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+{"srcContext": "/-\nCopyright (c) 2016 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Mario Carneiro\n-/\nimport Mathlib.Logic.Nonempty\nimport Mathlib.Init.Set\nimport Mathlib.Logic.Basic\n\n#align_import logic.function.basic from \"leanprover-community/mathlib\"@\"29cb56a7b35f72758b05a30490e1f10bd62c35c1\"\n\n/-!\n# Miscellaneous function constructions and lemmas\n-/\n\nset_option autoImplicit true\n\nopen Function\n\nuniverse u v w\n\nnamespace Function\n\nsection\n\nvariable {\u03b1 \u03b2 \u03b3 : Sort*} {f : \u03b1 \u2192 \u03b2}\n\n/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied\n  `Function.eval x : (\u2200 x, \u03b2 x) \u2192 \u03b2 x`. -/\n@[reducible, simp] def eval {\u03b2 : \u03b1 \u2192 Sort*} (x : \u03b1) (f : \u2200 x, \u03b2 x) : \u03b2 x := f x\n#align function.eval Function.eval\n\ntheorem eval_apply {\u03b2 : \u03b1 \u2192 Sort*} (x : \u03b1) (f : \u2200 x, \u03b2 x) : eval x f = f x :=\n  rfl\n#align function.eval_apply Function.eval_apply\n\ntheorem const_def {y : \u03b2} : (fun _ : \u03b1 \u21a6 y) = const \u03b1 y :=\n  rfl\n#align function.const_def Function.const_def\n\ntheorem const_injective [Nonempty \u03b1] : Injective (const \u03b1 : \u03b2 \u2192 \u03b1 \u2192 \u03b2) := fun y\u2081 y\u2082 h \u21a6\n  let \u27e8x\u27e9 := \u2039Nonempty \u03b1\u203a\n  congr_fun h x\n#align function.const_injective Function.const_injective\n\n@[simp]\ntheorem const_inj [Nonempty \u03b1] {y\u2081 y\u2082 : \u03b2} : const \u03b1 y\u2081 = const \u03b1 y\u2082 \u2194 y\u2081 = y\u2082 :=\n  \u27e8fun h \u21a6 const_injective h, fun h \u21a6 h \u25b8 rfl\u27e9\n#align function.const_inj Function.const_inj\n\ntheorem id_def : @id \u03b1 = fun x \u21a6 x :=\n  rfl\n#align function.id_def Function.id_def\n\n-- Porting note: `Function.onFun` is now reducible\n-- @[simp]\ntheorem onFun_apply (f : \u03b2 \u2192 \u03b2 \u2192 \u03b3) (g : \u03b1 \u2192 \u03b2) (a b : \u03b1) : onFun f g a b = f (g a) (g b) :=\n  rfl\n#align function.on_fun_apply Function.onFun_apply\n\nlemma hfunext {\u03b1 \u03b1' : Sort u} {\u03b2 : \u03b1 \u2192 Sort v} {\u03b2' : \u03b1' \u2192 Sort v} {f : \u2200a, \u03b2 a} {f' : \u2200a, \u03b2' a}\n    (h\u03b1 : \u03b1 = \u03b1') (h : \u2200a a', HEq a a' \u2192 HEq (f a) (f' a')) : HEq f f' := by\n  subst h\u03b1\n  have : \u2200a, HEq (f a) (f' a) := fun a \u21a6 h a a (HEq.refl a)\n  have : \u03b2 = \u03b2' := by funext a; exact type_eq_of_heq (this a)\n  subst this\n  apply heq_of_eq\n  funext a\n  exact eq_of_heq (this a)\n#align function.hfunext Function.hfunext\n\ntheorem funext_iff {\u03b2 : \u03b1 \u2192 Sort*} {f\u2081 f\u2082 : \u2200 x : \u03b1, \u03b2 x} : f\u2081 = f\u2082 \u2194 \u2200 a, f\u2081 a = f\u2082 a :=\n  Iff.intro (fun h _ \u21a6 h \u25b8 rfl) funext\n#align function.funext_iff Function.funext_iff\n\ntheorem ne_iff {\u03b2 : \u03b1 \u2192 Sort*} {f\u2081 f\u2082 : \u2200 a, \u03b2 a} : f\u2081 \u2260 f\u2082 \u2194 \u2203 a, f\u2081 a \u2260 f\u2082 a :=\n  funext_iff.not.trans not_forall\n#align function.ne_iff Function.ne_iff\n\nlemma funext_iff_of_subsingleton [Subsingleton \u03b1] {g : \u03b1 \u2192 \u03b2} (x y : \u03b1) :\n    f x = g y \u2194 f = g := by\n  refine \u27e8fun h \u21a6 funext fun z \u21a6 ?_, fun h \u21a6 ?_\u27e9\n  \u00b7 rwa [Subsingleton.elim x z, Subsingleton.elim y z] at h\n  \u00b7 rw [h, Subsingleton.elim x y]\n\nprotected theorem Bijective.injective {f : \u03b1 \u2192 \u03b2} (hf : Bijective f) : Injective f := hf.1\n#align function.bijective.injective Function.Bijective.injective\nprotected theorem Bijective.surjective {f : \u03b1 \u2192 \u03b2} (hf : Bijective f) : Surjective f := hf.2\n#align function.bijective.surjective Function.Bijective.surjective\n\ntheorem Injective.eq_iff (I : Injective f) {a b : \u03b1} : f a = f b \u2194 a = b :=\n  \u27e8@I _ _, congr_arg f\u27e9\n#align function.injective.eq_iff Function.Injective.eq_iff\n\ntheorem Injective.beq_eq {\u03b1 \u03b2 : Type*} [BEq \u03b1] [LawfulBEq \u03b1] [BEq \u03b2] [LawfulBEq \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (I : Injective f) {a b : \u03b1} : (f a == f b) = (a == b) := by\n  by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h\n\ntheorem Injective.eq_iff' (I : Injective f) {a b : \u03b1} {c : \u03b2} (h : f b = c) : f a = c \u2194 a = b :=\n  h \u25b8 I.eq_iff\n#align function.injective.eq_iff' Function.Injective.eq_iff'\n\ntheorem Injective.ne (hf : Injective f) {a\u2081 a\u2082 : \u03b1} : a\u2081 \u2260 a\u2082 \u2192 f a\u2081 \u2260 f a\u2082 :=\n  mt fun h \u21a6 hf h\n#align function.injective.ne Function.Injective.ne\n\ntheorem Injective.ne_iff (hf : Injective f) {x y : \u03b1} : f x \u2260 f y \u2194 x \u2260 y :=\n  \u27e8mt <| congr_arg f, hf.ne\u27e9\n#align function.injective.ne_iff Function.Injective.ne_iff\n\ntheorem Injective.ne_iff' (hf : Injective f) {x y : \u03b1} {z : \u03b2} (h : f y = z) : f x \u2260 z \u2194 x \u2260 y :=\n  h \u25b8 hf.ne_iff\n#align function.injective.ne_iff' Function.Injective.ne_iff'\n\ntheorem not_injective_iff : \u00ac Injective f \u2194 \u2203 a b, f a = f b \u2227 a \u2260 b := by\n  simp only [Injective, not_forall, exists_prop]\n\n/-- If the co-domain `\u03b2` of an injective function `f : \u03b1 \u2192 \u03b2` has decidable equality, then\nthe domain `\u03b1` also has decidable equality. -/\nprotected def Injective.decidableEq [DecidableEq \u03b2] (I : Injective f) : DecidableEq \u03b1 :=\n  fun _ _ \u21a6 decidable_of_iff _ I.eq_iff\n#align function.injective.decidable_eq Function.Injective.decidableEq\n\ntheorem Injective.of_comp {g : \u03b3 \u2192 \u03b1} (I : Injective (f \u2218 g)) : Injective g :=\n  fun _ _ h \u21a6 I <| congr_arg f h\n#align function.injective.of_comp Function.Injective.of_comp\n\n@[simp]\ntheorem Injective.of_comp_iff (hf : Injective f) (g : \u03b3 \u2192 \u03b1) :\n    Injective (f \u2218 g) \u2194 Injective g :=\n  \u27e8Injective.of_comp, hf.comp\u27e9\n#align function.injective.of_comp_iff Function.Injective.of_comp_iff\n\ntheorem Injective.of_comp_right {g : \u03b3 \u2192 \u03b1} (I : Injective (f \u2218 g)) (hg : Surjective g) :\n    Injective f := fun x y h \u21a6 by\n  obtain \u27e8x, rfl\u27e9 := hg x\n  obtain \u27e8y, rfl\u27e9 := hg y\n  exact congr_arg g (I h)\n\ntheorem Surjective.bijective\u2082_of_injective {g : \u03b3 \u2192 \u03b1} (hf : Surjective f) (hg : Surjective g)\n    (I : Injective (f \u2218 g)) : Bijective f \u2227 Bijective g :=\n  \u27e8\u27e8I.of_comp_right hg, hf\u27e9, I.of_comp, hg\u27e9\n\n@[simp]\ntheorem Injective.of_comp_iff' (f : \u03b1 \u2192 \u03b2) {g : \u03b3 \u2192 \u03b1} (hg : Bijective g) :\n    Injective (f \u2218 g) \u2194 Injective f :=\n  \u27e8fun I \u21a6 I.of_comp_right hg.2, fun h \u21a6 h.comp hg.injective\u27e9\n#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'\n\n/-- Composition by an injective function on the left is itself injective. -/\ntheorem Injective.comp_left {g : \u03b2 \u2192 \u03b3} (hg : Function.Injective g) :\n    Function.Injective (g \u2218 \u00b7 : (\u03b1 \u2192 \u03b2) \u2192 \u03b1 \u2192 \u03b3) :=\n  fun _ _ hgf \u21a6 funext fun i \u21a6 hg <| (congr_fun hgf i : _)\n#align function.injective.comp_left Function.Injective.comp_left\n\ntheorem injective_of_subsingleton [Subsingleton \u03b1] (f : \u03b1 \u2192 \u03b2) : Injective f :=\n  fun _ _ _ \u21a6 Subsingleton.elim _ _\n#align function.injective_of_subsingleton Function.injective_of_subsingleton\n\nlemma Injective.dite (p : \u03b1 \u2192 Prop) [DecidablePred p]\n    {f : {a : \u03b1 // p a} \u2192 \u03b2} {f' : {a : \u03b1 // \u00ac p a} \u2192 \u03b2}\n    (hf : Injective f) (hf' : Injective f')\n    (im_disj : \u2200 {x x' : \u03b1} {hx : p x} {hx' : \u00ac p x'}, f \u27e8x, hx\u27e9 \u2260 f' \u27e8x', hx'\u27e9) :\n    Function.Injective (fun x \u21a6 if h : p x then f \u27e8x, h\u27e9 else f' \u27e8x, h\u27e9) := fun x\u2081 x\u2082 h => by\n dsimp only at h\n by_cases h\u2081 : p x\u2081 <;> by_cases h\u2082 : p x\u2082\n \u00b7 rw [dif_pos h\u2081, dif_pos h\u2082] at h; injection (hf h)\n \u00b7 rw [dif_pos h\u2081, dif_neg h\u2082] at h; exact (im_disj h).elim\n \u00b7 rw [dif_neg h\u2081, dif_pos h\u2082] at h; exact (im_disj h.symm).elim\n \u00b7 rw [dif_neg h\u2081, dif_neg h\u2082] at h; injection (hf' h)\n#align function.injective.dite Function.Injective.dite\n\ntheorem Surjective.of_comp {g : \u03b3 \u2192 \u03b1} (S : Surjective (f \u2218 g)) : Surjective f := fun y \u21a6\n  let \u27e8x, h\u27e9 := S y\n  \u27e8g x, h\u27e9\n#align function.surjective.of_comp Function.Surjective.of_comp\n\n@[simp]\ntheorem Surjective.of_comp_iff (f : \u03b1 \u2192 \u03b2) {g : \u03b3 \u2192 \u03b1} (hg : Surjective g) :\n    Surjective (f \u2218 g) \u2194 Surjective f :=\n  \u27e8Surjective.of_comp, fun h \u21a6 h.comp hg\u27e9\n#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff\n\ntheorem Surjective.of_comp_left {g : \u03b3 \u2192 \u03b1} (S : Surjective (f \u2218 g)) (hf : Injective f) :\n    Surjective g := fun a \u21a6 let \u27e8c, hc\u27e9 := S (f a); \u27e8c, hf hc\u27e9\n\ntheorem Injective.bijective\u2082_of_surjective {g : \u03b3 \u2192 \u03b1} (hf : Injective f) (hg : Injective g)\n    (S : Surjective (f \u2218 g)) : Bijective f \u2227 Bijective g :=\n  \u27e8\u27e8hf, S.of_comp\u27e9, hg, S.of_comp_left hf\u27e9\n\n@[simp]\ntheorem Surjective.of_comp_iff' (hf : Bijective f) (g : \u03b3 \u2192 \u03b1) :\n    Surjective (f \u2218 g) \u2194 Surjective g :=\n  \u27e8fun S \u21a6 S.of_comp_left hf.1, hf.surjective.comp\u27e9\n#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'\n\ninstance decidableEqPFun (p : Prop) [Decidable p] (\u03b1 : p \u2192 Type*) [\u2200 hp, DecidableEq (\u03b1 hp)] :\n    DecidableEq (\u2200 hp, \u03b1 hp)\n  | f, g => decidable_of_iff (\u2200 hp, f hp = g hp) funext_iff.symm\n\nprotected theorem Surjective.forall (hf : Surjective f) {p : \u03b2 \u2192 Prop} :\n    (\u2200 y, p y) \u2194 \u2200 x, p (f x) :=\n  \u27e8fun h x \u21a6 h (f x), fun h y \u21a6\n    let \u27e8x, hx\u27e9 := hf y\n    hx \u25b8 h x\u27e9\n#align function.surjective.forall Function.Surjective.forall\n\nprotected theorem Surjective.forall\u2082 (hf : Surjective f) {p : \u03b2 \u2192 \u03b2 \u2192 Prop} :\n    (\u2200 y\u2081 y\u2082, p y\u2081 y\u2082) \u2194 \u2200 x\u2081 x\u2082, p (f x\u2081) (f x\u2082) :=\n  hf.forall.trans <| forall_congr' fun _ \u21a6 hf.forall\n#align function.surjective.forall\u2082 Function.Surjective.forall\u2082\n\nprotected theorem Surjective.forall\u2083 (hf : Surjective f) {p : \u03b2 \u2192 \u03b2 \u2192 \u03b2 \u2192 Prop} :\n    (\u2200 y\u2081 y\u2082 y\u2083, p y\u2081 y\u2082 y\u2083) \u2194 \u2200 x\u2081 x\u2082 x\u2083, p (f x\u2081) (f x\u2082) (f x\u2083) :=\n  hf.forall.trans <| forall_congr' fun _ \u21a6 hf.forall\u2082\n#align function.surjective.forall\u2083 Function.Surjective.forall\u2083\n\nprotected theorem Surjective.exists (hf : Surjective f) {p : \u03b2 \u2192 Prop} :\n    (\u2203 y, p y) \u2194 \u2203 x, p (f x) :=\n  \u27e8fun \u27e8y, hy\u27e9 \u21a6\n    let \u27e8x, hx\u27e9 := hf y\n    \u27e8x, hx.symm \u25b8 hy\u27e9,\n    fun \u27e8x, hx\u27e9 \u21a6 \u27e8f x, hx\u27e9\u27e9\n#align function.surjective.exists Function.Surjective.exists\n\nprotected theorem Surjective.exists\u2082 (hf : Surjective f) {p : \u03b2 \u2192 \u03b2 \u2192 Prop} :\n    (\u2203 y\u2081 y\u2082, p y\u2081 y\u2082) \u2194 \u2203 x\u2081 x\u2082, p (f x\u2081) (f x\u2082) :=\n  hf.exists.trans <| exists_congr fun _ \u21a6 hf.exists\n#align function.surjective.exists\u2082 Function.Surjective.exists\u2082\n\nprotected theorem Surjective.exists\u2083 (hf : Surjective f) {p : \u03b2 \u2192 \u03b2 \u2192 \u03b2 \u2192 Prop} :\n    (\u2203 y\u2081 y\u2082 y\u2083, p y\u2081 y\u2082 y\u2083) \u2194 \u2203 x\u2081 x\u2082 x\u2083, p (f x\u2081) (f x\u2082) (f x\u2083) :=\n  hf.exists.trans <| exists_congr fun _ \u21a6 hf.exists\u2082\n#align function.surjective.exists\u2083 Function.Surjective.exists\u2083\n\ntheorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : \u03b2 \u2192 \u03b3 \u21a6 g \u2218 f :=\n  fun _ _ h \u21a6 funext <| hf.forall.2 <| congr_fun h\n#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right\n\nprotected theorem Surjective.right_cancellable (hf : Surjective f) {g\u2081 g\u2082 : \u03b2 \u2192 \u03b3} :\n    g\u2081 \u2218 f = g\u2082 \u2218 f \u2194 g\u2081 = g\u2082 :=\n  hf.injective_comp_right.eq_iff\n#align function.surjective.right_cancellable Function.Surjective.right_cancellable\n\ntheorem surjective_of_right_cancellable_Prop (h : \u2200 g\u2081 g\u2082 : \u03b2 \u2192 Prop, g\u2081 \u2218 f = g\u2082 \u2218 f \u2192 g\u2081 = g\u2082) :\n    Surjective f := by\n  specialize h (fun y \u21a6 \u2203 x, f x = y) (fun _ \u21a6 True) (funext fun x \u21a6 eq_true \u27e8_, rfl\u27e9)\n  intro y; rw [congr_fun h y]; trivial\n#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop\n\ntheorem bijective_iff_existsUnique (f : \u03b1 \u2192 \u03b2) : Bijective f \u2194 \u2200 b : \u03b2, \u2203! a : \u03b1, f a = b :=\n  \u27e8fun hf b \u21a6\n      let \u27e8a, ha\u27e9 := hf.surjective b\n      \u27e8a, ha, fun _ ha' \u21a6 hf.injective (ha'.trans ha.symm)\u27e9,\n    fun he \u21a6 \u27e8fun {_a a'} h \u21a6 (he (f a')).unique h rfl, fun b \u21a6 (he b).exists\u27e9\u27e9\n#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique\n\n/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/\nprotected theorem Bijective.existsUnique {f : \u03b1 \u2192 \u03b2} (hf : Bijective f) (b : \u03b2) :\n    \u2203! a : \u03b1, f a = b :=\n  (bijective_iff_existsUnique f).mp hf b\n#align function.bijective.exists_unique Function.Bijective.existsUnique\n\ntheorem Bijective.existsUnique_iff {f : \u03b1 \u2192 \u03b2} (hf : Bijective f) {p : \u03b2 \u2192 Prop} :\n    (\u2203! y, p y) \u2194 \u2203! x, p (f x) :=\n  \u27e8fun \u27e8y, hpy, hy\u27e9 \u21a6\n    let \u27e8x, hx\u27e9 := hf.surjective y\n    \u27e8x, by simpa [hx], fun z (hz : p (f z)) \u21a6 hf.injective <| hx.symm \u25b8 hy _ hz\u27e9,\n    fun \u27e8x, hpx, hx\u27e9 \u21a6\n    \u27e8f x, hpx, fun y hy \u21a6\n      let \u27e8z, hz\u27e9 := hf.surjective y\n      hz \u25b8 congr_arg f (hx _ (by simpa [hz]))\u27e9\u27e9\n#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff\n\ntheorem Bijective.of_comp_iff (f : \u03b1 \u2192 \u03b2) {g : \u03b3 \u2192 \u03b1} (hg : Bijective g) :\n    Bijective (f \u2218 g) \u2194 Bijective f :=\n  and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)\n#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff\n\ntheorem Bijective.of_comp_iff' {f : \u03b1 \u2192 \u03b2} (hf : Bijective f) (g : \u03b3 \u2192 \u03b1) :\n    Function.Bijective (f \u2218 g) \u2194 Function.Bijective g :=\n  and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)\n#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'\n\n/-- **Cantor's diagonal argument** implies that there are no surjective functions from `\u03b1`\nto `Set \u03b1`. -/\ntheorem cantor_surjective {\u03b1} (f : \u03b1 \u2192 Set \u03b1) : \u00acSurjective f\n  | h => let \u27e8D, e\u27e9 := h {a | \u00ac f a a}\n        @iff_not_self (D \u2208 f D) <| iff_of_eq <| congr_arg (D \u2208 \u00b7) e\n#align function.cantor_surjective Function.cantor_surjective\n\n/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set \u03b1`\nto `\u03b1`. -/\ntheorem cantor_injective {\u03b1 : Type*} (f : Set \u03b1 \u2192 \u03b1) : \u00acInjective f\n  | i => cantor_surjective (fun a \u21a6 {b | \u2200 U, a = f U \u2192 U b}) <|\n         RightInverse.surjective (fun U \u21a6 Set.ext fun _ \u21a6 \u27e8fun h \u21a6 h U rfl, fun h _ e \u21a6 i e \u25b8 h\u27e9)\n#align function.cantor_injective Function.cantor_injective\n\n/-- There is no surjection from `\u03b1 : Type u` into `Type (max u v)`. This theorem\n  demonstrates why `Type : Type` would be inconsistent in Lean. -/\ntheorem not_surjective_Type {\u03b1 : Type u} (f : \u03b1 \u2192 Type max u v) : \u00acSurjective f := by\n  intro hf\n  let T : Type max u v := Sigma f\n  cases hf (Set T) with | intro U hU =>\n  let g : Set T \u2192 T := fun s \u21a6 \u27e8U, cast hU.symm s\u27e9\n  have hg : Injective g := by\n    intro s t h\n    suffices cast hU (g s).2 = cast hU (g t).2 by\n      simp only [cast_cast, cast_eq] at this\n      assumption\n    \u00b7 congr\n  exact cantor_injective g hg\n#align function.not_surjective_Type Function.not_surjective_Type\n\n/-- `g` is a partial inverse to `f` (an injective but not necessarily\n  surjective function) if `g y = some x` implies `f x = y`, and `g y = none`\n  implies that `y` is not in the range of `f`. -/\ndef IsPartialInv {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 Option \u03b1) : Prop :=\n  \u2200 x y, g y = some x \u2194 f x = y\n#align function.is_partial_inv Function.IsPartialInv\n\ntheorem isPartialInv_left {\u03b1 \u03b2} {f : \u03b1 \u2192 \u03b2} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=\n  (H _ _).2 rfl\n#align function.is_partial_inv_left Function.isPartialInv_left\n\ntheorem injective_of_isPartialInv {\u03b1 \u03b2} {f : \u03b1 \u2192 \u03b2} {g} (H : IsPartialInv f g) :\n    Injective f := fun _ _ h \u21a6\n  Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)\n#align function.injective_of_partial_inv Function.injective_of_isPartialInv\n\ntheorem injective_of_isPartialInv_right {\u03b1 \u03b2} {f : \u03b1 \u2192 \u03b2} {g} (H : IsPartialInv f g) (x y b)\n    (h\u2081 : b \u2208 g x) (h\u2082 : b \u2208 g y) : x = y :=\n  ((H _ _).1 h\u2081).symm.trans ((H _ _).1 h\u2082)\n#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right\n\ntheorem LeftInverse.comp_eq_id {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1} (h : LeftInverse f g) : f \u2218 g = id :=\n  funext h\n#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id\n\ntheorem leftInverse_iff_comp {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1} : LeftInverse f g \u2194 f \u2218 g = id :=\n  \u27e8LeftInverse.comp_eq_id, congr_fun\u27e9\n#align function.left_inverse_iff_comp Function.leftInverse_iff_comp\n\ntheorem RightInverse.comp_eq_id {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1} (h : RightInverse f g) : g \u2218 f = id :=\n  funext h\n#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id\n\ntheorem rightInverse_iff_comp {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1} : RightInverse f g \u2194 g \u2218 f = id :=\n  \u27e8RightInverse.comp_eq_id, congr_fun\u27e9\n#align function.right_inverse_iff_comp Function.rightInverse_iff_comp\n\ntheorem LeftInverse.comp {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1} {h : \u03b2 \u2192 \u03b3} {i : \u03b3 \u2192 \u03b2} (hf : LeftInverse f g)\n    (hh : LeftInverse h i) : LeftInverse (h \u2218 f) (g \u2218 i) :=\n  fun a \u21a6 show h (f (g (i a))) = a by rw [hf (i a), hh a]\n#align function.left_inverse.comp Function.LeftInverse.comp\n\ntheorem RightInverse.comp {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1} {h : \u03b2 \u2192 \u03b3} {i : \u03b3 \u2192 \u03b2} (hf : RightInverse f g)\n    (hh : RightInverse h i) : RightInverse (h \u2218 f) (g \u2218 i) :=\n  LeftInverse.comp hh hf\n#align function.right_inverse.comp Function.RightInverse.comp\n\ntheorem LeftInverse.rightInverse {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1} (h : LeftInverse g f) : RightInverse f g :=\n  h\n#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse\n\ntheorem RightInverse.leftInverse {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1} (h : RightInverse g f) : LeftInverse f g :=\n  h\n#align function.right_inverse.left_inverse Function.RightInverse.leftInverse\n\ntheorem LeftInverse.surjective {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1} (h : LeftInverse f g) : Surjective f :=\n  h.rightInverse.surjective\n#align function.left_inverse.surjective Function.LeftInverse.surjective\n\ntheorem RightInverse.injective {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1} (h : RightInverse f g) : Injective f :=\n  h.leftInverse.injective\n#align function.right_inverse.injective Function.RightInverse.injective\n\ntheorem LeftInverse.rightInverse_of_injective {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1} (h : LeftInverse f g)\n    (hf : Injective f) : RightInverse f g :=\n  fun x \u21a6 hf <| h (f x)\n#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective\n\ntheorem LeftInverse.rightInverse_of_surjective {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1} (h : LeftInverse f g)\n    (hg : Surjective g) : RightInverse f g :=\n  fun x \u21a6 let \u27e8y, hy\u27e9 := hg x; hy \u25b8 congr_arg g (h y)\n#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective\n\ntheorem RightInverse.leftInverse_of_surjective {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1} :\n    RightInverse f g \u2192 Surjective f \u2192 LeftInverse f g :=\n  LeftInverse.rightInverse_of_surjective\n#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective\n\ntheorem RightInverse.leftInverse_of_injective {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1} :\n    RightInverse f g \u2192 Injective g \u2192 LeftInverse f g :=\n  LeftInverse.rightInverse_of_injective\n#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective\n\ntheorem LeftInverse.eq_rightInverse {f : \u03b1 \u2192 \u03b2} {g\u2081 g\u2082 : \u03b2 \u2192 \u03b1} (h\u2081 : LeftInverse g\u2081 f)\n    (h\u2082 : RightInverse g\u2082 f) : g\u2081 = g\u2082 :=\n  calc\n    g\u2081 = g\u2081 \u2218 f \u2218 g\u2082 := by rw [h\u2082.comp_eq_id, comp_id]\n     _ = g\u2082 := by rw [\u2190 comp.assoc, h\u2081.comp_eq_id, id_comp]\n#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse\n\nattribute [local instance] Classical.propDecidable\n\n/-- We can use choice to construct explicitly a partial inverse for\n  a given injective function `f`. -/\nnoncomputable def partialInv {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) (b : \u03b2) : Option \u03b1 :=\n  if h : \u2203 a, f a = b then some (Classical.choose h) else none\n#align function.partial_inv Function.partialInv\n\ntheorem partialInv_of_injective {\u03b1 \u03b2} {f : \u03b1 \u2192 \u03b2} (I : Injective f) : IsPartialInv f (partialInv f)\n  | a, b =>\n  \u27e8fun h =>\n    have hpi : partialInv f b = if h : \u2203 a, f a = b then some (Classical.choose h) else none :=\n      rfl\n    if h' : \u2203 a, f a = b\n    then by rw [hpi, dif_pos h'] at h\n            injection h with h\n            subst h\n            apply Classical.choose_spec h'\n    else by rw [hpi, dif_neg h'] at h; contradiction,\n  fun e => e \u25b8 have h : \u2203 a', f a' = f a := \u27e8_, rfl\u27e9\n              (dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))\u27e9\n#align function.partial_inv_of_injective Function.partialInv_of_injective\n\ntheorem partialInv_left {\u03b1 \u03b2} {f : \u03b1 \u2192 \u03b2} (I : Injective f) : \u2200 x, partialInv f (f x) = some x :=\n  isPartialInv_left (partialInv_of_injective I)\n#align function.partial_inv_left Function.partialInv_left\n\nend\n\nsection InvFun\n\nvariable {\u03b1 \u03b2 : Sort*} [Nonempty \u03b1] {f : \u03b1 \u2192 \u03b2} {a : \u03b1} {b : \u03b2}\n\nattribute [local instance] Classical.propDecidable\n\n/-- The inverse of a function (which is a left inverse if `f` is injective\n  and a right inverse if `f` is surjective). -/\n-- Explicit Sort so that `\u03b1` isn't inferred to be Prop via `exists_prop_decidable`\nnoncomputable def invFun {\u03b1 : Sort u} {\u03b2} [Nonempty \u03b1] (f : \u03b1 \u2192 \u03b2) : \u03b2 \u2192 \u03b1 :=\n  fun y \u21a6 if h : (\u2203 x, f x = y) then h.choose else Classical.arbitrary \u03b1\n#align function.inv_fun Function.invFun\n\ntheorem invFun_eq (h : \u2203 a, f a = b) : f (invFun f b) = b :=\n  by simp only [invFun, dif_pos h, h.choose_spec]\n#align function.inv_fun_eq Function.invFun_eq\n\ntheorem apply_invFun_apply {\u03b1 : Type u\u2081} {\u03b2 : Type u\u2082} {f : \u03b1 \u2192 \u03b2} {a : \u03b1} :\n    f (@invFun _ _ \u27e8a\u27e9 f (f a)) = f a :=\n  @invFun_eq _ _ \u27e8a\u27e9 _ _ \u27e8_, rfl\u27e9\n\ntheorem invFun_neg (h : \u00ac\u2203 a, f a = b) : invFun f b = Classical.choice \u2039_\u203a :=\n  dif_neg h\n#align function.inv_fun_neg Function.invFun_neg\n\ntheorem invFun_eq_of_injective_of_rightInverse {g : \u03b2 \u2192 \u03b1} (hf : Injective f)\n    (hg : RightInverse g f) : invFun f = g :=\n  funext fun b \u21a6\n    hf\n      (by\n        rw [hg b]\n        exact invFun_eq \u27e8g b, hg b\u27e9)\n#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse\n\ntheorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=\n  fun b \u21a6 invFun_eq <| hf b\n#align function.right_inverse_inv_fun Function.rightInverse_invFun\n\ntheorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=\n  fun b \u21a6 hf <| invFun_eq \u27e8b, rfl\u27e9\n#align function.left_inverse_inv_fun Function.leftInverse_invFun\n\ntheorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=\n  (leftInverse_invFun hf).surjective\n#align function.inv_fun_surjective Function.invFun_surjective\n\ntheorem invFun_comp (hf : Injective f) : invFun f \u2218 f = id :=\n  funext <| leftInverse_invFun hf\n#align function.inv_fun_comp Function.invFun_comp\n\ntheorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=\n  \u27e8invFun f, leftInverse_invFun hf\u27e9\n#align function.injective.has_left_inverse Function.Injective.hasLeftInverse\n\ntheorem injective_iff_hasLeftInverse : Injective f \u2194 HasLeftInverse f :=\n  \u27e8Injective.hasLeftInverse, HasLeftInverse.injective\u27e9\n#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse\n\nend InvFun\n\nsection SurjInv\n\nvariable {\u03b1 : Sort u} {\u03b2 : Sort v} {\u03b3 : Sort w} {f : \u03b1 \u2192 \u03b2}\n\n/-- The inverse of a surjective function. (Unlike `invFun`, this does not require\n  `\u03b1` to be inhabited.) -/\nnoncomputable def surjInv {f : \u03b1 \u2192 \u03b2} (h : Surjective f) (b : \u03b2) : \u03b1 :=\n  Classical.choose (h b)\n#align function.surj_inv Function.surjInv\n\ntheorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=\n  Classical.choose_spec (h b)\n#align function.surj_inv_eq Function.surjInv_eq\n\ntheorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=\n  surjInv_eq hf\n#align function.right_inverse_surj_inv Function.rightInverse_surjInv\n\ntheorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=\n  rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)\n#align function.left_inverse_surj_inv Function.leftInverse_surjInv\n\ntheorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=\n  \u27e8_, rightInverse_surjInv hf\u27e9\n#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse\n\ntheorem surjective_iff_hasRightInverse : Surjective f \u2194 HasRightInverse f :=\n  \u27e8Surjective.hasRightInverse, HasRightInverse.surjective\u27e9\n#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse\n\ntheorem bijective_iff_has_inverse : Bijective f \u2194 \u2203 g, LeftInverse g f \u2227 RightInverse g f :=\n  \u27e8fun hf \u21a6 \u27e8_, leftInverse_surjInv hf, rightInverse_surjInv hf.2\u27e9, fun \u27e8_, gl, gr\u27e9 \u21a6\n    \u27e8gl.injective, gr.surjective\u27e9\u27e9\n#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse\n\ntheorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=\n  (rightInverse_surjInv h).injective\n#align function.injective_surj_inv Function.injective_surjInv\n\ntheorem surjective_to_subsingleton [na : Nonempty \u03b1] [Subsingleton \u03b2] (f : \u03b1 \u2192 \u03b2) :\n    Surjective f :=\n  fun _ \u21a6 let \u27e8a\u27e9 := na; \u27e8a, Subsingleton.elim _ _\u27e9\n#align function.surjective_to_subsingleton Function.surjective_to_subsingleton\n\n/-- Composition by a surjective function on the left is itself surjective. -/\ntheorem Surjective.comp_left {g : \u03b2 \u2192 \u03b3} (hg : Surjective g) :\n    Surjective (g \u2218 \u00b7 : (\u03b1 \u2192 \u03b2) \u2192 \u03b1 \u2192 \u03b3) := fun f \u21a6\n  \u27e8surjInv hg \u2218 f, funext fun _ \u21a6 rightInverse_surjInv _ _\u27e9\n#align function.surjective.comp_left Function.Surjective.comp_left\n\n/-- Composition by a bijective function on the left is itself bijective. -/\ntheorem Bijective.comp_left {g : \u03b2 \u2192 \u03b3} (hg : Bijective g) :\n    Bijective (g \u2218 \u00b7 : (\u03b1 \u2192 \u03b2) \u2192 \u03b1 \u2192 \u03b3) :=\n  \u27e8hg.injective.comp_left, hg.surjective.comp_left\u27e9\n#align function.bijective.comp_left Function.Bijective.comp_left\n\nend SurjInv\n\nsection Update\n\nvariable {\u03b1 : Sort u} {\u03b2 : \u03b1 \u2192 Sort v} {\u03b1' : Sort w} [DecidableEq \u03b1] [DecidableEq \u03b1']\n  {f g : (a : \u03b1) \u2192 \u03b2 a} {a : \u03b1} {b : \u03b2 a}\n\n\n/-- Replacing the value of a function at a given point by a given value. -/\ndef update (f : \u2200 a, \u03b2 a) (a' : \u03b1) (v : \u03b2 a') (a : \u03b1) : \u03b2 a :=\n  if h : a = a' then Eq.ndrec v h.symm else f a\n#align function.update Function.update\n\n@[simp]\ntheorem update_same (a : \u03b1) (v : \u03b2 a) (f : \u2200 a, \u03b2 a) : update f a v a = v :=\n  dif_pos rfl\n#align function.update_same Function.update_same\n\n@[simp]\ntheorem update_noteq {a a' : \u03b1} (h : a \u2260 a') (v : \u03b2 a') (f : \u2200 a, \u03b2 a) : update f a' v a = f a :=\n  dif_neg h\n#align function.update_noteq Function.update_noteq\n\n/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/\ntheorem update_apply {\u03b2 : Sort*} (f : \u03b1 \u2192 \u03b2) (a' : \u03b1) (b : \u03b2) (a : \u03b1) :\n    update f a' b a = if a = a' then b else f a := by\n  rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]\n#align function.update_apply Function.update_apply\n\n@[nontriviality]\ntheorem update_eq_const_of_subsingleton [Subsingleton \u03b1] (a : \u03b1) (v : \u03b1') (f : \u03b1 \u2192 \u03b1') :\n    update f a v = const \u03b1 v :=\n  funext fun a' \u21a6 Subsingleton.elim a a' \u25b8 update_same _ _ _\n\ntheorem surjective_eval {\u03b1 : Sort u} {\u03b2 : \u03b1 \u2192 Sort v} [h : \u2200 a, Nonempty (\u03b2 a)] (a : \u03b1) :\n    Surjective (eval a : (\u2200 a, \u03b2 a) \u2192 \u03b2 a) := fun b \u21a6\n  \u27e8@update _ _ (Classical.decEq \u03b1) (fun a \u21a6 (h a).some) a b,\n   @update_same _ _ (Classical.decEq \u03b1) _ _ _\u27e9\n#align function.surjective_eval Function.surjective_eval\n\ntheorem update_injective (f : \u2200 a, \u03b2 a) (a' : \u03b1) : Injective (update f a') := fun v v' h \u21a6 by\n  have := congr_fun h a'\n  rwa [update_same, update_same] at this\n#align function.update_injective Function.update_injective\n\nlemma forall_update_iff (f : \u2200a, \u03b2 a) {a : \u03b1} {b : \u03b2 a} (p : \u2200a, \u03b2 a \u2192 Prop) :\n    (\u2200 x, p x (update f a b x)) \u2194 p a b \u2227 \u2200 x, x \u2260 a \u2192 p x (f x) := by\n  rw [\u2190 and_forall_ne a, update_same]\n  simp (config := { contextual := true })\n#align function.forall_update_iff Function.forall_update_iff\n\ntheorem exists_update_iff (f : \u2200 a, \u03b2 a) {a : \u03b1} {b : \u03b2 a} (p : \u2200 a, \u03b2 a \u2192 Prop) :\n    (\u2203 x, p x (update f a b x)) \u2194 p a b \u2228 \u2203 x \u2260 a, p x (f x) := by\n  rw [\u2190 not_forall_not, forall_update_iff f fun a b \u21a6 \u00acp a b]\n  simp [-not_and, not_and_or]\n#align function.exists_update_iff Function.exists_update_iff\n\ntheorem update_eq_iff {a : \u03b1} {b : \u03b2 a} {f g : \u2200 a, \u03b2 a} :\n    update f a b = g \u2194 b = g a \u2227 \u2200 x \u2260 a, f x = g x :=\n  funext_iff.trans <| forall_update_iff _ fun x y \u21a6 y = g x\n#align function.update_eq_iff Function.update_eq_iff\n\ntheorem eq_update_iff {a : \u03b1} {b : \u03b2 a} {f g : \u2200 a, \u03b2 a} :\n    g = update f a b \u2194 g a = b \u2227 \u2200 x \u2260 a, g x = f x :=\n  funext_iff.trans <| forall_update_iff _ fun x y \u21a6 g x = y\n#align function.eq_update_iff Function.eq_update_iff\n\n@[simp] lemma update_eq_self_iff : update f a b = f \u2194 b = f a := by simp [update_eq_iff]\n#align function.update_eq_self_iff Function.update_eq_self_iff\n\n@[simp] lemma eq_update_self_iff : f = update f a b \u2194 f a = b := by simp [eq_update_iff]\n#align function.eq_update_self_iff Function.eq_update_self_iff\n\nlemma ne_update_self_iff : f \u2260 update f a b \u2194 f a \u2260 b := eq_update_self_iff.not\n#align function.ne_update_self_iff Function.ne_update_self_iff\n\nlemma update_ne_self_iff : update f a b \u2260 f \u2194 b \u2260 f a := update_eq_self_iff.not\n#align function.update_ne_self_iff Function.update_ne_self_iff\n\n@[simp]\ntheorem update_eq_self (a : \u03b1) (f : \u2200 a, \u03b2 a) : update f a (f a) = f :=\n  update_eq_iff.2 \u27e8rfl, fun _ _ \u21a6 rfl\u27e9\n#align function.update_eq_self Function.update_eq_self\n\ntheorem update_comp_eq_of_forall_ne' {\u03b1'} (g : \u2200 a, \u03b2 a) {f : \u03b1' \u2192 \u03b1} {i : \u03b1} (a : \u03b2 i)\n    (h : \u2200 x, f x \u2260 i) : (fun j \u21a6 (update g i a) (f j)) = fun j \u21a6 g (f j) :=\n  funext fun _ \u21a6 update_noteq (h _) _ _\n#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'\n\n/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/\ntheorem update_comp_eq_of_forall_ne {\u03b1 \u03b2 : Sort*} (g : \u03b1' \u2192 \u03b2) {f : \u03b1 \u2192 \u03b1'} {i : \u03b1'} (a : \u03b2)\n    (h : \u2200 x, f x \u2260 i) : update g i a \u2218 f = g \u2218 f :=\n  update_comp_eq_of_forall_ne' g a h\n#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne\n\ntheorem update_comp_eq_of_injective' (g : \u2200 a, \u03b2 a) {f : \u03b1' \u2192 \u03b1} (hf : Function.Injective f)\n    (i : \u03b1') (a : \u03b2 (f i)) : (fun j \u21a6 update g (f i) a (f j)) = update (fun i \u21a6 g (f i)) i a :=\n  eq_update_iff.2 \u27e8update_same _ _ _, fun _ hj \u21a6 update_noteq (hf.ne hj) _ _\u27e9\n#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'\n\n/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/\ntheorem update_comp_eq_of_injective {\u03b2 : Sort*} (g : \u03b1' \u2192 \u03b2) {f : \u03b1 \u2192 \u03b1'}\n    (hf : Function.Injective f) (i : \u03b1) (a : \u03b2) :\n    Function.update g (f i) a \u2218 f = Function.update (g \u2218 f) i a :=\n  update_comp_eq_of_injective' g hf i a\n#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective\n\ntheorem apply_update {\u03b9 : Sort*} [DecidableEq \u03b9] {\u03b1 \u03b2 : \u03b9 \u2192 Sort*} (f : \u2200 i, \u03b1 i \u2192 \u03b2 i)\n    (g : \u2200 i, \u03b1 i) (i : \u03b9) (v : \u03b1 i) (j : \u03b9) :\n    f j (update g i v j) = update (fun k \u21a6 f k (g k)) i (f i v) j := by\n  by_cases h:j = i\n  \u00b7 subst j\n    simp\n  \u00b7 simp [h]\n#align function.apply_update Function.apply_update\n\ntheorem apply_update\u2082 {\u03b9 : Sort*} [DecidableEq \u03b9] {\u03b1 \u03b2 \u03b3 : \u03b9 \u2192 Sort*} (f : \u2200 i, \u03b1 i \u2192 \u03b2 i \u2192 \u03b3 i)\n    (g : \u2200 i, \u03b1 i) (h : \u2200 i, \u03b2 i) (i : \u03b9) (v : \u03b1 i) (w : \u03b2 i) (j : \u03b9) :\n    f j (update g i v j) (update h i w j) = update (fun k \u21a6 f k (g k) (h k)) i (f i v w) j := by\n  by_cases h:j = i\n  \u00b7 subst j\n    simp\n  \u00b7 simp [h]\n#align function.apply_update\u2082 Function.apply_update\u2082\n\ntheorem pred_update (P : \u2200 \u2983a\u2984, \u03b2 a \u2192 Prop) (f : \u2200 a, \u03b2 a) (a' : \u03b1) (v : \u03b2 a') (a : \u03b1) :\n    P (update f a' v a) \u2194 a = a' \u2227 P v \u2228 a \u2260 a' \u2227 P (f a) := by\n  rw [apply_update P, update_apply, ite_prop_iff_or]\n\ntheorem comp_update {\u03b1' : Sort*} {\u03b2 : Sort*} (f : \u03b1' \u2192 \u03b2) (g : \u03b1 \u2192 \u03b1') (i : \u03b1) (v : \u03b1') :\n    f \u2218 update g i v = update (f \u2218 g) i (f v) :=\n  funext <| apply_update _ _ _ _\n#align function.comp_update Function.comp_update\n\ntheorem update_comm {\u03b1} [DecidableEq \u03b1] {\u03b2 : \u03b1 \u2192 Sort*} {a b : \u03b1} (h : a \u2260 b) (v : \u03b2 a) (w : \u03b2 b)\n    (f : \u2200 a, \u03b2 a) : update (update f a v) b w = update (update f b w) a v := by\n  funext c\n  simp only [update]\n  by_cases h\u2081 : c = b <;> by_cases h\u2082 : c = a\n  \u00b7 rw [dif_pos h\u2081, dif_pos h\u2082]\n    cases h (h\u2082.symm.trans h\u2081)\n  \u00b7 rw [dif_pos h\u2081, dif_pos h\u2081, dif_neg h\u2082]\n  \u00b7 rw [dif_neg h\u2081, dif_neg h\u2081]\n  \u00b7 rw [dif_neg h\u2081, dif_neg h\u2081]\n#align function.update_comm Function.update_comm\n\n@[simp]\ntheorem update_idem {\u03b1} [DecidableEq \u03b1] {\u03b2 : \u03b1 \u2192 Sort*} {a : \u03b1} (v w : \u03b2 a) (f : \u2200 a, \u03b2 a) :\n    update (update f a v) a w = update f a w := by\n  funext b\n  by_cases h : b = a <;> simp [update, h]\n#align function.update_idem Function.update_idem\n\nend Update\n\nnoncomputable section Extend\n\nattribute [local instance] Classical.propDecidable\n\nvariable {\u03b1 \u03b2 \u03b3 : Sort*} {f : \u03b1 \u2192 \u03b2}\n\n/-- Extension of a function `g : \u03b1 \u2192 \u03b3` along a function `f : \u03b1 \u2192 \u03b2`.\n\nFor every `a : \u03b1`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary\nfunction `j : \u03b2 \u2192 \u03b3` by sending `b : \u03b2` not in the range of `f` to `j b`. If you do not care about\nthe behavior outside the range, `j` can be used as a junk value by setting it to be `0` or\n`Classical.arbitrary` (assuming `\u03b3` is nonempty).\n\nThis definition is mathematically meaningful only when `f a\u2081 = f a\u2082 \u2192 g a\u2081 = g a\u2082` (spelled\n`g.FactorsThrough f`). In particular this holds if `f` is injective.\n\nA typical use case is extending a function from a subtype to the entire type. If you wish to extend\n`g : {b : \u03b2 // p b} \u2192 \u03b3` to a function `\u03b2 \u2192 \u03b3`, you should use `Function.extend Subtype.val g j`. -/\ndef extend (f : \u03b1 \u2192 \u03b2) (g : \u03b1 \u2192 \u03b3) (j : \u03b2 \u2192 \u03b3) : \u03b2 \u2192 \u03b3 := fun b \u21a6\n  if h : \u2203 a, f a = b then g (Classical.choose h) else j b\n#align function.extend Function.extend\n\n/-- g factors through f : `f a = f b \u2192 g a = g b` -/\ndef FactorsThrough (g : \u03b1 \u2192 \u03b3) (f : \u03b1 \u2192 \u03b2) : Prop :=\n  \u2200 \u2983a b\u2984, f a = f b \u2192 g a = g b\n#align function.factors_through Function.FactorsThrough\n\ntheorem extend_def (f : \u03b1 \u2192 \u03b2) (g : \u03b1 \u2192 \u03b3) (e' : \u03b2 \u2192 \u03b3) (b : \u03b2) [Decidable (\u2203 a, f a = b)] :\n    extend f g e' b = if h : \u2203 a, f a = b then g (Classical.choose h) else e' b := by\n  unfold extend\n  congr\n#align function.extend_def Function.extend_def\n\nlemma Injective.factorsThrough (hf : Injective f) (g : \u03b1 \u2192 \u03b3) : g.FactorsThrough f :=\n  fun _ _ h => congr_arg g (hf h)\n#align function.injective.factors_through Function.Injective.factorsThrough\n\nlemma FactorsThrough.extend_apply {g : \u03b1 \u2192 \u03b3} (hf : g.FactorsThrough f) (e' : \u03b2 \u2192 \u03b3) (a : \u03b1) :\n    extend f g e' (f a) = g a := by\n  simp only [extend_def, dif_pos, exists_apply_eq_apply]\n  exact hf (Classical.choose_spec (exists_apply_eq_apply f a))\n#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply\n\n@[simp]\ntheorem Injective.extend_apply (hf : Injective f) (g : \u03b1 \u2192 \u03b3) (e' : \u03b2 \u2192 \u03b3) (a : \u03b1) :\n    extend f g e' (f a) = g a :=\n  (hf.factorsThrough g).extend_apply e' a\n#align function.injective.extend_apply Function.Injective.extend_apply\n\n@[simp]\ntheorem extend_apply' (g : \u03b1 \u2192 \u03b3) (e' : \u03b2 \u2192 \u03b3) (b : \u03b2) (hb : \u00ac\u2203 a, f a = b) :\n    extend f g e' b = e' b := by\n  simp [Function.extend_def, hb]\n#align function.extend_apply' Function.extend_apply'\n\nlemma factorsThrough_iff (g : \u03b1 \u2192 \u03b3) [Nonempty \u03b3] : g.FactorsThrough f \u2194 \u2203 (e : \u03b2 \u2192 \u03b3), g = e \u2218 f :=\n\u27e8fun hf => \u27e8extend f g (const \u03b2 (Classical.arbitrary \u03b3)),\n      funext (fun x => by simp only [comp_apply, hf.extend_apply])\u27e9,\n  fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]\u27e9\n#align function.factors_through_iff Function.factorsThrough_iff\n\nlemma apply_extend {\u03b4} {g : \u03b1 \u2192 \u03b3} (F : \u03b3 \u2192 \u03b4) (f : \u03b1 \u2192 \u03b2) (e' : \u03b2 \u2192 \u03b3) (b : \u03b2) :\n    F (extend f g e' b) = extend f (F \u2218 g) (F \u2218 e') b :=\n  apply_dite F _ _ _\n#align function.factors_through.apply_extend Function.apply_extend\n#align function.injective.apply_extend Function.apply_extend\n\ntheorem extend_injective (hf : Injective f) (e' : \u03b2 \u2192 \u03b3) : Injective fun g \u21a6 extend f g e' := by\n  intro g\u2081 g\u2082 hg\n  refine' funext fun x \u21a6 _\n  have H := congr_fun hg (f x)\n  simp only [hf.extend_apply] at H\n  exact H\n#align function.extend_injective Function.extend_injective\n\nlemma FactorsThrough.extend_comp {g : \u03b1 \u2192 \u03b3} (e' : \u03b2 \u2192 \u03b3) (hf : FactorsThrough g f) :\n    extend f g e' \u2218 f = g :=\n  funext fun a => hf.extend_apply e' a\n#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp\n\n@[simp]\ntheorem extend_comp (hf : Injective f) (g : \u03b1 \u2192 \u03b3) (e' : \u03b2 \u2192 \u03b3) : extend f g e' \u2218 f = g :=\n  funext fun a \u21a6 hf.extend_apply g e' a\n#align function.extend_comp Function.extend_comp\n\ntheorem Injective.surjective_comp_right' (hf : Injective f) (g\u2080 : \u03b2 \u2192 \u03b3) :\n    Surjective fun g : \u03b2 \u2192 \u03b3 \u21a6 g \u2218 f :=\n  fun g \u21a6 \u27e8extend f g g\u2080, extend_comp hf _ _\u27e9\n#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'\n\ntheorem Injective.surjective_comp_right [Nonempty \u03b3] (hf : Injective f) :\n    Surjective fun g : \u03b2 \u2192 \u03b3 \u21a6 g \u2218 f :=\n  hf.surjective_comp_right' fun _ \u21a6 Classical.choice \u2039_\u203a\n#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right\n\ntheorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : \u03b2 \u2192 \u03b3 \u21a6 g \u2218 f :=\n  \u27e8hf.surjective.injective_comp_right, fun g \u21a6\n    \u27e8g \u2218 surjInv hf.surjective,\n     by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp_id]\u27e9\u27e9\n#align function.bijective.comp_right Function.Bijective.comp_right\n\nend Extend\n\nnamespace FactorsThrough\n\nprotected theorem rfl {f : \u03b1 \u2192 \u03b2} : FactorsThrough f f := fun _ _ \u21a6 id\n\ntheorem comp_left {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b3} (h : FactorsThrough g f) (g' : \u03b3 \u2192 \u03b4) :\n    FactorsThrough (g' \u2218 g) f := fun _x _y hxy \u21a6\n  congr_arg g' (h hxy)\n\ntheorem comp_right {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b3} (h : FactorsThrough g f) (g' : \u03b4 \u2192 \u03b1) :\n    FactorsThrough (g \u2218 g') (f \u2218 g') := fun _x _y hxy \u21a6\n  h hxy\n\nend FactorsThrough\n\ntheorem uncurry_def {\u03b1 \u03b2 \u03b3} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) : uncurry f = fun p \u21a6 f p.1 p.2 :=\n  rfl\n#align function.uncurry_def Function.uncurry_def\n\n@[simp]\ntheorem uncurry_apply_pair {\u03b1 \u03b2 \u03b3} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (x : \u03b1) (y : \u03b2) : uncurry f (x, y) = f x y :=\n  rfl\n#align function.uncurry_apply_pair Function.uncurry_apply_pair\n\n@[simp]\ntheorem curry_apply {\u03b1 \u03b2 \u03b3} (f : \u03b1 \u00d7 \u03b2 \u2192 \u03b3) (x : \u03b1) (y : \u03b2) : curry f x y = f (x, y) :=\n  rfl\n#align function.curry_apply Function.curry_apply\n\nsection Bicomp\n\nvariable {\u03b1 \u03b2 \u03b3 \u03b4 \u03b5 : Type*}\n\n/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.\nIf both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/\ndef bicompl (f : \u03b3 \u2192 \u03b4 \u2192 \u03b5) (g : \u03b1 \u2192 \u03b3) (h : \u03b2 \u2192 \u03b4) (a b) :=\n  f (g a) (h b)\n#align function.bicompl Function.bicompl\n\n/-- Compose a unary function `f` with a binary function `g`. -/\ndef bicompr (f : \u03b3 \u2192 \u03b4) (g : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (a b) :=\n  f (g a b)\n#align function.bicompr Function.bicompr\n\n-- Suggested local notation:\nlocal notation f \" \u2218\u2082 \" g => bicompr f g\n\ntheorem uncurry_bicompr (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (g : \u03b3 \u2192 \u03b4) : uncurry (g \u2218\u2082 f) = g \u2218 uncurry f :=\n  rfl\n#align function.uncurry_bicompr Function.uncurry_bicompr\n\ntheorem uncurry_bicompl (f : \u03b3 \u2192 \u03b4 \u2192 \u03b5) (g : \u03b1 \u2192 \u03b3) (h : \u03b2 \u2192 \u03b4) :\n    uncurry (bicompl f g h) = uncurry f \u2218 Prod.map g h :=\n  rfl\n#align function.uncurry_bicompl Function.uncurry_bicompl\n\nend Bicomp\n\nsection Uncurry\n\nvariable {\u03b1 \u03b2 \u03b3 \u03b4 : Type*}\n\n/-- Records a way to turn an element of `\u03b1` into a function from `\u03b2` to `\u03b3`. The most generic use\nis to recursively uncurry. For instance `f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4` will be turned into\n`\u21bff : \u03b1 \u00d7 \u03b2 \u00d7 \u03b3 \u2192 \u03b4`. One can also add instances for bundled maps. -/\nclass HasUncurry (\u03b1 : Type*) (\u03b2 : outParam (Type*)) (\u03b3 : outParam (Type*)) where\n  /-- Uncurrying operator. The most generic use is to recursively uncurry. For instance\n  `f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4` will be turned into `\u21bff : \u03b1 \u00d7 \u03b2 \u00d7 \u03b3 \u2192 \u03b4`. One can also add instances\n  for bundled maps. -/\n  uncurry : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n#align function.has_uncurry Function.HasUncurry\n\n@[inherit_doc] notation:arg \"\u21bf\" x:arg => HasUncurry.uncurry x\n\ninstance hasUncurryBase : HasUncurry (\u03b1 \u2192 \u03b2) \u03b1 \u03b2 :=\n  \u27e8id\u27e9\n\ninstance hasUncurryInduction [HasUncurry \u03b2 \u03b3 \u03b4] : HasUncurry (\u03b1 \u2192 \u03b2) (\u03b1 \u00d7 \u03b3) \u03b4 :=\n  \u27e8fun f p \u21a6 (\u21bf(f p.1)) p.2\u27e9\n\nend Uncurry\n\n/-- A function is involutive, if `f \u2218 f = id`. -/\ndef Involutive {\u03b1} (f : \u03b1 \u2192 \u03b1) : Prop :=\n  \u2200 x, f (f x) = x\n#align function.involutive Function.Involutive\n\ntheorem _root_.Bool.involutive_not : Involutive not :=\n  Bool.not_not\n\nnamespace Involutive\n\nvariable {\u03b1 : Sort u} {f : \u03b1 \u2192 \u03b1} (h : Involutive f)\n\n@[simp]\ntheorem comp_self : f \u2218 f = id :=\n  funext h\n#align function.involutive.comp_self Function.Involutive.comp_self\n\nprotected theorem leftInverse : LeftInverse f f := h\n#align function.involutive.left_inverse Function.Involutive.leftInverse\n\nprotected theorem rightInverse : RightInverse f f := h\n#align function.involutive.right_inverse Function.Involutive.rightInverse\n\nprotected theorem injective : Injective f := h.leftInverse.injective\n#align function.involutive.injective Function.Involutive.injective\n\nprotected theorem surjective : Surjective f := fun x \u21a6 \u27e8f x, h x\u27e9\n#align function.involutive.surjective Function.Involutive.surjective\n\nprotected theorem bijective : Bijective f := \u27e8h.injective, h.surjective\u27e9\n#align function.involutive.bijective Function.Involutive.bijective\n\n/-- Involuting an `ite` of an involuted value `x : \u03b1` negates the `Prop` condition in the `ite`. -/\nprotected theorem ite_not (P : Prop) [Decidable P] (x : \u03b1) : f (ite P x (f x)) = ite (\u00acP) x (f x) :=\n  by rw [apply_ite f, h, ite_not]\n#align function.involutive.ite_not Function.Involutive.ite_not\n\n/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/\nprotected theorem eq_iff {x y : \u03b1} : f x = y \u2194 x = f y :=\n  h.injective.eq_iff' (h y)\n#align function.involutive.eq_iff Function.Involutive.eq_iff\n\nend Involutive\n\n", "theoremStatement": "lemma not_involutive : Involutive Not ", "theoremName": "Function.not_involutive", "fileCreated": {"commit": "f8294a67548f8f3d1f5913677b070a2ef5bcf120", "date": "2021-06-09"}, "theoremCreated": {"commit": "160fb23c2fdddeaf4b24ac532134de4535bc1dae", "date": "2024-03-29"}, "file": "mathlib/Mathlib/Logic/Function/Basic.lean", "module": "Mathlib.Logic.Function.Basic", "jsonFile": "Mathlib.Logic.Function.Basic.jsonl", "positionMetadata": {"lineInFile": 933, "tokenPositionInFile": 39958, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 4, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", 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"Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":= fun _ \u21a6 propext not_not", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 26}}
