \u22a4 \u226b TopCat.Presheaf.germ (TopCat.Sheaf.presheaf (structureSheaf \u2191S)) { val := p, property := trivial }\n[PROOFSTEP]\nrfl\n[GOAL]\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191S\nx : \u2191R\n\u22a2 \u2191(((stalkIso (\u2191R) (\u2191(PrimeSpectrum.comap f) p)).inv \u226b PresheafedSpace.stalkMap (Spec.sheafedSpaceMap f) p) \u226b\n (stalkIso (\u2191S) p).hom)\n (\u2191(algebraMap (\u2191R) (Localization.AtPrime (\u2191(PrimeSpectrum.comap f) p).asIdeal)) x) =\n \u2191(algebraMap ((fun x => \u2191S) x) (Localization.AtPrime p.asIdeal)) (\u2191f x)\n[PROOFSTEP]\nrw [stalkIso_hom, stalkIso_inv, comp_apply, comp_apply, localizationToStalk_of]\n[GOAL]\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191S\nx : \u2191R\n\u22a2 \u2191(stalkToFiberRingHom (\u2191S) p)\n (\u2191(PresheafedSpace.stalkMap (Spec.sheafedSpaceMap f) p) (\u2191(toStalk (\u2191R) (\u2191(PrimeSpectrum.comap f) p)) x)) =\n \u2191(algebraMap ((fun x => \u2191S) x) (Localization.AtPrime p.asIdeal)) (\u2191f x)\n[PROOFSTEP]\nerw [stalkMap_toStalk_apply f p x, stalkToFiberRingHom_toStalk]\n[GOAL]\nR S : CommRingCat\nf : R \u27f6 S\np : \u2191\u2191(locallyRingedSpaceObj S).toSheafedSpace.toPresheafedSpace\na : \u2191(PresheafedSpace.stalk (locallyRingedSpaceObj R).toSheafedSpace.toPresheafedSpace (\u2191(sheafedSpaceMap f).base p))\nha : IsUnit (\u2191(PresheafedSpace.stalkMap (sheafedSpaceMap f) p) a)\n\u22a2 IsUnit a\n[PROOFSTEP]\nerw [\u2190 localRingHom_comp_stalkIso_apply] at ha \n[GOAL]\nR S : CommRingCat\nf : R \u27f6 S\np : \u2191\u2191(locallyRingedSpaceObj S).toSheafedSpace.toPresheafedSpace\na : \u2191(PresheafedSpace.stalk (locallyRingedSpaceObj R).toSheafedSpace.toPresheafedSpace (\u2191(sheafedSpaceMap f).base p))\nha :\n IsUnit\n (\u2191(localizationToStalk (\u2191S) p)\n (\u2191(Localization.localRingHom (\u2191(PrimeSpectrum.comap f) p).asIdeal p.asIdeal f\n (_ : (\u2191(PrimeSpectrum.comap f) p).asIdeal = (\u2191(PrimeSpectrum.comap f) p).asIdeal))\n (\u2191(stalkToFiberRingHom (\u2191R) (\u2191(PrimeSpectrum.comap f) p)) a)))\n\u22a2 IsUnit a\n[PROOFSTEP]\nreplace ha := (stalkIso S p).hom.isUnit_map ha\n[GOAL]\nR S : CommRingCat\nf : R \u27f6 S\np : \u2191\u2191(locallyRingedSpaceObj S).toSheafedSpace.toPresheafedSpace\na : \u2191(PresheafedSpace.stalk (locallyRingedSpaceObj R).toSheafedSpace.toPresheafedSpace (\u2191(sheafedSpaceMap f).base p))\nha :\n IsUnit\n (\u2191(stalkIso (\u2191S) p).hom\n (\u2191(localizationToStalk (\u2191S) p)\n (\u2191(Localization.localRingHom (\u2191(PrimeSpectrum.comap f) p).asIdeal p.asIdeal f\n (_ : (\u2191(PrimeSpectrum.comap f) p).asIdeal = (\u2191(PrimeSpectrum.comap f) p).asIdeal))\n (\u2191(stalkToFiberRingHom (\u2191R) (\u2191(PrimeSpectrum.comap f) p)) a))))\n\u22a2 IsUnit a\n[PROOFSTEP]\nrw [\u2190 comp_apply, show localizationToStalk S p = (stalkIso S p).inv from rfl, Iso.inv_hom_id, id_apply] at ha \n[GOAL]\nR S : CommRingCat\nf : R \u27f6 S\np : \u2191\u2191(locallyRingedSpaceObj S).toSheafedSpace.toPresheafedSpace\na : \u2191(PresheafedSpace.stalk (locallyRingedSpaceObj R).toSheafedSpace.toPresheafedSpace (\u2191(sheafedSpaceMap f).base p))\nha :\n IsUnit\n (\u2191(Localization.localRingHom (\u2191(PrimeSpectrum.comap f) p).asIdeal p.asIdeal f\n (_ : (\u2191(PrimeSpectrum.comap f) p).asIdeal = (\u2191(PrimeSpectrum.comap f) p).asIdeal))\n (\u2191(stalkToFiberRingHom (\u2191R) (\u2191(PrimeSpectrum.comap f) p)) a))\n\u22a2 IsUnit a\n[PROOFSTEP]\nreplace ha := IsLocalRingHom.map_nonunit (R := Localization.AtPrime (PrimeSpectrum.comap f p).asIdeal) _ ha\n[GOAL]\nR S : CommRingCat\nf : R \u27f6 S\np : \u2191\u2191(locallyRingedSpaceObj S).toSheafedSpace.toPresheafedSpace\na : \u2191(PresheafedSpace.stalk (locallyRingedSpaceObj R).toSheafedSpace.toPresheafedSpace (\u2191(sheafedSpaceMap f).base p))\nha : IsUnit (\u2191(stalkToFiberRingHom (\u2191R) (\u2191(PrimeSpectrum.comap f) p)) a)\n\u22a2 IsUnit a\n[PROOFSTEP]\nconvert RingHom.isUnit_map (stalkIso R (PrimeSpectrum.comap f p)).inv ha\n[GOAL]\ncase h.e'_3.h\nR S : CommRingCat\nf : R \u27f6 S\np : \u2191\u2191(locallyRingedSpaceObj S).toSheafedSpace.toPresheafedSpace\na : \u2191(PresheafedSpace.stalk (locallyRingedSpaceObj R).toSheafedSpace.toPresheafedSpace (\u2191(sheafedSpaceMap f).base p))\nha : IsUnit (\u2191(stalkToFiberRingHom (\u2191R) (\u2191(PrimeSpectrum.comap f) p)) a)\ne_1\u271d :\n \u2191(PresheafedSpace.stalk (locallyRingedSpaceObj R).toSheafedSpace.toPresheafedSpace (\u2191(sheafedSpaceMap f).base p)) =\n (fun x => \u2191(TopCat.Presheaf.stalk (TopCat.Sheaf.presheaf (structureSheaf \u2191R)) (\u2191(PrimeSpectrum.comap f) p)))\n (\u2191(stalkToFiberRingHom (\u2191R) (\u2191(PrimeSpectrum.comap f) p)) a)\n\u22a2 a = \u2191(stalkIso (\u2191R) (\u2191(PrimeSpectrum.comap f) p)).inv (\u2191(stalkToFiberRingHom (\u2191R) (\u2191(PrimeSpectrum.comap f) p)) a)\n[PROOFSTEP]\nerw [\u2190 comp_apply, show stalkToFiberRingHom R _ = (stalkIso _ _).hom from rfl, Iso.hom_inv_id, id_apply]\n[GOAL]\nR : CommRingCat\n\u22a2 (locallyRingedSpaceMap (\ud835\udfd9 R)).val = (\ud835\udfd9 (locallyRingedSpaceObj R)).val\n[PROOFSTEP]\nrw [Spec.locallyRingedSpaceMap_val, Spec.sheafedSpaceMap_id]\n[GOAL]\nR : CommRingCat\n\u22a2 \ud835\udfd9 (sheafedSpaceObj R) = (\ud835\udfd9 (locallyRingedSpaceObj R)).val\n[PROOFSTEP]\nrfl\n[GOAL]\nR S T : CommRingCat\nf : R \u27f6 S\ng : S \u27f6 T\n\u22a2 (locallyRingedSpaceMap (f \u226b g)).val = (locallyRingedSpaceMap g \u226b locallyRingedSpaceMap f).val\n[PROOFSTEP]\nrw [Spec.locallyRingedSpaceMap_val, Spec.sheafedSpaceMap_comp]\n[GOAL]\nR S T : CommRingCat\nf : R \u27f6 S\ng : S \u27f6 T\n\u22a2 sheafedSpaceMap g \u226b sheafedSpaceMap f = (locallyRingedSpaceMap g \u226b locallyRingedSpaceMap f).val\n[PROOFSTEP]\nrfl\n[GOAL]\nR : CommRingCat\u1d52\u1d56\n\u22a2 { obj := fun R => locallyRingedSpaceObj R.unop, map := fun {X Y} f => locallyRingedSpaceMap f.unop }.map (\ud835\udfd9 R) =\n \ud835\udfd9 ({ obj := fun R => locallyRingedSpaceObj R.unop, map := fun {X Y} f => locallyRingedSpaceMap f.unop }.obj R)\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : CommRingCat\u1d52\u1d56\n\u22a2 locallyRingedSpaceMap (\ud835\udfd9 R.unop) = \ud835\udfd9 (locallyRingedSpaceObj R.unop)\n[PROOFSTEP]\nrw [Spec.locallyRingedSpaceMap_id]\n[GOAL]\nX\u271d Y\u271d Z\u271d : CommRingCat\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun R => locallyRingedSpaceObj R.unop, map := fun {X Y} f => locallyRingedSpaceMap f.unop }.map (f \u226b g) =\n { obj := fun R => locallyRingedSpaceObj R.unop, map := fun {X Y} f => locallyRingedSpaceMap f.unop }.map f \u226b\n { obj := fun R => locallyRingedSpaceObj R.unop, map := fun {X Y} f => locallyRingedSpaceMap f.unop }.map g\n[PROOFSTEP]\ndsimp\n[GOAL]\nX\u271d Y\u271d Z\u271d : CommRingCat\u1d52\u1d56\nf : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\u271d\n\u22a2 locallyRingedSpaceMap (g.unop \u226b f.unop) = locallyRingedSpaceMap f.unop \u226b locallyRingedSpaceMap g.unop\n[PROOFSTEP]\nrw [Spec.locallyRingedSpaceMap_comp]\n[GOAL]\nR : CommRingCat\n\u22a2 IsIso (toSpec\u0393 R)\n[PROOFSTEP]\ncases R\n[GOAL]\ncase mk\n\u03b1\u271d : Type ?u.412830\nstr\u271d : CommRing \u03b1\u271d\n\u22a2 IsIso (toSpec\u0393 (Bundled.mk \u03b1\u271d))\n[PROOFSTEP]\napply StructureSheaf.isIso_to_global\n[GOAL]\nR S : CommRingCat\nf : R \u27f6 S\n\u22a2 f \u226b toSpec\u0393 S = toSpec\u0393 R \u226b \u0393.map (Spec.toLocallyRingedSpace.map f.op).op\n[PROOFSTEP]\nrefine RingHom.ext fun x => Subtype.ext <| funext fun x' => ?_\n[GOAL]\nR S : CommRingCat\nf : R \u27f6 S\nx : \u2191R\nx' : { x // x \u2208 (op \u22a4).unop }\n\u22a2 \u2191(\u2191(f \u226b toSpec\u0393 S) x) x' = \u2191(\u2191(toSpec\u0393 R \u226b \u0393.map (Spec.toLocallyRingedSpace.map f.op).op) x) x'\n[PROOFSTEP]\nsymm\n[GOAL]\nR S : CommRingCat\nf : R \u27f6 S\nx : \u2191R\nx' : { x // x \u2208 (op \u22a4).unop }\n\u22a2 \u2191(\u2191(toSpec\u0393 R \u226b \u0393.map (Spec.toLocallyRingedSpace.map f.op).op) x) x' = \u2191(\u2191(f \u226b toSpec\u0393 S) x) x'\n[PROOFSTEP]\napply Localization.localRingHom_to_map\n[GOAL]\nX Y : CommRingCat\nf : X \u27f6 Y\n\u22a2 (\ud835\udfed CommRingCat).map f \u226b ((fun R => asIso (toSpec\u0393 R)) Y).hom =\n ((fun R => asIso (toSpec\u0393 R)) X).hom \u226b (Spec.toLocallyRingedSpace.rightOp \u22d9 \u0393).map f\n[PROOFSTEP]\nexact Spec_\u0393_naturality (R := X) (S := Y) f\n[GOAL]\nR : CommRingCat\nM : Submonoid \u2191R\nx : PrimeSpectrum (Localization M)\n\u22a2 IsIso\n (PresheafedSpace.stalkMap (Spec.toPresheafedSpace.map (CommRingCat.ofHom (algebraMap (\u2191R) (Localization M))).op) x)\n[PROOFSTEP]\nerw [\u2190 localRingHom_comp_stalkIso]\n -- Porting note: replaced `apply (config := { instances := false })`.\n -- See https://github.com/leanprover/lean4/issues/2273\n[GOAL]\nR : CommRingCat\nM : Submonoid \u2191R\nx : PrimeSpectrum (Localization M)\n\u22a2 IsIso\n ((stalkIso (\u2191(op (CommRingCat.of \u2191R)).unop)\n (\u2191(PrimeSpectrum.comap (CommRingCat.ofHom (algebraMap (\u2191R) (Localization M))).op.unop) x)).hom \u226b\n Localization.localRingHom\n (\u2191(PrimeSpectrum.comap (CommRingCat.ofHom (algebraMap (\u2191R) (Localization M))).op.unop) x).asIdeal x.asIdeal\n (CommRingCat.ofHom (algebraMap (\u2191R) (Localization M))).op.unop\n (_ :\n (\u2191(PrimeSpectrum.comap (CommRingCat.ofHom (algebraMap (\u2191R) (Localization M))).op.unop) x).asIdeal =\n (\u2191(PrimeSpectrum.comap (CommRingCat.ofHom (algebraMap (\u2191R) (Localization M))).op.unop) x).asIdeal) \u226b\n (stalkIso (\u2191(op (CommRingCat.of (Localization M))).unop) x).inv)\n[PROOFSTEP]\nrefine @IsIso.comp_isIso _ _ _ _ _ _ _ _ (?_)\n[GOAL]\nR : CommRingCat\nM : Submonoid \u2191R\nx : PrimeSpectrum (Localization M)\n\u22a2 IsIso\n (Localization.localRingHom\n (\u2191(PrimeSpectrum.comap (CommRingCat.ofHom (algebraMap (\u2191R) (Localization M))).op.unop) x).asIdeal x.asIdeal\n (CommRingCat.ofHom (algebraMap (\u2191R) (Localization M))).op.unop\n (_ :\n (\u2191(PrimeSpectrum.comap (CommRingCat.ofHom (algebraMap (\u2191R) (Localization M))).op.unop) x).asIdeal =\n (\u2191(PrimeSpectrum.comap (CommRingCat.ofHom (algebraMap (\u2191R) (Localization M))).op.unop) x).asIdeal) \u226b\n (stalkIso (\u2191(op (CommRingCat.of (Localization M))).unop) x).inv)\n[PROOFSTEP]\nrefine\n @IsIso.comp_isIso _ _ _ _ _ _ _ (?_)\n _\n /- I do not know why this is defeq to the goal, but I'm happy to accept that it is. -/\n[GOAL]\nR : CommRingCat\nM : Submonoid \u2191R\nx : PrimeSpectrum (Localization M)\n\u22a2 IsIso\n (Localization.localRingHom\n (\u2191(PrimeSpectrum.comap (CommRingCat.ofHom (algebraMap (\u2191R) (Localization M))).op.unop) x).asIdeal x.asIdeal\n (CommRingCat.ofHom (algebraMap (\u2191R) (Localization M))).op.unop\n (_ :\n (\u2191(PrimeSpectrum.comap (CommRingCat.ofHom (algebraMap (\u2191R) (Localization M))).op.unop) x).asIdeal =\n (\u2191(PrimeSpectrum.comap (CommRingCat.ofHom (algebraMap (\u2191R) (Localization M))).op.unop) x).asIdeal))\n[PROOFSTEP]\nshow IsIso (IsLocalization.localizationLocalizationAtPrimeIsoLocalization M x.asIdeal).toRingEquiv.toCommRingCatIso.hom\n[GOAL]\nR : CommRingCat\nM : Submonoid \u2191R\nx : PrimeSpectrum (Localization M)\n\u22a2 IsIso\n (RingEquiv.toCommRingCatIso\n (AlgEquiv.toRingEquiv (IsLocalization.localizationLocalizationAtPrimeIsoLocalization M x.asIdeal))).hom\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\n\u22a2 f \u226b toPushforwardStalk f p =\n toStalk (\u2191R) p \u226b (TopCat.Presheaf.stalkFunctor CommRingCat p).map (Spec.sheafedSpaceMap f).c\n[PROOFSTEP]\nrw [StructureSheaf.toStalk]\n[GOAL]\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\n\u22a2 f \u226b toPushforwardStalk f p =\n (toOpen \u2191R \u22a4 \u226b\n TopCat.Presheaf.germ (TopCat.Sheaf.presheaf (structureSheaf \u2191R)) { val := p, property := True.intro }) \u226b\n (TopCat.Presheaf.stalkFunctor CommRingCat p).map (Spec.sheafedSpaceMap f).c\n[PROOFSTEP]\nerw [Category.assoc]\n[GOAL]\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\n\u22a2 f \u226b toPushforwardStalk f p =\n toOpen \u2191R \u22a4 \u226b\n TopCat.Presheaf.germ (TopCat.Sheaf.presheaf (structureSheaf \u2191R)) { val := p, property := True.intro } \u226b\n (TopCat.Presheaf.stalkFunctor CommRingCat p).map (Spec.sheafedSpaceMap f).c\n[PROOFSTEP]\nrw [TopCat.Presheaf.stalkFunctor_map_germ]\n[GOAL]\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\n\u22a2 f \u226b toPushforwardStalk f p =\n toOpen \u2191R \u22a4 \u226b\n NatTrans.app (Spec.sheafedSpaceMap f).c (op \u22a4) \u226b\n TopCat.Presheaf.germ ((Spec.sheafedSpaceMap f).base _* (Spec.sheafedSpaceObj S).toPresheafedSpace.presheaf)\n { val := p, property := True.intro }\n[PROOFSTEP]\nexact Spec_\u0393_naturality_assoc f _\n[GOAL]\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\n\u22a2 \u2203 x, x.snd \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.fst\n[PROOFSTEP]\nobtain \u27e8U, hp, s, e\u27e9 := TopCat.Presheaf.germ_exist (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf S).val) _ y\n[GOAL]\ncase intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hp }) s = y\n\u22a2 \u2203 x, x.snd \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.fst\n[PROOFSTEP]\nobtain \u27e8_, \u27e8r, rfl\u27e9, hpr : p \u2208 PrimeSpectrum.basicOpen r, hrU : PrimeSpectrum.basicOpen r \u2264 U\u27e9 :=\n PrimeSpectrum.isTopologicalBasis_basic_opens.exists_subset_of_mem_open (show p \u2208 U from hp) U.2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hp }) s = y\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\n\u22a2 \u2203 x, x.snd \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.fst\n[PROOFSTEP]\nchange PrimeSpectrum.basicOpen r \u2264 U at hrU \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hp }) s = y\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\n\u22a2 \u2203 x, x.snd \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.fst\n[PROOFSTEP]\nreplace e := ((Spec.topMap (algebraMap R S) _* (structureSheaf S).1).germ_res_apply (homOfLE hrU) \u27e8p, hpr\u27e9 _).trans e\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s) =\n y\n\u22a2 \u2203 x, x.snd \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.fst\n[PROOFSTEP]\nset s' := (Spec.topMap (algebraMap R S) _* (structureSheaf S).1).map (homOfLE hrU).op s with h\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ns' : (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r))) :=\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\nh : s' = \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\n\u22a2 \u2203 x, x.snd \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.fst\n[PROOFSTEP]\nreplace