+{"srcContext": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Mario Carneiro\n-/\nimport Mathlib.Algebra.Module.Submodule.Bilinear\nimport Mathlib.GroupTheory.Congruence\nimport Mathlib.LinearAlgebra.Basic\nimport Mathlib.Tactic.SuppressCompilation\n\n#align_import linear_algebra.tensor_product from \"leanprover-community/mathlib\"@\"88fcdc3da43943f5b01925deddaa5bf0c0e85e4e\"\n\n/-!\n# Tensor product of modules over commutative semirings.\n\nThis file constructs the tensor product of modules over commutative semirings. Given a semiring\n`R` and modules over it `M` and `N`, the standard construction of the tensor product is\n`TensorProduct R M N`. It is also a module over `R`.\n\nIt comes with a canonical bilinear map `M \u2192 N \u2192 TensorProduct R M N`.\n\nGiven any bilinear map `M \u2192 N \u2192 P`, there is a unique linear map `TensorProduct R M N \u2192 P` whose\ncomposition with the canonical bilinear map `M \u2192 N \u2192 TensorProduct R M N` is the given bilinear\nmap `M \u2192 N \u2192 P`.\n\nWe start by proving basic lemmas about bilinear maps.\n\n## Notations\n\nThis file uses the localized notation `M \u2297 N` and `M \u2297[R] N` for `TensorProduct R M N`, as well\nas `m \u2297\u209c n` and `m \u2297\u209c[R] n` for `TensorProduct.tmul R m n`.\n\n## Tags\n\nbilinear, tensor, tensor product\n-/\n\nsuppress_compilation\n\nsection Semiring\n\nvariable {R : Type*} [CommSemiring R]\nvariable {R' : Type*} [Monoid R']\nvariable {R'' : Type*} [Semiring R'']\nvariable {M : Type*} {N : Type*} {P : Type*} {Q : Type*} {S : Type*} {T : Type*}\nvariable [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P]\nvariable [AddCommMonoid Q] [AddCommMonoid S] [AddCommMonoid T]\nvariable [Module R M] [Module R N] [Module R P] [Module R Q] [Module R S] [Module R T]\nvariable [DistribMulAction R' M]\nvariable [Module R'' M]\nvariable (M N)\n\nnamespace TensorProduct\n\nsection\n\nvariable (R)\n\n/-- The relation on `FreeAddMonoid (M \u00d7 N)` that generates a congruence whose quotient is\nthe tensor product. -/\ninductive Eqv : FreeAddMonoid (M \u00d7 N) \u2192 FreeAddMonoid (M \u00d7 N) \u2192 Prop\n  | of_zero_left : \u2200 n : N, Eqv (.of (0, n)) 0\n  | of_zero_right : \u2200 m : M, Eqv (.of (m, 0)) 0\n  | of_add_left : \u2200 (m\u2081 m\u2082 : M) (n : N), Eqv (.of (m\u2081, n) + .of (m\u2082, n)) (.of (m\u2081 + m\u2082, n))\n  | of_add_right : \u2200 (m : M) (n\u2081 n\u2082 : N), Eqv (.of (m, n\u2081) + .of (m, n\u2082)) (.of (m, n\u2081 + n\u2082))\n  | of_smul : \u2200 (r : R) (m : M) (n : N), Eqv (.of (r \u2022 m, n)) (.of (m, r \u2022 n))\n  | add_comm : \u2200 x y, Eqv (x + y) (y + x)\n#align tensor_product.eqv TensorProduct.Eqv\n\nend\n\nend TensorProduct\n\nvariable (R)\n\n/-- The tensor product of two modules `M` and `N` over the same commutative semiring `R`.\nThe localized notations are `M \u2297 N` and `M \u2297[R] N`, accessed by `open scoped TensorProduct`. -/\ndef TensorProduct : Type _ :=\n  (addConGen (TensorProduct.Eqv R M N)).Quotient\n#align tensor_product TensorProduct\n\nvariable {R}\n\nset_option quotPrecheck false in\n@[inherit_doc TensorProduct] scoped[TensorProduct] infixl:100 \" \u2297 \" => TensorProduct _\n\n@[inherit_doc] scoped[TensorProduct] notation:100 M \" \u2297[\" R \"] \" N:100 => TensorProduct R M N\n\nnamespace TensorProduct\n\nsection Module\n\nprotected instance add : Add (M \u2297[R] N) :=\n  (addConGen (TensorProduct.Eqv R M N)).hasAdd\n\ninstance addZeroClass : AddZeroClass (M \u2297[R] N) :=\n  { (addConGen (TensorProduct.Eqv R M N)).addMonoid with\n    /- The `toAdd` field is given explicitly as `TensorProduct.add` for performance reasons.\n    This avoids any need to unfold `Con.addMonoid` when the type checker is checking\n    that instance diagrams commute -/\n    toAdd := TensorProduct.add _ _ }\n\ninstance addSemigroup : AddSemigroup (M \u2297[R] N) :=\n  { (addConGen (TensorProduct.Eqv R M N)).addMonoid with\n    toAdd := TensorProduct.add _ _ }\n\ninstance addCommSemigroup : AddCommSemigroup (M \u2297[R] N) :=\n  { (addConGen (TensorProduct.Eqv R M N)).addMonoid with\n    toAddSemigroup := TensorProduct.addSemigroup _ _\n    add_comm := fun x y =>\n      AddCon.induction_on\u2082 x y fun _ _ =>\n        Quotient.sound' <| AddConGen.Rel.of _ _ <| Eqv.add_comm _ _ }\n\ninstance : Inhabited (M \u2297[R] N) :=\n  \u27e80\u27e9\n\nvariable (R) {M N}\n\n/-- The canonical function `M \u2192 N \u2192 M \u2297 N`. The localized notations are `m \u2297\u209c n` and `m \u2297\u209c[R] n`,\naccessed by `open scoped TensorProduct`. -/\ndef tmul (m : M) (n : N) : M \u2297[R] N :=\n  AddCon.mk' _ <| FreeAddMonoid.of (m, n)\n#align tensor_product.tmul TensorProduct.tmul\n\nvariable {R}\n\n/-- The canonical function `M \u2192 N \u2192 M \u2297 N`. -/\ninfixl:100 \" \u2297\u209c \" => tmul _\n\n/-- The canonical function `M \u2192 N \u2192 M \u2297 N`. -/\nnotation:100 x \" \u2297\u209c[\" R \"] \" y:100 => tmul R x y\n\n-- Porting note: make the arguments of induction_on explicit\n@[elab_as_elim]\nprotected theorem induction_on {motive : M \u2297[R] N \u2192 Prop} (z : M \u2297[R] N)\n    (zero : motive 0)\n    (tmul : \u2200 x y, motive <| x \u2297\u209c[R] y)\n    (add : \u2200 x y, motive x \u2192 motive y \u2192 motive (x + y)) : motive z :=\n  AddCon.induction_on z fun x =>\n    FreeAddMonoid.recOn x zero fun \u27e8m, n\u27e9 y ih => by\n      rw [AddCon.coe_add]\n      exact add _ _ (tmul ..) ih\n#align tensor_product.induction_on TensorProduct.induction_on\n\n/-- Lift an `R`-balanced map to the tensor product.\n\nA map `f : M \u2192+ N \u2192+ P` additive in both components is `R`-balanced, or middle linear with respect\nto `R`, if scalar multiplication in either argument is equivalent, `f (r \u2022 m) n = f m (r \u2022 n)`.\n\nNote that strictly the first action should be a right-action by `R`, but for now `R` is commutative\nso it doesn't matter. -/\n-- TODO: use this to implement `lift` and `SMul.aux`. For now we do not do this as it causes\n-- performance issues elsewhere.\ndef liftAddHom (f : M \u2192+ N \u2192+ P)\n    (hf : \u2200 (r : R) (m : M) (n : N), f (r \u2022 m) n = f m (r \u2022 n)) :\n    M \u2297[R] N \u2192+ P :=\n  (addConGen (TensorProduct.Eqv R M N)).lift (FreeAddMonoid.lift (fun mn : M \u00d7 N => f mn.1 mn.2)) <|\n    AddCon.addConGen_le fun x y hxy =>\n      match x, y, hxy with\n      | _, _, .of_zero_left n =>\n        (AddCon.ker_rel _).2 <| by simp_rw [map_zero, FreeAddMonoid.lift_eval_of, map_zero,\n          AddMonoidHom.zero_apply]\n      | _, _, .of_zero_right m =>\n        (AddCon.ker_rel _).2 <| by simp_rw [map_zero, FreeAddMonoid.lift_eval_of, map_zero]\n      | _, _, .of_add_left m\u2081 m\u2082 n =>\n        (AddCon.ker_rel _).2 <| by simp_rw [map_add, FreeAddMonoid.lift_eval_of, map_add,\n          AddMonoidHom.add_apply]\n      | _, _, .of_add_right m n\u2081 n\u2082 =>\n        (AddCon.ker_rel _).2 <| by simp_rw [map_add, FreeAddMonoid.lift_eval_of, map_add]\n      | _, _, .of_smul s m n =>\n        (AddCon.ker_rel _).2 <| by rw [FreeAddMonoid.lift_eval_of, FreeAddMonoid.lift_eval_of, hf]\n      | _, _, .add_comm x y =>\n        (AddCon.ker_rel _).2 <| by simp_rw [map_add, add_comm]\n\n@[simp]\ntheorem liftAddHom_tmul (f : M \u2192+ N \u2192+ P)\n    (hf : \u2200 (r : R) (m : M) (n : N), f (r \u2022 m) n = f m (r \u2022 n)) (m : M) (n : N) :\n    liftAddHom f hf (m \u2297\u209c n) = f m n :=\n  rfl\n\nvariable (M)\n\n@[simp]\ntheorem zero_tmul (n : N) : (0 : M) \u2297\u209c[R] n = 0 :=\n  Quotient.sound' <| AddConGen.Rel.of _ _ <| Eqv.of_zero_left _\n#align tensor_product.zero_tmul TensorProduct.zero_tmul\n\nvariable {M}\n\ntheorem add_tmul (m\u2081 m\u2082 : M) (n : N) : (m\u2081 + m\u2082) \u2297\u209c n = m\u2081 \u2297\u209c n + m\u2082 \u2297\u209c[R] n :=\n  Eq.symm <| Quotient.sound' <| AddConGen.Rel.of _ _ <| Eqv.of_add_left _ _ _\n#align tensor_product.add_tmul TensorProduct.add_tmul\n\nvariable (N)\n\n@[simp]\ntheorem tmul_zero (m : M) : m \u2297\u209c[R] (0 : N) = 0 :=\n  Quotient.sound' <| AddConGen.Rel.of _ _ <| Eqv.of_zero_right _\n#align tensor_product.tmul_zero TensorProduct.tmul_zero\n\nvariable {N}\n\ntheorem tmul_add (m : M) (n\u2081 n\u2082 : N) : m \u2297\u209c (n\u2081 + n\u2082) = m \u2297\u209c n\u2081 + m \u2297\u209c[R] n\u2082 :=\n  Eq.symm <| Quotient.sound' <| AddConGen.Rel.of _ _ <| Eqv.of_add_right _ _ _\n#align tensor_product.tmul_add TensorProduct.tmul_add\n\ninstance uniqueLeft [Subsingleton M] : Unique (M \u2297[R] N) where\n  default := 0\n  uniq z := z.induction_on rfl (fun x y \u21a6 by rw [Subsingleton.elim x 0, zero_tmul]; rfl) <| by\n    rintro _ _ rfl rfl; apply add_zero\n\ninstance uniqueRight [Subsingleton N] : Unique (M \u2297[R] N) where\n  default := 0\n  uniq z := z.induction_on rfl (fun x y \u21a6 by rw [Subsingleton.elim y 0, tmul_zero]; rfl) <| by\n    rintro _ _ rfl rfl; apply add_zero\n\nsection\n\nvariable (R R' M N)\n\n/-- A typeclass for `SMul` structures which can be moved across a tensor product.\n\nThis typeclass is generated automatically from an `IsScalarTower` instance, but exists so that\nwe can also add an instance for `AddCommGroup.intModule`, allowing `z \u2022` to be moved even if\n`R` does not support negation.\n\nNote that `Module R' (M \u2297[R] N)` is available even without this typeclass on `R'`; it's only\nneeded if `TensorProduct.smul_tmul`, `TensorProduct.smul_tmul'`, or `TensorProduct.tmul_smul` is\nused.\n-/\nclass CompatibleSMul [DistribMulAction R' N] : Prop where\n  smul_tmul : \u2200 (r : R') (m : M) (n : N), (r \u2022 m) \u2297\u209c n = m \u2297\u209c[R] (r \u2022 n)\n#align tensor_product.compatible_smul TensorProduct.CompatibleSMul\n\nend\n\n/-- Note that this provides the default `compatible_smul R R M N` instance through\n`IsScalarTower.left`. -/\ninstance (priority := 100) CompatibleSMul.isScalarTower [SMul R' R] [IsScalarTower R' R M]\n    [DistribMulAction R' N] [IsScalarTower R' R N] : CompatibleSMul R R' M N :=\n  \u27e8fun r m n => by\n    conv_lhs => rw [\u2190 one_smul R m]\n    conv_rhs => rw [\u2190 one_smul R n]\n    rw [\u2190 smul_assoc, \u2190 smul_assoc]\n    exact Quotient.sound' <| AddConGen.Rel.of _ _ <| Eqv.of_smul _ _ _\u27e9\n#align tensor_product.compatible_smul.is_scalar_tower TensorProduct.CompatibleSMul.isScalarTower\n\n/-- `smul` can be moved from one side of the product to the other . -/\ntheorem smul_tmul [DistribMulAction R' N] [CompatibleSMul R R' M N] (r : R') (m : M) (n : N) :\n    (r \u2022 m) \u2297\u209c n = m \u2297\u209c[R] (r \u2022 n) :=\n  CompatibleSMul.smul_tmul _ _ _\n#align tensor_product.smul_tmul TensorProduct.smul_tmul\n\n-- Porting note: This is added as a local instance for `SMul.aux`.\n-- For some reason type-class inference in Lean 3 unfolded this definition.\nprivate def addMonoidWithWrongNSMul : AddMonoid (M \u2297[R] N) :=\n  { (addConGen (TensorProduct.Eqv R M N)).addMonoid with }\n\nattribute [local instance] addMonoidWithWrongNSMul in\n/-- Auxiliary function to defining scalar multiplication on tensor product. -/\ndef SMul.aux {R' : Type*} [SMul R' M] (r : R') : FreeAddMonoid (M \u00d7 N) \u2192+ M \u2297[R] N :=\n  FreeAddMonoid.lift fun p : M \u00d7 N => (r \u2022 p.1) \u2297\u209c p.2\n#align tensor_product.smul.aux TensorProduct.SMul.aux\n\ntheorem SMul.aux_of {R' : Type*} [SMul R' M] (r : R') (m : M) (n : N) :\n    SMul.aux r (.of (m, n)) = (r \u2022 m) \u2297\u209c[R] n :=\n  rfl\n#align tensor_product.smul.aux_of TensorProduct.SMul.aux_of\n\nvariable [SMulCommClass R R' M] [SMulCommClass R R'' M]\n\n/-- Given two modules over a commutative semiring `R`, if one of the factors carries a\n(distributive) action of a second type of scalars `R'`, which commutes with the action of `R`, then\nthe tensor product (over `R`) carries an action of `R'`.\n\nThis instance defines this `R'` action in the case that it is the left module which has the `R'`\naction. Two natural ways in which this situation arises are:\n * Extension of scalars\n * A tensor product of a group representation with a module not carrying an action\n\nNote that in the special case that `R = R'`, since `R` is commutative, we just get the usual scalar\naction on a tensor product of two modules. This special case is important enough that, for\nperformance reasons, we define it explicitly below. -/\ninstance leftHasSMul : SMul R' (M \u2297[R] N) :=\n  \u27e8fun r =>\n    (addConGen (TensorProduct.Eqv R M N)).lift (SMul.aux r : _ \u2192+ M \u2297[R] N) <|\n      AddCon.addConGen_le fun x y hxy =>\n        match x, y, hxy with\n        | _, _, .of_zero_left n =>\n          (AddCon.ker_rel _).2 <| by simp_rw [map_zero, SMul.aux_of, smul_zero, zero_tmul]\n        | _, _, .of_zero_right m =>\n          (AddCon.ker_rel _).2 <| by simp_rw [map_zero, SMul.aux_of, tmul_zero]\n        | _, _, .of_add_left m\u2081 m\u2082 n =>\n          (AddCon.ker_rel _).2 <| by simp_rw [map_add, SMul.aux_of, smul_add, add_tmul]\n        | _, _, .of_add_right m n\u2081 n\u2082 =>\n          (AddCon.ker_rel _).2 <| by simp_rw [map_add, SMul.aux_of, tmul_add]\n        | _, _, .of_smul s m n =>\n          (AddCon.ker_rel _).2 <| by rw [SMul.aux_of, SMul.aux_of, \u2190 smul_comm, smul_tmul]\n        | _, _, .add_comm x y =>\n          (AddCon.ker_rel _).2 <| by simp_rw [map_add, add_comm]\u27e9\n#align tensor_product.left_has_smul TensorProduct.leftHasSMul\n\ninstance : SMul R (M \u2297[R] N) :=\n  TensorProduct.leftHasSMul\n\nprotected theorem smul_zero (r : R') : r \u2022 (0 : M \u2297[R] N) = 0 :=\n  AddMonoidHom.map_zero _\n#align tensor_product.smul_zero TensorProduct.smul_zero\n\nprotected theorem smul_add (r : R') (x y : M \u2297[R] N) : r \u2022 (x + y) = r \u2022 x + r \u2022 y :=\n  AddMonoidHom.map_add _ _ _\n#align tensor_product.smul_add TensorProduct.smul_add\n\nprotected theorem zero_smul (x : M \u2297[R] N) : (0 : R'') \u2022 x = 0 :=\n  have : \u2200 (r : R'') (m : M) (n : N), r \u2022 m \u2297\u209c[R] n = (r \u2022 m) \u2297\u209c n := fun _ _ _ => rfl\n  x.induction_on (by rw [TensorProduct.smul_zero])\n    (fun m n => by rw [this, zero_smul, zero_tmul]) fun x y ihx ihy => by\n    rw [TensorProduct.smul_add, ihx, ihy, add_zero]\n#align tensor_product.zero_smul TensorProduct.zero_smul\n\nprotected theorem one_smul (x : M \u2297[R] N) : (1 : R') \u2022 x = x :=\n  have : \u2200 (r : R') (m : M) (n : N), r \u2022 m \u2297\u209c[R] n = (r \u2022 m) \u2297\u209c n := fun _ _ _ => rfl\n  x.induction_on (by rw [TensorProduct.smul_zero])\n    (fun m n => by rw [this, one_smul])\n    fun x y ihx ihy => by rw [TensorProduct.smul_add, ihx, ihy]\n#align tensor_product.one_smul TensorProduct.one_smul\n\nprotected theorem add_smul (r s : R'') (x : M \u2297[R] N) : (r + s) \u2022 x = r \u2022 x + s \u2022 x :=\n  have : \u2200 (r : R'') (m : M) (n : N), r \u2022 m \u2297\u209c[R] n = (r \u2022 m) \u2297\u209c n := fun _ _ _ => rfl\n  x.induction_on (by simp_rw [TensorProduct.smul_zero, add_zero])\n    (fun m n => by simp_rw [this, add_smul, add_tmul]) fun x y ihx ihy => by\n    simp_rw [TensorProduct.smul_add]\n    rw [ihx, ihy, add_add_add_comm]\n#align tensor_product.add_smul TensorProduct.add_smul\n\ninstance addMonoid : AddMonoid (M \u2297[R] N) :=\n  { TensorProduct.addZeroClass _ _ with\n    toAddSemigroup := TensorProduct.addSemigroup _ _\n    toZero := (TensorProduct.addZeroClass _ _).toZero\n    nsmul := fun n v => n \u2022 v\n    nsmul_zero := by simp [TensorProduct.zero_smul]\n    nsmul_succ := by simp only [TensorProduct.one_smul, TensorProduct.add_smul, add_comm,\n      forall_const] }\n\ninstance addCommMonoid : AddCommMonoid (M \u2297[R] N) :=\n  { TensorProduct.addCommSemigroup _ _ with\n    toAddMonoid := TensorProduct.addMonoid }\n\ninstance leftDistribMulAction : DistribMulAction R' (M \u2297[R] N) :=\n  have : \u2200 (r : R') (m : M) (n : N), r \u2022 m \u2297\u209c[R] n = (r \u2022 m) \u2297\u209c n := fun _ _ _ => rfl\n  { smul_add := fun r x y => TensorProduct.smul_add r x y\n    mul_smul := fun r s x =>\n      x.induction_on (by simp_rw [TensorProduct.smul_zero])\n        (fun m n => by simp_rw [this, mul_smul]) fun x y ihx ihy => by\n        simp_rw [TensorProduct.smul_add]\n        rw [ihx, ihy]\n    one_smul := TensorProduct.one_smul\n    smul_zero := TensorProduct.smul_zero }\n#align tensor_product.left_distrib_mul_action TensorProduct.leftDistribMulAction\n\ninstance : DistribMulAction R (M \u2297[R] N) :=\n  TensorProduct.leftDistribMulAction\n\ntheorem smul_tmul' (r : R') (m : M) (n : N) : r \u2022 m \u2297\u209c[R] n = (r \u2022 m) \u2297\u209c n :=\n  rfl\n#align tensor_product.smul_tmul' TensorProduct.smul_tmul'\n\n@[simp]\ntheorem tmul_smul [DistribMulAction R' N] [CompatibleSMul R R' M N] (r : R') (x : M) (y : N) :\n    x \u2297\u209c (r \u2022 y) = r \u2022 x \u2297\u209c[R] y :=\n  (smul_tmul _ _ _).symm\n#align tensor_product.tmul_smul TensorProduct.tmul_smul\n\ntheorem smul_tmul_smul (r s : R) (m : M) (n : N) : (r \u2022 m) \u2297\u209c[R] (s \u2022 n) = (r * s) \u2022 m \u2297\u209c[R] n := by\n  simp_rw [smul_tmul, tmul_smul, mul_smul]\n#align tensor_product.smul_tmul_smul TensorProduct.smul_tmul_smul\n\ninstance leftModule : Module R'' (M \u2297[R] N) :=\n  { add_smul := TensorProduct.add_smul\n    zero_smul := TensorProduct.zero_smul }\n#align tensor_product.left_module TensorProduct.leftModule\n\ninstance : Module R (M \u2297[R] N) :=\n  TensorProduct.leftModule\n\ninstance [Module R''\u1d50\u1d52\u1d56 M] [IsCentralScalar R'' M] : IsCentralScalar R'' (M \u2297[R] N) where\n  op_smul_eq_smul r x :=\n    x.induction_on (by rw [smul_zero, smul_zero])\n      (fun x y => by rw [smul_tmul', smul_tmul', op_smul_eq_smul]) fun x y hx hy => by\n      rw [smul_add, smul_add, hx, hy]\n\nsection\n\n-- Like `R'`, `R'\u2082` provides a `DistribMulAction R'\u2082 (M \u2297[R] N)`\nvariable {R'\u2082 : Type*} [Monoid R'\u2082] [DistribMulAction R'\u2082 M]\nvariable [SMulCommClass R R'\u2082 M]\n\n/-- `SMulCommClass R' R'\u2082 M` implies `SMulCommClass R' R'\u2082 (M \u2297[R] N)` -/\ninstance smulCommClass_left [SMulCommClass R' R'\u2082 M] : SMulCommClass R' R'\u2082 (M \u2297[R] N) where\n  smul_comm r' r'\u2082 x :=\n    TensorProduct.induction_on x (by simp_rw [TensorProduct.smul_zero])\n      (fun m n => by simp_rw [smul_tmul', smul_comm]) fun x y ihx ihy => by\n      simp_rw [TensorProduct.smul_add]; rw [ihx, ihy]\n#align tensor_product.smul_comm_class_left TensorProduct.smulCommClass_left\n\nvariable [SMul R'\u2082 R']\n\n/-- `IsScalarTower R'\u2082 R' M` implies `IsScalarTower R'\u2082 R' (M \u2297[R] N)` -/\ninstance isScalarTower_left [IsScalarTower R'\u2082 R' M] : IsScalarTower R'\u2082 R' (M \u2297[R] N) :=\n  \u27e8fun s r x =>\n    x.induction_on (by simp)\n      (fun m n => by rw [smul_tmul', smul_tmul', smul_tmul', smul_assoc]) fun x y ihx ihy => by\n      rw [smul_add, smul_add, smul_add, ihx, ihy]\u27e9\n#align tensor_product.is_scalar_tower_left TensorProduct.isScalarTower_left\n\nvariable [DistribMulAction R'\u2082 N] [DistribMulAction R' N]\nvariable [CompatibleSMul R R'\u2082 M N] [CompatibleSMul R R' M N]\n\n/-- `IsScalarTower R'\u2082 R' N` implies `IsScalarTower R'\u2082 R' (M \u2297[R] N)` -/\ninstance isScalarTower_right [IsScalarTower R'\u2082 R' N] : IsScalarTower R'\u2082 R' (M \u2297[R] N) :=\n  \u27e8fun s r x =>\n    x.induction_on (by simp)\n      (fun m n => by rw [\u2190 tmul_smul, \u2190 tmul_smul, \u2190 tmul_smul, smul_assoc]) fun x y ihx ihy => by\n      rw [smul_add, smul_add, smul_add, ihx, ihy]\u27e9\n#align tensor_product.is_scalar_tower_right TensorProduct.isScalarTower_right\n\nend\n\n/-- A short-cut instance for the common case, where the requirements for the `compatible_smul`\ninstances are sufficient. -/\ninstance isScalarTower [SMul R' R] [IsScalarTower R' R M] : IsScalarTower R' R (M \u2297[R] N) :=\n  TensorProduct.isScalarTower_left\n#align tensor_product.is_scalar_tower TensorProduct.isScalarTower\n\n-- or right\nvariable (R M N)\n\n/-- The canonical bilinear map `M \u2192 N \u2192 M \u2297[R] N`. -/\ndef mk : M \u2192\u2097[R] N \u2192\u2097[R] M \u2297[R] N :=\n  LinearMap.mk\u2082 R (\u00b7 \u2297\u209c \u00b7) add_tmul (fun c m n => by simp_rw [smul_tmul, tmul_smul])\n    tmul_add tmul_smul\n#align tensor_product.mk TensorProduct.mk\n\nvariable {R M N}\n\n@[simp]\ntheorem mk_apply (m : M) (n : N) : mk R M N m n = m \u2297\u209c n :=\n  rfl\n#align tensor_product.mk_apply TensorProduct.mk_apply\n\ntheorem ite_tmul (x\u2081 : M) (x\u2082 : N) (P : Prop) [Decidable P] :\n    (if P then x\u2081 else 0) \u2297\u209c[R] x\u2082 = if P then x\u2081 \u2297\u209c x\u2082 else 0 := by split_ifs <;> simp\n#align tensor_product.ite_tmul TensorProduct.ite_tmul\n\ntheorem tmul_ite (x\u2081 : M) (x\u2082 : N) (P : Prop) [Decidable P] :\n    (x\u2081 \u2297\u209c[R] if P then x\u2082 else 0) = if P then x\u2081 \u2297\u209c x\u2082 else 0 := by split_ifs <;> simp\n#align tensor_product.tmul_ite TensorProduct.tmul_ite\n\nsection\n\nopen BigOperators\n\ntheorem sum_tmul {\u03b1 : Type*} (s : Finset \u03b1) (m : \u03b1 \u2192 M) (n : N) :\n    (\u2211 a in s, m a) \u2297\u209c[R] n = \u2211 a in s, m a \u2297\u209c[R] n := by\n  classical\n    induction' s using Finset.induction with a s has ih h\n    \u00b7 simp\n    \u00b7 simp [Finset.sum_insert has, add_tmul, ih]\n#align tensor_product.sum_tmul TensorProduct.sum_tmul\n\ntheorem tmul_sum (m : M) {\u03b1 : Type*} (s : Finset \u03b1) (n : \u03b1 \u2192 N) :\n    (m \u2297\u209c[R] \u2211 a in s, n a) = \u2211 a in s, m \u2297\u209c[R] n a := by\n  classical\n    induction' s using Finset.induction with a s has ih h\n    \u00b7 simp\n    \u00b7 simp [Finset.sum_insert has, tmul_add, ih]\n#align tensor_product.tmul_sum TensorProduct.tmul_sum\n\nend\n\nvariable (R M N)\n\n/-- The simple (aka pure) elements span the tensor product. -/\ntheorem span_tmul_eq_top : Submodule.span R { t : M \u2297[R] N | \u2203 m n, m \u2297\u209c n = t } = \u22a4 := by\n  ext t; simp only [Submodule.mem_top, iff_true_iff]\n  refine t.induction_on ?_ ?_ ?_\n  \u00b7 exact Submodule.zero_mem _\n  \u00b7 intro m n\n    apply Submodule.subset_span\n    use m, n\n  \u00b7 intro t\u2081 t\u2082 ht\u2081 ht\u2082\n    exact Submodule.add_mem _ ht\u2081 ht\u2082\n#align tensor_product.span_tmul_eq_top TensorProduct.span_tmul_eq_top\n\n@[simp]\ntheorem map\u2082_mk_top_top_eq_top : Submodule.map\u2082 (mk R M N) \u22a4 \u22a4 = \u22a4 := by\n  rw [\u2190 top_le_iff, \u2190 span_tmul_eq_top, Submodule.map\u2082_eq_span_image2]\n  exact Submodule.span_mono fun _ \u27e8m, n, h\u27e9 => \u27e8m, trivial, n, trivial, h\u27e9\n#align tensor_product.map\u2082_mk_top_top_eq_top TensorProduct.map\u2082_mk_top_top_eq_top\n\ntheorem exists_eq_tmul_of_forall (x : TensorProduct R M N)\n    (h : \u2200 (m\u2081 m\u2082 : M) (n\u2081 n\u2082 : N), \u2203 m n, m\u2081 \u2297\u209c n\u2081 + m\u2082 \u2297\u209c n\u2082 = m \u2297\u209c[R] n) :\n    \u2203 m n, x = m \u2297\u209c n := by\n  induction x using TensorProduct.induction_on with\n  | zero =>\n    use 0, 0\n    rw [TensorProduct.zero_tmul]\n  | tmul m n => use m, n\n  | add x y h\u2081 h\u2082 =>\n    obtain \u27e8m\u2081, n\u2081, rfl\u27e9 := h\u2081\n    obtain \u27e8m\u2082, n\u2082, rfl\u27e9 := h\u2082\n    apply h\n\nend Module\n\nsection UMP\n\nvariable {M N}\nvariable (f : M \u2192\u2097[R] N \u2192\u2097[R] P)\n\n/-- Auxiliary function to constructing a linear map `M \u2297 N \u2192 P` given a bilinear map `M \u2192 N \u2192 P`\nwith the property that its composition with the canonical bilinear map `M \u2192 N \u2192 M \u2297 N` is\nthe given bilinear map `M \u2192 N \u2192 P`. -/\ndef liftAux : M \u2297[R] N \u2192+ P :=\n  liftAddHom (LinearMap.toAddMonoidHom'.comp <| f.toAddMonoidHom)\n    fun r m n => by dsimp; rw [LinearMap.map_smul\u2082, map_smul]\n#align tensor_product.lift_aux TensorProduct.liftAux\n\ntheorem liftAux_tmul (m n) : liftAux f (m \u2297\u209c n) = f m n :=\n  rfl\n#align tensor_product.lift_aux_tmul TensorProduct.liftAux_tmul\n\nvariable {f}\n\n@[simp]\ntheorem liftAux.smul (r : R) (x) : liftAux f (r \u2022 x) = r \u2022 liftAux f x :=\n  TensorProduct.induction_on x (smul_zero _).symm\n    (fun p q => by simp_rw [\u2190 tmul_smul, liftAux_tmul, (f p).map_smul])\n    fun p q ih1 ih2 => by simp_rw [smul_add, (liftAux f).map_add, ih1, ih2, smul_add]\n#align tensor_product.lift_aux.smul TensorProduct.liftAux.smul\n\nvariable (f)\n\n/-- Constructing a linear map `M \u2297 N \u2192 P` given a bilinear map `M \u2192 N \u2192 P` with the property that\nits composition with the canonical bilinear map `M \u2192 N \u2192 M \u2297 N` is\nthe given bilinear map `M \u2192 N \u2192 P`. -/\ndef lift : M \u2297[R] N \u2192\u2097[R] P :=\n  { liftAux f with map_smul' := liftAux.smul }\n#align tensor_product.lift TensorProduct.lift\n\nvariable {f}\n\n@[simp]\ntheorem lift.tmul (x y) : lift f (x \u2297\u209c y) = f x y :=\n  rfl\n#align tensor_product.lift.tmul TensorProduct.lift.tmul\n\n@[simp]\ntheorem lift.tmul' (x y) : (lift f).1 (x \u2297\u209c y) = f x y :=\n  rfl\n#align tensor_product.lift.tmul' TensorProduct.lift.tmul'\n\ntheorem ext' {g h : M \u2297[R] N \u2192\u2097[R] P} (H : \u2200 x y, g (x \u2297\u209c y) = h (x \u2297\u209c y)) : g = h :=\n  LinearMap.ext fun z =>\n    TensorProduct.induction_on z (by simp_rw [LinearMap.map_zero]) H fun x y ihx ihy => by\n      rw [g.map_add, h.map_add, ihx, ihy]\n#align tensor_product.ext' TensorProduct.ext'\n\ntheorem lift.unique {g : M \u2297[R] N \u2192\u2097[R] P} (H : \u2200 x y, g (x \u2297\u209c y) = f x y) : g = lift f :=\n  ext' fun m n => by rw [H, lift.tmul]\n#align tensor_product.lift.unique TensorProduct.lift.unique\n\ntheorem lift_mk : lift (mk R M N) = LinearMap.id :=\n  Eq.symm <| lift.unique fun _ _ => rfl\n#align tensor_product.lift_mk TensorProduct.lift_mk\n\ntheorem lift_compr\u2082 (g : P \u2192\u2097[R] Q) : lift (f.compr\u2082 g) = g.comp (lift f) :=\n  Eq.symm <| lift.unique fun _ _ => by simp\n#align tensor_product.lift_compr\u2082 TensorProduct.lift_compr\u2082\n\ntheorem lift_mk_compr\u2082 (f : M \u2297 N \u2192\u2097[R] P) : lift ((mk R M N).compr\u2082 f) = f := by\n  rw [lift_compr\u2082 f, lift_mk, LinearMap.comp_id]\n#align tensor_product.lift_mk_compr\u2082 TensorProduct.lift_mk_compr\u2082\n\n/-- This used to be an `@[ext]` lemma, but it fails very slowly when the `ext` tactic tries to apply\nit in some cases, notably when one wants to show equality of two linear maps. The `@[ext]`\nattribute is now added locally where it is needed. Using this as the `@[ext]` lemma instead of\n`TensorProduct.ext'` allows `ext` to apply lemmas specific to `M \u2192\u2097 _` and `N \u2192\u2097 _`.\n\nSee note [partially-applied ext lemmas]. -/\ntheorem ext {g h : M \u2297 N \u2192\u2097[R] P} (H : (mk R M N).compr\u2082 g = (mk R M N).compr\u2082 h) : g = h := by\n  rw [\u2190 lift_mk_compr\u2082 g, H, lift_mk_compr\u2082]\n#align tensor_product.ext TensorProduct.ext\n\nattribute [local ext high] ext\n\nexample : M \u2192 N \u2192 (M \u2192 N \u2192 P) \u2192 P := fun m => flip fun f => f m\n\nvariable (R M N P)\n\n/-- Linearly constructing a linear map `M \u2297 N \u2192 P` given a bilinear map `M \u2192 N \u2192 P`\nwith the property that its composition with the canonical bilinear map `M \u2192 N \u2192 M \u2297 N` is\nthe given bilinear map `M \u2192 N \u2192 P`. -/\ndef uncurry : (M \u2192\u2097[R] N \u2192\u2097[R] P) \u2192\u2097[R] M \u2297[R] N \u2192\u2097[R] P :=\n  LinearMap.flip <| lift <| LinearMap.lflip.comp (LinearMap.flip LinearMap.id)\n#align tensor_product.uncurry TensorProduct.uncurry\n\nvariable {R M N P}\n\n@[simp]\ntheorem uncurry_apply (f : M \u2192\u2097[R] N \u2192\u2097[R] P) (m : M) (n : N) :\n    uncurry R M N P f (m \u2297\u209c n) = f m n := by rw [uncurry, LinearMap.flip_apply, lift.tmul]; rfl\n#align tensor_product.uncurry_apply TensorProduct.uncurry_apply\n\nvariable (R M N P)\n\n/-- A linear equivalence constructing a linear map `M \u2297 N \u2192 P` given a bilinear map `M \u2192 N \u2192 P`\nwith the property that its composition with the canonical bilinear map `M \u2192 N \u2192 M \u2297 N` is\nthe given bilinear map `M \u2192 N \u2192 P`. -/\ndef lift.equiv : (M \u2192\u2097[R] N \u2192\u2097[R] P) \u2243\u2097[R] M \u2297[R] N \u2192\u2097[R] P :=\n  { uncurry R M N P with\n    invFun := fun f => (mk R M N).compr\u2082 f\n    left_inv := fun _ => LinearMap.ext\u2082 fun _ _ => lift.tmul _ _\n    right_inv := fun _ => ext' fun _ _ => lift.tmul _ _ }\n#align tensor_product.lift.equiv TensorProduct.lift.equiv\n\n@[simp]\ntheorem lift.equiv_apply (f : M \u2192\u2097[R] N \u2192\u2097[R] P) (m : M) (n : N) :\n    lift.equiv R M N P f (m \u2297\u209c n) = f m n :=\n  uncurry_apply f m n\n#align tensor_product.lift.equiv_apply TensorProduct.lift.equiv_apply\n\n@[simp]\ntheorem lift.equiv_symm_apply (f : M \u2297[R] N \u2192\u2097[R] P) (m : M) (n : N) :\n    (lift.equiv R M N P).symm f m n = f (m \u2297\u209c n) :=\n  rfl\n#align tensor_product.lift.equiv_symm_apply TensorProduct.lift.equiv_symm_apply\n\n/-- Given a linear map `M \u2297 N \u2192 P`, compose it with the canonical bilinear map `M \u2192 N \u2192 M \u2297 N` to\nform a bilinear map `M \u2192 N \u2192 P`. -/\ndef lcurry : (M \u2297[R] N \u2192\u2097[R] P) \u2192\u2097[R] M \u2192\u2097[R] N \u2192\u2097[R] P :=\n  (lift.equiv R M N P).symm\n#align tensor_product.lcurry TensorProduct.lcurry\n\nvariable {R M N P}\n\n@[simp]\ntheorem lcurry_apply (f : M \u2297[R] N \u2192\u2097[R] P) (m : M) (n : N) : lcurry R M N P f m n = f (m \u2297\u209c n) :=\n  rfl\n#align tensor_product.lcurry_apply TensorProduct.lcurry_apply\n\n/-- Given a linear map `M \u2297 N \u2192 P`, compose it with the canonical bilinear map `M \u2192 N \u2192 M \u2297 N` to\nform a bilinear map `M \u2192 N \u2192 P`. -/\ndef curry (f : M \u2297[R] N \u2192\u2097[R] P) : M \u2192\u2097[R] N \u2192\u2097[R] P :=\n  lcurry R M N P f\n#align tensor_product.curry TensorProduct.curry\n\n@[simp]\ntheorem curry_apply (f : M \u2297 N \u2192\u2097[R] P) (m : M) (n : N) : curry f m n = f (m \u2297\u209c n) :=\n  rfl\n#align tensor_product.curry_apply TensorProduct.curry_apply\n\ntheorem curry_injective : Function.Injective (curry : (M \u2297[R] N \u2192\u2097[R] P) \u2192 M \u2192\u2097[R] N \u2192\u2097[R] P) :=\n  fun _ _ H => ext H\n#align tensor_product.curry_injective TensorProduct.curry_injective\n\ntheorem ext_threefold {g h : (M \u2297[R] N) \u2297[R] P \u2192\u2097[R] Q}\n    (H : \u2200 x y z, g (x \u2297\u209c y \u2297\u209c z) = h (x \u2297\u209c y \u2297\u209c z)) : g = h := by\n  ext x y z\n  exact H x y z\n#align tensor_product.ext_threefold TensorProduct.ext_threefold\n\n-- We'll need this one for checking the pentagon identity!\ntheorem ext_fourfold {g h : ((M \u2297[R] N) \u2297[R] P) \u2297[R] Q \u2192\u2097[R] S}\n    (H : \u2200 w x y z, g (w \u2297\u209c x \u2297\u209c y \u2297\u209c z) = h (w \u2297\u209c x \u2297\u209c y \u2297\u209c z)) : g = h := by\n  ext w x y z\n  exact H w x y z\n#align tensor_product.ext_fourfold TensorProduct.ext_fourfold\n\n/-- Two linear maps (M \u2297 N) \u2297 (P \u2297 Q) \u2192 S which agree on all elements of the\nform (m \u2297\u209c n) \u2297\u209c (p \u2297\u209c q) are equal. -/\ntheorem ext_fourfold' {\u03c6 \u03c8 : (M \u2297[R] N) \u2297[R] P \u2297[R] Q \u2192\u2097[R] S}\n    (H : \u2200 w x y z, \u03c6 (w \u2297\u209c x \u2297\u209c (y \u2297\u209c z)) = \u03c8 (w \u2297\u209c x \u2297\u209c (y \u2297\u209c z))) : \u03c6 = \u03c8 := by\n  ext m n p q\n  exact H m n p q\n#align tensor_product.ext_fourfold' TensorProduct.ext_fourfold'\n\nend UMP\n\nvariable {M N}\n\nsection\n\nvariable (R M)\n\n/-- The base ring is a left identity for the tensor product of modules, up to linear equivalence.\n-/\nprotected def lid : R \u2297[R] M \u2243\u2097[R] M :=\n  LinearEquiv.ofLinear (lift <| LinearMap.lsmul R M) (mk R R M 1) (LinearMap.ext fun _ => by simp)\n    (ext' fun r m => by simp; rw [\u2190 tmul_smul, \u2190 smul_tmul, smul_eq_mul, mul_one])\n#align tensor_product.lid TensorProduct.lid\n\nend\n\n@[simp]\ntheorem lid_tmul (m : M) (r : R) : (TensorProduct.lid R M : R \u2297 M \u2192 M) (r \u2297\u209c m) = r \u2022 m :=\n  rfl\n#align tensor_product.lid_tmul TensorProduct.lid_tmul\n\n@[simp]\ntheorem lid_symm_apply (m : M) : (TensorProduct.lid R M).symm m = 1 \u2297\u209c m :=\n  rfl\n#align tensor_product.lid_symm_apply TensorProduct.lid_symm_apply\n\nsection\n\nvariable (R M N)\n\n/-- The tensor product of modules is commutative, up to linear equivalence.\n-/\nprotected def comm : M \u2297[R] N \u2243\u2097[R] N \u2297[R] M :=\n  LinearEquiv.ofLinear (lift (mk R N M).flip) (lift (mk R M N).flip) (ext' fun _ _ => rfl)\n    (ext' fun _ _ => rfl)\n#align tensor_product.comm TensorProduct.comm\n\n@[simp]\ntheorem comm_tmul (m : M) (n : N) : (TensorProduct.comm R M N) (m \u2297\u209c n) = n \u2297\u209c m :=\n  rfl\n#align tensor_product.comm_tmul TensorProduct.comm_tmul\n\n@[simp]\ntheorem comm_symm_tmul (m : M) (n : N) : (TensorProduct.comm R M N).symm (n \u2297\u209c m) = m \u2297\u209c n :=\n  rfl\n#align tensor_product.comm_symm_tmul TensorProduct.comm_symm_tmul\n\nlemma lift_comp_comm_eq  (f : M \u2192\u2097[R] N \u2192\u2097[R] P) :\n    lift f \u2218\u2097 TensorProduct.comm R N M = lift f.flip :=\n  ext rfl\nend\n\nsection\n\nvariable (R M)\n\n/-- The base ring is a right identity for the tensor product of modules, up to linear equivalence.\n-/\nprotected def rid : M \u2297[R] R \u2243\u2097[R] M :=\n  LinearEquiv.trans (TensorProduct.comm R M R) (TensorProduct.lid R M)\n#align tensor_product.rid TensorProduct.rid\n\nend\n\n@[simp]\ntheorem rid_tmul (m : M) (r : R) : (TensorProduct.rid R M) (m \u2297\u209c r) = r \u2022 m :=\n  rfl\n#align tensor_product.rid_tmul TensorProduct.rid_tmul\n\n@[simp]\ntheorem rid_symm_apply (m : M) : (TensorProduct.rid R M).symm m = m \u2297\u209c 1 :=\n  rfl\n#align tensor_product.rid_symm_apply TensorProduct.rid_symm_apply\n\nvariable (R) in\ntheorem lid_eq_rid : TensorProduct.lid R R = TensorProduct.rid R R :=\n  LinearEquiv.toLinearMap_injective <| ext' mul_comm\n\nopen LinearMap\n\nsection\n\nvariable (R M N P)\n\n/-- The associator for tensor product of R-modules, as a linear equivalence. -/\nprotected def assoc : (M \u2297[R] N) \u2297[R] P \u2243\u2097[R] M \u2297[R] N \u2297[R] P := by\n  refine\n      LinearEquiv.ofLinear (lift <| lift <| comp (lcurry R _ _ _) <| mk _ _ _)\n        (lift <| comp (uncurry R _ _ _) <| curry <| mk _ _ _)\n        (ext <| LinearMap.ext fun m => ext' fun n p => ?_)\n        (ext <| flip_inj <| LinearMap.ext fun p => ext' fun m n => ?_) <;>\n    repeat'\n      first\n        |rw [lift.tmul]|rw [compr\u2082_apply]|rw [comp_apply]|rw [mk_apply]|rw [flip_apply]\n        |rw [lcurry_apply]|rw [uncurry_apply]|rw [curry_apply]|rw [id_apply]\n#align tensor_product.assoc TensorProduct.assoc\n\nend\n\n@[simp]\ntheorem assoc_tmul (m : M) (n : N) (p : P) :\n    (TensorProduct.assoc R M N P) (m \u2297\u209c n \u2297\u209c p) = m \u2297\u209c (n \u2297\u209c p) :=\n  rfl\n#align tensor_product.assoc_tmul TensorProduct.assoc_tmul\n\n@[simp]\ntheorem assoc_symm_tmul (m : M) (n : N) (p : P) :\n    (TensorProduct.assoc R M N P).symm (m \u2297\u209c (n \u2297\u209c p)) = m \u2297\u209c n \u2297\u209c p :=\n  rfl\n#align tensor_product.assoc_symm_tmul TensorProduct.assoc_symm_tmul\n\n/-- The tensor product of a pair of linear maps between modules. -/\ndef map (f : M \u2192\u2097[R] P) (g : N \u2192\u2097[R] Q) : M \u2297[R] N \u2192\u2097[R] P \u2297[R] Q :=\n  lift <| comp (compl\u2082 (mk _ _ _) g) f\n#align tensor_product.map TensorProduct.map\n\n@[simp]\ntheorem map_tmul (f : M \u2192\u2097[R] P) (g : N \u2192\u2097[R] Q) (m : M) (n : N) : map f g (m \u2297\u209c n) = f m \u2297\u209c g n :=\n  rfl\n#align tensor_product.map_tmul TensorProduct.map_tmul\n\n/-- Given linear maps `f : M \u2192 P`, `g : N \u2192 Q`, if we identify `M \u2297 N` with `N \u2297 M` and `P \u2297 Q`\nwith `Q \u2297 P`, then this lemma states that `f \u2297 g = g \u2297 f`. -/\nlemma map_comp_comm_eq (f : M \u2192\u2097[R] P) (g : N \u2192\u2097[R] Q) :\n    map f g \u2218\u2097 TensorProduct.comm R N M = TensorProduct.comm R Q P \u2218\u2097 map g f :=\n  ext rfl\n\nlemma map_comm (f : M \u2192\u2097[R] P) (g : N \u2192\u2097[R] Q) (x : N \u2297[R] M):\n    map f g (TensorProduct.comm R N M x) = TensorProduct.comm R Q P (map g f x) :=\n  DFunLike.congr_fun (map_comp_comm_eq _ _) _\n\n/-- Given linear maps `f : M \u2192 Q`, `g : N \u2192 S`, and `h : P \u2192 T`, if we identify `(M \u2297 N) \u2297 P`\nwith `M \u2297 (N \u2297 P)` and `(Q \u2297 S) \u2297 T` with `Q \u2297 (S \u2297 T)`, then this lemma states that\n`f \u2297 (g \u2297 h) = (f \u2297 g) \u2297 h`. -/\nlemma map_map_comp_assoc_eq (f : M \u2192\u2097[R] Q) (g : N \u2192\u2097[R] S) (h : P \u2192\u2097[R] T) :\n    map f (map g h) \u2218\u2097 TensorProduct.assoc R M N P =\n      TensorProduct.assoc R Q S T \u2218\u2097 map (map f g) h :=\n  ext <| ext <| LinearMap.ext fun _ => LinearMap.ext fun _ => LinearMap.ext fun _ => rfl\n\nlemma map_map_assoc (f : M \u2192\u2097[R] Q) (g : N \u2192\u2097[R] S) (h : P \u2192\u2097[R] T) (x : (M \u2297[R] N) \u2297[R] P) :\n    map f (map g h) (TensorProduct.assoc R M N P x) =\n      TensorProduct.assoc R Q S T (map (map f g) h x) :=\n  DFunLike.congr_fun (map_map_comp_assoc_eq _ _ _) _\n\n/-- Given linear maps `f : M \u2192 Q`, `g : N \u2192 S`, and `h : P \u2192 T`, if we identify `M \u2297 (N \u2297 P)`\nwith `(M \u2297 N) \u2297 P` and `Q \u2297 (S \u2297 T)` with `(Q \u2297 S) \u2297 T`, then this lemma states that\n`(f \u2297 g) \u2297 h = f \u2297 (g \u2297 h)`. -/\nlemma map_map_comp_assoc_symm_eq (f : M \u2192\u2097[R] Q) (g : N \u2192\u2097[R] S) (h : P \u2192\u2097[R] T) :\n    map (map f g) h \u2218\u2097 (TensorProduct.assoc R M N P).symm =\n      (TensorProduct.assoc R Q S T).symm \u2218\u2097 map f (map g h) :=\n  ext <| LinearMap.ext fun _ => ext <| LinearMap.ext fun _ => LinearMap.ext fun _ => rfl\n\nlemma map_map_assoc_symm (f : M \u2192\u2097[R] Q) (g : N \u2192\u2097[R] S) (h : P \u2192\u2097[R] T) (x : M \u2297[R] (N \u2297[R] P)) :\n    map (map f g) h ((TensorProduct.assoc R M N P).symm x) =\n      (TensorProduct.assoc R Q S T).symm (map f (map g h) x) :=\n  DFunLike.congr_fun (map_map_comp_assoc_symm_eq _ _ _) _\n\ntheorem map_range_eq_span_tmul (f : M \u2192\u2097[R] P) (g : N \u2192\u2097[R] Q) :\n    range (map f g) = Submodule.span R { t | \u2203 m n, f m \u2297\u209c g n = t } := by\n  simp only [\u2190 Submodule.map_top, \u2190 span_tmul_eq_top, Submodule.map_span, Set.mem_image,\n    Set.mem_setOf_eq]\n  congr; ext t\n  constructor\n  \u00b7 rintro \u27e8_, \u27e8\u27e8m, n, rfl\u27e9, rfl\u27e9\u27e9\n    use m, n\n    simp only [map_tmul]\n  \u00b7 rintro \u27e8m, n, rfl\u27e9\n    refine \u27e8_, \u27e8\u27e8m, n, rfl\u27e9, ?_\u27e9\u27e9\n    simp only [map_tmul]\n#align tensor_product.map_range_eq_span_tmul TensorProduct.map_range_eq_span_tmul\n\n/-- Given submodules `p \u2286 P` and `q \u2286 Q`, this is the natural map: `p \u2297 q \u2192 P \u2297 Q`. -/\n@[simp]\ndef mapIncl (p : Submodule R P) (q : Submodule R Q) : p \u2297[R] q \u2192\u2097[R] P \u2297[R] Q :=\n  map p.subtype q.subtype\n#align tensor_product.map_incl TensorProduct.mapIncl\n\nlemma range_mapIncl (p : Submodule R P) (q : Submodule R Q) :\n    LinearMap.range (mapIncl p q) = Submodule.span R (Set.image2 (\u00b7 \u2297\u209c \u00b7) p q) := by\n  rw [mapIncl, map_range_eq_span_tmul]\n  congr; ext; simp\n\nsection\n\nvariable {P' Q' : Type*}\nvariable [AddCommMonoid P'] [Module R P']\nvariable [AddCommMonoid Q'] [Module R Q']\n\ntheorem map_comp (f\u2082 : P \u2192\u2097[R] P') (f\u2081 : M \u2192\u2097[R] P) (g\u2082 : Q \u2192\u2097[R] Q') (g\u2081 : N \u2192\u2097[R] Q) :\n    map (f\u2082.comp f\u2081) (g\u2082.comp g\u2081) = (map f\u2082 g\u2082).comp (map f\u2081 g\u2081) :=\n  ext' fun _ _ => rfl\n#align tensor_product.map_comp TensorProduct.map_comp\n\nlemma range_mapIncl_mono {p p' : Submodule R P} {q q' : Submodule R Q} (hp : p \u2264 p') (hq : q \u2264 q') :\n    LinearMap.range (mapIncl p q) \u2264 LinearMap.range (mapIncl p' q') := by\n  simp_rw [range_mapIncl]\n  exact Submodule.span_mono (Set.image2_subset hp hq)\n\ntheorem lift_comp_map (i : P \u2192\u2097[R] Q \u2192\u2097[R] Q') (f : M \u2192\u2097[R] P) (g : N \u2192\u2097[R] Q) :\n    (lift i).comp (map f g) = lift ((i.comp f).compl\u2082 g) :=\n  ext' fun _ _ => rfl\n#align tensor_product.lift_comp_map TensorProduct.lift_comp_map\n\nattribute [local ext high] ext\n\n@[simp]\ntheorem map_id : map (id : M \u2192\u2097[R] M) (id : N \u2192\u2097[R] N) = .id := by\n  ext\n  simp only [mk_apply, id_coe, compr\u2082_apply, _root_.id, map_tmul]\n#align tensor_product.map_id TensorProduct.map_id\n\n@[simp]\ntheorem map_one : map (1 : M \u2192\u2097[R] M) (1 : N \u2192\u2097[R] N) = 1 :=\n  map_id\n#align tensor_product.map_one TensorProduct.map_one\n\ntheorem map_mul (f\u2081 f\u2082 : M \u2192\u2097[R] M) (g\u2081 g\u2082 : N \u2192\u2097[R] N) :\n    map (f\u2081 * f\u2082) (g\u2081 * g\u2082) = map f\u2081 g\u2081 * map f\u2082 g\u2082 :=\n  map_comp f\u2081 f\u2082 g\u2081 g\u2082\n#align tensor_product.map_mul TensorProduct.map_mul\n\n@[simp]\nprotected theorem map_pow (f : M \u2192\u2097[R] M) (g : N \u2192\u2097[R] N) (n : \u2115) :\n    map f g ^ n = map (f ^ n) (g ^ n) := by\n  induction' n with n ih\n  \u00b7 simp only [Nat.zero_eq, pow_zero, map_one]\n  \u00b7 simp only [pow_succ', ih, map_mul]\n#align tensor_product.map_pow TensorProduct.map_pow\n\ntheorem map_add_left (f\u2081 f\u2082 : M \u2192\u2097[R] P) (g : N \u2192\u2097[R] Q) :\n    map (f\u2081 + f\u2082) g = map f\u2081 g + map f\u2082 g := by\n  ext\n  simp only [add_tmul, compr\u2082_apply, mk_apply, map_tmul, add_apply]\n#align tensor_product.map_add_left TensorProduct.map_add_left\n\ntheorem map_add_right (f : M \u2192\u2097[R] P) (g\u2081 g\u2082 : N \u2192\u2097[R] Q) :\n    map f (g\u2081 + g\u2082) = map f g\u2081 + map f g\u2082 := by\n  ext\n  simp only [tmul_add, compr\u2082_apply, mk_apply, map_tmul, add_apply]\n#align tensor_product.map_add_right TensorProduct.map_add_right\n\ntheorem map_smul_left (r : R) (f : M \u2192\u2097[R] P) (g : N \u2192\u2097[R] Q) : map (r \u2022 f) g = r \u2022 map f g := by\n  ext\n  simp only [smul_tmul, compr\u2082_apply, mk_apply, map_tmul, smul_apply, tmul_smul]\n#align tensor_product.map_smul_left TensorProduct.map_smul_left\n\ntheorem map_smul_right (r : R) (f : M \u2192\u2097[R] P) (g : N \u2192\u2097[R] Q) : map f (r \u2022 g) = r \u2022 map f g := by\n  ext\n  simp only [smul_tmul, compr\u2082_apply, mk_apply, map_tmul, smul_apply, tmul_smul]\n#align tensor_product.map_smul_right TensorProduct.map_smul_right\n\nvariable (R M N P Q)\n\n/-- The tensor product of a pair of linear maps between modules, bilinear in both maps. -/\ndef mapBilinear : (M \u2192\u2097[R] P) \u2192\u2097[R] (N \u2192\u2097[R] Q) \u2192\u2097[R] M \u2297[R] N \u2192\u2097[R] P \u2297[R] Q :=\n  LinearMap.mk\u2082 R map map_add_left map_smul_left map_add_right map_smul_right\n#align tensor_product.map_bilinear TensorProduct.mapBilinear\n\n/-- The canonical linear map from `P \u2297[R] (M \u2192\u2097[R] Q)` to `(M \u2192\u2097[R] P \u2297[R] Q)` -/\ndef lTensorHomToHomLTensor : P \u2297[R] (M \u2192\u2097[R] Q) \u2192\u2097[R] M \u2192\u2097[R] P \u2297[R] Q :=\n  TensorProduct.lift (llcomp R M Q _ \u2218\u2097 mk R P Q)\n#align tensor_product.ltensor_hom_to_hom_ltensor TensorProduct.lTensorHomToHomLTensor\n\n/-- The canonical linear map from `(M \u2192\u2097[R] P) \u2297[R] Q` to `(M \u2192\u2097[R] P \u2297[R] Q)` -/\ndef rTensorHomToHomRTensor : (M \u2192\u2097[R] P) \u2297[R] Q \u2192\u2097[R] M \u2192\u2097[R] P \u2297[R] Q :=\n  TensorProduct.lift (llcomp R M P _ \u2218\u2097 (mk R P Q).flip).flip\n#align tensor_product.rtensor_hom_to_hom_rtensor TensorProduct.rTensorHomToHomRTensor\n\n/-- The linear map from `(M \u2192\u2097 P) \u2297 (N \u2192\u2097 Q)` to `(M \u2297 N \u2192\u2097 P \u2297 Q)` sending `f \u2297\u209c g` to\nthe `TensorProduct.map f g`, the tensor product of the two maps. -/\ndef homTensorHomMap : (M \u2192\u2097[R] P) \u2297[R] (N \u2192\u2097[R] Q) \u2192\u2097[R] M \u2297[R] N \u2192\u2097[R] P \u2297[R] Q :=\n  lift (mapBilinear R M N P Q)\n#align tensor_product.hom_tensor_hom_map TensorProduct.homTensorHomMap\n\nvariable {R M N P Q}\n\n/--\nThis is a binary version of `TensorProduct.map`: Given a bilinear map `f : M \u27f6 P \u27f6 Q` and a\nbilinear map `g : N \u27f6 S \u27f6 T`, if we think `f` and `g` as linear maps with two inputs, then\n`map\u2082 f g` is a bilinear map taking two inputs `M \u2297 N \u2192 P \u2297 S \u2192 Q \u2297 S` defined by\n`map\u2082 f g (m \u2297 n) (p \u2297 s) = f m p \u2297 g n s`.\n\nMathematically, `TensorProduct.map\u2082` is defined as the composition\n`M \u2297 N -map\u2192 Hom(P, Q) \u2297 Hom(S, T) -homTensorHomMap\u2192 Hom(P \u2297 S, Q \u2297 T)`.\n-/\ndef map\u2082 (f : M \u2192\u2097[R] P \u2192\u2097[R] Q) (g : N \u2192\u2097[R] S \u2192\u2097[R] T) :\n    M \u2297[R] N \u2192\u2097[R] P \u2297[R] S \u2192\u2097[R] Q \u2297[R] T :=\n  homTensorHomMap R _ _ _ _ \u2218\u2097 map f g\n\n@[simp]\ntheorem mapBilinear_apply (f : M \u2192\u2097[R] P) (g : N \u2192\u2097[R] Q) : mapBilinear R M N P Q f g = map f g :=\n  rfl\n#align tensor_product.map_bilinear_apply TensorProduct.mapBilinear_apply\n\n@[simp]\ntheorem lTensorHomToHomLTensor_apply (p : P) (f : M \u2192\u2097[R] Q) (m : M) :\n    lTensorHomToHomLTensor R M P Q (p \u2297\u209c f) m = p \u2297\u209c f m :=\n  rfl\n#align tensor_product.ltensor_hom_to_hom_ltensor_apply TensorProduct.lTensorHomToHomLTensor_apply\n\n@[simp]\ntheorem rTensorHomToHomRTensor_apply (f : M \u2192\u2097[R] P) (q : Q) (m : M) :\n    rTensorHomToHomRTensor R M P Q (f \u2297\u209c q) m = f m \u2297\u209c q :=\n  rfl\n#align tensor_product.rtensor_hom_to_hom_rtensor_apply TensorProduct.rTensorHomToHomRTensor_apply\n\n@[simp]\ntheorem homTensorHomMap_apply (f : M \u2192\u2097[R] P) (g : N \u2192\u2097[R] Q) :\n    homTensorHomMap R M N P Q (f \u2297\u209c g) = map f g :=\n  rfl\n#align tensor_product.hom_tensor_hom_map_apply TensorProduct.homTensorHomMap_apply\n\n@[simp]\ntheorem map\u2082_apply_tmul (f : M \u2192\u2097[R] P \u2192\u2097[R] Q) (g : N \u2192\u2097[R] S \u2192\u2097[R] T) (m : M) (n : N) :\n    map\u2082 f g (m \u2297\u209c n) = map (f m) (g n) := rfl\n\n@[simp]\ntheorem map_zero_left (g : N \u2192\u2097[R] Q) : map (0 : M \u2192\u2097[R] P) g = 0 :=\n  (mapBilinear R M N P Q).map_zero\u2082 _\n\n@[simp]\ntheorem map_zero_right (f : M \u2192\u2097[R] P) : map f (0 : N \u2192\u2097[R] Q) = 0 :=\n  (mapBilinear R M N P Q _).map_zero\n\nend\n\n/-- If `M` and `P` are linearly equivalent and `N` and `Q` are linearly equivalent\nthen `M \u2297 N` and `P \u2297 Q` are linearly equivalent. -/\ndef congr (f : M \u2243\u2097[R] P) (g : N \u2243\u2097[R] Q) : M \u2297[R] N \u2243\u2097[R] P \u2297[R] Q :=\n  LinearEquiv.ofLinear (map f g) (map f.symm g.symm)\n    (ext' fun m n => by simp)\n    (ext' fun m n => by simp)\n#align tensor_product.congr TensorProduct.congr\n\n@[simp]\ntheorem congr_tmul (f : M \u2243\u2097[R] P) (g : N \u2243\u2097[R] Q) (m : M) (n : N) :\n    congr f g (m \u2297\u209c n) = f m \u2297\u209c g n :=\n  rfl\n#align tensor_product.congr_tmul TensorProduct.congr_tmul\n\n@[simp]\ntheorem congr_symm_tmul (f : M \u2243\u2097[R] P) (g : N \u2243\u2097[R] Q) (p : P) (q : Q) :\n    (congr f g).symm (p \u2297\u209c q) = f.symm p \u2297\u209c g.symm q :=\n  rfl\n#align tensor_product.congr_symm_tmul TensorProduct.congr_symm_tmul\n\nvariable (R M N P Q)\n\n/-- A tensor product analogue of `mul_left_comm`. -/\ndef leftComm : M \u2297[R] N \u2297[R] P \u2243\u2097[R] N \u2297[R] M \u2297[R] P :=\n  let e\u2081 := (TensorProduct.assoc R M N P).symm\n  let e\u2082 := congr (TensorProduct.comm R M N) (1 : P \u2243\u2097[R] P)\n  let e\u2083 := TensorProduct.assoc R N M P\n  e\u2081 \u226a\u226b\u2097 (e\u2082 \u226a\u226b\u2097 e\u2083)\n#align tensor_product.left_comm TensorProduct.leftComm\n\nvariable {M N P Q}\n\n@[simp]\ntheorem leftComm_tmul (m : M) (n : N) (p : P) : leftComm R M N P (m \u2297\u209c (n \u2297\u209c p)) = n \u2297\u209c (m \u2297\u209c p) :=\n  rfl\n#align tensor_product.left_comm_tmul TensorProduct.leftComm_tmul\n\n@[simp]\ntheorem leftComm_symm_tmul (m : M) (n : N) (p : P) :\n    (leftComm R M N P).symm (n \u2297\u209c (m \u2297\u209c p)) = m \u2297\u209c (n \u2297\u209c p) :=\n  rfl\n#align tensor_product.left_comm_symm_tmul TensorProduct.leftComm_symm_tmul\n\nvariable (M N P Q)\n\n/-- This special case is worth defining explicitly since it is useful for defining multiplication\non tensor products of modules carrying multiplications (e.g., associative rings, Lie rings, ...).\n\nE.g., suppose `M = P` and `N = Q` and that `M` and `N` carry bilinear multiplications:\n`M \u2297 M \u2192 M` and `N \u2297 N \u2192 N`. Using `map`, we can define `(M \u2297 M) \u2297 (N \u2297 N) \u2192 M \u2297 N` which, when\ncombined with this definition, yields a bilinear multiplication on `M \u2297 N`:\n`(M \u2297 N) \u2297 (M \u2297 N) \u2192 M \u2297 N`. In particular we could use this to define the multiplication in\nthe `TensorProduct.semiring` instance (currently defined \"by hand\" using `TensorProduct.mul`).\n\nSee also `mul_mul_mul_comm`. -/\ndef tensorTensorTensorComm : (M \u2297[R] N) \u2297[R] P \u2297[R] Q \u2243\u2097[R] (M \u2297[R] P) \u2297[R] N \u2297[R] Q :=\n  let e\u2081 := TensorProduct.assoc R M N (P \u2297[R] Q)\n  let e\u2082 := congr (1 : M \u2243\u2097[R] M) (leftComm R N P Q)\n  let e\u2083 := (TensorProduct.assoc R M P (N \u2297[R] Q)).symm\n  e\u2081 \u226a\u226b\u2097 (e\u2082 \u226a\u226b\u2097 e\u2083)\n#align tensor_product.tensor_tensor_tensor_comm TensorProduct.tensorTensorTensorComm\n\nvariable {M N P Q}\n\n@[simp]\ntheorem tensorTensorTensorComm_tmul (m : M) (n : N) (p : P) (q : Q) :\n    tensorTensorTensorComm R M N P Q (m \u2297\u209c n \u2297\u209c (p \u2297\u209c q)) = m \u2297\u209c p \u2297\u209c (n \u2297\u209c q) :=\n  rfl\n#align tensor_product.tensor_tensor_tensor_comm_tmul TensorProduct.tensorTensorTensorComm_tmul\n\n-- Porting note: the proof here was `rfl` but that caused a timeout.\n@[simp]\ntheorem tensorTensorTensorComm_symm :\n    (tensorTensorTensorComm R M N P Q).symm = tensorTensorTensorComm R M P N Q :=\n  by ext; rfl\n#align tensor_product.tensor_tensor_tensor_comm_symm TensorProduct.tensorTensorTensorComm_symm\n\nvariable (M N P Q)\n\n/-- This special case is useful for describing the interplay between `dualTensorHomEquiv` and\ncomposition of linear maps.\n\nE.g., composition of linear maps gives a map `(M \u2192 N) \u2297 (N \u2192 P) \u2192 (M \u2192 P)`, and applying\n`dual_tensor_hom_equiv.symm` to the three hom-modules gives a map\n`(M.dual \u2297 N) \u2297 (N.dual \u2297 P) \u2192 (M.dual \u2297 P)`, which agrees with the application of `contractRight`\non `N \u2297 N.dual` after the suitable rebracketting.\n-/\ndef tensorTensorTensorAssoc : (M \u2297[R] N) \u2297[R] P \u2297[R] Q \u2243\u2097[R] (M \u2297[R] N \u2297[R] P) \u2297[R] Q :=\n  (TensorProduct.assoc R (M \u2297[R] N) P Q).symm \u226a\u226b\u2097\n    congr (TensorProduct.assoc R M N P) (1 : Q \u2243\u2097[R] Q)\n#align tensor_product.tensor_tensor_tensor_assoc TensorProduct.tensorTensorTensorAssoc\n\nvariable {M N P Q}\n\n@[simp]\ntheorem tensorTensorTensorAssoc_tmul (m : M) (n : N) (p : P) (q : Q) :\n    tensorTensorTensorAssoc R M N P Q (m \u2297\u209c n \u2297\u209c (p \u2297\u209c q)) = m \u2297\u209c (n \u2297\u209c p) \u2297\u209c q :=\n  rfl\n#align tensor_product.tensor_tensor_tensor_assoc_tmul TensorProduct.tensorTensorTensorAssoc_tmul\n\n@[simp]\ntheorem tensorTensorTensorAssoc_symm_tmul (m : M) (n : N) (p : P) (q : Q) :\n    (tensorTensorTensorAssoc R M N P Q).symm (m \u2297\u209c (n \u2297\u209c p) \u2297\u209c q) = m \u2297\u209c n \u2297\u209c (p \u2297\u209c q) :=\n  rfl\n#align tensor_product.tensor_tensor_tensor_assoc_symm_tmul TensorProduct.tensorTensorTensorAssoc_symm_tmul\n\nend TensorProduct\n\nopen scoped TensorProduct\nnamespace LinearMap\n\nvariable {N}\n\n/-- `lTensor M f : M \u2297 N \u2192\u2097 M \u2297 P` is the natural linear map induced by `f : N \u2192\u2097 P`. -/\ndef lTensor (f : N \u2192\u2097[R] P) : M \u2297[R] N \u2192\u2097[R] M \u2297[R] P :=\n  TensorProduct.map id f\n#align linear_map.ltensor LinearMap.lTensor\n\n/-- `rTensor f M : N\u2081 \u2297 M \u2192\u2097 N\u2082 \u2297 M` is the natural linear map induced by `f : N\u2081 \u2192\u2097 N\u2082`. -/\ndef rTensor (f : N \u2192\u2097[R] P) : N \u2297[R] M \u2192\u2097[R] P \u2297[R] M :=\n  TensorProduct.map f id\n#align linear_map.rtensor LinearMap.rTensor\n\nvariable (g : P \u2192\u2097[R] Q) (f : N \u2192\u2097[R] P)\n\n@[simp]\ntheorem lTensor_tmul (m : M) (n : N) : f.lTensor M (m \u2297\u209c n) = m \u2297\u209c f n :=\n  rfl\n#align linear_map.ltensor_tmul LinearMap.lTensor_tmul\n\n@[simp]\ntheorem rTensor_tmul (m : M) (n : N) : f.rTensor M (n \u2297\u209c m) = f n \u2297\u209c m :=\n  rfl\n#align linear_map.rtensor_tmul LinearMap.rTensor_tmul\n\n@[simp]\ntheorem lTensor_comp_mk (m : M) :\n    f.lTensor M \u2218\u2097 TensorProduct.mk R M N m = TensorProduct.mk R M P m \u2218\u2097 f :=\n  rfl\n\n@[simp]\ntheorem rTensor_comp_flip_mk (m : M) :\n    f.rTensor M \u2218\u2097 (TensorProduct.mk R N M).flip m = (TensorProduct.mk R P M).flip m \u2218\u2097 f :=\n  rfl\n\nlemma comm_comp_rTensor_comp_comm_eq (g : N \u2192\u2097[R] P) :\n    TensorProduct.comm R P Q \u2218\u2097 rTensor Q g \u2218\u2097 TensorProduct.comm R Q N =\n      lTensor Q g :=\n  TensorProduct.ext rfl\n\nlemma comm_comp_lTensor_comp_comm_eq (g : N \u2192\u2097[R] P) :\n    TensorProduct.comm R Q P \u2218\u2097 lTensor Q g \u2218\u2097 TensorProduct.comm R N Q =\n      rTensor Q g :=\n  TensorProduct.ext rfl\n\n/-- Given a linear map `f : N \u2192 P`, `f \u2297 M` is injective if and only if `M \u2297 f` is injective. -/\ntheorem lTensor_inj_iff_rTensor_inj :\n    Function.Injective (lTensor M f) \u2194 Function.Injective (rTensor M f) := by\n  simp [\u2190 comm_comp_rTensor_comp_comm_eq]\n\n/-- Given a linear map `f : N \u2192 P`, `f \u2297 M` is surjective if and only if `M \u2297 f` is surjective. -/\ntheorem lTensor_surj_iff_rTensor_surj :\n    Function.Surjective (lTensor M f) \u2194 Function.Surjective (rTensor M f) := by\n  simp [\u2190 comm_comp_rTensor_comp_comm_eq]\n\n", "theoremStatement": "/-- Given a linear map `f : N \u2192 P`, `f \u2297 M` is bijective if and only if `M \u2297 f` is bijective. -/\ntheorem lTensor_bij_iff_rTensor_bij :\n    Function.Bijective (lTensor M f) \u2194 Function.Bijective (rTensor M f) ", "theoremName": "LinearMap.lTensor_bij_iff_rTensor_bij", "fileCreated": {"commit": "0ee5f8231fc294d4ee59712faf6b20ecd0d2c6bf", "date": "2024-03-11"}, "theoremCreated": {"commit": "6bd93cd5e5b9163e2ce7f04a5c6cc816a302b580", "date": "2024-03-29"}, "file": "mathlib/Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "module": "Mathlib.LinearAlgebra.TensorProduct.Basic", "jsonFile": 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"Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", 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+{"srcContext": "/-\nCopyright (c) 2024 Jeremy Tan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Tan\n-/\nimport Mathlib.Combinatorics.SimpleGraph.Clique\n\n/-!\n# The Tur\u00e1n graph\n\nThis file defines the Tur\u00e1n graph and proves some of its basic properties.\n\n## Main declarations\n\n* `SimpleGraph.IsTuranMaximal`: `G.IsTuranMaximal r` means that `G` has the most number of edges for\n  its number of vertices while still being `r + 1`-cliquefree.\n* `SimpleGraph.turanGraph n r`: The canonical `r + 1`-cliquefree Tur\u00e1n graph on `n` vertices.\n\n## TODO\n\n* Port the rest of Tur\u00e1n's theorem from https://github.com/leanprover-community/mathlib4/pull/9317\n-/\n\nopen Finset\n\nnamespace SimpleGraph\nvariable {V : Type*} [Fintype V] [DecidableEq V] (G H : SimpleGraph V) [DecidableRel G.Adj]\n  {n r : \u2115}\n\n/-- An `r + 1`-cliquefree graph is `r`-Tur\u00e1n-maximal if any other `r + 1`-cliquefree graph on\nthe same vertex set has the same or fewer number of edges. -/\ndef IsTuranMaximal (r : \u2115) : Prop :=\n  G.CliqueFree (r + 1) \u2227 \u2200 (H : SimpleGraph V) [DecidableRel H.Adj],\n    H.CliqueFree (r + 1) \u2192 H.edgeFinset.card \u2264 G.edgeFinset.card\n\nvariable {G H}\n\nlemma IsTuranMaximal.le_iff_eq (hG : G.IsTuranMaximal r) (hH : H.CliqueFree (r + 1)) :\n    G \u2264 H \u2194 G = H := by\n  classical exact \u27e8fun hGH \u21a6 edgeFinset_inj.1 <| eq_of_subset_of_card_le\n    (edgeFinset_subset_edgeFinset.2 hGH) (hG.2 _ hH), le_of_eq\u27e9\n\n/-- The canonical `r + 1`-cliquefree Tur\u00e1n graph on `n` vertices. -/\ndef turanGraph (n r : \u2115) : SimpleGraph (Fin n) where Adj v w := v % r \u2260 w % r\n\ninstance turanGraph.instDecidableRelAdj : DecidableRel (turanGraph n r).Adj := by\n  dsimp only [turanGraph]; infer_instance\n\n@[simp]\nlemma turanGraph_zero : turanGraph n 0 = \u22a4 := by\n  ext a b; simp_rw [turanGraph, top_adj, Nat.mod_zero, not_iff_not, Fin.val_inj]\n\n", "theoremStatement": "@[simp]\ntheorem 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"Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Combinatorics.SimpleGraph.Init", "Std.WF", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Tactic.Lemma", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Tactic.TypeStar", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Data.Subtype", "Mathlib.Order.Notation", "Mathlib.Data.Set.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Cases", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Basic", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Tactic.PushNeg", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Tactic.CasesM", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.ByContra", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Data.Set.Prod", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.MinMax", "Mathlib.Tactic.Says", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Util.AssertExists", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Logic.Pairwise", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Data.Rel", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Nat.Defs", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", 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"Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Data.Sym.Sym2", "Mathlib.Combinatorics.SimpleGraph.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Sym.Card", "Mathlib.Combinatorics.SimpleGraph.Finite", "Mathlib.Combinatorics.SimpleGraph.Dart", "Mathlib.Data.FunLike.Fintype", "Mathlib.Combinatorics.SimpleGraph.Maps", "Mathlib.Combinatorics.SimpleGraph.Subgraph", "Mathlib.Combinatorics.SimpleGraph.Connectivity", "Mathlib.Combinatorics.SimpleGraph.Operations", "Mathlib.Data.Finset.Pairwise", "Mathlib.Combinatorics.SimpleGraph.Clique"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  simp_rw [SimpleGraph.ext_iff, Function.funext_iff, turanGraph, top_adj, eq_iff_iff, not_iff_not]\n  refine' \u27e8fun h \u21a6 _, _\u27e9\n  \u00b7 contrapose! h\n    use \u27e80, (Nat.pos_of_ne_zero h.1).trans h.2\u27e9, \u27e8r, h.2\u27e9\n    simp [h.1.symm]\n  \u00b7 rintro (rfl | h) a b\n    \u00b7 simp [Fin.val_inj]\n    \u00b7 rw [Nat.mod_eq_of_lt (a.2.trans_le h), Nat.mod_eq_of_lt (b.2.trans_le h), Fin.val_inj]", "proofType": "tactic", "proofLengthLines": 8, "proofLengthTokens": 368}}