e : ((Spec.topMap (algebraMap R S) _* (structureSheaf S).val).germ \u27e8p, hpr\u27e9) s' = y\n[GOAL]\ncase e\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ns' : (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r))) :=\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\nh : s' = \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase e\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ns' : (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r))) :=\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\nh : s' = \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s) =\n y\n[PROOFSTEP]\nexact e\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ns' : (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r))) :=\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\nh : s' = \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\n\u22a2 \u2203 x, x.snd \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.fst\n[PROOFSTEP]\nclear_value s'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhp : p \u2208 U\ns : (forget CommRingCat).obj ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op U))\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\nh : s' = \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op) s\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\n\u22a2 \u2203 x, x.snd \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.fst\n[PROOFSTEP]\nclear! U\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\n\u22a2 \u2203 x, x.snd \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.fst\n[PROOFSTEP]\nobtain \u27e8\u27e8s, \u27e8_, n, rfl\u27e9\u27e9, hsn\u27e9 :=\n @IsLocalization.surj _ _ _ _ _ _ (StructureSheaf.IsLocalization.to_basicOpen S <| algebraMap R S r) s'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).snd =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).fst\n\u22a2 \u2203 x, x.snd \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.fst\n[PROOFSTEP]\nrefine' \u27e8\u27e8s, \u27e8r, hpr\u27e9 ^ n\u27e9, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).snd =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).fst\n\u22a2 (s, { val := r, property := hpr } ^ n).snd \u2022 y =\n \u2191(toPushforwardStalkAlgHom R S p) (s, { val := r, property := hpr } ^ n).fst\n[PROOFSTEP]\nrw [Submonoid.smul_def, Algebra.smul_def, algebraMap_pushforward_stalk, toPushforwardStalk, comp_apply, comp_apply]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).snd =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).fst\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).snd)) *\n y =\n \u2191(toPushforwardStalkAlgHom R S p) (s, { val := r, property := hpr } ^ n).fst\n[PROOFSTEP]\niterate 2 erw [\u2190 (Spec.topMap (algebraMap R S) _* (structureSheaf S).1).germ_res_apply (homOfLE le_top) \u27e8p, hpr\u27e9]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).snd =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).fst\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).snd)) *\n y =\n \u2191(toPushforwardStalkAlgHom R S p) (s, { val := r, property := hpr } ^ n).fst\n[PROOFSTEP]\nerw [\u2190 (Spec.topMap (algebraMap R S) _* (structureSheaf S).1).germ_res_apply (homOfLE le_top) \u27e8p, hpr\u27e9]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).snd =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).fst\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).snd))) *\n y =\n \u2191(toPushforwardStalkAlgHom R S p) (s, { val := r, property := hpr } ^ n).fst\n[PROOFSTEP]\nerw [\u2190 (Spec.topMap (algebraMap R S) _* (structureSheaf S).1).germ_res_apply (homOfLE le_top) \u27e8p, hpr\u27e9]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).snd =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).fst\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).snd))) *\n y =\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).fst))\n[PROOFSTEP]\nrw [\u2190 e]\n -- Porting note : without this `change`, Lean doesn't know how to rewrite `map_mul`\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).snd =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).fst\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).snd))) *\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n s' =\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).fst))\n[PROOFSTEP]\nlet f := TopCat.Presheaf.germ (Spec.topMap (algebraMap R S) _* (structureSheaf S).val) \u27e8p, hpr\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf\u271d : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).snd =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).fst\nf : (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)) \u27f6\n TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) \u2191{ val := p, property := hpr } :=\n TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }\n\u22a2 \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).snd))) *\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n s' =\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr })\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).fst))\n[PROOFSTEP]\nchange f _ * f _ = f _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf\u271d : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).snd =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).fst\nf : (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)) \u27f6\n TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) \u2191{ val := p, property := hpr } :=\n TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }\n\u22a2 \u2191f\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).snd))) *\n \u2191f s' =\n \u2191f\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).fst))\n[PROOFSTEP]\nrw [\u2190 map_mul, mul_comm]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf\u271d : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n \u2191(s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).snd =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (s,\n { val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y =\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) }).fst\nf : (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)) \u27f6\n TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) \u2191{ val := p, property := hpr } :=\n TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }\n\u22a2 \u2191f\n (s' *\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).snd))) =\n \u2191f\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).fst))\n[PROOFSTEP]\ndsimp only [Subtype.coe_mk] at hsn \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf\u271d : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (\u2191(algebraMap \u2191R \u2191S) r ^ n) =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))))) s\nf : (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)) \u27f6\n TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) \u2191{ val := p, property := hpr } :=\n TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }\n\u22a2 \u2191f\n (s' *\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).snd))) =\n \u2191f\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).fst))\n[PROOFSTEP]\nrw [\u2190 map_pow (algebraMap R S)] at hsn \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.mk.mk.intro\nR S : CommRingCat\nf\u271d : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\ny : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)\nr : \u2191R\nhpr : p \u2208 PrimeSpectrum.basicOpen r\ns' :\n (forget CommRingCat).obj\n ((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)))\ne :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }) s' =\n y\ns : \u2191S\nn : \u2115\nhsn :\n s' *\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r)))))\n (\u2191(algebraMap \u2191R \u2191S) (r ^ n)) =\n \u2191(algebraMap \u2191S \u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))))) s\nf : (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).obj (op (PrimeSpectrum.basicOpen r)) \u27f6\n TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) \u2191{ val := p, property := hpr } :=\n TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := hpr }\n\u22a2 \u2191f\n (s' *\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (\u2191(algebraMap \u2191R \u2191S) \u2191(s, { val := r, property := hpr } ^ n).snd))) =\n \u2191f\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map\n (homOfLE (_ : PrimeSpectrum.basicOpen r \u2264 \u22a4)).op)\n (\u2191(toOpen \u2191S \u22a4) (s, { val := r, property := hpr } ^ n).fst))\n[PROOFSTEP]\ncongr 1\n[GOAL]\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\n\u22a2 IsLocalizedModule (Ideal.primeCompl p.asIdeal) (AlgHom.toLinearMap (toPushforwardStalkAlgHom R S p))\n[PROOFSTEP]\napply IsLocalizedModule.mkOfAlgebra\n[GOAL]\ncase h\u2081\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\n\u22a2 \u2200 (x : \u2191R),\n x \u2208 Ideal.primeCompl p.asIdeal \u2192\n IsUnit (\u2191(algebraMap \u2191R \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)) x)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h\u2081\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191R\nhx : x \u2208 Ideal.primeCompl p.asIdeal\n\u22a2 IsUnit (\u2191(algebraMap \u2191R \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)) x)\n[PROOFSTEP]\nrw [algebraMap_pushforward_stalk, toPushforwardStalk_comp]\n[GOAL]\ncase h\u2081\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191R\nhx : x \u2208 Ideal.primeCompl p.asIdeal\n\u22a2 IsUnit\n (\u2191(toStalk (\u2191R) p \u226b (TopCat.Presheaf.stalkFunctor CommRingCat p).map (Spec.sheafedSpaceMap (algebraMap \u2191R \u2191S)).c) x)\n[PROOFSTEP]\nchange IsUnit ((TopCat.Presheaf.stalkFunctor CommRingCat p).map (Spec.sheafedSpaceMap (algebraMap \u2191R \u2191S)).c _)\n[GOAL]\ncase h\u2081\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191R\nhx : x \u2208 Ideal.primeCompl p.asIdeal\n\u22a2 IsUnit\n (\u2191((TopCat.Presheaf.stalkFunctor CommRingCat p).map (Spec.sheafedSpaceMap (algebraMap \u2191R \u2191S)).c)\n (\u2191(toStalk (\u2191R) p) x))\n[PROOFSTEP]\nexact (IsLocalization.map_units ((structureSheaf R).presheaf.stalk p) \u27e8x, hx\u27e9).map _\n[GOAL]\ncase h\u2082\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\n\u22a2 \u2200 (y : \u2191(TopCat.Presheaf.stalk (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) p)),\n \u2203 x, x.snd \u2022 y = \u2191(toPushforwardStalkAlgHom R S p) x.fst\n[PROOFSTEP]\napply isLocalizedModule_toPushforwardStalkAlgHom_aux\n[GOAL]\ncase h\u2083\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\n\u22a2 \u2200 (x : \u2191S), \u2191(toPushforwardStalkAlgHom R S p) x = 0 \u2192 \u2203 m, m \u2022 x = 0\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h\u2083\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191S\nhx : \u2191(toPushforwardStalkAlgHom R S p) x = 0\n\u22a2 \u2203 m, m \u2022 x = 0\n[PROOFSTEP]\nrw [toPushforwardStalkAlgHom_apply, \u2190 (toPushforwardStalk (algebraMap R S) p).map_zero, toPushforwardStalk] at hx \n[GOAL]\ncase h\u2083\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191S\nhx :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) x) =\n \u2191(toOpen \u2191S \u22a4 \u226b\n TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n 0\n\u22a2 \u2203 m, m \u2022 x = 0\n[PROOFSTEP]\nchange\n _ =\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n (\u27e8p, trivial\u27e9 : (\u22a4 : TopologicalSpace.Opens (PrimeSpectrum R))) (toOpen S \u22a4 0)) at\n hx \n[GOAL]\ncase h\u2083\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191S\nhx :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) x) =\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) 0)\n\u22a2 \u2203 m, m \u2022 x = 0\n[PROOFSTEP]\nrw [map_zero] at hx \n[GOAL]\ncase h\u2083\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191S\nhx :\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val) { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) x) =\n \u2191(TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n 0\n\u22a2 \u2203 m, m \u2022 x = 0\n[PROOFSTEP]\nchange (forget CommRingCat).map _ _ = (forget _).map _ _ at hx \n[GOAL]\ncase h\u2083\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191S\nhx :\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) x) =\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n 0\n\u22a2 \u2203 m, m \u2022 x = 0\n[PROOFSTEP]\nobtain \u27e8U, hpU, i\u2081, i\u2082, e\u27e9 := TopCat.Presheaf.germ_eq _ _ _ _ _ _ hx\n[GOAL]\ncase h\u2083.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191S\nhx :\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) x) =\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n 0\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhpU : p \u2208 U\ni\u2081 : U \u27f6 \u22a4\ni\u2082 : U \u27f6 \u22a4\ne :\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map i\u2081.op) (\u2191(toOpen \u2191S \u22a4) x) =\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map i\u2082.op) 0\n\u22a2 \u2203 m, m \u2022 x = 0\n[PROOFSTEP]\nobtain \u27e8_, \u27e8r, rfl\u27e9, hpr, hrU\u27e9 :=\n PrimeSpectrum.isTopologicalBasis_basic_opens.exists_subset_of_mem_open (show p \u2208 U.1 from hpU) U.2\n[GOAL]\ncase h\u2083.intro.intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191S\nhx :\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) x) =\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n 0\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhpU : p \u2208 U\ni\u2081 : U \u27f6 \u22a4\ni\u2082 : U \u27f6 \u22a4\ne :\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map i\u2081.op) (\u2191(toOpen \u2191S \u22a4) x) =\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map i\u2082.op) 0\nr : \u2191R\nhpr : p \u2208 (fun r => \u2191(PrimeSpectrum.basicOpen r)) r\nhrU : (fun r => \u2191(PrimeSpectrum.basicOpen r)) r \u2286 U.carrier\n\u22a2 \u2203 m, m \u2022 x = 0\n[PROOFSTEP]\nchange PrimeSpectrum.basicOpen r \u2264 U at hrU \n[GOAL]\ncase h\u2083.intro.intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191S\nhx :\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) x) =\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n 0\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhpU : p \u2208 U\ni\u2081 : U \u27f6 \u22a4\ni\u2082 : U \u27f6 \u22a4\ne :\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map i\u2081.op) (\u2191(toOpen \u2191S \u22a4) x) =\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map i\u2082.op) 0\nr : \u2191R\nhpr : p \u2208 (fun r => \u2191(PrimeSpectrum.basicOpen r)) r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\n\u22a2 \u2203 m, m \u2022 x = 0\n[PROOFSTEP]\napply_fun (Spec.topMap (algebraMap R S) _* (structureSheaf S).1).map (homOfLE hrU).op at e \n[GOAL]\ncase h\u2083.intro.intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191S\nhx :\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) x) =\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n 0\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhpU : p \u2208 U\ni\u2081 : U \u27f6 \u22a4\ni\u2082 : U \u27f6 \u22a4\nr : \u2191R\nhpr : p \u2208 (fun r => \u2191(PrimeSpectrum.basicOpen r)) r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ne :\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op)\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map i\u2081.op) (\u2191(toOpen \u2191S \u22a4) x)) =\n \u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map (homOfLE hrU).op)\n (\u2191((Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val).map i\u2082.op) 0)\n\u22a2 \u2203 m, m \u2022 x = 0\n[PROOFSTEP]\nsimp only [TopCat.Presheaf.pushforwardObj_map, Functor.op_map, map_zero, \u2190 comp_apply, toOpen_res] at e \n[GOAL]\ncase h\u2083.intro.intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191S\nhx :\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) x) =\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n 0\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhpU : p \u2208 U\ni\u2081 : U \u27f6 \u22a4\ni\u2082 : U \u27f6 \u22a4\nr : \u2191R\nhpr : p \u2208 (fun r => \u2191(PrimeSpectrum.basicOpen r)) r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ne :\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2081.op.unop).op)\n (\u2191(toOpen \u2191S \u22a4) x)) =\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2082.op.unop).op)\n 0)\n\u22a2 \u2203 m, m \u2022 x = 0\n[PROOFSTEP]\nhave : toOpen S (PrimeSpectrum.basicOpen <| algebraMap R S r) x = 0 := by refine' Eq.trans _ e; rfl\n[GOAL]\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191S\nhx :\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) x) =\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n 0\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhpU : p \u2208 U\ni\u2081 : U \u27f6 \u22a4\ni\u2082 : U \u27f6 \u22a4\nr : \u2191R\nhpr : p \u2208 (fun r => \u2191(PrimeSpectrum.basicOpen r)) r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ne :\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2081.op.unop).op)\n (\u2191(toOpen \u2191S \u22a4) x)) =\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2082.op.unop).op)\n 0)\n\u22a2 \u2191(toOpen (\u2191S) (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))) x = 0\n[PROOFSTEP]\nrefine' Eq.trans _ e\n[GOAL]\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191S\nhx :\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) x) =\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n 0\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhpU : p \u2208 U\ni\u2081 : U \u27f6 \u22a4\ni\u2082 : U \u27f6 \u22a4\nr : \u2191R\nhpr : p \u2208 (fun r => \u2191(PrimeSpectrum.basicOpen r)) r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ne :\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2081.op.unop).op)\n (\u2191(toOpen \u2191S \u22a4) x)) =\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2082.op.unop).op)\n 0)\n\u22a2 \u2191(toOpen (\u2191S) (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))) x =\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2081.op.unop).op)\n (\u2191(toOpen \u2191S \u22a4) x))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\u2083.intro.intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191S\nhx :\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) x) =\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n 0\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhpU : p \u2208 U\ni\u2081 : U \u27f6 \u22a4\ni\u2082 : U \u27f6 \u22a4\nr : \u2191R\nhpr : p \u2208 (fun r => \u2191(PrimeSpectrum.basicOpen r)) r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ne :\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2081.op.unop).op)\n (\u2191(toOpen \u2191S \u22a4) x)) =\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2082.op.unop).op)\n 0)\nthis : \u2191(toOpen (\u2191S) (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))) x = 0\n\u22a2 \u2203 m, m \u2022 x = 0\n[PROOFSTEP]\nhave :=\n (@IsLocalization.mk'_one _ _ _ _ _ _ (StructureSheaf.IsLocalization.to_basicOpen S <| algebraMap R S r) x).trans this\n[GOAL]\ncase h\u2083.intro.intro.intro.intro.intro.intro.intro.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191S\nhx :\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) x) =\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n 0\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhpU : p \u2208 U\ni\u2081 : U \u27f6 \u22a4\ni\u2082 : U \u27f6 \u22a4\nr : \u2191R\nhpr : p \u2208 (fun r => \u2191(PrimeSpectrum.basicOpen r)) r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ne :\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2081.op.unop).op)\n (\u2191(toOpen \u2191S \u22a4) x)) =\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2082.op.unop).op)\n 0)\nthis\u271d : \u2191(toOpen (\u2191S) (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))) x = 0\nthis :\n IsLocalization.mk' (\u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))))) x 1 = 0\n\u22a2 \u2203 m, m \u2022 x = 0\n[PROOFSTEP]\nobtain \u27e8\u27e8_, n, rfl\u27e9, e\u27e9 := (IsLocalization.mk'_eq_zero_iff _ _).mp this\n[GOAL]\ncase h\u2083.intro.intro.intro.intro.intro.intro.intro.intro.intro.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191S\nhx :\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) x) =\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n 0\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhpU : p \u2208 U\ni\u2081 : U \u27f6 \u22a4\ni\u2082 : U \u27f6 \u22a4\nr : \u2191R\nhpr : p \u2208 (fun r => \u2191(PrimeSpectrum.basicOpen r)) r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ne\u271d :\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2081.op.unop).op)\n (\u2191(toOpen \u2191S \u22a4) x)) =\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2082.op.unop).op)\n 0)\nthis\u271d : \u2191(toOpen (\u2191S) (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))) x = 0\nthis :\n IsLocalization.mk' (\u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))))) x 1 = 0\nn : \u2115\ne :\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y = (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) } *\n x =\n 0\n\u22a2 \u2203 m, m \u2022 x = 0\n[PROOFSTEP]\nrefine' \u27e8\u27e8r, hpr\u27e9 ^ n, _\u27e9\n[GOAL]\ncase h\u2083.intro.intro.intro.intro.intro.intro.intro.intro.intro.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191S\nhx :\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) x) =\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n 0\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhpU : p \u2208 U\ni\u2081 : U \u27f6 \u22a4\ni\u2082 : U \u27f6 \u22a4\nr : \u2191R\nhpr : p \u2208 (fun r => \u2191(PrimeSpectrum.basicOpen r)) r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ne\u271d :\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2081.op.unop).op)\n (\u2191(toOpen \u2191S \u22a4) x)) =\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2082.op.unop).op)\n 0)\nthis\u271d : \u2191(toOpen (\u2191S) (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))) x = 0\nthis :\n IsLocalization.mk' (\u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))))) x 1 = 0\nn : \u2115\ne :\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y = (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) } *\n x =\n 0\n\u22a2 { val := r, property := hpr } ^ n \u2022 x = 0\n[PROOFSTEP]\nrw [Submonoid.smul_def, Algebra.smul_def]\n -- Porting note : manually rewrite `Submonoid.coe_pow`\n[GOAL]\ncase h\u2083.intro.intro.intro.intro.intro.intro.intro.intro.intro.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191S\nhx :\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) x) =\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n 0\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhpU : p \u2208 U\ni\u2081 : U \u27f6 \u22a4\ni\u2082 : U \u27f6 \u22a4\nr : \u2191R\nhpr : p \u2208 (fun r => \u2191(PrimeSpectrum.basicOpen r)) r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ne\u271d :\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2081.op.unop).op)\n (\u2191(toOpen \u2191S \u22a4) x)) =\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2082.op.unop).op)\n 0)\nthis\u271d : \u2191(toOpen (\u2191S) (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))) x = 0\nthis :\n IsLocalization.mk' (\u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))))) x 1 = 0\nn : \u2115\ne :\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y = (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) } *\n x =\n 0\n\u22a2 \u2191(algebraMap \u2191R \u2191S) \u2191({ val := r, property := hpr } ^ n) * x = 0\n[PROOFSTEP]\nchange (algebraMap R S) (r ^ n) * x = 0\n[GOAL]\ncase h\u2083.intro.intro.intro.intro.intro.intro.intro.intro.intro.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191S\nhx :\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) x) =\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n 0\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhpU : p \u2208 U\ni\u2081 : U \u27f6 \u22a4\ni\u2082 : U \u27f6 \u22a4\nr : \u2191R\nhpr : p \u2208 (fun r => \u2191(PrimeSpectrum.basicOpen r)) r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ne\u271d :\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2081.op.unop).op)\n (\u2191(toOpen \u2191S \u22a4) x)) =\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2082.op.unop).op)\n 0)\nthis\u271d : \u2191(toOpen (\u2191S) (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))) x = 0\nthis :\n IsLocalization.mk' (\u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))))) x 1 = 0\nn : \u2115\ne :\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y = (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) } *\n x =\n 0\n\u22a2 \u2191(algebraMap \u2191R \u2191S) (r ^ n) * x = 0\n[PROOFSTEP]\nrw [map_pow]\n[GOAL]\ncase h\u2083.intro.intro.intro.intro.intro.intro.intro.intro.intro.mk.intro\nR S : CommRingCat\nf : R \u27f6 S\np : PrimeSpectrum \u2191R\ninst\u271d : Algebra \u2191R \u2191S\nx : \u2191S\nhx :\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n (\u2191(toOpen \u2191S \u22a4) x) =\n (forget CommRingCat).map\n (TopCat.Presheaf.germ (Spec.topMap (algebraMap \u2191R \u2191S) _* (structureSheaf \u2191S).val)\n { val := p, property := trivial })\n 0\nU : TopologicalSpace.Opens \u2191(Spec.topObj R)\nhpU : p \u2208 U\ni\u2081 : U \u27f6 \u22a4\ni\u2082 : U \u27f6 \u22a4\nr : \u2191R\nhpr : p \u2208 (fun r => \u2191(PrimeSpectrum.basicOpen r)) r\nhrU : PrimeSpectrum.basicOpen r \u2264 U\ne\u271d :\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2081.op.unop).op)\n (\u2191(toOpen \u2191S \u22a4) x)) =\n \u2191((structureSheaf \u2191S).val.map\n ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map (homOfLE hrU).op.unop).op)\n (\u2191((structureSheaf \u2191S).val.map ((TopologicalSpace.Opens.map (Spec.topMap (algebraMap \u2191R \u2191S))).map i\u2082.op.unop).op)\n 0)\nthis\u271d : \u2191(toOpen (\u2191S) (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))) x = 0\nthis :\n IsLocalization.mk' (\u2191((structureSheaf \u2191S).val.obj (op (PrimeSpectrum.basicOpen (\u2191(algebraMap \u2191R \u2191S) r))))) x 1 = 0\nn : \u2115\ne :\n \u2191{ val := (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n,\n property :=\n (_ :\n \u2203 y,\n (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) y = (fun x x_1 => x ^ x_1) (\u2191(algebraMap \u2191R \u2191S) r) n) } *\n x =\n 0\n\u22a2 \u2191(algebraMap \u2191R \u2191S) r ^ n * x = 0\n[PROOFSTEP]\nexact e\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.Spec", "llama_tokens": 38961, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.4882833952958347, "lm_q2_score": 0.014281933925111154, "lm_q1q2_score": 0.006973631188344042}} {"text": "[GOAL]\nn m : WithBot \u2115\n\u22a2 n + m = 0 \u2194 n = 0 \u2227 m = 0\n[PROOFSTEP]\nrcases n, m with \u27e8_ | _, _ | _\u27e9\n[GOAL]\ncase none.none\n\u22a2 none + none = 0 \u2194 none = 0 \u2227 none = 0\ncase none.some\nval\u271d : \u2115\n\u22a2 none + some val\u271d = 0 \u2194 none = 0 \u2227 some val\u271d = 0\ncase some.none\nval\u271d : \u2115\n\u22a2 some val\u271d + none = 0 \u2194 some val\u271d = 0 \u2227 none = 0\ncase some.some val\u271d\u00b9 val\u271d : \u2115 \u22a2 some val\u271d\u00b9 + some val\u271d = 0 \u2194 some val\u271d\u00b9 = 0 \u2227 some val\u271d = 0\n[PROOFSTEP]\nany_goals (exact \u27e8fun h => Option.noConfusion h, fun h => Option.noConfusion h.1\u27e9)\n[GOAL]\ncase none.none\n\u22a2 none + none = 0 \u2194 none = 0 \u2227 none = 0\n[PROOFSTEP]\nexact \u27e8fun h => Option.noConfusion h, fun h => Option.noConfusion h.1\u27e9\n[GOAL]\ncase none.some\nval\u271d : \u2115\n\u22a2 none + some val\u271d = 0 \u2194 none = 0 \u2227 some val\u271d = 0\n[PROOFSTEP]\nexact \u27e8fun h => Option.noConfusion h, fun h => Option.noConfusion h.1\u27e9\n[GOAL]\ncase some.none\nval\u271d : \u2115\n\u22a2 some val\u271d + none = 0 \u2194 some val\u271d = 0 \u2227 none = 0\n[PROOFSTEP]\nexact \u27e8fun h => Option.noConfusion h, fun h => Option.noConfusion h.1\u27e9\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 0 \u2194 some val\u271d\u00b9 = 0 \u2227 some val\u271d = 0\n[PROOFSTEP]\nexact \u27e8fun h => Option.noConfusion h, fun h => Option.noConfusion h.1\u27e9\n[GOAL]\ncase some.none\nval\u271d : \u2115\n\u22a2 some val\u271d + none = 0 \u2194 some val\u271d = 0 \u2227 none = 0\ncase some.some val\u271d\u00b9 val\u271d : \u2115 \u22a2 some val\u271d\u00b9 + some val\u271d = 0 \u2194 some val\u271d\u00b9 = 0 \u2227 some val\u271d = 0\n[PROOFSTEP]\nexact \u27e8fun h => Option.noConfusion h, fun h => Option.noConfusion h.2\u27e9\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 0 \u2194 some val\u271d\u00b9 = 0 \u2227 some val\u271d = 0\n[PROOFSTEP]\nrepeat' erw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 0 \u2194 some val\u271d\u00b9 = 0 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = 0 \u2194 some val\u271d\u00b9 = 0 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = 0 \u2194 val\u271d\u00b9 = 0 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = 0 \u2194 val\u271d\u00b9 = 0 \u2227 val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = 0 \u2194 val\u271d\u00b9 = 0 \u2227 val\u271d = 0\n[PROOFSTEP]\nexact add_eq_zero_iff' (zero_le _) (zero_le _)\n[GOAL]\nn m : WithBot \u2115\n\u22a2 n + m = 1 \u2194 n = 0 \u2227 m = 1 \u2228 n = 1 \u2227 m = 0\n[PROOFSTEP]\nrcases n, m with \u27e8_ | _, _ | _\u27e9\n[GOAL]\ncase none.none\n\u22a2 none + none = 1 \u2194 none = 0 \u2227 none = 1 \u2228 none = 1 \u2227 none = 0\ncase none.some\nval\u271d : \u2115\n\u22a2 none + some val\u271d = 1 \u2194 none = 0 \u2227 some val\u271d = 1 \u2228 none = 1 \u2227 some val\u271d = 0\ncase some.none\nval\u271d : \u2115\n\u22a2 some val\u271d + none = 1 \u2194 some val\u271d = 0 \u2227 none = 1 \u2228 some val\u271d = 1 \u2227 none = 0\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 1 \u2194 some val\u271d\u00b9 = 0 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 1 \u2227 some val\u271d = 0\n[PROOFSTEP]\nany_goals refine' \u27e8fun h => Option.noConfusion h, fun h => _\u27e9; aesop\n[GOAL]\ncase none.none\n\u22a2 none + none = 1 \u2194 none = 0 \u2227 none = 1 \u2228 none = 1 \u2227 none = 0\n[PROOFSTEP]\nrefine' \u27e8fun h => Option.noConfusion h, fun h => _\u27e9\n[GOAL]\ncase none.none\nh : none = 0 \u2227 none = 1 \u2228 none = 1 \u2227 none = 0\n\u22a2 none + none = 1\n[PROOFSTEP]\naesop\n[GOAL]\ncase none.some\nval\u271d : \u2115\n\u22a2 none + some val\u271d = 1 \u2194 none = 0 \u2227 some val\u271d = 1 \u2228 none = 1 \u2227 some val\u271d = 0\n[PROOFSTEP]\nrefine' \u27e8fun h => Option.noConfusion h, fun h => _\u27e9\n[GOAL]\ncase none.some\nval\u271d : \u2115\nh : none = 0 \u2227 some val\u271d = 1 \u2228 none = 1 \u2227 some val\u271d = 0\n\u22a2 none + some val\u271d = 1\n[PROOFSTEP]\naesop\n[GOAL]\ncase some.none\nval\u271d : \u2115\n\u22a2 some val\u271d + none = 1 \u2194 some val\u271d = 0 \u2227 none = 1 \u2228 some val\u271d = 1 \u2227 none = 0\n[PROOFSTEP]\nrefine' \u27e8fun h => Option.noConfusion h, fun h => _\u27e9\n[GOAL]\ncase some.none\nval\u271d : \u2115\nh : some val\u271d = 0 \u2227 none = 1 \u2228 some val\u271d = 1 \u2227 none = 0\n\u22a2 some val\u271d + none = 1\n[PROOFSTEP]\naesop\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 1 \u2194 some val\u271d\u00b9 = 0 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 1 \u2227 some val\u271d = 0\n[PROOFSTEP]\nrefine' \u27e8fun h => Option.noConfusion h, fun h => _\u27e9\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 1 \u2194 some val\u271d\u00b9 = 0 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 1 \u2227 some val\u271d = 0\n[PROOFSTEP]\nrepeat' erw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 1 \u2194 some val\u271d\u00b9 = 0 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 1 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = 1 \u2194 some val\u271d\u00b9 = 0 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 1 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = 1 \u2194 val\u271d\u00b9 = 0 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 1 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = 1 \u2194 val\u271d\u00b9 = 0 \u2227 val\u271d = 1 \u2228 some val\u271d\u00b9 = 1 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = 1 \u2194 val\u271d\u00b9 = 0 \u2227 val\u271d = 1 \u2228 val\u271d\u00b9 = 1 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = 1 \u2194 val\u271d\u00b9 = 0 \u2227 val\u271d = 1 \u2228 val\u271d\u00b9 = 1 \u2227 val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = 1 \u2194 val\u271d\u00b9 = 0 \u2227 val\u271d = 1 \u2228 val\u271d\u00b9 = 1 \u2227 val\u271d = 0\n[PROOFSTEP]\nexact Nat.add_eq_one_iff\n[GOAL]\nn m : WithBot \u2115\n\u22a2 n + m = 2 \u2194 n = 0 \u2227 m = 2 \u2228 n = 1 \u2227 m = 1 \u2228 n = 2 \u2227 m = 0\n[PROOFSTEP]\nrcases n, m with \u27e8_ | _, _ | _\u27e9\n[GOAL]\ncase none.none\n\u22a2 none + none = 2 \u2194 none = 0 \u2227 none = 2 \u2228 none = 1 \u2227 none = 1 \u2228 none = 2 \u2227 none = 0\ncase none.some\nval\u271d : \u2115\n\u22a2 none + some val\u271d = 2 \u2194 none = 0 \u2227 some val\u271d = 2 \u2228 none = 1 \u2227 some val\u271d = 1 \u2228 none = 2 \u2227 some val\u271d = 0\ncase some.none\nval\u271d : \u2115\n\u22a2 some val\u271d + none = 2 \u2194 some val\u271d = 0 \u2227 none = 2 \u2228 some val\u271d = 1 \u2227 none = 1 \u2228 some val\u271d = 2 \u2227 none = 0\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 2 \u2194\n some val\u271d\u00b9 = 0 \u2227 some val\u271d = 2 \u2228 some val\u271d\u00b9 = 1 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 2 \u2227 some val\u271d = 0\n[PROOFSTEP]\nany_goals refine' \u27e8fun h => Option.noConfusion h, fun h => _\u27e9; aesop\n[GOAL]\ncase none.none\n\u22a2 none + none = 2 \u2194 none = 0 \u2227 none = 2 \u2228 none = 1 \u2227 none = 1 \u2228 none = 2 \u2227 none = 0\n[PROOFSTEP]\nrefine' \u27e8fun h => Option.noConfusion h, fun h => _\u27e9\n[GOAL]\ncase none.none\nh : none = 0 \u2227 none = 2 \u2228 none = 1 \u2227 none = 1 \u2228 none = 2 \u2227 none = 0\n\u22a2 none + none = 2\n[PROOFSTEP]\naesop\n[GOAL]\ncase none.some\nval\u271d : \u2115\n\u22a2 none + some val\u271d = 2 \u2194 none = 0 \u2227 some val\u271d = 2 \u2228 none = 1 \u2227 some val\u271d = 1 \u2228 none = 2 \u2227 some val\u271d = 0\n[PROOFSTEP]\nrefine' \u27e8fun h => Option.noConfusion h, fun h => _\u27e9\n[GOAL]\ncase none.some\nval\u271d : \u2115\nh : none = 0 \u2227 some val\u271d = 2 \u2228 none = 1 \u2227 some val\u271d = 1 \u2228 none = 2 \u2227 some val\u271d = 0\n\u22a2 none + some val\u271d = 2\n[PROOFSTEP]\naesop\n[GOAL]\ncase some.none\nval\u271d : \u2115\n\u22a2 some val\u271d + none = 2 \u2194 some val\u271d = 0 \u2227 none = 2 \u2228 some val\u271d = 1 \u2227 none = 1 \u2228 some val\u271d = 2 \u2227 none = 0\n[PROOFSTEP]\nrefine' \u27e8fun h => Option.noConfusion h, fun h => _\u27e9\n[GOAL]\ncase some.none\nval\u271d : \u2115\nh : some val\u271d = 0 \u2227 none = 2 \u2228 some val\u271d = 1 \u2227 none = 1 \u2228 some val\u271d = 2 \u2227 none = 0\n\u22a2 some val\u271d + none = 2\n[PROOFSTEP]\naesop\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 2 \u2194\n some val\u271d\u00b9 = 0 \u2227 some val\u271d = 2 \u2228 some val\u271d\u00b9 = 1 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 2 \u2227 some val\u271d = 0\n[PROOFSTEP]\nrefine' \u27e8fun h => Option.noConfusion h, fun h => _\u27e9\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 2 \u2194\n some val\u271d\u00b9 = 0 \u2227 some val\u271d = 2 \u2228 some val\u271d\u00b9 = 1 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 2 \u2227 some val\u271d = 0\n[PROOFSTEP]\nrepeat' erw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 2 \u2194\n some val\u271d\u00b9 = 0 \u2227 some val\u271d = 2 \u2228 some val\u271d\u00b9 = 1 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 2 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = \u21912 \u2194\n some val\u271d\u00b9 = 0 \u2227 some val\u271d = 2 \u2228 some val\u271d\u00b9 = 1 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 2 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = \u21912 \u2194\n val\u271d\u00b9 = 0 \u2227 some val\u271d = 2 \u2228 some val\u271d\u00b9 = 1 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 2 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = \u21912 \u2194\n val\u271d\u00b9 = 0 \u2227 val\u271d = \u21912 \u2228 some val\u271d\u00b9 = 1 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 2 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = \u21912 \u2194\n val\u271d\u00b9 = 0 \u2227 val\u271d = \u21912 \u2228 val\u271d\u00b9 = 1 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 2 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = \u21912 \u2194 val\u271d\u00b9 = 0 \u2227 val\u271d = \u21912 \u2228 val\u271d\u00b9 = 1 \u2227 val\u271d = 1 \u2228 some val\u271d\u00b9 = 2 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = \u21912 \u2194 val\u271d\u00b9 = 0 \u2227 val\u271d = \u21912 \u2228 val\u271d\u00b9 = 1 \u2227 val\u271d = 1 \u2228 val\u271d\u00b9 = \u21912 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = \u21912 \u2194 val\u271d\u00b9 = 0 \u2227 val\u271d = \u21912 \u2228 val\u271d\u00b9 = 1 \u2227 val\u271d = 1 \u2228 val\u271d\u00b9 = \u21912 \u2227 val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = \u21912 \u2194 val\u271d\u00b9 = 0 \u2227 val\u271d = \u21912 \u2228 val\u271d\u00b9 = 1 \u2227 val\u271d = 1 \u2228 val\u271d\u00b9 = \u21912 \u2227 val\u271d = 0\n[PROOFSTEP]\nexact Nat.add_eq_two_iff\n[GOAL]\nn m : WithBot \u2115\n\u22a2 n + m = 3 \u2194 n = 0 \u2227 m = 3 \u2228 n = 1 \u2227 m = 2 \u2228 n = 2 \u2227 m = 1 \u2228 n = 3 \u2227 m = 0\n[PROOFSTEP]\nrcases n, m with \u27e8_ | _, _ | _\u27e9\n[GOAL]\ncase none.none\n\u22a2 none + none = 3 \u2194 none = 0 \u2227 none = 3 \u2228 none = 1 \u2227 none = 2 \u2228 none = 2 \u2227 none = 1 \u2228 none = 3 \u2227 none = 0\ncase none.some\nval\u271d : \u2115\n\u22a2 none + some val\u271d = 3 \u2194\n none = 0 \u2227 some val\u271d = 3 \u2228 none = 1 \u2227 some val\u271d = 2 \u2228 none = 2 \u2227 some val\u271d = 1 \u2228 none = 3 \u2227 some val\u271d = 0\ncase some.none\nval\u271d : \u2115\n\u22a2 some val\u271d + none = 3 \u2194\n some val\u271d = 0 \u2227 none = 3 \u2228 some val\u271d = 1 \u2227 none = 2 \u2228 some val\u271d = 2 \u2227 none = 1 \u2228 some val\u271d = 3 \u2227 none = 0\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 3 \u2194\n some val\u271d\u00b9 = 0 \u2227 some val\u271d = 3 \u2228\n some val\u271d\u00b9 = 1 \u2227 some val\u271d = 2 \u2228 some val\u271d\u00b9 = 2 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 3 \u2227 some val\u271d = 0\n[PROOFSTEP]\nany_goals refine' \u27e8fun h => Option.noConfusion h, fun h => _\u27e9; aesop\n[GOAL]\ncase none.none\n\u22a2 none + none = 3 \u2194 none = 0 \u2227 none = 3 \u2228 none = 1 \u2227 none = 2 \u2228 none = 2 \u2227 none = 1 \u2228 none = 3 \u2227 none = 0\n[PROOFSTEP]\nrefine' \u27e8fun h => Option.noConfusion h, fun h => _\u27e9\n[GOAL]\ncase none.none\nh : none = 0 \u2227 none = 3 \u2228 none = 1 \u2227 none = 2 \u2228 none = 2 \u2227 none = 1 \u2228 none = 3 \u2227 none = 0\n\u22a2 none + none = 3\n[PROOFSTEP]\naesop\n[GOAL]\ncase none.some\nval\u271d : \u2115\n\u22a2 none + some val\u271d = 3 \u2194\n none = 0 \u2227 some val\u271d = 3 \u2228 none = 1 \u2227 some val\u271d = 2 \u2228 none = 2 \u2227 some val\u271d = 1 \u2228 none = 3 \u2227 some val\u271d = 0\n[PROOFSTEP]\nrefine' \u27e8fun h => Option.noConfusion h, fun h => _\u27e9\n[GOAL]\ncase none.some\nval\u271d : \u2115\nh : none = 0 \u2227 some val\u271d = 3 \u2228 none = 1 \u2227 some val\u271d = 2 \u2228 none = 2 \u2227 some val\u271d = 1 \u2228 none = 3 \u2227 some val\u271d = 0\n\u22a2 none + some val\u271d = 3\n[PROOFSTEP]\naesop\n[GOAL]\ncase some.none\nval\u271d : \u2115\n\u22a2 some val\u271d + none = 3 \u2194\n some val\u271d = 0 \u2227 none = 3 \u2228 some val\u271d = 1 \u2227 none = 2 \u2228 some val\u271d = 2 \u2227 none = 1 \u2228 some val\u271d = 3 \u2227 none = 0\n[PROOFSTEP]\nrefine' \u27e8fun h => Option.noConfusion h, fun h => _\u27e9\n[GOAL]\ncase some.none\nval\u271d : \u2115\nh : some val\u271d = 0 \u2227 none = 3 \u2228 some val\u271d = 1 \u2227 none = 2 \u2228 some val\u271d = 2 \u2227 none = 1 \u2228 some val\u271d = 3 \u2227 none = 0\n\u22a2 some val\u271d + none = 3\n[PROOFSTEP]\naesop\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 3 \u2194\n some val\u271d\u00b9 = 0 \u2227 some val\u271d = 3 \u2228\n some val\u271d\u00b9 = 1 \u2227 some val\u271d = 2 \u2228 some val\u271d\u00b9 = 2 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 3 \u2227 some val\u271d = 0\n[PROOFSTEP]\nrefine' \u27e8fun h => Option.noConfusion h, fun h => _\u27e9\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 3 \u2194\n some val\u271d\u00b9 = 0 \u2227 some val\u271d = 3 \u2228\n some val\u271d\u00b9 = 1 \u2227 some val\u271d = 2 \u2228 some val\u271d\u00b9 = 2 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 3 \u2227 some val\u271d = 0\n[PROOFSTEP]\nrepeat' erw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 3 \u2194\n some val\u271d\u00b9 = 0 \u2227 some val\u271d = 3 \u2228\n some val\u271d\u00b9 = 1 \u2227 some val\u271d = 2 \u2228 some val\u271d\u00b9 = 2 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 3 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = \u21913 \u2194\n some val\u271d\u00b9 = 0 \u2227 some val\u271d = 3 \u2228\n some val\u271d\u00b9 = 1 \u2227 some val\u271d = 2 \u2228 some val\u271d\u00b9 = 2 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 3 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = \u21913 \u2194\n val\u271d\u00b9 = 0 \u2227 some val\u271d = 3 \u2228\n some val\u271d\u00b9 = 1 \u2227 some val\u271d = 2 \u2228 some val\u271d\u00b9 = 2 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 3 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = \u21913 \u2194\n val\u271d\u00b9 = 0 \u2227 val\u271d = \u21913 \u2228\n some val\u271d\u00b9 = 1 \u2227 some val\u271d = 2 \u2228 some val\u271d\u00b9 = 2 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 3 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = \u21913 \u2194\n val\u271d\u00b9 = 0 \u2227 val\u271d = \u21913 \u2228 val\u271d\u00b9 = 1 \u2227 some val\u271d = 2 \u2228 some val\u271d\u00b9 = 2 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 3 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = \u21913 \u2194\n val\u271d\u00b9 = 0 \u2227 val\u271d = \u21913 \u2228 val\u271d\u00b9 = 1 \u2227 val\u271d = \u21912 \u2228 some val\u271d\u00b9 = 2 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 3 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = \u21913 \u2194\n val\u271d\u00b9 = 0 \u2227 val\u271d = \u21913 \u2228 val\u271d\u00b9 = 1 \u2227 val\u271d = \u21912 \u2228 val\u271d\u00b9 = \u21912 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 3 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = \u21913 \u2194\n val\u271d\u00b9 = 0 \u2227 val\u271d = \u21913 \u2228 val\u271d\u00b9 = 1 \u2227 val\u271d = \u21912 \u2228 val\u271d\u00b9 = \u21912 \u2227 val\u271d = 1 \u2228 some val\u271d\u00b9 = 3 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = \u21913 \u2194\n val\u271d\u00b9 = 0 \u2227 val\u271d = \u21913 \u2228 val\u271d\u00b9 = 1 \u2227 val\u271d = \u21912 \u2228 val\u271d\u00b9 = \u21912 \u2227 val\u271d = 1 \u2228 val\u271d\u00b9 = \u21913 \u2227 some val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = \u21913 \u2194\n val\u271d\u00b9 = 0 \u2227 val\u271d = \u21913 \u2228 val\u271d\u00b9 = 1 \u2227 val\u271d = \u21912 \u2228 val\u271d\u00b9 = \u21912 \u2227 val\u271d = 1 \u2228 val\u271d\u00b9 = \u21913 \u2227 val\u271d = 0\n[PROOFSTEP]\nerw [WithBot.coe_eq_coe]\n[GOAL]\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = \u21913 \u2194\n val\u271d\u00b9 = 0 \u2227 val\u271d = \u21913 \u2228 val\u271d\u00b9 = 1 \u2227 val\u271d = \u21912 \u2228 val\u271d\u00b9 = \u21912 \u2227 val\u271d = 1 \u2228 val\u271d\u00b9 = \u21913 \u2227 val\u271d = 0\n[PROOFSTEP]\nexact Nat.add_eq_three_iff\n[GOAL]\nn : \u2115\n\u22a2 0 \u2264 \u2191n\n[PROOFSTEP]\nrw [\u2190 WithBot.coe_zero]\n[GOAL]\nn : \u2115\n\u22a2 \u21910 \u2264 \u2191n\n[PROOFSTEP]\nexact WithBot.coe_le_coe.mpr (Nat.zero_le n)\n[GOAL]\nn : WithBot \u2115\n\u22a2 n < 0 \u2194 n = \u22a5\n[PROOFSTEP]\nrefine' Option.casesOn n _ _\n[GOAL]\ncase refine'_1\nn : WithBot \u2115\n\u22a2 none < 0 \u2194 none = \u22a5\ncase refine'_2 n : WithBot \u2115 \u22a2 \u2200 (val : \u2115), some val < 0 \u2194 some val = \u22a5\n[PROOFSTEP]\nexact of_eq_true (eq_true_of_decide (Eq.refl true))\n[GOAL]\ncase refine'_2\nn : WithBot \u2115\n\u22a2 \u2200 (val : \u2115), some val < 0 \u2194 some val = \u22a5\n[PROOFSTEP]\nintro n\n[GOAL]\ncase refine'_2\nn\u271d : WithBot \u2115\nn : \u2115\n\u22a2 some n < 0 \u2194 some n = \u22a5\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_2.refine'_1\nn\u271d : WithBot \u2115\nn : \u2115\nh : some n < 0\n\u22a2 some n = \u22a5\ncase refine'_2.refine'_2 n\u271d : WithBot \u2115 n : \u2115 h : some n = \u22a5 \u22a2 some n < 0\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase refine'_2.refine'_1.h\nn\u271d : WithBot \u2115\nn : \u2115\nh : some n < 0\n\u22a2 False\n[PROOFSTEP]\nrw [WithBot.some_eq_coe] at h \n[GOAL]\ncase refine'_2.refine'_1.h\nn\u271d : WithBot \u2115\nn : \u2115\nh : \u2191n < 0\n\u22a2 False\n[PROOFSTEP]\nexact not_le_of_lt h WithBot.coe_nonneg\n[GOAL]\ncase refine'_2.refine'_2\nn\u271d : WithBot \u2115\nn : \u2115\nh : some n = \u22a5\n\u22a2 some n < 0\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase refine'_2.refine'_2\nn\u271d : WithBot \u2115\nn : \u2115\nh : some n = \u22a5\n\u22a2 \u22a5 < 0\n[PROOFSTEP]\nexact of_eq_true (eq_true_of_decide (Eq.refl true))\n[GOAL]\nx : WithBot \u2115\n\u22a2 1 \u2264 x \u2194 0 < x\n[PROOFSTEP]\nrefine' \u27e8fun h => lt_of_lt_of_le (WithBot.coe_lt_coe.mpr zero_lt_one) h, fun h => _\u27e9\n[GOAL]\nx : WithBot \u2115\nh : 0 < x\n\u22a2 1 \u2264 x\n[PROOFSTEP]\ninduction x using WithBot.recBotCoe\n[GOAL]\ncase bot\nh : 0 < \u22a5\n\u22a2 1 \u2264 \u22a5\n[PROOFSTEP]\nexact (not_lt_bot h).elim\n[GOAL]\ncase coe\na\u271d : \u2115\nh : 0 < \u2191a\u271d\n\u22a2 1 \u2264 \u2191a\u271d\n[PROOFSTEP]\nexact WithBot.coe_le_coe.mpr (Nat.succ_le_iff.mpr (WithBot.coe_lt_coe.mp h))\n[GOAL]\nx : WithBot \u2115\n\u22a2 \u00acx < 1 \u2194 \u00acx \u2264 0\n[PROOFSTEP]\nsimpa using one_le_iff_zero_lt\n[GOAL]\nn m : WithBot \u2115\nh : n < m\n\u22a2 n + 1 \u2264 m\n[PROOFSTEP]\ncases n\n[GOAL]\ncase none\nm : WithBot \u2115\nh : none < m\n\u22a2 none + 1 \u2264 m\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase some\nm : WithBot \u2115\nval\u271d : \u2115\nh : some val\u271d < m\n\u22a2 some val\u271d + 1 \u2264 m\n[PROOFSTEP]\ncases m\n[GOAL]\ncase some.none\nval\u271d : \u2115\nh : some val\u271d < none\n\u22a2 some val\u271d + 1 \u2264 none\ncase some.some val\u271d\u00b9 val\u271d : \u2115 h : some val\u271d\u00b9 < some val\u271d \u22a2 some val\u271d\u00b9 + 1 \u2264 some val\u271d\n[PROOFSTEP]\nexacts [(not_lt_bot h).elim, WithBot.some_le_some.2 (WithBot.some_lt_some.1 h)]\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.WithBot", "llama_tokens": 9953, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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NO", "lm_q1_score": 0.20434189993684582, "lm_q2_score": 0.02976009276274224, "lm_q1q2_score": 0.0060812338974355245}} {"text": "", "meta": {"mathlib_filename": "Mathlib.Lean.Meta.DiscrTree", "llama_tokens": 0, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.20434188055505068, "lm_q2_score": 0.02976009459135805, "lm_q1q2_score": 0.006081233694294297}} {"text": "", "meta": {"mathlib_filename": "Mathlib.Lean.IO.Process", "llama_tokens": 0, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.20434188055505068, "lm_q2_score": 0.02976009459135805, "lm_q1q2_score": 0.006081233694294297}} {"text": "", "meta": {"mathlib_filename": "Mathlib.Lean.Meta.Basic", "llama_tokens": 0, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. 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NO", "lm_q1_score": 0.20434188055505068, "lm_q2_score": 0.02976009459135805, "lm_q1q2_score": 0.006081233694294297}} {"text": "", "meta": {"mathlib_filename": "Mathlib.Tactic.NormCast", "llama_tokens": 0, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.20689405859634896, "lm_q2_score": 0.02931222979739437, "lm_q1q2_score": 0.006064526189291757}} {"text": "", "meta": {"mathlib_filename": "Mathlib.Lean.Data.NameMap", "llama_tokens": 0, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.20689405859634893, "lm_q2_score": 0.02931222947940706, "lm_q1q2_score": 0.006064526123502071}} {"text": "", "meta": {"mathlib_filename": "Mathlib.Data.Array.Basic", "llama_tokens": 0, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. 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NO", "lm_q1_score": 0.20181322226037884, "lm_q2_score": 0.02887090412620902, "lm_q1q2_score": 0.00582653019128071}} {"text": "", "meta": {"mathlib_filename": "Mathlib.Tactic.Measurability.Init", "llama_tokens": 0, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.20181320305759337, "lm_q2_score": 0.028870905170682647, "lm_q1q2_score": 0.005826529847667499}} {"text": "[GOAL]\n\u03b1 : Type u_1\na : Lists \u03b1\nl : Lists' \u03b1 true\n\u22a2 toList (cons a l) = a :: toList l\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nl : List (Lists \u03b1)\n\u22a2 toList (ofList l) = l\n[PROOFSTEP]\ninduction l\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u22a2 toList (ofList []) = []\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nhead\u271d : Lists \u03b1\ntail\u271d : List (Lists \u03b1)\ntail_ih\u271d : toList (ofList tail\u271d) = tail\u271d\n\u22a2 toList (ofList (head\u271d :: tail\u271d)) = head\u271d :: tail\u271d\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\nb : Bool\nh : true = b\nl : Lists' \u03b1 b\n\u22a2 Lists' \u03b1 true\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\nb : Bool\nh : true = b\nl : Lists' \u03b1 b\n\u22a2 Lists' \u03b1 b\n[PROOFSTEP]\nexact l\n[GOAL]\n\u03b1 : Type u_1\nb : Bool\nh : true = b\nl : Lists' \u03b1 b\n\u22a2 let l' := Eq.mpr (_ : Lists' \u03b1 true = Lists' \u03b1 b) l;\n ofList (toList l') = l'\n[PROOFSTEP]\ninduction l with\n| atom => cases h\n| nil => simp\n| cons' b a _ IH =>\n intro l'\n simpa [cons] using IH rfl\n[GOAL]\n\u03b1 : Type u_1\nb : Bool\nh : true = b\nl : Lists' \u03b1 b\n\u22a2 let l' := Eq.mpr (_ : Lists' \u03b1 true = Lists' \u03b1 b) l;\n ofList (toList l') = l'\n[PROOFSTEP]\ninduction l with\n| atom => cases h\n| nil => simp\n| cons' b a _ IH =>\n intro l'\n simpa [cons] using IH rfl\n[GOAL]\ncase atom\n\u03b1 : Type u_1\nb : Bool\na\u271d : \u03b1\nh : true = false\n\u22a2 let l' := Eq.mpr (_ : Lists' \u03b1 true = Lists' \u03b1 false) (atom a\u271d);\n ofList (toList l') = l'\n[PROOFSTEP]\n\n| atom => cases h\n[GOAL]\ncase atom\n\u03b1 : Type u_1\nb : Bool\na\u271d : \u03b1\nh : true = false\n\u22a2 let l' := Eq.mpr (_ : Lists' \u03b1 true = Lists' \u03b1 false) (atom a\u271d);\n ofList (toList l') = l'\n[PROOFSTEP]\ncases h\n[GOAL]\ncase nil\n\u03b1 : Type u_1\nb : Bool\nh : true = true\n\u22a2 let l' := Eq.mpr (_ : Lists' \u03b1 true = Lists' \u03b1 true) nil;\n ofList (toList l') = l'\n[PROOFSTEP]\n\n| nil => simp\n[GOAL]\ncase nil\n\u03b1 : Type u_1\nb : Bool\nh : true = true\n\u22a2 let l' := Eq.mpr (_ : Lists' \u03b1 true = Lists' \u03b1 true) nil;\n ofList (toList l') = l'\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons'\n\u03b1 : Type u_1\nb\u271d\u00b9 b\u271d : Bool\nb : Lists' \u03b1 b\u271d\na : Lists' \u03b1 true\na_ih\u271d :\n \u2200 (h : true = b\u271d),\n let l' := Eq.mpr (_ : Lists' \u03b1 true = Lists' \u03b1 b\u271d) b;\n ofList (toList l') = l'\nIH :\n \u2200 (h : true = true),\n let l' := Eq.mpr (_ : Lists' \u03b1 true = Lists' \u03b1 true) a;\n ofList (toList l') = l'\nh : true = true\n\u22a2 let l' := Eq.mpr (_ : Lists' \u03b1 true = Lists' \u03b1 true) (cons' b a);\n ofList (toList l') = l'\n[PROOFSTEP]\n\n| cons' b a _ IH =>\n intro l'\n simpa [cons] using IH rfl\n[GOAL]\ncase cons'\n\u03b1 : Type u_1\nb\u271d\u00b9 b\u271d : Bool\nb : Lists' \u03b1 b\u271d\na : Lists' \u03b1 true\na_ih\u271d :\n \u2200 (h : true = b\u271d),\n let l' := Eq.mpr (_ : Lists' \u03b1 true = Lists' \u03b1 b\u271d) b;\n ofList (toList l') = l'\nIH :\n \u2200 (h : true = true),\n let l' := Eq.mpr (_ : Lists' \u03b1 true = Lists' \u03b1 true) a;\n ofList (toList l') = l'\nh : true = true\n\u22a2 let l' := Eq.mpr (_ : Lists' \u03b1 true = Lists' \u03b1 true) (cons' b a);\n ofList (toList l') = l'\n[PROOFSTEP]\nintro l'\n[GOAL]\ncase cons'\n\u03b1 : Type u_1\nb\u271d\u00b9 b\u271d : Bool\nb : Lists' \u03b1 b\u271d\na : Lists' \u03b1 true\na_ih\u271d :\n \u2200 (h : true = b\u271d),\n let l' := Eq.mpr (_ : Lists' \u03b1 true = Lists' \u03b1 b\u271d) b;\n ofList (toList l') = l'\nIH :\n \u2200 (h : true = true),\n let l' := Eq.mpr (_ : Lists' \u03b1 true = Lists' \u03b1 true) a;\n ofList (toList l') = l'\nh : true = true\nl' : Lists' \u03b1 true := Eq.mpr (_ : Lists' \u03b1 true = Lists' \u03b1 true) (cons' b a)\n\u22a2 ofList (toList l') = l'\n[PROOFSTEP]\nsimpa [cons] using IH rfl\n[GOAL]\n\u03b1 : Type u_1\na y : Lists \u03b1\nl : Lists' \u03b1 true\n\u22a2 a \u2208 cons y l \u2194 a ~ y \u2228 a \u2208 l\n[PROOFSTEP]\nsimp [mem_def, or_and_right, exists_or]\n[GOAL]\n\u03b1 : Type u_1\na : Lists \u03b1\nl\u2081 l\u2082 : Lists' \u03b1 true\n\u22a2 cons a l\u2081 \u2286 l\u2082 \u2194 a \u2208 l\u2082 \u2227 l\u2081 \u2286 l\u2082\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun \u27e8\u27e8a', m, e\u27e9, s\u27e9 => Subset.cons e m s\u27e9\n[GOAL]\n\u03b1 : Type u_1\na : Lists \u03b1\nl\u2081 l\u2082 : Lists' \u03b1 true\nh : cons a l\u2081 \u2286 l\u2082\n\u22a2 a \u2208 l\u2082 \u2227 l\u2081 \u2286 l\u2082\n[PROOFSTEP]\ngeneralize h' : Lists'.cons a l\u2081 = l\u2081' at h \n[GOAL]\n\u03b1 : Type u_1\na : Lists \u03b1\nl\u2081 l\u2082 l\u2081' : Lists' \u03b1 true\nh' : cons a l\u2081 = l\u2081'\nh : l\u2081' \u2286 l\u2082\n\u22a2 a \u2208 l\u2082 \u2227 l\u2081 \u2286 l\u2082\n[PROOFSTEP]\ncases' h with l a' a'' l l' e m s\n[GOAL]\ncase nil\n\u03b1 : Type u_1\na : Lists \u03b1\nl\u2081 l\u2082 : Lists' \u03b1 true\nh' : cons a l\u2081 = nil\n\u22a2 a \u2208 l\u2082 \u2227 l\u2081 \u2286 l\u2082\n[PROOFSTEP]\ncases a\n[GOAL]\ncase nil.mk\n\u03b1 : Type u_1\nl\u2081 l\u2082 : Lists' \u03b1 true\nfst\u271d : Bool\nsnd\u271d : Lists' \u03b1 fst\u271d\nh' : cons { fst := fst\u271d, snd := snd\u271d } l\u2081 = nil\n\u22a2 { fst := fst\u271d, snd := snd\u271d } \u2208 l\u2082 \u2227 l\u2081 \u2286 l\u2082\n[PROOFSTEP]\ncases h'\n[GOAL]\ncase cons\n\u03b1 : Type u_1\na : Lists \u03b1\nl\u2081 l\u2082 : Lists' \u03b1 true\na' a'' : Lists \u03b1\nl : Lists' \u03b1 true\ne : a' ~ a''\nh' : cons a l\u2081 = cons a' l\nm : a'' \u2208 toList l\u2082\ns : Subset l l\u2082\n\u22a2 a \u2208 l\u2082 \u2227 l\u2081 \u2286 l\u2082\n[PROOFSTEP]\ncases a\n[GOAL]\ncase cons.mk\n\u03b1 : Type u_1\nl\u2081 l\u2082 : Lists' \u03b1 true\na' a'' : Lists \u03b1\nl : Lists' \u03b1 true\ne : a' ~ a''\nm : a'' \u2208 toList l\u2082\ns : Subset l l\u2082\nfst\u271d : Bool\nsnd\u271d : Lists' \u03b1 fst\u271d\nh' : cons { fst := fst\u271d, snd := snd\u271d } l\u2081 = cons a' l\n\u22a2 { fst := fst\u271d, snd := snd\u271d } \u2208 l\u2082 \u2227 l\u2081 \u2286 l\u2082\n[PROOFSTEP]\ncases a'\n[GOAL]\ncase cons.mk.mk\n\u03b1 : Type u_1\nl\u2081 l\u2082 : Lists' \u03b1 true\na'' : Lists \u03b1\nl : Lists' \u03b1 true\nm : a'' \u2208 toList l\u2082\ns : Subset l l\u2082\nfst\u271d\u00b9 : Bool\nsnd\u271d\u00b9 : Lists' \u03b1 fst\u271d\u00b9\nfst\u271d : Bool\nsnd\u271d : Lists' \u03b1 fst\u271d\ne : { fst := fst\u271d, snd := snd\u271d } ~ a''\nh' : cons { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 } l\u2081 = cons { fst := fst\u271d, snd := snd\u271d } l\n\u22a2 { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 } \u2208 l\u2082 \u2227 l\u2081 \u2286 l\u2082\n[PROOFSTEP]\ncases h'\n[GOAL]\ncase cons.mk.mk.refl\n\u03b1 : Type u_1\nl\u2081 l\u2082 : Lists' \u03b1 true\na'' : Lists \u03b1\nm : a'' \u2208 toList l\u2082\nfst\u271d : Bool\nsnd\u271d : Lists' \u03b1 fst\u271d\ne : { fst := fst\u271d, snd := snd\u271d } ~ a''\ns : Subset l\u2081 l\u2082\n\u22a2 { fst := fst\u271d, snd := snd\u271d } \u2208 l\u2082 \u2227 l\u2081 \u2286 l\u2082\n[PROOFSTEP]\nexact \u27e8\u27e8_, m, e\u27e9, s\u27e9\n[GOAL]\n\u03b1 : Type u_1\nl\u2081 l\u2082 : List (Lists \u03b1)\nh : l\u2081 \u2286 l\u2082\n\u22a2 ofList l\u2081 \u2286 ofList l\u2082\n[PROOFSTEP]\ninduction' l\u2081 with _ _ l\u2081_ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\nl\u2081 l\u2082 : List (Lists \u03b1)\nh\u271d : l\u2081 \u2286 l\u2082\nh : [] \u2286 l\u2082\n\u22a2 ofList [] \u2286 ofList l\u2082\n[PROOFSTEP]\nexact Subset.nil\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nl\u2081 l\u2082 : List (Lists \u03b1)\nh\u271d : l\u2081 \u2286 l\u2082\nhead\u271d : Lists \u03b1\ntail\u271d : List (Lists \u03b1)\nl\u2081_ih : tail\u271d \u2286 l\u2082 \u2192 ofList tail\u271d \u2286 ofList l\u2082\nh : head\u271d :: tail\u271d \u2286 l\u2082\n\u22a2 ofList (head\u271d :: tail\u271d) \u2286 ofList l\u2082\n[PROOFSTEP]\nrefine' Subset.cons (Lists.Equiv.refl _) _ (l\u2081_ih (List.subset_of_cons_subset h))\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nl\u2081 l\u2082 : List (Lists \u03b1)\nh\u271d : l\u2081 \u2286 l\u2082\nhead\u271d : Lists \u03b1\ntail\u271d : List (Lists \u03b1)\nl\u2081_ih : tail\u271d \u2286 l\u2082 \u2192 ofList tail\u271d \u2286 ofList l\u2082\nh : head\u271d :: tail\u271d \u2286 l\u2082\n\u22a2 head\u271d \u2208 toList (ofList l\u2082)\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase cons\n\u03b1 : Type u_1\nl\u2081 l\u2082 : List (Lists \u03b1)\nh\u271d : l\u2081 \u2286 l\u2082\nhead\u271d : Lists \u03b1\ntail\u271d : List (Lists \u03b1)\nl\u2081_ih : tail\u271d \u2286 l\u2082 \u2192 ofList tail\u271d \u2286 ofList l\u2082\nh : head\u271d \u2208 l\u2082 \u2227 tail\u271d \u2286 l\u2082\n\u22a2 head\u271d \u2208 toList (ofList l\u2082)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\nl : Lists' \u03b1 true\n\u22a2 l \u2286 l\n[PROOFSTEP]\nrw [\u2190 Lists'.of_toList l]\n[GOAL]\n\u03b1 : Type