+{"srcContext": "/-\nCopyright (c) 2024 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.CategoryTheory.Elements\nimport Mathlib.CategoryTheory.Limits.Types\nimport Mathlib.CategoryTheory.Limits.Creates\nimport Mathlib.CategoryTheory.Limits.Preserves.Limits\n\n/-!\n# Limits in the category of elements\n\nWe show that if `C` has limits of shape `I` and `A : C \u2964 Type w` preserves limits of shape `I`, then\nthe category of elements of `A` has limits of shape `I` and the forgetful functor\n`\u03c0 : A.Elements \u2964 C` creates them.\n\n-/\n\nuniverse w v\u2081 v u\u2081 u\n\nnamespace CategoryTheory\n\nopen Limits Opposite\n\nvariable {C : Type u} [Category.{v} C]\n\nnamespace CategoryOfElements\n\nvariable {A : C \u2964 Type w} {I : Type u\u2081} [Category.{v\u2081} I] [Small.{w} I] [HasLimitsOfShape I C]\n  [PreservesLimitsOfShape I A]\n\nnamespace CreatesLimitsAux\n\nvariable (F : I \u2964 A.Elements)\n\n/-- (implementation) A system `(Fi, fi)_i` of elements induces an element in `lim_i A(Fi)`. -/\nnoncomputable def liftedConeElement' : limit ((F \u22d9 \u03c0 A) \u22d9 A) :=\n  Types.Limit.mk _ (fun i => (F.obj i).2) (by simp)\n\n@[simp]\nlemma \u03c0_liftedConeElement' (i : I) :\n    limit.\u03c0 ((F \u22d9 \u03c0 A) \u22d9 A) i (liftedConeElement' F) = (F.obj i).2 :=\n  Types.Limit.\u03c0_mk _ _ _ _\n\n/-- (implementation) A system `(Fi, fi)_i` of elements induces an element in `A(lim_i Fi)`. -/\nnoncomputable def liftedConeElement : A.obj (limit (F \u22d9 \u03c0 A)) :=\n  (preservesLimitIso A (F \u22d9 \u03c0 A)).inv (liftedConeElement' F)\n\n@[simp]\nlemma map_lift_mapCone (c : Cone F) :\n    A.map (limit.lift (F \u22d9 \u03c0 A) ((\u03c0 A).mapCone c)) c.pt.snd = liftedConeElement F := by\n  apply (preservesLimitIso A (F \u22d9 \u03c0 A)).toEquiv.injective\n  ext i\n  have h\u2081 := congrFun (preservesLimitsIso_hom_\u03c0 A (F \u22d9 \u03c0 A) i)\n    (A.map (limit.lift (F \u22d9 \u03c0 A) ((\u03c0 A).mapCone c)) c.pt.snd)\n  have h\u2082 := (c.\u03c0.app i).property\n  simp_all [\u2190 FunctorToTypes.map_comp_apply, liftedConeElement]\n\n", "theoremStatement": "@[simp]\nlemma map_\u03c0_liftedConeElement (i : I) :\n    A.map (limit.\u03c0 (F \u22d9 \u03c0 A) i) (liftedConeElement F) = (F.obj i).snd ", "theoremName": "CategoryTheory.CategoryOfElements.CreatesLimitsAux.map_\u03c0_liftedConeElement", "fileCreated": {"commit": "dda2b3f9d6c525a42d4ff328a37036d5ced89f44", "date": "2024-03-26"}, "theoremCreated": {"commit": "dda2b3f9d6c525a42d4ff328a37036d5ced89f44", "date": "2024-03-26"}, "file": "mathlib/Mathlib/CategoryTheory/Limits/Elements.lean", "module": "Mathlib.CategoryTheory.Limits.Elements", "jsonFile": "Mathlib.CategoryTheory.Limits.Elements.jsonl", "positionMetadata": {"lineInFile": 60, "tokenPositionInFile": 1926, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, 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"Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Opposite", "Mathlib.Tactic.Cases", "Mathlib.Combinatorics.Quiver.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.Convert", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Spread", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.AssertExists", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.Data.Prod.Basic", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Control.ULift", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Order.Notation", "Mathlib.Data.Set.Defs", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.MinMax", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Elements", "Mathlib.Data.TypeMax", "Mathlib.CategoryTheory.Limits.Shapes.Equalizers", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Shapes.Images", "Mathlib.CategoryTheory.Limits.Types", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Creates", "Mathlib.CategoryTheory.Limits.Preserves.Limits"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have := congrFun\n    (preservesLimitsIso_inv_\u03c0 A (F \u22d9 \u03c0 A) i) (liftedConeElement' F)\n  simp_all [liftedConeElement]", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 123}}
+{"srcContext": "/-\nCopyright (c) 2020 Hanting Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Hanting Zhang, Johan Commelin\n-/\nimport Mathlib.Algebra.Algebra.Subalgebra.Basic\nimport Mathlib.Algebra.MvPolynomial.Rename\nimport Mathlib.Algebra.MvPolynomial.CommRing\n\n#align_import ring_theory.mv_polynomial.symmetric from \"leanprover-community/mathlib\"@\"2f5b500a507264de86d666a5f87ddb976e2d8de4\"\n\n/-!\n# Symmetric Polynomials and Elementary Symmetric Polynomials\n\nThis file defines symmetric `MvPolynomial`s and elementary symmetric `MvPolynomial`s.\nWe also prove some basic facts about them.\n\n## Main declarations\n\n* `MvPolynomial.IsSymmetric`\n\n* `MvPolynomial.symmetricSubalgebra`\n\n* `MvPolynomial.esymm`\n\n* `MvPolynomial.psum`\n\n## Notation\n\n+ `esymm \u03c3 R n` is the `n`th elementary symmetric polynomial in `MvPolynomial \u03c3 R`.\n\n+ `psum \u03c3 R n` is the degree-`n` power sum in `MvPolynomial \u03c3 R`, i.e. the sum of monomials\n  `(X i)^n` over `i \u2208 \u03c3`.\n\nAs in other polynomial files, we typically use the notation:\n\n+ `\u03c3 \u03c4 : Type*` (indexing the variables)\n\n+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)\n\n+ `r : R` elements of the coefficient ring\n\n+ `i : \u03c3`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians\n\n+ `\u03c6 \u03c8 : MvPolynomial \u03c3 R`\n\n-/\n\n\nopen Equiv (Perm)\n\nopen BigOperators\n\nnoncomputable section\n\nnamespace Multiset\n\nvariable {R : Type*} [CommSemiring R]\n\n/-- The `n`th elementary symmetric function evaluated at the elements of `s` -/\ndef esymm (s : Multiset R) (n : \u2115) : R :=\n  ((s.powersetCard n).map Multiset.prod).sum\n#align multiset.esymm Multiset.esymm\n\ntheorem _root_.Finset.esymm_map_val {\u03c3} (f : \u03c3 \u2192 R) (s : Finset \u03c3) (n : \u2115) :\n    (s.val.map f).esymm n = (s.powersetCard n).sum fun t => t.prod f := by\n  simp only [esymm, powersetCard_map, \u2190 Finset.map_val_val_powersetCard, map_map]\n  rfl\n#align finset.esymm_map_val Finset.esymm_map_val\n\nend Multiset\n\nnamespace MvPolynomial\n\nvariable {\u03c3 : Type*} {R : Type*}\nvariable {\u03c4 : Type*} {S : Type*}\n\n/-- A `MvPolynomial \u03c6` is symmetric if it is invariant under\npermutations of its variables by the `rename` operation -/\ndef IsSymmetric [CommSemiring R] (\u03c6 : MvPolynomial \u03c3 R) : Prop :=\n  \u2200 e : Perm \u03c3, rename e \u03c6 = \u03c6\n#align mv_polynomial.is_symmetric MvPolynomial.IsSymmetric\n\nvariable (\u03c3 R)\n\n/-- The subalgebra of symmetric `MvPolynomial`s. -/\ndef symmetricSubalgebra [CommSemiring R] : Subalgebra R (MvPolynomial \u03c3 R) where\n  carrier := setOf IsSymmetric\n  algebraMap_mem' r e := rename_C e r\n  mul_mem' ha hb e := by rw [AlgHom.map_mul, ha, hb]\n  add_mem' ha hb e := by rw [AlgHom.map_add, ha, hb]\n#align mv_polynomial.symmetric_subalgebra MvPolynomial.symmetricSubalgebra\n\nvariable {\u03c3 R}\n\n@[simp]\ntheorem mem_symmetricSubalgebra [CommSemiring R] (p : MvPolynomial \u03c3 R) :\n    p \u2208 symmetricSubalgebra \u03c3 R \u2194 p.IsSymmetric :=\n  Iff.rfl\n#align mv_polynomial.mem_symmetric_subalgebra MvPolynomial.mem_symmetricSubalgebra\n\nnamespace IsSymmetric\n\nsection CommSemiring\n\nvariable [CommSemiring R] [CommSemiring S] {\u03c6 \u03c8 : MvPolynomial \u03c3 R}\n\n@[simp]\ntheorem C (r : R) : IsSymmetric (C r : MvPolynomial \u03c3 R) :=\n  (symmetricSubalgebra \u03c3 R).algebraMap_mem r\nset_option linter.uppercaseLean3 false in\n#align mv_polynomial.is_symmetric.C MvPolynomial.IsSymmetric.C\n\n@[simp]\ntheorem zero : IsSymmetric (0 : MvPolynomial \u03c3 R) :=\n  (symmetricSubalgebra \u03c3 R).zero_mem\n#align mv_polynomial.is_symmetric.zero MvPolynomial.IsSymmetric.zero\n\n@[simp]\ntheorem one : IsSymmetric (1 : MvPolynomial \u03c3 R) :=\n  (symmetricSubalgebra \u03c3 R).one_mem\n#align mv_polynomial.is_symmetric.one MvPolynomial.IsSymmetric.one\n\ntheorem add (h\u03c6 : IsSymmetric \u03c6) (h\u03c8 : IsSymmetric \u03c8) : IsSymmetric (\u03c6 + \u03c8) :=\n  (symmetricSubalgebra \u03c3 R).add_mem h\u03c6 h\u03c8\n#align mv_polynomial.is_symmetric.add MvPolynomial.IsSymmetric.add\n\ntheorem mul (h\u03c6 : IsSymmetric \u03c6) (h\u03c8 : IsSymmetric \u03c8) : IsSymmetric (\u03c6 * \u03c8) :=\n  (symmetricSubalgebra \u03c3 R).mul_mem h\u03c6 h\u03c8\n#align mv_polynomial.is_symmetric.mul MvPolynomial.IsSymmetric.mul\n\ntheorem smul (r : R) (h\u03c6 : IsSymmetric \u03c6) : IsSymmetric (r \u2022 \u03c6) :=\n  (symmetricSubalgebra \u03c3 R).smul_mem h\u03c6 r\n#align mv_polynomial.is_symmetric.smul MvPolynomial.IsSymmetric.smul\n\n@[simp]\ntheorem map (h\u03c6 : IsSymmetric \u03c6) (f : R \u2192+* S) : IsSymmetric (map f \u03c6) := fun e => by\n  rw [\u2190 map_rename, h\u03c6]\n#align mv_polynomial.is_symmetric.map MvPolynomial.IsSymmetric.map\n\nprotected theorem rename (h\u03c6 : \u03c6.IsSymmetric) (e : \u03c3 \u2243 \u03c4) : (rename e \u03c6).IsSymmetric := fun _ => by\n  apply rename_injective _ e.symm.injective\n  simp_rw [rename_rename, \u2190 Equiv.coe_trans, Equiv.self_trans_symm, Equiv.coe_refl, rename_id]\n  rw [h\u03c6]\n\n", "theoremStatement": "@[simp]\ntheorem _root_.MvPolynomial.isSymmetric_rename {e : \u03c3 \u2243 \u03c4} :\n    (MvPolynomial.rename e \u03c6).IsSymmetric \u2194 \u03c6.IsSymmetric ", "theoremName": "MvPolynomial.isSymmetric_rename", "fileCreated": {"commit": "0989d4a8d7b4f7b95d9190f34d6a57be9587c522", "date": "2023-03-19"}, "theoremCreated": {"commit": "11e323efe514844587305802203328ef3898037c", "date": "2024-03-24"}, "file": "mathlib/Mathlib/RingTheory/MvPolynomial/Symmetric.lean", "module": "Mathlib.RingTheory.MvPolynomial.Symmetric", "jsonFile": "Mathlib.RingTheory.MvPolynomial.Symmetric.jsonl", "positionMetadata": {"lineInFile": 146, "tokenPositionInFile": 4623, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": 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"Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Set.UnionLift", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Tactic.FinCases", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Regular.Pow", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n  \u27e8fun h => by simpa using (IsSymmetric.rename (R := R) h e.symm), (IsSymmetric.rename \u00b7 e)\u27e9", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 95}}
+{"srcContext": "/-\nCopyright (c) 2022 Ya\u00ebl Dillies, Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ya\u00ebl Dillies, Bhavik Mehta\n-/\nimport Mathlib.Combinatorics.SimpleGraph.Connectivity\nimport Mathlib.Combinatorics.SimpleGraph.Operations\nimport Mathlib.Data.Finset.Pairwise\n\n#align_import combinatorics.simple_graph.clique from \"leanprover-community/mathlib\"@\"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe\"\n\n/-!\n# Graph cliques\n\nThis file defines cliques in simple graphs. A clique is a set of vertices that are pairwise\nadjacent.\n\n## Main declarations\n\n* `SimpleGraph.IsClique`: Predicate for a set of vertices to be a clique.\n* `SimpleGraph.IsNClique`: Predicate for a set of vertices to be an `n`-clique.\n* `SimpleGraph.cliqueFinset`: Finset of `n`-cliques of a graph.\n* `SimpleGraph.CliqueFree`: Predicate for a graph to have no `n`-cliques.\n\n## TODO\n\n* Clique numbers\n* Dualise all the API to get independent sets\n-/\n\n\nopen Finset Fintype Function SimpleGraph.Walk\n\nnamespace SimpleGraph\n\nvariable {\u03b1 \u03b2 : Type*} (G H : SimpleGraph \u03b1)\n\n/-! ### Cliques -/\n\n\nsection Clique\n\nvariable {s t : Set \u03b1}\n\n/-- A clique in a graph is a set of vertices that are pairwise adjacent. -/\nabbrev IsClique (s : Set \u03b1) : Prop :=\n  s.Pairwise G.Adj\n#align simple_graph.is_clique SimpleGraph.IsClique\n\ntheorem isClique_iff : G.IsClique s \u2194 s.Pairwise G.Adj :=\n  Iff.rfl\n#align simple_graph.is_clique_iff SimpleGraph.isClique_iff\n\n/-- A clique is a set of vertices whose induced graph is complete. -/\ntheorem isClique_iff_induce_eq : G.IsClique s \u2194 G.induce s = \u22a4 := by\n  rw [isClique_iff]\n  constructor\n  \u00b7 intro h\n    ext \u27e8v, hv\u27e9 \u27e8w, hw\u27e9\n    simp only [comap_adj, Subtype.coe_mk, top_adj, Ne, Subtype.mk_eq_mk]\n    exact \u27e8Adj.ne, h hv hw\u27e9\n  \u00b7 intro h v hv w hw hne\n    have h2 : (G.induce s).Adj \u27e8v, hv\u27e9 \u27e8w, hw\u27e9 = _ := rfl\n    conv_lhs at h2 => rw [h]\n    simp only [top_adj, ne_eq, Subtype.mk.injEq, eq_iff_iff] at h2\n    exact h2.1 hne\n#align simple_graph.is_clique_iff_induce_eq SimpleGraph.isClique_iff_induce_eq\n\ninstance [DecidableEq \u03b1] [DecidableRel G.Adj] {s : Finset \u03b1} : Decidable (G.IsClique s) :=\n  decidable_of_iff' _ G.isClique_iff\n\nvariable {G H} {a b : \u03b1}\n\nlemma isClique_empty : G.IsClique \u2205 := by simp\n#align simple_graph.is_clique_empty SimpleGraph.isClique_empty\n\nlemma isClique_singleton (a : \u03b1) : G.IsClique {a} := by simp\n#align simple_graph.is_clique_singleton SimpleGraph.isClique_singleton\n\nlemma isClique_pair : G.IsClique {a, b} \u2194 a \u2260 b \u2192 G.Adj a b := Set.pairwise_pair_of_symmetric G.symm\n#align simple_graph.is_clique_pair SimpleGraph.isClique_pair\n\n@[simp]\nlemma isClique_insert : G.IsClique (insert a s) \u2194 G.IsClique s \u2227 \u2200 b \u2208 s, a \u2260 b \u2192 G.Adj a b :=\n  Set.pairwise_insert_of_symmetric G.symm\n#align simple_graph.is_clique_insert SimpleGraph.isClique_insert\n\nlemma isClique_insert_of_not_mem (ha : a \u2209 s) :\n    G.IsClique (insert a s) \u2194 G.IsClique s \u2227 \u2200 b \u2208 s, G.Adj a b :=\n  Set.pairwise_insert_of_symmetric_of_not_mem G.symm ha\n#align simple_graph.is_clique_insert_of_not_mem SimpleGraph.isClique_insert_of_not_mem\n\nlemma IsClique.insert (hs : G.IsClique s) (h : \u2200 b \u2208 s, a \u2260 b \u2192 G.Adj a b) :\n    G.IsClique (insert a s) := hs.insert_of_symmetric G.symm h\n#align simple_graph.is_clique.insert SimpleGraph.IsClique.insert\n\ntheorem IsClique.mono (h : G \u2264 H) : G.IsClique s \u2192 H.IsClique s := Set.Pairwise.mono' h\n#align simple_graph.is_clique.mono SimpleGraph.IsClique.mono\n\ntheorem IsClique.subset (h : t \u2286 s) : G.IsClique s \u2192 G.IsClique t := Set.Pairwise.mono h\n#align simple_graph.is_clique.subset SimpleGraph.IsClique.subset\n\nprotected theorem IsClique.map {s : Set \u03b1} (h : G.IsClique s) {f : \u03b1 \u21aa \u03b2} :\n    (G.map f).IsClique (f '' s) := by\n  rintro _ \u27e8a, ha, rfl\u27e9 _ \u27e8b, hb, rfl\u27e9 hab\n  exact \u27e8a, b, h ha hb <| ne_of_apply_ne _ hab, rfl, rfl\u27e9\n#align simple_graph.is_clique.map SimpleGraph.IsClique.map\n\n@[simp]\ntheorem isClique_bot_iff : (\u22a5 : SimpleGraph \u03b1).IsClique s \u2194 (s : Set \u03b1).Subsingleton :=\n  Set.pairwise_bot_iff\n#align simple_graph.is_clique_bot_iff SimpleGraph.isClique_bot_iff\n\nalias \u27e8IsClique.subsingleton, _\u27e9 := isClique_bot_iff\n#align simple_graph.is_clique.subsingleton SimpleGraph.IsClique.subsingleton\n\nend Clique\n\n/-! ### `n`-cliques -/\n\n\nsection NClique\n\nvariable {n : \u2115} {s : Finset \u03b1}\n\n/-- An `n`-clique in a graph is a set of `n` vertices which are pairwise connected. -/\nstructure IsNClique (n : \u2115) (s : Finset \u03b1) : Prop where\n  clique : G.IsClique s\n  card_eq : s.card = n\n#align simple_graph.is_n_clique SimpleGraph.IsNClique\n\ntheorem isNClique_iff : G.IsNClique n s \u2194 G.IsClique s \u2227 s.card = n :=\n  \u27e8fun h \u21a6 \u27e8h.1, h.2\u27e9, fun h \u21a6 \u27e8h.1, h.2\u27e9\u27e9\n#align simple_graph.is_n_clique_iff SimpleGraph.isNClique_iff\n\ninstance [DecidableEq \u03b1] [DecidableRel G.Adj] {n : \u2115} {s : Finset \u03b1} :\n    Decidable (G.IsNClique n s) :=\n  decidable_of_iff' _ G.isNClique_iff\n\nvariable {G H} {a b c : \u03b1}\n\n@[simp] lemma isNClique_empty : G.IsNClique n \u2205 \u2194 n = 0 := by simp [isNClique_iff, eq_comm]\n#align simple_graph.is_n_clique_empty SimpleGraph.isNClique_empty\n\n@[simp]\nlemma isNClique_singleton : G.IsNClique n {a} \u2194 n = 1 := by simp [isNClique_iff, eq_comm]\n#align simple_graph.is_n_clique_singleton SimpleGraph.isNClique_singleton\n\ntheorem IsNClique.mono (h : G \u2264 H) : G.IsNClique n s \u2192 H.IsNClique n s := by\n  simp_rw [isNClique_iff]\n  exact And.imp_left (IsClique.mono h)\n#align simple_graph.is_n_clique.mono SimpleGraph.IsNClique.mono\n\nprotected theorem IsNClique.map (h : G.IsNClique n s) {f : \u03b1 \u21aa \u03b2} :\n    (G.map f).IsNClique n (s.map f) :=\n  \u27e8by rw [coe_map]; exact h.1.map, (card_map _).trans h.2\u27e9\n#align simple_graph.is_n_clique.map SimpleGraph.IsNClique.map\n\n@[simp]\ntheorem isNClique_bot_iff : (\u22a5 : SimpleGraph \u03b1).IsNClique n s \u2194 n \u2264 1 \u2227 s.card = n := by\n  rw [isNClique_iff, isClique_bot_iff]\n  refine' and_congr_left _\n  rintro rfl\n  exact card_le_one.symm\n#align simple_graph.is_n_clique_bot_iff SimpleGraph.isNClique_bot_iff\n\n@[simp]\ntheorem isNClique_zero : G.IsNClique 0 s \u2194 s = \u2205 := by\n  simp only [isNClique_iff, Finset.card_eq_zero, and_iff_right_iff_imp]; rintro rfl; simp\n#align simple_graph.is_n_clique_zero SimpleGraph.isNClique_zero\n\n@[simp]\ntheorem isNClique_one : G.IsNClique 1 s \u2194 \u2203 a, s = {a} := by\n  simp only [isNClique_iff, card_eq_one, and_iff_right_iff_imp]; rintro \u27e8a, rfl\u27e9; simp\n#align simple_graph.is_n_clique_one SimpleGraph.isNClique_one\n\nsection DecidableEq\n\nvariable [DecidableEq \u03b1]\n\ntheorem IsNClique.insert (hs : G.IsNClique n s) (h : \u2200 b \u2208 s, G.Adj a b) :\n    G.IsNClique (n + 1) (insert a s) := by\n  constructor\n  \u00b7 push_cast\n    exact hs.1.insert fun b hb _ => h _ hb\n  \u00b7 rw [card_insert_of_not_mem fun ha => (h _ ha).ne rfl, hs.2]\n#align simple_graph.is_n_clique.insert SimpleGraph.IsNClique.insert\n\ntheorem is3Clique_triple_iff : G.IsNClique 3 {a, b, c} \u2194 G.Adj a b \u2227 G.Adj a c \u2227 G.Adj b c := by\n  simp only [isNClique_iff, isClique_iff, Set.pairwise_insert_of_symmetric G.symm, coe_insert]\n  by_cases hab : a = b <;> by_cases hbc : b = c <;> by_cases hac : a = c <;> subst_vars <;>\n    simp [G.ne_of_adj, and_rotate, *]\n#align simple_graph.is_3_clique_triple_iff SimpleGraph.is3Clique_triple_iff\n\ntheorem is3Clique_iff :\n    G.IsNClique 3 s \u2194 \u2203 a b c, G.Adj a b \u2227 G.Adj a c \u2227 G.Adj b c \u2227 s = {a, b, c} := by\n  refine' \u27e8fun h \u21a6 _, _\u27e9\n  \u00b7 obtain \u27e8a, b, c, -, -, -, hs\u27e9 := card_eq_three.1 h.card_eq\n    refine' \u27e8a, b, c, _\u27e9\n    rwa [hs, eq_self_iff_true, and_true, is3Clique_triple_iff.symm, \u2190 hs]\n  \u00b7 rintro \u27e8a, b, c, hab, hbc, hca, rfl\u27e9\n    exact is3Clique_triple_iff.2 \u27e8hab, hbc, hca\u27e9\n#align simple_graph.is_3_clique_iff SimpleGraph.is3Clique_iff\n\nend DecidableEq\n\n", 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"Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Data.Subtype", "Mathlib.Order.Notation", "Mathlib.Data.Set.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Cases", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Basic", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Tactic.PushNeg", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Tactic.CasesM", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.ByContra", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Data.Set.Prod", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.MinMax", "Mathlib.Tactic.Says", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Util.AssertExists", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Logic.Pairwise", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Data.Rel", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Nat.Defs", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Data.Sym.Sym2", "Mathlib.Combinatorics.SimpleGraph.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Sym.Card", "Mathlib.Combinatorics.SimpleGraph.Finite", "Mathlib.Combinatorics.SimpleGraph.Dart", "Mathlib.Data.FunLike.Fintype", "Mathlib.Combinatorics.SimpleGraph.Maps", "Mathlib.Combinatorics.SimpleGraph.Subgraph", "Mathlib.Combinatorics.SimpleGraph.Connectivity", "Mathlib.Combinatorics.SimpleGraph.Operations", "Mathlib.Data.Finset.Pairwise"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  classical\n  simp_rw [is3Clique_iff, isCycle_def]\n  exact\n    \u27e8(fun \u27e8_, a, _, _, hab, hac, hbc, _\u27e9 => \u27e8a, cons hab (cons hbc (cons hac.symm nil)), by aesop\u27e9),\n    (fun \u27e8_, .cons hab (.cons hbc (.cons hca nil)), _, _\u27e9 => \u27e8_, _, _, _, hab, hca.symm, hbc, rfl\u27e9)\u27e9", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 266}}
+{"srcContext": "/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, S\u00e9bastien Gou\u00ebzel, Heather Macbeth\n-/\nimport Mathlib.Analysis.Convex.Slope\nimport Mathlib.Analysis.SpecialFunctions.Pow.Real\nimport Mathlib.Tactic.LinearCombination\n\n#align_import analysis.convex.specific_functions.basic from \"leanprover-community/mathlib\"@\"8f9fea08977f7e450770933ee6abb20733b47c92\"\n\n/-!\n# Collection of convex functions\n\nIn this file we prove that the following functions are convex or strictly convex:\n\n* `strictConvexOn_exp` : The exponential function is strictly convex.\n* `strictConcaveOn_log_Ioi`, `strictConcaveOn_log_Iio`: `Real.log` is strictly concave on\n  $(0, +\u221e)$ and $(-\u221e, 0)$ respectively.\n* `convexOn_rpow`, `strictConvexOn_rpow` : For `p : \u211d`, `fun x \u21a6 x ^ p` is convex on $[0, +\u221e)$ when\n  `1 \u2264 p` and strictly convex when `1 < p`.\n\nThe proofs in this file are deliberately elementary, *not* by appealing to the sign of the second\nderivative. This is in order to keep this file early in the import hierarchy, since it is on the\npath to H\u00f6lder's and Minkowski's inequalities and after that to Lp spaces and most of measure\ntheory.\n\n(Strict) concavity of `fun x \u21a6 x ^ p` for `0 < p < 1` (`0 \u2264 p \u2264 1`) can be found in\n`Analysis.Convex.SpecificFunctions.Pow`.\n\n## See also\n\n`Analysis.Convex.Mul` for convexity of `x \u21a6 x ^ n`\n-/\n\nopen Real Set BigOperators NNReal\n\n/-- `Real.exp` is strictly convex on the whole real line. -/\ntheorem strictConvexOn_exp : StrictConvexOn \u211d univ exp := by\n  apply strictConvexOn_of_slope_strict_mono_adjacent convex_univ\n  rintro x y z - - hxy hyz\n  trans exp y\n  \u00b7 have h1 : 0 < y - x := by linarith\n    have h2 : x - y < 0 := by linarith\n    rw [div_lt_iff h1]\n    calc\n      exp y - exp x = exp y - exp y * exp (x - y) := by rw [\u2190 exp_add]; ring_nf\n      _ = exp y * (1 - exp (x - y)) := by ring\n      _ < exp y * -(x - y) := by gcongr; linarith [add_one_lt_exp h2.ne]\n      _ = exp y * (y - x) := by ring\n  \u00b7 have h1 : 0 < z - y := by linarith\n    rw [lt_div_iff h1]\n    calc\n      exp y * (z - y) < exp y * (exp (z - y) - 1) := by\n        gcongr _ * ?_\n        linarith [add_one_lt_exp h1.ne']\n      _ = exp (z - y) * exp y - exp y := by ring\n      _ \u2264 exp z - exp y := by rw [\u2190 exp_add]; ring_nf; rfl\n#align strict_convex_on_exp strictConvexOn_exp\n\n/-- `Real.exp` is convex on the whole real line. -/\ntheorem convexOn_exp : ConvexOn \u211d univ exp :=\n  strictConvexOn_exp.convexOn\n#align convex_on_exp convexOn_exp\n\n/-- `Real.log` is strictly concave on `(0, +\u221e)`. -/\ntheorem strictConcaveOn_log_Ioi : StrictConcaveOn \u211d (Ioi 0) log := by\n  apply strictConcaveOn_of_slope_strict_anti_adjacent (convex_Ioi (0 : \u211d))\n  intro x y z (hx : 0 < x) (hz : 0 < z) hxy hyz\n  have hy : 0 < y := hx.trans hxy\n  trans y\u207b\u00b9\n  \u00b7 have h : 0 < z - y := by linarith\n    rw [div_lt_iff h]\n    have hyz' : 0 < z / y := by positivity\n    have hyz'' : z / y \u2260 1 := by\n      contrapose! h\n      rw [div_eq_one_iff_eq hy.ne'] at h\n      simp [h]\n    calc\n      log z - log y = log (z / y) := by rw [\u2190 log_div hz.ne' hy.ne']\n      _ < z / y - 1 := log_lt_sub_one_of_pos hyz' hyz''\n      _ = y\u207b\u00b9 * (z - y) := by field_simp\n  \u00b7 have h : 0 < y - x := by linarith\n    rw [lt_div_iff h]\n    have hxy' : 0 < x / y := by positivity\n    have hxy'' : x / y \u2260 1 := by\n      contrapose! h\n      rw [div_eq_one_iff_eq hy.ne'] at h\n      simp [h]\n    calc\n      y\u207b\u00b9 * (y - x) = 1 - x / y := by field_simp\n      _ < -log (x / y) := by linarith [log_lt_sub_one_of_pos hxy' hxy'']\n      _ = -(log x - log y) := by rw [log_div hx.ne' hy.ne']\n      _ = log y - log x := by ring\n#align strict_concave_on_log_Ioi strictConcaveOn_log_Ioi\n\n/-- **Bernoulli's inequality** for real exponents, strict version: for `1 < p` and `-1 \u2264 s`, with\n`s \u2260 0`, we have `1 + p * s < (1 + s) ^ p`. -/\ntheorem one_add_mul_self_lt_rpow_one_add {s : \u211d} (hs : -1 \u2264 s) (hs' : s \u2260 0) {p : \u211d} (hp : 1 < p) :\n    1 + p * s < (1 + s) ^ p := by\n  have hp' : 0 < p := zero_lt_one.trans hp\n  rcases eq_or_lt_of_le hs with rfl | hs\n  \u00b7 rwa [add_right_neg, zero_rpow hp'.ne', mul_neg_one, add_neg_lt_iff_lt_add, zero_add]\n  have hs1 : 0 < 1 + s := neg_lt_iff_pos_add'.mp hs\n  rcases le_or_lt (1 + p * s) 0 with hs2 | hs2\n  \u00b7 exact hs2.trans_lt (rpow_pos_of_pos hs1 _)\n  have hs3 : 1 + s \u2260 1 := hs' \u2218 add_right_eq_self.mp\n  have hs4 : 1 + p * s \u2260 1 := by\n    contrapose! hs'; rwa [add_right_eq_self, mul_eq_zero, eq_false_intro hp'.ne', false_or] at hs'\n  rw [rpow_def_of_pos hs1, \u2190 exp_log hs2]\n  apply exp_strictMono\n  cases' lt_or_gt_of_ne hs' with hs' hs'\n  \u00b7 rw [\u2190 div_lt_iff hp', \u2190 div_lt_div_right_of_neg hs']\n    convert strictConcaveOn_log_Ioi.secant_strict_mono (zero_lt_one' \u211d) hs2 hs1 hs4 hs3 _ using 1\n    \u00b7 rw [add_sub_cancel_left, log_one, sub_zero]\n    \u00b7 rw [add_sub_cancel_left, div_div, log_one, sub_zero]\n    \u00b7 apply add_lt_add_left (mul_lt_of_one_lt_left hs' hp)\n  \u00b7 rw [\u2190 div_lt_iff hp', \u2190 div_lt_div_right hs']\n    convert strictConcaveOn_log_Ioi.secant_strict_mono (zero_lt_one' \u211d) hs1 hs2 hs3 hs4 _ using 1\n    \u00b7 rw [add_sub_cancel_left, div_div, log_one, sub_zero]\n    \u00b7 rw [add_sub_cancel_left, log_one, sub_zero]\n    \u00b7 apply add_lt_add_left (lt_mul_of_one_lt_left hs' hp)\n#align one_add_mul_self_lt_rpow_one_add one_add_mul_self_lt_rpow_one_add\n\n/-- **Bernoulli's inequality** for real exponents, non-strict version: for `1 \u2264 p` and `-1 \u2264 s`\nwe have `1 + p * s \u2264 (1 + s) ^ p`. -/\ntheorem one_add_mul_self_le_rpow_one_add {s : \u211d} (hs : -1 \u2264 s) {p : \u211d} (hp : 1 \u2264 p) :\n    1 + p * s \u2264 (1 + s) ^ p := by\n  rcases eq_or_lt_of_le hp with (rfl | hp)\n  \u00b7 simp\n  by_cases hs' : s = 0\n  \u00b7 simp [hs']\n  exact (one_add_mul_self_lt_rpow_one_add hs hs' hp).le\n#align one_add_mul_self_le_rpow_one_add one_add_mul_self_le_rpow_one_add\n\n/-- **Bernoulli's inequality** for real exponents, strict version: for `0 < p < 1` and `-1 \u2264 s`,\nwith `s \u2260 0`, we have `(1 + s) ^ p < 1 + p * s`. -/\ntheorem rpow_one_add_lt_one_add_mul_self {s : \u211d} (hs : -1 \u2264 s) (hs' : s \u2260 0) {p : \u211d} (hp1 : 0 < p)\n    (hp2 : p < 1) : (1 + s) ^ p < 1 + p * s := by\n  rcases eq_or_lt_of_le hs with rfl | hs\n  \u00b7 rwa [add_right_neg, zero_rpow hp1.ne', mul_neg_one, lt_add_neg_iff_add_lt, zero_add]\n  have hs1 : 0 < 1 + s := neg_lt_iff_pos_add'.mp hs\n  have hs2 : 0 < 1 + p * s := by\n    rw [\u2190 neg_lt_iff_pos_add']\n    rcases lt_or_gt_of_ne hs' with h | h\n    \u00b7 exact hs.trans (lt_mul_of_lt_one_left h hp2)\n    \u00b7 exact neg_one_lt_zero.trans (mul_pos hp1 h)\n  have hs3 : 1 + s \u2260 1 := hs' \u2218 add_right_eq_self.mp\n  have hs4 : 1 + p * s \u2260 1 := by\n    contrapose! hs'; rwa [add_right_eq_self, mul_eq_zero, eq_false_intro hp1.ne', false_or] at hs'\n  rw [rpow_def_of_pos hs1, \u2190 exp_log hs2]\n  apply exp_strictMono\n  cases' lt_or_gt_of_ne hs' with hs' hs'\n  \u00b7 rw [\u2190 lt_div_iff hp1, \u2190 div_lt_div_right_of_neg hs']\n    convert strictConcaveOn_log_Ioi.secant_strict_mono (zero_lt_one' \u211d) hs1 hs2 hs3 hs4 _ using 1\n    \u00b7 rw [add_sub_cancel_left, div_div, log_one, sub_zero]\n    \u00b7 rw [add_sub_cancel_left, log_one, sub_zero]\n    \u00b7 apply add_lt_add_left (lt_mul_of_lt_one_left hs' hp2)\n  \u00b7 rw [\u2190 lt_div_iff hp1, \u2190 div_lt_div_right hs']\n    convert strictConcaveOn_log_Ioi.secant_strict_mono (zero_lt_one' \u211d) hs2 hs1 hs4 hs3 _ using 1\n    \u00b7 rw [add_sub_cancel_left, log_one, sub_zero]\n    \u00b7 rw [add_sub_cancel_left, div_div, log_one, sub_zero]\n    \u00b7 apply add_lt_add_left (mul_lt_of_lt_one_left hs' hp2)\n\n", "theoremStatement": "/-- **Bernoulli's inequality** for real exponents, non-strict version: for `0 \u2264 p \u2264 1` and `-1 \u2264 s`\nwe have `(1 + s) ^ p \u2264 1 + p * s`. -/\ntheorem rpow_one_add_le_one_add_mul_self {s : \u211d} (hs : -1 \u2264 s) {p : \u211d} (hp1 : 0 \u2264 p) (hp2 : p \u2264 1) :\n    (1 + s) ^ p \u2264 1 + p * s ", "theoremName": "rpow_one_add_le_one_add_mul_self", "fileCreated": {"commit": "1b43f3b72d6b53709fa98d078498c6eba3562fb7", "date": "2023-05-25"}, "theoremCreated": {"commit": "2c429c3c379cff812f3418403399ca9798b47469", "date": "2024-03-25"}, "file": "mathlib/Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean", "module": "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "jsonFile": "Mathlib.Analysis.Convex.SpecificFunctions.Basic.jsonl", "positionMetadata": {"lineInFile": 165, "tokenPositionInFile": 7401, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, 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"Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Extr", "Mathlib.Analysis.Convex.Function", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Analysis.Convex.Slope", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  rcases eq_or_lt_of_le hp1 with (rfl | hp1)\n  \u00b7 simp\n  rcases eq_or_lt_of_le hp2 with (rfl | hp2)\n  \u00b7 simp\n  by_cases hs' : s = 0\n  \u00b7 simp [hs']\n  exact (rpow_one_add_lt_one_add_mul_self hs hs' hp1 hp2).le", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 212}}
+{"srcContext": "/-\nCopyright (c) 2024 David Loeffler. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: David Loeffler\n-/\n\nimport Mathlib.Analysis.SpecialFunctions.Gamma.Deligne\n/-!\n# Dirichlet series as Mellin transforms\n\nHere we prove general results of the form \"the Mellin transform of a power series in exp (-t) is\na Dirichlet series\".\n-/\n\nopen Filter Topology Asymptotics Real Set MeasureTheory\nopen Complex\n\nvariable {\u03b9 : Type*} [Countable \u03b9]\n\n/-- Most basic version of the \"Mellin transform = Dirichlet series\" argument. -/\nlemma hasSum_mellin {a : \u03b9 \u2192 \u2102} {p : \u03b9 \u2192 \u211d} {F : \u211d \u2192 \u2102} {s : \u2102}\n    (hp : \u2200 i, a i = 0 \u2228 0 < p i) (hs : 0 < s.re)\n    (hF : \u2200 t \u2208 Ioi 0, HasSum (fun i \u21a6 a i * rexp (-p i * t)) (F t))\n    (h_sum : Summable fun i \u21a6 \u2016a i\u2016 / (p i) ^ s.re) :\n    HasSum (fun i \u21a6 Gamma s * a i / p i ^ s) (mellin F s) := by\n  simp_rw [mellin, smul_eq_mul, \u2190 setIntegral_congr measurableSet_Ioi\n    (fun t ht \u21a6 congr_arg _ (hF t ht).tsum_eq), \u2190 tsum_mul_left]\n  convert hasSum_integral_of_summable_integral_norm\n    (F := fun i t \u21a6 t ^ (s - 1) * (a i * rexp (-p i * t))) (fun i \u21a6 ?_) ?_ using 2 with i\n  \u00b7 simp_rw [\u2190 mul_assoc, mul_comm _ (a _), mul_assoc (a _), mul_div_assoc, integral_mul_left]\n    rcases hp i with hai | hpi\n    \u00b7 rw [hai, zero_mul, zero_mul]\n    have := integral_cpow_mul_exp_neg_mul_Ioi hs hpi\n    simp_rw [\u2190 ofReal_mul, \u2190 ofReal_neg, \u2190 ofReal_exp, \u2190 neg_mul (p i)] at this\n    rw [this, one_div, inv_cpow _ _ (arg_ofReal_of_nonneg hpi.le \u25b8 pi_pos.ne), div_eq_inv_mul]\n  \u00b7 -- integrability of terms\n    rcases hp i with hai | hpi\n    \u00b7 simpa only [hai, zero_mul, mul_zero] using integrable_zero _ _ _\n    simp_rw [\u2190 mul_assoc, mul_comm _ (a i), mul_assoc]\n    have := Complex.GammaIntegral_convergent hs\n    rw [\u2190 mul_zero (p i), \u2190 integrableOn_Ioi_comp_mul_left_iff _ _ hpi] at this\n    refine (IntegrableOn.congr_fun (this.const_mul (1 / p i ^ (s - 1)))\n      (fun t (ht : 0 < t) \u21a6 ?_) measurableSet_Ioi).const_mul _\n    simp_rw [mul_comm (\u2191(rexp _) : \u2102), \u2190 mul_assoc, neg_mul, ofReal_mul]\n    rw [mul_cpow_ofReal_nonneg hpi.le ht.le, \u2190 mul_assoc, one_div, inv_mul_cancel, one_mul]\n    \u00b7 rw [Ne, cpow_eq_zero_iff, not_and_or]\n      exact Or.inl (ofReal_ne_zero.mpr hpi.ne')\n  \u00b7 -- summability of integrals of norms\n    apply Summable.of_norm\n    convert h_sum.mul_left (Real.Gamma s.re) using 2 with i\n    simp_rw [\u2190 mul_assoc, mul_comm _ (a i), mul_assoc, norm_mul (a i), integral_mul_left]\n    rw [\u2190 mul_div_assoc, mul_comm (Real.Gamma _), mul_div_assoc, norm_mul \u2016a i\u2016, norm_norm]\n    rcases hp i with hai | hpi\n    \u00b7 simp only [hai, norm_zero, zero_mul]\n    congr 1\n    have := Real.integral_rpow_mul_exp_neg_mul_Ioi hs hpi\n    simp_rw [\u2190 neg_mul (p i), one_div, inv_rpow hpi.le, \u2190 div_eq_inv_mul] at this\n    rw [norm_of_nonneg (integral_nonneg (fun _ \u21a6 norm_nonneg _)), \u2190 this]\n    refine setIntegral_congr measurableSet_Ioi (fun t ht \u21a6 ?_)\n    rw [norm_mul, norm_real, Real.norm_eq_abs, Real.abs_exp, Complex.norm_eq_abs,\n      abs_cpow_eq_rpow_re_of_pos ht, sub_re, one_re]\n\n/-- Shortcut version for the commonly arising special case when `p i = \u03c0 * q i` for some other\nsequence `q`. -/\nlemma hasSum_mellin_pi_mul {a : \u03b9 \u2192 \u2102} {q : \u03b9 \u2192 \u211d} {F : \u211d \u2192 \u2102} {s : \u2102}\n    (hq : \u2200 i, a i = 0 \u2228 0 < q i) (hs : 0 < s.re)\n    (hF : \u2200 t \u2208 Ioi 0, HasSum (fun i \u21a6 a i * rexp (-\u03c0 * q i * t)) (F t))\n    (h_sum : Summable fun i \u21a6 \u2016a i\u2016 / (q i) ^ s.re) :\n    HasSum (fun i \u21a6 \u03c0 ^ (-s) * Gamma s * a i / q i ^ s) (mellin F s) := by\n  have hp i : a i = 0 \u2228 0 < \u03c0 * q i := by rcases hq i with h | h <;> simp [h, pi_pos]\n  convert hasSum_mellin hp hs (by simpa using hF) ?_ using 2 with i\n  \u00b7 have : a i / \u2191(\u03c0 * q i) ^ s = \u03c0 ^ (-s) * a i / q i ^ s := by\n      rcases hq i with h | h\n      \u00b7 simp [h]\n      \u00b7 rw [ofReal_mul, mul_cpow_ofReal_nonneg pi_pos.le h.le, \u2190 div_div, cpow_neg,\n          \u2190 div_eq_inv_mul]\n    simp_rw [mul_div_assoc, this]\n    ring_nf\n  \u00b7 have (i) : \u2016a i\u2016 / \u2191(\u03c0 * q i) ^ s.re = \u03c0 ^ (-s.re) * \u2016a i\u2016 / q i ^ s.re := by\n      rcases hq i with h | h\n      \u00b7 simp [h]\n      \u00b7 rw [mul_rpow pi_pos.le h.le, \u2190 div_div, rpow_neg pi_pos.le, \u2190 div_eq_inv_mul]\n    simpa only [this, mul_div_assoc] using h_sum.mul_left _\n\n/-- Version allowing some constant terms (which are omitted from the sums). -/\nlemma hasSum_mellin_pi_mul\u2080 {a : \u03b9 \u2192 \u2102} {p : \u03b9 \u2192 \u211d} {F : \u211d \u2192 \u2102} {s : \u2102}\n    (hp : \u2200 i, 0 \u2264 p i) (hs : 0 < s.re)\n    (hF : \u2200 t \u2208 Ioi 0, HasSum (fun i \u21a6 if p i = 0 then 0 else a i * rexp (-\u03c0 * p i * t)) (F t))\n    (h_sum : Summable fun i \u21a6 \u2016a i\u2016 / (p i) ^ s.re) :\n    HasSum (fun i \u21a6 \u03c0 ^ (-s) * Gamma s * a i / p i ^ s) (mellin F s) := by\n  have hs' : s \u2260 0 := fun h \u21a6 lt_irrefl _ (zero_re \u25b8 h \u25b8 hs)\n  let a' i := if p i = 0 then 0 else a i\n  have hp' i : a' i = 0 \u2228 0 < p i := by\n    simp only [a']\n    split_ifs with h <;> tauto\n    exact Or.inr (lt_of_le_of_ne (hp i) (Ne.symm h))\n  have (i t) : (if p i = 0 then 0 else a i * rexp (-\u03c0 * p i * t)) =\n      a' i * rexp (-\u03c0 * p i * t) := by\n    simp only [a', ite_mul, zero_mul]\n  simp_rw [this] at hF\n  convert hasSum_mellin_pi_mul hp' hs hF ?_ using 2 with i\n  \u00b7 rcases eq_or_ne (p i) 0 with h | h <;>\n    simp [a', h, if_false, ofReal_zero, zero_cpow hs', div_zero]\n  \u00b7 refine h_sum.of_norm_bounded _ (fun i \u21a6 ?_)\n    simp only [a']\n    split_ifs\n    \u00b7 simp only [norm_zero, zero_div]\n      positivity\n    \u00b7 have := hp i\n      rw [norm_of_nonneg (by positivity)]\n\n/-- Tailored version for even Jacobi theta functions. -/\nlemma hasSum_mellin_pi_mul_sq {a : \u03b9 \u2192 \u2102} {r : \u03b9 \u2192 \u211d} {F : \u211d \u2192 \u2102} {s : \u2102} (hs : 0 < s.re)\n    (hF : \u2200 t \u2208 Ioi 0, HasSum (fun i \u21a6 if r i = 0 then 0 else a i * rexp (-\u03c0 * r i ^ 2 * t)) (F t))\n    (h_sum : Summable fun i \u21a6 \u2016a i\u2016 / |r i| ^ s.re) :\n    HasSum (fun i \u21a6 Gamma\u211d s * a i / |r i| ^ s) (mellin F (s / 2)) := by\n  have hs' : 0 < (s / 2).re := by rw [div_ofNat_re]; positivity\n  simp_rw [\u2190 sq_eq_zero_iff (a := r _)] at hF\n  convert hasSum_mellin_pi_mul\u2080 (fun i \u21a6 sq_nonneg (r i)) hs' hF ?_ using 3 with i\n  \u00b7 rw [\u2190 neg_div, Gamma\u211d_def]\n  \u00b7 rw [\u2190 _root_.sq_abs, ofReal_pow, \u2190 cpow_nat_mul']\n    ring_nf\n    all_goals rw [arg_ofReal_of_nonneg (abs_nonneg _)]; linarith [pi_pos]\n  \u00b7 convert h_sum using 3 with i\n    rw [\u2190 _root_.sq_abs, \u2190 rpow_natCast_mul (abs_nonneg _), div_ofNat_re, Nat.cast_ofNat,\n      mul_div_cancel\u2080 _ two_pos.ne']\n\n", "theoremStatement": "/-- Tailored version for odd Jacobi theta functions. -/\nlemma hasSum_mellin_pi_mul_sq' {a : \u03b9 \u2192 \u2102} {r : \u03b9 \u2192 \u211d} {F : \u211d \u2192 \u2102} {s : \u2102} (hs : 0 < s.re)\n    (hF : \u2200 t \u2208 Ioi 0, HasSum (fun i \u21a6 a i * r i * rexp (-\u03c0 * r i ^ 2 * t)) (F t))\n    (h_sum : Summable fun i \u21a6 \u2016a i\u2016 / |r i| ^ s.re) :\n    HasSum (fun i \u21a6 Gamma\u211d (s + 1) * a i * SignType.sign (r i) / |r i| ^ s)\n    (mellin F ((s + 1) / 2)) ", "theoremName": "hasSum_mellin_pi_mul_sq'", "fileCreated": {"commit": "ea083105e5450c53be6ba297f51ff1aa04f8ce7e", "date": "2024-03-28"}, "theoremCreated": {"commit": "ea083105e5450c53be6ba297f51ff1aa04f8ce7e", "date": "2024-03-28"}, "file": 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"Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Algebra.CharP.Invertible", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.Topology.CountableSeparatingOn", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.Analysis.SpecialFunctions.Gamma.Deligne"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have hs\u2081 : s \u2260 0 := fun h \u21a6 lt_irrefl _ (zero_re \u25b8 h \u25b8 hs)\n  have hs\u2082 : 0 < (s + 1).re := by rw [add_re, one_re]; positivity\n  have hs\u2083 : s + 1 \u2260 0 := fun h \u21a6 lt_irrefl _ (zero_re \u25b8 h \u25b8 hs\u2082)\n  have (i t) : (a i * r i * rexp (-\u03c0 * r i ^ 2 * t)) =\n      if r i = 0 then 0 else (a i * r i * rexp (-\u03c0 * r i ^ 2 * t)) := by\n    split_ifs with h <;> simp [h]\n  conv at hF => enter [t, ht, 1, i]; rw [this]\n  convert hasSum_mellin_pi_mul_sq hs\u2082 hF ?_ using 2 with i\n  \u00b7 rcases eq_or_ne (r i) 0 with h | h\n    \u00b7 rw [h, abs_zero, ofReal_zero, zero_cpow hs\u2081, zero_cpow hs\u2083, div_zero, div_zero]\n    \u00b7 rw [cpow_add _ _ (ofReal_ne_zero.mpr <| abs_ne_zero.mpr h), cpow_one]\n      conv_rhs => enter [1]; rw [\u2190 sign_mul_abs (r i), ofReal_mul, \u2190 ofReal_eq_coe,\n        SignType.map_cast]\n      field_simp [h]\n      ring_nf\n  \u00b7 convert h_sum using 2 with i\n    rcases eq_or_ne (r i) 0 with h | h\n    \u00b7 rw [h, abs_zero, ofReal_zero, zero_rpow hs\u2082.ne', zero_rpow hs.ne', div_zero, div_zero]\n    \u00b7 rw [add_re, one_re, rpow_add (abs_pos.mpr h), rpow_one, norm_mul, norm_real,\n        Real.norm_eq_abs, \u2190 div_div, div_right_comm, mul_div_cancel_right\u2080 _ (abs_ne_zero.mpr h)]", "proofType": "tactic", "proofLengthLines": 20, "proofLengthTokens": 1159}}
+{"srcContext": "/-\nCopyright (c) 2023 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz\n-/\nimport Mathlib.CategoryTheory.Limits.Shapes.Products\n/-!\n\n# Effective epimorphisms\n\nWe define the notion of effective epimorphism and effective epimorphic family of morphisms.\n\nA morphism is an *effective epi* if it is a joint coequalizer of all pairs of\nmorphisms which it coequalizes.\n\nA family of morphisms with fixed target is *effective epimorphic* if it is initial among families\nof morphisms with its sources and a general fixed target, coequalizing every pair of morphisms it\ncoequalizes (here, the pair of morphisms coequalized can have different targets among the sources\nof the family).\n\nWe have defined the notion of effective epi for morphisms and families of morphisms in such a\nway that avoids requiring the existence of pullbacks. However, if the relevant pullbacks exist\nthen these definitions are equivalent, see the file\n`CategoryTheory/EffectiveEpi/RegularEpi.lean`\nSee [nlab: *Effective Epimorphism*](https://ncatlab.org/nlab/show/effective+epimorphism) and\n[Stacks 00WP](https://stacks.math.columbia.edu/tag/00WP) for the standard definitions. Note that\nour notion of `EffectiveEpi`\u00a0is often called \"strict epi\" in the literature.\n\n## References\n- [Elephant]: *Sketches of an Elephant*, P. T. Johnstone: C2.1, Example 2.1.12.\n- [nlab: *Effective Epimorphism*](https://ncatlab.org/nlab/show/effective+epimorphism) and\n- [Stacks 00WP](https://stacks.math.columbia.edu/tag/00WP) for the standard definitions.\n\n-/\n\nnamespace CategoryTheory\n\nopen Limits\n\nvariable {C : Type*} [Category C]\n\n/--\nThis structure encodes the data required for a morphism to be an effective epimorphism.\n-/\nstructure EffectiveEpiStruct {X Y : C} (f : Y \u27f6 X) where\n  desc : \u2200 {W : C} (e : Y \u27f6 W),\n    (\u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) \u2192 (X \u27f6 W)\n  fac : \u2200 {W : C} (e : Y \u27f6 W)\n    (h : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e),\n    f \u226b desc e h = e\n  uniq : \u2200 {W : C} (e : Y \u27f6 W)\n    (h : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e)\n    (m : X \u27f6 W), f \u226b m = e \u2192 m = desc e h\n\nattribute [nolint docBlame]\n  EffectiveEpiStruct.desc\n  EffectiveEpiStruct.fac\n  EffectiveEpiStruct.uniq\n\n/--\nA morphism `f : Y \u27f6 X` is an effective epimorphism provided that `f` exhibits `X` as a colimit\nof the diagram of all \"relations\" `R \u21c9 Y`.\nIf `f` has a kernel pair, then this is equivalent to showing that the corresponding cofork is\na colimit.\n-/\nclass EffectiveEpi {X Y : C} (f : Y \u27f6 X) : Prop where\n  effectiveEpi : Nonempty (EffectiveEpiStruct f)\n\nattribute [nolint docBlame] EffectiveEpi.effectiveEpi\n\n/-- Some chosen `EffectiveEpiStruct` associated to an effective epi. -/\nnoncomputable\ndef EffectiveEpi.getStruct {X Y : C} (f : Y \u27f6 X) [EffectiveEpi f] : EffectiveEpiStruct f :=\n  EffectiveEpi.effectiveEpi.some\n\n/-- Descend along an effective epi. -/\nnoncomputable\ndef EffectiveEpi.desc {X Y W : C} (f : Y \u27f6 X) [EffectiveEpi f]\n    (e : Y \u27f6 W) (h : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) :\n    X \u27f6 W := (EffectiveEpi.getStruct f).desc e h\n\n@[reassoc (attr := simp)]\nlemma EffectiveEpi.fac {X Y W : C} (f : Y \u27f6 X) [EffectiveEpi f]\n    (e : Y \u27f6 W) (h : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e) :\n    f \u226b EffectiveEpi.desc f e h = e :=\n  (EffectiveEpi.getStruct f).fac e h\n\nlemma EffectiveEpi.uniq {X Y W : C} (f : Y \u27f6 X) [EffectiveEpi f]\n    (e : Y \u27f6 W) (h : \u2200 {Z : C} (g\u2081 g\u2082 : Z \u27f6 Y), g\u2081 \u226b f = g\u2082 \u226b f \u2192 g\u2081 \u226b e = g\u2082 \u226b e)\n    (m : X \u27f6 W) (hm : f \u226b m = e) :\n    m = EffectiveEpi.desc f e h :=\n  (EffectiveEpi.getStruct f).uniq e h _ hm\n\ninstance epiOfEffectiveEpi {X Y : C} (f : Y \u27f6 X) [EffectiveEpi f] : Epi f := by\n  constructor\n  intro W m\u2081 m\u2082 h\n  have : m\u2082 = EffectiveEpi.desc f (f \u226b m\u2082)\n    (fun {Z} g\u2081 g\u2082 h => by simp only [\u2190 Category.assoc, h]) := EffectiveEpi.uniq _ _ _ _ rfl\n  rw [this]\n  exact EffectiveEpi.uniq _ _ _ _ h\n\n/--\nThis structure encodes the data required for a family of morphisms to be effective epimorphic.\n-/\nstructure EffectiveEpiFamilyStruct {B : C} {\u03b1 : Type*}\n    (X : \u03b1 \u2192 C) (\u03c0 : (a : \u03b1) \u2192 (X a \u27f6 B)) where\n  desc : \u2200 {W} (e : (a : \u03b1) \u2192 (X a \u27f6 W)),\n          (\u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082),\n      g\u2081 \u226b \u03c0 _ = g\u2082 \u226b \u03c0 _ \u2192 g\u2081 \u226b e _ = g\u2082 \u226b e _) \u2192 (B \u27f6 W)\n  fac : \u2200 {W} (e : (a : \u03b1) \u2192 (X a \u27f6 W))\n          (h : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082),\n            g\u2081 \u226b \u03c0 _ = g\u2082 \u226b \u03c0 _ \u2192 g\u2081 \u226b e _ = g\u2082 \u226b e _)\n          (a : \u03b1), \u03c0 a \u226b desc e h = e a\n  uniq : \u2200 {W} (e : (a : \u03b1) \u2192 (X a \u27f6 W))\n          (h : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082),\n            g\u2081 \u226b \u03c0 _ = g\u2082 \u226b \u03c0 _ \u2192 g\u2081 \u226b e _ = g\u2082 \u226b e _)\n          (m : B \u27f6 W), (\u2200 (a : \u03b1), \u03c0 a \u226b m = e a) \u2192 m = desc e h\n\nattribute [nolint docBlame]\n  EffectiveEpiFamilyStruct.desc\n  EffectiveEpiFamilyStruct.fac\n  EffectiveEpiFamilyStruct.uniq\n\n/--\nA family of morphisms `f a : X a \u27f6 B` indexed by `\u03b1` is effective epimorphic\nprovided that the `f a` exhibit `B` as a colimit of the diagram of all \"relations\"\n`R \u2192 X a\u2081`, `R \u27f6 X a\u2082` for all `a\u2081 a\u2082 : \u03b1`.\n-/\nclass EffectiveEpiFamily {B : C} {\u03b1 : Type*} (X : \u03b1 \u2192 C) (\u03c0 : (a : \u03b1) \u2192 (X a \u27f6 B)) : Prop where\n  effectiveEpiFamily : Nonempty (EffectiveEpiFamilyStruct X \u03c0)\n\nattribute [nolint docBlame] EffectiveEpiFamily.effectiveEpiFamily\n\n/-- Some chosen `EffectiveEpiFamilyStruct` associated to an effective epi family. -/\nnoncomputable\ndef EffectiveEpiFamily.getStruct {B : C} {\u03b1 : Type*} (X : \u03b1 \u2192 C) (\u03c0 : (a : \u03b1) \u2192 (X a \u27f6 B))\n    [EffectiveEpiFamily X \u03c0] : EffectiveEpiFamilyStruct X \u03c0 :=\n  EffectiveEpiFamily.effectiveEpiFamily.some\n\n/-- Descend along an effective epi family. -/\nnoncomputable\ndef EffectiveEpiFamily.desc {B W : C} {\u03b1 : Type*} (X : \u03b1 \u2192 C) (\u03c0 : (a : \u03b1) \u2192 (X a \u27f6 B))\n    [EffectiveEpiFamily X \u03c0] (e : (a : \u03b1) \u2192 (X a \u27f6 W))\n    (h : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082),\n      g\u2081 \u226b \u03c0 _ = g\u2082 \u226b \u03c0 _ \u2192 g\u2081 \u226b e _ = g\u2082 \u226b e _) : B \u27f6 W :=\n  (EffectiveEpiFamily.getStruct X \u03c0).desc e h\n\n@[reassoc (attr := simp)]\nlemma EffectiveEpiFamily.fac {B W : C} {\u03b1 : Type*} (X : \u03b1 \u2192 C) (\u03c0 : (a : \u03b1) \u2192 (X a \u27f6 B))\n    [EffectiveEpiFamily X \u03c0] (e : (a : \u03b1) \u2192 (X a \u27f6 W))\n    (h : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082),\n      g\u2081 \u226b \u03c0 _ = g\u2082 \u226b \u03c0 _ \u2192 g\u2081 \u226b e _ = g\u2082 \u226b e _) (a : \u03b1) :\n    \u03c0 a \u226b EffectiveEpiFamily.desc X \u03c0 e h = e a :=\n  (EffectiveEpiFamily.getStruct X \u03c0).fac e h a\n\n/-\nNOTE: The `simpNF` linter complains for some reason. See the two examples below.\nZulip discussion:\nhttps://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/simpNF.20bug.3F\n-/\nattribute [nolint simpNF]\n  EffectiveEpiFamily.fac\n  EffectiveEpiFamily.fac_assoc\n\nexample {B W : C} {\u03b1 : Type*} (X : \u03b1 \u2192 C) (\u03c0 : (a : \u03b1) \u2192 (X a \u27f6 B))\n    [EffectiveEpiFamily X \u03c0] (e : (a : \u03b1) \u2192 (X a \u27f6 W))\n    (h : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082),\n      g\u2081 \u226b \u03c0 _ = g\u2082 \u226b \u03c0 _ \u2192 g\u2081 \u226b e _ = g\u2082 \u226b e _) (a : \u03b1) :\n    \u03c0 a \u226b EffectiveEpiFamily.desc X \u03c0 e h = e a :=\n  by simp\n\nexample {B W Q : C} {\u03b1 : Type*} (X : \u03b1 \u2192 C) (\u03c0 : (a : \u03b1) \u2192 (X a \u27f6 B))\n    [EffectiveEpiFamily X \u03c0] (e : (a : \u03b1) \u2192 (X a \u27f6 W))\n    (h : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082),\n      g\u2081 \u226b \u03c0 _ = g\u2082 \u226b \u03c0 _ \u2192 g\u2081 \u226b e _ = g\u2082 \u226b e _) (a : \u03b1)\n    (q : W \u27f6 Q) :\n    \u03c0 a \u226b EffectiveEpiFamily.desc X \u03c0 e h \u226b q = e a \u226b q :=\n  by simp\n\nlemma EffectiveEpiFamily.uniq {B W : C} {\u03b1 : Type*} (X : \u03b1 \u2192 C) (\u03c0 : (a : \u03b1) \u2192 (X a \u27f6 B))\n    [EffectiveEpiFamily X \u03c0] (e : (a : \u03b1) \u2192 (X a \u27f6 W))\n    (h : \u2200 {Z : C} (a\u2081 a\u2082 : \u03b1) (g\u2081 : Z \u27f6 X a\u2081) (g\u2082 : Z \u27f6 X a\u2082),\n      g\u2081 \u226b \u03c0 _ = g\u2082 \u226b \u03c0 _ \u2192 g\u2081 \u226b e _ = g\u2082 \u226b e _)\n    (m : B \u27f6 W) (hm : \u2200 a, \u03c0 a \u226b m = e a) :\n    m = EffectiveEpiFamily.desc X \u03c0 e h :=\n  (EffectiveEpiFamily.getStruct X \u03c0).uniq e h m hm\n\n-- TODO: Once we have \"jointly epimorphic families\", we could rephrase this as such a property.\nlemma EffectiveEpiFamily.hom_ext {B W : C} {\u03b1 : Type*} (X : \u03b1 \u2192 C) (\u03c0 : (a : \u03b1) \u2192 (X a \u27f6 B))\n    [EffectiveEpiFamily X \u03c0] (m\u2081 m\u2082 : B \u27f6 W) (h : \u2200 a, \u03c0 a \u226b m\u2081 = \u03c0 a \u226b m\u2082) :\n    m\u2081 = m\u2082 := by\n  have : m\u2082 = EffectiveEpiFamily.desc X \u03c0 (fun a => \u03c0 a \u226b m\u2082)\n      (fun a\u2081 a\u2082 g\u2081 g\u2082 h => by simp only [\u2190 Category.assoc, h]) := by\n    apply EffectiveEpiFamily.uniq; intro; rfl\n  rw [this]\n  exact EffectiveEpiFamily.uniq _ _ _ _ _ h\n\n/--\nAn `EffectiveEpiFamily` consisting of a single `EffectiveEpi`\n-/\nnoncomputable\ndef effectiveEpiFamilyStructSingletonOfEffectiveEpi {B X : C} (f : X \u27f6 B) [EffectiveEpi f] :\n    EffectiveEpiFamilyStruct (fun () \u21a6 X) (fun () \u21a6 f) where\n  desc e h := EffectiveEpi.desc f (e ()) (fun g\u2081 g\u2082 hg \u21a6 h () () g\u2081 g\u2082 hg)\n  fac e h := fun _ \u21a6 EffectiveEpi.fac f (e ()) (fun g\u2081 g\u2082 hg \u21a6 h () () g\u2081 g\u2082 hg)\n  uniq e h m hm := by apply EffectiveEpi.uniq f (e ()) (h () ()); exact hm ()\n\ninstance {B X : C} (f : X \u27f6 B) [EffectiveEpi f] : EffectiveEpiFamily (fun () \u21a6 X) (fun () \u21a6 f) :=\n  \u27e8\u27e8effectiveEpiFamilyStructSingletonOfEffectiveEpi f\u27e9\u27e9\n\n/--\nA single element `EffectiveEpiFamily`\u00a0constists of an `EffectiveEpi`\n-/\nnoncomputable\ndef effectiveEpiStructOfEffectiveEpiFamilySingleton {B X : C} (f : X \u27f6 B)\n    [EffectiveEpiFamily (fun () \u21a6 X) (fun () \u21a6 f)] :\n    EffectiveEpiStruct f where\n  desc e h := EffectiveEpiFamily.desc\n    (fun () \u21a6 X) (fun () \u21a6 f) (fun () \u21a6 e) (fun _ _ g\u2081 g\u2082 hg \u21a6 h g\u2081 g\u2082 hg)\n  fac e h := EffectiveEpiFamily.fac\n    (fun () \u21a6 X) (fun () \u21a6 f) (fun () \u21a6 e) (fun _ _ g\u2081 g\u2082 hg \u21a6 h g\u2081 g\u2082 hg) ()\n  uniq e h m hm := EffectiveEpiFamily.uniq\n    (fun () \u21a6 X) (fun () \u21a6 f) (fun () \u21a6 e) (fun _ _ g\u2081 g\u2082 hg \u21a6 h g\u2081 g\u2082 hg) m (fun _ \u21a6 hm)\n\ninstance {B X : C} (f : X \u27f6 B) [EffectiveEpiFamily (fun () \u21a6 X) (fun () \u21a6 f)] :\n    EffectiveEpi f :=\n  \u27e8\u27e8effectiveEpiStructOfEffectiveEpiFamilySingleton f\u27e9\u27e9\n\ntheorem effectiveEpi_iff_effectiveEpiFamily {B X : C} (f : X \u27f6 B) :\n    EffectiveEpi f \u2194 EffectiveEpiFamily (fun () \u21a6 X) (fun () \u21a6 f) :=\n  \u27e8fun _ \u21a6 inferInstance, fun _ \u21a6 inferInstance\u27e9\n\n/--\nA family of morphisms with the same target inducing an isomorphism from the coproduct to the target\nis an `EffectiveEpiFamily`.\n-/\nnoncomputable\ndef effectiveEpiFamilyStructOfIsIsoDesc {B : C} {\u03b1 : Type*} (X : \u03b1 \u2192 C)\n    (\u03c0 : (a : \u03b1) \u2192 (X a \u27f6 B)) [HasCoproduct X] [IsIso (Sigma.desc \u03c0)] :\n    EffectiveEpiFamilyStruct X \u03c0 where\n  desc e _ := (asIso (Sigma.desc \u03c0)).inv \u226b (Sigma.desc e)\n  fac e h := by\n    intro a\n    have : \u03c0 a = Sigma.\u03b9 X a \u226b (asIso (Sigma.desc \u03c0)).hom := by simp only [asIso_hom,\n      colimit.\u03b9_desc, Cofan.mk_pt, Cofan.mk_\u03b9_app]\n    rw [this, Category.assoc]\n    simp only [asIso_hom, asIso_inv, IsIso.hom_inv_id_assoc, colimit.\u03b9_desc, Cofan.mk_pt,\n      Cofan.mk_\u03b9_app]\n  uniq e h m hm := by\n    simp only [asIso_inv, IsIso.eq_inv_comp]\n    ext a\n    simp only [colimit.\u03b9_desc_assoc, Discrete.functor_obj, Cofan.mk_pt, Cofan.mk_\u03b9_app,\n      colimit.\u03b9_desc]\n    exact hm a\n\ninstance {B : C} {\u03b1 : Type*} (X : \u03b1 \u2192 C) (\u03c0 : (a : \u03b1) \u2192 (X a \u27f6 B)) [HasCoproduct X]\n    [IsIso (Sigma.desc \u03c0)] : EffectiveEpiFamily X \u03c0 :=\n  \u27e8\u27e8effectiveEpiFamilyStructOfIsIsoDesc X \u03c0\u27e9\u27e9\n\n/-- Any isomorphism is an effective epi. -/\nnoncomputable\ndef effectiveEpiStructOfIsIso {X Y : C} (f : X \u27f6 Y) [IsIso f] : EffectiveEpiStruct f where\n  desc e _ := inv f \u226b e\n  fac _ _ := by simp\n  uniq _ _ _ h := by simpa using h\n\ninstance {X Y : C} (f : X \u27f6 Y) [IsIso f] : EffectiveEpi f := \u27e8\u27e8effectiveEpiStructOfIsIso f\u27e9\u27e9\n\nexample {X : C} : EffectiveEpiFamily (fun _ => X : Unit \u2192 C) (fun _ => \ud835\udfd9 X) := inferInstance\n\n/--\nReindex the indexing type of an effective epi family struct.\n-/\ndef EffectiveEpiFamilyStruct.reindex\n    {B : C} {\u03b1 \u03b1' : Type*}\n    (X : \u03b1 \u2192 C)\n    (\u03c0 : (a : \u03b1) \u2192 (X a \u27f6 B))\n    (e : \u03b1' \u2243 \u03b1)\n    (P : EffectiveEpiFamilyStruct (fun a => X (e a)) (fun a => \u03c0 (e a))) :\n    EffectiveEpiFamilyStruct X \u03c0 where\n  desc := fun f h => P.desc (fun a => f _) (fun a\u2081 a\u2082 => h _ _)\n  fac _ _ a := by\n    obtain \u27e8a,rfl\u27e9 := e.surjective a\n    apply P.fac\n  uniq _ _ m hm := P.uniq _ _ _ fun a => hm _\n\n", "theoremStatement": "/--\nReindex the indexing type of an effective epi family.\n-/\nlemma EffectiveEpiFamily.reindex\n    {B : C} {\u03b1 \u03b1' : Type*}\n    (X : \u03b1 \u2192 C)\n    (\u03c0 : (a : \u03b1) \u2192 (X a \u27f6 B))\n    (e : \u03b1' \u2243 \u03b1)\n    (h : EffectiveEpiFamily (fun a => X (e a)) (fun a => \u03c0 (e a))) :\n    EffectiveEpiFamily X \u03c0 ", "theoremName": "CategoryTheory.EffectiveEpiFamily.reindex", "fileCreated": {"commit": "6d4365dba4d36ff9ceafd8ace95b560f01bdd4c7", "date": "2024-03-19"}, "theoremCreated": {"commit": "e8abc61dba28e3cec860de064eaf7c14e33c6b00", "date": "2024-03-23"}, "file": "mathlib/Mathlib/CategoryTheory/EffectiveEpi/Basic.lean", "module": "Mathlib.CategoryTheory.EffectiveEpi.Basic", "jsonFile": "Mathlib.CategoryTheory.EffectiveEpi.Basic.jsonl", "positionMetadata": {"lineInFile": 287, "tokenPositionInFile": 11817, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 13, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", 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"Mathlib.Logic.IsEmpty", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Opposite", "Mathlib.Tactic.Cases", "Mathlib.Combinatorics.Quiver.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.Convert", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Spread", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.AssertExists", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.EqToHom", "Mathlib.Data.Prod.Basic", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Control.ULift", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Order.Notation", "Mathlib.Data.Set.Defs", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.Products"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n  .mk <| .intro <| @EffectiveEpiFamily.getStruct _ _ _ _ _ _ h |>.reindex _ _ e", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 82}}