u_1\nl : Lists' \u03b1 true\n\u22a2 ofList (toList l) \u2286 ofList (toList l)\n[PROOFSTEP]\nexact ofList_subset (List.Subset.refl _)\n[GOAL]\n\u03b1 : Type u_1\nl : Lists' \u03b1 true\n\u22a2 l \u2286 nil \u2192 l = nil\n[PROOFSTEP]\nrw [\u2190 of_toList l]\n[GOAL]\n\u03b1 : Type u_1\nl : Lists' \u03b1 true\n\u22a2 ofList (toList l) \u2286 nil \u2192 ofList (toList l) = nil\n[PROOFSTEP]\ninduction toList l\n[GOAL]\ncase nil\n\u03b1 : Type u_1\nl : Lists' \u03b1 true\n\u22a2 ofList [] \u2286 nil \u2192 ofList [] = nil\n[PROOFSTEP]\nintro h\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nl : Lists' \u03b1 true\nhead\u271d : Lists \u03b1\ntail\u271d : List (Lists \u03b1)\ntail_ih\u271d : ofList tail\u271d \u2286 nil \u2192 ofList tail\u271d = nil\n\u22a2 ofList (head\u271d :: tail\u271d) \u2286 nil \u2192 ofList (head\u271d :: tail\u271d) = nil\n[PROOFSTEP]\nintro h\n[GOAL]\ncase nil\n\u03b1 : Type u_1\nl : Lists' \u03b1 true\nh : ofList [] \u2286 nil\n\u22a2 ofList [] = nil\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nl : Lists' \u03b1 true\nhead\u271d : Lists \u03b1\ntail\u271d : List (Lists \u03b1)\ntail_ih\u271d : ofList tail\u271d \u2286 nil \u2192 ofList tail\u271d = nil\nh : ofList (head\u271d :: tail\u271d) \u2286 nil\n\u22a2 ofList (head\u271d :: tail\u271d) = nil\n[PROOFSTEP]\nrcases cons_subset.1 h with \u27e8\u27e8_, \u27e8\u27e9, _\u27e9, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\na : Lists \u03b1\nx\u271d : Lists' \u03b1 true\nh : a \u2208 toList nil\n\u22a2 a \u2208 x\u271d\n[PROOFSTEP]\ncases h\n[GOAL]\n\u03b1 : Type u_1\na : Lists \u03b1\nb\u271d : Bool\na0 : Lists' \u03b1 b\u271d\nl0 l\u2082 : Lists' \u03b1 true\ns : cons' a0 l0 \u2286 l\u2082\nh : a \u2208 toList (cons' a0 l0)\n\u22a2 a \u2208 l\u2082\n[PROOFSTEP]\ncases' s with _ _ _ _ _ e m s\n[GOAL]\ncase cons\n\u03b1 : Type u_1\na : Lists \u03b1\nl0 l\u2082 : Lists' \u03b1 true\na\u271d a'\u271d : Lists \u03b1\ne : a\u271d ~ a'\u271d\nh : a \u2208 toList (cons' a\u271d.snd l0)\nm : a'\u271d \u2208 toList l\u2082\ns : Subset l0 l\u2082\n\u22a2 a \u2208 l\u2082\n[PROOFSTEP]\nsimp only [toList, Sigma.eta, List.find?, List.mem_cons] at h \n[GOAL]\ncase cons\n\u03b1 : Type u_1\na : Lists \u03b1\nl0 l\u2082 : Lists' \u03b1 true\na\u271d a'\u271d : Lists \u03b1\ne : a\u271d ~ a'\u271d\nm : a'\u271d \u2208 toList l\u2082\ns : Subset l0 l\u2082\nh : a = a\u271d \u2228 a \u2208 toList l0\n\u22a2 a \u2208 l\u2082\n[PROOFSTEP]\nrcases h with (rfl | h)\n[GOAL]\ncase cons.inl\n\u03b1 : Type u_1\na : Lists \u03b1\nl0 l\u2082 : Lists' \u03b1 true\na'\u271d : Lists \u03b1\nm : a'\u271d \u2208 toList l\u2082\ns : Subset l0 l\u2082\ne : a ~ a'\u271d\n\u22a2 a \u2208 l\u2082\n[PROOFSTEP]\nexact \u27e8_, m, e\u27e9\n[GOAL]\ncase cons.inr\n\u03b1 : Type u_1\na : Lists \u03b1\nl0 l\u2082 : Lists' \u03b1 true\na\u271d a'\u271d : Lists \u03b1\ne : a\u271d ~ a'\u271d\nm : a'\u271d \u2208 toList l\u2082\ns : Subset l0 l\u2082\nh : a \u2208 toList l0\n\u22a2 a \u2208 l\u2082\n[PROOFSTEP]\nexact mem_of_subset' s h\n[GOAL]\n\u03b1 : Type u_1\nl\u2081 l\u2082 : Lists' \u03b1 true\nH : \u2200 (a : Lists \u03b1), a \u2208 toList l\u2081 \u2192 a \u2208 l\u2082\n\u22a2 l\u2081 \u2286 l\u2082\n[PROOFSTEP]\nrw [\u2190 of_toList l\u2081]\n[GOAL]\n\u03b1 : Type u_1\nl\u2081 l\u2082 : Lists' \u03b1 true\nH : \u2200 (a : Lists \u03b1), a \u2208 toList l\u2081 \u2192 a \u2208 l\u2082\n\u22a2 ofList (toList l\u2081) \u2286 l\u2082\n[PROOFSTEP]\nrevert H\n[GOAL]\n\u03b1 : Type u_1\nl\u2081 l\u2082 : Lists' \u03b1 true\n\u22a2 (\u2200 (a : Lists \u03b1), a \u2208 toList l\u2081 \u2192 a \u2208 l\u2082) \u2192 ofList (toList l\u2081) \u2286 l\u2082\n[PROOFSTEP]\ninduction' toList l\u2081 with h t t_ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\nl\u2081 l\u2082 : Lists' \u03b1 true\n\u22a2 (\u2200 (a : Lists \u03b1), a \u2208 [] \u2192 a \u2208 l\u2082) \u2192 ofList [] \u2286 l\u2082\n[PROOFSTEP]\nintro H\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nl\u2081 l\u2082 : Lists' \u03b1 true\nh : Lists \u03b1\nt : List (Lists \u03b1)\nt_ih : (\u2200 (a : Lists \u03b1), a \u2208 t \u2192 a \u2208 l\u2082) \u2192 ofList t \u2286 l\u2082\n\u22a2 (\u2200 (a : Lists \u03b1), a \u2208 h :: t \u2192 a \u2208 l\u2082) \u2192 ofList (h :: t) \u2286 l\u2082\n[PROOFSTEP]\nintro H\n[GOAL]\ncase nil\n\u03b1 : Type u_1\nl\u2081 l\u2082 : Lists' \u03b1 true\nH : \u2200 (a : Lists \u03b1), a \u2208 [] \u2192 a \u2208 l\u2082\n\u22a2 ofList [] \u2286 l\u2082\n[PROOFSTEP]\nexact Subset.nil\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nl\u2081 l\u2082 : Lists' \u03b1 true\nh : Lists \u03b1\nt : List (Lists \u03b1)\nt_ih : (\u2200 (a : Lists \u03b1), a \u2208 t \u2192 a \u2208 l\u2082) \u2192 ofList t \u2286 l\u2082\nH : \u2200 (a : Lists \u03b1), a \u2208 h :: t \u2192 a \u2208 l\u2082\n\u22a2 ofList (h :: t) \u2286 l\u2082\n[PROOFSTEP]\nsimp only [ofList, List.find?, List.mem_cons, forall_eq_or_imp] at *\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nl\u2081 l\u2082 : Lists' \u03b1 true\nh : Lists \u03b1\nt : List (Lists \u03b1)\nt_ih : (\u2200 (a : Lists \u03b1), a \u2208 t \u2192 a \u2208 l\u2082) \u2192 ofList t \u2286 l\u2082\nH : h \u2208 l\u2082 \u2227 \u2200 (a : Lists \u03b1), a \u2208 t \u2192 a \u2208 l\u2082\n\u22a2 cons h (ofList t) \u2286 l\u2082\n[PROOFSTEP]\nexact cons_subset.2 \u27e8H.1, t_ih H.2\u27e9\n[GOAL]\n\u03b1 : Type u_1\nl : List (Lists \u03b1)\n\u22a2 toList (ofList l) = l\n[PROOFSTEP]\nsimp [ofList, of']\n[GOAL]\n\u03b1 : Type u_1\nl : Lists' \u03b1 true\nx\u271d : IsList { fst := true, snd := l }\n\u22a2 ofList (toList { fst := true, snd := l }) = { fst := true, snd := l }\n[PROOFSTEP]\nsimp_all [ofList, of']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u22a2 DecidableEq (Lists \u03b1)\n[PROOFSTEP]\nunfold Lists\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u22a2 DecidableEq ((b : Bool) \u00d7 Lists' \u03b1 b)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : SizeOf \u03b1\n\u22a2 SizeOf (Lists \u03b1)\n[PROOFSTEP]\nunfold Lists\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : SizeOf \u03b1\n\u22a2 SizeOf ((b : Bool) \u00d7 Lists' \u03b1 b)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\nC : Lists \u03b1 \u2192 Sort u_2\nD : Lists' \u03b1 true \u2192 Sort u_3\nC0 : (a : \u03b1) \u2192 C (atom a)\nC1 : (l : Lists' \u03b1 true) \u2192 D l \u2192 C (of' l)\nD0 : D Lists'.nil\nD1 : (a : Lists \u03b1) \u2192 (l : Lists' \u03b1 true) \u2192 C a \u2192 D l \u2192 D (Lists'.cons a l)\n\u22a2 PProd ((l : Lists \u03b1) \u2192 C l) ((l : Lists' \u03b1 true) \u2192 D l)\n[PROOFSTEP]\nsuffices\n \u2200 {b} (l : Lists' \u03b1 b),\n PProd (C \u27e8_, l\u27e9)\n (match b, l with\n | true, l => D l\n | false, _ => PUnit)\n by exact \u27e8fun \u27e8b, l\u27e9 => (this _).1, fun l => (this l).2\u27e9\n[GOAL]\n\u03b1 : Type u_1\nC : Lists \u03b1 \u2192 Sort u_2\nD : Lists' \u03b1 true \u2192 Sort u_3\nC0 : (a : \u03b1) \u2192 C (atom a)\nC1 : (l : Lists' \u03b1 true) \u2192 D l \u2192 C (of' l)\nD0 : D Lists'.nil\nD1 : (a : Lists \u03b1) \u2192 (l : Lists' \u03b1 true) \u2192 C a \u2192 D l \u2192 D (Lists'.cons a l)\nthis :\n {b : Bool} \u2192\n (l : Lists' \u03b1 b) \u2192\n PProd (C { fst := b, snd := l })\n (match b, l with\n | true, l => D l\n | false, x => PUnit)\n\u22a2 PProd ((l : Lists \u03b1) \u2192 C l) ((l : Lists' \u03b1 true) \u2192 D l)\n[PROOFSTEP]\nexact \u27e8fun \u27e8b, l\u27e9 => (this _).1, fun l => (this l).2\u27e9\n[GOAL]\n\u03b1 : Type u_1\nC : Lists \u03b1 \u2192 Sort u_2\nD : Lists' \u03b1 true \u2192 Sort u_3\nC0 : (a : \u03b1) \u2192 C (atom a)\nC1 : (l : Lists' \u03b1 true) \u2192 D l \u2192 C (of' l)\nD0 : D Lists'.nil\nD1 : (a : Lists \u03b1) \u2192 (l : Lists' \u03b1 true) \u2192 C a \u2192 D l \u2192 D (Lists'.cons a l)\n\u22a2 {b : Bool} \u2192\n (l : Lists' \u03b1 b) \u2192\n PProd (C { fst := b, snd := l })\n (match b, l with\n | true, l => D l\n | false, x => PUnit)\n[PROOFSTEP]\nintros b l\n[GOAL]\n\u03b1 : Type u_1\nC : Lists \u03b1 \u2192 Sort u_2\nD : Lists' \u03b1 true \u2192 Sort u_3\nC0 : (a : \u03b1) \u2192 C (atom a)\nC1 : (l : Lists' \u03b1 true) \u2192 D l \u2192 C (of' l)\nD0 : D Lists'.nil\nD1 : (a : Lists \u03b1) \u2192 (l : Lists' \u03b1 true) \u2192 C a \u2192 D l \u2192 D (Lists'.cons a l)\nb : Bool\nl : Lists' \u03b1 b\n\u22a2 PProd (C { fst := b, snd := l })\n (match b, l with\n | true, l => D l\n | false, x => PUnit)\n[PROOFSTEP]\ninduction' l with a b a l IH\u2081 IH\n[GOAL]\ncase atom\n\u03b1 : Type u_1\nC : Lists \u03b1 \u2192 Sort u_2\nD : Lists' \u03b1 true \u2192 Sort u_3\nC0 : (a : \u03b1) \u2192 C (atom a)\nC1 : (l : Lists' \u03b1 true) \u2192 D l \u2192 C (of' l)\nD0 : D Lists'.nil\nD1 : (a : Lists \u03b1) \u2192 (l : Lists' \u03b1 true) \u2192 C a \u2192 D l \u2192 D (Lists'.cons a l)\nb : Bool\na : \u03b1\n\u22a2 PProd (C { fst := false, snd := Lists'.atom a })\n (match false, Lists'.atom a with\n | true, l => D l\n | false, x => PUnit)\n[PROOFSTEP]\nexact \u27e8C0 _, \u27e8\u27e9\u27e9\n[GOAL]\ncase nil\n\u03b1 : Type u_1\nC : Lists \u03b1 \u2192 Sort u_2\nD : Lists' \u03b1 true \u2192 Sort u_3\nC0 : (a : \u03b1) \u2192 C (atom a)\nC1 : (l : Lists' \u03b1 true) \u2192 D l \u2192 C (of' l)\nD0 : D Lists'.nil\nD1 : (a : Lists \u03b1) \u2192 (l : Lists' \u03b1 true) \u2192 C a \u2192 D l \u2192 D (Lists'.cons a l)\nb : Bool\n\u22a2 PProd (C { fst := true, snd := Lists'.nil })\n (match true, Lists'.nil with\n | true, l => D l\n | false, x => PUnit)\n[PROOFSTEP]\nexact \u27e8C1 _ D0, D0\u27e9\n[GOAL]\ncase cons'\n\u03b1 : Type u_1\nC : Lists \u03b1 \u2192 Sort u_2\nD : Lists' \u03b1 true \u2192 Sort u_3\nC0 : (a : \u03b1) \u2192 C (atom a)\nC1 : (l : Lists' \u03b1 true) \u2192 D l \u2192 C (of' l)\nD0 : D Lists'.nil\nD1 : (a : Lists \u03b1) \u2192 (l : Lists' \u03b1 true) \u2192 C a \u2192 D l \u2192 D (Lists'.cons a l)\nb\u271d b : Bool\na : Lists' \u03b1 b\nl : Lists' \u03b1 true\nIH\u2081 :\n PProd (C { fst := b, snd := a })\n (match b, a with\n | true, l => D l\n | false, x => PUnit)\nIH :\n PProd (C { fst := true, snd := l })\n (match true, l with\n | true, l => D l\n | false, x => PUnit)\n\u22a2 PProd (C { fst := true, snd := Lists'.cons' a l })\n (match true, Lists'.cons' a l with\n | true, l => D l\n | false, x => PUnit)\n[PROOFSTEP]\nhave : D (Lists'.cons' a l) := D1 \u27e8_, _\u27e9 _ IH\u2081.1 IH.2\n[GOAL]\ncase cons'\n\u03b1 : Type u_1\nC : Lists \u03b1 \u2192 Sort u_2\nD : Lists' \u03b1 true \u2192 Sort u_3\nC0 : (a : \u03b1) \u2192 C (atom a)\nC1 : (l : Lists' \u03b1 true) \u2192 D l \u2192 C (of' l)\nD0 : D Lists'.nil\nD1 : (a : Lists \u03b1) \u2192 (l : Lists' \u03b1 true) \u2192 C a \u2192 D l \u2192 D (Lists'.cons a l)\nb\u271d b : Bool\na : Lists' \u03b1 b\nl : Lists' \u03b1 true\nIH\u2081 :\n PProd (C { fst := b, snd := a })\n (match b, a with\n | true, l => D l\n | false, x => PUnit)\nIH :\n PProd (C { fst := true, snd := l })\n (match true, l with\n | true, l => D l\n | false, x => PUnit)\nthis : D (Lists'.cons' a l)\n\u22a2 PProd (C { fst := true, snd := Lists'.cons' a l })\n (match true, Lists'.cons' a l with\n | true, l => D l\n | false, x => PUnit)\n[PROOFSTEP]\nexact \u27e8C1 _ this, this\u27e9\n[GOAL]\n\u03b1 : Type u_1\nl\u2081 l\u2082 : Lists' \u03b1 true\n\u22a2 of' l\u2081 ~ of' l\u2082 \u2194 l\u2081 \u2286 l\u2082 \u2227 l\u2082 \u2286 l\u2081\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun \u27e8h\u2081, h\u2082\u27e9 => Equiv.antisymm h\u2081 h\u2082\u27e9\n[GOAL]\n\u03b1 : Type u_1\nl\u2081 l\u2082 : Lists' \u03b1 true\nh : of' l\u2081 ~ of' l\u2082\n\u22a2 l\u2081 \u2286 l\u2082 \u2227 l\u2082 \u2286 l\u2081\n[PROOFSTEP]\ncases' h with _ _ _ h\u2081 h\u2082\n[GOAL]\ncase refl\n\u03b1 : Type u_1\nl\u2081 : Lists' \u03b1 true\n\u22a2 l\u2081 \u2286 l\u2081 \u2227 l\u2081 \u2286 l\u2081\n[PROOFSTEP]\nsimp [Lists'.Subset.refl]\n[GOAL]\ncase antisymm\n\u03b1 : Type u_1\nl\u2081 l\u2082 : Lists' \u03b1 true\nh\u2081 : Lists'.Subset l\u2081 l\u2082\nh\u2082 : Lists'.Subset l\u2082 l\u2081\n\u22a2 l\u2081 \u2286 l\u2082 \u2227 l\u2082 \u2286 l\u2081\n[PROOFSTEP]\nexact \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\n\u03b1 : Type u_1\na : \u03b1\nl : Lists \u03b1\nh : atom a ~ l\n\u22a2 atom a = l\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\n\u03b1 : Type u_1\na : \u03b1\n\u22a2 atom a = atom a\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nl\u2081 l\u2082 : Lists \u03b1\nh : l\u2081 ~ l\u2082\n\u22a2 l\u2082 ~ l\u2081\n[PROOFSTEP]\ncases' h with _ _ _ h\u2081 h\u2082 <;> [rfl; exact Equiv.antisymm h\u2082 h\u2081]\n[GOAL]\n\u03b1 : Type u_1\nl\u2081 l\u2082 : Lists \u03b1\nh : l\u2081 ~ l\u2082\n\u22a2 l\u2082 ~ l\u2081\n[PROOFSTEP]\ncases' h with _ _ _ h\u2081 h\u2082\n[GOAL]\ncase refl\n\u03b1 : Type u_1\nl\u2081 : Lists \u03b1\n\u22a2 l\u2081 ~ l\u2081\n[PROOFSTEP]\nrfl\n[GOAL]\ncase antisymm\n\u03b1 : Type u_1\nl\u2081\u271d l\u2082\u271d : Lists' \u03b1 true\nh\u2081 : Lists'.Subset l\u2081\u271d l\u2082\u271d\nh\u2082 : Lists'.Subset l\u2082\u271d l\u2081\u271d\n\u22a2 { fst := true, snd := l\u2082\u271d } ~ { fst := true, snd := l\u2081\u271d }\n[PROOFSTEP]\nexact Equiv.antisymm h\u2082 h\u2081\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 {l\u2081 l\u2082 l\u2083 : Lists \u03b1}, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\n[PROOFSTEP]\nlet trans := fun 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[GOAL]\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\n\u22a2 \u2200 {l\u2081 l\u2082 l\u2083 : Lists \u03b1}, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\n[PROOFSTEP]\nsuffices PProd (\u2200 l\u2081, trans l\u2081) (\u2200 (l : Lists' \u03b1 true), \u2200 l' \u2208 l.toList, trans l') by exact this.1\n[GOAL]\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nthis : PProd (\u2200 (l\u2081 : Lists \u03b1), trans l\u2081) (\u2200 (l : Lists' \u03b1 true) (l' : Lists \u03b1), l' \u2208 Lists'.toList l \u2192 trans l')\n\u22a2 \u2200 {l\u2081 l\u2082 l\u2083 : Lists \u03b1}, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\n[PROOFSTEP]\nexact this.1\n[GOAL]\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\n\u22a2 PProd (\u2200 (l\u2081 : Lists \u03b1), trans l\u2081) (\u2200 (l : Lists' \u03b1 true) (l' : Lists \u03b1), l' \u2208 Lists'.toList l \u2192 trans l')\n[PROOFSTEP]\napply inductionMut\n[GOAL]\ncase C0\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\n\u22a2 \u2200 (a : \u03b1), trans (atom a)\n[PROOFSTEP]\nintro a l\u2082 l\u2083 h\u2081 h\u2082\n[GOAL]\ncase C0\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\na : \u03b1\nl\u2082 l\u2083 : Lists \u03b1\nh\u2081 : atom a ~ l\u2082\nh\u2082 : l\u2082 ~ l\u2083\n\u22a2 atom a ~ l\u2083\n[PROOFSTEP]\nrwa [\u2190 equiv_atom.1 h\u2081] at h\u2082 \n[GOAL]\ncase C1\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\n\u22a2 \u2200 (l : Lists' \u03b1 true), (\u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l \u2192 trans l') \u2192 trans (of' l)\n[PROOFSTEP]\nintro l\u2081 IH l\u2082 l\u2083 h\u2081 h\u2082\n[GOAL]\ncase C1\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nl\u2081 : Lists' \u03b1 true\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l\u2081 \u2192 trans l'\nl\u2082 l\u2083 : Lists \u03b1\nh\u2081 : of' l\u2081 ~ l\u2082\nh\u2082 : l\u2082 ~ l\u2083\n\u22a2 of' l\u2081 ~ l\u2083\n[PROOFSTEP]\nhave h\u2081' := h\u2081\n[GOAL]\ncase C1\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nl\u2081 : Lists' \u03b1 true\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l\u2081 \u2192 trans l'\nl\u2082 l\u2083 : Lists \u03b1\nh\u2081 : of' l\u2081 ~ l\u2082\nh\u2082 : l\u2082 ~ l\u2083\nh\u2081' : of' l\u2081 ~ l\u2082\n\u22a2 of' l\u2081 ~ l\u2083\n[PROOFSTEP]\nhave h\u2082' := h\u2082\n[GOAL]\ncase C1\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nl\u2081 : Lists' \u03b1 true\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l\u2081 \u2192 trans l'\nl\u2082 l\u2083 : Lists \u03b1\nh\u2081 : of' l\u2081 ~ l\u2082\nh\u2082 : l\u2082 ~ l\u2083\nh\u2081' : of' l\u2081 ~ l\u2082\nh\u2082' : l\u2082 ~ l\u2083\n\u22a2 of' l\u2081 ~ l\u2083\n[PROOFSTEP]\ncases' h\u2081 with _ _ l\u2082\n[GOAL]\ncase C1.refl\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nl\u2081 : Lists' \u03b1 true\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l\u2081 \u2192 trans l'\nl\u2083 : Lists \u03b1\nh\u2082 : of' l\u2081 ~ l\u2083\nh\u2081' : of' l\u2081 ~ of' l\u2081\nh\u2082' : of' l\u2081 ~ l\u2083\n\u22a2 of' l\u2081 ~ l\u2083\n[PROOFSTEP]\nexact h\u2082\n[GOAL]\ncase C1.antisymm\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nl\u2081 : Lists' \u03b1 true\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l\u2081 \u2192 trans l'\nl\u2083 : Lists \u03b1\nl\u2082 : Lists' \u03b1 true\na\u271d\u00b9 : Lists'.Subset l\u2081 l\u2082\na\u271d : Lists'.Subset l\u2082 l\u2081\nh\u2082 : { fst := true, snd := l\u2082 } ~ l\u2083\nh\u2081' : of' l\u2081 ~ { fst := true, snd := l\u2082 }\nh\u2082' : { fst := true, snd := l\u2082 } ~ l\u2083\n\u22a2 of' l\u2081 ~ l\u2083\n[PROOFSTEP]\ncases' h\u2082 with _ _ l\u2083\n[GOAL]\ncase C1.antisymm.refl\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nl\u2081 : Lists' \u03b1 true\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l\u2081 \u2192 trans l'\nl\u2082 : Lists' \u03b1 true\na\u271d\u00b9 : Lists'.Subset l\u2081 l\u2082\na\u271d : Lists'.Subset l\u2082 l\u2081\nh\u2081' : of' l\u2081 ~ { fst := true, snd := l\u2082 }\nh\u2082' : { fst := true, snd := l\u2082 } ~ { fst := true, snd := l\u2082 }\n\u22a2 of' l\u2081 ~ { fst := true, snd := l\u2082 }\n[PROOFSTEP]\nexact h\u2081'\n[GOAL]\ncase C1.antisymm.antisymm\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nl\u2081 : Lists' \u03b1 true\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l\u2081 \u2192 trans l'\nl\u2082 : Lists' \u03b1 true\na\u271d\u00b3 : Lists'.Subset l\u2081 l\u2082\na\u271d\u00b2 : Lists'.Subset l\u2082 l\u2081\nh\u2081' : of' l\u2081 ~ { fst := true, snd := l\u2082 }\nl\u2083 : Lists' \u03b1 true\na\u271d\u00b9 : Lists'.Subset l\u2082 l\u2083\na\u271d : Lists'.Subset l\u2083 l\u2082\nh\u2082' : { fst := true, snd := l\u2082 } ~ { fst := true, snd := l\u2083 }\n\u22a2 of' l\u2081 ~ { fst := true, snd := l\u2083 }\n[PROOFSTEP]\ncases' Equiv.antisymm_iff.1 h\u2081' with hl\u2081 hr\u2081\n[GOAL]\ncase C1.antisymm.antisymm.intro\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nl\u2081 : Lists' \u03b1 true\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l\u2081 \u2192 trans l'\nl\u2082 : Lists' \u03b1 true\na\u271d\u00b3 : Lists'.Subset l\u2081 l\u2082\na\u271d\u00b2 : Lists'.Subset l\u2082 l\u2081\nh\u2081' : of' l\u2081 ~ { fst := true, snd := l\u2082 }\nl\u2083 : Lists' \u03b1 true\na\u271d\u00b9 : Lists'.Subset l\u2082 l\u2083\na\u271d : Lists'.Subset l\u2083 l\u2082\nh\u2082' : { fst := true, snd := l\u2082 } ~ { fst := true, snd := l\u2083 }\nhl\u2081 : l\u2081 \u2286 l\u2082\nhr\u2081 : l\u2082 \u2286 l\u2081\n\u22a2 of' l\u2081 ~ { fst := true, snd := l\u2083 }\n[PROOFSTEP]\ncases' Equiv.antisymm_iff.1 h\u2082' with hl\u2082 hr\u2082\n[GOAL]\ncase C1.antisymm.antisymm.intro.intro\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nl\u2081 : Lists' \u03b1 true\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l\u2081 \u2192 trans l'\nl\u2082 : Lists' \u03b1 true\na\u271d\u00b3 : Lists'.Subset l\u2081 l\u2082\na\u271d\u00b2 : Lists'.Subset l\u2082 l\u2081\nh\u2081' : of' l\u2081 ~ { fst := true, snd := l\u2082 }\nl\u2083 : Lists' \u03b1 true\na\u271d\u00b9 : Lists'.Subset l\u2082 l\u2083\na\u271d : Lists'.Subset l\u2083 l\u2082\nh\u2082' : { fst := true, snd := l\u2082 } ~ { fst := true, snd := l\u2083 }\nhl\u2081 : l\u2081 \u2286 l\u2082\nhr\u2081 : l\u2082 \u2286 l\u2081\nhl\u2082 : l\u2082 \u2286 l\u2083\nhr\u2082 : l\u2083 \u2286 l\u2082\n\u22a2 of' l\u2081 ~ { fst := true, snd := l\u2083 }\n[PROOFSTEP]\napply Equiv.antisymm_iff.2\n[GOAL]\ncase C1.antisymm.antisymm.intro.intro\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nl\u2081 : Lists' \u03b1 true\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l\u2081 \u2192 trans l'\nl\u2082 : Lists' \u03b1 true\na\u271d\u00b3 : Lists'.Subset l\u2081 l\u2082\na\u271d\u00b2 : Lists'.Subset l\u2082 l\u2081\nh\u2081' : of' l\u2081 ~ { fst := true, snd := l\u2082 }\nl\u2083 : Lists' \u03b1 true\na\u271d\u00b9 : Lists'.Subset l\u2082 l\u2083\na\u271d : Lists'.Subset l\u2083 l\u2082\nh\u2082' : { fst := true, snd := l\u2082 } ~ { fst := true, snd := l\u2083 }\nhl\u2081 : l\u2081 \u2286 l\u2082\nhr\u2081 : l\u2082 \u2286 l\u2081\nhl\u2082 : l\u2082 \u2286 l\u2083\nhr\u2082 : l\u2083 \u2286 l\u2082\n\u22a2 l\u2081 \u2286 l\u2083 \u2227 l\u2083 \u2286 l\u2081\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase C1.antisymm.antisymm.intro.intro.left\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nl\u2081 : Lists' \u03b1 true\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l\u2081 \u2192 trans l'\nl\u2082 : Lists' \u03b1 true\na\u271d\u00b3 : Lists'.Subset l\u2081 l\u2082\na\u271d\u00b2 : Lists'.Subset l\u2082 l\u2081\nh\u2081' : of' l\u2081 ~ { fst := true, snd := l\u2082 }\nl\u2083 : Lists' \u03b1 true\na\u271d\u00b9 : Lists'.Subset l\u2082 l\u2083\na\u271d : Lists'.Subset l\u2083 l\u2082\nh\u2082' : { fst := true, snd := l\u2082 } ~ { fst := true, snd := l\u2083 }\nhl\u2081 : l\u2081 \u2286 l\u2082\nhr\u2081 : l\u2082 \u2286 l\u2081\nhl\u2082 : l\u2082 \u2286 l\u2083\nhr\u2082 : l\u2083 \u2286 l\u2082\n\u22a2 l\u2081 \u2286 l\u2083\n[PROOFSTEP]\napply Lists'.subset_def.2\n[GOAL]\ncase C1.antisymm.antisymm.intro.intro.right\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nl\u2081 : Lists' \u03b1 true\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l\u2081 \u2192 trans l'\nl\u2082 : Lists' \u03b1 true\na\u271d\u00b3 : Lists'.Subset l\u2081 l\u2082\na\u271d\u00b2 : Lists'.Subset l\u2082 l\u2081\nh\u2081' : of' l\u2081 ~ { fst := true, snd := l\u2082 }\nl\u2083 : Lists' \u03b1 true\na\u271d\u00b9 : Lists'.Subset l\u2082 l\u2083\na\u271d : Lists'.Subset l\u2083 l\u2082\nh\u2082' : { fst := true, snd := l\u2082 } ~ { fst := true, snd := l\u2083 }\nhl\u2081 : l\u2081 \u2286 l\u2082\nhr\u2081 : l\u2082 \u2286 l\u2081\nhl\u2082 : l\u2082 \u2286 l\u2083\nhr\u2082 : l\u2083 \u2286 l\u2082\n\u22a2 l\u2083 \u2286 l\u2081\n[PROOFSTEP]\napply Lists'.subset_def.2\n[GOAL]\ncase C1.antisymm.antisymm.intro.intro.left\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nl\u2081 : Lists' \u03b1 true\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l\u2081 \u2192 trans l'\nl\u2082 : Lists' \u03b1 true\na\u271d\u00b3 : Lists'.Subset l\u2081 l\u2082\na\u271d\u00b2 : Lists'.Subset l\u2082 l\u2081\nh\u2081' : of' l\u2081 ~ { fst := true, snd := l\u2082 }\nl\u2083 : Lists' \u03b1 true\na\u271d\u00b9 : Lists'.Subset l\u2082 l\u2083\na\u271d : Lists'.Subset l\u2083 l\u2082\nh\u2082' : { fst := true, snd := l\u2082 } ~ { fst := true, snd := l\u2083 }\nhl\u2081 : l\u2081 \u2286 l\u2082\nhr\u2081 : l\u2082 \u2286 l\u2081\nhl\u2082 : l\u2082 \u2286 l\u2083\nhr\u2082 : l\u2083 \u2286 l\u2082\n\u22a2 \u2200 (a : Lists \u03b1), a \u2208 Lists'.toList l\u2081 \u2192 a \u2208 l\u2083\n[PROOFSTEP]\nintro a\u2081 m\u2081\n[GOAL]\ncase C1.antisymm.antisymm.intro.intro.left\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nl\u2081 : Lists' \u03b1 true\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l\u2081 \u2192 trans l'\nl\u2082 : Lists' \u03b1 true\na\u271d\u00b3 : Lists'.Subset l\u2081 l\u2082\na\u271d\u00b2 : Lists'.Subset l\u2082 l\u2081\nh\u2081' : of' l\u2081 ~ { fst := true, snd := l\u2082 }\nl\u2083 : Lists' \u03b1 true\na\u271d\u00b9 : Lists'.Subset l\u2082 l\u2083\na\u271d : Lists'.Subset l\u2083 l\u2082\nh\u2082' : { fst := true, snd := l\u2082 } ~ { fst := true, snd := l\u2083 }\nhl\u2081 : l\u2081 \u2286 l\u2082\nhr\u2081 : l\u2082 \u2286 l\u2081\nhl\u2082 : l\u2082 \u2286 l\u2083\nhr\u2082 : l\u2083 \u2286 l\u2082\na\u2081 : Lists \u03b1\nm\u2081 : a\u2081 \u2208 Lists'.toList l\u2081\n\u22a2 a\u2081 \u2208 l\u2083\n[PROOFSTEP]\nrcases Lists'.mem_of_subset' hl\u2081 m\u2081 with \u27e8a\u2082, m\u2082, e\u2081\u2082\u27e9\n[GOAL]\ncase C1.antisymm.antisymm.intro.intro.left.intro.intro\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nl\u2081 : Lists' \u03b1 true\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l\u2081 \u2192 trans l'\nl\u2082 : Lists' \u03b1 true\na\u271d\u00b3 : Lists'.Subset l\u2081 l\u2082\na\u271d\u00b2 : Lists'.Subset l\u2082 l\u2081\nh\u2081' : of' l\u2081 ~ { fst := true, snd := l\u2082 }\nl\u2083 : Lists' \u03b1 true\na\u271d\u00b9 : Lists'.Subset l\u2082 l\u2083\na\u271d : Lists'.Subset l\u2083 l\u2082\nh\u2082' : { fst := true, snd := l\u2082 } ~ { fst := true, snd := l\u2083 }\nhl\u2081 : l\u2081 \u2286 l\u2082\nhr\u2081 : l\u2082 \u2286 l\u2081\nhl\u2082 : l\u2082 \u2286 l\u2083\nhr\u2082 : l\u2083 \u2286 l\u2082\na\u2081 : Lists \u03b1\nm\u2081 : a\u2081 \u2208 Lists'.toList l\u2081\na\u2082 : Lists \u03b1\nm\u2082 : a\u2082 \u2208 Lists'.toList l\u2082\ne\u2081\u2082 : a\u2081 ~ a\u2082\n\u22a2 a\u2081 \u2208 l\u2083\n[PROOFSTEP]\nrcases Lists'.mem_of_subset' hl\u2082 m\u2082 with \u27e8a\u2083, m\u2083, e\u2082\u2083\u27e9\n[GOAL]\ncase C1.antisymm.antisymm.intro.intro.left.intro.intro.intro.intro\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nl\u2081 : Lists' \u03b1 true\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l\u2081 \u2192 trans l'\nl\u2082 : Lists' \u03b1 true\na\u271d\u00b3 : Lists'.Subset l\u2081 l\u2082\na\u271d\u00b2 : Lists'.Subset l\u2082 l\u2081\nh\u2081' : of' l\u2081 ~ { fst := true, snd := l\u2082 }\nl\u2083 : Lists' \u03b1 true\na\u271d\u00b9 : Lists'.Subset l\u2082 l\u2083\na\u271d : Lists'.Subset l\u2083 l\u2082\nh\u2082' : { fst := true, snd := l\u2082 } ~ { fst := true, snd := l\u2083 }\nhl\u2081 : l\u2081 \u2286 l\u2082\nhr\u2081 : l\u2082 \u2286 l\u2081\nhl\u2082 : l\u2082 \u2286 l\u2083\nhr\u2082 : l\u2083 \u2286 l\u2082\na\u2081 : Lists \u03b1\nm\u2081 : a\u2081 \u2208 Lists'.toList l\u2081\na\u2082 : Lists \u03b1\nm\u2082 : a\u2082 \u2208 Lists'.toList l\u2082\ne\u2081\u2082 : a\u2081 ~ a\u2082\na\u2083 : Lists \u03b1\nm\u2083 : a\u2083 \u2208 Lists'.toList l\u2083\ne\u2082\u2083 : a\u2082 ~ a\u2083\n\u22a2 a\u2081 \u2208 l\u2083\n[PROOFSTEP]\nexact \u27e8a\u2083, m\u2083, IH _ m\u2081 e\u2081\u2082 e\u2082\u2083\u27e9\n[GOAL]\ncase C1.antisymm.antisymm.intro.intro.right\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nl\u2081 : Lists' \u03b1 true\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l\u2081 \u2192 trans l'\nl\u2082 : Lists' \u03b1 true\na\u271d\u00b3 : Lists'.Subset l\u2081 l\u2082\na\u271d\u00b2 : Lists'.Subset l\u2082 l\u2081\nh\u2081' : of' l\u2081 ~ { fst := true, snd := l\u2082 }\nl\u2083 : Lists' \u03b1 true\na\u271d\u00b9 : Lists'.Subset l\u2082 l\u2083\na\u271d : Lists'.Subset l\u2083 l\u2082\nh\u2082' : { fst := true, snd := l\u2082 } ~ { fst := true, snd := l\u2083 }\nhl\u2081 : l\u2081 \u2286 l\u2082\nhr\u2081 : l\u2082 \u2286 l\u2081\nhl\u2082 : l\u2082 \u2286 l\u2083\nhr\u2082 : l\u2083 \u2286 l\u2082\n\u22a2 \u2200 (a : Lists \u03b1), a \u2208 Lists'.toList l\u2083 \u2192 a \u2208 l\u2081\n[PROOFSTEP]\nintro a\u2083 m\u2083\n[GOAL]\ncase C1.antisymm.antisymm.intro.intro.right\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nl\u2081 : Lists' \u03b1 true\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l\u2081 \u2192 trans l'\nl\u2082 : Lists' \u03b1 true\na\u271d\u00b3 : Lists'.Subset l\u2081 l\u2082\na\u271d\u00b2 : Lists'.Subset l\u2082 l\u2081\nh\u2081' : of' l\u2081 ~ { fst := true, snd := l\u2082 }\nl\u2083 : Lists' \u03b1 true\na\u271d\u00b9 : Lists'.Subset l\u2082 l\u2083\na\u271d : Lists'.Subset l\u2083 l\u2082\nh\u2082' : { fst := true, snd := l\u2082 } ~ { fst := true, snd := l\u2083 }\nhl\u2081 : l\u2081 \u2286 l\u2082\nhr\u2081 : l\u2082 \u2286 l\u2081\nhl\u2082 : l\u2082 \u2286 l\u2083\nhr\u2082 : l\u2083 \u2286 l\u2082\na\u2083 : Lists \u03b1\nm\u2083 : a\u2083 \u2208 Lists'.toList l\u2083\n\u22a2 a\u2083 \u2208 l\u2081\n[PROOFSTEP]\nrcases Lists'.mem_of_subset' hr\u2082 m\u2083 with \u27e8a\u2082, m\u2082, e\u2083\u2082\u27e9\n[GOAL]\ncase C1.antisymm.antisymm.intro.intro.right.intro.intro\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nl\u2081 : Lists' \u03b1 true\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l\u2081 \u2192 trans l'\nl\u2082 : Lists' \u03b1 true\na\u271d\u00b3 : Lists'.Subset l\u2081 l\u2082\na\u271d\u00b2 : Lists'.Subset l\u2082 l\u2081\nh\u2081' : of' l\u2081 ~ { fst := true, snd := l\u2082 }\nl\u2083 : Lists' \u03b1 true\na\u271d\u00b9 : Lists'.Subset l\u2082 l\u2083\na\u271d : Lists'.Subset l\u2083 l\u2082\nh\u2082' : { fst := true, snd := l\u2082 } ~ { fst := true, snd := l\u2083 }\nhl\u2081 : l\u2081 \u2286 l\u2082\nhr\u2081 : l\u2082 \u2286 l\u2081\nhl\u2082 : l\u2082 \u2286 l\u2083\nhr\u2082 : l\u2083 \u2286 l\u2082\na\u2083 : Lists \u03b1\nm\u2083 : a\u2083 \u2208 Lists'.toList l\u2083\na\u2082 : Lists \u03b1\nm\u2082 : a\u2082 \u2208 Lists'.toList l\u2082\ne\u2083\u2082 : a\u2083 ~ a\u2082\n\u22a2 a\u2083 \u2208 l\u2081\n[PROOFSTEP]\nrcases Lists'.mem_of_subset' hr\u2081 m\u2082 with \u27e8a\u2081, m\u2081, e\u2082\u2081\u27e9\n[GOAL]\ncase