+{"srcContext": "/-\nCopyright (c) 2019 S\u00e9bastien Gou\u00ebzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: S\u00e9bastien Gou\u00ebzel\n-/\nimport Mathlib.Analysis.Calculus.FDeriv.Equiv\nimport Mathlib.Analysis.Calculus.FormalMultilinearSeries\n\n#align_import analysis.calculus.cont_diff_def from \"leanprover-community/mathlib\"@\"3a69562db5a458db8322b190ec8d9a8bbd8a5b14\"\n\n/-!\n# Higher differentiability\n\nA function is `C^1` on a domain if it is differentiable there, and its derivative is continuous.\nBy induction, it is `C^n` if it is `C^{n-1}` and its (n-1)-th derivative is `C^1` there or,\nequivalently, if it is `C^1` and its derivative is `C^{n-1}`.\nFinally, it is `C^\u221e` if it is `C^n` for all n.\n\nWe formalize these notions by defining iteratively the `n+1`-th derivative of a function as the\nderivative of the `n`-th derivative. It is called `iteratedFDeriv \ud835\udd5c n f x` where `\ud835\udd5c` is the\nfield, `n` is the number of iterations, `f` is the function and `x` is the point, and it is given\nas an `n`-multilinear map. We also define a version `iteratedFDerivWithin` relative to a domain,\nas well as predicates `ContDiffWithinAt`, `ContDiffAt`, `ContDiffOn` and\n`ContDiff` saying that the function is `C^n` within a set at a point, at a point, on a set\nand on the whole space respectively.\n\nTo avoid the issue of choice when choosing a derivative in sets where the derivative is not\nnecessarily unique, `ContDiffOn` is not defined directly in terms of the\nregularity of the specific choice `iteratedFDerivWithin \ud835\udd5c n f s` inside `s`, but in terms of the\nexistence of a nice sequence of derivatives, expressed with a predicate\n`HasFTaylorSeriesUpToOn`.\n\nWe prove basic properties of these notions.\n\n## Main definitions and results\nLet `f : E \u2192 F` be a map between normed vector spaces over a nontrivially normed field `\ud835\udd5c`.\n\n* `HasFTaylorSeriesUpTo n f p`: expresses that the formal multilinear series `p` is a sequence\n  of iterated derivatives of `f`, up to the `n`-th term (where `n` is a natural number or `\u221e`).\n* `HasFTaylorSeriesUpToOn n f p s`: same thing, but inside a set `s`. The notion of derivative\n  is now taken inside `s`. In particular, derivatives don't have to be unique.\n* `ContDiff \ud835\udd5c n f`: expresses that `f` is `C^n`, i.e., it admits a Taylor series up to\n  rank `n`.\n* `ContDiffOn \ud835\udd5c n f s`: expresses that `f` is `C^n` in `s`.\n* `ContDiffAt \ud835\udd5c n f x`: expresses that `f` is `C^n` around `x`.\n* `ContDiffWithinAt \ud835\udd5c n f s x`: expresses that `f` is `C^n` around `x` within the set `s`.\n* `iteratedFDerivWithin \ud835\udd5c n f s x` is an `n`-th derivative of `f` over the field `\ud835\udd5c` on the\n  set `s` at the point `x`. It is a continuous multilinear map from `E^n` to `F`, defined as a\n  derivative within `s` of `iteratedFDerivWithin \ud835\udd5c (n-1) f s` if one exists, and `0` otherwise.\n* `iteratedFDeriv \ud835\udd5c n f x` is the `n`-th derivative of `f` over the field `\ud835\udd5c` at the point `x`.\n  It is a continuous multilinear map from `E^n` to `F`, defined as a derivative of\n  `iteratedFDeriv \ud835\udd5c (n-1) f` if one exists, and `0` otherwise.\n\nIn sets of unique differentiability, `ContDiffOn \ud835\udd5c n f s` can be expressed in terms of the\nproperties of `iteratedFDerivWithin \ud835\udd5c m f s` for `m \u2264 n`. In the whole space,\n`ContDiff \ud835\udd5c n f` can be expressed in terms of the properties of `iteratedFDeriv \ud835\udd5c m f`\nfor `m \u2264 n`.\n\n## Implementation notes\n\nThe definitions in this file are designed to work on any field `\ud835\udd5c`. They are sometimes slightly more\ncomplicated than the naive definitions one would guess from the intuition over the real or complex\nnumbers, but they are designed to circumvent the lack of gluing properties and partitions of unity\nin general. In the usual situations, they coincide with the usual definitions.\n\n### Definition of `C^n` functions in domains\n\nOne could define `C^n` functions in a domain `s` by fixing an arbitrary choice of derivatives (this\nis what we do with `iteratedFDerivWithin`) and requiring that all these derivatives up to `n` are\ncontinuous. If the derivative is not unique, this could lead to strange behavior like two `C^n`\nfunctions `f` and `g` on `s` whose sum is not `C^n`. A better definition is thus to say that a\nfunction is `C^n` inside `s` if it admits a sequence of derivatives up to `n` inside `s`.\n\nThis definition still has the problem that a function which is locally `C^n` would not need to\nbe `C^n`, as different choices of sequences of derivatives around different points might possibly\nnot be glued together to give a globally defined sequence of derivatives. (Note that this issue\ncan not happen over reals, thanks to partition of unity, but the behavior over a general field is\nnot so clear, and we want a definition for general fields). Also, there are locality\nproblems for the order parameter: one could image a function which, for each `n`, has a nice\nsequence of derivatives up to order `n`, but they do not coincide for varying `n` and can therefore\nnot be glued to give rise to an infinite sequence of derivatives. This would give a function\nwhich is `C^n` for all `n`, but not `C^\u221e`. We solve this issue by putting locality conditions\nin space and order in our definition of `ContDiffWithinAt` and `ContDiffOn`.\nThe resulting definition is slightly more complicated to work with (in fact not so much), but it\ngives rise to completely satisfactory theorems.\n\nFor instance, with this definition, a real function which is `C^m` (but not better) on `(-1/m, 1/m)`\nfor each natural `m` is by definition `C^\u221e` at `0`.\n\nThere is another issue with the definition of `ContDiffWithinAt \ud835\udd5c n f s x`. We can\nrequire the existence and good behavior of derivatives up to order `n` on a neighborhood of `x`\nwithin `s`. However, this does not imply continuity or differentiability within `s` of the function\nat `x` when `x` does not belong to `s`. Therefore, we require such existence and good behavior on\na neighborhood of `x` within `s \u222a {x}` (which appears as `insert x s` in this file).\n\n### Side of the composition, and universe issues\n\nWith a na\u00efve direct definition, the `n`-th derivative of a function belongs to the space\n`E \u2192L[\ud835\udd5c] (E \u2192L[\ud835\udd5c] (E ... F)...)))` where there are n iterations of `E \u2192L[\ud835\udd5c]`. This space\nmay also be seen as the space of continuous multilinear functions on `n` copies of `E` with\nvalues in `F`, by uncurrying. This is the point of view that is usually adopted in textbooks,\nand that we also use. This means that the definition and the first proofs are slightly involved,\nas one has to keep track of the uncurrying operation. The uncurrying can be done from the\nleft or from the right, amounting to defining the `n+1`-th derivative either as the derivative of\nthe `n`-th derivative, or as the `n`-th derivative of the derivative.\nFor proofs, it would be more convenient to use the latter approach (from the right),\nas it means to prove things at the `n+1`-th step we only need to understand well enough the\nderivative in `E \u2192L[\ud835\udd5c] F` (contrary to the approach from the left, where one would need to know\nenough on the `n`-th derivative to deduce things on the `n+1`-th derivative).\n\nHowever, the definition from the right leads to a universe polymorphism problem: if we define\n`iteratedFDeriv \ud835\udd5c (n + 1) f x = iteratedFDeriv \ud835\udd5c n (fderiv \ud835\udd5c f) x` by induction, we need to\ngeneralize over all spaces (as `f` and `fderiv \ud835\udd5c f` don't take values in the same space). It is\nonly possible to generalize over all spaces in some fixed universe in an inductive definition.\nFor `f : E \u2192 F`, then `fderiv \ud835\udd5c f` is a map `E \u2192 (E \u2192L[\ud835\udd5c] F)`. Therefore, the definition will only\nwork if `F` and `E \u2192L[\ud835\udd5c] F` are in the same universe.\n\nThis issue does not appear with the definition from the left, where one does not need to generalize\nover all spaces. Therefore, we use the definition from the left. This means some proofs later on\nbecome a little bit more complicated: to prove that a function is `C^n`, the most efficient approach\nis to exhibit a formula for its `n`-th derivative and prove it is continuous (contrary to the\ninductive approach where one would prove smoothness statements without giving a formula for the\nderivative). In the end, this approach is still satisfactory as it is good to have formulas for the\niterated derivatives in various constructions.\n\nOne point where we depart from this explicit approach is in the proof of smoothness of a\ncomposition: there is a formula for the `n`-th derivative of a composition (Fa\u00e0 di Bruno's formula),\nbut it is very complicated and barely usable, while the inductive proof is very simple. Thus, we\ngive the inductive proof. As explained above, it works by generalizing over the target space, hence\nit only works well if all spaces belong to the same universe. To get the general version, we lift\nthings to a common universe using a trick.\n\n### Variables management\n\nThe textbook definitions and proofs use various identifications and abuse of notations, for instance\nwhen saying that the natural space in which the derivative lives, i.e.,\n`E \u2192L[\ud835\udd5c] (E \u2192L[\ud835\udd5c] ( ... \u2192L[\ud835\udd5c] F))`, is the same as a space of multilinear maps. When doing things\nformally, we need to provide explicit maps for these identifications, and chase some diagrams to see\neverything is compatible with the identifications. In particular, one needs to check that taking the\nderivative and then doing the identification, or first doing the identification and then taking the\nderivative, gives the same result. The key point for this is that taking the derivative commutes\nwith continuous linear equivalences. Therefore, we need to implement all our identifications with\ncontinuous linear equivs.\n\n## Notations\n\nWe use the notation `E [\u00d7n]\u2192L[\ud835\udd5c] F` for the space of continuous multilinear maps on `E^n` with\nvalues in `F`. This is the space in which the `n`-th derivative of a function from `E` to `F` lives.\n\nIn this file, we denote `\u22a4 : \u2115\u221e` with `\u221e`.\n\n## Tags\n\nderivative, differentiability, higher derivative, `C^n`, multilinear, Taylor series, formal series\n-/\n\nnoncomputable section\n\nopen scoped Classical\nopen NNReal Topology Filter\n\nlocal notation \"\u221e\" => (\u22a4 : \u2115\u221e)\n\n/-\nPorting note: These lines are not required in Mathlib4.\nattribute [local instance 1001]\n  NormedAddCommGroup.toAddCommGroup NormedSpace.toModule' AddCommGroup.toAddCommMonoid\n-/\n\nopen Set Fin Filter Function\n\nuniverse u uE uF uG uX\n\nvariable {\ud835\udd5c : Type u} [NontriviallyNormedField \ud835\udd5c] {E : Type uE} [NormedAddCommGroup E]\n  [NormedSpace \ud835\udd5c E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace \ud835\udd5c F] {G : Type uG}\n  [NormedAddCommGroup G] [NormedSpace \ud835\udd5c G] {X : Type uX} [NormedAddCommGroup X] [NormedSpace \ud835\udd5c X]\n  {s s\u2081 t u : Set E} {f f\u2081 : E \u2192 F} {g : F \u2192 G} {x x\u2080 : E} {c : F} {m n : \u2115\u221e}\n  {p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F}\n\n/-! ### Functions with a Taylor series on a domain -/\n\n/-- `HasFTaylorSeriesUpToOn n f p s` registers the fact that `p 0 = f` and `p (m+1)` is a\nderivative of `p m` for `m < n`, and is continuous for `m \u2264 n`. This is a predicate analogous to\n`HasFDerivWithinAt` but for higher order derivatives.\n\nNotice that `p` does not sum up to `f` on the diagonal (`FormalMultilinearSeries.sum`), even if\n`f` is analytic and `n = \u221e`: an additional `1/m!` factor on the `m`th term is necessary for that. -/\nstructure HasFTaylorSeriesUpToOn (n : \u2115\u221e) (f : E \u2192 F) (p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F)\n  (s : Set E) : Prop where\n  zero_eq : \u2200 x \u2208 s, (p x 0).uncurry0 = f x\n  protected fderivWithin : \u2200 m : \u2115, (m : \u2115\u221e) < n \u2192 \u2200 x \u2208 s,\n    HasFDerivWithinAt (p \u00b7 m) (p x m.succ).curryLeft s x\n  cont : \u2200 m : \u2115, (m : \u2115\u221e) \u2264 n \u2192 ContinuousOn (p \u00b7 m) s\n#align has_ftaylor_series_up_to_on HasFTaylorSeriesUpToOn\n\ntheorem HasFTaylorSeriesUpToOn.zero_eq' (h : HasFTaylorSeriesUpToOn n f p s) {x : E} (hx : x \u2208 s) :\n    p x 0 = (continuousMultilinearCurryFin0 \ud835\udd5c E F).symm (f x) := by\n  rw [\u2190 h.zero_eq x hx]\n  exact (p x 0).uncurry0_curry0.symm\n#align has_ftaylor_series_up_to_on.zero_eq' HasFTaylorSeriesUpToOn.zero_eq'\n\n/-- If two functions coincide on a set `s`, then a Taylor series for the first one is as well a\nTaylor series for the second one. -/\ntheorem HasFTaylorSeriesUpToOn.congr (h : HasFTaylorSeriesUpToOn n f p s)\n    (h\u2081 : \u2200 x \u2208 s, f\u2081 x = f x) : HasFTaylorSeriesUpToOn n f\u2081 p s := by\n  refine' \u27e8fun x hx => _, h.fderivWithin, h.cont\u27e9\n  rw [h\u2081 x hx]\n  exact h.zero_eq x hx\n#align has_ftaylor_series_up_to_on.congr HasFTaylorSeriesUpToOn.congr\n\ntheorem HasFTaylorSeriesUpToOn.mono (h : HasFTaylorSeriesUpToOn n f p s) {t : Set E} (hst : t \u2286 s) :\n    HasFTaylorSeriesUpToOn n f p t :=\n  \u27e8fun x hx => h.zero_eq x (hst hx), fun m hm x hx => (h.fderivWithin m hm x (hst hx)).mono hst,\n    fun m hm => (h.cont m hm).mono hst\u27e9\n#align has_ftaylor_series_up_to_on.mono HasFTaylorSeriesUpToOn.mono\n\ntheorem HasFTaylorSeriesUpToOn.of_le (h : HasFTaylorSeriesUpToOn n f p s) (hmn : m \u2264 n) :\n    HasFTaylorSeriesUpToOn m f p s :=\n  \u27e8h.zero_eq, fun k hk x hx => h.fderivWithin k (lt_of_lt_of_le hk hmn) x hx, fun k hk =>\n    h.cont k (le_trans hk hmn)\u27e9\n#align has_ftaylor_series_up_to_on.of_le HasFTaylorSeriesUpToOn.of_le\n\ntheorem HasFTaylorSeriesUpToOn.continuousOn (h : HasFTaylorSeriesUpToOn n f p s) :\n    ContinuousOn f s := by\n  have := (h.cont 0 bot_le).congr fun x hx => (h.zero_eq' hx).symm\n  rwa [\u2190 (continuousMultilinearCurryFin0 \ud835\udd5c E F).symm.comp_continuousOn_iff]\n#align has_ftaylor_series_up_to_on.continuous_on HasFTaylorSeriesUpToOn.continuousOn\n\ntheorem hasFTaylorSeriesUpToOn_zero_iff :\n    HasFTaylorSeriesUpToOn 0 f p s \u2194 ContinuousOn f s \u2227 \u2200 x \u2208 s, (p x 0).uncurry0 = f x := by\n  refine \u27e8fun H => \u27e8H.continuousOn, H.zero_eq\u27e9, fun H =>\n      \u27e8H.2, fun m hm => False.elim (not_le.2 hm bot_le), fun m hm \u21a6 ?_\u27e9\u27e9\n  obtain rfl : m = 0 := mod_cast hm.antisymm (zero_le _)\n  have : EqOn (p \u00b7 0) ((continuousMultilinearCurryFin0 \ud835\udd5c E F).symm \u2218 f) s := fun x hx \u21a6\n    (continuousMultilinearCurryFin0 \ud835\udd5c E F).eq_symm_apply.2 (H.2 x hx)\n  rw [continuousOn_congr this, LinearIsometryEquiv.comp_continuousOn_iff]\n  exact H.1\n#align has_ftaylor_series_up_to_on_zero_iff hasFTaylorSeriesUpToOn_zero_iff\n\ntheorem hasFTaylorSeriesUpToOn_top_iff :\n    HasFTaylorSeriesUpToOn \u221e f p s \u2194 \u2200 n : \u2115, HasFTaylorSeriesUpToOn n f p s := by\n  constructor\n  \u00b7 intro H n; exact H.of_le le_top\n  \u00b7 intro H\n    constructor\n    \u00b7 exact (H 0).zero_eq\n    \u00b7 intro m _\n      apply (H m.succ).fderivWithin m (WithTop.coe_lt_coe.2 (lt_add_one m))\n    \u00b7 intro m _\n      apply (H m).cont m le_rfl\n#align has_ftaylor_series_up_to_on_top_iff hasFTaylorSeriesUpToOn_top_iff\n\n/-- In the case that `n = \u221e` we don't need the continuity assumption in\n`HasFTaylorSeriesUpToOn`. -/\ntheorem hasFTaylorSeriesUpToOn_top_iff' :\n    HasFTaylorSeriesUpToOn \u221e f p s \u2194\n      (\u2200 x \u2208 s, (p x 0).uncurry0 = f x) \u2227\n        \u2200 m : \u2115, \u2200 x \u2208 s, HasFDerivWithinAt (fun y => p y m) (p x m.succ).curryLeft s x :=\n  -- Everything except for the continuity is trivial:\n  \u27e8fun h => \u27e8h.1, fun m => h.2 m (WithTop.coe_lt_top m)\u27e9, fun h =>\n    \u27e8h.1, fun m _ => h.2 m, fun m _ x hx =>\n      -- The continuity follows from the existence of a derivative:\n      (h.2 m x hx).continuousWithinAt\u27e9\u27e9\n#align has_ftaylor_series_up_to_on_top_iff' hasFTaylorSeriesUpToOn_top_iff'\n\n/-- If a function has a Taylor series at order at least `1`, then the term of order `1` of this\nseries is a derivative of `f`. -/\ntheorem HasFTaylorSeriesUpToOn.hasFDerivWithinAt (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 \u2264 n)\n    (hx : x \u2208 s) : HasFDerivWithinAt f (continuousMultilinearCurryFin1 \ud835\udd5c E F (p x 1)) s x := by\n  have A : \u2200 y \u2208 s, f y = (continuousMultilinearCurryFin0 \ud835\udd5c E F) (p y 0) := fun y hy \u21a6\n    (h.zero_eq y hy).symm\n  suffices H : HasFDerivWithinAt (continuousMultilinearCurryFin0 \ud835\udd5c E F \u2218 (p \u00b7 0))\n    (continuousMultilinearCurryFin1 \ud835\udd5c E F (p x 1)) s x from H.congr A (A x hx)\n  rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff']\n  have : ((0 : \u2115) : \u2115\u221e) < n := zero_lt_one.trans_le hn\n  convert h.fderivWithin _ this x hx\n  ext y v\n  change (p x 1) (snoc 0 y) = (p x 1) (cons y v)\n  congr with i\n  rw [Unique.eq_default (\u03b1 := Fin 1) i]\n  rfl\n#align has_ftaylor_series_up_to_on.has_fderiv_within_at HasFTaylorSeriesUpToOn.hasFDerivWithinAt\n\ntheorem HasFTaylorSeriesUpToOn.differentiableOn (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 \u2264 n) :\n    DifferentiableOn \ud835\udd5c f s := fun _x hx => (h.hasFDerivWithinAt hn hx).differentiableWithinAt\n#align has_ftaylor_series_up_to_on.differentiable_on HasFTaylorSeriesUpToOn.differentiableOn\n\n/-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then the term\nof order `1` of this series is a derivative of `f` at `x`. -/\ntheorem HasFTaylorSeriesUpToOn.hasFDerivAt (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 \u2264 n)\n    (hx : s \u2208 \ud835\udcdd x) : HasFDerivAt f (continuousMultilinearCurryFin1 \ud835\udd5c E F (p x 1)) x :=\n  (h.hasFDerivWithinAt hn (mem_of_mem_nhds hx)).hasFDerivAt hx\n#align has_ftaylor_series_up_to_on.has_fderiv_at HasFTaylorSeriesUpToOn.hasFDerivAt\n\n/-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then\nin a neighborhood of `x`, the term of order `1` of this series is a derivative of `f`. -/\ntheorem HasFTaylorSeriesUpToOn.eventually_hasFDerivAt (h : HasFTaylorSeriesUpToOn n f p s)\n    (hn : 1 \u2264 n) (hx : s \u2208 \ud835\udcdd x) :\n    \u2200\u1da0 y in \ud835\udcdd x, HasFDerivAt f (continuousMultilinearCurryFin1 \ud835\udd5c E F (p y 1)) y :=\n  (eventually_eventually_nhds.2 hx).mono fun _y hy => h.hasFDerivAt hn hy\n#align has_ftaylor_series_up_to_on.eventually_has_fderiv_at HasFTaylorSeriesUpToOn.eventually_hasFDerivAt\n\n/-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then\nit is differentiable at `x`. -/\ntheorem HasFTaylorSeriesUpToOn.differentiableAt (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 \u2264 n)\n    (hx : s \u2208 \ud835\udcdd x) : DifferentiableAt \ud835\udd5c f x :=\n  (h.hasFDerivAt hn hx).differentiableAt\n#align has_ftaylor_series_up_to_on.differentiable_at HasFTaylorSeriesUpToOn.differentiableAt\n\n/-- `p` is a Taylor series of `f` up to `n+1` if and only if `p` is a Taylor series up to `n`, and\n`p (n + 1)` is a derivative of `p n`. -/\ntheorem hasFTaylorSeriesUpToOn_succ_iff_left {n : \u2115} :\n    HasFTaylorSeriesUpToOn (n + 1) f p s \u2194\n      HasFTaylorSeriesUpToOn n f p s \u2227\n        (\u2200 x \u2208 s, HasFDerivWithinAt (fun y => p y n) (p x n.succ).curryLeft s x) \u2227\n          ContinuousOn (fun x => p x (n + 1)) s := by\n  constructor\n  \u00b7 exact fun h \u21a6 \u27e8h.of_le (WithTop.coe_le_coe.2 (Nat.le_succ n)),\n      h.fderivWithin _ (WithTop.coe_lt_coe.2 (lt_add_one n)), h.cont (n + 1) le_rfl\u27e9\n  \u00b7 intro h\n    constructor\n    \u00b7 exact h.1.zero_eq\n    \u00b7 intro m hm\n      by_cases h' : m < n\n      \u00b7 exact h.1.fderivWithin m (WithTop.coe_lt_coe.2 h')\n      \u00b7 have : m = n := Nat.eq_of_lt_succ_of_not_lt (WithTop.coe_lt_coe.1 hm) h'\n        rw [this]\n        exact h.2.1\n    \u00b7 intro m hm\n      by_cases h' : m \u2264 n\n      \u00b7 apply h.1.cont m (WithTop.coe_le_coe.2 h')\n      \u00b7 have : m = n + 1 := le_antisymm (WithTop.coe_le_coe.1 hm) (not_le.1 h')\n        rw [this]\n        exact h.2.2\n#align has_ftaylor_series_up_to_on_succ_iff_left hasFTaylorSeriesUpToOn_succ_iff_left\n\n-- Porting note: this was split out from `hasFTaylorSeriesUpToOn_succ_iff_right` to avoid a timeout.\ntheorem HasFTaylorSeriesUpToOn.shift_of_succ\n    {n : \u2115} (H : HasFTaylorSeriesUpToOn (n + 1 : \u2115) f p s) :\n    (HasFTaylorSeriesUpToOn n (fun x => continuousMultilinearCurryFin1 \ud835\udd5c E F (p x 1))\n      (fun x => (p x).shift)) s := by\n  constructor\n  \u00b7 intro x _\n    rfl\n  \u00b7 intro m (hm : (m : \u2115\u221e) < n) x (hx : x \u2208 s)\n    have A : (m.succ : \u2115\u221e) < n.succ := by\n      rw [Nat.cast_lt] at hm \u22a2\n      exact Nat.succ_lt_succ hm\n    change HasFDerivWithinAt ((continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F).symm \u2218 (p \u00b7 m.succ))\n      (p x m.succ.succ).curryRight.curryLeft s x\n    rw [((continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F).symm).comp_hasFDerivWithinAt_iff']\n    convert H.fderivWithin _ A x hx\n    ext y v\n    change p x (m + 2) (snoc (cons y (init v)) (v (last _))) = p x (m + 2) (cons y v)\n    rw [\u2190 cons_snoc_eq_snoc_cons, snoc_init_self]\n  \u00b7 intro m (hm : (m : \u2115\u221e) \u2264 n)\n    suffices A : ContinuousOn (p \u00b7 (m + 1)) s from\n      ((continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F).symm).continuous.comp_continuousOn A\n    refine H.cont _ ?_\n    rw [Nat.cast_le] at hm \u22a2\n    exact Nat.succ_le_succ hm\n\n/-- `p` is a Taylor series of `f` up to `n+1` if and only if `p.shift` is a Taylor series up to `n`\nfor `p 1`, which is a derivative of `f`. -/\ntheorem hasFTaylorSeriesUpToOn_succ_iff_right {n : \u2115} :\n    HasFTaylorSeriesUpToOn (n + 1 : \u2115) f p s \u2194\n      (\u2200 x \u2208 s, (p x 0).uncurry0 = f x) \u2227\n        (\u2200 x \u2208 s, HasFDerivWithinAt (fun y => p y 0) (p x 1).curryLeft s x) \u2227\n          HasFTaylorSeriesUpToOn n (fun x => continuousMultilinearCurryFin1 \ud835\udd5c E F (p x 1))\n            (fun x => (p x).shift) s := by\n  constructor\n  \u00b7 intro H\n    refine' \u27e8H.zero_eq, H.fderivWithin 0 (Nat.cast_lt.2 (Nat.succ_pos n)), _\u27e9\n    exact H.shift_of_succ\n  \u00b7 rintro \u27e8Hzero_eq, Hfderiv_zero, Htaylor\u27e9\n    constructor\n    \u00b7 exact Hzero_eq\n    \u00b7 intro m (hm : (m : \u2115\u221e) < n.succ) x (hx : x \u2208 s)\n      cases' m with m\n      \u00b7 exact Hfderiv_zero x hx\n      \u00b7 have A : (m : \u2115\u221e) < n := by\n          rw [Nat.cast_lt] at hm \u22a2\n          exact Nat.lt_of_succ_lt_succ hm\n        have :\n          HasFDerivWithinAt ((continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F).symm \u2218 (p \u00b7 m.succ))\n            ((p x).shift m.succ).curryLeft s x := Htaylor.fderivWithin _ A x hx\n        rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff'] at this\n        convert this\n        ext y v\n        change\n          (p x (Nat.succ (Nat.succ m))) (cons y v) =\n            (p x m.succ.succ) (snoc (cons y (init v)) (v (last _)))\n        rw [\u2190 cons_snoc_eq_snoc_cons, snoc_init_self]\n    \u00b7 intro m (hm : (m : \u2115\u221e) \u2264 n.succ)\n      cases' m with m\n      \u00b7 have : DifferentiableOn \ud835\udd5c (fun x => p x 0) s := fun x hx =>\n          (Hfderiv_zero x hx).differentiableWithinAt\n        exact this.continuousOn\n      \u00b7 refine (continuousMultilinearCurryRightEquiv' \ud835\udd5c m E F).symm.comp_continuousOn_iff.mp ?_\n        refine Htaylor.cont _ ?_\n        rw [Nat.cast_le] at hm \u22a2\n        exact Nat.lt_succ_iff.mp hm\n#align has_ftaylor_series_up_to_on_succ_iff_right hasFTaylorSeriesUpToOn_succ_iff_right\n\n/-! ### Smooth functions within a set around a point -/\n\nvariable (\ud835\udd5c)\n\n/-- A function is continuously differentiable up to order `n` within a set `s` at a point `x` if\nit admits continuous derivatives up to order `n` in a neighborhood of `x` in `s \u222a {x}`.\nFor `n = \u221e`, we only require that this holds up to any finite order (where the neighborhood may\ndepend on the finite order we consider).\n\nFor instance, a real function which is `C^m` on `(-1/m, 1/m)` for each natural `m`, but not\nbetter, is `C^\u221e` at `0` within `univ`.\n-/\ndef ContDiffWithinAt (n : \u2115\u221e) (f : E \u2192 F) (s : Set E) (x : E) : Prop :=\n  \u2200 m : \u2115, (m : \u2115\u221e) \u2264 n \u2192 \u2203 u \u2208 \ud835\udcdd[insert x s] x,\n    \u2203 p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F, HasFTaylorSeriesUpToOn m f p u\n#align cont_diff_within_at ContDiffWithinAt\n\nvariable {\ud835\udd5c}\n\ntheorem contDiffWithinAt_nat {n : \u2115} :\n    ContDiffWithinAt \ud835\udd5c n f s x \u2194 \u2203 u \u2208 \ud835\udcdd[insert x s] x,\n      \u2203 p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F, HasFTaylorSeriesUpToOn n f p u :=\n  \u27e8fun H => H n le_rfl, fun \u27e8u, hu, p, hp\u27e9 _m hm => \u27e8u, hu, p, hp.of_le hm\u27e9\u27e9\n#align cont_diff_within_at_nat contDiffWithinAt_nat\n\ntheorem ContDiffWithinAt.of_le (h : ContDiffWithinAt \ud835\udd5c n f s x) (hmn : m \u2264 n) :\n    ContDiffWithinAt \ud835\udd5c m f s x := fun k hk => h k (le_trans hk hmn)\n#align cont_diff_within_at.of_le ContDiffWithinAt.of_le\n\ntheorem contDiffWithinAt_iff_forall_nat_le :\n    ContDiffWithinAt \ud835\udd5c n f s x \u2194 \u2200 m : \u2115, \u2191m \u2264 n \u2192 ContDiffWithinAt \ud835\udd5c m f s x :=\n  \u27e8fun H _m hm => H.of_le hm, fun H m hm => H m hm _ le_rfl\u27e9\n#align cont_diff_within_at_iff_forall_nat_le contDiffWithinAt_iff_forall_nat_le\n\ntheorem contDiffWithinAt_top : ContDiffWithinAt \ud835\udd5c \u221e f s x \u2194 \u2200 n : \u2115, ContDiffWithinAt \ud835\udd5c n f s x :=\n  contDiffWithinAt_iff_forall_nat_le.trans <| by simp only [forall_prop_of_true, le_top]\n#align cont_diff_within_at_top contDiffWithinAt_top\n\ntheorem ContDiffWithinAt.continuousWithinAt (h : ContDiffWithinAt \ud835\udd5c n f s x) :\n    ContinuousWithinAt f s x := by\n  rcases h 0 bot_le with \u27e8u, hu, p, H\u27e9\n  rw [mem_nhdsWithin_insert] at hu\n  exact (H.continuousOn.continuousWithinAt hu.1).mono_of_mem hu.2\n#align cont_diff_within_at.continuous_within_at ContDiffWithinAt.continuousWithinAt\n\ntheorem ContDiffWithinAt.congr_of_eventuallyEq (h : ContDiffWithinAt \ud835\udd5c n f s x)\n    (h\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f) (hx : f\u2081 x = f x) : ContDiffWithinAt \ud835\udd5c n f\u2081 s x := fun m hm =>\n  let \u27e8u, hu, p, H\u27e9 := h m hm\n  \u27e8{ x \u2208 u | f\u2081 x = f x }, Filter.inter_mem hu (mem_nhdsWithin_insert.2 \u27e8hx, h\u2081\u27e9), p,\n    (H.mono (sep_subset _ _)).congr fun _ => And.right\u27e9\n#align cont_diff_within_at.congr_of_eventually_eq ContDiffWithinAt.congr_of_eventuallyEq\n\ntheorem ContDiffWithinAt.congr_of_eventuallyEq_insert (h : ContDiffWithinAt \ud835\udd5c n f s x)\n    (h\u2081 : f\u2081 =\u1da0[\ud835\udcdd[insert x s] x] f) : ContDiffWithinAt \ud835\udd5c n f\u2081 s x :=\n  h.congr_of_eventuallyEq (nhdsWithin_mono x (subset_insert x s) h\u2081)\n    (mem_of_mem_nhdsWithin (mem_insert x s) h\u2081 : _)\n#align cont_diff_within_at.congr_of_eventually_eq_insert ContDiffWithinAt.congr_of_eventuallyEq_insert\n\ntheorem ContDiffWithinAt.congr_of_eventually_eq' (h : ContDiffWithinAt \ud835\udd5c n f s x)\n    (h\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f) (hx : x \u2208 s) : ContDiffWithinAt \ud835\udd5c n f\u2081 s x :=\n  h.congr_of_eventuallyEq h\u2081 <| h\u2081.self_of_nhdsWithin hx\n#align cont_diff_within_at.congr_of_eventually_eq' ContDiffWithinAt.congr_of_eventually_eq'\n\ntheorem Filter.EventuallyEq.contDiffWithinAt_iff (h\u2081 : f\u2081 =\u1da0[\ud835\udcdd[s] x] f) (hx : f\u2081 x = f x) :\n    ContDiffWithinAt \ud835\udd5c n f\u2081 s x \u2194 ContDiffWithinAt \ud835\udd5c n f s x :=\n  \u27e8fun H => ContDiffWithinAt.congr_of_eventuallyEq H h\u2081.symm hx.symm, fun H =>\n    H.congr_of_eventuallyEq h\u2081 hx\u27e9\n#align filter.eventually_eq.cont_diff_within_at_iff Filter.EventuallyEq.contDiffWithinAt_iff\n\ntheorem ContDiffWithinAt.congr (h : ContDiffWithinAt \ud835\udd5c n f s x) (h\u2081 : \u2200 y \u2208 s, f\u2081 y = f y)\n    (hx : f\u2081 x = f x) : ContDiffWithinAt \ud835\udd5c n f\u2081 s x :=\n  h.congr_of_eventuallyEq (Filter.eventuallyEq_of_mem self_mem_nhdsWithin h\u2081) hx\n#align cont_diff_within_at.congr ContDiffWithinAt.congr\n\ntheorem ContDiffWithinAt.congr' (h : ContDiffWithinAt \ud835\udd5c n f s x) (h\u2081 : \u2200 y \u2208 s, f\u2081 y = f y)\n    (hx : x \u2208 s) : ContDiffWithinAt \ud835\udd5c n f\u2081 s x :=\n  h.congr h\u2081 (h\u2081 _ hx)\n#align cont_diff_within_at.congr' ContDiffWithinAt.congr'\n\ntheorem ContDiffWithinAt.mono_of_mem (h : ContDiffWithinAt \ud835\udd5c n f s x) {t : Set E}\n    (hst : s \u2208 \ud835\udcdd[t] x) : ContDiffWithinAt \ud835\udd5c n f t x := by\n  intro m hm\n  rcases h m hm with \u27e8u, hu, p, H\u27e9\n  exact \u27e8u, nhdsWithin_le_of_mem (insert_mem_nhdsWithin_insert hst) hu, p, H\u27e9\n#align cont_diff_within_at.mono_of_mem ContDiffWithinAt.mono_of_mem\n\ntheorem ContDiffWithinAt.mono (h : ContDiffWithinAt \ud835\udd5c n f s x) {t : Set E} (hst : t \u2286 s) :\n    ContDiffWithinAt \ud835\udd5c n f t x :=\n  h.mono_of_mem <| Filter.mem_of_superset self_mem_nhdsWithin hst\n#align cont_diff_within_at.mono ContDiffWithinAt.mono\n\ntheorem ContDiffWithinAt.congr_nhds (h : ContDiffWithinAt \ud835\udd5c n f s x) {t : Set E}\n    (hst : \ud835\udcdd[s] x = \ud835\udcdd[t] x) : ContDiffWithinAt \ud835\udd5c n f t x :=\n  h.mono_of_mem <| hst \u25b8 self_mem_nhdsWithin\n#align cont_diff_within_at.congr_nhds ContDiffWithinAt.congr_nhds\n\ntheorem contDiffWithinAt_congr_nhds {t : Set E} (hst : \ud835\udcdd[s] x = \ud835\udcdd[t] x) :\n    ContDiffWithinAt \ud835\udd5c n f s x \u2194 ContDiffWithinAt \ud835\udd5c n f t x :=\n  \u27e8fun h => h.congr_nhds hst, fun h => h.congr_nhds hst.symm\u27e9\n#align cont_diff_within_at_congr_nhds contDiffWithinAt_congr_nhds\n\ntheorem contDiffWithinAt_inter' (h : t \u2208 \ud835\udcdd[s] x) :\n    ContDiffWithinAt \ud835\udd5c n f (s \u2229 t) x \u2194 ContDiffWithinAt \ud835\udd5c n f s x :=\n  contDiffWithinAt_congr_nhds <| Eq.symm <| nhdsWithin_restrict'' _ h\n#align cont_diff_within_at_inter' contDiffWithinAt_inter'\n\ntheorem contDiffWithinAt_inter (h : t \u2208 \ud835\udcdd x) :\n    ContDiffWithinAt \ud835\udd5c n f (s \u2229 t) x \u2194 ContDiffWithinAt \ud835\udd5c n f s x :=\n  contDiffWithinAt_inter' (mem_nhdsWithin_of_mem_nhds h)\n#align cont_diff_within_at_inter contDiffWithinAt_inter\n\ntheorem contDiffWithinAt_insert_self :\n    ContDiffWithinAt \ud835\udd5c n f (insert x s) x \u2194 ContDiffWithinAt \ud835\udd5c n f s x := by\n  simp_rw [ContDiffWithinAt, insert_idem]\n\ntheorem contDiffWithinAt_insert {y : E} :\n    ContDiffWithinAt \ud835\udd5c n f (insert y s) x \u2194 ContDiffWithinAt \ud835\udd5c n f s x := by\n  rcases eq_or_ne x y with (rfl | h)\n  \u00b7 exact contDiffWithinAt_insert_self\n  simp_rw [ContDiffWithinAt, insert_comm x y, nhdsWithin_insert_of_ne h]\n#align cont_diff_within_at_insert contDiffWithinAt_insert\n\nalias \u27e8ContDiffWithinAt.of_insert, ContDiffWithinAt.insert'\u27e9 := contDiffWithinAt_insert\n#align cont_diff_within_at.of_insert ContDiffWithinAt.of_insert\n#align cont_diff_within_at.insert' ContDiffWithinAt.insert'\n\nprotected theorem ContDiffWithinAt.insert (h : ContDiffWithinAt \ud835\udd5c n f s x) :\n    ContDiffWithinAt \ud835\udd5c n f (insert x s) x :=\n  h.insert'\n#align cont_diff_within_at.insert ContDiffWithinAt.insert\n\n/-- If a function is `C^n` within a set at a point, with `n \u2265 1`, then it is differentiable\nwithin this set at this point. -/\ntheorem ContDiffWithinAt.differentiable_within_at' (h : ContDiffWithinAt \ud835\udd5c n f s x) (hn : 1 \u2264 n) :\n    DifferentiableWithinAt \ud835\udd5c f (insert x s) x := by\n  rcases h 1 hn with \u27e8u, hu, p, H\u27e9\n  rcases mem_nhdsWithin.1 hu with \u27e8t, t_open, xt, tu\u27e9\n  rw [inter_comm] at tu\n  have := ((H.mono tu).differentiableOn le_rfl) x \u27e8mem_insert x s, xt\u27e9\n  exact (differentiableWithinAt_inter (IsOpen.mem_nhds t_open xt)).1 this\n#align cont_diff_within_at.differentiable_within_at' ContDiffWithinAt.differentiable_within_at'\n\ntheorem ContDiffWithinAt.differentiableWithinAt (h : ContDiffWithinAt \ud835\udd5c n f s x) (hn : 1 \u2264 n) :\n    DifferentiableWithinAt \ud835\udd5c f s x :=\n  (h.differentiable_within_at' hn).mono (subset_insert x s)\n#align cont_diff_within_at.differentiable_within_at ContDiffWithinAt.differentiableWithinAt\n\n/-- A function is `C^(n + 1)` on a domain iff locally, it has a derivative which is `C^n`. -/\ntheorem contDiffWithinAt_succ_iff_hasFDerivWithinAt {n : \u2115} :\n    ContDiffWithinAt \ud835\udd5c (n + 1 : \u2115) f s x \u2194 \u2203 u \u2208 \ud835\udcdd[insert x s] x, \u2203 f' : E \u2192 E \u2192L[\ud835\udd5c] F,\n      (\u2200 x \u2208 u, HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffWithinAt \ud835\udd5c n f' u x := by\n  constructor\n  \u00b7 intro h\n    rcases h n.succ le_rfl with \u27e8u, hu, p, Hp\u27e9\n    refine'\n      \u27e8u, hu, fun y => (continuousMultilinearCurryFin1 \ud835\udd5c E F) (p y 1), fun y hy =>\n        Hp.hasFDerivWithinAt (WithTop.coe_le_coe.2 (Nat.le_add_left 1 n)) hy, _\u27e9\n    intro m hm\n    refine' \u27e8u, _, fun y : E => (p y).shift, _\u27e9\n    \u00b7 -- Porting note: without the explicit argument Lean is not sure of the type.\n      convert @self_mem_nhdsWithin _ _ x u\n      have : x \u2208 insert x s := by simp\n      exact insert_eq_of_mem (mem_of_mem_nhdsWithin this hu)\n    \u00b7 rw [hasFTaylorSeriesUpToOn_succ_iff_right] at Hp\n      exact Hp.2.2.of_le hm\n  \u00b7 rintro \u27e8u, hu, f', f'_eq_deriv, Hf'\u27e9\n    rw [contDiffWithinAt_nat]\n    rcases Hf' n le_rfl with \u27e8v, hv, p', Hp'\u27e9\n    refine' \u27e8v \u2229 u, _, fun x => (p' x).unshift (f x), _\u27e9\n    \u00b7 apply Filter.inter_mem _ hu\n      apply nhdsWithin_le_of_mem hu\n      exact nhdsWithin_mono _ (subset_insert x u) hv\n    \u00b7 rw [hasFTaylorSeriesUpToOn_succ_iff_right]\n      refine' \u27e8fun y _ => rfl, fun y hy => _, _\u27e9\n      \u00b7 change\n          HasFDerivWithinAt (fun z => (continuousMultilinearCurryFin0 \ud835\udd5c E F).symm (f z))\n            (FormalMultilinearSeries.unshift (p' y) (f y) 1).curryLeft (v \u2229 u) y\n        -- Porting note: needed `erw` here.\n        -- https://github.com/leanprover-community/mathlib4/issues/5164\n        erw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff']\n        convert (f'_eq_deriv y hy.2).mono (inter_subset_right v u)\n        rw [\u2190 Hp'.zero_eq y hy.1]\n        ext z\n        change ((p' y 0) (init (@cons 0 (fun _ => E) z 0))) (@cons 0 (fun _ => E) z 0 (last 0)) =\n          ((p' y 0) 0) z\n        congr\n        norm_num [eq_iff_true_of_subsingleton]\n      \u00b7 convert (Hp'.mono (inter_subset_left v u)).congr fun x hx => Hp'.zero_eq x hx.1 using 1\n        \u00b7 ext x y\n          change p' x 0 (init (@snoc 0 (fun _ : Fin 1 => E) 0 y)) y = p' x 0 0 y\n          rw [init_snoc]\n        \u00b7 ext x k v y\n          change p' x k (init (@snoc k (fun _ : Fin k.succ => E) v y))\n            (@snoc k (fun _ : Fin k.succ => E) v y (last k)) = p' x k v y\n          rw [snoc_last, init_snoc]\n#align cont_diff_within_at_succ_iff_has_fderiv_within_at contDiffWithinAt_succ_iff_hasFDerivWithinAt\n\n/-- A version of `contDiffWithinAt_succ_iff_hasFDerivWithinAt` where all derivatives\n  are taken within the same set. -/\ntheorem contDiffWithinAt_succ_iff_hasFDerivWithinAt' {n : \u2115} :\n    ContDiffWithinAt \ud835\udd5c (n + 1 : \u2115) f s x \u2194\n      \u2203 u \u2208 \ud835\udcdd[insert x s] x, u \u2286 insert x s \u2227 \u2203 f' : E \u2192 E \u2192L[\ud835\udd5c] F,\n        (\u2200 x \u2208 u, HasFDerivWithinAt f (f' x) s x) \u2227 ContDiffWithinAt \ud835\udd5c n f' s x := by\n  refine' \u27e8fun hf => _, _\u27e9\n  \u00b7 obtain \u27e8u, hu, f', huf', hf'\u27e9 := contDiffWithinAt_succ_iff_hasFDerivWithinAt.mp hf\n    obtain \u27e8w, hw, hxw, hwu\u27e9 := mem_nhdsWithin.mp hu\n    rw [inter_comm] at hwu\n    refine' \u27e8insert x s \u2229 w, inter_mem_nhdsWithin _ (hw.mem_nhds hxw), inter_subset_left _ _, f',\n      fun y hy => _, _\u27e9\n    \u00b7 refine' ((huf' y <| hwu hy).mono hwu).mono_of_mem _\n      refine' mem_of_superset _ (inter_subset_inter_left _ (subset_insert _ _))\n      exact inter_mem_nhdsWithin _ (hw.mem_nhds hy.2)\n    \u00b7 exact hf'.mono_of_mem (nhdsWithin_mono _ (subset_insert _ _) hu)\n  \u00b7 rw [\u2190 contDiffWithinAt_insert, contDiffWithinAt_succ_iff_hasFDerivWithinAt,\n      insert_eq_of_mem (mem_insert _ _)]\n    rintro \u27e8u, hu, hus, f', huf', hf'\u27e9\n    exact \u27e8u, hu, f', fun y hy => (huf' y hy).insert'.mono hus, hf'.insert.mono hus\u27e9\n#align cont_diff_within_at_succ_iff_has_fderiv_within_at' contDiffWithinAt_succ_iff_hasFDerivWithinAt'\n\n/-! ### Smooth functions within a set -/\n\nvariable (\ud835\udd5c)\n\n/-- A function is continuously differentiable up to `n` on `s` if, for any point `x` in `s`, it\nadmits continuous derivatives up to order `n` on a neighborhood of `x` in `s`.\n\nFor `n = \u221e`, we only require that this holds up to any finite order (where the neighborhood may\ndepend on the finite order we consider).\n-/\ndef ContDiffOn (n : \u2115\u221e) (f : E \u2192 F) (s : Set E) : Prop :=\n  \u2200 x \u2208 s, ContDiffWithinAt \ud835\udd5c n f s x\n#align cont_diff_on ContDiffOn\n\nvariable {\ud835\udd5c}\n\ntheorem HasFTaylorSeriesUpToOn.contDiffOn {f' : E \u2192 FormalMultilinearSeries \ud835\udd5c E F}\n    (hf : HasFTaylorSeriesUpToOn n f f' s) : ContDiffOn \ud835\udd5c n f s := by\n  intro x hx m hm\n  use s\n  simp only [Set.insert_eq_of_mem hx, self_mem_nhdsWithin, true_and_iff]\n  exact \u27e8f', hf.of_le hm\u27e9\n#align has_ftaylor_series_up_to_on.cont_diff_on HasFTaylorSeriesUpToOn.contDiffOn\n\ntheorem ContDiffOn.contDiffWithinAt (h : ContDiffOn \ud835\udd5c n f s) (hx : x \u2208 s) :\n    ContDiffWithinAt \ud835\udd5c n f s x :=\n  h x hx\n#align cont_diff_on.cont_diff_within_at ContDiffOn.contDiffWithinAt\n\ntheorem ContDiffWithinAt.contDiffOn' {m : \u2115} (hm : (m : \u2115\u221e) \u2264 n)\n    (h : ContDiffWithinAt \ud835\udd5c n f s x) :\n    \u2203 u, IsOpen u \u2227 x \u2208 u \u2227 ContDiffOn \ud835\udd5c m f (insert x s \u2229 u) := by\n  rcases h m hm with \u27e8t, ht, p, hp\u27e9\n  rcases mem_nhdsWithin.1 ht with \u27e8u, huo, hxu, hut\u27e9\n  rw [inter_comm] at hut\n  exact \u27e8u, huo, hxu, (hp.mono hut).contDiffOn\u27e9\n#align cont_diff_within_at.cont_diff_on' ContDiffWithinAt.contDiffOn'\n\ntheorem ContDiffWithinAt.contDiffOn {m : \u2115} (hm : (m : \u2115\u221e) \u2264 n) (h : ContDiffWithinAt \ud835\udd5c n f s x) :\n    \u2203 u \u2208 \ud835\udcdd[insert x s] x, u \u2286 insert x s \u2227 ContDiffOn \ud835\udd5c m f u :=\n  let \u27e8_u, uo, xu, h\u27e9 := h.contDiffOn' hm\n  \u27e8_, inter_mem_nhdsWithin _ (uo.mem_nhds xu), inter_subset_left _ _, h\u27e9\n#align cont_diff_within_at.cont_diff_on ContDiffWithinAt.contDiffOn\n\nprotected theorem ContDiffWithinAt.eventually {n : \u2115} (h : ContDiffWithinAt \ud835\udd5c n f s x) :\n    \u2200\u1da0 y in \ud835\udcdd[insert x s] x, ContDiffWithinAt \ud835\udd5c n f s y := by\n  rcases h.contDiffOn le_rfl with \u27e8u, hu, _, hd\u27e9\n  have : \u2200\u1da0 y : E in \ud835\udcdd[insert x s] x, u \u2208 \ud835\udcdd[insert x s] y \u2227 y \u2208 u :=\n    (eventually_nhdsWithin_nhdsWithin.2 hu).and hu\n  refine' this.mono fun y hy => (hd y hy.2).mono_of_mem _\n  exact nhdsWithin_mono y (subset_insert _ _) hy.1\n#align cont_diff_within_at.eventually ContDiffWithinAt.eventually\n\ntheorem ContDiffOn.of_le (h : ContDiffOn \ud835\udd5c n f s) (hmn : m \u2264 n) : ContDiffOn \ud835\udd5c m f s := fun x hx =>\n  (h x hx).of_le hmn\n#align cont_diff_on.of_le ContDiffOn.of_le\n\ntheorem ContDiffOn.of_succ {n : \u2115} (h : ContDiffOn \ud835\udd5c (n + 1) f s) : ContDiffOn \ud835\udd5c n f s :=\n  h.of_le <| WithTop.coe_le_coe.mpr le_self_add\n#align cont_diff_on.of_succ ContDiffOn.of_succ\n\ntheorem ContDiffOn.one_of_succ {n : \u2115} (h : ContDiffOn \ud835\udd5c (n + 1) f s) : ContDiffOn \ud835\udd5c 1 f s :=\n  h.of_le <| WithTop.coe_le_coe.mpr le_add_self\n#align cont_diff_on.one_of_succ ContDiffOn.one_of_succ\n\ntheorem contDiffOn_iff_forall_nat_le : ContDiffOn \ud835\udd5c n f s \u2194 \u2200 m : \u2115, \u2191m \u2264 n \u2192 ContDiffOn \ud835\udd5c m f s :=\n  \u27e8fun H _ hm => H.of_le hm, fun H x hx m hm => H m hm x hx m le_rfl\u27e9\n#align cont_diff_on_iff_forall_nat_le contDiffOn_iff_forall_nat_le\n\ntheorem contDiffOn_top : ContDiffOn \ud835\udd5c \u221e f s \u2194 \u2200 n : \u2115, ContDiffOn \ud835\udd5c n f s :=\n  contDiffOn_iff_forall_nat_le.trans <| by simp only [le_top, forall_prop_of_true]\n#align cont_diff_on_top contDiffOn_top\n\ntheorem contDiffOn_all_iff_nat : (\u2200 n, ContDiffOn \ud835\udd5c n f s) \u2194 \u2200 n : \u2115, ContDiffOn \ud835\udd5c n f s := by\n  refine' \u27e8fun H n => H n, _\u27e9\n  rintro H (_ | n)\n  exacts [contDiffOn_top.2 H, H n]\n#align cont_diff_on_all_iff_nat contDiffOn_all_iff_nat\n\ntheorem ContDiffOn.continuousOn (h : ContDiffOn \ud835\udd5c n f s) : ContinuousOn f s := fun x hx =>\n  (h x hx).continuousWithinAt\n#align cont_diff_on.continuous_on ContDiffOn.continuousOn\n\ntheorem ContDiffOn.congr (h : ContDiffOn \ud835\udd5c n f s) (h\u2081 : \u2200 x \u2208 s, f\u2081 x = f x) :\n    ContDiffOn \ud835\udd5c n f\u2081 s := fun x hx => (h x hx).congr h\u2081 (h\u2081 x hx)\n#align cont_diff_on.congr ContDiffOn.congr\n\ntheorem contDiffOn_congr (h\u2081 : \u2200 x \u2208 s, f\u2081 x = f x) : ContDiffOn \ud835\udd5c n f\u2081 s \u2194 ContDiffOn \ud835\udd5c n f s :=\n  \u27e8fun H => H.congr fun x hx => (h\u2081 x hx).symm, fun H => H.congr h\u2081\u27e9\n#align cont_diff_on_congr contDiffOn_congr\n\ntheorem ContDiffOn.mono (h : ContDiffOn \ud835\udd5c n f s) {t : Set E} (hst : t \u2286 s) : ContDiffOn \ud835\udd5c n f t :=\n  fun x hx => (h x (hst hx)).mono hst\n#align cont_diff_on.mono ContDiffOn.mono\n\ntheorem ContDiffOn.congr_mono (hf : ContDiffOn \ud835\udd5c n f s) (h\u2081 : \u2200 x \u2208 s\u2081, f\u2081 x = f x) (hs : s\u2081 \u2286 s) :\n    ContDiffOn \ud835\udd5c n f\u2081 s\u2081 :=\n  (hf.mono hs).congr h\u2081\n#align cont_diff_on.congr_mono ContDiffOn.congr_mono\n\n/-- If a function is `C^n` on a set with `n \u2265 1`, then it is differentiable there. -/\ntheorem ContDiffOn.differentiableOn (h : ContDiffOn \ud835\udd5c n f s) (hn : 1 \u2264 n) :\n    DifferentiableOn \ud835\udd5c f s := fun x hx => (h x hx).differentiableWithinAt hn\n#align cont_diff_on.differentiable_on ContDiffOn.differentiableOn\n\n/-- If a function is `C^n` around each point in a set, then it is `C^n` on the set. -/\ntheorem contDiffOn_of_locally_contDiffOn\n    (h : \u2200 x \u2208 s, \u2203 u, IsOpen u \u2227 x \u2208 u \u2227 ContDiffOn \ud835\udd5c n f (s \u2229 u)) : ContDiffOn \ud835\udd5c n f s := by\n  intro x xs\n  rcases h x xs with \u27e8u, u_open, xu, hu\u27e9\n  apply (contDiffWithinAt_inter _).1 (hu x \u27e8xs, xu\u27e9)\n  exact IsOpen.mem_nhds u_open xu\n#align cont_diff_on_of_locally_cont_diff_on contDiffOn_of_locally_contDiffOn\n\n/-- A function is `C^(n + 1)` on a domain iff locally, it has a derivative which is `C^n`. -/\ntheorem contDiffOn_succ_iff_hasFDerivWithinAt {n : \u2115} :\n    ContDiffOn \ud835\udd5c (n + 1 : \u2115) f s \u2194\n      \u2200 x \u2208 s, \u2203 u \u2208 \ud835\udcdd[insert x s] x, \u2203 f' : E \u2192 E \u2192L[\ud835\udd5c] F,\n        (\u2200 x \u2208 u, HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffOn \ud835\udd5c n f' u := by\n  constructor\n  \u00b7 intro h x hx\n    rcases (h x hx) n.succ le_rfl with \u27e8u, hu, p, Hp\u27e9\n    refine'\n      \u27e8u, hu, fun y => (continuousMultilinearCurryFin1 \ud835\udd5c E F) (p y 1), fun y hy =>\n        Hp.hasFDerivWithinAt (WithTop.coe_le_coe.2 (Nat.le_add_left 1 n)) hy, _\u27e9\n    rw [hasFTaylorSeriesUpToOn_succ_iff_right] at Hp\n    intro z hz m hm\n    refine' \u27e8u, _, fun x : E => (p x).shift, Hp.2.2.of_le hm\u27e9\n    -- Porting note: without the explicit arguments `convert` can not determine the type.\n    convert @self_mem_nhdsWithin _ _ z u\n    exact insert_eq_of_mem hz\n  \u00b7 intro h x hx\n    rw [contDiffWithinAt_succ_iff_hasFDerivWithinAt]\n    rcases h x hx with \u27e8u, u_nhbd, f', hu, hf'\u27e9\n    have : x \u2208 u := mem_of_mem_nhdsWithin (mem_insert _ _) u_nhbd\n    exact \u27e8u, u_nhbd, f', hu, hf' x this\u27e9\n#align cont_diff_on_succ_iff_has_fderiv_within_at contDiffOn_succ_iff_hasFDerivWithinAt\n\n/-! ### Iterated derivative within a set -/\n\n\nvariable (\ud835\udd5c)\n\n/-- The `n`-th derivative of a function along a set, defined inductively by saying that the `n+1`-th\nderivative of `f` is the derivative of the `n`-th derivative of `f` along this set, together with\nan uncurrying step to see it as a multilinear map in `n+1` variables..\n-/\nnoncomputable def iteratedFDerivWithin (n : \u2115) (f : E \u2192 F) (s : Set E) : E \u2192 E[\u00d7n]\u2192L[\ud835\udd5c] F :=\n  Nat.recOn n (fun x => ContinuousMultilinearMap.curry0 \ud835\udd5c E (f x)) fun _ rec x =>\n    ContinuousLinearMap.uncurryLeft (fderivWithin \ud835\udd5c rec s x)\n#align iterated_fderiv_within iteratedFDerivWithin\n\n/-- Formal Taylor series associated to a function within a set. -/\ndef ftaylorSeriesWithin (f : E \u2192 F) (s : Set E) (x : E) : FormalMultilinearSeries \ud835\udd5c E F := fun n =>\n  iteratedFDerivWithin \ud835\udd5c n f s x\n#align ftaylor_series_within ftaylorSeriesWithin\n\nvariable {\ud835\udd5c}\n\n@[simp]\ntheorem iteratedFDerivWithin_zero_apply (m : Fin 0 \u2192 E) :\n    (iteratedFDerivWithin \ud835\udd5c 0 f s x : (Fin 0 \u2192 E) \u2192 F) m = f x :=\n  rfl\n#align iterated_fderiv_within_zero_apply iteratedFDerivWithin_zero_apply\n\ntheorem iteratedFDerivWithin_zero_eq_comp :\n    iteratedFDerivWithin \ud835\udd5c 0 f s = (continuousMultilinearCurryFin0 \ud835\udd5c E F).symm \u2218 f :=\n  rfl\n#align iterated_fderiv_within_zero_eq_comp iteratedFDerivWithin_zero_eq_comp\n\n@[simp]\ntheorem norm_iteratedFDerivWithin_zero : \u2016iteratedFDerivWithin \ud835\udd5c 0 f s x\u2016 = \u2016f x\u2016 := by\n  -- Porting note: added `comp_apply`.\n  rw [iteratedFDerivWithin_zero_eq_comp, comp_apply, LinearIsometryEquiv.norm_map]\n#align norm_iterated_fderiv_within_zero norm_iteratedFDerivWithin_zero\n\ntheorem iteratedFDerivWithin_succ_apply_left {n : \u2115} (m : Fin (n + 1) \u2192 E) :\n    (iteratedFDerivWithin \ud835\udd5c (n + 1) f s x : (Fin (n + 1) \u2192 E) \u2192 F) m =\n      (fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f s) s x : E \u2192 E[\u00d7n]\u2192L[\ud835\udd5c] F) (m 0) (tail m) :=\n  rfl\n#align iterated_fderiv_within_succ_apply_left iteratedFDerivWithin_succ_apply_left\n\n/-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,\nand the derivative of the `n`-th derivative. -/\ntheorem iteratedFDerivWithin_succ_eq_comp_left {n : \u2115} :\n    iteratedFDerivWithin \ud835\udd5c (n + 1) f s =\n      (continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun _ : Fin (n + 1) => E) F :\n          (E \u2192L[\ud835\udd5c] (E [\u00d7n]\u2192L[\ud835\udd5c] F)) \u2192 (E [\u00d7n.succ]\u2192L[\ud835\udd5c] F)) \u2218\n        fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f s) s :=\n  rfl\n#align iterated_fderiv_within_succ_eq_comp_left iteratedFDerivWithin_succ_eq_comp_left\n\ntheorem fderivWithin_iteratedFDerivWithin {s : Set E} {n : \u2115} :\n    fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f s) s =\n      (continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun _ : Fin (n + 1) => E) F).symm \u2218\n        iteratedFDerivWithin \ud835\udd5c (n + 1) f s := by\n  rw [iteratedFDerivWithin_succ_eq_comp_left]\n  ext1 x\n  simp only [Function.comp_apply, LinearIsometryEquiv.symm_apply_apply]\n\ntheorem norm_fderivWithin_iteratedFDerivWithin {n : \u2115} :\n    \u2016fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f s) s x\u2016 =\n      \u2016iteratedFDerivWithin \ud835\udd5c (n + 1) f s x\u2016 := by\n  -- Porting note: added `comp_apply`.\n  rw [iteratedFDerivWithin_succ_eq_comp_left, comp_apply, LinearIsometryEquiv.norm_map]\n#align norm_fderiv_within_iterated_fderiv_within norm_fderivWithin_iteratedFDerivWithin\n\ntheorem iteratedFDerivWithin_succ_apply_right {n : \u2115} (hs : UniqueDiffOn \ud835\udd5c s) (hx : x \u2208 s)\n    (m : Fin (n + 1) \u2192 E) :\n    (iteratedFDerivWithin \ud835\udd5c (n + 1) f s x : (Fin (n + 1) \u2192 E) \u2192 F) m =\n      iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s x (init m) (m (last n)) := by\n  induction' n with n IH generalizing x\n  \u00b7 rw [iteratedFDerivWithin_succ_eq_comp_left, iteratedFDerivWithin_zero_eq_comp,\n      iteratedFDerivWithin_zero_apply, Function.comp_apply,\n      LinearIsometryEquiv.comp_fderivWithin _ (hs x hx)]\n    rfl\n  \u00b7 let I := continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F\n    have A : \u2200 y \u2208 s, iteratedFDerivWithin \ud835\udd5c n.succ f s y =\n        (I \u2218 iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s) y := fun y hy \u21a6 by\n      ext m\n      rw [@IH y hy m]\n      rfl\n    calc\n      (iteratedFDerivWithin \ud835\udd5c (n + 2) f s x : (Fin (n + 2) \u2192 E) \u2192 F) m =\n          (fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n.succ f s) s x : E \u2192 E[\u00d7n + 1]\u2192L[\ud835\udd5c] F) (m 0)\n            (tail m) :=\n        rfl\n      _ = (fderivWithin \ud835\udd5c (I \u2218 iteratedFDerivWithin \ud835\udd5c n (fderivWithin \ud835\udd5c f s) s) s x :\n              E \u2192 E[\u00d7n + 1]\u2192L[\ud835\udd5c] F) (m 0) (tail m) := by\n        rw [fderivWithin_congr A (A x hx)]\n      _ = (I \u2218 fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n (fderivWithin \ud835\udd5c f s) s) s x :\n              E \u2192 E[\u00d7n + 1]\u2192L[\ud835\udd5c] F) (m 0) (tail m) := by\n        simp only [LinearIsometryEquiv.comp_fderivWithin _ (hs x hx)]; rfl\n      _ = (fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s) s x :\n              E \u2192 E[\u00d7n]\u2192L[\ud835\udd5c] E \u2192L[\ud835\udd5c] F) (m 0) (init (tail m)) ((tail m) (last n)) := rfl\n      _ = iteratedFDerivWithin \ud835\udd5c (Nat.succ n) (fun y => fderivWithin \ud835\udd5c f s y) s x (init m)\n            (m (last (n + 1))) := by\n        rw [iteratedFDerivWithin_succ_apply_left, tail_init_eq_init_tail]\n        rfl\n#align iterated_fderiv_within_succ_apply_right iteratedFDerivWithin_succ_apply_right\n\n/-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,\nand the `n`-th derivative of the derivative. -/\ntheorem iteratedFDerivWithin_succ_eq_comp_right {n : \u2115} (hs : UniqueDiffOn \ud835\udd5c s) (hx : x \u2208 s) :\n    iteratedFDerivWithin \ud835\udd5c (n + 1) f s x =\n      (continuousMultilinearCurryRightEquiv' \ud835\udd5c n E F \u2218\n          iteratedFDerivWithin \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s)\n        x :=\n  by ext m; rw [iteratedFDerivWithin_succ_apply_right hs hx]; rfl\n#align iterated_fderiv_within_succ_eq_comp_right iteratedFDerivWithin_succ_eq_comp_right\n\ntheorem norm_iteratedFDerivWithin_fderivWithin {n : \u2115} (hs : UniqueDiffOn \ud835\udd5c s) (hx : x \u2208 s) :\n    \u2016iteratedFDerivWithin \ud835\udd5c n (fderivWithin \ud835\udd5c f s) s x\u2016 =\n      \u2016iteratedFDerivWithin \ud835\udd5c (n + 1) f s x\u2016 := by\n  -- Porting note: added `comp_apply`.\n  rw [iteratedFDerivWithin_succ_eq_comp_right hs hx, comp_apply, LinearIsometryEquiv.norm_map]\n#align norm_iterated_fderiv_within_fderiv_within norm_iteratedFDerivWithin_fderivWithin\n\n@[simp]\ntheorem iteratedFDerivWithin_one_apply (h : UniqueDiffWithinAt \ud835\udd5c s x) (m : Fin 1 \u2192 E) :\n    (iteratedFDerivWithin \ud835\udd5c 1 f s x : (Fin 1 \u2192 E) \u2192 F) m =\n      (fderivWithin \ud835\udd5c f s x : E \u2192 F) (m 0) := by\n  simp only [iteratedFDerivWithin_succ_apply_left, iteratedFDerivWithin_zero_eq_comp,\n    (continuousMultilinearCurryFin0 \ud835\udd5c E F).symm.comp_fderivWithin h]\n  rfl\n#align iterated_fderiv_within_one_apply iteratedFDerivWithin_one_apply\n\ntheorem Filter.EventuallyEq.iteratedFDerivWithin' (h : f\u2081 =\u1da0[\ud835\udcdd[s] x] f) (ht : t \u2286 s) (n : \u2115) :\n    iteratedFDerivWithin \ud835\udd5c n f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c n f t := by\n  induction' n with n ihn\n  \u00b7 exact h.mono fun y hy => DFunLike.ext _ _ fun _ => hy\n  \u00b7 have : fderivWithin \ud835\udd5c _ t =\u1da0[\ud835\udcdd[s] x] fderivWithin \ud835\udd5c _ t := ihn.fderivWithin' ht\n    apply this.mono\n    intro y hy\n    simp only [iteratedFDerivWithin_succ_eq_comp_left, hy, (\u00b7 \u2218 \u00b7)]\n#align filter.eventually_eq.iterated_fderiv_within' Filter.EventuallyEq.iteratedFDerivWithin'\n\nprotected theorem Filter.EventuallyEq.iteratedFDerivWithin (h : f\u2081 =\u1da0[\ud835\udcdd[s] x] f) (n : \u2115) :\n    iteratedFDerivWithin \ud835\udd5c n f\u2081 s =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c n f s :=\n  h.iteratedFDerivWithin' Subset.rfl n\n#align filter.eventually_eq.iterated_fderiv_within Filter.EventuallyEq.iteratedFDerivWithin\n\n/-- If two functions coincide in a neighborhood of `x` within a set `s` and at `x`, then their\niterated differentials within this set at `x` coincide. -/\ntheorem Filter.EventuallyEq.iteratedFDerivWithin_eq (h : f\u2081 =\u1da0[\ud835\udcdd[s] x] f) (hx : f\u2081 x = f x)\n    (n : \u2115) : iteratedFDerivWithin \ud835\udd5c n f\u2081 s x = iteratedFDerivWithin \ud835\udd5c n f s x :=\n  have : f\u2081 =\u1da0[\ud835\udcdd[insert x s] x] f := by simpa [EventuallyEq, hx]\n  (this.iteratedFDerivWithin' (subset_insert _ _) n).self_of_nhdsWithin (mem_insert _ _)\n#align filter.eventually_eq.iterated_fderiv_within_eq Filter.EventuallyEq.iteratedFDerivWithin_eq\n\n/-- If two functions coincide on a set `s`, then their iterated differentials within this set\ncoincide. See also `Filter.EventuallyEq.iteratedFDerivWithin_eq` and\n`Filter.EventuallyEq.iteratedFDerivWithin`. -/\ntheorem iteratedFDerivWithin_congr (hs : EqOn f\u2081 f s) (hx : x \u2208 s) (n : \u2115) :\n    iteratedFDerivWithin \ud835\udd5c n f\u2081 s x = iteratedFDerivWithin \ud835\udd5c n f s x :=\n  (hs.eventuallyEq.filter_mono inf_le_right).iteratedFDerivWithin_eq (hs hx) _\n#align iterated_fderiv_within_congr iteratedFDerivWithin_congr\n\n/-- If two functions coincide on a set `s`, then their iterated differentials within this set\ncoincide. See also `Filter.EventuallyEq.iteratedFDerivWithin_eq` and\n`Filter.EventuallyEq.iteratedFDerivWithin`. -/\nprotected theorem Set.EqOn.iteratedFDerivWithin (hs : EqOn f\u2081 f s) (n : \u2115) :\n    EqOn (iteratedFDerivWithin \ud835\udd5c n f\u2081 s) (iteratedFDerivWithin \ud835\udd5c n f s) s := fun _x hx =>\n  iteratedFDerivWithin_congr hs hx n\n#align set.eq_on.iterated_fderiv_within Set.EqOn.iteratedFDerivWithin\n\ntheorem iteratedFDerivWithin_eventually_congr_set' (y : E) (h : s =\u1da0[\ud835\udcdd[{y}\u1d9c] x] t) (n : \u2115) :\n    iteratedFDerivWithin \ud835\udd5c n f s =\u1da0[\ud835\udcdd x] iteratedFDerivWithin \ud835\udd5c n f t := by\n  induction' n with n ihn generalizing x\n  \u00b7 rfl\n  \u00b7 refine' (eventually_nhds_nhdsWithin.2 h).mono fun y hy => _\n    simp only [iteratedFDerivWithin_succ_eq_comp_left, (\u00b7 \u2218 \u00b7)]\n    rw [(ihn hy).fderivWithin_eq_nhds, fderivWithin_congr_set' _ hy]\n#align iterated_fderiv_within_eventually_congr_set' iteratedFDerivWithin_eventually_congr_set'\n\ntheorem iteratedFDerivWithin_eventually_congr_set (h : s =\u1da0[\ud835\udcdd x] t) (n : \u2115) :\n    iteratedFDerivWithin \ud835\udd5c n f s =\u1da0[\ud835\udcdd x] iteratedFDerivWithin \ud835\udd5c n f t :=\n  iteratedFDerivWithin_eventually_congr_set' x (h.filter_mono inf_le_left) n\n#align iterated_fderiv_within_eventually_congr_set iteratedFDerivWithin_eventually_congr_set\n\ntheorem iteratedFDerivWithin_congr_set (h : s =\u1da0[\ud835\udcdd x] t) (n : \u2115) :\n    iteratedFDerivWithin \ud835\udd5c n f s x = iteratedFDerivWithin \ud835\udd5c n f t x :=\n  (iteratedFDerivWithin_eventually_congr_set h n).self_of_nhds\n#align iterated_fderiv_within_congr_set iteratedFDerivWithin_congr_set\n\n/-- The iterated differential within a set `s` at a point `x` is not modified if one intersects\n`s` with a neighborhood of `x` within `s`. -/\ntheorem iteratedFDerivWithin_inter' {n : \u2115} (hu : u \u2208 \ud835\udcdd[s] x) :\n    iteratedFDerivWithin \ud835\udd5c n f (s \u2229 u) x = iteratedFDerivWithin \ud835\udd5c n f s x :=\n  iteratedFDerivWithin_congr_set (nhdsWithin_eq_iff_eventuallyEq.1 <| nhdsWithin_inter_of_mem' hu) _\n#align iterated_fderiv_within_inter' iteratedFDerivWithin_inter'\n\n/-- The iterated differential within a set `s` at a point `x` is not modified if one intersects\n`s` with a neighborhood of `x`. -/\ntheorem iteratedFDerivWithin_inter {n : \u2115} (hu : u \u2208 \ud835\udcdd x) :\n    iteratedFDerivWithin \ud835\udd5c n f (s \u2229 u) x = iteratedFDerivWithin \ud835\udd5c n f s x :=\n  iteratedFDerivWithin_inter' (mem_nhdsWithin_of_mem_nhds hu)\n#align iterated_fderiv_within_inter iteratedFDerivWithin_inter\n\n/-- The iterated differential within a set `s` at a point `x` is not modified if one intersects\n`s` with an open set containing `x`. -/\ntheorem iteratedFDerivWithin_inter_open {n : \u2115} (hu : IsOpen u) (hx : x \u2208 u) :\n    iteratedFDerivWithin \ud835\udd5c n f (s \u2229 u) x = iteratedFDerivWithin \ud835\udd5c n f s x :=\n  iteratedFDerivWithin_inter (hu.mem_nhds hx)\n#align iterated_fderiv_within_inter_open iteratedFDerivWithin_inter_open\n\n@[simp]\ntheorem contDiffOn_zero : ContDiffOn \ud835\udd5c 0 f s \u2194 ContinuousOn f s := by\n  refine' \u27e8fun H => H.continuousOn, fun H => _\u27e9\n  intro x hx m hm\n  have : (m : \u2115\u221e) = 0 := le_antisymm hm bot_le\n  rw [this]\n  refine' \u27e8insert x s, self_mem_nhdsWithin, ftaylorSeriesWithin \ud835\udd5c f s, _\u27e9\n  rw [hasFTaylorSeriesUpToOn_zero_iff]\n  exact \u27e8by rwa [insert_eq_of_mem hx], fun x _ => by simp [ftaylorSeriesWithin]\u27e9\n#align cont_diff_on_zero contDiffOn_zero\n\ntheorem contDiffWithinAt_zero (hx : x \u2208 s) :\n    ContDiffWithinAt \ud835\udd5c 0 f s x \u2194 \u2203 u \u2208 \ud835\udcdd[s] x, ContinuousOn f (s \u2229 u) := by\n  constructor\n  \u00b7 intro h\n    obtain \u27e8u, H, p, hp\u27e9 := h 0 le_rfl\n    refine' \u27e8u, _, _\u27e9\n    \u00b7 simpa [hx] using H\n    \u00b7 simp only [Nat.cast_zero, hasFTaylorSeriesUpToOn_zero_iff] at hp\n      exact hp.1.mono (inter_subset_right s u)\n  \u00b7 rintro \u27e8u, H, hu\u27e9\n    rw [\u2190 contDiffWithinAt_inter' H]\n    have h' : x \u2208 s \u2229 u := \u27e8hx, mem_of_mem_nhdsWithin hx H\u27e9\n    exact (contDiffOn_zero.mpr hu).contDiffWithinAt h'\n#align cont_diff_within_at_zero contDiffWithinAt_zero\n\n/-- On a set with unique differentiability, any choice of iterated differential has to coincide\nwith the one we have chosen in `iteratedFDerivWithin \ud835\udd5c m f s`. -/\ntheorem HasFTaylorSeriesUpToOn.eq_iteratedFDerivWithin_of_uniqueDiffOn\n    (h : HasFTaylorSeriesUpToOn n f p s) {m : \u2115} (hmn : (m : \u2115\u221e) \u2264 n) (hs : UniqueDiffOn \ud835\udd5c s)\n    (hx : x \u2208 s) : p x m = iteratedFDerivWithin \ud835\udd5c m f s x := by\n  induction' m with m IH generalizing x\n  \u00b7 rw [Nat.zero_eq, h.zero_eq' hx, iteratedFDerivWithin_zero_eq_comp]; rfl\n  \u00b7 have A : (m : \u2115\u221e) < n := lt_of_lt_of_le (WithTop.coe_lt_coe.2 (lt_add_one m)) hmn\n    have :\n      HasFDerivWithinAt (fun y : E => iteratedFDerivWithin \ud835\udd5c m f s y)\n        (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x :=\n      (h.fderivWithin m A x hx).congr (fun y hy => (IH (le_of_lt A) hy).symm)\n        (IH (le_of_lt A) hx).symm\n    rw [iteratedFDerivWithin_succ_eq_comp_left, Function.comp_apply, this.fderivWithin (hs x hx)]\n    exact (ContinuousMultilinearMap.uncurry_curryLeft _).symm\n#align has_ftaylor_series_up_to_on.eq_ftaylor_series_of_unique_diff_on HasFTaylorSeriesUpToOn.eq_iteratedFDerivWithin_of_uniqueDiffOn\n\n@[deprecated] alias HasFTaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn :=\n  HasFTaylorSeriesUpToOn.eq_iteratedFDerivWithin_of_uniqueDiffOn -- 2024-03-28\n\n/-- When a function is `C^n` in a set `s` of unique differentiability, it admits\n`ftaylorSeriesWithin \ud835\udd5c f s` as a Taylor series up to order `n` in `s`. -/\nprotected theorem ContDiffOn.ftaylorSeriesWithin (h : ContDiffOn \ud835\udd5c n f s) (hs : UniqueDiffOn \ud835\udd5c s) :\n    HasFTaylorSeriesUpToOn n f (ftaylorSeriesWithin \ud835\udd5c f s) s := by\n  constructor\n  \u00b7 intro x _\n    simp only [ftaylorSeriesWithin, ContinuousMultilinearMap.uncurry0_apply,\n      iteratedFDerivWithin_zero_apply]\n  \u00b7 intro m hm x hx\n    rcases (h x hx) m.succ (ENat.add_one_le_of_lt hm) with \u27e8u, hu, p, Hp\u27e9\n    rw [insert_eq_of_mem hx] at hu\n    rcases mem_nhdsWithin.1 hu with \u27e8o, o_open, xo, ho\u27e9\n    rw [inter_comm] at ho\n    have : p x m.succ = ftaylorSeriesWithin \ud835\udd5c f s x m.succ := by\n      change p x m.succ = iteratedFDerivWithin \ud835\udd5c m.succ f s x\n      rw [\u2190 iteratedFDerivWithin_inter_open o_open xo]\n      exact (Hp.mono ho).eq_ftaylor_series_of_uniqueDiffOn le_rfl (hs.inter o_open) \u27e8hx, xo\u27e9\n    rw [\u2190 this, \u2190 hasFDerivWithinAt_inter (IsOpen.mem_nhds o_open xo)]\n    have A : \u2200 y \u2208 s \u2229 o, p y m = ftaylorSeriesWithin \ud835\udd5c f s y m := by\n      rintro y \u27e8hy, yo\u27e9\n      change p y m = iteratedFDerivWithin \ud835\udd5c m f s y\n      rw [\u2190 iteratedFDerivWithin_inter_open o_open yo]\n      exact\n        (Hp.mono ho).eq_ftaylor_series_of_uniqueDiffOn (WithTop.coe_le_coe.2 (Nat.le_succ m))\n          (hs.inter o_open) \u27e8hy, yo\u27e9\n    exact\n      ((Hp.mono ho).fderivWithin m (WithTop.coe_lt_coe.2 (lt_add_one m)) x \u27e8hx, xo\u27e9).congr\n        (fun y hy => (A y hy).symm) (A x \u27e8hx, xo\u27e9).symm\n  \u00b7 intro m hm\n    apply continuousOn_of_locally_continuousOn\n    intro x hx\n    rcases h x hx m hm with \u27e8u, hu, p, Hp\u27e9\n    rcases mem_nhdsWithin.1 hu with \u27e8o, o_open, xo, ho\u27e9\n    rw [insert_eq_of_mem hx] at ho\n    rw [inter_comm] at ho\n    refine' \u27e8o, o_open, xo, _\u27e9\n    have A : \u2200 y \u2208 s \u2229 o, p y m = ftaylorSeriesWithin \ud835\udd5c f s y m := by\n      rintro y \u27e8hy, yo\u27e9\n      change p y m = iteratedFDerivWithin \ud835\udd5c m f s y\n      rw [\u2190 iteratedFDerivWithin_inter_open o_open yo]\n      exact (Hp.mono ho).eq_ftaylor_series_of_uniqueDiffOn le_rfl (hs.inter o_open) \u27e8hy, yo\u27e9\n    exact ((Hp.mono ho).cont m le_rfl).congr fun y hy => (A y hy).symm\n#align cont_diff_on.ftaylor_series_within ContDiffOn.ftaylorSeriesWithin\n\ntheorem contDiffOn_of_continuousOn_differentiableOn\n    (Hcont : \u2200 m : \u2115, (m : \u2115\u221e) \u2264 n \u2192 ContinuousOn (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s)\n    (Hdiff : \u2200 m : \u2115, (m : \u2115\u221e) < n \u2192\n      DifferentiableOn \ud835\udd5c (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s) :\n    ContDiffOn \ud835\udd5c n f s := by\n  intro x hx m hm\n  rw [insert_eq_of_mem hx]\n  refine' \u27e8s, self_mem_nhdsWithin, ftaylorSeriesWithin \ud835\udd5c f s, _\u27e9\n  constructor\n  \u00b7 intro y _\n    simp only [ftaylorSeriesWithin, ContinuousMultilinearMap.uncurry0_apply,\n      iteratedFDerivWithin_zero_apply]\n  \u00b7 intro k hk y hy\n    convert (Hdiff k (lt_of_lt_of_le hk hm) y hy).hasFDerivWithinAt\n  \u00b7 intro k hk\n    exact Hcont k (le_trans hk hm)\n#align cont_diff_on_of_continuous_on_differentiable_on contDiffOn_of_continuousOn_differentiableOn\n\ntheorem contDiffOn_of_differentiableOn\n    (h : \u2200 m : \u2115, (m : \u2115\u221e) \u2264 n \u2192 DifferentiableOn \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c m f s) s) :\n    ContDiffOn \ud835\udd5c n f s :=\n  contDiffOn_of_continuousOn_differentiableOn (fun m hm => (h m hm).continuousOn) fun m hm =>\n    h m (le_of_lt hm)\n#align cont_diff_on_of_differentiable_on contDiffOn_of_differentiableOn\n\ntheorem ContDiffOn.continuousOn_iteratedFDerivWithin {m : \u2115} (h : ContDiffOn \ud835\udd5c n f s)\n    (hmn : (m : \u2115\u221e) \u2264 n) (hs : UniqueDiffOn \ud835\udd5c s) : ContinuousOn (iteratedFDerivWithin \ud835\udd5c m f s) s :=\n  (h.ftaylorSeriesWithin hs).cont m hmn\n#align cont_diff_on.continuous_on_iterated_fderiv_within ContDiffOn.continuousOn_iteratedFDerivWithin\n\ntheorem ContDiffOn.differentiableOn_iteratedFDerivWithin {m : \u2115} (h : ContDiffOn \ud835\udd5c n f s)\n    (hmn : (m : \u2115\u221e) < n) (hs : UniqueDiffOn \ud835\udd5c s) :\n    DifferentiableOn \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c m f s) s := fun x hx =>\n  ((h.ftaylorSeriesWithin hs).fderivWithin m hmn x hx).differentiableWithinAt\n#align cont_diff_on.differentiable_on_iterated_fderiv_within ContDiffOn.differentiableOn_iteratedFDerivWithin\n\ntheorem ContDiffWithinAt.differentiableWithinAt_iteratedFDerivWithin {m : \u2115}\n    (h : ContDiffWithinAt \ud835\udd5c n f s x) (hmn : (m : \u2115\u221e) < n) (hs : UniqueDiffOn \ud835\udd5c (insert x s)) :\n    DifferentiableWithinAt \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c m f s) s x := by\n  rcases h.contDiffOn' (ENat.add_one_le_of_lt hmn) with \u27e8u, uo, xu, hu\u27e9\n  set t := insert x s \u2229 u\n  have A : t =\u1da0[\ud835\udcdd[\u2260] x] s := by\n    simp only [set_eventuallyEq_iff_inf_principal, \u2190 nhdsWithin_inter']\n    rw [\u2190 inter_assoc, nhdsWithin_inter_of_mem', \u2190 diff_eq_compl_inter, insert_diff_of_mem,\n      diff_eq_compl_inter]\n    exacts [rfl, mem_nhdsWithin_of_mem_nhds (uo.mem_nhds xu)]\n  have B : iteratedFDerivWithin \ud835\udd5c m f s =\u1da0[\ud835\udcdd x] iteratedFDerivWithin \ud835\udd5c m f t :=\n    iteratedFDerivWithin_eventually_congr_set' _ A.symm _\n  have C : DifferentiableWithinAt \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c m f t) t x :=\n    hu.differentiableOn_iteratedFDerivWithin (Nat.cast_lt.2 m.lt_succ_self) (hs.inter uo) x\n      \u27e8mem_insert _ _, xu\u27e9\n  rw [differentiableWithinAt_congr_set' _ A] at C\n  exact C.congr_of_eventuallyEq (B.filter_mono inf_le_left) B.self_of_nhds\n#align cont_diff_within_at.differentiable_within_at_iterated_fderiv_within ContDiffWithinAt.differentiableWithinAt_iteratedFDerivWithin\n\ntheorem contDiffOn_iff_continuousOn_differentiableOn (hs : UniqueDiffOn \ud835\udd5c s) :\n    ContDiffOn \ud835\udd5c n f s \u2194\n      (\u2200 m : \u2115, (m : \u2115\u221e) \u2264 n \u2192 ContinuousOn (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s) \u2227\n        \u2200 m : \u2115, (m : \u2115\u221e) < n \u2192 DifferentiableOn \ud835\udd5c (fun x => iteratedFDerivWithin \ud835\udd5c m f s x) s :=\n  \u27e8fun h => \u27e8fun _m hm => h.continuousOn_iteratedFDerivWithin hm hs, fun _m hm =>\n      h.differentiableOn_iteratedFDerivWithin hm hs\u27e9,\n    fun h => contDiffOn_of_continuousOn_differentiableOn h.1 h.2\u27e9\n#align cont_diff_on_iff_continuous_on_differentiable_on contDiffOn_iff_continuousOn_differentiableOn\n\ntheorem contDiffOn_succ_of_fderivWithin {n : \u2115} (hf : DifferentiableOn \ud835\udd5c f s)\n    (h : ContDiffOn \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s) : ContDiffOn \ud835\udd5c (n + 1 : \u2115) f s := by\n  intro x hx\n  rw [contDiffWithinAt_succ_iff_hasFDerivWithinAt, insert_eq_of_mem hx]\n  exact\n    \u27e8s, self_mem_nhdsWithin, fderivWithin \ud835\udd5c f s, fun y hy => (hf y hy).hasFDerivWithinAt, h x hx\u27e9\n#align cont_diff_on_succ_of_fderiv_within contDiffOn_succ_of_fderivWithin\n\n/-- A function is `C^(n + 1)` on a domain with unique derivatives if and only if it is\ndifferentiable there, and its derivative (expressed with `fderivWithin`) is `C^n`. -/\ntheorem contDiffOn_succ_iff_fderivWithin {n : \u2115} (hs : UniqueDiffOn \ud835\udd5c s) :\n    ContDiffOn \ud835\udd5c (n + 1 : \u2115) f s \u2194\n      DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c n (fun y => fderivWithin \ud835\udd5c f s y) s := by\n  refine' \u27e8fun H => _, fun h => contDiffOn_succ_of_fderivWithin h.1 h.2\u27e9\n  refine' \u27e8H.differentiableOn (WithTop.coe_le_coe.2 (Nat.le_add_left 1 n)), fun x hx => _\u27e9\n  rcases contDiffWithinAt_succ_iff_hasFDerivWithinAt.1 (H x hx) with \u27e8u, hu, f', hff', hf'\u27e9\n  rcases mem_nhdsWithin.1 hu with \u27e8o, o_open, xo, ho\u27e9\n  rw [inter_comm, insert_eq_of_mem hx] at ho\n  have := hf'.mono ho\n  rw [contDiffWithinAt_inter' (mem_nhdsWithin_of_mem_nhds (IsOpen.mem_nhds o_open xo))] at this\n  apply this.congr_of_eventually_eq' _ hx\n  have : o \u2229 s \u2208 \ud835\udcdd[s] x := mem_nhdsWithin.2 \u27e8o, o_open, xo, Subset.refl _\u27e9\n  rw [inter_comm] at this\n  refine Filter.eventuallyEq_of_mem this fun y hy => ?_\n  have A : fderivWithin \ud835\udd5c f (s \u2229 o) y = f' y :=\n    ((hff' y (ho hy)).mono ho).fderivWithin (hs.inter o_open y hy)\n  rwa [fderivWithin_inter (o_open.mem_nhds hy.2)] at A\n#align cont_diff_on_succ_iff_fderiv_within contDiffOn_succ_iff_fderivWithin\n\ntheorem contDiffOn_succ_iff_hasFDerivWithin {n : \u2115} (hs : UniqueDiffOn \ud835\udd5c s) :\n    ContDiffOn \ud835\udd5c (n + 1 : \u2115) f s \u2194\n      \u2203 f' : E \u2192 E \u2192L[\ud835\udd5c] F, ContDiffOn \ud835\udd5c n f' s \u2227 \u2200 x, x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x := by\n  rw [contDiffOn_succ_iff_fderivWithin hs]\n  refine' \u27e8fun h => \u27e8fderivWithin \ud835\udd5c f s, h.2, fun x hx => (h.1 x hx).hasFDerivWithinAt\u27e9, fun h => _\u27e9\n  rcases h with \u27e8f', h1, h2\u27e9\n  refine' \u27e8fun x hx => (h2 x hx).differentiableWithinAt, fun x hx => _\u27e9\n  exact (h1 x hx).congr' (fun y hy => (h2 y hy).fderivWithin (hs y hy)) hx\n#align cont_diff_on_succ_iff_has_fderiv_within contDiffOn_succ_iff_hasFDerivWithin\n\n/-- A function is `C^(n + 1)` on an open domain if and only if it is\ndifferentiable there, and its derivative (expressed with `fderiv`) is `C^n`. -/\ntheorem contDiffOn_succ_iff_fderiv_of_isOpen {n : \u2115} (hs : IsOpen s) :\n    ContDiffOn \ud835\udd5c (n + 1 : \u2115) f s \u2194\n      DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c n (fun y => fderiv \ud835\udd5c f y) s := by\n  rw [contDiffOn_succ_iff_fderivWithin hs.uniqueDiffOn]\n  exact Iff.rfl.and (contDiffOn_congr fun x hx \u21a6 fderivWithin_of_isOpen hs hx)\n#align cont_diff_on_succ_iff_fderiv_of_open contDiffOn_succ_iff_fderiv_of_isOpen\n\n/-- A function is `C^\u221e` on a domain with unique derivatives if and only if it is differentiable\nthere, and its derivative (expressed with `fderivWithin`) is `C^\u221e`. -/\ntheorem contDiffOn_top_iff_fderivWithin (hs : UniqueDiffOn \ud835\udd5c s) :\n    ContDiffOn \ud835\udd5c \u221e f s \u2194\n      DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c \u221e (fun y => fderivWithin \ud835\udd5c f s y) s := by\n  constructor\n  \u00b7 intro h\n    refine' \u27e8h.differentiableOn le_top, _\u27e9\n    refine' contDiffOn_top.2 fun n => ((contDiffOn_succ_iff_fderivWithin hs).1 _).2\n    exact h.of_le le_top\n  \u00b7 intro h\n    refine' contDiffOn_top.2 fun n => _\n    have A : (n : \u2115\u221e) \u2264 \u221e := le_top\n    apply ((contDiffOn_succ_iff_fderivWithin hs).2 \u27e8h.1, h.2.of_le A\u27e9).of_le\n    exact WithTop.coe_le_coe.2 (Nat.le_succ n)\n#align cont_diff_on_top_iff_fderiv_within contDiffOn_top_iff_fderivWithin\n\n/-- A function is `C^\u221e` on an open domain if and only if it is differentiable there, and its\nderivative (expressed with `fderiv`) is `C^\u221e`. -/\ntheorem contDiffOn_top_iff_fderiv_of_isOpen (hs : IsOpen s) :\n    ContDiffOn \ud835\udd5c \u221e f s \u2194 DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c \u221e (fun y => fderiv \ud835\udd5c f y) s := by\n  rw [contDiffOn_top_iff_fderivWithin hs.uniqueDiffOn]\n  exact Iff.rfl.and <| contDiffOn_congr fun x hx \u21a6 fderivWithin_of_isOpen hs hx\n#align cont_diff_on_top_iff_fderiv_of_open contDiffOn_top_iff_fderiv_of_isOpen\n\nprotected theorem ContDiffOn.fderivWithin (hf : ContDiffOn \ud835\udd5c n f s) (hs : UniqueDiffOn \ud835\udd5c s)\n    (hmn : m + 1 \u2264 n) : ContDiffOn \ud835\udd5c m (fun y => fderivWithin \ud835\udd5c f s y) s := by\n  cases' m with m\n  \u00b7 change \u221e + 1 \u2264 n at hmn\n    have : n = \u221e := by simpa using hmn\n    rw [this] at hf\n    exact ((contDiffOn_top_iff_fderivWithin hs).1 hf).2\n  \u00b7 change (m.succ : \u2115\u221e) \u2264 n at hmn\n    exact ((contDiffOn_succ_iff_fderivWithin hs).1 (hf.of_le hmn)).2\n#align cont_diff_on.fderiv_within ContDiffOn.fderivWithin\n\ntheorem ContDiffOn.fderiv_of_isOpen (hf : ContDiffOn \ud835\udd5c n f s) (hs : IsOpen s) (hmn : m + 1 \u2264 n) :\n    ContDiffOn \ud835\udd5c m (fun y => fderiv \ud835\udd5c f y) s :=\n  (hf.fderivWithin hs.uniqueDiffOn hmn).congr fun _ hx => (fderivWithin_of_isOpen hs hx).symm\n#align cont_diff_on.fderiv_of_open ContDiffOn.fderiv_of_isOpen\n\ntheorem ContDiffOn.continuousOn_fderivWithin (h : ContDiffOn \ud835\udd5c n f s) (hs : UniqueDiffOn \ud835\udd5c s)\n    (hn : 1 \u2264 n) : ContinuousOn (fun x => fderivWithin \ud835\udd5c f s x) s :=\n  ((contDiffOn_succ_iff_fderivWithin hs).1 (h.of_le hn)).2.continuousOn\n#align cont_diff_on.continuous_on_fderiv_within ContDiffOn.continuousOn_fderivWithin\n\ntheorem ContDiffOn.continuousOn_fderiv_of_isOpen (h : ContDiffOn \ud835\udd5c n f s) (hs : IsOpen s)\n    (hn : 1 \u2264 n) : ContinuousOn (fun x => fderiv \ud835\udd5c f x) s :=\n  ((contDiffOn_succ_iff_fderiv_of_isOpen hs).1 (h.of_le hn)).2.continuousOn\n#align cont_diff_on.continuous_on_fderiv_of_open ContDiffOn.continuousOn_fderiv_of_isOpen\n\n/-! ### Functions with a Taylor series on the whole space -/\n\n/-- `HasFTaylorSeriesUpTo n f p` registers the fact that `p 0 = f` and `p (m+1)` is a\nderivative of `p m` for `m < n`, and is continuous for `m \u2264 n`. This is a predicate analogous to\n`HasFDerivAt` but for higher order derivatives.\n\nNotice that `p` does not sum up to `f` on the diagonal (`FormalMultilinearSeries.sum`), even if\n`f` is analytic and `n = \u221e`: an addition `1/m!` factor on the `m`th term is necessary for that. -/\nstructure HasFTaylorSeriesUpTo (n : \u2115\u221e) (f : E \u2192 F) (p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F) :\n  Prop where\n  zero_eq : \u2200 x, (p x 0).uncurry0 = f x\n  fderiv : \u2200 m : \u2115, (m : \u2115\u221e) < n \u2192 \u2200 x, HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x\n  cont : \u2200 m : \u2115, (m : \u2115\u221e) \u2264 n \u2192 Continuous fun x => p x m\n#align has_ftaylor_series_up_to HasFTaylorSeriesUpTo\n\ntheorem HasFTaylorSeriesUpTo.zero_eq' (h : HasFTaylorSeriesUpTo n f p) (x : E) :\n    p x 0 = (continuousMultilinearCurryFin0 \ud835\udd5c E F).symm (f x) := by\n  rw [\u2190 h.zero_eq x]\n  exact (p x 0).uncurry0_curry0.symm\n#align has_ftaylor_series_up_to.zero_eq' HasFTaylorSeriesUpTo.zero_eq'\n\ntheorem hasFTaylorSeriesUpToOn_univ_iff :\n    HasFTaylorSeriesUpToOn n f p univ \u2194 HasFTaylorSeriesUpTo n f p := by\n  constructor\n  \u00b7 intro H\n    constructor\n    \u00b7 exact fun x => H.zero_eq x (mem_univ x)\n    \u00b7 intro m hm x\n      rw [\u2190 hasFDerivWithinAt_univ]\n      exact H.fderivWithin m hm x (mem_univ x)\n    \u00b7 intro m hm\n      rw [continuous_iff_continuousOn_univ]\n      exact H.cont m hm\n  \u00b7 intro H\n    constructor\n    \u00b7 exact fun x _ => H.zero_eq x\n    \u00b7 intro m hm x _\n      rw [hasFDerivWithinAt_univ]\n      exact H.fderiv m hm x\n    \u00b7 intro m hm\n      rw [\u2190 continuous_iff_continuousOn_univ]\n      exact H.cont m hm\n#align has_ftaylor_series_up_to_on_univ_iff hasFTaylorSeriesUpToOn_univ_iff\n\ntheorem HasFTaylorSeriesUpTo.hasFTaylorSeriesUpToOn (h : HasFTaylorSeriesUpTo n f p) (s : Set E) :\n    HasFTaylorSeriesUpToOn n f p s :=\n  (hasFTaylorSeriesUpToOn_univ_iff.2 h).mono (subset_univ _)\n#align has_ftaylor_series_up_to.has_ftaylor_series_up_to_on HasFTaylorSeriesUpTo.hasFTaylorSeriesUpToOn\n\ntheorem HasFTaylorSeriesUpTo.ofLe (h : HasFTaylorSeriesUpTo n f p) (hmn : m \u2264 n) :\n    HasFTaylorSeriesUpTo m f p := by\n  rw [\u2190 hasFTaylorSeriesUpToOn_univ_iff] at h \u22a2; exact h.of_le hmn\n#align has_ftaylor_series_up_to.of_le HasFTaylorSeriesUpTo.ofLe\n\ntheorem HasFTaylorSeriesUpTo.continuous (h : HasFTaylorSeriesUpTo n f p) : Continuous f := by\n  rw [\u2190 hasFTaylorSeriesUpToOn_univ_iff] at h\n  rw [continuous_iff_continuousOn_univ]\n  exact h.continuousOn\n#align has_ftaylor_series_up_to.continuous HasFTaylorSeriesUpTo.continuous\n\ntheorem hasFTaylorSeriesUpTo_zero_iff :\n    HasFTaylorSeriesUpTo 0 f p \u2194 Continuous f \u2227 \u2200 x, (p x 0).uncurry0 = f x := by\n  simp [hasFTaylorSeriesUpToOn_univ_iff.symm, continuous_iff_continuousOn_univ,\n    hasFTaylorSeriesUpToOn_zero_iff]\n#align has_ftaylor_series_up_to_zero_iff hasFTaylorSeriesUpTo_zero_iff\n\ntheorem hasFTaylorSeriesUpTo_top_iff :\n    HasFTaylorSeriesUpTo \u221e f p \u2194 \u2200 n : \u2115, HasFTaylorSeriesUpTo n f p := by\n  simp only [\u2190 hasFTaylorSeriesUpToOn_univ_iff, hasFTaylorSeriesUpToOn_top_iff]\n#align has_ftaylor_series_up_to_top_iff hasFTaylorSeriesUpTo_top_iff\n\n/-- In the case that `n = \u221e` we don't need the continuity assumption in\n`HasFTaylorSeriesUpTo`. -/\ntheorem hasFTaylorSeriesUpTo_top_iff' :\n    HasFTaylorSeriesUpTo \u221e f p \u2194\n      (\u2200 x, (p x 0).uncurry0 = f x) \u2227\n        \u2200 (m : \u2115) (x), HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x := by\n  simp only [\u2190 hasFTaylorSeriesUpToOn_univ_iff, hasFTaylorSeriesUpToOn_top_iff', mem_univ,\n    forall_true_left, hasFDerivWithinAt_univ]\n#align has_ftaylor_series_up_to_top_iff' hasFTaylorSeriesUpTo_top_iff'\n\n/-- If a function has a Taylor series at order at least `1`, then the term of order `1` of this\nseries is a derivative of `f`. -/\ntheorem HasFTaylorSeriesUpTo.hasFDerivAt (h : HasFTaylorSeriesUpTo n f p) (hn : 1 \u2264 n) (x : E) :\n    HasFDerivAt f (continuousMultilinearCurryFin1 \ud835\udd5c E F (p x 1)) x := by\n  rw [\u2190 hasFDerivWithinAt_univ]\n  exact (hasFTaylorSeriesUpToOn_univ_iff.2 h).hasFDerivWithinAt hn (mem_univ _)\n#align has_ftaylor_series_up_to.has_fderiv_at HasFTaylorSeriesUpTo.hasFDerivAt\n\ntheorem HasFTaylorSeriesUpTo.differentiable (h : HasFTaylorSeriesUpTo n f p) (hn : 1 \u2264 n) :\n    Differentiable \ud835\udd5c f := fun x => (h.hasFDerivAt hn x).differentiableAt\n#align has_ftaylor_series_up_to.differentiable HasFTaylorSeriesUpTo.differentiable\n\n/-- `p` is a Taylor series of `f` up to `n+1` if and only if `p.shift` is a Taylor series up to `n`\nfor `p 1`, which is a derivative of `f`. -/\ntheorem hasFTaylorSeriesUpTo_succ_iff_right {n : \u2115} :\n    HasFTaylorSeriesUpTo (n + 1 : \u2115) f p \u2194\n      (\u2200 x, (p x 0).uncurry0 = f x) \u2227\n        (\u2200 x, HasFDerivAt (fun y => p y 0) (p x 1).curryLeft x) \u2227\n          HasFTaylorSeriesUpTo n (fun x => continuousMultilinearCurryFin1 \ud835\udd5c E F (p x 1)) fun x =>\n            (p x).shift := by\n  simp only [hasFTaylorSeriesUpToOn_succ_iff_right, \u2190 hasFTaylorSeriesUpToOn_univ_iff, mem_univ,\n    forall_true_left, hasFDerivWithinAt_univ]\n#align has_ftaylor_series_up_to_succ_iff_right hasFTaylorSeriesUpTo_succ_iff_right\n\n/-! ### Smooth functions at a point -/\n\nvariable (\ud835\udd5c)\n\n/-- A function is continuously differentiable up to `n` at a point `x` if, for any integer `k \u2264 n`,\nthere is a neighborhood of `x` where `f` admits derivatives up to order `n`, which are continuous.\n-/\ndef ContDiffAt (n : \u2115\u221e) (f : E \u2192 F) (x : E) : Prop :=\n  ContDiffWithinAt \ud835\udd5c n f univ x\n#align cont_diff_at ContDiffAt\n\nvariable {\ud835\udd5c}\n\ntheorem contDiffWithinAt_univ : ContDiffWithinAt \ud835\udd5c n f univ x \u2194 ContDiffAt \ud835\udd5c n f x :=\n  Iff.rfl\n#align cont_diff_within_at_univ contDiffWithinAt_univ\n\ntheorem contDiffAt_top : ContDiffAt \ud835\udd5c \u221e f x \u2194 \u2200 n : \u2115, ContDiffAt \ud835\udd5c n f x := by\n  simp [\u2190 contDiffWithinAt_univ, contDiffWithinAt_top]\n#align cont_diff_at_top contDiffAt_top\n\ntheorem ContDiffAt.contDiffWithinAt (h : ContDiffAt \ud835\udd5c n f x) : ContDiffWithinAt \ud835\udd5c n f s x :=\n  h.mono (subset_univ _)\n#align cont_diff_at.cont_diff_within_at ContDiffAt.contDiffWithinAt\n\ntheorem ContDiffWithinAt.contDiffAt (h : ContDiffWithinAt \ud835\udd5c n f s x) (hx : s \u2208 \ud835\udcdd x) :\n    ContDiffAt \ud835\udd5c n f x := by rwa [ContDiffAt, \u2190 contDiffWithinAt_inter hx, univ_inter]\n#align cont_diff_within_at.cont_diff_at ContDiffWithinAt.contDiffAt\n\n-- Porting note (#10756): new lemma\ntheorem ContDiffOn.contDiffAt (h : ContDiffOn \ud835\udd5c n f s) (hx : s \u2208 \ud835\udcdd x) :\n    ContDiffAt \ud835\udd5c n f x :=\n  (h _ (mem_of_mem_nhds hx)).contDiffAt hx\n\ntheorem ContDiffAt.congr_of_eventuallyEq (h : ContDiffAt \ud835\udd5c n f x) (hg : f\u2081 =\u1da0[\ud835\udcdd x] f) :\n    ContDiffAt \ud835\udd5c n f\u2081 x :=\n  h.congr_of_eventually_eq' (by rwa [nhdsWithin_univ]) (mem_univ x)\n#align cont_diff_at.congr_of_eventually_eq ContDiffAt.congr_of_eventuallyEq\n\ntheorem ContDiffAt.of_le (h : ContDiffAt \ud835\udd5c n f x) (hmn : m \u2264 n) : ContDiffAt \ud835\udd5c m f x :=\n  ContDiffWithinAt.of_le h hmn\n#align cont_diff_at.of_le ContDiffAt.of_le\n\ntheorem ContDiffAt.continuousAt (h : ContDiffAt \ud835\udd5c n f x) : ContinuousAt f x := by\n  simpa [continuousWithinAt_univ] using h.continuousWithinAt\n#align cont_diff_at.continuous_at ContDiffAt.continuousAt\n\n/-- If a function is `C^n` with `n \u2265 1` at a point, then it is differentiable there. -/\ntheorem ContDiffAt.differentiableAt (h : ContDiffAt \ud835\udd5c n f x) (hn : 1 \u2264 n) :\n    DifferentiableAt \ud835\udd5c f x := by\n  simpa [hn, differentiableWithinAt_univ] using h.differentiableWithinAt\n#align cont_diff_at.differentiable_at ContDiffAt.differentiableAt\n\nnonrec lemma ContDiffAt.contDiffOn {m : \u2115} (h : ContDiffAt \ud835\udd5c n f x) (hm : m \u2264 n) :\n    \u2203 u \u2208 \ud835\udcdd x, ContDiffOn \ud835\udd5c m f u := by\n  simpa [nhdsWithin_univ] using h.contDiffOn hm\n\n/-- A function is `C^(n + 1)` at a point iff locally, it has a derivative which is `C^n`. -/\ntheorem contDiffAt_succ_iff_hasFDerivAt {n : \u2115} :\n    ContDiffAt \ud835\udd5c (n + 1 : \u2115) f x \u2194\n      \u2203 f' : E \u2192 E \u2192L[\ud835\udd5c] F, (\u2203 u \u2208 \ud835\udcdd x, \u2200 x \u2208 u, HasFDerivAt f (f' x) x) \u2227 ContDiffAt \ud835\udd5c n f' x := by\n  rw [\u2190 contDiffWithinAt_univ, contDiffWithinAt_succ_iff_hasFDerivWithinAt]\n  simp only [nhdsWithin_univ, exists_prop, mem_univ, insert_eq_of_mem]\n  constructor\n  \u00b7 rintro \u27e8u, H, f', h_fderiv, h_cont_diff\u27e9\n    rcases mem_nhds_iff.mp H with \u27e8t, htu, ht, hxt\u27e9\n    refine' \u27e8f', \u27e8t, _\u27e9, h_cont_diff.contDiffAt H\u27e9\n    refine' \u27e8mem_nhds_iff.mpr \u27e8t, Subset.rfl, ht, hxt\u27e9, _\u27e9\n    intro y hyt\n    refine' (h_fderiv y (htu hyt)).hasFDerivAt _\n    exact mem_nhds_iff.mpr \u27e8t, htu, ht, hyt\u27e9\n  \u00b7 rintro \u27e8f', \u27e8u, H, h_fderiv\u27e9, h_cont_diff\u27e9\n    refine' \u27e8u, H, f', _, h_cont_diff.contDiffWithinAt\u27e9\n    intro x hxu\n    exact (h_fderiv x hxu).hasFDerivWithinAt\n#align cont_diff_at_succ_iff_has_fderiv_at contDiffAt_succ_iff_hasFDerivAt\n\nprotected theorem ContDiffAt.eventually {n : \u2115} (h : ContDiffAt \ud835\udd5c n f x) :\n    \u2200\u1da0 y in \ud835\udcdd x, ContDiffAt \ud835\udd5c n f y := by\n  simpa [nhdsWithin_univ] using ContDiffWithinAt.eventually h\n#align cont_diff_at.eventually ContDiffAt.eventually\n\n/-! ### Smooth functions -/\n\nvariable (\ud835\udd5c)\n\n/-- A function is continuously differentiable up to `n` if it admits derivatives up to\norder `n`, which are continuous. Contrary to the case of definitions in domains (where derivatives\nmight not be unique) we do not need to localize the definition in space or time.\n-/\ndef ContDiff (n : \u2115\u221e) (f : E \u2192 F) : Prop :=\n  \u2203 p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F, HasFTaylorSeriesUpTo n f p\n#align cont_diff ContDiff\n\nvariable {\ud835\udd5c}\n\n/-- If `f` has a Taylor series up to `n`, then it is `C^n`. -/\ntheorem HasFTaylorSeriesUpTo.contDiff {f' : E \u2192 FormalMultilinearSeries \ud835\udd5c E F}\n    (hf : HasFTaylorSeriesUpTo n f f') : ContDiff \ud835\udd5c n f :=\n  \u27e8f', hf\u27e9\n#align has_ftaylor_series_up_to.cont_diff HasFTaylorSeriesUpTo.contDiff\n\ntheorem contDiffOn_univ : ContDiffOn \ud835\udd5c n f univ \u2194 ContDiff \ud835\udd5c n f := by\n  constructor\n  \u00b7 intro H\n    use ftaylorSeriesWithin \ud835\udd5c f univ\n    rw [\u2190 hasFTaylorSeriesUpToOn_univ_iff]\n    exact H.ftaylorSeriesWithin uniqueDiffOn_univ\n  \u00b7 rintro \u27e8p, hp\u27e9 x _ m hm\n    exact \u27e8univ, Filter.univ_sets _, p, (hp.hasFTaylorSeriesUpToOn univ).of_le hm\u27e9\n#align cont_diff_on_univ contDiffOn_univ\n\ntheorem contDiff_iff_contDiffAt : ContDiff \ud835\udd5c n f \u2194 \u2200 x, ContDiffAt \ud835\udd5c n f x := by\n  simp [\u2190 contDiffOn_univ, ContDiffOn, ContDiffAt]\n#align cont_diff_iff_cont_diff_at contDiff_iff_contDiffAt\n\ntheorem ContDiff.contDiffAt (h : ContDiff \ud835\udd5c n f) : ContDiffAt \ud835\udd5c n f x :=\n  contDiff_iff_contDiffAt.1 h x\n#align cont_diff.cont_diff_at ContDiff.contDiffAt\n\ntheorem ContDiff.contDiffWithinAt (h : ContDiff \ud835\udd5c n f) : ContDiffWithinAt \ud835\udd5c n f s x :=\n  h.contDiffAt.contDiffWithinAt\n#align cont_diff.cont_diff_within_at ContDiff.contDiffWithinAt\n\ntheorem contDiff_top : ContDiff \ud835\udd5c \u221e f \u2194 \u2200 n : \u2115, ContDiff \ud835\udd5c n f := by\n  simp [contDiffOn_univ.symm, contDiffOn_top]\n#align cont_diff_top contDiff_top\n\ntheorem contDiff_all_iff_nat : (\u2200 n, ContDiff \ud835\udd5c n f) \u2194 \u2200 n : \u2115, ContDiff \ud835\udd5c n f := by\n  simp only [\u2190 contDiffOn_univ, contDiffOn_all_iff_nat]\n#align cont_diff_all_iff_nat contDiff_all_iff_nat\n\ntheorem ContDiff.contDiffOn (h : ContDiff \ud835\udd5c n f) : ContDiffOn \ud835\udd5c n f s :=\n  (contDiffOn_univ.2 h).mono (subset_univ _)\n#align cont_diff.cont_diff_on ContDiff.contDiffOn\n\n@[simp]\ntheorem contDiff_zero : ContDiff \ud835\udd5c 0 f \u2194 Continuous f := by\n  rw [\u2190 contDiffOn_univ, continuous_iff_continuousOn_univ]\n  exact contDiffOn_zero\n#align cont_diff_zero contDiff_zero\n\ntheorem contDiffAt_zero : ContDiffAt \ud835\udd5c 0 f x \u2194 \u2203 u \u2208 \ud835\udcdd x, ContinuousOn f u := by\n  rw [\u2190 contDiffWithinAt_univ]; simp [contDiffWithinAt_zero, nhdsWithin_univ]\n#align cont_diff_at_zero contDiffAt_zero\n\ntheorem contDiffAt_one_iff :\n    ContDiffAt \ud835\udd5c 1 f x \u2194\n      \u2203 f' : E \u2192 E \u2192L[\ud835\udd5c] F, \u2203 u \u2208 \ud835\udcdd x, ContinuousOn f' u \u2227 \u2200 x \u2208 u, HasFDerivAt f (f' x) x := by\n  simp_rw [show (1 : \u2115\u221e) = (0 + 1 : \u2115) from (zero_add 1).symm, contDiffAt_succ_iff_hasFDerivAt,\n    show ((0 : \u2115) : \u2115\u221e) = 0 from rfl, contDiffAt_zero,\n    exists_mem_and_iff antitone_bforall antitone_continuousOn, and_comm]\n#align cont_diff_at_one_iff contDiffAt_one_iff\n\ntheorem ContDiff.of_le (h : ContDiff \ud835\udd5c n f) (hmn : m \u2264 n) : ContDiff \ud835\udd5c m f :=\n  contDiffOn_univ.1 <| (contDiffOn_univ.2 h).of_le hmn\n#align cont_diff.of_le ContDiff.of_le\n\ntheorem ContDiff.of_succ {n : \u2115} (h : ContDiff \ud835\udd5c (n + 1) f) : ContDiff \ud835\udd5c n f :=\n  h.of_le <| WithTop.coe_le_coe.mpr le_self_add\n#align cont_diff.of_succ ContDiff.of_succ\n\ntheorem ContDiff.one_of_succ {n : \u2115} (h : ContDiff \ud835\udd5c (n + 1) f) : ContDiff \ud835\udd5c 1 f :=\n  h.of_le <| WithTop.coe_le_coe.mpr le_add_self\n#align cont_diff.one_of_succ ContDiff.one_of_succ\n\ntheorem ContDiff.continuous (h : ContDiff \ud835\udd5c n f) : Continuous f :=\n  contDiff_zero.1 (h.of_le bot_le)\n#align cont_diff.continuous ContDiff.continuous\n\n/-- If a function is `C^n` with `n \u2265 1`, then it is differentiable. -/\ntheorem ContDiff.differentiable (h : ContDiff \ud835\udd5c n f) (hn : 1 \u2264 n) : Differentiable \ud835\udd5c f :=\n  differentiableOn_univ.1 <| (contDiffOn_univ.2 h).differentiableOn hn\n#align cont_diff.differentiable ContDiff.differentiable\n\ntheorem contDiff_iff_forall_nat_le : ContDiff \ud835\udd5c n f \u2194 \u2200 m : \u2115, \u2191m \u2264 n \u2192 ContDiff \ud835\udd5c m f := by\n  simp_rw [\u2190 contDiffOn_univ]; exact contDiffOn_iff_forall_nat_le\n#align cont_diff_iff_forall_nat_le contDiff_iff_forall_nat_le\n\n/-- A function is `C^(n+1)` iff it has a `C^n` derivative. -/\ntheorem contDiff_succ_iff_hasFDerivAt {n : \u2115} :\n    ContDiff \ud835\udd5c (n + 1 : \u2115) f \u2194\n      \u2203 f' : E \u2192 E \u2192L[\ud835\udd5c] F, ContDiff \ud835\udd5c n f' \u2227 \u2200 x, HasFDerivAt f (f' x) x := by\n  simp only [\u2190 contDiffOn_univ, \u2190 hasFDerivWithinAt_univ,\n    contDiffOn_succ_iff_hasFDerivWithin uniqueDiffOn_univ, Set.mem_univ, forall_true_left]\n#align cont_diff_succ_iff_has_fderiv contDiff_succ_iff_hasFDerivAt\n\n/-! ### Iterated derivative -/\n\n\nvariable (\ud835\udd5c)\n\n/-- The `n`-th derivative of a function, as a multilinear map, defined inductively. -/\nnoncomputable def iteratedFDeriv (n : \u2115) (f : E \u2192 F) : E \u2192 E[\u00d7n]\u2192L[\ud835\udd5c] F :=\n  Nat.recOn n (fun x => ContinuousMultilinearMap.curry0 \ud835\udd5c E (f x)) fun _ rec x =>\n    ContinuousLinearMap.uncurryLeft (fderiv \ud835\udd5c rec x)\n#align iterated_fderiv iteratedFDeriv\n\n/-- Formal Taylor series associated to a function. -/\ndef ftaylorSeries (f : E \u2192 F) (x : E) : FormalMultilinearSeries \ud835\udd5c E F := fun n =>\n  iteratedFDeriv \ud835\udd5c n f x\n#align ftaylor_series ftaylorSeries\n\nvariable {\ud835\udd5c}\n\n@[simp]\ntheorem iteratedFDeriv_zero_apply (m : Fin 0 \u2192 E) :\n    (iteratedFDeriv \ud835\udd5c 0 f x : (Fin 0 \u2192 E) \u2192 F) m = f x :=\n  rfl\n#align iterated_fderiv_zero_apply iteratedFDeriv_zero_apply\n\ntheorem iteratedFDeriv_zero_eq_comp :\n    iteratedFDeriv \ud835\udd5c 0 f = (continuousMultilinearCurryFin0 \ud835\udd5c E F).symm \u2218 f :=\n  rfl\n#align iterated_fderiv_zero_eq_comp iteratedFDeriv_zero_eq_comp\n\n@[simp]\ntheorem norm_iteratedFDeriv_zero : \u2016iteratedFDeriv \ud835\udd5c 0 f x\u2016 = \u2016f x\u2016 := by\n  -- Porting note: added `comp_apply`.\n  rw [iteratedFDeriv_zero_eq_comp, comp_apply, LinearIsometryEquiv.norm_map]\n#align norm_iterated_fderiv_zero norm_iteratedFDeriv_zero\n\ntheorem iteratedFDeriv_with_zero_eq : iteratedFDerivWithin \ud835\udd5c 0 f s = iteratedFDeriv \ud835\udd5c 0 f := rfl\n#align iterated_fderiv_with_zero_eq iteratedFDeriv_with_zero_eq\n\ntheorem iteratedFDeriv_succ_apply_left {n : \u2115} (m : Fin (n + 1) \u2192 E) :\n    (iteratedFDeriv \ud835\udd5c (n + 1) f x : (Fin (n + 1) \u2192 E) \u2192 F) m =\n      (fderiv \ud835\udd5c (iteratedFDeriv \ud835\udd5c n f) x : E \u2192 E[\u00d7n]\u2192L[\ud835\udd5c] F) (m 0) (tail m) :=\n  rfl\n#align iterated_fderiv_succ_apply_left iteratedFDeriv_succ_apply_left\n\n/-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,\nand the derivative of the `n`-th derivative. -/\ntheorem iteratedFDeriv_succ_eq_comp_left {n : \u2115} :\n    iteratedFDeriv \ud835\udd5c (n + 1) f =\n      continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun _ : Fin (n + 1) => E) F \u2218\n        fderiv \ud835\udd5c (iteratedFDeriv \ud835\udd5c n f) :=\n  rfl\n#align iterated_fderiv_succ_eq_comp_left iteratedFDeriv_succ_eq_comp_left\n\n/-- Writing explicitly the derivative of the `n`-th derivative as the composition of a currying\nlinear equiv, and the `n + 1`-th derivative. -/\ntheorem fderiv_iteratedFDeriv {n : \u2115} :\n    fderiv \ud835\udd5c (iteratedFDeriv \ud835\udd5c n f) =\n      (continuousMultilinearCurryLeftEquiv \ud835\udd5c (fun _ : Fin (n + 1) => E) F).symm \u2218\n        iteratedFDeriv \ud835\udd5c (n + 1) f := by\n  rw [iteratedFDeriv_succ_eq_comp_left]\n  ext1 x\n  simp only [Function.comp_apply, LinearIsometryEquiv.symm_apply_apply]\n#align fderiv_iterated_fderiv fderiv_iteratedFDeriv\n\ntheorem tsupport_iteratedFDeriv_subset (n : \u2115) : tsupport (iteratedFDeriv \ud835\udd5c n f) \u2286 tsupport f := by\n  induction' n with n IH\n  \u00b7 rw [iteratedFDeriv_zero_eq_comp]\n    exact closure_minimal ((support_comp_subset (LinearIsometryEquiv.map_zero _) _).trans\n      subset_closure) isClosed_closure\n  \u00b7 rw [iteratedFDeriv_succ_eq_comp_left]\n    exact closure_minimal ((support_comp_subset (LinearIsometryEquiv.map_zero _) _).trans\n      ((support_fderiv_subset \ud835\udd5c).trans IH)) isClosed_closure\n\ntheorem support_iteratedFDeriv_subset (n : \u2115) : support (iteratedFDeriv \ud835\udd5c n f) \u2286 tsupport f :=\n  subset_closure.trans (tsupport_iteratedFDeriv_subset n)\n\ntheorem HasCompactSupport.iteratedFDeriv (hf : HasCompactSupport f) (n : \u2115) :\n    HasCompactSupport (iteratedFDeriv \ud835\udd5c n f) :=\n  hf.of_isClosed_subset isClosed_closure (tsupport_iteratedFDeriv_subset n)\n#align has_compact_support.iterated_fderiv HasCompactSupport.iteratedFDeriv\n\ntheorem norm_fderiv_iteratedFDeriv {n : \u2115} :\n    \u2016fderiv \ud835\udd5c (iteratedFDeriv \ud835\udd5c n f) x\u2016 = \u2016iteratedFDeriv \ud835\udd5c (n + 1) f x\u2016 := by\n  -- Porting note: added `comp_apply`.\n  rw [iteratedFDeriv_succ_eq_comp_left, comp_apply, LinearIsometryEquiv.norm_map]\n#align norm_fderiv_iterated_fderiv norm_fderiv_iteratedFDeriv\n\ntheorem iteratedFDerivWithin_univ {n : \u2115} :\n    iteratedFDerivWithin \ud835\udd5c n f univ = iteratedFDeriv \ud835\udd5c n f := by\n  induction' n with n IH\n  \u00b7 ext x; simp\n  \u00b7 ext x m\n    rw [iteratedFDeriv_succ_apply_left, iteratedFDerivWithin_succ_apply_left, IH, fderivWithin_univ]\n#align iterated_fderiv_within_univ iteratedFDerivWithin_univ\n\n", "theoremStatement": "theorem HasFTaylorSeriesUpTo.eq_iteratedFDeriv\n    (h : HasFTaylorSeriesUpTo n f p) {m : \u2115} (hmn : (m : \u2115\u221e) \u2264 n) (x : E) :\n    p x m = iteratedFDeriv \ud835\udd5c m f x ", "theoremName": "HasFTaylorSeriesUpTo.eq_iteratedFDeriv", "fileCreated": {"commit": "7766bcc46ae69e55b32297edb59448f2e1fe1dde", "date": "2023-11-11"}, "theoremCreated": {"commit": "9b795560ce6e7bdf0aa2054b02bf2b18c5681bb0", "date": "2024-03-30"}, "file": "mathlib/Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "module": "Mathlib.Analysis.Calculus.ContDiff.Defs", "jsonFile": "Mathlib.Analysis.Calculus.ContDiff.Defs.jsonl", "positionMetadata": {"lineInFile": 1610, "tokenPositionInFile": 82713, "theoremPositionInFile": 0}, 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"Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  rw [\u2190 iteratedFDerivWithin_univ]\n  rw [\u2190 hasFTaylorSeriesUpToOn_univ_iff] at h\n  exact h.eq_iteratedFDerivWithin_of_uniqueDiffOn hmn uniqueDiffOn_univ (mem_univ _)", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 171}}