C1.antisymm.antisymm.intro.intro.right.intro.intro.intro.intro\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\nl\u2081 : Lists' \u03b1 true\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l\u2081 \u2192 trans l'\nl\u2082 : Lists' \u03b1 true\na\u271d\u00b3 : Lists'.Subset l\u2081 l\u2082\na\u271d\u00b2 : Lists'.Subset l\u2082 l\u2081\nh\u2081' : of' l\u2081 ~ { fst := true, snd := l\u2082 }\nl\u2083 : Lists' \u03b1 true\na\u271d\u00b9 : Lists'.Subset l\u2082 l\u2083\na\u271d : Lists'.Subset l\u2083 l\u2082\nh\u2082' : { fst := true, snd := l\u2082 } ~ { fst := true, snd := l\u2083 }\nhl\u2081 : l\u2081 \u2286 l\u2082\nhr\u2081 : l\u2082 \u2286 l\u2081\nhl\u2082 : l\u2082 \u2286 l\u2083\nhr\u2082 : l\u2083 \u2286 l\u2082\na\u2083 : Lists \u03b1\nm\u2083 : a\u2083 \u2208 Lists'.toList l\u2083\na\u2082 : Lists \u03b1\nm\u2082 : a\u2082 \u2208 Lists'.toList l\u2082\ne\u2083\u2082 : a\u2083 ~ a\u2082\na\u2081 : Lists \u03b1\nm\u2081 : a\u2081 \u2208 Lists'.toList l\u2081\ne\u2082\u2081 : a\u2082 ~ a\u2081\n\u22a2 a\u2083 \u2208 l\u2081\n[PROOFSTEP]\nexact \u27e8a\u2081, m\u2081, (IH _ m\u2081 e\u2082\u2081.symm e\u2083\u2082.symm).symm\u27e9\n[GOAL]\ncase D0\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\n\u22a2 \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList Lists'.nil \u2192 trans l'\n[PROOFSTEP]\nrintro _ \u27e8\u27e9\n[GOAL]\ncase D1\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\n\u22a2 \u2200 (a : Lists \u03b1) (l : Lists' \u03b1 true),\n trans a \u2192\n (\u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l \u2192 trans l') \u2192\n \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList (Lists'.cons a l) \u2192 trans l'\n[PROOFSTEP]\nintro a l IH\u2081 IH\n[GOAL]\ncase D1\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\na : Lists \u03b1\nl : Lists' \u03b1 true\nIH\u2081 : trans a\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l \u2192 trans l'\n\u22a2 \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList (Lists'.cons a l) \u2192 trans l'\n[PROOFSTEP]\nsimp only [Lists'.toList, Sigma.eta, List.find?, List.mem_cons, forall_eq_or_imp]\n[GOAL]\ncase D1\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\na : Lists \u03b1\nl : Lists' \u03b1 true\nIH\u2081 : trans a\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l \u2192 trans l'\n\u22a2 (\u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, a ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 a ~ l\u2083) \u2227\n \u2200 (a : Lists \u03b1), a \u2208 Lists'.toList l \u2192 \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, a ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 a ~ l\u2083\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase D1.left\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\na : Lists \u03b1\nl : Lists' \u03b1 true\nIH\u2081 : trans a\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l \u2192 trans l'\n\u22a2 \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, a ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 a ~ l\u2083\n[PROOFSTEP]\nintros l\u2082 l\u2083 h\u2081 h\u2082\n[GOAL]\ncase D1.left\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\na : Lists \u03b1\nl : Lists' \u03b1 true\nIH\u2081 : trans a\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l \u2192 trans l'\nl\u2082 l\u2083 : Lists \u03b1\nh\u2081 : a ~ l\u2082\nh\u2082 : l\u2082 ~ l\u2083\n\u22a2 a ~ l\u2083\n[PROOFSTEP]\nexact IH\u2081 h\u2081 h\u2082\n[GOAL]\ncase D1.right\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\na : Lists \u03b1\nl : Lists' \u03b1 true\nIH\u2081 : trans a\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l \u2192 trans l'\n\u22a2 \u2200 (a : Lists \u03b1), a \u2208 Lists'.toList l \u2192 \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, a ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 a ~ l\u2083\n[PROOFSTEP]\nintros a h\u2081 l\u2082 l\u2083 h\u2082 h\u2083\n[GOAL]\ncase D1.right\n\u03b1 : Type u_1\ntrans : Lists \u03b1 \u2192 Prop := fun l\u2081 => \u2200 \u2983l\u2082 l\u2083 : Lists \u03b1\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083\na\u271d : Lists \u03b1\nl : Lists' \u03b1 true\nIH\u2081 : trans a\u271d\nIH : \u2200 (l' : Lists \u03b1), l' \u2208 Lists'.toList l \u2192 trans l'\na : Lists \u03b1\nh\u2081 : a \u2208 Lists'.toList l\nl\u2082 l\u2083 : Lists \u03b1\nh\u2082 : a ~ l\u2082\nh\u2083 : l\u2082 ~ l\u2083\n\u22a2 a ~ l\u2083\n[PROOFSTEP]\nexact IH _ h\u2081 h\u2082 h\u2083\n[GOAL]\n\u03b1 : Type u_1\nb : Bool\nl : Lists' \u03b1 b\n\u22a2 0 < sizeOf l\n[PROOFSTEP]\ncases l\n[GOAL]\ncase atom\n\u03b1 : Type u_1\na\u271d : \u03b1\n\u22a2 0 < sizeOf (Lists'.atom a\u271d)\n[PROOFSTEP]\nsimp only [Lists'.atom.sizeOf_spec, Lists'.nil.sizeOf_spec, Lists'.cons'.sizeOf_spec, true_or, add_pos_iff]\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u22a2 0 < sizeOf Lists'.nil\n[PROOFSTEP]\nsimp only [Lists'.atom.sizeOf_spec, Lists'.nil.sizeOf_spec, Lists'.cons'.sizeOf_spec, true_or, add_pos_iff]\n[GOAL]\ncase cons'\n\u03b1 : Type u_1\nb\u271d : Bool\na\u271d\u00b9 : Lists' \u03b1 b\u271d\na\u271d : Lists' \u03b1 true\n\u22a2 0 < sizeOf (Lists'.cons' a\u271d\u00b9 a\u271d)\n[PROOFSTEP]\nsimp only [Lists'.atom.sizeOf_spec, Lists'.nil.sizeOf_spec, Lists'.cons'.sizeOf_spec, true_or, add_pos_iff]\n[GOAL]\n\u03b1 : Type u_1\nb : Bool\na : Lists' \u03b1 b\nl : Lists' \u03b1 true\n\u22a2 sizeOf { fst := b, snd := a } < sizeOf (Lists'.cons' a l)\n[PROOFSTEP]\nsimp only [Sigma.mk.sizeOf_spec, Lists'.cons'.sizeOf_spec, lt_add_iff_pos_right]\n[GOAL]\n\u03b1 : Type u_1\nb : Bool\na : Lists' \u03b1 b\nl : Lists' \u03b1 true\n\u22a2 0 < sizeOf l\n[PROOFSTEP]\napply sizeof_pos\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : Lists' \u03b1 false\n\u22a2 { fst := false, snd := l\u2081 } ~ { fst := false, snd := l\u2082 } \u2194 l\u2081 = l\u2082\n[PROOFSTEP]\ncases l\u2081\n[GOAL]\ncase atom\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u2082 : Lists' \u03b1 false\na\u271d : \u03b1\n\u22a2 { fst := false, snd := Lists'.atom a\u271d } ~ { fst := false, snd := l\u2082 } \u2194 Lists'.atom a\u271d = l\u2082\n[PROOFSTEP]\napply equiv_atom.trans\n[GOAL]\ncase atom\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u2082 : Lists' \u03b1 false\na\u271d : \u03b1\n\u22a2 atom a\u271d = { fst := false, snd := l\u2082 } \u2194 Lists'.atom a\u271d = l\u2082\n[PROOFSTEP]\nsimp [atom]\n[GOAL]\ncase atom\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u2082 : Lists' \u03b1 false\na\u271d : \u03b1\n\u22a2 { fst := false, snd := Lists'.atom a\u271d } = { fst := false, snd := l\u2082 } \u2194 Lists'.atom a\u271d = l\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase atom.mp\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u2082 : Lists' \u03b1 false\na\u271d : \u03b1\n\u22a2 { fst := false, snd := Lists'.atom a\u271d } = { fst := false, snd := l\u2082 } \u2192 Lists'.atom a\u271d = l\u2082\n[PROOFSTEP]\nrintro \u27e8rfl\u27e9\n[GOAL]\ncase atom.mp.refl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na\u271d : \u03b1\n\u22a2 Lists'.atom a\u271d = Lists'.atom a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase atom.mpr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u2082 : Lists' \u03b1 false\na\u271d : \u03b1\n\u22a2 Lists'.atom a\u271d = l\u2082 \u2192 { fst := false, snd := Lists'.atom a\u271d } = { fst := false, snd := l\u2082 }\n[PROOFSTEP]\nrintro \u27e8rfl\u27e9\n[GOAL]\ncase atom.mpr.refl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na\u271d : \u03b1\n\u22a2 { fst := false, snd := Lists'.atom a\u271d } = { fst := false, snd := Lists'.atom a\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u2081 : Lists' \u03b1 false\nl\u2082 : Lists' \u03b1 true\n\u22a2 \u00ac{ fst := false, snd := l\u2081 } ~ { fst := true, snd := l\u2082 }\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u2081 : Lists' \u03b1 true\nl\u2082 : Lists' \u03b1 false\n\u22a2 \u00ac{ fst := true, snd := l\u2081 } ~ { fst := false, snd := l\u2082 }\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : Lists' \u03b1 true\n\u22a2 Decidable ({ fst := true, snd := l\u2081 } ~ { fst := true, snd := l\u2082 })\n[PROOFSTEP]\nhaveI : Decidable (l\u2081 \u2286 l\u2082) :=\n have :\n SizeOf.sizeOf l\u2081 + SizeOf.sizeOf l\u2082 < SizeOf.sizeOf (\u27e8true, l\u2081\u27e9 : Lists \u03b1) + SizeOf.sizeOf (\u27e8true, l\u2082\u27e9 : Lists \u03b1) :=\n by decreasing_tactic\n Subset.decidable l\u2081 l\u2082\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : Lists' \u03b1 true\n\u22a2 sizeOf l\u2081 + sizeOf l\u2082 < sizeOf { fst := true, snd := l\u2081 } + sizeOf { fst := true, snd := l\u2082 }\n[PROOFSTEP]\ndecreasing_tactic\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : Lists' \u03b1 true\nthis : Decidable (l\u2081 \u2286 l\u2082)\n\u22a2 Decidable ({ fst := true, snd := l\u2081 } ~ { fst := true, snd := l\u2082 })\n[PROOFSTEP]\nhaveI : Decidable (l\u2082 \u2286 l\u2081) :=\n have :\n SizeOf.sizeOf l\u2082 + SizeOf.sizeOf l\u2081 < SizeOf.sizeOf (\u27e8true, l\u2081\u27e9 : Lists \u03b1) + SizeOf.sizeOf (\u27e8true, l\u2082\u27e9 : Lists \u03b1) :=\n by decreasing_tactic\n Subset.decidable l\u2082 l\u2081\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : Lists' \u03b1 true\nthis : Decidable (l\u2081 \u2286 l\u2082)\n\u22a2 sizeOf l\u2082 + sizeOf l\u2081 < sizeOf { fst := true, snd := l\u2081 } + sizeOf { fst := true, snd := l\u2082 }\n[PROOFSTEP]\ndecreasing_tactic\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : Lists' \u03b1 true\nthis\u271d : Decidable (l\u2081 \u2286 l\u2082)\nthis : Decidable (l\u2082 \u2286 l\u2081)\n\u22a2 Decidable ({ fst := true, snd := l\u2081 } ~ { fst := true, snd := l\u2082 })\n[PROOFSTEP]\nexact decidable_of_iff' _ Equiv.antisymm_iff\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nb : Bool\na : Lists' \u03b1 b\nl\u2081 l\u2082 : Lists' \u03b1 true\n\u22a2 Decidable (Lists'.cons' a l\u2081 \u2286 l\u2082)\n[PROOFSTEP]\nhaveI :=\n have : sizeOf (\u27e8b, a\u27e9 : Lists \u03b1) < 1 + 1 + sizeOf a + sizeOf l\u2081 := by simp [sizeof_pos]\n mem.decidable \u27e8b, a\u27e9 l\u2082\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nb : Bool\na : Lists' \u03b1 b\nl\u2081 l\u2082 : Lists' \u03b1 true\n\u22a2 sizeOf { fst := b, snd := a } < 1 + 1 + sizeOf a + sizeOf l\u2081\n[PROOFSTEP]\nsimp [sizeof_pos]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nb : Bool\na : Lists' \u03b1 b\nl\u2081 l\u2082 : Lists' \u03b1 true\nthis : Decidable ({ fst := b, snd := a } \u2208 l\u2082)\n\u22a2 Decidable (Lists'.cons' a l\u2081 \u2286 l\u2082)\n[PROOFSTEP]\nhaveI :=\n have : SizeOf.sizeOf l\u2081 + SizeOf.sizeOf l\u2082 < SizeOf.sizeOf (Lists'.cons' a l\u2081) + SizeOf.sizeOf l\u2082 := by\n decreasing_tactic\n Subset.decidable l\u2081 l\u2082\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nb : Bool\na : Lists' \u03b1 b\nl\u2081 l\u2082 : Lists' \u03b1 true\nthis : Decidable ({ fst := b, snd := a } \u2208 l\u2082)\n\u22a2 sizeOf l\u2081 + sizeOf l\u2082 < sizeOf (Lists'.cons' a l\u2081) + sizeOf l\u2082\n[PROOFSTEP]\ndecreasing_tactic\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nb : Bool\na : Lists' \u03b1 b\nl\u2081 l\u2082 : Lists' \u03b1 true\nthis\u271d : Decidable ({ fst := b, snd := a } \u2208 l\u2082)\nthis : Decidable (l\u2081 \u2286 l\u2082)\n\u22a2 Decidable (Lists'.cons' a l\u2081 \u2286 l\u2082)\n[PROOFSTEP]\nexact decidable_of_iff' _ (@Lists'.cons_subset _ \u27e8_, _\u27e9 _ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : Lists \u03b1\n\u22a2 \u00aca \u2208 Lists'.nil\n[PROOFSTEP]\nrintro \u27e8_, \u27e8\u27e9, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : Lists \u03b1\nb\u271d : Bool\nb : Lists' \u03b1 b\u271d\nl\u2082 : Lists' \u03b1 true\n\u22a2 Decidable (a \u2208 Lists'.cons' b l\u2082)\n[PROOFSTEP]\nhaveI :=\n have : sizeOf (\u27e8_, b\u27e9 : Lists \u03b1) < 1 + 1 + sizeOf b + sizeOf l\u2082 := by simp [sizeof_pos]\n Equiv.decidable a \u27e8_, b\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : Lists \u03b1\nb\u271d : Bool\nb : Lists' \u03b1 b\u271d\nl\u2082 : Lists' \u03b1 true\n\u22a2 sizeOf { fst := b\u271d, snd := b } < 1 + 1 + sizeOf b + sizeOf l\u2082\n[PROOFSTEP]\nsimp [sizeof_pos]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : Lists \u03b1\nb\u271d : Bool\nb : Lists' \u03b1 b\u271d\nl\u2082 : Lists' \u03b1 true\nthis : Decidable (a ~ { fst := b\u271d, snd := b })\n\u22a2 Decidable (a \u2208 Lists'.cons' b l\u2082)\n[PROOFSTEP]\nhaveI :=\n have : SizeOf.sizeOf a + SizeOf.sizeOf l\u2082 < SizeOf.sizeOf a + SizeOf.sizeOf (Lists'.cons' b l\u2082) := by\n decreasing_tactic\n mem.decidable a l\u2082\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : Lists \u03b1\nb\u271d : Bool\nb : Lists' \u03b1 b\u271d\nl\u2082 : Lists' \u03b1 true\nthis : Decidable (a ~ { fst := b\u271d, snd := b })\n\u22a2 sizeOf a + sizeOf l\u2082 < sizeOf a + sizeOf (Lists'.cons' b l\u2082)\n[PROOFSTEP]\ndecreasing_tactic\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : Lists \u03b1\nb\u271d : Bool\nb : Lists' \u03b1 b\u271d\nl\u2082 : Lists' \u03b1 true\nthis\u271d : Decidable (a ~ { fst := b\u271d, snd := b })\nthis : Decidable (a \u2208 l\u2082)\n\u22a2 Decidable (a \u2208 Lists'.cons' b l\u2082)\n[PROOFSTEP]\nrefine' decidable_of_iff' (a ~ \u27e8_, b\u27e9 \u2228 a \u2208 l\u2082) _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : Lists \u03b1\nb\u271d : Bool\nb : Lists' \u03b1 b\u271d\nl\u2082 : Lists' \u03b1 true\nthis\u271d : Decidable (a ~ { fst := b\u271d, snd := b })\nthis : Decidable (a \u2208 l\u2082)\n\u22a2 a \u2208 Lists'.cons' b l\u2082 \u2194 a ~ { fst := b\u271d, snd := b } \u2228 a \u2208 l\u2082\n[PROOFSTEP]\nrw [\u2190 Lists'.mem_cons]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : Lists \u03b1\nb\u271d : Bool\nb : Lists' \u03b1 b\u271d\nl\u2082 : Lists' \u03b1 true\nthis\u271d : Decidable (a ~ { fst := b\u271d, snd := b })\nthis : Decidable (a \u2208 l\u2082)\n\u22a2 a \u2208 Lists'.cons' b l\u2082 \u2194 a \u2208 Lists'.cons { fst := b\u271d, snd := b } l\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u22a2 DecidableEq (Finsets \u03b1)\n[PROOFSTEP]\nunfold Finsets\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\u22a2 DecidableEq (Quotient Lists.instSetoidLists)\n[PROOFSTEP]\nexact (Quotient.decidableEq (d := fun _ _ => Lists.Equiv.decidable _ _))\n", "meta": {"mathlib_filename": "Mathlib.SetTheory.Lists", "llama_tokens": 20459, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. NO\n2. NO", "lm_q1_score": 0.34864513533394575, "lm_q2_score": 0.014957087471327428, "lm_q1q2_score": 0.005214715785642615}}