+{"srcContext": "/-\nCopyright (c) 2020 Yury G. Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury G. Kudryashov\n-/\nimport Mathlib.Analysis.SpecialFunctions.Pow.NNReal\nimport Mathlib.Analysis.SpecialFunctions.Pow.Continuity\nimport Mathlib.Analysis.SumOverResidueClass\n\n#align_import analysis.p_series from \"leanprover-community/mathlib\"@\"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8\"\n\n/-!\n# Convergence of `p`-series\n\nIn this file we prove that the series `\u2211' k in \u2115, 1 / k ^ p` converges if and only if `p > 1`.\nThe proof is based on the\n[Cauchy condensation test](https://en.wikipedia.org/wiki/Cauchy_condensation_test): `\u2211 k, f k`\nconverges if and only if so does `\u2211 k, 2 ^ k f (2 ^ k)`. We prove this test in\n`NNReal.summable_condensed_iff` and `summable_condensed_iff_of_nonneg`, then use it to prove\n`summable_one_div_rpow`. After this transformation, a `p`-series turns into a geometric series.\n\n## TODO\n\nIt should be easy to generalize arguments to Schl\u00f6milch's generalization of the Cauchy condensation\ntest once we need it.\n\n## Tags\n\np-series, Cauchy condensation test\n-/\n\n/-!\n### Cauchy condensation test\n\nIn this section we prove the Cauchy condensation test: for an antitone `f : \u2115 \u2192 \u211d\u22650` or `f : \u2115 \u2192 \u211d`,\n`\u2211 k, f k` converges if and only if so does `\u2211 k, 2 ^ k f (2 ^ k)`. Instead of giving a monolithic\nproof, we split it into a series of lemmas with explicit estimates of partial sums of each series in\nterms of the partial sums of the other series.\n-/\n\n\nnamespace Finset\n\nopen BigOperators\n\nvariable {M : Type*} [OrderedAddCommMonoid M] {f : \u2115 \u2192 M}\n\ntheorem le_sum_condensed' (hf : \u2200 \u2983m n\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m) (n : \u2115) :\n    (\u2211 k in Ico 1 (2 ^ n), f k) \u2264 \u2211 k in range n, 2 ^ k \u2022 f (2 ^ k) := by\n  induction' n with n ihn\n  \u00b7 simp\n  suffices (\u2211 k in Ico (2 ^ n) (2 ^ (n + 1)), f k) \u2264 2 ^ n \u2022 f (2 ^ n) by\n    rw [sum_range_succ, \u2190 sum_Ico_consecutive]\n    exact add_le_add ihn this\n    exacts [n.one_le_two_pow, Nat.pow_le_pow_of_le_right zero_lt_two n.le_succ]\n  have : \u2200 k \u2208 Ico (2 ^ n) (2 ^ (n + 1)), f k \u2264 f (2 ^ n) := fun k hk =>\n    hf (pow_pos zero_lt_two _) (mem_Ico.mp hk).1\n  convert sum_le_sum this\n  simp [pow_succ, mul_two]\n#align finset.le_sum_condensed' Finset.le_sum_condensed'\n\ntheorem le_sum_condensed (hf : \u2200 \u2983m n\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m) (n : \u2115) :\n    (\u2211 k in range (2 ^ n), f k) \u2264 f 0 + \u2211 k in range n, 2 ^ k \u2022 f (2 ^ k) := by\n  convert add_le_add_left (le_sum_condensed' hf n) (f 0)\n  rw [\u2190 sum_range_add_sum_Ico _ n.one_le_two_pow, sum_range_succ, sum_range_zero, zero_add]\n#align finset.le_sum_condensed Finset.le_sum_condensed\n\ntheorem sum_condensed_le' (hf : \u2200 \u2983m n\u2984, 1 < m \u2192 m \u2264 n \u2192 f n \u2264 f m) (n : \u2115) :\n    (\u2211 k in range n, 2 ^ k \u2022 f (2 ^ (k + 1))) \u2264 \u2211 k in Ico 2 (2 ^ n + 1), f k := by\n  induction' n with n ihn\n  \u00b7 simp\n  suffices 2 ^ n \u2022 f (2 ^ (n + 1)) \u2264 \u2211 k in Ico (2 ^ n + 1) (2 ^ (n + 1) + 1), f k by\n    rw [sum_range_succ, \u2190 sum_Ico_consecutive]\n    exacts [add_le_add ihn this,\n      (add_le_add_right n.one_le_two_pow _ : 1 + 1 \u2264 2 ^ n + 1),\n      add_le_add_right (Nat.pow_le_pow_of_le_right zero_lt_two n.le_succ) _]\n  have : \u2200 k \u2208 Ico (2 ^ n + 1) (2 ^ (n + 1) + 1), f (2 ^ (n + 1)) \u2264 f k := by\n    -- Note(kmill): was `fun k hk => ...` but `mem_Ico.mp hk` was elaborating with some\n    -- delayed assignment metavariables that weren't resolved in time. `intro` fixes this.\n    intro k hk\n    exact hf (Nat.one_le_two_pow.trans_lt <| (Nat.lt_succ_of_le le_rfl).trans_le (mem_Ico.mp hk).1)\n      (Nat.le_of_lt_succ <| (mem_Ico.mp hk).2)\n  convert sum_le_sum this\n  simp [pow_succ, mul_two]\n#align finset.sum_condensed_le' Finset.sum_condensed_le'\n\ntheorem sum_condensed_le (hf : \u2200 \u2983m n\u2984, 1 < m \u2192 m \u2264 n \u2192 f n \u2264 f m) (n : \u2115) :\n    (\u2211 k in range (n + 1), 2 ^ k \u2022 f (2 ^ k)) \u2264 f 1 + 2 \u2022 \u2211 k in Ico 2 (2 ^ n + 1), f k := by\n  convert add_le_add_left (nsmul_le_nsmul_right (sum_condensed_le' hf n) 2) (f 1)\n  simp [sum_range_succ', add_comm, pow_succ', mul_nsmul', sum_nsmul]\n#align finset.sum_condensed_le Finset.sum_condensed_le\n\nend Finset\n\nnamespace ENNReal\n\nopen Filter BigOperators\n\nvariable {f : \u2115 \u2192 \u211d\u22650\u221e}\n\ntheorem le_tsum_condensed (hf : \u2200 \u2983m n\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m) :\n    \u2211' k, f k \u2264 f 0 + \u2211' k : \u2115, 2 ^ k * f (2 ^ k) := by\n  rw [ENNReal.tsum_eq_iSup_nat' (Nat.tendsto_pow_atTop_atTop_of_one_lt _root_.one_lt_two)]\n  refine' iSup_le fun n => (Finset.le_sum_condensed hf n).trans (add_le_add_left _ _)\n  simp only [nsmul_eq_mul, Nat.cast_pow, Nat.cast_two]\n  apply ENNReal.sum_le_tsum\n#align ennreal.le_tsum_condensed ENNReal.le_tsum_condensed\n\ntheorem tsum_condensed_le (hf : \u2200 \u2983m n\u2984, 1 < m \u2192 m \u2264 n \u2192 f n \u2264 f m) :\n    (\u2211' k : \u2115, 2 ^ k * f (2 ^ k)) \u2264 f 1 + 2 * \u2211' k, f k := by\n  rw [ENNReal.tsum_eq_iSup_nat' (tendsto_atTop_mono Nat.le_succ tendsto_id), two_mul, \u2190 two_nsmul]\n  refine'\n    iSup_le fun n =>\n      le_trans _\n        (add_le_add_left\n          (nsmul_le_nsmul_right (ENNReal.sum_le_tsum <| Finset.Ico 2 (2 ^ n + 1)) _) _)\n  simpa using Finset.sum_condensed_le hf n\n#align ennreal.tsum_condensed_le ENNReal.tsum_condensed_le\n\nend ENNReal\n\nnamespace NNReal\n\nopen ENNReal in\n/-- Cauchy condensation test for a series of `NNReal` version. -/\ntheorem summable_condensed_iff {f : \u2115 \u2192 \u211d\u22650} (hf : \u2200 \u2983m n\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m) :\n    (Summable fun k : \u2115 => (2 : \u211d\u22650) ^ k * f (2 ^ k)) \u2194 Summable f := by\n  simp only [\u2190 ENNReal.tsum_coe_ne_top_iff_summable, Ne, not_iff_not, ENNReal.coe_mul,\n    ENNReal.coe_pow, ENNReal.coe_two]\n  constructor <;> intro h\n  \u00b7 replace hf : \u2200 m n, 1 < m \u2192 m \u2264 n \u2192 (f n : \u211d\u22650\u221e) \u2264 f m := fun m n hm hmn =>\n      ENNReal.coe_le_coe.2 (hf (zero_lt_one.trans hm) hmn)\n    simpa [h, ENNReal.add_eq_top, ENNReal.mul_eq_top] using ENNReal.tsum_condensed_le hf\n  \u00b7 replace hf : \u2200 m n, 0 < m \u2192 m \u2264 n \u2192 (f n : \u211d\u22650\u221e) \u2264 f m := fun m n hm hmn =>\n      ENNReal.coe_le_coe.2 (hf hm hmn)\n    simpa [h, ENNReal.add_eq_top] using ENNReal.le_tsum_condensed hf\n#align nnreal.summable_condensed_iff NNReal.summable_condensed_iff\n\nend NNReal\n\nopen NNReal in\n/-- Cauchy condensation test for antitone series of nonnegative real numbers. -/\ntheorem summable_condensed_iff_of_nonneg {f : \u2115 \u2192 \u211d} (h_nonneg : \u2200 n, 0 \u2264 f n)\n    (h_mono : \u2200 \u2983m n\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m) :\n    (Summable fun k : \u2115 => (2 : \u211d) ^ k * f (2 ^ k)) \u2194 Summable f := by\n  lift f to \u2115 \u2192 \u211d\u22650 using h_nonneg\n  simp only [NNReal.coe_le_coe] at *\n  exact_mod_cast NNReal.summable_condensed_iff h_mono\n#align summable_condensed_iff_of_nonneg summable_condensed_iff_of_nonneg\n\nsection p_series\n\n/-!\n### Convergence of the `p`-series\n\nIn this section we prove that for a real number `p`, the series `\u2211' n : \u2115, 1 / (n ^ p)` converges if\nand only if `1 < p`. There are many different proofs of this fact. The proof in this file uses the\nCauchy condensation test we formalized above. This test implies that `\u2211 n, 1 / (n ^ p)` converges if\nand only if `\u2211 n, 2 ^ n / ((2 ^ n) ^ p)` converges, and the latter series is a geometric series with\ncommon ratio `2 ^ {1 - p}`. -/\n\nnamespace Real\n\nopen Filter BigOperators\n\n/-- Test for convergence of the `p`-series: the real-valued series `\u2211' n : \u2115, (n ^ p)\u207b\u00b9` converges\nif and only if `1 < p`. -/\n@[simp]\ntheorem summable_nat_rpow_inv {p : \u211d} :\n    Summable (fun n => ((n : \u211d) ^ p)\u207b\u00b9 : \u2115 \u2192 \u211d) \u2194 1 < p := by\n  rcases le_or_lt 0 p with hp | hp\n  /- Cauchy condensation test applies only to antitone sequences, so we consider the\n    cases `0 \u2264 p` and `p < 0` separately. -/\n  \u00b7 rw [\u2190 summable_condensed_iff_of_nonneg]\n    \u00b7 simp_rw [Nat.cast_pow, Nat.cast_two, \u2190 rpow_natCast, \u2190 rpow_mul zero_lt_two.le, mul_comm _ p,\n        rpow_mul zero_lt_two.le, rpow_natCast, \u2190 inv_pow, \u2190 mul_pow,\n        summable_geometric_iff_norm_lt_one]\n      nth_rw 1 [\u2190 rpow_one 2]\n      rw [\u2190 division_def, \u2190 rpow_sub zero_lt_two, norm_eq_abs,\n        abs_of_pos (rpow_pos_of_pos zero_lt_two _), rpow_lt_one_iff zero_lt_two.le]\n      norm_num\n    \u00b7 intro n\n      positivity\n    \u00b7 intro m n hm hmn\n      gcongr\n  -- If `p < 0`, then `1 / n ^ p` tends to infinity, thus the series diverges.\n  \u00b7 suffices \u00acSummable (fun n => ((n : \u211d) ^ p)\u207b\u00b9 : \u2115 \u2192 \u211d) by\n      have : \u00ac1 < p := fun hp\u2081 => hp.not_le (zero_le_one.trans hp\u2081.le)\n      simpa only [this, iff_false]\n    \u00b7 intro h\n      obtain \u27e8k : \u2115, hk\u2081 : ((k : \u211d) ^ p)\u207b\u00b9 < 1, hk\u2080 : k \u2260 0\u27e9 :=\n        ((h.tendsto_cofinite_zero.eventually (gt_mem_nhds zero_lt_one)).and\n            (eventually_cofinite_ne 0)).exists\n      apply hk\u2080\n      rw [\u2190 pos_iff_ne_zero, \u2190 @Nat.cast_pos \u211d] at hk\u2080\n      simpa [inv_lt_one_iff_of_pos (rpow_pos_of_pos hk\u2080 _), one_lt_rpow_iff_of_pos hk\u2080, hp,\n        hp.not_lt, hk\u2080] using hk\u2081\n#align real.summable_nat_rpow_inv Real.summable_nat_rpow_inv\n\n@[simp]\ntheorem summable_nat_rpow {p : \u211d} : Summable (fun n => (n : \u211d) ^ p : \u2115 \u2192 \u211d) \u2194 p < -1 := by\n  rcases neg_surjective p with \u27e8p, rfl\u27e9\n  simp [rpow_neg]\n#align real.summable_nat_rpow Real.summable_nat_rpow\n\n/-- Test for convergence of the `p`-series: the real-valued series `\u2211' n : \u2115, 1 / n ^ p` converges\nif and only if `1 < p`. -/\ntheorem summable_one_div_nat_rpow {p : \u211d} :\n    Summable (fun n => 1 / (n : \u211d) ^ p : \u2115 \u2192 \u211d) \u2194 1 < p := by\n  simp\n#align real.summable_one_div_nat_rpow Real.summable_one_div_nat_rpow\n\n/-- Test for convergence of the `p`-series: the real-valued series `\u2211' n : \u2115, (n ^ p)\u207b\u00b9` converges\nif and only if `1 < p`. -/\n-- Porting note: temporarily remove `@[simp]` because of a problem with `simp`\n-- see https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/looping.20in.20.60simp.60.20set/near/361134234\ntheorem summable_nat_pow_inv {p : \u2115} :\n    Summable (fun n => ((n : \u211d) ^ p)\u207b\u00b9 : \u2115 \u2192 \u211d) \u2194 1 < p := by\n  simp only [\u2190 rpow_natCast, summable_nat_rpow_inv, Nat.one_lt_cast]\n#align real.summable_nat_pow_inv Real.summable_nat_pow_inv\n\n/-- Test for convergence of the `p`-series: the real-valued series `\u2211' n : \u2115, 1 / n ^ p` converges\nif and only if `1 < p`. -/\ntheorem summable_one_div_nat_pow {p : \u2115} :\n    Summable (fun n => 1 / (n : \u211d) ^ p : \u2115 \u2192 \u211d) \u2194 1 < p := by\n  -- porting note (#10745): explicitly supplied two simp lemmas\n  simp only [one_div, Real.summable_nat_pow_inv]\n#align real.summable_one_div_nat_pow Real.summable_one_div_nat_pow\n\n/-- Summability of the `p`-series over `\u2124`. -/\ntheorem summable_one_div_int_pow {p : \u2115} :\n    (Summable fun n : \u2124 \u21a6 1 / (n : \u211d) ^ p) \u2194 1 < p := by\n  refine \u27e8fun h \u21a6 summable_one_div_nat_pow.mp (h.comp_injective Nat.cast_injective),\n    fun h \u21a6 .of_nat_of_neg (summable_one_div_nat_pow.mpr h)\n      (((summable_one_div_nat_pow.mpr h).mul_left <| 1 / (-1 : \u211d) ^ p).congr fun n \u21a6 ?_)\u27e9\n  rw [Int.cast_neg, Int.cast_natCast, neg_eq_neg_one_mul (n : \u211d), mul_pow, mul_one_div, div_div]\n#align real.summable_one_div_int_pow Real.summable_one_div_int_pow\n\ntheorem summable_abs_int_rpow {b : \u211d} (hb : 1 < b) :\n    Summable fun n : \u2124 => |(n : \u211d)| ^ (-b) := by\n  apply Summable.of_nat_of_neg\n  on_goal 2 => simp_rw [Int.cast_neg, abs_neg]\n  all_goals\n    simp_rw [Int.cast_natCast, fun n : \u2115 => abs_of_nonneg (n.cast_nonneg : 0 \u2264 (n : \u211d))]\n    rwa [summable_nat_rpow, neg_lt_neg_iff]\n#align real.summable_abs_int_rpow Real.summable_abs_int_rpow\n\n/-- Harmonic series is not unconditionally summable. -/\ntheorem not_summable_natCast_inv : \u00acSummable (fun n => n\u207b\u00b9 : \u2115 \u2192 \u211d) := by\n  have : \u00acSummable (fun n => ((n : \u211d) ^ 1)\u207b\u00b9 : \u2115 \u2192 \u211d) :=\n    mt (summable_nat_pow_inv (p := 1)).1 (lt_irrefl 1)\n  simpa\n#align real.not_summable_nat_cast_inv Real.not_summable_natCast_inv\n\n/-- Harmonic series is not unconditionally summable. -/\ntheorem not_summable_one_div_natCast : \u00acSummable (fun n => 1 / n : \u2115 \u2192 \u211d) := by\n  simpa only [inv_eq_one_div] using not_summable_natCast_inv\n#align real.not_summable_one_div_nat_cast Real.not_summable_one_div_natCast\n\n/-- **Divergence of the Harmonic Series** -/\ntheorem tendsto_sum_range_one_div_nat_succ_atTop :\n    Tendsto (fun n => \u2211 i in Finset.range n, (1 / (i + 1) : \u211d)) atTop atTop := by\n  rw [\u2190 not_summable_iff_tendsto_nat_atTop_of_nonneg]\n  \u00b7 exact_mod_cast mt (_root_.summable_nat_add_iff 1).1 not_summable_one_div_natCast\n  \u00b7 exact fun i => by positivity\n#align real.tendsto_sum_range_one_div_nat_succ_at_top Real.tendsto_sum_range_one_div_nat_succ_atTop\n\nend Real\n\nnamespace NNReal\n\n@[simp]\ntheorem summable_rpow_inv {p : \u211d} :\n    Summable (fun n => ((n : \u211d\u22650) ^ p)\u207b\u00b9 : \u2115 \u2192 \u211d\u22650) \u2194 1 < p := by\n  simp [\u2190 NNReal.summable_coe]\n#align nnreal.summable_rpow_inv NNReal.summable_rpow_inv\n\n@[simp]\ntheorem summable_rpow {p : \u211d} : Summable (fun n => (n : \u211d\u22650) ^ p : \u2115 \u2192 \u211d\u22650) \u2194 p < -1 := by\n  simp [\u2190 NNReal.summable_coe]\n#align nnreal.summable_rpow NNReal.summable_rpow\n\ntheorem summable_one_div_rpow {p : \u211d} :\n    Summable (fun n => 1 / (n : \u211d\u22650) ^ p : \u2115 \u2192 \u211d\u22650) \u2194 1 < p := by\n  simp\n#align nnreal.summable_one_div_rpow NNReal.summable_one_div_rpow\n\nend NNReal\n\nend p_series\n\nsection\n\nopen Finset BigOperators\n\nvariable {\u03b1 : Type*} [LinearOrderedField \u03b1]\n\nset_option tactic.skipAssignedInstances false in\ntheorem sum_Ioc_inv_sq_le_sub {k n : \u2115} (hk : k \u2260 0) (h : k \u2264 n) :\n    (\u2211 i in Ioc k n, ((i : \u03b1) ^ 2)\u207b\u00b9) \u2264 (k : \u03b1)\u207b\u00b9 - (n : \u03b1)\u207b\u00b9 := by\n  refine' Nat.le_induction _ _ n h\n  \u00b7 simp only [Ioc_self, sum_empty, sub_self, le_refl]\n  intro n hn IH\n  rw [sum_Ioc_succ_top hn]\n  apply (add_le_add IH le_rfl).trans\n  simp only [sub_eq_add_neg, add_assoc, Nat.cast_add, Nat.cast_one, le_add_neg_iff_add_le,\n    add_le_iff_nonpos_right, neg_add_le_iff_le_add, add_zero]\n  have A : 0 < (n : \u03b1) := by simpa using hk.bot_lt.trans_le hn\n  have B : 0 < (n : \u03b1) + 1 := by linarith\n  field_simp\n  rw [div_le_div_iff _ A, \u2190 sub_nonneg]\n  \u00b7 ring_nf\n    rw [add_comm]\n    exact B.le\n  \u00b7 -- Porting note: was `nlinarith`\n    positivity\n#align sum_Ioc_inv_sq_le_sub sum_Ioc_inv_sq_le_sub\n\ntheorem sum_Ioo_inv_sq_le (k n : \u2115) : (\u2211 i in Ioo k n, (i ^ 2 : \u03b1)\u207b\u00b9) \u2264 2 / (k + 1) :=\n  calc\n    (\u2211 i in Ioo k n, ((i : \u03b1) ^ 2)\u207b\u00b9) \u2264 \u2211 i in Ioc k (max (k + 1) n), ((i : \u03b1) ^ 2)\u207b\u00b9 := by\n      apply sum_le_sum_of_subset_of_nonneg\n      \u00b7 intro x hx\n        simp only [mem_Ioo] at hx\n        simp only [hx, hx.2.le, mem_Ioc, le_max_iff, or_true_iff, and_self_iff]\n      \u00b7 intro i _hi _hident\n        positivity\n    _ \u2264 ((k + 1 : \u03b1) ^ 2)\u207b\u00b9 + \u2211 i in Ioc k.succ (max (k + 1) n), ((i : \u03b1) ^ 2)\u207b\u00b9 := by\n      rw [\u2190 Nat.Icc_succ_left, \u2190 Nat.Ico_succ_right, sum_eq_sum_Ico_succ_bot]\n      swap; \u00b7 exact Nat.succ_lt_succ ((Nat.lt_succ_self k).trans_le (le_max_left _ _))\n      rw [Nat.Ico_succ_right, Nat.Icc_succ_left, Nat.cast_succ]\n    _ \u2264 ((k + 1 : \u03b1) ^ 2)\u207b\u00b9 + (k + 1 : \u03b1)\u207b\u00b9 := by\n      refine' add_le_add le_rfl ((sum_Ioc_inv_sq_le_sub _ (le_max_left _ _)).trans _)\n      \u00b7 simp only [Ne, Nat.succ_ne_zero, not_false_iff]\n      \u00b7 simp only [Nat.cast_succ, one_div, sub_le_self_iff, inv_nonneg, Nat.cast_nonneg]\n    _ \u2264 1 / (k + 1) + 1 / (k + 1) := by\n      have A : (1 : \u03b1) \u2264 k + 1 := by simp only [le_add_iff_nonneg_left, Nat.cast_nonneg]\n      simp_rw [\u2190 one_div]\n      gcongr\n      simpa using pow_le_pow_right A one_le_two\n    _ = 2 / (k + 1) := by ring\n#align sum_Ioo_inv_sq_le sum_Ioo_inv_sq_le\n\nend\n\nopen Set Nat in\n/-- The harmonic series restricted to a residue class is not summable. -/\nlemma Real.not_summable_indicator_one_div_natCast {m : \u2115} (hm : m \u2260 0) (k : ZMod m) :\n    \u00ac Summable ({n : \u2115 | (n : ZMod m) = k}.indicator fun n : \u2115 \u21a6 (1 / n : \u211d)) := by\n  have : NeZero m := \u27e8hm\u27e9 -- instance is needed below\n  rw [\u2190 summable_nat_add_iff 1] -- shift by one to avoid non-monotonicity at zero\n  have h (n : \u2115) : {n : \u2115 | (n : ZMod m) = k - 1}.indicator (fun n : \u2115 \u21a6 (1 / (n + 1 :) : \u211d)) n =\n      if (n : ZMod m) = k - 1 then (1 / (n + 1) : \u211d) else (0 : \u211d) := by\n    simp only [indicator_apply, mem_setOf_eq, cast_add, cast_one]\n  simp_rw [indicator_apply, mem_setOf, cast_add, cast_one, \u2190 eq_sub_iff_add_eq, \u2190 h]\n  rw [summable_indicator_mod_iff (fun n\u2081 n\u2082 h \u21a6 by gcongr) (k - 1)]\n  exact mt (summable_nat_add_iff (f := fun n : \u2115 \u21a6 1 / (n : \u211d)) 1).mp not_summable_one_div_natCast\n\n/-!\n## Translating the `p`-series by a real number\n-/\nsection shifted\n\nopen Filter Asymptotics Topology\n\nlemma Real.summable_one_div_nat_add_rpow (a : \u211d) (s : \u211d) :\n    Summable (fun n : \u2115 \u21a6 1 / |n + a| ^ s) \u2194 1 < s := by\n  suffices \u2200 (b c : \u211d), Summable (fun n : \u2115 \u21a6 1 / |n + b| ^ s) \u2192\n      Summable (fun n : \u2115 \u21a6 1 / |n + c| ^ s) by\n    simp_rw [\u2190 summable_one_div_nat_rpow, Iff.intro (this a 0) (this 0 a), add_zero, Nat.abs_cast]\n  refine fun b c h \u21a6 summable_of_isBigO_nat h (isBigO_of_div_tendsto_nhds ?_ 1 ?_)\n  \u00b7 filter_upwards [eventually_gt_atTop (Nat.ceil |b|)] with n hn hx\n    have hna : 0 < n + b := by linarith [lt_of_abs_lt ((abs_neg b).symm \u25b8 Nat.lt_of_ceil_lt hn)]\n    exfalso\n    revert hx\n    positivity\n  \u00b7 simp_rw [Pi.div_def, div_div, mul_one_div, one_div_div]\n    refine (?_ : Tendsto (fun x : \u211d \u21a6 |x + b| ^ s / |x + c| ^ s) atTop (\ud835\udcdd 1)).comp\n      tendsto_natCast_atTop_atTop\n    have : Tendsto (fun x : \u211d \u21a6 1 + (b - c) / x) atTop (\ud835\udcdd 1) := by\n      simpa using tendsto_const_nhds.add ((tendsto_const_nhds (X := \u211d)).div_atTop tendsto_id)\n    have : Tendsto (fun x \u21a6 (x + b) / (x + c)) atTop (\ud835\udcdd 1) := by\n      refine (this.comp (tendsto_id.atTop_add (tendsto_const_nhds (x := c)))).congr' ?_\n      filter_upwards [eventually_gt_atTop (-c)] with x hx\n      field_simp [(by linarith : 0 < x + c).ne']\n    apply (one_rpow s \u25b8 (continuousAt_rpow_const _ s (by simp)).tendsto.comp this).congr'\n    filter_upwards [eventually_gt_atTop (-b), eventually_gt_atTop (-c)] with x hb hc\n    rw [neg_lt_iff_pos_add] at hb hc\n    rw [Function.comp_apply, div_rpow hb.le hc.le, abs_of_pos hb, abs_of_pos hc]\n\n", "theoremStatement": "lemma Real.summable_one_div_int_add_rpow (a : \u211d) (s : \u211d) :\n    Summable (fun n : \u2124 \u21a6 1 / |n + a| ^ s) \u2194 1 < s ", "theoremName": "Real.summable_one_div_int_add_rpow", "fileCreated": {"commit": "3dde1ee7ebf54afe7d00971284f6c784d2a9d41b", "date": "2023-05-25"}, "theoremCreated": {"commit": "dff9042e145f907f200e01e56e15ebc046c146ba", "date": "2024-03-21"}, "file": "mathlib/Mathlib/Analysis/PSeries.lean", "module": "Mathlib.Analysis.PSeries", "jsonFile": "Mathlib.Analysis.PSeries.jsonl", "positionMetadata": {"lineInFile": 391, "tokenPositionInFile": 17410, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, 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"Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.SumOverResidueClass"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  simp_rw [summable_int_iff_summable_nat_and_neg, \u2190 abs_neg (\u2191(-_ : \u2124) + a), neg_add,\n    Int.cast_neg, neg_neg, Int.cast_natCast, summable_one_div_nat_add_rpow, and_self]", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 177}}
+{"srcContext": "/-\nCopyright (c) 2023 Kevin Buzzard. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Ya\u00ebl Dillies\n-/\nimport Mathlib.Algebra.GCDMonoid.Finset\nimport Mathlib.Algebra.GroupPower.Ring\nimport Mathlib.Data.Nat.Parity\nimport Mathlib.Data.Rat.Defs\nimport Mathlib.RingTheory.Int.Basic\nimport Mathlib.Tactic.NormNum\nimport Mathlib.Tactic.Positivity.Basic\nimport Mathlib.Tactic.TFAE\n\n/-!\n# Statement of Fermat's Last Theorem\n\nThis file states Fermat's Last Theorem. We provide a statement over a general semiring with\nspecific exponent, along with the usual statement over the naturals.\n-/\n\nopen List\n\n/-- Statement of Fermat's Last Theorem over a given semiring with a specific exponent. -/\ndef FermatLastTheoremWith (\u03b1 : Type*) [Semiring \u03b1] (n : \u2115) : Prop :=\n  \u2200 a b c : \u03b1, a \u2260 0 \u2192 b \u2260 0 \u2192 c \u2260 0 \u2192 a ^ n + b ^ n \u2260 c ^ n\n\n/-- Statement of Fermat's Last Theorem over the naturals for a given exponent. -/\ndef FermatLastTheoremFor (n : \u2115) : Prop := FermatLastTheoremWith \u2115 n\n\n/-- Statement of Fermat's Last Theorem: `a ^ n + b ^ n = c ^ n` has no nontrivial natural solution\nwhen `n \u2265 3`. -/\ndef FermatLastTheorem : Prop := \u2200 n \u2265 3, FermatLastTheoremFor n\n\nlemma fermatLastTheoremFor_zero : FermatLastTheoremFor 0 :=\n  fun _ _ _ _ _ _ \u21a6 by norm_num\n\nlemma not_fermatLastTheoremFor_one : \u00ac FermatLastTheoremFor 1 :=\n  fun h \u21a6 h 1 1 2 (by norm_num) (by norm_num) (by norm_num) (by norm_num)\n\nlemma not_fermatLastTheoremFor_two : \u00ac FermatLastTheoremFor 2 :=\n  fun h \u21a6 h 3 4 5 (by norm_num) (by norm_num) (by norm_num) (by norm_num)\n\nvariable {\u03b1 : Type*} [Semiring \u03b1] [NoZeroDivisors \u03b1] {m n : \u2115}\n\nlemma FermatLastTheoremWith.mono (hmn : m \u2223 n) (hm : FermatLastTheoremWith \u03b1 m) :\n    FermatLastTheoremWith \u03b1 n := by\n  rintro a b c ha hb hc\n  obtain \u27e8k, rfl\u27e9 := hmn\n  simp_rw [pow_mul']\n  refine hm _ _ _ ?_ ?_ ?_ <;> exact pow_ne_zero _ \u2039_\u203a\n\nlemma FermatLastTheoremFor.mono (hmn : m \u2223 n) (hm : FermatLastTheoremFor m) :\n    FermatLastTheoremFor n := by\n  exact FermatLastTheoremWith.mono hmn hm\n\nlemma fermatLastTheoremWith_nat_int_rat_tfae (n : \u2115) :\n    TFAE [FermatLastTheoremWith \u2115 n, FermatLastTheoremWith \u2124 n, FermatLastTheoremWith \u211a n] := by\n  tfae_have 1 \u2192 2\n  \u00b7 rintro h a b c ha hb hc habc\n    obtain hn | hn := n.even_or_odd\n    \u00b7 refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)\n        (Int.natCast_inj.1 _)\n      push_cast\n      simp only [hn.pow_abs, habc]\n    obtain ha | ha := ha.lt_or_lt <;> obtain hb | hb := hb.lt_or_lt <;>\n      obtain hc | hc := hc.lt_or_lt\n    \u00b7 refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)\n        (Int.natCast_inj.1 _)\n      push_cast\n      simp only [abs_of_neg, neg_pow a, neg_pow b, neg_pow c, \u2190 mul_add, habc, *]\n    \u00b7 exact (by positivity : 0 < c ^ n).not_lt <| habc.symm.trans_lt <| add_neg (hn.pow_neg ha) <|\n        hn.pow_neg hb\n    \u00b7 refine' h b.natAbs c.natAbs a.natAbs (by positivity) (by positivity) (by positivity)\n        (Int.natCast_inj.1 _)\n      push_cast\n      simp only [abs_of_pos, abs_of_neg, hn.neg_pow, habc, add_neg_eq_iff_eq_add,\n        eq_neg_add_iff_add_eq, *]\n    \u00b7 refine' h a.natAbs c.natAbs b.natAbs (by positivity) (by positivity) (by positivity)\n        (Int.natCast_inj.1 _)\n      push_cast\n      simp only [abs_of_pos, abs_of_neg, hn.neg_pow, habc, neg_add_eq_iff_eq_add,\n        eq_neg_add_iff_add_eq, *]\n    \u00b7 refine' h c.natAbs a.natAbs b.natAbs (by positivity) (by positivity) (by positivity)\n        (Int.natCast_inj.1 _)\n      push_cast\n      simp only [abs_of_pos, abs_of_neg, hn.neg_pow, habc, neg_add_eq_iff_eq_add,\n        eq_add_neg_iff_add_eq, *]\n    \u00b7 refine' h c.natAbs b.natAbs a.natAbs (by positivity) (by positivity) (by positivity)\n        (Int.natCast_inj.1 _)\n      push_cast\n      simp only [abs_of_pos, abs_of_neg, hn.neg_pow, habc, add_neg_eq_iff_eq_add,\n        eq_add_neg_iff_add_eq, *]\n    \u00b7 exact (by positivity : 0 < a ^ n + b ^ n).not_lt <| habc.trans_lt <| hn.pow_neg hc\n    \u00b7 refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)\n        (Int.natCast_inj.1 _)\n      push_cast\n      simp only [abs_of_pos, habc, *]\n  tfae_have 2 \u2192 3\n  \u00b7 rintro h a b c ha hb hc habc\n    rw [\u2190 Rat.num_ne_zero] at ha hb hc\n    refine' h (a.num * b.den * c.den) (a.den * b.num * c.den) (a.den * b.den * c.num)\n      (by positivity) (by positivity) (by positivity) _\n    have : (a.den * b.den * c.den : \u211a) ^ n \u2260 0 := by positivity\n    refine' Int.cast_injective <| (div_left_inj' this).1 _\n    push_cast\n    simp only [add_div, \u2190 div_pow, mul_div_mul_comm, div_self (by positivity : (a.den : \u211a) \u2260 0),\n      div_self (by positivity : (b.den : \u211a) \u2260 0), div_self (by positivity : (c.den : \u211a) \u2260 0),\n      one_mul, mul_one, Rat.num_div_den, habc]\n  tfae_have 3 \u2192 1\n  \u00b7 rintro h a b c\n    exact mod_cast h a b c\n  tfae_finish\n\nlemma fermatLastTheoremFor_iff_nat {n : \u2115} : FermatLastTheoremFor n \u2194 FermatLastTheoremWith \u2115 n :=\n  Iff.rfl\n\nlemma fermatLastTheoremFor_iff_int {n : \u2115} : FermatLastTheoremFor n \u2194 FermatLastTheoremWith \u2124 n :=\n  (fermatLastTheoremWith_nat_int_rat_tfae n).out 0 1\n\nlemma fermatLastTheoremFor_iff_rat {n : \u2115} : FermatLastTheoremFor n \u2194 FermatLastTheoremWith \u211a n :=\n  (fermatLastTheoremWith_nat_int_rat_tfae n).out 0 2\n\nopen Finset in\n", "theoremStatement": "/-- To prove Fermat Last Theorem in any semiring that is a `NormalizedGCDMonoid` one can assume\nthat the `gcd` of `{a, b, c}` is `1`. -/\nlemma fermatLastTheoremWith_of_fermatLastTheoremWith_coprime {n : \u2115} {R : Type*} [CommSemiring R]\n    [IsDomain R] [DecidableEq R] [NormalizedGCDMonoid R]\n    (hn : \u2200 a b c : R, a \u2260 0 \u2192 b \u2260 0 \u2192 c \u2260 0 \u2192 ({a, b, c} : Finset R).gcd id = 1 \u2192\n      a ^ n + b ^ n \u2260 c ^ n) :\n    FermatLastTheoremWith R n ", "theoremName": "fermatLastTheoremWith_of_fermatLastTheoremWith_coprime", "fileCreated": {"commit": "ca73a84a06ff3161f85d115801ce39ac88c7eed4", "date": "2023-09-12"}, "theoremCreated": {"commit": 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"Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Cases", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Function.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Subtype", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Tactic.PushNeg", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Tactic.Contrapose", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Tactic.Lift", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Prod.PProd", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Order.RelIso.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.Int.Units", "Mathlib.Data.Nat.Prime", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Data.List.Prime", "Mathlib.Data.Fin.OrderHom", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.Nat.Factors", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Regular.SMul", "Mathlib.Data.Rat.BigOperators", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Data.Finsupp.Basic", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Set.UnionLift", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Tactic.FinCases", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.GroupTheory.Finiteness", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.Multiplicity", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  intro a b c ha hb hc habc\n  let s : Finset R := {a, b, c}; let d := s.gcd id\n  obtain \u27e8A, hA\u27e9 : d \u2223 a := gcd_dvd (by simp [s])\n  obtain \u27e8B, hB\u27e9 : d \u2223 b := gcd_dvd (by simp [s])\n  obtain \u27e8C, hC\u27e9 : d \u2223 c := gcd_dvd (by simp [s])\n  simp only [hA, hB, hC, mul_ne_zero_iff, mul_pow] at ha hb hc habc\n  rw [\u2190 mul_add, mul_right_inj' (pow_ne_zero n ha.1)] at habc\n  refine hn A B C ha.2 hb.2 hc.2 ?_ habc\n  rw [\u2190 Finset.normalize_gcd, normalize_eq_one]\n  obtain \u27e8u, hu\u27e9 := normalize_associated d\n  refine \u27e8u, mul_left_cancel\u2080 (mt normalize_eq_zero.mp ha.1) (hu.symm \u25b8 ?_)\u27e9\n  rw [\u2190 Finset.gcd_mul_left, gcd_eq_gcd_image, image_insert, image_insert, image_singleton,\n      id_eq, id_eq, id_eq, \u2190 hA, \u2190 hB, \u2190 hC]", "proofType": "tactic", "proofLengthLines": 13, "proofLengthTokens": 710}}
+{"srcContext": "/-\nCopyright (c) 2024 Sophie Morel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sophie Morel\n-/\nimport Mathlib.Analysis.NormedSpace.Multilinear.Basic\nimport Mathlib.LinearAlgebra.PiTensorProduct\n\n/-!\n# Projective seminorm on the tensor of a finite family of normed spaces.\n\nLet `\ud835\udd5c` be a nontrivially normed field and `E` be a family of normed `\ud835\udd5c`-vector spaces `E\u1d62`,\nindexed by a finite type `\u03b9`. We define a seminorm on `\u2a02[\ud835\udd5c] i, E\u1d62`, which we call the\n\"projective seminorm\". For `x` an element of `\u2a02[\ud835\udd5c] i, E\u1d62`, its projective seminorm is the\ninfimum over all expressions of `x` as `\u2211 j, \u2a02\u209c[\ud835\udd5c] m\u2c7c i` (with the `m\u2c7c` in `\u03a0 i, E\u1d62`)\nof `\u2211 j, \u03a0 i, \u2016m\u2c7c i\u2016`.\n\nIn particular, every norm `\u2016.\u2016` on `\u2a02[\ud835\udd5c] i, E\u1d62` satisfying `\u2016\u2a02\u209c[\ud835\udd5c] i, m i\u2016 \u2264 \u03a0 i, \u2016m i\u2016`\nfor every `m` in `\u03a0 i, E\u1d62` is bounded above by the projective seminorm.\n\n## Main definitions\n\n* `PiTensorProduct.projectiveSeminorm`: The projective seminorm on `\u2a02[\ud835\udd5c] i, E\u1d62`.\n\n## Main results\n\n* `PiTensorProduct.norm_eval_le_projectiveSeminorm`: If `f` is a continuous multilinear map on\n`E = \u03a0 i, E\u1d62` and `x` is in `\u2a02[\ud835\udd5c] i, E\u1d62`, then `\u2016f.lift x\u2016 \u2264 projectiveSeminorm x * \u2016f\u2016`.\n\n## TODO\n* If the base field is `\u211d` or `\u2102` (or more generally if the injection of `E\u1d62` into its bidual is\nan isometry for every `i`), then we have `projectiveSeminorm \u2a02\u209c[\ud835\udd5c] i, m\u1d62 = \u03a0 i, \u2016m\u1d62\u2016`.\n\n* The functoriality.\n\n-/\n\nuniverse u\u03b9 u\ud835\udd5c uE uF\n\nvariable {\u03b9 : Type u\u03b9} [Fintype \u03b9]\nvariable {\ud835\udd5c : Type u\ud835\udd5c} [NontriviallyNormedField \ud835\udd5c]\nvariable {E : \u03b9 \u2192 Type uE} [\u2200 i, SeminormedAddCommGroup (E i)] [\u2200 i, NormedSpace \ud835\udd5c (E i)]\nvariable {F : Type uF} [SeminormedAddCommGroup F] [NormedSpace \ud835\udd5c F]\n\nopen scoped TensorProduct\n\nopen BigOperators\n\nnamespace PiTensorProduct\n\n/-- A lift of the projective seminorm to `FreeAddMonoid (\ud835\udd5c \u00d7 \u03a0 i, E\u1d62)`, useful to prove the\nproperties of `projectiveSeminorm`.\n-/\ndef projectiveSeminormAux : FreeAddMonoid (\ud835\udd5c \u00d7 \u03a0 i, E i) \u2192 \u211d :=\n  List.sum \u2218 (List.map (fun p \u21a6 \u2016p.1\u2016 * \u220f i, \u2016p.2 i\u2016))\n\ntheorem projectiveSeminormAux_nonneg (p : FreeAddMonoid (\ud835\udd5c \u00d7 \u03a0 i, E i)) :\n    0 \u2264 projectiveSeminormAux p := by\n  simp only [projectiveSeminormAux, Function.comp_apply]\n  refine List.sum_nonneg ?_\n  intro a\n  simp only [Multiset.map_coe, Multiset.mem_coe, List.mem_map, Prod.exists, forall_exists_index,\n    and_imp]\n  intro x m _ h\n  rw [\u2190 h]\n  exact mul_nonneg (norm_nonneg _) (Finset.prod_nonneg (fun _ _ \u21a6 norm_nonneg _))\n\ntheorem projectiveSeminormAux_add_le (p q : FreeAddMonoid (\ud835\udd5c \u00d7 \u03a0 i, E i)) :\n    projectiveSeminormAux (p + q) \u2264 projectiveSeminormAux p + projectiveSeminormAux q := by\n  simp only [projectiveSeminormAux, Function.comp_apply, Multiset.map_coe, Multiset.sum_coe]\n  erw [List.map_append]\n  rw [List.sum_append]\n  rfl\n\n", "theoremStatement": "theorem projectiveSeminormAux_smul (p : FreeAddMonoid (\ud835\udd5c \u00d7 \u03a0 i, E i)) (a : \ud835\udd5c) :\n    projectiveSeminormAux (List.map (fun (y : \ud835\udd5c \u00d7 \u03a0 i, E i) \u21a6 (a * y.1, y.2)) p) =\n    \u2016a\u2016 * projectiveSeminormAux p ", "theoremName": "PiTensorProduct.projectiveSeminormAux_smul", "fileCreated": {"commit": "0c297cbed86c7358191878ee7abd82465afabc96", "date": "2024-03-24"}, "theoremCreated": {"commit": "0c297cbed86c7358191878ee7abd82465afabc96", "date": "2024-03-24"}, "file": "mathlib/Mathlib/Analysis/NormedSpace/PiTensorProduct/ProjectiveSeminorm.lean", "module": "Mathlib.Analysis.NormedSpace.PiTensorProduct.ProjectiveSeminorm", "jsonFile": "Mathlib.Analysis.NormedSpace.PiTensorProduct.ProjectiveSeminorm.jsonl", "positionMetadata": {"lineInFile": 75, "tokenPositionInFile": 2727, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 79, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", 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"Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Extr", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Data.Set.UnionLift", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.TensorProduct", "Mathlib.LinearAlgebra.PiTensorProduct"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  simp only [projectiveSeminormAux, Function.comp_apply, Multiset.map_coe, List.map_map,\n    Multiset.sum_coe]\n  rw [\u2190 smul_eq_mul, List.smul_sum, \u2190 List.comp_map]\n  congr 2\n  ext x\n  simp only [Function.comp_apply, norm_mul, smul_eq_mul]\n  rw [mul_assoc]", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 261}}
+{"srcContext": "/-\nCopyright (c) 2019 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Antoine Chambert-Loir\n-/\nimport Mathlib.Algebra.DirectSum.Finsupp\nimport Mathlib.LinearAlgebra.Finsupp\nimport Mathlib.LinearAlgebra.DirectSum.TensorProduct\n\n#align_import linear_algebra.direct_sum.finsupp from \"leanprover-community/mathlib\"@\"9b9d125b7be0930f564a68f1d73ace10cf46064d\"\n\n/-!\n# Results on finitely supported functions.\n\n* `TensorProduct.finsuppLeft`, the tensor product of `\u03b9 \u2192\u2080 M` and `N`\n  is linearly equivalent to `\u03b9 \u2192\u2080 M \u2297[R] N`\n\n* `TensorProduct.finsuppScalarLeft`, the tensor product of `\u03b9 \u2192\u2080 R` and `N`\n  is linearly equivalent to `\u03b9 \u2192\u2080 N`\n\n* `TensorProduct.finsuppRight`, the tensor product of `M` and `\u03b9 \u2192\u2080 N`\n  is linearly equivalent to `\u03b9 \u2192\u2080 M \u2297[R] N`\n\n* `TensorProduct.finsuppScalarRight`, the tensor product of `M` and `\u03b9 \u2192\u2080 R`\n  is linearly equivalent to `\u03b9 \u2192\u2080 N`\n\n* `TensorProduct.finsuppLeft'`, if `M` is an `S`-module,\n  then the tensor product of `\u03b9 \u2192\u2080 M` and `N` is `S`-linearly equivalent\n  to `\u03b9 \u2192\u2080 M \u2297[R] N`\n\n* `finsuppTensorFinsupp`, the tensor product of `\u03b9 \u2192\u2080 M` and `\u03ba \u2192\u2080 N`\n  is linearly equivalent to `(\u03b9 \u00d7 \u03ba) \u2192\u2080 (M \u2297 N)`.\n\n## Case of MvPolynomial\n\nThese functions apply to `MvPolynomial`, one can define\n```\nnoncomputable def MvPolynomial.rTensor' :\n    MvPolynomial \u03c3 S \u2297[R] N \u2243\u2097[S] (\u03c3 \u2192\u2080 \u2115) \u2192\u2080 (S \u2297[R] N) :=\n  TensorProduct.finsuppLeft'\n\nnoncomputable def MvPolynomial.rTensor :\n    MvPolynomial \u03c3 R \u2297[R] N \u2243\u2097[R] (\u03c3 \u2192\u2080 \u2115) \u2192\u2080 N :=\n  TensorProduct.finsuppScalarLeft\n ```\n\nHowever, to be actually usable, these definitions need lemmas to be given in companion PR.\n\n## Case of `Polynomial`\n\n`Polynomial` is a structure containing a `Finsupp`, so these functions\ncan't be applied directly to `Polynomial`.\n\nSome linear equivs need to be added to mathlib for that.\nThis belongs to a companion PR.\n\n## TODO\n\n* generalize to `MonoidAlgebra`, `AlgHom `\n\n* reprove `TensorProduct.finsuppLeft'` using existing heterobasic version of `TensorProduct.congr`\n-/\n\n\nnoncomputable section\n\nopen DirectSum TensorProduct\n\nopen Set LinearMap Submodule\n\nsection TensorProduct\n\nvariable (R : Type*) [CommSemiring R]\n  (M : Type*) [AddCommMonoid M] [Module R M]\n  (N : Type*) [AddCommMonoid N] [Module R N]\n\nnamespace TensorProduct\n\nvariable (\u03b9 : Type*) [DecidableEq \u03b9]\n\n/-- The tensor product of `\u03b9 \u2192\u2080 M` and `N` is linearly equivalent to `\u03b9 \u2192\u2080 M \u2297[R] N` -/\nnoncomputable def finsuppLeft :\n    (\u03b9 \u2192\u2080 M) \u2297[R] N \u2243\u2097[R] \u03b9 \u2192\u2080 M \u2297[R] N :=\n  congr (finsuppLEquivDirectSum R M \u03b9) (.refl R N) \u226a\u226b\u2097\n    directSumLeft R (fun _ \u21a6 M) N \u226a\u226b\u2097 (finsuppLEquivDirectSum R _ \u03b9).symm\n\nvariable {R M N \u03b9}\n\nlemma finsuppLeft_apply_tmul (p : \u03b9 \u2192\u2080 M) (n : N) :\n    finsuppLeft R M N \u03b9 (p \u2297\u209c[R] n) = p.sum fun i m \u21a6 Finsupp.single i (m \u2297\u209c[R] n) := by\n  apply p.induction_linear\n  \u00b7 simp\n  \u00b7 intros f g hf hg; simp [add_tmul, map_add, hf, hg, Finsupp.sum_add_index]\n  \u00b7 simp [finsuppLeft]\n\n@[simp]\nlemma finsuppLeft_apply_tmul_apply (p : \u03b9 \u2192\u2080 M) (n : N) (i : \u03b9) :\n    finsuppLeft R M N \u03b9 (p \u2297\u209c[R] n) i = p i \u2297\u209c[R] n := by\n  rw [finsuppLeft_apply_tmul, Finsupp.sum_apply,\n    Finsupp.sum_eq_single i (fun _ _ \u21a6 Finsupp.single_eq_of_ne) (by simp), Finsupp.single_eq_same]\n\ntheorem finsuppLeft_apply (t : (\u03b9 \u2192\u2080 M) \u2297[R] N) (i : \u03b9) :\n    finsuppLeft R M N \u03b9 t i = rTensor N (Finsupp.lapply i) t := by\n  induction t using TensorProduct.induction_on with\n  | zero => simp\n  | tmul f n => simp only [finsuppLeft_apply_tmul_apply, rTensor_tmul, Finsupp.lapply_apply]\n  | add x y hx hy => simp [map_add, hx, hy]\n\n@[simp]\nlemma finsuppLeft_symm_apply_single (i : \u03b9) (m : M) (n : N) :\n    (finsuppLeft R M N \u03b9).symm (Finsupp.single i (m \u2297\u209c[R] n)) =\n      Finsupp.single i m \u2297\u209c[R] n := by\n  simp [finsuppLeft, Finsupp.lsum]\n\nvariable (R M N \u03b9)\n/-- The tensor product of `M` and `\u03b9 \u2192\u2080 N` is linearly equivalent to `\u03b9 \u2192\u2080 M \u2297[R] N` -/\nnoncomputable def finsuppRight :\n    M \u2297[R] (\u03b9 \u2192\u2080 N) \u2243\u2097[R] \u03b9 \u2192\u2080 M \u2297[R] N :=\n  congr (.refl R M) (finsuppLEquivDirectSum R N \u03b9) \u226a\u226b\u2097\n    directSumRight R M (fun _ : \u03b9 \u21a6 N) \u226a\u226b\u2097 (finsuppLEquivDirectSum R _ \u03b9).symm\n\nvariable {R M N \u03b9}\n\nlemma finsuppRight_apply_tmul (m : M) (p : \u03b9 \u2192\u2080 N) :\n    finsuppRight R M N \u03b9 (m \u2297\u209c[R] p) = p.sum fun i n \u21a6 Finsupp.single i (m \u2297\u209c[R] n) := by\n  apply p.induction_linear\n  \u00b7 simp\n  \u00b7 intros f g hf hg; simp [tmul_add, map_add, hf, hg, Finsupp.sum_add_index]\n  \u00b7 simp [finsuppRight]\n\n@[simp]\nlemma finsuppRight_apply_tmul_apply (m : M) (p : \u03b9 \u2192\u2080 N) (i : \u03b9) :\n    finsuppRight R M N \u03b9 (m \u2297\u209c[R] p) i = m \u2297\u209c[R] p i := by\n  rw [finsuppRight_apply_tmul, Finsupp.sum_apply,\n    Finsupp.sum_eq_single i (fun _ _ \u21a6 Finsupp.single_eq_of_ne) (by simp), Finsupp.single_eq_same]\n\ntheorem finsuppRight_apply (t : M \u2297[R] (\u03b9 \u2192\u2080 N)) (i : \u03b9) :\n    finsuppRight R M N \u03b9 t i = lTensor M (Finsupp.lapply i) t := by\n  induction t using TensorProduct.induction_on with\n  | zero => simp\n  | tmul m f => simp [finsuppRight_apply_tmul_apply]\n  | add x y hx hy => simp [map_add, hx, hy]\n\n@[simp]\nlemma finsuppRight_symm_apply_single (i : \u03b9) (m : M) (n : N) :\n    (finsuppRight R M N \u03b9).symm (Finsupp.single i (m \u2297\u209c[R] n)) =\n      m \u2297\u209c[R] Finsupp.single i n := by\n  simp [finsuppRight, Finsupp.lsum]\n\nvariable {S : Type*} [CommSemiring S] [Algebra R S]\n  [Module S M] [IsScalarTower R S M]\n\nlemma finsuppLeft_smul' (s : S) (t : (\u03b9 \u2192\u2080 M) \u2297[R] N) :\n    finsuppLeft R M N \u03b9 (s \u2022 t) = s \u2022 finsuppLeft R M N \u03b9 t := by\n  induction t using TensorProduct.induction_on with\n  | zero => simp\n  | add x y hx hy => simp [hx, hy]\n  | tmul p n => ext; simp [smul_tmul', finsuppLeft_apply_tmul_apply]\n\nvariable (R M N \u03b9 S)\n/-- When `M` is also an `S`-module, then `TensorProduct.finsuppLeft R M N``\n  is an `S`-linear equiv -/\nnoncomputable def finsuppLeft' :\n    (\u03b9 \u2192\u2080 M) \u2297[R] N \u2243\u2097[S] \u03b9 \u2192\u2080 M \u2297[R] N where\n  __ := finsuppLeft R M N \u03b9\n  map_smul' := finsuppLeft_smul'\n\nvariable {R M N \u03b9 S}\nlemma finsuppLeft'_apply (x : (\u03b9 \u2192\u2080 M) \u2297[R] N) :\n    finsuppLeft' R M N \u03b9 S x = finsuppLeft R M N \u03b9 x := rfl\n\n/- -- TODO : reprove using the existing heterobasic lemmas\nnoncomputable example :\n    (\u03b9 \u2192\u2080 M) \u2297[R] N \u2243\u2097[S] \u03b9 \u2192\u2080 (M \u2297[R] N) := by\n  have f : (\u2a01 (i\u2081 : \u03b9), M) \u2297[R] N \u2243\u2097[S] \u2a01 (i : \u03b9), M \u2297[R] N := sorry\n  exact (AlgebraTensorModule.congr\n    (finsuppLEquivDirectSum S M \u03b9) (.refl R N)).trans\n    (f.trans (finsuppLEquivDirectSum S (M \u2297[R] N) \u03b9).symm) -/\n\nvariable (R M N \u03b9)\n/-- The tensor product of `\u03b9 \u2192\u2080 R` and `N` is linearly equivalent to `\u03b9 \u2192\u2080 N` -/\nnoncomputable def finsuppScalarLeft :\n    (\u03b9 \u2192\u2080 R) \u2297[R] N \u2243\u2097[R] \u03b9 \u2192\u2080 N :=\n  finsuppLeft R R N \u03b9 \u226a\u226b\u2097 (Finsupp.mapRange.linearEquiv (TensorProduct.lid R N))\n\nvariable {R M N \u03b9}\n@[simp]\nlemma finsuppScalarLeft_apply_tmul_apply (p : \u03b9 \u2192\u2080 R) (n : N) (i : \u03b9) :\n    finsuppScalarLeft R N \u03b9 (p \u2297\u209c[R] n) i = p i \u2022 n := by\n  simp [finsuppScalarLeft]\n\nlemma finsuppScalarLeft_apply_tmul (p : \u03b9 \u2192\u2080 R) (n : N) :\n    finsuppScalarLeft R N \u03b9 (p \u2297\u209c[R] n) = p.sum fun i m \u21a6 Finsupp.single i (m \u2022 n) := by\n  ext i\n  rw [finsuppScalarLeft_apply_tmul_apply, Finsupp.sum_apply,\n    Finsupp.sum_eq_single i (fun _ _ \u21a6 Finsupp.single_eq_of_ne) (by simp), Finsupp.single_eq_same]\n\nlemma finsuppScalarLeft_apply (pn : (\u03b9 \u2192\u2080 R) \u2297[R] N) (i : \u03b9) :\n    finsuppScalarLeft R N \u03b9 pn i = TensorProduct.lid R N ((Finsupp.lapply i).rTensor N pn) := by\n  simp [finsuppScalarLeft, finsuppLeft_apply]\n\n@[simp]\nlemma finsuppScalarLeft_symm_apply_single (i : \u03b9) (n : N) :\n    (finsuppScalarLeft R N \u03b9).symm (Finsupp.single i n) =\n      (Finsupp.single i 1) \u2297\u209c[R] n := by\n  simp [finsuppScalarLeft, finsuppLeft_symm_apply_single]\n\nvariable (R M N \u03b9)\n\n/-- The tensor product of `M` and `\u03b9 \u2192\u2080 R` is linearly equivalent to `\u03b9 \u2192\u2080 N` -/\nnoncomputable def finsuppScalarRight :\n    M \u2297[R] (\u03b9 \u2192\u2080 R) \u2243\u2097[R] \u03b9 \u2192\u2080 M :=\n  finsuppRight R M R \u03b9 \u226a\u226b\u2097 Finsupp.mapRange.linearEquiv (TensorProduct.rid R M)\n\nvariable {R M N \u03b9}\n\n@[simp]\nlemma finsuppScalarRight_apply_tmul_apply (m : M) (p : \u03b9 \u2192\u2080 R) (i : \u03b9) :\n    finsuppScalarRight R M \u03b9 (m \u2297\u209c[R] p) i = p i \u2022 m := by\n  simp [finsuppScalarRight]\n\nlemma finsuppScalarRight_apply_tmul (m : M) (p : \u03b9 \u2192\u2080 R) :\n    finsuppScalarRight R M \u03b9 (m \u2297\u209c[R] p) = p.sum fun i n \u21a6 Finsupp.single i (n \u2022 m) := by\n  ext i\n  rw [finsuppScalarRight_apply_tmul_apply, Finsupp.sum_apply,\n    Finsupp.sum_eq_single i (fun _ _ \u21a6 Finsupp.single_eq_of_ne) (by simp), Finsupp.single_eq_same]\n\nlemma finsuppScalarRight_apply (t : M \u2297[R] (\u03b9 \u2192\u2080 R)) (i : \u03b9) :\n    finsuppScalarRight R M \u03b9 t i = TensorProduct.rid R M ((Finsupp.lapply i).lTensor M t) := by\n  simp [finsuppScalarRight, finsuppRight_apply]\n\n@[simp]\nlemma finsuppScalarRight_symm_apply_single (i : \u03b9) (m : M) :\n    (finsuppScalarRight R M \u03b9).symm (Finsupp.single i m) =\n      m \u2297\u209c[R] (Finsupp.single i 1) := by\n  simp [finsuppScalarRight, finsuppRight_symm_apply_single]\n\nend TensorProduct\n\nend TensorProduct\n\nvariable (R M N \u03b9 \u03ba : Type*)\n  [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N]\n\nopen scoped Classical in\n/-- The tensor product of `\u03b9 \u2192\u2080 M` and `\u03ba \u2192\u2080 N` is linearly equivalent to `(\u03b9 \u00d7 \u03ba) \u2192\u2080 (M \u2297 N)`. -/\nnoncomputable def finsuppTensorFinsupp :\n    (\u03b9 \u2192\u2080 M) \u2297[R] (\u03ba \u2192\u2080 N) \u2243\u2097[R] \u03b9 \u00d7 \u03ba \u2192\u2080 M \u2297[R] N :=\n  TensorProduct.congr (finsuppLEquivDirectSum R M \u03b9) (finsuppLEquivDirectSum R N \u03ba) \u226a\u226b\u2097\n    (TensorProduct.directSum R (fun _ \u21a6 M) fun _ \u21a6 N) \u226a\u226b\u2097 (finsuppLEquivDirectSum R _ _).symm\n#align finsupp_tensor_finsupp finsuppTensorFinsupp\n\n@[simp]\ntheorem finsuppTensorFinsupp_single (i : \u03b9) (m : M) (k : \u03ba) (n : N) :\n    finsuppTensorFinsupp R M N \u03b9 \u03ba (Finsupp.single i m \u2297\u209c Finsupp.single k n) =\n      Finsupp.single (i, k) (m \u2297\u209c n) := by\n  simp [finsuppTensorFinsupp]\n#align finsupp_tensor_finsupp_single finsuppTensorFinsupp_single\n\n@[simp]\ntheorem finsuppTensorFinsupp_apply (f : \u03b9 \u2192\u2080 M) (g : \u03ba \u2192\u2080 N) (i : \u03b9) (k : \u03ba) :\n    finsuppTensorFinsupp R M N \u03b9 \u03ba (f \u2297\u209c g) (i, k) = f i \u2297\u209c g k := by\n  apply Finsupp.induction_linear f\n  \u00b7 simp\n  \u00b7 intro f\u2081 f\u2082 hf\u2081 hf\u2082\n    simp [add_tmul, hf\u2081, hf\u2082]\n  intro i' m\n  apply Finsupp.induction_linear g\n  \u00b7 simp\n  \u00b7 intro g\u2081 g\u2082 hg\u2081 hg\u2082\n    simp [tmul_add, hg\u2081, hg\u2082]\n  intro k' n\n  classical\n  simp_rw [finsuppTensorFinsupp_single, Finsupp.single_apply, Prod.mk.inj_iff, ite_and]\n  split_ifs <;> simp\n#align finsupp_tensor_finsupp_apply finsuppTensorFinsupp_apply\n\n@[simp]\ntheorem finsuppTensorFinsupp_symm_single (i : \u03b9 \u00d7 \u03ba) (m : M) (n : N) :\n    (finsuppTensorFinsupp R M N \u03b9 \u03ba).symm (Finsupp.single i (m \u2297\u209c n)) =\n      Finsupp.single i.1 m \u2297\u209c Finsupp.single i.2 n :=\n  Prod.casesOn i fun _ _ =>\n    (LinearEquiv.symm_apply_eq _).2 (finsuppTensorFinsupp_single _ _ _ _ _ _ _ _ _).symm\n#align finsupp_tensor_finsupp_symm_single finsuppTensorFinsupp_symm_single\n\n/-- A variant of `finsuppTensorFinsupp` where the first module is the ground ring. -/\ndef finsuppTensorFinsuppLid : (\u03b9 \u2192\u2080 R) \u2297[R] (\u03ba \u2192\u2080 N) \u2243\u2097[R] \u03b9 \u00d7 \u03ba \u2192\u2080 N :=\n  finsuppTensorFinsupp R R N \u03b9 \u03ba \u226a\u226b\u2097 Finsupp.lcongr (Equiv.refl _) (TensorProduct.lid R N)\n\n@[simp]\ntheorem finsuppTensorFinsuppLid_apply_apply (f : \u03b9 \u2192\u2080 R) (g : \u03ba \u2192\u2080 N) (a : \u03b9) (b : \u03ba) :\n    finsuppTensorFinsuppLid R N \u03b9 \u03ba (f \u2297\u209c[R] g) (a, b) = f a \u2022 g b := by\n  simp [finsuppTensorFinsuppLid]\n\n@[simp]\ntheorem finsuppTensorFinsuppLid_single_tmul_single (a : \u03b9) (b : \u03ba) (r : R) (n : N) :\n    finsuppTensorFinsuppLid R N \u03b9 \u03ba (Finsupp.single a r \u2297\u209c[R] Finsupp.single b n) =\n      Finsupp.single (a, b) (r \u2022 n) := by\n  simp [finsuppTensorFinsuppLid]\n\n@[simp]\ntheorem finsuppTensorFinsuppLid_symm_single_smul (i : \u03b9 \u00d7 \u03ba) (r : R) (n : N) :\n    (finsuppTensorFinsuppLid R N \u03b9 \u03ba).symm (Finsupp.single i (r \u2022 n)) =\n      Finsupp.single i.1 r \u2297\u209c Finsupp.single i.2 n :=\n  Prod.casesOn i fun _ _ =>\n    (LinearEquiv.symm_apply_eq _).2 (finsuppTensorFinsuppLid_single_tmul_single ..).symm\n\n/-- A variant of `finsuppTensorFinsupp` where the second module is the ground ring. -/\ndef finsuppTensorFinsuppRid : (\u03b9 \u2192\u2080 M) \u2297[R] (\u03ba \u2192\u2080 R) \u2243\u2097[R] \u03b9 \u00d7 \u03ba \u2192\u2080 M :=\n  finsuppTensorFinsupp R M R \u03b9 \u03ba \u226a\u226b\u2097 Finsupp.lcongr (Equiv.refl _) (TensorProduct.rid R M)\n\n", "theoremStatement": "@[simp]\ntheorem finsuppTensorFinsuppRid_apply_apply (f : \u03b9 \u2192\u2080 M) (g : \u03ba \u2192\u2080 R) (a : \u03b9) (b : \u03ba) :\n    finsuppTensorFinsuppRid R M \u03b9 \u03ba (f \u2297\u209c[R] g) (a, b) = g b \u2022 f a ", "theoremName": 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"Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", 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"Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.LinearAlgebra.DirectSum.TensorProduct"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  simp [finsuppTensorFinsuppRid]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 38}}
+{"srcContext": "/-\nCopyright (c) 2024 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Stoll\n-/\nimport Mathlib.NumberTheory.DirichletCharacter.Bounds\nimport Mathlib.NumberTheory.LSeries.Convolution\nimport Mathlib.NumberTheory.LSeries.Deriv\nimport Mathlib.NumberTheory.SumPrimeReciprocals\nimport Mathlib.NumberTheory.VonMangoldt\nimport Mathlib.NumberTheory.ZetaFunction\n\n/-!\n# L-series of Dirichlet characters and arithmetic functions\n\nWe collect some results on L-series of specific (arithmetic) functions, for example,\nthe M\u00f6bius function `\u03bc` or the von Mangoldt function `\u039b`. In particular, we show that\n`L \u2197\u039b` is the negative of the logarithmic derivative of the Riemann zeta function\non `re s > 1`; see `LSeries_vonMangoldt_eq_deriv_riemannZeta_div`.\n\nWe also prove some general results on L-series associated to Dirichlet characters\n(i.e., Dirichlet L-series). For example, we show that the abscissa of absolute convergence\nequals `1` (see `DirichletCharacter.absicssaOfAbsConv`) and that the L-series does not\nvanish on the open half-plane `re s > 1` (see `DirichletCharacter.LSeries_ne_zero_of_one_lt_re`).\n\nWe deduce results on the Riemann zeta function (which is `L 1` or `L \u2197\u03b6` on `re s > 1`)\nas special cases.\n\n## Tags\n\nDirichlet L-series, M\u00f6bius function, von Mangoldt function, Riemann zeta function\n-/\n\nopen scoped LSeries.notation\n\n/-- `\u03b4` is the function underlying the arithmetic function `1`. -/\nlemma ArithmeticFunction.one_eq_delta : \u2197(1 : ArithmeticFunction \u2102) = \u03b4 := by\n  ext\n  simp only [one_apply, LSeries.delta]\n\n\nsection Moebius\n\n/-!\n### The L-series of the M\u00f6bius function\n\nWe show that `L \u03bc s` converges absolutely if and only if `re s > 1`.\n-/\n\nnamespace ArithmeticFunction\n\nopen LSeries Nat Complex\n\nlemma not_LSeriesSummable_moebius_at_one : \u00ac LSeriesSummable \u2197\u03bc 1 := by\n  intro h\n  refine not_summable_one_div_on_primes <| summable_ofReal.mp <| Summable.of_neg ?_\n  simp only [\u2190 Pi.neg_def, Set.indicator_comp_of_zero ofReal_zero, ofReal_inv, ofReal_natCast]\n  refine (h.indicator {n | n.Prime}).congr (fun n \u21a6 ?_)\n  by_cases hn : n \u2208 {p | p.Prime}\n  \u00b7 simp only [Pi.neg_apply, Set.indicator_of_mem hn, term_of_ne_zero hn.ne_zero,\n      moebius_apply_prime hn, cpow_one, push_cast, neg_div]\n  \u00b7 simp only [one_div, Pi.neg_apply, Set.indicator_of_not_mem hn, ofReal_zero, neg_zero]\n\n/-- The L-series of the M\u00f6bius function converges absolutely at `s` if and only if `re s > 1`. -/\nlemma LSeriesSummable_moebius_iff {s : \u2102} : LSeriesSummable \u2197\u03bc s \u2194 1 < s.re := by\n  refine \u27e8fun H \u21a6 ?_, LSeriesSummable_of_bounded_of_one_lt_re (m := 1) fun n _ \u21a6 ?_\u27e9\n  \u00b7 by_contra! h\n    have h' : s.re \u2264 (1 : \u2102).re := by simp only [one_re, h]\n    exact not_LSeriesSummable_moebius_at_one <| LSeriesSummable.of_re_le_re h' H\n  \u00b7 rw [abs_intCast] -- not done by `norm_cast`\n    norm_cast\n    exact abs_moebius_le_one\n\n/-- The abscissa of absolute convergence of the L-series of the M\u00f6bius function is `1`. -/\nlemma abscissaOfAbsConv_moebius : abscissaOfAbsConv \u2197\u03bc = 1 := by\n  simpa only [abscissaOfAbsConv, LSeriesSummable_moebius_iff, ofReal_re, Set.Ioi_def,\n    EReal.image_coe_Ioi, EReal.coe_one] using csInf_Ioo <| EReal.coe_lt_top _\n\nend ArithmeticFunction\n\nend Moebius\n\n\n/-!\n### L-series of Dirichlet characters\n-/\n\nopen Nat\n\nopen scoped ArithmeticFunction.zeta in\nlemma ArithmeticFunction.const_one_eq_zeta {R : Type*} [Semiring R] {n : \u2115} (hn : n \u2260 0) :\n    (1 : \u2115 \u2192 R) n = (\u03b6 \u00b7) n := by\n  simp only [Pi.one_apply, zeta_apply, hn, \u2193reduceIte, cast_one]\n\nlemma LSeries.one_convolution_eq_zeta_convolution {R : Type*} [Semiring R] (f : \u2115 \u2192 R):\n    (1 : \u2115 \u2192 R) \u235f f = ((ArithmeticFunction.zeta \u00b7) : \u2115 \u2192 R) \u235f f :=\n  convolution_congr ArithmeticFunction.const_one_eq_zeta fun _ \u21a6 rfl\n\nlemma LSeries.convolution_one_eq_convolution_zeta {R : Type*} [Semiring R] (f : \u2115 \u2192 R):\n    f \u235f (1 : \u2115 \u2192 R) = f \u235f ((ArithmeticFunction.zeta \u00b7) : \u2115 \u2192 R) :=\n  convolution_congr (fun _ \u21a6 rfl) ArithmeticFunction.const_one_eq_zeta\n\n/-- `\u03c7\u2081` is (local) notation for the (necessarily trivial) Dirichlet character modulo `1`. -/\nlocal notation (name := Dchar_one) \"\u03c7\u2081\" => (1 : DirichletCharacter \u2102 1)\n\nnamespace DirichletCharacter\n\nopen LSeries Nat Complex\n\n/-- Twisting by a Dirichlet character `\u03c7` distributes over convolution. -/\nlemma mul_convolution_distrib {R : Type*} [CommSemiring R] {n : \u2115} (\u03c7 : DirichletCharacter R n)\n    (f g : \u2115 \u2192 R) :\n    (((\u03c7 \u00b7) : \u2115 \u2192 R) * f) \u235f (((\u03c7 \u00b7) : \u2115 \u2192 R) * g) = ((\u03c7 \u00b7) : \u2115 \u2192 R) * (f \u235f g) := by\n  ext n\n  simp only [Pi.mul_apply, LSeries.convolution_def, Finset.mul_sum]\n  refine Finset.sum_congr rfl fun p hp \u21a6 ?_\n  rw [(mem_divisorsAntidiagonal.mp hp).1.symm, cast_mul, map_mul]\n  exact mul_mul_mul_comm ..\n\nlemma mul_delta {n : \u2115} (\u03c7 : DirichletCharacter \u2102 n) : \u2197\u03c7 * \u03b4 = \u03b4 :=\n  LSeries.mul_delta <| by rw [cast_one, map_one]\n\nlemma delta_mul {n : \u2115} (\u03c7 : DirichletCharacter \u2102 n) : \u03b4 * \u2197\u03c7 = \u03b4 :=\n  mul_comm \u03b4 _ \u25b8 mul_delta ..\n\nopen ArithmeticFunction in\n/-- The convolution of a Dirichlet character `\u03c7` with the twist `\u03c7 * \u03bc` is `\u03b4`,\nthe indicator function of `{1}`. -/\nlemma convolution_mul_moebius {n : \u2115} (\u03c7 : DirichletCharacter \u2102 n) : \u2197\u03c7 \u235f (\u2197\u03c7 * \u2197\u03bc) = \u03b4 := by\n  have : (1 : \u2115 \u2192 \u2102) \u235f (\u03bc \u00b7) = \u03b4 := by\n    rw [one_convolution_eq_zeta_convolution, \u2190 one_eq_delta]\n    simp_rw [\u2190 natCoe_apply, \u2190 intCoe_apply, coe_mul, coe_zeta_mul_coe_moebius]\n  nth_rewrite 1 [\u2190 mul_one \u2197\u03c7]\n  simpa only [mul_convolution_distrib \u03c7 1 \u2197\u03bc, this] using mul_delta _\n\n/-- The Dirichlet character mod `0` corresponds to `\u03b4`. -/\nlemma modZero_eq_delta {\u03c7 : DirichletCharacter \u2102 0} : \u2197\u03c7 = \u03b4 := by\n  ext n\n  rcases eq_or_ne n 0 with rfl | hn\n  \u00b7 simp_rw [cast_zero, \u03c7.map_nonunit not_isUnit_zero, delta, if_false]\n  rcases eq_or_ne n 1 with rfl | hn'\n  \u00b7 simp only [cast_one, map_one, delta, \u2193reduceIte]\n  have : \u00ac IsUnit (n : ZMod 0) := fun h \u21a6 hn' <| ZMod.eq_one_of_isUnit_natCast h\n  simp only [\u03c7.map_nonunit this, delta, hn', \u2193reduceIte]\n\n/-- The Dirichlet character mod `1` corresponds to the constant function `1`. -/\nlemma modOne_eq_one {R : Type*} [CommSemiring R] {\u03c7 : DirichletCharacter R 1} :\n    ((\u03c7 \u00b7) : \u2115 \u2192 R) = 1 := by\n  ext\n  rw [\u03c7.level_one, MulChar.one_apply (isUnit_of_subsingleton _), Pi.one_apply]\n\nlemma LSeries_modOne_eq : L \u2197\u03c7\u2081 = L 1 :=\n  congr_arg L modOne_eq_one\n\n/-- The L-series of a Dirichlet character mod `N > 0` does not converge absolutely at `s = 1`. -/\nlemma not_LSeriesSummable_at_one {N : \u2115} (hN : N \u2260 0) (\u03c7 : DirichletCharacter \u2102 N) :\n    \u00ac LSeriesSummable \u2197\u03c7 1 := by\n  refine fun h \u21a6 (Real.not_summable_indicator_one_div_natCast hN 1) ?_\n  refine h.norm.of_nonneg_of_le (fun m \u21a6 Set.indicator_apply_nonneg (fun _ \u21a6 by positivity))\n    (fun n \u21a6 ?_)\n  rw [norm_term_eq, one_re, Real.rpow_one, Set.indicator]\n  split_ifs with h\u2081 h\u2082\n  \u00b7 rw [h\u2082, cast_zero, div_zero]\n  \u00b7 rw [h\u2081, \u03c7.map_one, norm_one]\n  all_goals positivity\n\n/-- The L-series of a Dirichlet character converges absolutely at `s` if `re s > 1`. -/\nlemma LSeriesSummable_of_one_lt_re {N : \u2115} (\u03c7 : DirichletCharacter \u2102 N) {s : \u2102} (hs : 1 < s.re) :\n    LSeriesSummable \u2197\u03c7 s :=\n  LSeriesSummable_of_bounded_of_one_lt_re (fun _ _ \u21a6 \u03c7.norm_le_one _) hs\n\n/-- The L-series of a Dirichlet character mod `N > 0` converges absolutely at `s` if and only if\n`re s > 1`. -/\nlemma LSeriesSummable_iff {N : \u2115} (hN : N \u2260 0) (\u03c7 : DirichletCharacter \u2102 N) {s : \u2102} :\n    LSeriesSummable \u2197\u03c7 s \u2194 1 < s.re := by\n  refine \u27e8fun H \u21a6 ?_, LSeriesSummable_of_one_lt_re \u03c7\u27e9\n  by_contra! h\n  exact not_LSeriesSummable_at_one hN \u03c7 <| LSeriesSummable.of_re_le_re (by simp only [one_re, h]) H\n\n/-- The abscissa of absolute convergence of the L-series of a Dirichlet character mod `N > 0`\nis `1`. -/\nlemma absicssaOfAbsConv_eq_one {N : \u2115} (hn : N \u2260 0) (\u03c7 : DirichletCharacter \u2102 N) :\n    abscissaOfAbsConv \u2197\u03c7 = 1 := by\n  simpa only [abscissaOfAbsConv, LSeriesSummable_iff hn \u03c7, ofReal_re, Set.Ioi_def,\n    EReal.image_coe_Ioi, EReal.coe_one] using csInf_Ioo <| EReal.coe_lt_top _\n\n", "theoremStatement": "/-- The L-series of the twist of `f` by a Dirichlet character converges at `s` if the L-series\nof `f` does. -/\nlemma LSeriesSummable_mul {N : \u2115} (\u03c7 : DirichletCharacter \u2102 N) {f : \u2115 \u2192 \u2102} {s : \u2102}\n    (h : LSeriesSummable f s) :\n    LSeriesSummable (\u2197\u03c7 * f) s ", "theoremName": "DirichletCharacter.LSeriesSummable_mul", "fileCreated": {"commit": "fdb986a81847a2bbb6586468fc976beccd981f77", "date": "2024-04-03"}, "theoremCreated": {"commit": "a251ad4621fbba21ca0d49adfd48c212dcc8f98d", "date": "2024-03-25"}, "file": "mathlib/Mathlib/NumberTheory/LSeries/Dirichlet.lean", "module": "Mathlib.NumberTheory.LSeries.Dirichlet", "jsonFile": "Mathlib.NumberTheory.LSeries.Dirichlet.jsonl", "positionMetadata": {"lineInFile": 187, "tokenPositionInFile": 7901, 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"Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", 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"Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Tactic.FinCases", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Data.List.Prime", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.RingTheory.IntegralDomain", "Mathlib.Algebra.CharP.Reduced", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.Topology.CountableSeparatingOn", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Tactic.MoveAdd", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.Valuation.PrimeMultiplicity", "Mathlib.LinearAlgebra.SModEq", "Mathlib.LinearAlgebra.AdicCompletion", "Mathlib.RingTheory.DiscreteValuationRing.Basic", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Order", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  refine .of_norm <| h.norm.of_nonneg_of_le (fun _ \u21a6 norm_nonneg _) fun n \u21a6 norm_term_le s ?_\n  rw [Pi.mul_apply, norm_mul]\n  exact mul_le_of_le_one_left (norm_nonneg _) <| norm_le_one ..", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 193}}
+{"srcContext": "/-\nCopyright (c) 2024 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\nimport Mathlib.CategoryTheory.GradedObject.Bifunctor\nimport Mathlib.CategoryTheory.GradedObject.Single\n/-!\n# The left and right unitors\n\nGiven a bifunctor `F : C \u2964 D \u2964 D`, an object `X : C` such that `F.obj X \u2245 \ud835\udfed D` and a\nmap `p : I \u00d7 J \u2192 J` such that `hp : \u2200 (j : J), p \u27e80, j\u27e9 = j`,\nwe define an isomorphism of `J`-graded objects for any `Y : GradedObject J D`.\n`mapBifunctorLeftUnitor F X e p hp Y : mapBifunctorMapObj F p ((single\u2080 I).obj X) Y \u2245 Y`.\nUnder similar assumptions, we also obtain a right unitor isomorphism\n`mapBifunctorMapObj F p X ((single\u2080 I).obj Y) \u2245 X`.\n\nTODO (@joelriou): get the triangle identity.\n\n-/\n\nnamespace CategoryTheory\n\nopen Category Limits\n\nnamespace GradedObject\n\nsection LeftUnitor\n\nvariable {C D I J : Type*} [Category C] [Category D]\n  [Zero I] [DecidableEq I] [HasInitial C]\n  (F : C \u2964 D \u2964 D) (X : C) (e : F.obj X \u2245 \ud835\udfed D)\n  [\u2200 (Y : D), PreservesColimit (Functor.empty.{0} C) (F.flip.obj Y)]\n  (p : I \u00d7 J \u2192 J) (hp : \u2200 (j : J), p \u27e80, j\u27e9 = j)\n  (Y Y' : GradedObject J D) (\u03c6 : Y \u27f6 Y')\n\n/-- Given `F : C \u2964 D \u2964 D`, `X : C`, `e : F.obj X \u2245 \ud835\udfed D` and `Y : GradedObject J D`,\nthis is the isomorphism `((mapBifunctor F I J).obj ((single\u2080 I).obj X)).obj Y a \u2245 Y a.2`\nwhen `a : I \u00d7 J` is such that `a.1 = 0`. -/\n@[simps!]\nnoncomputable def mapBifunctorObjSingle\u2080ObjIso (a : I \u00d7 J) (ha : a.1 = 0) :\n    ((mapBifunctor F I J).obj ((single\u2080 I).obj X)).obj Y a \u2245 Y a.2 :=\n  (F.mapIso (singleObjApplyIsoOfEq _ X _ ha)).app _ \u226a\u226b e.app (Y a.2)\n\n/-- Given `F : C \u2964 D \u2964 D`, `X : C` and `Y : GradedObject J D`,\n`((mapBifunctor F I J).obj ((single\u2080 I).obj X)).obj Y a` is an initial when `a : I \u00d7 J`\nis such that `a.1 \u2260 0`. -/\nnoncomputable def mapBifunctorObjSingle\u2080ObjIsInitial (a : I \u00d7 J) (ha : a.1 \u2260 0) :\n    IsInitial (((mapBifunctor F I J).obj ((single\u2080 I).obj X)).obj Y a) :=\n  IsInitial.isInitialObj (F.flip.obj (Y a.2)) _ (isInitialSingleObjApply _ _ _ ha)\n\n/-- Given `F : C \u2964 D \u2964 D`, `X : C`, `e : F.obj X \u2245 \ud835\udfed D`, `Y : GradedObject J D` and\n`p : I \u00d7 J \u2192 J` such that `p \u27e80, j\u27e9 = j` for all `j`,\nthis is the (colimit) cofan which shall be used to construct the isomorphism\n`mapBifunctorMapObj F p ((single\u2080 I).obj X) Y \u2245 Y`, see `mapBifunctorLeftUnitor`. -/\nnoncomputable def mapBifunctorLeftUnitorCofan (j : J) :\n    (((mapBifunctor F I J).obj ((single\u2080 I).obj X)).obj Y).CofanMapObjFun p j :=\n  CofanMapObjFun.mk _ _ _ (Y j) (fun a ha =>\n    if ha : a.1 = 0 then\n      (mapBifunctorObjSingle\u2080ObjIso F X e Y a ha).hom \u226b eqToHom (by aesop)\n    else\n      (mapBifunctorObjSingle\u2080ObjIsInitial F X Y a ha).to _)\n\n@[simp, reassoc]\nlemma mapBifunctorLeftUnitorCofan_inj (j : J) :\n    (mapBifunctorLeftUnitorCofan F X e p hp Y j).inj \u27e8\u27e80, j\u27e9, hp j\u27e9 =\n      (F.map (singleObjApplyIso (0 : I) X).hom).app (Y j) \u226b e.hom.app (Y j) := by\n  simp [mapBifunctorLeftUnitorCofan]\n\n/-- The cofan `mapBifunctorLeftUnitorCofan F X e p hp Y j` is a colimit. -/\nnoncomputable def mapBifunctorLeftUnitorCofanIsColimit (j : J) :\n    IsColimit (mapBifunctorLeftUnitorCofan F X e p hp Y j) :=\n  mkCofanColimit _\n    (fun s => e.inv.app (Y j) \u226b\n      (F.map (singleObjApplyIso (0 : I) X).inv).app (Y j) \u226b s.inj \u27e8\u27e80, j\u27e9, hp j\u27e9)\n    (fun s => by\n      rintro \u27e8\u27e8i, j'\u27e9, h\u27e9\n      by_cases hi : i = 0\n      \u00b7 subst hi\n        simp only [Set.mem_preimage, hp, Set.mem_singleton_iff] at h\n        subst h\n        simp\n      \u00b7 apply IsInitial.hom_ext\n        exact mapBifunctorObjSingle\u2080ObjIsInitial _ _ _ _ hi)\n    (fun s m hm => by simp [\u2190 hm \u27e8\u27e80, j\u27e9, hp j\u27e9])\n\nlemma mapBifunctorLeftUnitor_hasMap :\n    HasMap (((mapBifunctor F I J).obj ((single\u2080 I).obj X)).obj Y) p :=\n  CofanMapObjFun.hasMap _ _ _ (mapBifunctorLeftUnitorCofanIsColimit F X e p hp Y)\n\nvariable [HasMap (((mapBifunctor F I J).obj ((single\u2080 I).obj X)).obj Y) p]\n  [HasMap (((mapBifunctor F I J).obj ((single\u2080 I).obj X)).obj Y') p]\n\n/-- Given `F : C \u2964 D \u2964 D`, `X : C`, `e : F.obj X \u2245 \ud835\udfed D`, `Y : GradedObject J D` and\n`p : I \u00d7 J \u2192 J` such that `p \u27e80, j\u27e9 = j` for all `j`,\nthis is the left unitor isomorphism `mapBifunctorMapObj F p ((single\u2080 I).obj X) Y \u2245 Y`. -/\nnoncomputable def mapBifunctorLeftUnitor : mapBifunctorMapObj F p ((single\u2080 I).obj X) Y \u2245 Y :=\n  isoMk _ _ (fun j => (CofanMapObjFun.iso\n    (mapBifunctorLeftUnitorCofanIsColimit F X e p hp Y j)).symm)\n\n@[reassoc (attr := simp)]\nlemma \u03b9_mapBifunctorLeftUnitor_hom_apply (j : J) :\n    \u03b9MapBifunctorMapObj F p ((single\u2080 I).obj X) Y 0 j j (hp j) \u226b\n      (mapBifunctorLeftUnitor F X e p hp Y).hom j =\n      (F.map (singleObjApplyIso (0 : I) X).hom).app _ \u226b e.hom.app (Y j) := by\n  dsimp [mapBifunctorLeftUnitor]\n  erw [CofanMapObjFun.\u03b9MapObj_iso_inv]\n  rw [mapBifunctorLeftUnitorCofan_inj]\n\nlemma mapBifunctorLeftUnitor_inv_apply (j : J) :\n    (mapBifunctorLeftUnitor F X e p hp Y).inv j =\n      e.inv.app (Y j) \u226b (F.map (singleObjApplyIso (0 : I) X).inv).app (Y j) \u226b\n      \u03b9MapBifunctorMapObj F p ((single\u2080 I).obj X) Y 0 j j (hp j) := rfl\n\nvariable {Y Y'}\n\n@[reassoc]\nlemma mapBifunctorLeftUnitor_inv_naturality :\n    \u03c6 \u226b (mapBifunctorLeftUnitor F X e p hp Y').inv =\n      (mapBifunctorLeftUnitor F X e p hp Y).inv \u226b mapBifunctorMapMap F p (\ud835\udfd9 _) \u03c6 := by\n  ext j\n  dsimp\n  rw [mapBifunctorLeftUnitor_inv_apply, mapBifunctorLeftUnitor_inv_apply, assoc, assoc,\n    \u03b9_mapBifunctorMapMap]\n  dsimp\n  rw [Functor.map_id, NatTrans.id_app, id_comp]\n  erw [\u2190 NatTrans.naturality_assoc, \u2190 NatTrans.naturality_assoc]\n  rfl\n\n@[reassoc]\nlemma mapBifunctorLeftUnitor_naturality :\n    mapBifunctorMapMap F p (\ud835\udfd9 _) \u03c6 \u226b (mapBifunctorLeftUnitor F X e p hp Y').hom =\n      (mapBifunctorLeftUnitor F X e p hp Y).hom \u226b \u03c6 := by\n  rw [\u2190 cancel_mono (mapBifunctorLeftUnitor F X e p hp Y').inv, assoc, assoc, Iso.hom_inv_id,\n    comp_id, mapBifunctorLeftUnitor_inv_naturality, Iso.hom_inv_id_assoc]\n\nend LeftUnitor\n\nsection RightUnitor\n\nvariable {C D I J : Type*} [Category C] [Category D]\n  [Zero I] [DecidableEq I] [HasInitial C]\n  (F : D \u2964 C \u2964 D) (Y : C) (e : F.flip.obj Y \u2245 \ud835\udfed D)\n  [\u2200 (X : D), PreservesColimit (Functor.empty.{0} C) (F.obj X)]\n  (p : J \u00d7 I \u2192 J)\n  (hp : \u2200 (j : J), p \u27e8j, 0\u27e9 = j) (X X' : GradedObject J D) (\u03c6 : X \u27f6 X')\n\n/-- Given `F : D \u2964 C \u2964 D`, `Y : C`, `e : F.flip.obj X \u2245 \ud835\udfed D` and `X : GradedObject J D`,\nthis is the isomorphism `((mapBifunctor F J I).obj X).obj ((single\u2080 I).obj Y) a \u2245 Y a.2`\nwhen `a : J \u00d7 I` is such that `a.2 = 0`. -/\n@[simps!]\nnoncomputable def mapBifunctorObjObjSingle\u2080Iso (a : J \u00d7 I) (ha : a.2 = 0) :\n    ((mapBifunctor F J I).obj X).obj ((single\u2080 I).obj Y) a \u2245 X a.1 :=\n  Functor.mapIso _ (singleObjApplyIsoOfEq _ Y _ ha) \u226a\u226b e.app (X a.1)\n\n/-- Given `F : D \u2964 C \u2964 D`, `Y : C` and `X : GradedObject J D`,\n`((mapBifunctor F J I).obj X).obj ((single\u2080 I).obj X) a` is an initial when `a : J \u00d7 I`\nis such that `a.2 \u2260 0`. -/\nnoncomputable def mapBifunctorObjObjSingle\u2080IsInitial (a : J \u00d7 I) (ha : a.2 \u2260 0) :\n    IsInitial (((mapBifunctor F J I).obj X).obj ((single\u2080 I).obj Y) a) :=\n  IsInitial.isInitialObj (F.obj (X a.1)) _ (isInitialSingleObjApply _ _ _ ha)\n\n/-- Given `F : D \u2964 C \u2964 D`, `Y : C`, `e : F.flip.obj Y \u2245 \ud835\udfed D`, `X : GradedObject J D` and\n`p : J \u00d7 I \u2192 J` such that `p \u27e8j, 0\u27e9 = j` for all `j`,\nthis is the (colimit) cofan which shall be used to construct the isomorphism\n`mapBifunctorMapObj F p X ((single\u2080 I).obj Y) \u2245 X`, see `mapBifunctorRightUnitor`. -/\nnoncomputable def mapBifunctorRightUnitorCofan (j : J) :\n    (((mapBifunctor F J I).obj X).obj ((single\u2080 I).obj Y)).CofanMapObjFun p j :=\n  CofanMapObjFun.mk _ _ _ (X j) (fun a ha =>\n    if ha : a.2 = 0 then\n      (mapBifunctorObjObjSingle\u2080Iso F Y e X a ha).hom \u226b eqToHom (by aesop)\n    else\n      (mapBifunctorObjObjSingle\u2080IsInitial F Y X a ha).to _)\n\n@[simp, reassoc]\nlemma mapBifunctorRightUnitorCofan_inj (j : J) :\n    (mapBifunctorRightUnitorCofan F Y e p hp X j).inj \u27e8\u27e8j, 0\u27e9, hp j\u27e9 =\n      (F.obj (X j)).map (singleObjApplyIso (0 : I) Y).hom \u226b e.hom.app (X j) := by\n  simp [mapBifunctorRightUnitorCofan]\n\n/-- The cofan `mapBifunctorRightUnitorCofan F Y e p hp X j` is a colimit. -/\nnoncomputable def mapBifunctorRightUnitorCofanIsColimit (j : J) :\n    IsColimit (mapBifunctorRightUnitorCofan F Y e p hp X j) :=\n  mkCofanColimit _\n    (fun s => e.inv.app (X j) \u226b\n      (F.obj (X j)).map (singleObjApplyIso (0 : I) Y).inv \u226b s.inj \u27e8\u27e8j, 0\u27e9, hp j\u27e9)\n    (fun s => by\n      rintro \u27e8\u27e8j', i\u27e9, h\u27e9\n      by_cases hi : i = 0\n      \u00b7 subst hi\n        simp only [Set.mem_preimage, hp, Set.mem_singleton_iff] at h\n        subst h\n        dsimp\n        rw [mapBifunctorRightUnitorCofan_inj, assoc, Iso.hom_inv_id_app_assoc,\n          \u2190 Functor.map_comp_assoc, Iso.hom_inv_id, Functor.map_id, id_comp]\n      \u00b7 apply IsInitial.hom_ext\n        exact mapBifunctorObjObjSingle\u2080IsInitial _ _ _ _ hi)\n    (fun s m hm => by\n      dsimp\n      rw [\u2190 hm \u27e8\u27e8j, 0\u27e9, hp j\u27e9, mapBifunctorRightUnitorCofan_inj, assoc, \u2190 Functor.map_comp_assoc,\n        Iso.inv_hom_id, Functor.map_id, id_comp, Iso.inv_hom_id_app_assoc])\n\nlemma mapBifunctorRightUnitor_hasMap :\n    HasMap (((mapBifunctor F J I).obj X).obj ((single\u2080 I).obj Y)) p :=\n  CofanMapObjFun.hasMap _ _ _ (mapBifunctorRightUnitorCofanIsColimit F Y e p hp X)\n\nvariable [HasMap (((mapBifunctor F J I).obj X).obj ((single\u2080 I).obj Y)) p]\n  [HasMap (((mapBifunctor F J I).obj X').obj ((single\u2080 I).obj Y)) p]\n\n/-- Given `F : D \u2964 C \u2964 D`, `Y : C`, `e : F.flip.obj Y \u2245 \ud835\udfed D`, `X : GradedObject J D` and\n`p : J \u00d7 I \u2192 J` such that `p \u27e8j, 0\u27e9 = j` for all `j`,\nthis is the right unitor isomorphism `mapBifunctorMapObj F p X ((single\u2080 I).obj Y) \u2245 X`. -/\nnoncomputable def mapBifunctorRightUnitor : mapBifunctorMapObj F p X ((single\u2080 I).obj Y) \u2245 X :=\n  isoMk _ _ (fun j => (CofanMapObjFun.iso\n    (mapBifunctorRightUnitorCofanIsColimit F Y e p hp X j)).symm)\n\n@[reassoc (attr := simp)]\nlemma \u03b9_mapBifunctorRightUnitor_hom_apply (j : J) :\n    \u03b9MapBifunctorMapObj F p X ((single\u2080 I).obj Y) j 0 j (hp j) \u226b\n        (mapBifunctorRightUnitor F Y e p hp X).hom j =\n      (F.obj (X j)).map (singleObjApplyIso (0 : I) Y).hom \u226b e.hom.app (X j) := by\n  dsimp [mapBifunctorRightUnitor]\n  erw [CofanMapObjFun.\u03b9MapObj_iso_inv]\n  rw [mapBifunctorRightUnitorCofan_inj]\n\nlemma mapBifunctorRightUnitor_inv_apply (j : J) :\n    (mapBifunctorRightUnitor F Y e p hp X).inv j =\n      e.inv.app (X j) \u226b (F.obj (X j)).map (singleObjApplyIso (0 : I) Y).inv \u226b\n        \u03b9MapBifunctorMapObj F p X ((single\u2080 I).obj Y) j 0 j (hp j) := rfl\n\nvariable {Y Y'}\n\n@[reassoc]\nlemma mapBifunctorRightUnitor_inv_naturality :\n    \u03c6 \u226b (mapBifunctorRightUnitor F Y e p hp X').inv =\n      (mapBifunctorRightUnitor F Y e p hp X).inv \u226b mapBifunctorMapMap F p \u03c6 (\ud835\udfd9 _):= by\n  ext j\n  dsimp\n  rw [mapBifunctorRightUnitor_inv_apply, mapBifunctorRightUnitor_inv_apply, assoc, assoc,\n    \u03b9_mapBifunctorMapMap]\n  dsimp\n  rw [Functor.map_id, id_comp, NatTrans.naturality_assoc]\n  erw [\u2190 NatTrans.naturality_assoc]\n  rfl\n\n", "theoremStatement": "@[reassoc]\nlemma mapBifunctorRightUnitor_naturality :\n    mapBifunctorMapMap F p \u03c6 (\ud835\udfd9 _) \u226b (mapBifunctorRightUnitor F Y e p hp X').hom =\n      (mapBifunctorRightUnitor F Y e p hp X).hom \u226b \u03c6 ", "theoremName": "CategoryTheory.GradedObject.mapBifunctorRightUnitor_naturality", "fileCreated": {"commit": "babb8ed93ea297649c9e447d90364976ee7b8cfd", "date": "2024-03-05"}, "theoremCreated": {"commit": "dca027f0817b6a7cdd6954e965680f28252e53f8", "date": "2024-03-26"}, "file": "mathlib/Mathlib/CategoryTheory/GradedObject/Unitor.lean", "module": "Mathlib.CategoryTheory.GradedObject.Unitor", "jsonFile": 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"Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Cases", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.MinMax", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Terminal", "Mathlib.CategoryTheory.Limits.Shapes.Products", "Mathlib.CategoryTheory.Limits.Shapes.Equalizers", "Mathlib.CategoryTheory.Limits.Shapes.Images", "Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects", "Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.End", "Mathlib.CategoryTheory.Monoidal.NaturalTransformation", "Mathlib.CategoryTheory.Monoidal.Discrete", "Mathlib.CategoryTheory.Shift.Basic", "Mathlib.CategoryTheory.GradedObject", "Mathlib.CategoryTheory.GradedObject.Bifunctor", "Mathlib.CategoryTheory.GradedObject.Single"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  rw [\u2190 cancel_mono (mapBifunctorRightUnitor F Y e p hp X').inv, assoc, assoc, Iso.hom_inv_id,\n    comp_id, mapBifunctorRightUnitor_inv_naturality, Iso.hom_inv_id_assoc]", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 175}}
+{"srcContext": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Mario Carneiro, Jeremy Avigad\n-/\nimport Mathlib.Algebra.Function.Support\nimport Mathlib.Order.Filter.Lift\nimport Mathlib.Topology.Defs.Filter\n\n#align_import topology.basic from \"leanprover-community/mathlib\"@\"e354e865255654389cc46e6032160238df2e0f40\"\n\n/-!\n# Basic theory of topological spaces.\n\nThe main definition is the type class `TopologicalSpace X` which endows a type `X` with a topology.\nThen `Set X` gets predicates `IsOpen`, `IsClosed` and functions `interior`, `closure` and\n`frontier`. Each point `x` of `X` gets a neighborhood filter `\ud835\udcdd x`. A filter `F` on `X` has\n`x` as a cluster point if `ClusterPt x F : \ud835\udcdd x \u2293 F \u2260 \u22a5`. A map `f : \u03b1 \u2192 X` clusters at `x`\nalong `F : Filter \u03b1` if `MapClusterPt x F f : ClusterPt x (map f F)`. In particular\nthe notion of cluster point of a sequence `u` is `MapClusterPt x atTop u`.\n\nFor topological spaces `X` and `Y`, a function `f : X \u2192 Y` and a point `x : X`,\n`ContinuousAt f x` means `f` is continuous at `x`, and global continuity is\n`Continuous f`. There is also a version of continuity `PContinuous` for\npartially defined functions.\n\n## Notation\n\nThe following notation is introduced elsewhere and it heavily used in this file.\n\n* `\ud835\udcdd x`: the filter `nhds x` of neighborhoods of a point `x`;\n* `\ud835\udcdf s`: the principal filter of a set `s`;\n* `\ud835\udcdd[s] x`: the filter `nhdsWithin x s` of neighborhoods of a point `x` within a set `s`;\n* `\ud835\udcdd[\u2260] x`: the filter `nhdsWithin x {x}\u1d9c` of punctured neighborhoods of `x`.\n\n## Implementation notes\n\nTopology in mathlib heavily uses filters (even more than in Bourbaki). See explanations in\n<https://leanprover-community.github.io/theories/topology.html>.\n\n## References\n\n* [N. Bourbaki, *General Topology*][bourbaki1966]\n* [I. M. James, *Topologies and Uniformities*][james1999]\n\n## Tags\n\ntopological space, interior, closure, frontier, neighborhood, continuity, continuous function\n-/\n\nnoncomputable section\n\nopen Set Filter\n\nuniverse u v w x\n\n/-!\n### Topological spaces\n-/\n\n/-- A constructor for topologies by specifying the closed sets,\nand showing that they satisfy the appropriate conditions. -/\ndef TopologicalSpace.ofClosed {X : Type u} (T : Set (Set X)) (empty_mem : \u2205 \u2208 T)\n    (sInter_mem : \u2200 A, A \u2286 T \u2192 \u22c2\u2080 A \u2208 T)\n    (union_mem : \u2200 A, A \u2208 T \u2192 \u2200 B, B \u2208 T \u2192 A \u222a B \u2208 T) : TopologicalSpace X where\n  IsOpen X := X\u1d9c \u2208 T\n  isOpen_univ := by simp [empty_mem]\n  isOpen_inter s t hs ht := by simpa only [compl_inter] using union_mem s\u1d9c hs t\u1d9c ht\n  isOpen_sUnion s hs := by\n    simp only [Set.compl_sUnion]\n    exact sInter_mem (compl '' s) fun z \u27e8y, hy, hz\u27e9 => hz \u25b8 hs y hy\n#align topological_space.of_closed TopologicalSpace.ofClosed\n\nsection TopologicalSpace\n\nvariable {X : Type u} {Y : Type v} {\u03b9 : Sort w} {\u03b1 \u03b2 : Type*}\n  {x : X} {s s\u2081 s\u2082 t : Set X} {p p\u2081 p\u2082 : X \u2192 Prop}\n\nopen Topology\n\nlemma isOpen_mk {p h\u2081 h\u2082 h\u2083} : IsOpen[\u27e8p, h\u2081, h\u2082, h\u2083\u27e9] s \u2194 p s := Iff.rfl\n#align is_open_mk isOpen_mk\n\n@[ext]\nprotected theorem TopologicalSpace.ext :\n    \u2200 {f g : TopologicalSpace X}, IsOpen[f] = IsOpen[g] \u2192 f = g\n  | \u27e8_, _, _, _\u27e9, \u27e8_, _, _, _\u27e9, rfl => rfl\n#align topological_space_eq TopologicalSpace.ext\n\nsection\n\nvariable [TopologicalSpace X]\n\nend\n\nprotected theorem TopologicalSpace.ext_iff {t t' : TopologicalSpace X} :\n    t = t' \u2194 \u2200 s, IsOpen[t] s \u2194 IsOpen[t'] s :=\n  \u27e8fun h s => h \u25b8 Iff.rfl, fun h => by ext; exact h _\u27e9\n#align topological_space_eq_iff TopologicalSpace.ext_iff\n\ntheorem isOpen_fold {t : TopologicalSpace X} : t.IsOpen s = IsOpen[t] s :=\n  rfl\n#align is_open_fold isOpen_fold\n\nvariable [TopologicalSpace X]\n\ntheorem isOpen_iUnion {f : \u03b9 \u2192 Set X} (h : \u2200 i, IsOpen (f i)) : IsOpen (\u22c3 i, f i) :=\n  isOpen_sUnion (forall_mem_range.2 h)\n#align is_open_Union isOpen_iUnion\n\ntheorem isOpen_biUnion {s : Set \u03b1} {f : \u03b1 \u2192 Set X} (h : \u2200 i \u2208 s, IsOpen (f i)) :\n    IsOpen (\u22c3 i \u2208 s, f i) :=\n  isOpen_iUnion fun i => isOpen_iUnion fun hi => h i hi\n#align is_open_bUnion isOpen_biUnion\n\ntheorem IsOpen.union (h\u2081 : IsOpen s\u2081) (h\u2082 : IsOpen s\u2082) : IsOpen (s\u2081 \u222a s\u2082) := by\n  rw [union_eq_iUnion]; exact isOpen_iUnion (Bool.forall_bool.2 \u27e8h\u2082, h\u2081\u27e9)\n#align is_open.union IsOpen.union\n\nlemma isOpen_iff_of_cover {f : \u03b1 \u2192 Set X} (ho : \u2200 i, IsOpen (f i)) (hU : (\u22c3 i, f i) = univ) :\n    IsOpen s \u2194 \u2200 i, IsOpen (f i \u2229 s) := by\n  refine \u27e8fun h i \u21a6 (ho i).inter h, fun h \u21a6 ?_\u27e9\n  rw [\u2190 s.inter_univ, inter_comm, \u2190 hU, iUnion_inter]\n  exact isOpen_iUnion fun i \u21a6 h i\n\n@[simp] theorem isOpen_empty : IsOpen (\u2205 : Set X) := by\n  rw [\u2190 sUnion_empty]; exact isOpen_sUnion fun a => False.elim\n#align is_open_empty isOpen_empty\n\ntheorem Set.Finite.isOpen_sInter {s : Set (Set X)} (hs : s.Finite) :\n    (\u2200 t \u2208 s, IsOpen t) \u2192 IsOpen (\u22c2\u2080 s) :=\n  Finite.induction_on hs (fun _ => by rw [sInter_empty]; exact isOpen_univ) fun _ _ ih h => by\n    simp only [sInter_insert, forall_mem_insert] at h \u22a2\n    exact h.1.inter (ih h.2)\n#align is_open_sInter Set.Finite.isOpen_sInter\n\ntheorem Set.Finite.isOpen_biInter {s : Set \u03b1} {f : \u03b1 \u2192 Set X} (hs : s.Finite)\n    (h : \u2200 i \u2208 s, IsOpen (f i)) :\n    IsOpen (\u22c2 i \u2208 s, f i) :=\n  sInter_image f s \u25b8 (hs.image _).isOpen_sInter (forall_mem_image.2 h)\n#align is_open_bInter Set.Finite.isOpen_biInter\n\ntheorem isOpen_iInter_of_finite [Finite \u03b9] {s : \u03b9 \u2192 Set X} (h : \u2200 i, IsOpen (s i)) :\n    IsOpen (\u22c2 i, s i) :=\n  (finite_range _).isOpen_sInter  (forall_mem_range.2 h)\n#align is_open_Inter isOpen_iInter_of_finite\n\ntheorem isOpen_biInter_finset {s : Finset \u03b1} {f : \u03b1 \u2192 Set X} (h : \u2200 i \u2208 s, IsOpen (f i)) :\n    IsOpen (\u22c2 i \u2208 s, f i) :=\n  s.finite_toSet.isOpen_biInter h\n#align is_open_bInter_finset isOpen_biInter_finset\n\n@[simp] -- Porting note: added `simp`\ntheorem isOpen_const {p : Prop} : IsOpen { _x : X | p } := by by_cases p <;> simp [*]\n#align is_open_const isOpen_const\n\ntheorem IsOpen.and : IsOpen { x | p\u2081 x } \u2192 IsOpen { x | p\u2082 x } \u2192 IsOpen { x | p\u2081 x \u2227 p\u2082 x } :=\n  IsOpen.inter\n#align is_open.and IsOpen.and\n\n@[simp] theorem isOpen_compl_iff : IsOpen s\u1d9c \u2194 IsClosed s :=\n  \u27e8fun h => \u27e8h\u27e9, fun h => h.isOpen_compl\u27e9\n#align is_open_compl_iff isOpen_compl_iff\n\ntheorem TopologicalSpace.ext_iff_isClosed {t\u2081 t\u2082 : TopologicalSpace X} :\n    t\u2081 = t\u2082 \u2194 \u2200 s, IsClosed[t\u2081] s \u2194 IsClosed[t\u2082] s := by\n  rw [TopologicalSpace.ext_iff, compl_surjective.forall]\n  simp only [@isOpen_compl_iff _ _ t\u2081, @isOpen_compl_iff _ _ t\u2082]\n\nalias \u27e8_, TopologicalSpace.ext_isClosed\u27e9 := TopologicalSpace.ext_iff_isClosed\n\n-- Porting note (#10756): new lemma\ntheorem isClosed_const {p : Prop} : IsClosed { _x : X | p } := \u27e8isOpen_const (p := \u00acp)\u27e9\n\n@[simp] theorem isClosed_empty : IsClosed (\u2205 : Set X) := isClosed_const\n#align is_closed_empty isClosed_empty\n\n@[simp] theorem isClosed_univ : IsClosed (univ : Set X) := isClosed_const\n#align is_closed_univ isClosed_univ\n\ntheorem IsClosed.union : IsClosed s\u2081 \u2192 IsClosed s\u2082 \u2192 IsClosed (s\u2081 \u222a s\u2082) := by\n  simpa only [\u2190 isOpen_compl_iff, compl_union] using IsOpen.inter\n#align is_closed.union IsClosed.union\n\ntheorem isClosed_sInter {s : Set (Set X)} : (\u2200 t \u2208 s, IsClosed t) \u2192 IsClosed (\u22c2\u2080 s) := by\n  simpa only [\u2190 isOpen_compl_iff, compl_sInter, sUnion_image] using isOpen_biUnion\n#align is_closed_sInter isClosed_sInter\n\ntheorem isClosed_iInter {f : \u03b9 \u2192 Set X} (h : \u2200 i, IsClosed (f i)) : IsClosed (\u22c2 i, f i) :=\n  isClosed_sInter <| forall_mem_range.2 h\n#align is_closed_Inter isClosed_iInter\n\ntheorem isClosed_biInter {s : Set \u03b1} {f : \u03b1 \u2192 Set X} (h : \u2200 i \u2208 s, IsClosed (f i)) :\n    IsClosed (\u22c2 i \u2208 s, f i) :=\n  isClosed_iInter fun i => isClosed_iInter <| h i\n#align is_closed_bInter isClosed_biInter\n\n@[simp]\ntheorem isClosed_compl_iff {s : Set X} : IsClosed s\u1d9c \u2194 IsOpen s := by\n  rw [\u2190 isOpen_compl_iff, compl_compl]\n#align is_closed_compl_iff isClosed_compl_iff\n\nalias \u27e8_, IsOpen.isClosed_compl\u27e9 := isClosed_compl_iff\n#align is_open.is_closed_compl IsOpen.isClosed_compl\n\ntheorem IsOpen.sdiff (h\u2081 : IsOpen s) (h\u2082 : IsClosed t) : IsOpen (s \\ t) :=\n  IsOpen.inter h\u2081 h\u2082.isOpen_compl\n#align is_open.sdiff IsOpen.sdiff\n\ntheorem IsClosed.inter (h\u2081 : IsClosed s\u2081) (h\u2082 : IsClosed s\u2082) : IsClosed (s\u2081 \u2229 s\u2082) := by\n  rw [\u2190 isOpen_compl_iff] at *\n  rw [compl_inter]\n  exact IsOpen.union h\u2081 h\u2082\n#align is_closed.inter IsClosed.inter\n\ntheorem IsClosed.sdiff (h\u2081 : IsClosed s) (h\u2082 : IsOpen t) : IsClosed (s \\ t) :=\n  IsClosed.inter h\u2081 (isClosed_compl_iff.mpr h\u2082)\n#align is_closed.sdiff IsClosed.sdiff\n\ntheorem Set.Finite.isClosed_biUnion {s : Set \u03b1} {f : \u03b1 \u2192 Set X} (hs : s.Finite)\n    (h : \u2200 i \u2208 s, IsClosed (f i)) :\n    IsClosed (\u22c3 i \u2208 s, f i) := by\n  simp only [\u2190 isOpen_compl_iff, compl_iUnion] at *\n  exact hs.isOpen_biInter h\n#align is_closed_bUnion Set.Finite.isClosed_biUnion\n\nlemma isClosed_biUnion_finset {s : Finset \u03b1} {f : \u03b1 \u2192 Set X} (h : \u2200 i \u2208 s, IsClosed (f i)) :\n    IsClosed (\u22c3 i \u2208 s, f i) :=\n  s.finite_toSet.isClosed_biUnion h\n\ntheorem isClosed_iUnion_of_finite [Finite \u03b9] {s : \u03b9 \u2192 Set X} (h : \u2200 i, IsClosed (s i)) :\n    IsClosed (\u22c3 i, s i) := by\n  simp only [\u2190 isOpen_compl_iff, compl_iUnion] at *\n  exact isOpen_iInter_of_finite h\n#align is_closed_Union isClosed_iUnion_of_finite\n\ntheorem isClosed_imp {p q : X \u2192 Prop} (hp : IsOpen { x | p x }) (hq : IsClosed { x | q x }) :\n    IsClosed { x | p x \u2192 q x } := by\n  simpa only [imp_iff_not_or] using hp.isClosed_compl.union hq\n#align is_closed_imp isClosed_imp\n\ntheorem IsClosed.not : IsClosed { a | p a } \u2192 IsOpen { a | \u00acp a } :=\n  isOpen_compl_iff.mpr\n#align is_closed.not IsClosed.not\n\n/-!\n### Interior of a set\n-/\n\n-- Porting note: use `\u2203 t, t \u2286 s \u2227 _` instead of `\u2203 t \u2286 s, _`\ntheorem mem_interior : x \u2208 interior s \u2194 \u2203 t, t \u2286 s \u2227 IsOpen t \u2227 x \u2208 t := by\n  simp only [interior, mem_sUnion, mem_setOf_eq, and_assoc, and_left_comm]\n#align mem_interior mem_interior\u2093\n\n@[simp]\ntheorem isOpen_interior : IsOpen (interior s) :=\n  isOpen_sUnion fun _ => And.left\n#align is_open_interior isOpen_interior\n\ntheorem interior_subset : interior s \u2286 s :=\n  sUnion_subset fun _ => And.right\n#align interior_subset interior_subset\n\ntheorem interior_maximal (h\u2081 : t \u2286 s) (h\u2082 : IsOpen t) : t \u2286 interior s :=\n  subset_sUnion_of_mem \u27e8h\u2082, h\u2081\u27e9\n#align interior_maximal interior_maximal\n\ntheorem IsOpen.interior_eq (h : IsOpen s) : interior s = s :=\n  interior_subset.antisymm (interior_maximal (Subset.refl s) h)\n#align is_open.interior_eq IsOpen.interior_eq\n\ntheorem interior_eq_iff_isOpen : interior s = s \u2194 IsOpen s :=\n  \u27e8fun h => h \u25b8 isOpen_interior, IsOpen.interior_eq\u27e9\n#align interior_eq_iff_is_open interior_eq_iff_isOpen\n\ntheorem subset_interior_iff_isOpen : s \u2286 interior s \u2194 IsOpen s := by\n  simp only [interior_eq_iff_isOpen.symm, Subset.antisymm_iff, interior_subset, true_and]\n#align subset_interior_iff_is_open subset_interior_iff_isOpen\n\ntheorem IsOpen.subset_interior_iff (h\u2081 : IsOpen s) : s \u2286 interior t \u2194 s \u2286 t :=\n  \u27e8fun h => Subset.trans h interior_subset, fun h\u2082 => interior_maximal h\u2082 h\u2081\u27e9\n#align is_open.subset_interior_iff IsOpen.subset_interior_iff\n\ntheorem subset_interior_iff : t \u2286 interior s \u2194 \u2203 U, IsOpen U \u2227 t \u2286 U \u2227 U \u2286 s :=\n  \u27e8fun h => \u27e8interior s, isOpen_interior, h, interior_subset\u27e9, fun \u27e8_U, hU, htU, hUs\u27e9 =>\n    htU.trans (interior_maximal hUs hU)\u27e9\n#align subset_interior_iff subset_interior_iff\n\nlemma interior_subset_iff : interior s \u2286 t \u2194 \u2200 U, IsOpen U \u2192 U \u2286 s \u2192 U \u2286 t := by\n  simp [interior]\n\n@[mono, gcongr]\ntheorem interior_mono (h : s \u2286 t) : interior s \u2286 interior t :=\n  interior_maximal (Subset.trans interior_subset h) isOpen_interior\n#align interior_mono interior_mono\n\n@[simp]\ntheorem interior_empty : interior (\u2205 : Set X) = \u2205 :=\n  isOpen_empty.interior_eq\n#align interior_empty interior_empty\n\n@[simp]\ntheorem interior_univ : interior (univ : Set X) = univ :=\n  isOpen_univ.interior_eq\n#align interior_univ interior_univ\n\n@[simp]\ntheorem interior_eq_univ : interior s = univ \u2194 s = univ :=\n  \u27e8fun h => univ_subset_iff.mp <| h.symm.trans_le interior_subset, fun h => h.symm \u25b8 interior_univ\u27e9\n#align interior_eq_univ interior_eq_univ\n\n@[simp]\ntheorem interior_interior : interior (interior s) = interior s :=\n  isOpen_interior.interior_eq\n#align interior_interior interior_interior\n\n@[simp]\ntheorem interior_inter : interior (s \u2229 t) = interior s \u2229 interior t :=\n  (Monotone.map_inf_le (fun _ _ \u21a6 interior_mono) s t).antisymm <|\n    interior_maximal (inter_subset_inter interior_subset interior_subset) <|\n      isOpen_interior.inter isOpen_interior\n#align interior_inter interior_inter\n\ntheorem Set.Finite.interior_biInter {\u03b9 : Type*} {s : Set \u03b9} (hs : s.Finite) (f : \u03b9 \u2192 Set X) :\n    interior (\u22c2 i \u2208 s, f i) = \u22c2 i \u2208 s, interior (f i) :=\n  hs.induction_on (by simp) <| by intros; simp [*]\n\ntheorem Set.Finite.interior_sInter {S : Set (Set X)} (hS : S.Finite) :\n    interior (\u22c2\u2080 S) = \u22c2 s \u2208 S, interior s := by\n  rw [sInter_eq_biInter, hS.interior_biInter]\n\n@[simp]\ntheorem Finset.interior_iInter {\u03b9 : Type*} (s : Finset \u03b9) (f : \u03b9 \u2192 Set X) :\n    interior (\u22c2 i \u2208 s, f i) = \u22c2 i \u2208 s, interior (f i) :=\n  s.finite_toSet.interior_biInter f\n#align finset.interior_Inter Finset.interior_iInter\n\n@[simp]\ntheorem interior_iInter_of_finite [Finite \u03b9] (f : \u03b9 \u2192 Set X) :\n    interior (\u22c2 i, f i) = \u22c2 i, interior (f i) := by\n  rw [\u2190 sInter_range, (finite_range f).interior_sInter, biInter_range]\n#align interior_Inter interior_iInter_of_finite\n\ntheorem interior_union_isClosed_of_interior_empty (h\u2081 : IsClosed s)\n    (h\u2082 : interior t = \u2205) : interior (s \u222a t) = interior s :=\n  have : interior (s \u222a t) \u2286 s := fun x \u27e8u, \u27e8(hu\u2081 : IsOpen u), (hu\u2082 : u \u2286 s \u222a t)\u27e9, (hx\u2081 : x \u2208 u)\u27e9 =>\n    by_contradiction fun hx\u2082 : x \u2209 s =>\n      have : u \\ s \u2286 t := fun x \u27e8h\u2081, h\u2082\u27e9 => Or.resolve_left (hu\u2082 h\u2081) h\u2082\n      have : u \\ s \u2286 interior t := by rwa [(IsOpen.sdiff hu\u2081 h\u2081).subset_interior_iff]\n      have : u \\ s \u2286 \u2205 := by rwa [h\u2082] at this\n      this \u27e8hx\u2081, hx\u2082\u27e9\n  Subset.antisymm (interior_maximal this isOpen_interior) (interior_mono <| subset_union_left _ _)\n#align interior_union_is_closed_of_interior_empty interior_union_isClosed_of_interior_empty\n\ntheorem isOpen_iff_forall_mem_open : IsOpen s \u2194 \u2200 x \u2208 s, \u2203 t, t \u2286 s \u2227 IsOpen t \u2227 x \u2208 t := by\n  rw [\u2190 subset_interior_iff_isOpen]\n  simp only [subset_def, mem_interior]\n#align is_open_iff_forall_mem_open isOpen_iff_forall_mem_open\n\ntheorem interior_iInter_subset (s : \u03b9 \u2192 Set X) : interior (\u22c2 i, s i) \u2286 \u22c2 i, interior (s i) :=\n  subset_iInter fun _ => interior_mono <| iInter_subset _ _\n#align interior_Inter_subset interior_iInter_subset\n\ntheorem interior_iInter\u2082_subset (p : \u03b9 \u2192 Sort*) (s : \u2200 i, p i \u2192 Set X) :\n    interior (\u22c2 (i) (j), s i j) \u2286 \u22c2 (i) (j), interior (s i j) :=\n  (interior_iInter_subset _).trans <| iInter_mono fun _ => interior_iInter_subset _\n#align interior_Inter\u2082_subset interior_iInter\u2082_subset\n\ntheorem interior_sInter_subset (S : Set (Set X)) : interior (\u22c2\u2080 S) \u2286 \u22c2 s \u2208 S, interior s :=\n  calc\n    interior (\u22c2\u2080 S) = interior (\u22c2 s \u2208 S, s) := by rw [sInter_eq_biInter]\n    _ \u2286 \u22c2 s \u2208 S, interior s := interior_iInter\u2082_subset _ _\n#align interior_sInter_subset interior_sInter_subset\n\ntheorem Filter.HasBasis.lift'_interior {l : Filter X} {p : \u03b9 \u2192 Prop} {s : \u03b9 \u2192 Set X}\n    (h : l.HasBasis p s) : (l.lift' interior).HasBasis p fun i => interior (s i) :=\n  h.lift' fun _ _ \u21a6 interior_mono\n\ntheorem Filter.lift'_interior_le (l : Filter X) : l.lift' interior \u2264 l := fun _s hs \u21a6\n  mem_of_superset (mem_lift' hs) interior_subset\n\ntheorem Filter.HasBasis.lift'_interior_eq_self {l : Filter X} {p : \u03b9 \u2192 Prop} {s : \u03b9 \u2192 Set X}\n    (h : l.HasBasis p s) (ho : \u2200 i, p i \u2192 IsOpen (s i)) : l.lift' interior = l :=\n  le_antisymm l.lift'_interior_le <| h.lift'_interior.ge_iff.2 fun i hi \u21a6 by\n    simpa only [(ho i hi).interior_eq] using h.mem_of_mem hi\n\n/-!\n### Closure of a set\n-/\n\n@[simp]\ntheorem isClosed_closure : IsClosed (closure s) :=\n  isClosed_sInter fun _ => And.left\n#align is_closed_closure isClosed_closure\n\ntheorem subset_closure : s \u2286 closure s :=\n  subset_sInter fun _ => And.right\n#align subset_closure subset_closure\n\ntheorem not_mem_of_not_mem_closure {P : X} (hP : P \u2209 closure s) : P \u2209 s := fun h =>\n  hP (subset_closure h)\n#align not_mem_of_not_mem_closure not_mem_of_not_mem_closure\n\ntheorem closure_minimal (h\u2081 : s \u2286 t) (h\u2082 : IsClosed t) : closure s \u2286 t :=\n  sInter_subset_of_mem \u27e8h\u2082, h\u2081\u27e9\n#align closure_minimal closure_minimal\n\ntheorem Disjoint.closure_left (hd : Disjoint s t) (ht : IsOpen t) :\n    Disjoint (closure s) t :=\n  disjoint_compl_left.mono_left <| closure_minimal hd.subset_compl_right ht.isClosed_compl\n#align disjoint.closure_left Disjoint.closure_left\n\ntheorem Disjoint.closure_right (hd : Disjoint s t) (hs : IsOpen s) :\n    Disjoint s (closure t) :=\n  (hd.symm.closure_left hs).symm\n#align disjoint.closure_right Disjoint.closure_right\n\ntheorem IsClosed.closure_eq (h : IsClosed s) : closure s = s :=\n  Subset.antisymm (closure_minimal (Subset.refl s) h) subset_closure\n#align is_closed.closure_eq IsClosed.closure_eq\n\ntheorem IsClosed.closure_subset (hs : IsClosed s) : closure s \u2286 s :=\n  closure_minimal (Subset.refl _) hs\n#align is_closed.closure_subset IsClosed.closure_subset\n\ntheorem IsClosed.closure_subset_iff (h\u2081 : IsClosed t) : closure s \u2286 t \u2194 s \u2286 t :=\n  \u27e8Subset.trans subset_closure, fun h => closure_minimal h h\u2081\u27e9\n#align is_closed.closure_subset_iff IsClosed.closure_subset_iff\n\ntheorem IsClosed.mem_iff_closure_subset (hs : IsClosed s) :\n    x \u2208 s \u2194 closure ({x} : Set X) \u2286 s :=\n  (hs.closure_subset_iff.trans Set.singleton_subset_iff).symm\n#align is_closed.mem_iff_closure_subset IsClosed.mem_iff_closure_subset\n\n@[mono, gcongr]\ntheorem closure_mono (h : s \u2286 t) : closure s \u2286 closure t :=\n  closure_minimal (Subset.trans h subset_closure) isClosed_closure\n#align closure_mono closure_mono\n\ntheorem monotone_closure (X : Type*) [TopologicalSpace X] : Monotone (@closure X _) := fun _ _ =>\n  closure_mono\n#align monotone_closure monotone_closure\n\ntheorem diff_subset_closure_iff : s \\ t \u2286 closure t \u2194 s \u2286 closure t := by\n  rw [diff_subset_iff, union_eq_self_of_subset_left subset_closure]\n#align diff_subset_closure_iff diff_subset_closure_iff\n\ntheorem closure_inter_subset_inter_closure (s t : Set X) :\n    closure (s \u2229 t) \u2286 closure s \u2229 closure t :=\n  (monotone_closure X).map_inf_le s t\n#align closure_inter_subset_inter_closure closure_inter_subset_inter_closure\n\ntheorem isClosed_of_closure_subset (h : closure s \u2286 s) : IsClosed s := by\n  rw [subset_closure.antisymm h]; exact isClosed_closure\n#align is_closed_of_closure_subset isClosed_of_closure_subset\n\ntheorem closure_eq_iff_isClosed : closure s = s \u2194 IsClosed s :=\n  \u27e8fun h => h \u25b8 isClosed_closure, IsClosed.closure_eq\u27e9\n#align closure_eq_iff_is_closed closure_eq_iff_isClosed\n\ntheorem closure_subset_iff_isClosed : closure s \u2286 s \u2194 IsClosed s :=\n  \u27e8isClosed_of_closure_subset, IsClosed.closure_subset\u27e9\n#align closure_subset_iff_is_closed closure_subset_iff_isClosed\n\n@[simp]\ntheorem closure_empty : closure (\u2205 : Set X) = \u2205 :=\n  isClosed_empty.closure_eq\n#align closure_empty closure_empty\n\n@[simp]\ntheorem closure_empty_iff (s : Set X) : closure s = \u2205 \u2194 s = \u2205 :=\n  \u27e8subset_eq_empty subset_closure, fun h => h.symm \u25b8 closure_empty\u27e9\n#align closure_empty_iff closure_empty_iff\n\n@[simp]\ntheorem closure_nonempty_iff : (closure s).Nonempty \u2194 s.Nonempty := by\n  simp only [nonempty_iff_ne_empty, Ne, closure_empty_iff]\n#align closure_nonempty_iff closure_nonempty_iff\n\nalias \u27e8Set.Nonempty.of_closure, Set.Nonempty.closure\u27e9 := closure_nonempty_iff\n#align set.nonempty.of_closure Set.Nonempty.of_closure\n#align set.nonempty.closure Set.Nonempty.closure\n\n@[simp]\ntheorem closure_univ : closure (univ : Set X) = univ :=\n  isClosed_univ.closure_eq\n#align closure_univ closure_univ\n\n@[simp]\ntheorem closure_closure : closure (closure s) = closure s :=\n  isClosed_closure.closure_eq\n#align closure_closure closure_closure\n\ntheorem closure_eq_compl_interior_compl : closure s = (interior s\u1d9c)\u1d9c := by\n  rw [interior, closure, compl_sUnion, compl_image_set_of]\n  simp only [compl_subset_compl, isOpen_compl_iff]\n#align closure_eq_compl_interior_compl closure_eq_compl_interior_compl\n\n@[simp]\ntheorem closure_union : closure (s \u222a t) = closure s \u222a closure t := by\n  simp [closure_eq_compl_interior_compl, compl_inter]\n#align closure_union closure_union\n\ntheorem Set.Finite.closure_biUnion {\u03b9 : Type*} {s : Set \u03b9} (hs : s.Finite) (f : \u03b9 \u2192 Set X) :\n    closure (\u22c3 i \u2208 s, f i) = \u22c3 i \u2208 s, closure (f i) := by\n  simp [closure_eq_compl_interior_compl, hs.interior_biInter]\n\ntheorem Set.Finite.closure_sUnion {S : Set (Set X)} (hS : S.Finite) :\n    closure (\u22c3\u2080 S) = \u22c3 s \u2208 S, closure s := by\n  rw [sUnion_eq_biUnion, hS.closure_biUnion]\n\n@[simp]\ntheorem Finset.closure_biUnion {\u03b9 : Type*} (s : Finset \u03b9) (f : \u03b9 \u2192 Set X) :\n    closure (\u22c3 i \u2208 s, f i) = \u22c3 i \u2208 s, closure (f i) :=\n  s.finite_toSet.closure_biUnion f\n#align finset.closure_bUnion Finset.closure_biUnion\n\n@[simp]\ntheorem closure_iUnion_of_finite [Finite \u03b9] (f : \u03b9 \u2192 Set X) :\n    closure (\u22c3 i, f i) = \u22c3 i, closure (f i) := by\n  rw [\u2190 sUnion_range, (finite_range _).closure_sUnion, biUnion_range]\n#align closure_Union closure_iUnion_of_finite\n\ntheorem interior_subset_closure : interior s \u2286 closure s :=\n  Subset.trans interior_subset subset_closure\n#align interior_subset_closure interior_subset_closure\n\n@[simp]\ntheorem interior_compl : interior s\u1d9c = (closure s)\u1d9c := by\n  simp [closure_eq_compl_interior_compl]\n#align interior_compl interior_compl\n\n@[simp]\ntheorem closure_compl : closure s\u1d9c = (interior s)\u1d9c := by\n  simp [closure_eq_compl_interior_compl]\n#align closure_compl closure_compl\n\ntheorem mem_closure_iff :\n    x \u2208 closure s \u2194 \u2200 o, IsOpen o \u2192 x \u2208 o \u2192 (o \u2229 s).Nonempty :=\n  \u27e8fun h o oo ao =>\n    by_contradiction fun os =>\n      have : s \u2286 o\u1d9c := fun x xs xo => os \u27e8x, xo, xs\u27e9\n      closure_minimal this (isClosed_compl_iff.2 oo) h ao,\n    fun H _ \u27e8h\u2081, h\u2082\u27e9 =>\n    by_contradiction fun nc =>\n      let \u27e8_, hc, hs\u27e9 := H _ h\u2081.isOpen_compl nc\n      hc (h\u2082 hs)\u27e9\n#align mem_closure_iff mem_closure_iff\n\ntheorem closure_inter_open_nonempty_iff (h : IsOpen t) :\n    (closure s \u2229 t).Nonempty \u2194 (s \u2229 t).Nonempty :=\n  \u27e8fun \u27e8_x, hxcs, hxt\u27e9 => inter_comm t s \u25b8 mem_closure_iff.1 hxcs t h hxt, fun h =>\n    h.mono <| inf_le_inf_right t subset_closure\u27e9\n#align closure_inter_open_nonempty_iff closure_inter_open_nonempty_iff\n\ntheorem Filter.le_lift'_closure (l : Filter X) : l \u2264 l.lift' closure :=\n  le_lift'.2 fun _ h => mem_of_superset h subset_closure\n#align filter.le_lift'_closure Filter.le_lift'_closure\n\ntheorem Filter.HasBasis.lift'_closure {l : Filter X} {p : \u03b9 \u2192 Prop} {s : \u03b9 \u2192 Set X}\n    (h : l.HasBasis p s) : (l.lift' closure).HasBasis p fun i => closure (s i) :=\n  h.lift' (monotone_closure X)\n#align filter.has_basis.lift'_closure Filter.HasBasis.lift'_closure\n\ntheorem Filter.HasBasis.lift'_closure_eq_self {l : Filter X} {p : \u03b9 \u2192 Prop} {s : \u03b9 \u2192 Set X}\n    (h : l.HasBasis p s) (hc : \u2200 i, p i \u2192 IsClosed (s i)) : l.lift' closure = l :=\n  le_antisymm (h.ge_iff.2 fun i hi => (hc i hi).closure_eq \u25b8 mem_lift' (h.mem_of_mem hi))\n    l.le_lift'_closure\n#align filter.has_basis.lift'_closure_eq_self Filter.HasBasis.lift'_closure_eq_self\n\n@[simp]\ntheorem Filter.lift'_closure_eq_bot {l : Filter X} : l.lift' closure = \u22a5 \u2194 l = \u22a5 :=\n  \u27e8fun h => bot_unique <| h \u25b8 l.le_lift'_closure, fun h =>\n    h.symm \u25b8 by rw [lift'_bot (monotone_closure _), closure_empty, principal_empty]\u27e9\n#align filter.lift'_closure_eq_bot Filter.lift'_closure_eq_bot\n\ntheorem dense_iff_closure_eq : Dense s \u2194 closure s = univ :=\n  eq_univ_iff_forall.symm\n#align dense_iff_closure_eq dense_iff_closure_eq\n\nalias \u27e8Dense.closure_eq, _\u27e9 := dense_iff_closure_eq\n#align dense.closure_eq Dense.closure_eq\n\ntheorem interior_eq_empty_iff_dense_compl : interior s = \u2205 \u2194 Dense s\u1d9c := by\n  rw [dense_iff_closure_eq, closure_compl, compl_univ_iff]\n#align interior_eq_empty_iff_dense_compl interior_eq_empty_iff_dense_compl\n\ntheorem Dense.interior_compl (h : Dense s) : interior s\u1d9c = \u2205 :=\n  interior_eq_empty_iff_dense_compl.2 <| by rwa [compl_compl]\n#align dense.interior_compl Dense.interior_compl\n\n/-- The closure of a set `s` is dense if and only if `s` is dense. -/\n@[simp]\ntheorem dense_closure : Dense (closure s) \u2194 Dense s := by\n  rw [Dense, Dense, closure_closure]\n#align dense_closure dense_closure\n\nprotected alias \u27e8_, Dense.closure\u27e9 := dense_closure\nalias \u27e8Dense.of_closure, _\u27e9 := dense_closure\n#align dense.of_closure Dense.of_closure\n#align dense.closure Dense.closure\n\n@[simp]\ntheorem dense_univ : Dense (univ : Set X) := fun _ => subset_closure trivial\n#align dense_univ dense_univ\n\n/-- A set is dense if and only if it has a nonempty intersection with each nonempty open set. -/\ntheorem dense_iff_inter_open :\n    Dense s \u2194 \u2200 U, IsOpen U \u2192 U.Nonempty \u2192 (U \u2229 s).Nonempty := by\n  constructor <;> intro h\n  \u00b7 rintro U U_op \u27e8x, x_in\u27e9\n    exact mem_closure_iff.1 (h _) U U_op x_in\n  \u00b7 intro x\n    rw [mem_closure_iff]\n    intro U U_op x_in\n    exact h U U_op \u27e8_, x_in\u27e9\n#align dense_iff_inter_open dense_iff_inter_open\n\nalias \u27e8Dense.inter_open_nonempty, _\u27e9 := dense_iff_inter_open\n#align dense.inter_open_nonempty Dense.inter_open_nonempty\n\ntheorem Dense.exists_mem_open (hs : Dense s) {U : Set X} (ho : IsOpen U)\n    (hne : U.Nonempty) : \u2203 x \u2208 s, x \u2208 U :=\n  let \u27e8x, hx\u27e9 := hs.inter_open_nonempty U ho hne\n  \u27e8x, hx.2, hx.1\u27e9\n#align dense.exists_mem_open Dense.exists_mem_open\n\ntheorem Dense.nonempty_iff (hs : Dense s) : s.Nonempty \u2194 Nonempty X :=\n  \u27e8fun \u27e8x, _\u27e9 => \u27e8x\u27e9, fun \u27e8x\u27e9 =>\n    let \u27e8y, hy\u27e9 := hs.inter_open_nonempty _ isOpen_univ \u27e8x, trivial\u27e9\n    \u27e8y, hy.2\u27e9\u27e9\n#align dense.nonempty_iff Dense.nonempty_iff\n\ntheorem Dense.nonempty [h : Nonempty X] (hs : Dense s) : s.Nonempty :=\n  hs.nonempty_iff.2 h\n#align dense.nonempty Dense.nonempty\n\n@[mono]\ntheorem Dense.mono (h : s\u2081 \u2286 s\u2082) (hd : Dense s\u2081) : Dense s\u2082 := fun x =>\n  closure_mono h (hd x)\n#align dense.mono Dense.mono\n\n/-- Complement to a singleton is dense if and only if the singleton is not an open set. -/\ntheorem dense_compl_singleton_iff_not_open :\n    Dense ({x}\u1d9c : Set X) \u2194 \u00acIsOpen ({x} : Set X) := by\n  constructor\n  \u00b7 intro hd ho\n    exact (hd.inter_open_nonempty _ ho (singleton_nonempty _)).ne_empty (inter_compl_self _)\n  \u00b7 refine' fun ho => dense_iff_inter_open.2 fun U hU hne => inter_compl_nonempty_iff.2 fun hUx => _\n    obtain rfl : U = {x}\n    exact eq_singleton_iff_nonempty_unique_mem.2 \u27e8hne, hUx\u27e9\n    exact ho hU\n#align dense_compl_singleton_iff_not_open dense_compl_singleton_iff_not_open\n\n/-!\n### Frontier of a set\n-/\n\n@[simp]\ntheorem closure_diff_interior (s : Set X) : closure s \\ interior s = frontier s :=\n  rfl\n#align closure_diff_interior closure_diff_interior\n\n/-- Interior and frontier are disjoint. -/\nlemma disjoint_interior_frontier : Disjoint (interior s) (frontier s) := by\n  rw [disjoint_iff_inter_eq_empty, \u2190 closure_diff_interior, diff_eq,\n    \u2190 inter_assoc, inter_comm, \u2190 inter_assoc, compl_inter_self, empty_inter]\n\n@[simp]\ntheorem closure_diff_frontier (s : Set X) : closure s \\ frontier s = interior s := by\n  rw [frontier, diff_diff_right_self, inter_eq_self_of_subset_right interior_subset_closure]\n#align closure_diff_frontier closure_diff_frontier\n\n@[simp]\ntheorem self_diff_frontier (s : Set X) : s \\ frontier s = interior s := by\n  rw [frontier, diff_diff_right, diff_eq_empty.2 subset_closure,\n    inter_eq_self_of_subset_right interior_subset, empty_union]\n#align self_diff_frontier self_diff_frontier\n\ntheorem frontier_eq_closure_inter_closure : frontier s = closure s \u2229 closure s\u1d9c := by\n  rw [closure_compl, frontier, diff_eq]\n#align frontier_eq_closure_inter_closure frontier_eq_closure_inter_closure\n\ntheorem frontier_subset_closure : frontier s \u2286 closure s :=\n  diff_subset _ _\n#align frontier_subset_closure frontier_subset_closure\n\ntheorem IsClosed.frontier_subset (hs : IsClosed s) : frontier s \u2286 s :=\n  frontier_subset_closure.trans hs.closure_eq.subset\n#align is_closed.frontier_subset IsClosed.frontier_subset\n\ntheorem frontier_closure_subset : frontier (closure s) \u2286 frontier s :=\n  diff_subset_diff closure_closure.subset <| interior_mono subset_closure\n#align frontier_closure_subset frontier_closure_subset\n\ntheorem frontier_interior_subset : frontier (interior s) \u2286 frontier s :=\n  diff_subset_diff (closure_mono interior_subset) interior_interior.symm.subset\n#align frontier_interior_subset frontier_interior_subset\n\n/-- The complement of a set has the same frontier as the original set. -/\n@[simp]\ntheorem frontier_compl (s : Set X) : frontier s\u1d9c = frontier s := by\n  simp only [frontier_eq_closure_inter_closure, compl_compl, inter_comm]\n#align frontier_compl frontier_compl\n\n@[simp]\ntheorem frontier_univ : frontier (univ : Set X) = \u2205 := by simp [frontier]\n#align frontier_univ frontier_univ\n\n@[simp]\ntheorem frontier_empty : frontier (\u2205 : Set X) = \u2205 := by simp [frontier]\n#align frontier_empty frontier_empty\n\ntheorem frontier_inter_subset (s t : Set X) :\n    frontier (s \u2229 t) \u2286 frontier s \u2229 closure t \u222a closure s \u2229 frontier t := by\n  simp only [frontier_eq_closure_inter_closure, compl_inter, closure_union]\n  refine' (inter_subset_inter_left _ (closure_inter_subset_inter_closure s t)).trans_eq _\n  simp only [inter_union_distrib_left, union_inter_distrib_right, inter_assoc,\n    inter_comm (closure t)]\n#align frontier_inter_subset frontier_inter_subset\n\ntheorem frontier_union_subset (s t : Set X) :\n    frontier (s \u222a t) \u2286 frontier s \u2229 closure t\u1d9c \u222a closure s\u1d9c \u2229 frontier t := by\n  simpa only [frontier_compl, \u2190 compl_union] using frontier_inter_subset s\u1d9c t\u1d9c\n#align frontier_union_subset frontier_union_subset\n\ntheorem IsClosed.frontier_eq (hs : IsClosed s) : frontier s = s \\ interior s := by\n  rw [frontier, hs.closure_eq]\n#align is_closed.frontier_eq IsClosed.frontier_eq\n\ntheorem IsOpen.frontier_eq (hs : IsOpen s) : frontier s = closure s \\ s := by\n  rw [frontier, hs.interior_eq]\n#align is_open.frontier_eq IsOpen.frontier_eq\n\ntheorem IsOpen.inter_frontier_eq (hs : IsOpen s) : s \u2229 frontier s = \u2205 := by\n  rw [hs.frontier_eq, inter_diff_self]\n#align is_open.inter_frontier_eq IsOpen.inter_frontier_eq\n\n/-- The frontier of a set is closed. -/\ntheorem isClosed_frontier : IsClosed (frontier s) := by\n  rw [frontier_eq_closure_inter_closure]; exact IsClosed.inter isClosed_closure isClosed_closure\n#align is_closed_frontier isClosed_frontier\n\n/-- The frontier of a closed set has no interior point. -/\ntheorem interior_frontier (h : IsClosed s) : interior (frontier s) = \u2205 := by\n  have A : frontier s = s \\ interior s := h.frontier_eq\n  have B : interior (frontier s) \u2286 interior s := by rw [A]; exact interior_mono (diff_subset _ _)\n  have C : interior (frontier s) \u2286 frontier s := interior_subset\n  have : interior (frontier s) \u2286 interior s \u2229 (s \\ interior s) :=\n    subset_inter B (by simpa [A] using C)\n  rwa [inter_diff_self, subset_empty_iff] at this\n#align interior_frontier interior_frontier\n\ntheorem closure_eq_interior_union_frontier (s : Set X) : closure s = interior s \u222a frontier s :=\n  (union_diff_cancel interior_subset_closure).symm\n#align closure_eq_interior_union_frontier closure_eq_interior_union_frontier\n\ntheorem closure_eq_self_union_frontier (s : Set X) : closure s = s \u222a frontier s :=\n  (union_diff_cancel' interior_subset subset_closure).symm\n#align closure_eq_self_union_frontier closure_eq_self_union_frontier\n\ntheorem Disjoint.frontier_left (ht : IsOpen t) (hd : Disjoint s t) : Disjoint (frontier s) t :=\n  subset_compl_iff_disjoint_right.1 <|\n    frontier_subset_closure.trans <| closure_minimal (disjoint_left.1 hd) <| isClosed_compl_iff.2 ht\n#align disjoint.frontier_left Disjoint.frontier_left\n\ntheorem Disjoint.frontier_right (hs : IsOpen s) (hd : Disjoint s t) : Disjoint s (frontier t) :=\n  (hd.symm.frontier_left hs).symm\n#align disjoint.frontier_right Disjoint.frontier_right\n\ntheorem frontier_eq_inter_compl_interior :\n    frontier s = (interior s)\u1d9c \u2229 (interior s\u1d9c)\u1d9c := by\n  rw [\u2190 frontier_compl, \u2190 closure_compl, \u2190 diff_eq, closure_diff_interior]\n#align frontier_eq_inter_compl_interior frontier_eq_inter_compl_interior\n\ntheorem compl_frontier_eq_union_interior :\n    (frontier s)\u1d9c = interior s \u222a interior s\u1d9c := by\n  rw [frontier_eq_inter_compl_interior]\n  simp only [compl_inter, compl_compl]\n#align compl_frontier_eq_union_interior compl_frontier_eq_union_interior\n\n/-!\n### Neighborhoods\n-/\n\ntheorem nhds_def' (x : X) : \ud835\udcdd x = \u2a05 (s : Set X) (_ : IsOpen s) (_ : x \u2208 s), \ud835\udcdf s := by\n  simp only [nhds_def, mem_setOf_eq, @and_comm (x \u2208 _), iInf_and]\n#align nhds_def' nhds_def'\n\n/-- The open sets containing `x` are a basis for the neighborhood filter. See `nhds_basis_opens'`\nfor a variant using open neighborhoods instead. -/\ntheorem nhds_basis_opens (x : X) :\n    (\ud835\udcdd x).HasBasis (fun s : Set X => x \u2208 s \u2227 IsOpen s) fun s => s := by\n  rw [nhds_def]\n  exact hasBasis_biInf_principal\n    (fun s \u27e8has, hs\u27e9 t \u27e8hat, ht\u27e9 =>\n      \u27e8s \u2229 t, \u27e8\u27e8has, hat\u27e9, IsOpen.inter hs ht\u27e9, \u27e8inter_subset_left _ _, inter_subset_right _ _\u27e9\u27e9)\n    \u27e8univ, \u27e8mem_univ x, isOpen_univ\u27e9\u27e9\n#align nhds_basis_opens nhds_basis_opens\n\ntheorem nhds_basis_closeds (x : X) : (\ud835\udcdd x).HasBasis (fun s : Set X => x \u2209 s \u2227 IsClosed s) compl :=\n  \u27e8fun t => (nhds_basis_opens x).mem_iff.trans <|\n    compl_surjective.exists.trans <| by simp only [isOpen_compl_iff, mem_compl_iff]\u27e9\n#align nhds_basis_closeds nhds_basis_closeds\n\n@[simp]\ntheorem lift'_nhds_interior (x : X) : (\ud835\udcdd x).lift' interior = \ud835\udcdd x :=\n  (nhds_basis_opens x).lift'_interior_eq_self fun _ \u21a6 And.right\n\ntheorem Filter.HasBasis.nhds_interior {x : X} {p : \u03b9 \u2192 Prop} {s : \u03b9 \u2192 Set X}\n    (h : (\ud835\udcdd x).HasBasis p s) : (\ud835\udcdd x).HasBasis p (interior <| s \u00b7) :=\n  lift'_nhds_interior x \u25b8 h.lift'_interior\n\n/-- A filter lies below the neighborhood filter at `x` iff it contains every open set around `x`. -/\ntheorem le_nhds_iff {f} : f \u2264 \ud835\udcdd x \u2194 \u2200 s : Set X, x \u2208 s \u2192 IsOpen s \u2192 s \u2208 f := by simp [nhds_def]\n#align le_nhds_iff le_nhds_iff\n\n/-- To show a filter is above the neighborhood filter at `x`, it suffices to show that it is above\nthe principal filter of some open set `s` containing `x`. -/\ntheorem nhds_le_of_le {f} (h : x \u2208 s) (o : IsOpen s) (sf : \ud835\udcdf s \u2264 f) : \ud835\udcdd x \u2264 f := by\n  rw [nhds_def]; exact iInf\u2082_le_of_le s \u27e8h, o\u27e9 sf\n#align nhds_le_of_le nhds_le_of_le\n\n-- Porting note: use `\u2203 t, t \u2286 s \u2227 _` instead of `\u2203 t \u2286 s, _`\ntheorem mem_nhds_iff : s \u2208 \ud835\udcdd x \u2194 \u2203 t, t \u2286 s \u2227 IsOpen t \u2227 x \u2208 t :=\n  (nhds_basis_opens x).mem_iff.trans <| exists_congr fun _ =>\n    \u27e8fun h => \u27e8h.2, h.1.2, h.1.1\u27e9, fun h => \u27e8\u27e8h.2.2, h.2.1\u27e9, h.1\u27e9\u27e9\n#align mem_nhds_iff mem_nhds_iff\u2093\n\n/-- A predicate is true in a neighborhood of `x` iff it is true for all the points in an open set\ncontaining `x`. -/\ntheorem eventually_nhds_iff {p : X \u2192 Prop} :\n    (\u2200\u1da0 x in \ud835\udcdd x, p x) \u2194 \u2203 t : Set X, (\u2200 x \u2208 t, p x) \u2227 IsOpen t \u2227 x \u2208 t :=\n  mem_nhds_iff.trans <| by simp only [subset_def, exists_prop, mem_setOf_eq]\n#align eventually_nhds_iff eventually_nhds_iff\n\ntheorem mem_interior_iff_mem_nhds : x \u2208 interior s \u2194 s \u2208 \ud835\udcdd x :=\n  mem_interior.trans mem_nhds_iff.symm\n#align mem_interior_iff_mem_nhds mem_interior_iff_mem_nhds\n\ntheorem map_nhds {f : X \u2192 \u03b1} :\n    map f (\ud835\udcdd x) = \u2a05 s \u2208 { s : Set X | x \u2208 s \u2227 IsOpen s }, \ud835\udcdf (f '' s) :=\n  ((nhds_basis_opens x).map f).eq_biInf\n#align map_nhds map_nhds\n\ntheorem mem_of_mem_nhds : s \u2208 \ud835\udcdd x \u2192 x \u2208 s := fun H =>\n  let \u27e8_t, ht, _, hs\u27e9 := mem_nhds_iff.1 H; ht hs\n#align mem_of_mem_nhds mem_of_mem_nhds\n\n/-- If a predicate is true in a neighborhood of `x`, then it is true for `x`. -/\ntheorem Filter.Eventually.self_of_nhds {p : X \u2192 Prop} (h : \u2200\u1da0 y in \ud835\udcdd x, p y) : p x :=\n  mem_of_mem_nhds h\n#align filter.eventually.self_of_nhds Filter.Eventually.self_of_nhds\n\ntheorem IsOpen.mem_nhds (hs : IsOpen s) (hx : x \u2208 s) : s \u2208 \ud835\udcdd x :=\n  mem_nhds_iff.2 \u27e8s, Subset.refl _, hs, hx\u27e9\n#align is_open.mem_nhds IsOpen.mem_nhds\n\nprotected theorem IsOpen.mem_nhds_iff (hs : IsOpen s) : s \u2208 \ud835\udcdd x \u2194 x \u2208 s :=\n  \u27e8mem_of_mem_nhds, fun hx => mem_nhds_iff.2 \u27e8s, Subset.rfl, hs, hx\u27e9\u27e9\n#align is_open.mem_nhds_iff IsOpen.mem_nhds_iff\n\ntheorem IsClosed.compl_mem_nhds (hs : IsClosed s) (hx : x \u2209 s) : s\u1d9c \u2208 \ud835\udcdd x :=\n  hs.isOpen_compl.mem_nhds (mem_compl hx)\n#align is_closed.compl_mem_nhds IsClosed.compl_mem_nhds\n\ntheorem IsOpen.eventually_mem (hs : IsOpen s) (hx : x \u2208 s) :\n    \u2200\u1da0 x in \ud835\udcdd x, x \u2208 s :=\n  IsOpen.mem_nhds hs hx\n#align is_open.eventually_mem IsOpen.eventually_mem\n\n/-- The open neighborhoods of `x` are a basis for the neighborhood filter. See `nhds_basis_opens`\nfor a variant using open sets around `x` instead. -/\ntheorem nhds_basis_opens' (x : X) :\n    (\ud835\udcdd x).HasBasis (fun s : Set X => s \u2208 \ud835\udcdd x \u2227 IsOpen s) fun x => x := by\n  convert nhds_basis_opens x using 2\n  exact and_congr_left_iff.2 IsOpen.mem_nhds_iff\n#align nhds_basis_opens' nhds_basis_opens'\n\n/-- If `U` is a neighborhood of each point of a set `s` then it is a neighborhood of `s`:\nit contains an open set containing `s`. -/\ntheorem exists_open_set_nhds {U : Set X} (h : \u2200 x \u2208 s, U \u2208 \ud835\udcdd x) :\n    \u2203 V : Set X, s \u2286 V \u2227 IsOpen V \u2227 V \u2286 U :=\n  \u27e8interior U, fun x hx => mem_interior_iff_mem_nhds.2 <| h x hx, isOpen_interior, interior_subset\u27e9\n#align exists_open_set_nhds exists_open_set_nhds\n\n/-- If `U` is a neighborhood of each point of a set `s` then it is a neighborhood of s:\nit contains an open set containing `s`. -/\ntheorem exists_open_set_nhds' {U : Set X} (h : U \u2208 \u2a06 x \u2208 s, \ud835\udcdd x) :\n    \u2203 V : Set X, s \u2286 V \u2227 IsOpen V \u2227 V \u2286 U :=\n  exists_open_set_nhds (by simpa using h)\n#align exists_open_set_nhds' exists_open_set_nhds'\n\n/-- If a predicate is true in a neighbourhood of `x`, then for `y` sufficiently close\nto `x` this predicate is true in a neighbourhood of `y`. -/\ntheorem Filter.Eventually.eventually_nhds {p : X \u2192 Prop} (h : \u2200\u1da0 y in \ud835\udcdd x, p y) :\n    \u2200\u1da0 y in \ud835\udcdd x, \u2200\u1da0 x in \ud835\udcdd y, p x :=\n  let \u27e8t, htp, hto, ha\u27e9 := eventually_nhds_iff.1 h\n  eventually_nhds_iff.2 \u27e8t, fun _x hx => eventually_nhds_iff.2 \u27e8t, htp, hto, hx\u27e9, hto, ha\u27e9\n#align filter.eventually.eventually_nhds Filter.Eventually.eventually_nhds\n\n@[simp]\ntheorem eventually_eventually_nhds {p : X \u2192 Prop} :\n    (\u2200\u1da0 y in \ud835\udcdd x, \u2200\u1da0 x in \ud835\udcdd y, p x) \u2194 \u2200\u1da0 x in \ud835\udcdd x, p x :=\n  \u27e8fun h => h.self_of_nhds, fun h => h.eventually_nhds\u27e9\n#align eventually_eventually_nhds eventually_eventually_nhds\n\n@[simp]\ntheorem frequently_frequently_nhds {p : X \u2192 Prop} :\n    (\u2203\u1da0 x' in \ud835\udcdd x, \u2203\u1da0 x'' in \ud835\udcdd x', p x'') \u2194 \u2203\u1da0 x in \ud835\udcdd x, p x := by\n  rw [\u2190 not_iff_not]\n  simp only [not_frequently, eventually_eventually_nhds]\n#align frequently_frequently_nhds frequently_frequently_nhds\n\n@[simp]\ntheorem eventually_mem_nhds : (\u2200\u1da0 x' in \ud835\udcdd x, s \u2208 \ud835\udcdd x') \u2194 s \u2208 \ud835\udcdd x :=\n  eventually_eventually_nhds\n#align eventually_mem_nhds eventually_mem_nhds\n\n@[simp]\ntheorem nhds_bind_nhds : (\ud835\udcdd x).bind \ud835\udcdd = \ud835\udcdd x :=\n  Filter.ext fun _ => eventually_eventually_nhds\n#align nhds_bind_nhds nhds_bind_nhds\n\n@[simp]\ntheorem eventually_eventuallyEq_nhds {f g : X \u2192 \u03b1} :\n    (\u2200\u1da0 y in \ud835\udcdd x, f =\u1da0[\ud835\udcdd y] g) \u2194 f =\u1da0[\ud835\udcdd x] g :=\n  eventually_eventually_nhds\n#align eventually_eventually_eq_nhds eventually_eventuallyEq_nhds\n\ntheorem Filter.EventuallyEq.eq_of_nhds {f g : X \u2192 \u03b1} (h : f =\u1da0[\ud835\udcdd x] g) : f x = g x :=\n  h.self_of_nhds\n#align filter.eventually_eq.eq_of_nhds Filter.EventuallyEq.eq_of_nhds\n\n@[simp]\ntheorem eventually_eventuallyLE_nhds [LE \u03b1] {f g : X \u2192 \u03b1} :\n    (\u2200\u1da0 y in \ud835\udcdd x, f \u2264\u1da0[\ud835\udcdd y] g) \u2194 f \u2264\u1da0[\ud835\udcdd x] g :=\n  eventually_eventually_nhds\n#align eventually_eventually_le_nhds eventually_eventuallyLE_nhds\n\n/-- If two functions are equal in a neighbourhood of `x`, then for `y` sufficiently close\nto `x` these functions are equal in a neighbourhood of `y`. -/\ntheorem Filter.EventuallyEq.eventuallyEq_nhds {f g : X \u2192 \u03b1} (h : f =\u1da0[\ud835\udcdd x] g) :\n    \u2200\u1da0 y in \ud835\udcdd x, f =\u1da0[\ud835\udcdd y] g :=\n  h.eventually_nhds\n#align filter.eventually_eq.eventually_eq_nhds Filter.EventuallyEq.eventuallyEq_nhds\n\n/-- If `f x \u2264 g x` in a neighbourhood of `x`, then for `y` sufficiently close to `x` we have\n`f x \u2264 g x` in a neighbourhood of `y`. -/\ntheorem Filter.EventuallyLE.eventuallyLE_nhds [LE \u03b1] {f g : X \u2192 \u03b1} (h : f \u2264\u1da0[\ud835\udcdd x] g) :\n    \u2200\u1da0 y in \ud835\udcdd x, f \u2264\u1da0[\ud835\udcdd y] g :=\n  h.eventually_nhds\n#align filter.eventually_le.eventually_le_nhds Filter.EventuallyLE.eventuallyLE_nhds\n\ntheorem all_mem_nhds (x : X) (P : Set X \u2192 Prop) (hP : \u2200 s t, s \u2286 t \u2192 P s \u2192 P t) :\n    (\u2200 s \u2208 \ud835\udcdd x, P s) \u2194 \u2200 s, IsOpen s \u2192 x \u2208 s \u2192 P s :=\n  ((nhds_basis_opens x).forall_iff hP).trans <| by simp only [@and_comm (x \u2208 _), and_imp]\n#align all_mem_nhds all_mem_nhds\n\ntheorem all_mem_nhds_filter (x : X) (f : Set X \u2192 Set \u03b1) (hf : \u2200 s t, s \u2286 t \u2192 f s \u2286 f t)\n    (l : Filter \u03b1) : (\u2200 s \u2208 \ud835\udcdd x, f s \u2208 l) \u2194 \u2200 s, IsOpen s \u2192 x \u2208 s \u2192 f s \u2208 l :=\n  all_mem_nhds _ _ fun s t ssubt h => mem_of_superset h (hf s t ssubt)\n#align all_mem_nhds_filter all_mem_nhds_filter\n\ntheorem tendsto_nhds {f : \u03b1 \u2192 X} {l : Filter \u03b1} :\n    Tendsto f l (\ud835\udcdd x) \u2194 \u2200 s, IsOpen s \u2192 x \u2208 s \u2192 f \u207b\u00b9' s \u2208 l :=\n  all_mem_nhds_filter _ _ (fun _ _ h => preimage_mono h) _\n#align tendsto_nhds tendsto_nhds\n\ntheorem tendsto_atTop_nhds [Nonempty \u03b1] [SemilatticeSup \u03b1] {f : \u03b1 \u2192 X} :\n    Tendsto f atTop (\ud835\udcdd x) \u2194 \u2200 U : Set X, x \u2208 U \u2192 IsOpen U \u2192 \u2203 N, \u2200 n, N \u2264 n \u2192 f n \u2208 U :=\n  (atTop_basis.tendsto_iff (nhds_basis_opens x)).trans <| by\n    simp only [and_imp, exists_prop, true_and_iff, mem_Ici, ge_iff_le]\n#align tendsto_at_top_nhds tendsto_atTop_nhds\n\ntheorem tendsto_const_nhds {f : Filter \u03b1} : Tendsto (fun _ : \u03b1 => x) f (\ud835\udcdd x) :=\n  tendsto_nhds.mpr fun _ _ ha => univ_mem' fun _ => ha\n#align tendsto_const_nhds tendsto_const_nhds\n\ntheorem tendsto_atTop_of_eventually_const {\u03b9 : Type*} [SemilatticeSup \u03b9] [Nonempty \u03b9]\n    {u : \u03b9 \u2192 X} {i\u2080 : \u03b9} (h : \u2200 i \u2265 i\u2080, u i = x) : Tendsto u atTop (\ud835\udcdd x) :=\n  Tendsto.congr' (EventuallyEq.symm (eventually_atTop.mpr \u27e8i\u2080, h\u27e9)) tendsto_const_nhds\n#align tendsto_at_top_of_eventually_const tendsto_atTop_of_eventually_const\n\ntheorem tendsto_atBot_of_eventually_const {\u03b9 : Type*} [SemilatticeInf \u03b9] [Nonempty \u03b9]\n    {u : \u03b9 \u2192 X} {i\u2080 : \u03b9} (h : \u2200 i \u2264 i\u2080, u i = x) : Tendsto u atBot (\ud835\udcdd x) :=\n  Tendsto.congr' (EventuallyEq.symm (eventually_atBot.mpr \u27e8i\u2080, h\u27e9)) tendsto_const_nhds\n#align tendsto_at_bot_of_eventually_const tendsto_atBot_of_eventually_const\n\ntheorem pure_le_nhds : pure \u2264 (\ud835\udcdd : X \u2192 Filter X) := fun _ _ hs => mem_pure.2 <| mem_of_mem_nhds hs\n#align pure_le_nhds pure_le_nhds\n\ntheorem tendsto_pure_nhds (f : \u03b1 \u2192 X) (a : \u03b1) : Tendsto f (pure a) (\ud835\udcdd (f a)) :=\n  (tendsto_pure_pure f a).mono_right (pure_le_nhds _)\n#align tendsto_pure_nhds tendsto_pure_nhds\n\ntheorem OrderTop.tendsto_atTop_nhds [PartialOrder \u03b1] [OrderTop \u03b1] (f : \u03b1 \u2192 X) :\n    Tendsto f atTop (\ud835\udcdd (f \u22a4)) :=\n  (tendsto_atTop_pure f).mono_right (pure_le_nhds _)\n#align order_top.tendsto_at_top_nhds OrderTop.tendsto_atTop_nhds\n\n@[simp]\ninstance nhds_neBot : NeBot (\ud835\udcdd x) :=\n  neBot_of_le (pure_le_nhds x)\n#align nhds_ne_bot nhds_neBot\n\ntheorem tendsto_nhds_of_eventually_eq {l : Filter \u03b1} {f : \u03b1 \u2192 X} (h : \u2200\u1da0 x' in l, f x' = x) :\n    Tendsto f l (\ud835\udcdd x) :=\n  tendsto_const_nhds.congr' (.symm h)\n\ntheorem Filter.EventuallyEq.tendsto {l : Filter \u03b1} {f : \u03b1 \u2192 X} (hf : f =\u1da0[l] fun _ \u21a6 x) :\n    Tendsto f l (\ud835\udcdd x) :=\n  tendsto_nhds_of_eventually_eq hf\n\n/-!\n### Cluster points\n\nIn this section we define [cluster points](https://en.wikipedia.org/wiki/Limit_point)\n(also known as limit points and accumulation points) of a filter and of a sequence.\n-/\n\n\ntheorem ClusterPt.neBot {F : Filter X} (h : ClusterPt x F) : NeBot (\ud835\udcdd x \u2293 F) :=\n  h\n#align cluster_pt.ne_bot ClusterPt.neBot\n\ntheorem Filter.HasBasis.clusterPt_iff {\u03b9X \u03b9F} {pX : \u03b9X \u2192 Prop} {sX : \u03b9X \u2192 Set X} {pF : \u03b9F \u2192 Prop}\n    {sF : \u03b9F \u2192 Set X} {F : Filter X} (hX : (\ud835\udcdd x).HasBasis pX sX) (hF : F.HasBasis pF sF) :\n    ClusterPt x F \u2194 \u2200 \u2983i\u2984, pX i \u2192 \u2200 \u2983j\u2984, pF j \u2192 (sX i \u2229 sF j).Nonempty :=\n  hX.inf_basis_neBot_iff hF\n#align filter.has_basis.cluster_pt_iff Filter.HasBasis.clusterPt_iff\n\ntheorem clusterPt_iff {F : Filter X} :\n    ClusterPt x F \u2194 \u2200 \u2983U : Set X\u2984, U \u2208 \ud835\udcdd x \u2192 \u2200 \u2983V\u2984, V \u2208 F \u2192 (U \u2229 V).Nonempty :=\n  inf_neBot_iff\n#align cluster_pt_iff clusterPt_iff\n\ntheorem clusterPt_iff_not_disjoint {F : Filter X} :\n    ClusterPt x F \u2194 \u00acDisjoint (\ud835\udcdd x) F := by\n  rw [disjoint_iff, ClusterPt, neBot_iff]\n\n/-- `x` is a cluster point of a set `s` if every neighbourhood of `x` meets `s` on a nonempty\nset. See also `mem_closure_iff_clusterPt`. -/\ntheorem clusterPt_principal_iff :\n    ClusterPt x (\ud835\udcdf s) \u2194 \u2200 U \u2208 \ud835\udcdd x, (U \u2229 s).Nonempty :=\n  inf_principal_neBot_iff\n#align cluster_pt_principal_iff clusterPt_principal_iff\n\ntheorem clusterPt_principal_iff_frequently :\n    ClusterPt x (\ud835\udcdf s) \u2194 \u2203\u1da0 y in \ud835\udcdd x, y \u2208 s := by\n  simp only [clusterPt_principal_iff, frequently_iff, Set.Nonempty, exists_prop, mem_inter_iff]\n#align cluster_pt_principal_iff_frequently clusterPt_principal_iff_frequently\n\ntheorem ClusterPt.of_le_nhds {f : Filter X} (H : f \u2264 \ud835\udcdd x) [NeBot f] : ClusterPt x f := by\n  rwa [ClusterPt, inf_eq_right.mpr H]\n#align cluster_pt.of_le_nhds ClusterPt.of_le_nhds\n\ntheorem ClusterPt.of_le_nhds' {f : Filter X} (H : f \u2264 \ud835\udcdd x) (_hf : NeBot f) :\n    ClusterPt x f :=\n  ClusterPt.of_le_nhds H\n#align cluster_pt.of_le_nhds' ClusterPt.of_le_nhds'\n\ntheorem ClusterPt.of_nhds_le {f : Filter X} (H : \ud835\udcdd x \u2264 f) : ClusterPt x f := by\n  simp only [ClusterPt, inf_eq_left.mpr H, nhds_neBot]\n#align cluster_pt.of_nhds_le ClusterPt.of_nhds_le\n\ntheorem ClusterPt.mono {f g : Filter X} (H : ClusterPt x f) (h : f \u2264 g) : ClusterPt x g :=\n  NeBot.mono H <| inf_le_inf_left _ h\n#align cluster_pt.mono ClusterPt.mono\n\ntheorem ClusterPt.of_inf_left {f g : Filter X} (H : ClusterPt x <| f \u2293 g) : ClusterPt x f :=\n  H.mono inf_le_left\n#align cluster_pt.of_inf_left ClusterPt.of_inf_left\n\ntheorem ClusterPt.of_inf_right {f g : Filter X} (H : ClusterPt x <| f \u2293 g) :\n    ClusterPt x g :=\n  H.mono inf_le_right\n#align cluster_pt.of_inf_right ClusterPt.of_inf_right\n\ntheorem Ultrafilter.clusterPt_iff {f : Ultrafilter X} : ClusterPt x f \u2194 \u2191f \u2264 \ud835\udcdd x :=\n  \u27e8f.le_of_inf_neBot', fun h => ClusterPt.of_le_nhds h\u27e9\n#align ultrafilter.cluster_pt_iff Ultrafilter.clusterPt_iff\n\ntheorem clusterPt_iff_ultrafilter {f : Filter X} : ClusterPt x f \u2194\n    \u2203 u : Ultrafilter X, u \u2264 f \u2227 u \u2264 \ud835\udcdd x := by\n  simp_rw [ClusterPt, \u2190 le_inf_iff, exists_ultrafilter_iff, inf_comm]\n\ntheorem mapClusterPt_def {\u03b9 : Type*} (x : X) (F : Filter \u03b9) (u : \u03b9 \u2192 X) :\n    MapClusterPt x F u \u2194 ClusterPt x (map u F) := Iff.rfl\n\ntheorem mapClusterPt_iff {\u03b9 : Type*} (x : X) (F : Filter \u03b9) (u : \u03b9 \u2192 X) :\n    MapClusterPt x F u \u2194 \u2200 s \u2208 \ud835\udcdd x, \u2203\u1da0 a in F, u a \u2208 s := by\n  simp_rw [MapClusterPt, ClusterPt, inf_neBot_iff_frequently_left, frequently_map]\n  rfl\n#align map_cluster_pt_iff mapClusterPt_iff\n\ntheorem mapClusterPt_iff_ultrafilter {\u03b9 : Type*} (x : X) (F : Filter \u03b9) (u : \u03b9 \u2192 X) :\n    MapClusterPt x F u \u2194 \u2203 U : Ultrafilter \u03b9, U \u2264 F \u2227 Tendsto u U (\ud835\udcdd x) := by\n  simp_rw [MapClusterPt, ClusterPt, \u2190 Filter.push_pull', map_neBot_iff, tendsto_iff_comap,\n    \u2190 le_inf_iff, exists_ultrafilter_iff, inf_comm]\n\n", "theoremStatement": "theorem mapClusterPt_comp {X \u03b1 \u03b2 : Type*} {x : X} [TopologicalSpace X] {F : Filter \u03b1} {\u03c6 : \u03b1 \u2192 \u03b2}\n    {u : \u03b2 \u2192 X} : MapClusterPt x F (u \u2218 \u03c6) \u2194 MapClusterPt x (map \u03c6 F) u ", "theoremName": "mapClusterPt_comp", "fileCreated": {"commit": "e8f582b40d8a1896a3f518438fbe4aa3c791b705", "date": "2023-01-26"}, "theoremCreated": {"commit": "345188e4a9f9e68552737b8e2ea2816eb54ddb6a", "date": "2024-03-30"}, "file": "mathlib/Mathlib/Topology/Basic.lean", "module": "Mathlib.Topology.Basic", "jsonFile": "Mathlib.Topology.Basic.jsonl", "positionMetadata": {"lineInFile": 1102, "tokenPositionInFile": 46207, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 7, "importedModules": 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"Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", 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+{"srcContext": "/-\nCopyright (c) 2021 Arthur Paulino. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Arthur Paulino, Kyle Miller\n-/\nimport Mathlib.Combinatorics.SimpleGraph.Clique\nimport Mathlib.Data.ENat.Lattice\nimport Mathlib.Data.Nat.Lattice\nimport Mathlib.Data.Setoid.Partition\nimport Mathlib.Order.Antichain\n\n#align_import combinatorics.simple_graph.coloring from \"leanprover-community/mathlib\"@\"70fd9563a21e7b963887c9360bd29b2393e6225a\"\n\n/-!\n# Graph Coloring\n\nThis module defines colorings of simple graphs (also known as proper\ncolorings in the literature). A graph coloring is the attribution of\n\"colors\" to all of its vertices such that adjacent vertices have\ndifferent colors. A coloring can be represented as a homomorphism into\na complete graph, whose vertices represent the colors.\n\n## Main definitions\n\n* `G.Coloring \u03b1` is the type of `\u03b1`-colorings of a simple graph `G`,\n  with `\u03b1` being the set of available colors. The type is defined to\n  be homomorphisms from `G` into the complete graph on `\u03b1`, and\n  colorings have a coercion to `V \u2192 \u03b1`.\n\n* `G.Colorable n` is the proposition that `G` is `n`-colorable, which\n  is whether there exists a coloring with at most *n* colors.\n\n* `G.chromaticNumber` is the minimal `n` such that `G` is\n  `n`-colorable, or `\u22a4` if it cannot be colored with finitely many\n  colors.\n  (Cardinal-valued chromatic numbers are more niche, so we stick to `\u2115\u221e`.)\n  We write `G.chromaticNumber \u2260 \u22a4` to mean a graph is colorable with finitely many colors.\n\n* `C.colorClass c` is the set of vertices colored by `c : \u03b1` in the\n  coloring `C : G.Coloring \u03b1`.\n\n* `C.colorClasses` is the set containing all color classes.\n\n## Todo:\n\n  * Gather material from:\n    * https://github.com/leanprover-community/mathlib/blob/simple_graph_matching/src/combinatorics/simple_graph/coloring.lean\n    * https://github.com/kmill/lean-graphcoloring/blob/master/src/graph.lean\n\n  * Trees\n\n  * Planar graphs\n\n  * Chromatic polynomials\n\n  * develop API for partial colorings, likely as colorings of subgraphs (`H.coe.Coloring \u03b1`)\n-/\n\nopen Fintype Function\n\nuniverse u v\n\nnamespace SimpleGraph\n\nvariable {V : Type u} (G : SimpleGraph V) {n : \u2115}\n/-- An `\u03b1`-coloring of a simple graph `G` is a homomorphism of `G` into the complete graph on `\u03b1`.\nThis is also known as a proper coloring.\n-/\nabbrev Coloring (\u03b1 : Type v) := G \u2192g (\u22a4 : SimpleGraph \u03b1)\n#align simple_graph.coloring SimpleGraph.Coloring\n\nvariable {G} {\u03b1 \u03b2 : Type*} (C : G.Coloring \u03b1)\n\ntheorem Coloring.valid {v w : V} (h : G.Adj v w) : C v \u2260 C w :=\n  C.map_rel h\n#align simple_graph.coloring.valid SimpleGraph.Coloring.valid\n\n/-- Construct a term of `SimpleGraph.Coloring` using a function that\nassigns vertices to colors and a proof that it is as proper coloring.\n\n(Note: this is a definitionally the constructor for `SimpleGraph.Hom`,\nbut with a syntactically better proper coloring hypothesis.)\n-/\n@[match_pattern]\ndef Coloring.mk (color : V \u2192 \u03b1) (valid : \u2200 {v w : V}, G.Adj v w \u2192 color v \u2260 color w) :\n    G.Coloring \u03b1 :=\n  \u27e8color, @valid\u27e9\n#align simple_graph.coloring.mk SimpleGraph.Coloring.mk\n\n/-- The color class of a given color.\n-/\ndef Coloring.colorClass (c : \u03b1) : Set V := { v : V | C v = c }\n#align simple_graph.coloring.color_class SimpleGraph.Coloring.colorClass\n\n/-- The set containing all color classes. -/\ndef Coloring.colorClasses : Set (Set V) := (Setoid.ker C).classes\n#align simple_graph.coloring.color_classes SimpleGraph.Coloring.colorClasses\n\ntheorem Coloring.mem_colorClass (v : V) : v \u2208 C.colorClass (C v) := rfl\n#align simple_graph.coloring.mem_color_class SimpleGraph.Coloring.mem_colorClass\n\ntheorem Coloring.colorClasses_isPartition : Setoid.IsPartition C.colorClasses :=\n  Setoid.isPartition_classes (Setoid.ker C)\n#align simple_graph.coloring.color_classes_is_partition SimpleGraph.Coloring.colorClasses_isPartition\n\ntheorem Coloring.mem_colorClasses {v : V} : C.colorClass (C v) \u2208 C.colorClasses :=\n  \u27e8v, rfl\u27e9\n#align simple_graph.coloring.mem_color_classes SimpleGraph.Coloring.mem_colorClasses\n\ntheorem Coloring.colorClasses_finite [Finite \u03b1] : C.colorClasses.Finite :=\n  Setoid.finite_classes_ker _\n#align simple_graph.coloring.color_classes_finite SimpleGraph.Coloring.colorClasses_finite\n\ntheorem Coloring.card_colorClasses_le [Fintype \u03b1] [Fintype C.colorClasses] :\n    Fintype.card C.colorClasses \u2264 Fintype.card \u03b1 := by\n  simp [colorClasses]\n  -- Porting note: brute force instance declaration `[Fintype (Setoid.classes (Setoid.ker C))]`\n  haveI : Fintype (Setoid.classes (Setoid.ker C)) := by assumption\n  convert Setoid.card_classes_ker_le C\n#align simple_graph.coloring.card_color_classes_le SimpleGraph.Coloring.card_colorClasses_le\n\ntheorem Coloring.not_adj_of_mem_colorClass {c : \u03b1} {v w : V} (hv : v \u2208 C.colorClass c)\n    (hw : w \u2208 C.colorClass c) : \u00acG.Adj v w := fun h => C.valid h (Eq.trans hv (Eq.symm hw))\n#align simple_graph.coloring.not_adj_of_mem_color_class SimpleGraph.Coloring.not_adj_of_mem_colorClass\n\ntheorem Coloring.color_classes_independent (c : \u03b1) : IsAntichain G.Adj (C.colorClass c) :=\n  fun _ hv _ hw _ => C.not_adj_of_mem_colorClass hv hw\n#align simple_graph.coloring.color_classes_independent SimpleGraph.Coloring.color_classes_independent\n\n-- TODO make this computable\nnoncomputable instance [Fintype V] [Fintype \u03b1] : Fintype (Coloring G \u03b1) := by\n  classical\n  change Fintype (RelHom G.Adj (\u22a4 : SimpleGraph \u03b1).Adj)\n  apply Fintype.ofInjective _ RelHom.coe_fn_injective\n\nvariable (G)\n\n/-- Whether a graph can be colored by at most `n` colors. -/\ndef Colorable (n : \u2115) : Prop := Nonempty (G.Coloring (Fin n))\n#align simple_graph.colorable SimpleGraph.Colorable\n\n/-- The coloring of an empty graph. -/\ndef coloringOfIsEmpty [IsEmpty V] : G.Coloring \u03b1 :=\n  Coloring.mk isEmptyElim fun {v} => isEmptyElim v\n#align simple_graph.coloring_of_is_empty SimpleGraph.coloringOfIsEmpty\n\ntheorem colorable_of_isEmpty [IsEmpty V] (n : \u2115) : G.Colorable n :=\n  \u27e8G.coloringOfIsEmpty\u27e9\n#align simple_graph.colorable_of_is_empty SimpleGraph.colorable_of_isEmpty\n\ntheorem isEmpty_of_colorable_zero (h : G.Colorable 0) : IsEmpty V := by\n  constructor\n  intro v\n  obtain \u27e8i, hi\u27e9 := h.some v\n  exact Nat.not_lt_zero _ hi\n#align simple_graph.is_empty_of_colorable_zero SimpleGraph.isEmpty_of_colorable_zero\n\n/-- The \"tautological\" coloring of a graph, using the vertices of the graph as colors. -/\ndef selfColoring : G.Coloring V := Coloring.mk id fun {_ _} => G.ne_of_adj\n#align simple_graph.self_coloring SimpleGraph.selfColoring\n\n/-- The chromatic number of a graph is the minimal number of colors needed to color it.\nThis is `\u22a4` (infinity) iff `G` isn't colorable with finitely many colors.\n\nIf `G` is colorable, then `ENat.toNat G.chromaticNumber` is the `\u2115`-valued chromatic number. -/\nnoncomputable def chromaticNumber : \u2115\u221e := \u2a05 n \u2208 setOf G.Colorable, (n : \u2115\u221e)\n#align simple_graph.chromatic_number SimpleGraph.chromaticNumber\n\nlemma chromaticNumber_eq_biInf {G : SimpleGraph V} :\n    G.chromaticNumber = \u2a05 n \u2208 setOf G.Colorable, (n : \u2115\u221e) := rfl\n\nlemma chromaticNumber_eq_iInf {G : SimpleGraph V} :\n    G.chromaticNumber = \u2a05 n : {m | G.Colorable m}, (n : \u2115\u221e) := by\n  rw [chromaticNumber, iInf_subtype]\n\nlemma Colorable.chromaticNumber_eq_sInf {G : SimpleGraph V} {n} (h : G.Colorable n) :\n    G.chromaticNumber = sInf {n' : \u2115 | G.Colorable n'} := by\n  rw [ENat.coe_sInf, chromaticNumber]\n  exact \u27e8_, h\u27e9\n\n/-- Given an embedding, there is an induced embedding of colorings. -/\ndef recolorOfEmbedding {\u03b1 \u03b2 : Type*} (f : \u03b1 \u21aa \u03b2) : G.Coloring \u03b1 \u21aa G.Coloring \u03b2 where\n  toFun C := (Embedding.completeGraph f).toHom.comp C\n  inj' := by -- this was strangely painful; seems like missing lemmas about embeddings\n    intro C C' h\n    dsimp only at h\n    ext v\n    apply (Embedding.completeGraph f).inj'\n    change ((Embedding.completeGraph f).toHom.comp C) v = _\n    rw [h]\n    rfl\n#align simple_graph.recolor_of_embedding SimpleGraph.recolorOfEmbedding\n\n@[simp] lemma coe_recolorOfEmbedding (f : \u03b1 \u21aa \u03b2) :\n    \u21d1(G.recolorOfEmbedding f) = (Embedding.completeGraph f).toHom.comp := rfl\n\n/-- Given an equivalence, there is an induced equivalence between colorings. -/\ndef recolorOfEquiv {\u03b1 \u03b2 : Type*} (f : \u03b1 \u2243 \u03b2) : G.Coloring \u03b1 \u2243 G.Coloring \u03b2 where\n  toFun := G.recolorOfEmbedding f.toEmbedding\n  invFun := G.recolorOfEmbedding f.symm.toEmbedding\n  left_inv C := by\n    ext v\n    apply Equiv.symm_apply_apply\n  right_inv C := by\n    ext v\n    apply Equiv.apply_symm_apply\n#align simple_graph.recolor_of_equiv SimpleGraph.recolorOfEquiv\n\n@[simp] lemma coe_recolorOfEquiv (f : \u03b1 \u2243 \u03b2) :\n    \u21d1(G.recolorOfEquiv f) = (Embedding.completeGraph f).toHom.comp := rfl\n\n/-- There is a noncomputable embedding of `\u03b1`-colorings to `\u03b2`-colorings if\n`\u03b2` has at least as large a cardinality as `\u03b1`. -/\nnoncomputable def recolorOfCardLE {\u03b1 \u03b2 : Type*} [Fintype \u03b1] [Fintype \u03b2]\n    (hn : Fintype.card \u03b1 \u2264 Fintype.card \u03b2) : G.Coloring \u03b1 \u21aa G.Coloring \u03b2 :=\n  G.recolorOfEmbedding <| (Function.Embedding.nonempty_of_card_le hn).some\n#align simple_graph.recolor_of_card_le SimpleGraph.recolorOfCardLE\n\n", "theoremStatement": "@[simp] lemma coe_recolorOfCardLE [Fintype \u03b1] [Fintype \u03b2] 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"Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Data.Subtype", "Mathlib.Order.Notation", "Mathlib.Data.Set.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Cases", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Basic", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Tactic.PushNeg", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Tactic.CasesM", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.ByContra", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Data.Set.Prod", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.MinMax", "Mathlib.Tactic.Says", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Util.AssertExists", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Logic.Pairwise", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Data.Rel", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Nat.Defs", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Data.Sym.Sym2", "Mathlib.Combinatorics.SimpleGraph.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Sym.Card", "Mathlib.Combinatorics.SimpleGraph.Finite", "Mathlib.Combinatorics.SimpleGraph.Dart", "Mathlib.Data.FunLike.Fintype", "Mathlib.Combinatorics.SimpleGraph.Maps", "Mathlib.Combinatorics.SimpleGraph.Subgraph", "Mathlib.Combinatorics.SimpleGraph.Connectivity", "Mathlib.Combinatorics.SimpleGraph.Operations", "Mathlib.Data.Finset.Pairwise", "Mathlib.Combinatorics.SimpleGraph.Clique", 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"Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Order.SupIndep", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.Partition.Finpartition", "Mathlib.Data.Setoid.Partition"]}, "proofMetadata": {"hasProof": true, "proof": ":= rfl", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 6}}
+{"srcContext": "/-\nCopyright (c) 2024 David Loeffler. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: David Loeffler\n-/\n\nimport Mathlib.Data.Complex.ExponentialBounds\nimport Mathlib.NumberTheory.Harmonic.Defs\nimport Mathlib.Analysis.Normed.Order.Lattice\nimport Mathlib.Analysis.SpecialFunctions.Pow.Real\n\n/-!\n# The Euler-Mascheroni constant `\u03b3`\n\nWe define the constant `\u03b3`, and give upper and lower bounds for it.\n\n##\u00a0Main definitions and results\n\n* `Real.eulerMascheroniConstant`: the constant `\u03b3`\n* `Real.tendsto_harmonic_sub_log`: the sequence `n \u21a6 harmonic n - log n` tends to `\u03b3` as `n \u2192 \u221e`\n* `one_half_lt_eulerMascheroniConstant` and `eulerMascheroniConstant_lt_two_thirds`: upper and\n  lower bounds.\n\n## Outline of proofs\n\nWe show that\n\n* the sequence `eulerMascheroniSeq` given by `n \u21a6 harmonic n - log (n + 1)` is strictly increasing;\n* the sequence `eulerMascheroniSeq'` given by `n \u21a6 harmonic n - log n`, modified with a junk value\n  for `n = 0`, is strictly decreasing;\n* the difference `eulerMascheroniSeq' n - eulerMascheroniSeq n` is non-negative and tends to 0.\n\nIt follows that both sequences tend to a common limit `\u03b3`, and we have the inequality\n`eulerMascheroniSeq n < \u03b3 < eulerMascheroniSeq' n` for all `n`. Taking `n = 6` gives the bounds\n`1 / 2 < \u03b3 < 2 / 3`.\n-/\n\nopen Filter Topology\n\nnamespace Real\n\nsection LowerSequence\n\n/-- The sequence with `n`-th term `harmonic n - log (n + 1)`. -/\nnoncomputable def eulerMascheroniSeq (n : \u2115) : \u211d := harmonic n - log (n + 1)\n\nlemma eulerMascheroniSeq_zero : eulerMascheroniSeq 0 = 0 := by\n  simp [eulerMascheroniSeq, harmonic_zero]\n\nlemma strictMono_eulerMascheroniSeq : StrictMono eulerMascheroniSeq := by\n  refine strictMono_nat_of_lt_succ (fun n \u21a6 ?_)\n  rw [eulerMascheroniSeq, eulerMascheroniSeq, \u2190 sub_pos, sub_sub_sub_comm,\n    harmonic_succ, add_comm, Rat.cast_add, add_sub_cancel_right,\n    \u2190 log_div (by positivity) (by positivity), add_div, Nat.cast_add_one,\n    Nat.cast_add_one, div_self (by positivity), sub_pos, one_div, Rat.cast_inv, Rat.cast_add,\n    Rat.cast_one, Rat.cast_natCast]\n  refine (log_lt_sub_one_of_pos ?_ (ne_of_gt <| lt_add_of_pos_right _ ?_)).trans_le (le_of_eq ?_)\n  \u00b7 positivity\n  \u00b7 positivity\n  \u00b7 simp only [add_sub_cancel_left]\n\nlemma one_half_lt_eulerMascheroniSeq_six : 1 / 2 < eulerMascheroniSeq 6 := by\n  have : eulerMascheroniSeq 6 = 49 / 20 - log 7 := by\n    rw [eulerMascheroniSeq]\n    norm_num\n  rw [this, lt_sub_iff_add_lt, \u2190 lt_sub_iff_add_lt', log_lt_iff_lt_exp (by positivity)]\n  refine lt_of_lt_of_le ?_ (Real.sum_le_exp_of_nonneg (by norm_num) 7)\n  simp_rw [Finset.sum_range_succ, Nat.factorial_succ]\n  norm_num\n\nend LowerSequence\n\nsection UpperSequence\n\n/-- The sequence with `n`-th term `harmonic n - log n`. We use a junk value for `n = 0`, in order\nto have the sequence be strictly decreasing. -/\nnoncomputable def eulerMascheroniSeq' (n : \u2115) : \u211d :=\n  if n = 0 then 2 else \u2191(harmonic n) - log n\n\nlemma eulerMascheroniSeq'_one : eulerMascheroniSeq' 1 = 1 := by\n  simp [eulerMascheroniSeq']\n\nlemma strictAnti_eulerMascheroniSeq' : StrictAnti eulerMascheroniSeq' := by\n  refine strictAnti_nat_of_succ_lt (fun n \u21a6 ?_)\n  rcases Nat.eq_zero_or_pos n with rfl | hn\n  \u00b7 simp [eulerMascheroniSeq']\n  simp_rw [eulerMascheroniSeq', (by simp : (n + 1 = 0) = False), eq_false_intro hn.ne', if_false]\n  rw [\u2190 sub_pos, sub_sub_sub_comm,\n    harmonic_succ, Rat.cast_add, \u2190 sub_sub, sub_self, zero_sub, sub_eq_add_neg, neg_sub,\n    \u2190 sub_eq_neg_add, sub_pos, \u2190 log_div (by positivity) (by positivity), \u2190 neg_lt_neg_iff,\n    \u2190 log_inv]\n  refine (log_lt_sub_one_of_pos ?_ ?_).trans_le (le_of_eq ?_)\n  \u00b7 positivity\n  \u00b7 field_simp\n  \u00b7 field_simp\n\nlemma eulerMascheroniSeq'_six_lt_two_thirds : eulerMascheroniSeq' 6 < 2 / 3 := by\n  have h1 : eulerMascheroniSeq' 6 = 49 / 20 - log 6 := by\n    rw [eulerMascheroniSeq']\n    norm_num\n  rw [h1, sub_lt_iff_lt_add, \u2190 sub_lt_iff_lt_add', lt_log_iff_exp_lt (by positivity)]\n  norm_num\n  have := rpow_lt_rpow (exp_pos _).le exp_one_lt_d9 (by norm_num : (0 : \u211d) < 107 / 60)\n  rw [exp_one_rpow] at this\n  refine lt_trans this ?_\n  rw [\u2190 rpow_lt_rpow_iff (z := 60), \u2190 rpow_mul, div_mul_cancel\u2080, \u2190 Nat.cast_ofNat,\n    \u2190 Nat.cast_ofNat, rpow_natCast, Nat.cast_ofNat, \u2190 Nat.cast_ofNat (n := 60), rpow_natCast]\n  norm_num\n  all_goals positivity\n\nlemma eulerMascheroniSeq_lt_eulerMascheroniSeq' (m n : \u2115) :\n    eulerMascheroniSeq m < eulerMascheroniSeq' n := by\n  have (r : \u2115) : eulerMascheroniSeq r < eulerMascheroniSeq' r := by\n    rcases eq_zero_or_pos r with rfl | hr\n    \u00b7 simp [eulerMascheroniSeq, eulerMascheroniSeq']\n    simp only [eulerMascheroniSeq, eulerMascheroniSeq', hr.ne', if_false]\n    gcongr\n    linarith\n  apply (strictMono_eulerMascheroniSeq.monotone (le_max_left m n)).trans_lt\n  exact (this _).trans_le (strictAnti_eulerMascheroniSeq'.antitone (le_max_right m n))\n\nend UpperSequence\n\n/-- The Euler-Mascheroni constant `\u03b3`. -/\nnoncomputable def eulerMascheroniConstant : \u211d := limUnder atTop eulerMascheroniSeq\n\nlemma tendsto_eulerMascheroniSeq :\n    Tendsto eulerMascheroniSeq atTop (\ud835\udcdd eulerMascheroniConstant) := by\n  have := tendsto_atTop_ciSup strictMono_eulerMascheroniSeq.monotone ?_\n  rwa [eulerMascheroniConstant, this.limUnder_eq]\n  exact \u27e8_, fun _ \u27e8_, hn\u27e9 \u21a6 hn \u25b8 (eulerMascheroniSeq_lt_eulerMascheroniSeq' _ 1).le\u27e9\n\nlemma tendsto_harmonic_sub_log_add_one :\n    Tendsto (fun n : \u2115 \u21a6 harmonic n - log (n + 1)) atTop (\ud835\udcdd eulerMascheroniConstant) :=\n  tendsto_eulerMascheroniSeq\n\nlemma tendsto_eulerMascheroniSeq' :\n    Tendsto eulerMascheroniSeq' atTop (\ud835\udcdd eulerMascheroniConstant) := by\n  suffices Tendsto (fun n \u21a6 eulerMascheroniSeq' n - eulerMascheroniSeq n) atTop (\ud835\udcdd 0) by\n    simpa using this.add tendsto_eulerMascheroniSeq\n  suffices Tendsto (fun x : \u211d \u21a6 log (x + 1) - log x) atTop (\ud835\udcdd 0) by\n    apply (this.comp tendsto_natCast_atTop_atTop).congr'\n    filter_upwards [eventually_ne_atTop 0] with n hn\n    simp [eulerMascheroniSeq, eulerMascheroniSeq', eq_false_intro hn]\n  suffices Tendsto (fun x : \u211d \u21a6 log (1 + 1 / x)) atTop (\ud835\udcdd 0) by\n    apply this.congr'\n    filter_upwards [eventually_gt_atTop 0] with x hx\n    rw [\u2190 log_div (by positivity) (by positivity), add_div, div_self hx.ne']\n  simpa only [add_zero, log_one] using\n    ((tendsto_const_nhds.div_atTop tendsto_id).const_add 1).log (by positivity)\n\nlemma tendsto_harmonic_sub_log :\n    Tendsto (fun n : \u2115 \u21a6 harmonic n - log n) atTop (\ud835\udcdd eulerMascheroniConstant) := by\n  apply tendsto_eulerMascheroniSeq'.congr'\n  filter_upwards [eventually_ne_atTop 0] with n hn\n  simp_rw [eulerMascheroniSeq', hn, if_false]\n\n", "theoremStatement": "lemma eulerMascheroniSeq_lt_eulerMascheroniConstant (n : \u2115) :\n    eulerMascheroniSeq n < eulerMascheroniConstant ", "theoremName": "Real.eulerMascheroniSeq_lt_eulerMascheroniConstant", "fileCreated": {"commit": "28e1bd49018baff3563fc4b31ab786aa8abc1208", "date": "2024-03-24"}, "theoremCreated": {"commit": "28e1bd49018baff3563fc4b31ab786aa8abc1208", "date": "2024-03-24"}, "file": "mathlib/Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean", "module": "Mathlib.NumberTheory.Harmonic.EulerMascheroni", "jsonFile": "Mathlib.NumberTheory.Harmonic.EulerMascheroni.jsonl", "positionMetadata": {"lineInFile": 158, "tokenPositionInFile": 6559, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": 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"Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Field.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Matrix.Invertible", "Mathlib.Algebra.Regular.Pow", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Tactic.ComputeDegree", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Data.Complex.ExponentialBounds", "Mathlib.NumberTheory.Harmonic.Defs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  refine (strictMono_eulerMascheroniSeq (Nat.lt_succ_self n)).trans_le ?_\n  apply strictMono_eulerMascheroniSeq.monotone.ge_of_tendsto tendsto_eulerMascheroniSeq", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 167}}
+{"srcContext": "/-\nCopyright (c) 2023 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\nimport Mathlib.CategoryTheory.Localization.Opposite\n\n/-!\n# Calculus of fractions\n\nFollowing the definitions by [Gabriel and Zisman][gabriel-zisman-1967],\ngiven a morphism property `W : MorphismProperty C` on a category `C`,\nwe introduce the class `W.HasLeftCalculusOfFractions`. The main\nresult `Localization.exists_leftFraction` is that if `L : C \u2964 D`\nis a localization functor for `W`, then for any morphism `L.obj X \u27f6 L.obj Y` in `D`,\nthere exists an auxiliary object `Y' : C` and morphisms `g : X \u27f6 Y'` and `s : Y \u27f6 Y'`,\nwith `W s`, such that the given morphism is a sort of fraction `g / s`,\nor more precisely of the form `L.map g \u226b (Localization.isoOfHom L W s hs).inv`.\nWe also show that the functor `L.mapArrow : Arrow C \u2964 Arrow D` is essentially surjective.\n\nSimilar results are obtained when `W` has a right calculus of fractions.\n\n## References\n\n* [P. Gabriel, M. Zisman, *Calculus of fractions and homotopy theory*][gabriel-zisman-1967]\n\n-/\n\nnamespace CategoryTheory\n\nvariable {C D : Type*} [Category C] [Category D]\n\nopen Category\n\nnamespace MorphismProperty\n\n/-- A left fraction from `X : C` to `Y : C` for `W : MorphismProperty C` consists of the\ndatum of an object `Y' : C` and maps `f : X \u27f6 Y'` and `s : Y \u27f6 Y'` such that `W s`. -/\nstructure LeftFraction (W : MorphismProperty C) (X Y : C) where\n  /-- the auxiliary object of a left fraction -/\n  {Y' : C}\n  /-- the numerator of a left fraction -/\n  f : X \u27f6 Y'\n  /-- the denominator of a left fraction -/\n  s : Y \u27f6 Y'\n  /-- the condition that the denominator belongs to the given morphism property -/\n  hs : W s\n\nnamespace LeftFraction\n\nvariable (W : MorphismProperty C) {X Y : C}\n\n/-- The left fraction from `X` to `Y` given by a morphism `f : X \u27f6 Y`. -/\n@[simps]\ndef ofHom (f : X \u27f6 Y) [W.ContainsIdentities] :\n    W.LeftFraction X Y := mk f (\ud835\udfd9 Y) (W.id_mem Y)\n\nvariable {W}\n\n/-- The left fraction from `X` to `Y` given by a morphism `s : Y \u27f6 X` such that `W s`. -/\n@[simps]\ndef ofInv (s : Y \u27f6 X) (hs : W s) :\n    W.LeftFraction X Y := mk (\ud835\udfd9 X) s hs\n\n/-- If `\u03c6 : W.LeftFraction X Y` and `L` is a functor which inverts `W`, this is the\ninduced morphism `L.obj X \u27f6 L.obj Y`  -/\nnoncomputable def map (\u03c6 : W.LeftFraction X Y) (L : C \u2964 D) (hL : W.IsInvertedBy L) :\n    L.obj X \u27f6 L.obj Y :=\n  have := hL _ \u03c6.hs\n  L.map \u03c6.f \u226b inv (L.map \u03c6.s)\n\n@[reassoc (attr := simp)]\nlemma map_comp_map_s (\u03c6 : W.LeftFraction X Y) (L : C \u2964 D) (hL : W.IsInvertedBy L) :\n    \u03c6.map L hL \u226b L.map \u03c6.s = L.map \u03c6.f := by\n  letI := hL _ \u03c6.hs\n  simp [map]\n\nvariable (W)\n\nlemma map_ofHom (f : X \u27f6 Y) (L : C \u2964 D) (hL : W.IsInvertedBy L) [W.ContainsIdentities] :\n    (ofHom W f).map L hL = L.map f := by\n  simp [map]\n\n@[reassoc (attr := simp)]\nlemma map_ofInv_hom_id (s : Y \u27f6 X) (hs : W s) (L : C \u2964 D) (hL : W.IsInvertedBy L) :\n    (ofInv s hs).map L hL \u226b L.map s = \ud835\udfd9 _ := by\n  letI := hL _ hs\n  simp [map]\n\n@[reassoc (attr := simp)]\nlemma map_hom_ofInv_id (s : Y \u27f6 X) (hs : W s) (L : C \u2964 D) (hL : W.IsInvertedBy L) :\n    L.map s \u226b (ofInv s hs).map L hL = \ud835\udfd9 _ := by\n  letI := hL _ hs\n  simp [map]\n\nvariable {W}\n\nlemma cases (\u03b1 : W.LeftFraction X Y) :\n    \u2203 (Y' : C) (f : X \u27f6 Y') (s : Y \u27f6 Y') (hs : W s), \u03b1 = LeftFraction.mk f s hs :=\n  \u27e8_, _, _, _, rfl\u27e9\n\nend LeftFraction\n\n/-- A right fraction from `X : C` to `Y : C` for `W : MorphismProperty C` consists of the\ndatum of an object `X' : C` and maps `s : X' \u27f6 X` and `f : X' \u27f6 Y` such that `W s`. -/\nstructure RightFraction (W : MorphismProperty C) (X Y : C) where\n  /-- the auxiliary object of a right fraction -/\n  {X' : C}\n  /-- the denominator of a right fraction -/\n  s : X' \u27f6 X\n  /-- the condition that the denominator belongs to the given morphism property -/\n  hs : W s\n  /-- the numerator of a right fraction -/\n  f : X' \u27f6 Y\n\nnamespace RightFraction\n\nvariable (W : MorphismProperty C)\nvariable {X Y : C}\n\n/-- The right fraction from `X` to `Y` given by a morphism `f : X \u27f6 Y`. -/\n@[simps]\ndef ofHom (f : X \u27f6 Y) [W.ContainsIdentities] :\n    W.RightFraction X Y := mk (\ud835\udfd9 X) (W.id_mem X) f\n\nvariable {W}\n\n/-- The right fraction from `X` to `Y` given by a morphism `s : Y \u27f6 X` such that `W s`. -/\n@[simps]\ndef ofInv (s : Y \u27f6 X) (hs : W s) :\n    W.RightFraction X Y := mk s hs (\ud835\udfd9 Y)\n\n/-- If `\u03c6 : W.RightFraction X Y` and `L` is a functor which inverts `W`, this is the\ninduced morphism `L.obj X \u27f6 L.obj Y`  -/\nnoncomputable def map (\u03c6 : W.RightFraction X Y) (L : C \u2964 D) (hL : W.IsInvertedBy L) :\n    L.obj X \u27f6 L.obj Y :=\n  have := hL _ \u03c6.hs\n  inv (L.map \u03c6.s) \u226b L.map \u03c6.f\n\n@[reassoc (attr := simp)]\nlemma map_s_comp_map (\u03c6 : W.RightFraction X Y) (L : C \u2964 D) (hL : W.IsInvertedBy L) :\n    L.map \u03c6.s \u226b \u03c6.map L hL = L.map \u03c6.f := by\n  letI := hL _ \u03c6.hs\n  simp [map]\n\nvariable (W)\n\n@[simp]\nlemma map_ofHom (f : X \u27f6 Y) (L : C \u2964 D) (hL : W.IsInvertedBy L) [W.ContainsIdentities] :\n    (ofHom W f).map L hL = L.map f := by\n  simp [map]\n\n@[reassoc (attr := simp)]\nlemma map_ofInv_hom_id (s : Y \u27f6 X) (hs : W s) (L : C \u2964 D) (hL : W.IsInvertedBy L) :\n    (ofInv s hs).map L hL \u226b L.map s = \ud835\udfd9 _ := by\n  letI := hL _ hs\n  simp [map]\n\n@[reassoc (attr := simp)]\nlemma map_hom_ofInv_id (s : Y \u27f6 X) (hs : W s) (L : C \u2964 D) (hL : W.IsInvertedBy L) :\n    L.map s \u226b (ofInv s hs).map L hL = \ud835\udfd9 _ := by\n  letI := hL _ hs\n  simp [map]\n\nvariable {W}\n\nlemma cases (\u03b1 : W.RightFraction X Y) :\n    \u2203 (X' : C) (s : X' \u27f6 X) (hs : W s) (f : X' \u27f6 Y) , \u03b1 = RightFraction.mk s hs f :=\n  \u27e8_, _, _, _, rfl\u27e9\n\nend RightFraction\n\nvariable (W : MorphismProperty C)\n\n/-- A multiplicative morphism property `W` has left calculus of fractions if\nany right fraction can be turned into a left fraction and that two morphisms\nthat can be equalized by precomposition with a morphism in `W` can also\nbe equalized by postcomposition with a morphism in `W`. -/\nclass HasLeftCalculusOfFractions extends W.IsMultiplicative : Prop where\n  exists_leftFraction \u2983X Y : C\u2984 (\u03c6 : W.RightFraction X Y) :\n    \u2203 (\u03c8 : W.LeftFraction X Y), \u03c6.f \u226b \u03c8.s = \u03c6.s \u226b \u03c8.f\n  ext : \u2200 \u2983X' X Y : C\u2984 (f\u2081 f\u2082 : X \u27f6 Y) (s : X' \u27f6 X) (_ : W s)\n    (_ : s \u226b f\u2081 = s \u226b f\u2082), \u2203 (Y' : C) (t : Y \u27f6 Y') (_ : W t), f\u2081 \u226b t = f\u2082 \u226b t\n\n/-- A multiplicative morphism property `W` has right calculus of fractions if\nany left fraction can be turned into a right fraction and that two morphisms\nthat can be equalized by postcomposition with a morphism in `W` can also\nbe equalized by precomposition with a morphism in `W`. -/\nclass HasRightCalculusOfFractions extends W.IsMultiplicative : Prop where\n  exists_rightFraction \u2983X Y : C\u2984 (\u03c6 : W.LeftFraction X Y) :\n    \u2203 (\u03c8 : W.RightFraction X Y), \u03c8.s \u226b \u03c6.f = \u03c8.f \u226b \u03c6.s\n  ext : \u2200 \u2983X Y Y' : C\u2984 (f\u2081 f\u2082 : X \u27f6 Y) (s : Y \u27f6 Y') (_ : W s)\n    (_ : f\u2081 \u226b s = f\u2082 \u226b s), \u2203 (X' : C) (t : X' \u27f6 X) (_ : W t), t \u226b f\u2081 = t \u226b f\u2082\n\nvariable {W}\n\nlemma RightFraction.exists_leftFraction [W.HasLeftCalculusOfFractions] {X Y : C}\n    (\u03c6 : W.RightFraction X Y) : \u2203 (\u03c8 : W.LeftFraction X Y), \u03c6.f \u226b \u03c8.s = \u03c6.s \u226b \u03c8.f :=\n  HasLeftCalculusOfFractions.exists_leftFraction \u03c6\n\n/-- A choice of a left fraction deduced from a right fraction for a morphism property `W`\nwhen `W` has left calculus of fractions. -/\nnoncomputable def RightFraction.leftFraction [W.HasLeftCalculusOfFractions] {X Y : C}\n    (\u03c6 : W.RightFraction X Y) : W.LeftFraction X Y :=\n  \u03c6.exists_leftFraction.choose\n\n@[reassoc]\nlemma RightFraction.leftFraction_fac [W.HasLeftCalculusOfFractions] {X Y : C}\n    (\u03c6 : W.RightFraction X Y) : \u03c6.f \u226b \u03c6.leftFraction.s = \u03c6.s \u226b \u03c6.leftFraction.f :=\n  \u03c6.exists_leftFraction.choose_spec\n\nlemma LeftFraction.exists_rightFraction [W.HasRightCalculusOfFractions] {X Y : C}\n    (\u03c6 : W.LeftFraction X Y) : \u2203 (\u03c8 : W.RightFraction X Y), \u03c8.s \u226b \u03c6.f = \u03c8.f \u226b \u03c6.s :=\n  HasRightCalculusOfFractions.exists_rightFraction \u03c6\n\n/-- A choice of a right fraction deduced from a left fraction for a morphism property `W`\nwhen `W` has right calculus of fractions. -/\nnoncomputable def LeftFraction.rightFraction [W.HasRightCalculusOfFractions] {X Y : C}\n    (\u03c6 : W.LeftFraction X Y) : W.RightFraction X Y :=\n  \u03c6.exists_rightFraction.choose\n\n@[reassoc]\nlemma LeftFraction.rightFraction_fac [W.HasRightCalculusOfFractions] {X Y : C}\n    (\u03c6 : W.LeftFraction X Y) : \u03c6.rightFraction.s \u226b \u03c6.f = \u03c6.rightFraction.f \u226b \u03c6.s :=\n  \u03c6.exists_rightFraction.choose_spec\n\n/-- The equivalence relation on left fractions for a morphism property `W`. -/\ndef LeftFractionRel {X Y : C} (z\u2081 z\u2082 : W.LeftFraction X Y) : Prop :=\n  \u2203 (Z : C)  (t\u2081 : z\u2081.Y' \u27f6 Z) (t\u2082 : z\u2082.Y' \u27f6 Z) (_ : z\u2081.s \u226b t\u2081 = z\u2082.s \u226b t\u2082)\n    (_ : z\u2081.f \u226b t\u2081 = z\u2082.f \u226b t\u2082), W (z\u2081.s \u226b t\u2081)\n\nnamespace LeftFractionRel\n\nlemma refl {X Y : C} (z : W.LeftFraction X Y) : LeftFractionRel z z :=\n  \u27e8z.Y', \ud835\udfd9 _, \ud835\udfd9 _, rfl, rfl, by simpa only [Category.comp_id] using z.hs\u27e9\n\nlemma symm {X Y : C} {z\u2081 z\u2082 : W.LeftFraction X Y} (h : LeftFractionRel z\u2081 z\u2082) :\n    LeftFractionRel z\u2082 z\u2081 := by\n  obtain \u27e8Z, t\u2081, t\u2082, hst, hft, ht\u27e9 := h\n  exact \u27e8Z, t\u2082, t\u2081, hst.symm, hft.symm, by simpa only [\u2190 hst] using ht\u27e9\n\nlemma trans {X Y : C} {z\u2081 z\u2082 z\u2083 : W.LeftFraction X Y}\n    [HasLeftCalculusOfFractions W]\n    (h\u2081\u2082 : LeftFractionRel z\u2081 z\u2082) (h\u2082\u2083 : LeftFractionRel z\u2082 z\u2083) :\n    LeftFractionRel z\u2081 z\u2083 := by\n  obtain \u27e8Z\u2084, t\u2081, t\u2082, hst, hft, ht\u27e9 := h\u2081\u2082\n  obtain \u27e8Z\u2085, u\u2082, u\u2083, hsu, hfu, hu\u27e9 := h\u2082\u2083\n  obtain \u27e8\u27e8v\u2084, v\u2085, hv\u2085\u27e9, fac\u27e9 := HasLeftCalculusOfFractions.exists_leftFraction\n    (RightFraction.mk (z\u2081.s \u226b t\u2081) ht (z\u2083.s \u226b u\u2083))\n  simp only [Category.assoc] at fac\n  have eq : z\u2082.s \u226b u\u2082 \u226b v\u2085  = z\u2082.s \u226b t\u2082 \u226b v\u2084 := by\n    simpa only [\u2190 reassoc_of% hsu, reassoc_of% hst] using fac\n  obtain \u27e8Z\u2087, w, hw, fac'\u27e9 := HasLeftCalculusOfFractions.ext _ _ _ z\u2082.hs eq\n  simp only [Category.assoc] at fac'\n  refine' \u27e8Z\u2087, t\u2081 \u226b v\u2084 \u226b w, u\u2083 \u226b v\u2085 \u226b w, _, _, _\u27e9\n  \u00b7 rw [reassoc_of% fac]\n  \u00b7 rw [reassoc_of% hft, \u2190 fac', reassoc_of% hfu]\n  \u00b7 rw [\u2190 reassoc_of% fac, \u2190 reassoc_of% hsu, \u2190 Category.assoc]\n    exact W.comp_mem _ _ hu (W.comp_mem _ _ hv\u2085 hw)\n\nend LeftFractionRel\n\nsection\n\nvariable [W.HasLeftCalculusOfFractions] (W)\n\nlemma equivalenceLeftFractionRel (X Y : C) :\n    @_root_.Equivalence (W.LeftFraction X Y) LeftFractionRel where\n  refl := LeftFractionRel.refl\n  symm := LeftFractionRel.symm\n  trans := LeftFractionRel.trans\n\nvariable {W}\n\nnamespace LeftFraction\n\nopen HasLeftCalculusOfFractions\n\n/-- Auxiliary definition for the composition of left fractions. -/\n@[simp]\ndef comp\u2080 {X Y Z : C} (z\u2081 : W.LeftFraction X Y) (z\u2082 : W.LeftFraction Y Z)\n    (z\u2083 : W.LeftFraction z\u2081.Y' z\u2082.Y') :\n    W.LeftFraction X Z :=\n  mk (z\u2081.f \u226b z\u2083.f) (z\u2082.s \u226b z\u2083.s) (W.comp_mem _ _ z\u2082.hs z\u2083.hs)\n\n/-- The equivalence class of `z\u2081.comp\u2080 z\u2082 z\u2083` does not depend on the choice of `z\u2083` provided\nthey satisfy the compatibility `z\u2082.f \u226b z\u2083.s = z\u2081.s \u226b z\u2083.f`. -/\nlemma comp\u2080_rel {X Y Z : C} (z\u2081 : W.LeftFraction X Y) (z\u2082 : W.LeftFraction Y Z)\n    (z\u2083 z\u2083' : W.LeftFraction z\u2081.Y' z\u2082.Y') (h\u2083 : z\u2082.f \u226b z\u2083.s = z\u2081.s \u226b z\u2083.f)\n    (h\u2083' : z\u2082.f \u226b z\u2083'.s = z\u2081.s \u226b z\u2083'.f) :\n    LeftFractionRel (z\u2081.comp\u2080 z\u2082 z\u2083) (z\u2081.comp\u2080 z\u2082 z\u2083') := by\n  obtain \u27e8z\u2084, fac\u27e9 := exists_leftFraction (RightFraction.mk z\u2083.s z\u2083.hs z\u2083'.s)\n  dsimp at fac\n  have eq : z\u2081.s \u226b z\u2083.f \u226b z\u2084.f = z\u2081.s \u226b z\u2083'.f \u226b z\u2084.s := by\n    rw [\u2190 reassoc_of% h\u2083, \u2190 reassoc_of% h\u2083', fac]\n  obtain \u27e8Y, t, ht, fac'\u27e9 := HasLeftCalculusOfFractions.ext _ _ _ z\u2081.hs eq\n  simp only [assoc] at fac'\n  refine' \u27e8Y, z\u2084.f \u226b t, z\u2084.s \u226b t, _, _, _\u27e9\n  \u00b7 simp only [comp\u2080, assoc, reassoc_of% fac]\n  \u00b7 simp only [comp\u2080, assoc, fac']\n  \u00b7 simp only [comp\u2080, assoc, \u2190 reassoc_of% fac]\n    exact W.comp_mem _ _ z\u2082.hs (W.comp_mem _ _ z\u2083'.hs (W.comp_mem _ _ z\u2084.hs ht))\n\nvariable (W)\n\n/-- The morphisms in the constructed localized category for a morphism property `W`\nthat has left calculus of fractions are equivalence classes of left fractions. -/\ndef Localization.Hom (X Y : C) :=\n  Quot (LeftFractionRel : W.LeftFraction X Y \u2192 W.LeftFraction X Y \u2192 Prop)\n\nvariable {W}\n\n/-- The morphism in the constructed localized category that is induced by a left fraction. -/\ndef Localization.Hom.mk {X Y : C} (z : W.LeftFraction X Y) : Localization.Hom W X Y :=\n  Quot.mk _ z\n\nlemma Localization.Hom.mk_surjective {X Y : C} (f : Localization.Hom W X Y) :\n    \u2203 (z : W.LeftFraction X Y), f = mk z := by\n  obtain \u27e8z\u27e9 := f\n  exact \u27e8z, rfl\u27e9\n\n/-- Auxiliary definition towards the definition of the composition of morphisms\nin the constructed localized category for a morphism property that has\nleft calculus of fractions. -/\nnoncomputable def comp {X Y Z : C} (z\u2081 : W.LeftFraction X Y) (z\u2082 : W.LeftFraction Y Z) :\n    Localization.Hom W X Z :=\n  Localization.Hom.mk (z\u2081.comp\u2080 z\u2082 (RightFraction.mk z\u2081.s z\u2081.hs z\u2082.f).leftFraction)\n\nlemma comp_eq {X Y Z : C} (z\u2081 : W.LeftFraction X Y) (z\u2082 : W.LeftFraction Y Z)\n    (z\u2083 : W.LeftFraction z\u2081.Y' z\u2082.Y') (h\u2083 : z\u2082.f \u226b z\u2083.s = z\u2081.s \u226b z\u2083.f) :\n    z\u2081.comp z\u2082 = Localization.Hom.mk (z\u2081.comp\u2080 z\u2082 z\u2083) :=\n  Quot.sound (LeftFraction.comp\u2080_rel _ _ _ _\n    (RightFraction.leftFraction_fac (RightFraction.mk z\u2081.s z\u2081.hs z\u2082.f)) h\u2083)\n\nnamespace Localization\n\n/-- Composition of morphisms in the constructed localized category\nfor a morphism property that has left calculus of fractions. -/\nnoncomputable def Hom.comp {X Y Z : C} (z\u2081 : Hom W X Y) (z\u2082 : Hom W Y Z) : Hom W X Z := by\n  refine' Quot.lift\u2082 (fun a b => a.comp b) _ _ z\u2081 z\u2082\n  \u00b7 rintro a b\u2081 b\u2082 \u27e8U, t\u2081, t\u2082, hst, hft, ht\u27e9\n    obtain \u27e8z\u2081, fac\u2081\u27e9 := exists_leftFraction (RightFraction.mk a.s a.hs b\u2081.f)\n    obtain \u27e8z\u2082, fac\u2082\u27e9 := exists_leftFraction (RightFraction.mk a.s a.hs b\u2082.f)\n    obtain \u27e8w\u2081, fac\u2081'\u27e9 := exists_leftFraction (RightFraction.mk z\u2081.s z\u2081.hs t\u2081)\n    obtain \u27e8w\u2082, fac\u2082'\u27e9 := exists_leftFraction (RightFraction.mk z\u2082.s z\u2082.hs t\u2082)\n    obtain \u27e8u, fac\u2083\u27e9 := exists_leftFraction (RightFraction.mk w\u2081.s w\u2081.hs w\u2082.s)\n    dsimp at fac\u2081 fac\u2082 fac\u2081' fac\u2082' fac\u2083 \u22a2\n    have eq : a.s \u226b z\u2081.f \u226b w\u2081.f \u226b u.f = a.s \u226b z\u2082.f \u226b w\u2082.f \u226b u.s := by\n      rw [\u2190 reassoc_of% fac\u2081, \u2190 reassoc_of% fac\u2082, \u2190 reassoc_of% fac\u2081', \u2190 reassoc_of% fac\u2082',\n        reassoc_of% hft, fac\u2083]\n    obtain \u27e8Z, p, hp, fac\u2084\u27e9 := HasLeftCalculusOfFractions.ext _ _ _ a.hs eq\n    simp only [assoc] at fac\u2084\n    rw [comp_eq _ _ z\u2081 fac\u2081, comp_eq _ _ z\u2082 fac\u2082]\n    apply Quot.sound\n    refine' \u27e8Z, w\u2081.f \u226b u.f \u226b p, w\u2082.f \u226b u.s \u226b p, _, _, _\u27e9\n    \u00b7 dsimp\n      simp only [assoc, \u2190 reassoc_of% fac\u2081', \u2190 reassoc_of% fac\u2082',\n        reassoc_of% hst, reassoc_of% fac\u2083]\n    \u00b7 dsimp\n      simp only [assoc, fac\u2084]\n    \u00b7 dsimp\n      simp only [assoc]\n      rw [\u2190 reassoc_of% fac\u2081', \u2190 reassoc_of% fac\u2083, \u2190 assoc]\n      exact W.comp_mem _ _ ht (W.comp_mem _ _ w\u2082.hs (W.comp_mem _ _ u.hs hp))\n  \u00b7 rintro a\u2081 a\u2082 b \u27e8U, t\u2081, t\u2082, hst, hft, ht\u27e9\n    obtain \u27e8z\u2081, fac\u2081\u27e9 := exists_leftFraction (RightFraction.mk a\u2081.s a\u2081.hs b.f)\n    obtain \u27e8z\u2082, fac\u2082\u27e9 := exists_leftFraction (RightFraction.mk a\u2082.s a\u2082.hs b.f)\n    obtain \u27e8w\u2081, fac\u2081'\u27e9 := exists_leftFraction (RightFraction.mk (a\u2081.s \u226b t\u2081) ht (b.f \u226b z\u2081.s))\n    obtain \u27e8w\u2082, fac\u2082'\u27e9 := exists_leftFraction (RightFraction.mk (a\u2082.s \u226b t\u2082)\n      (show W _ by rw [\u2190 hst]; exact ht) (b.f \u226b z\u2082.s))\n    let p\u2081 : W.LeftFraction X Z := LeftFraction.mk (a\u2081.f \u226b t\u2081 \u226b w\u2081.f) (b.s \u226b z\u2081.s \u226b w\u2081.s)\n      (W.comp_mem _ _ b.hs (W.comp_mem _ _ z\u2081.hs w\u2081.hs))\n    let p\u2082 : W.LeftFraction X Z := LeftFraction.mk (a\u2082.f \u226b t\u2082 \u226b w\u2082.f) (b.s \u226b z\u2082.s \u226b w\u2082.s)\n      (W.comp_mem _ _ b.hs (W.comp_mem _ _ z\u2082.hs w\u2082.hs))\n    dsimp at fac\u2081 fac\u2082 fac\u2081' fac\u2082' \u22a2\n    simp only [assoc] at fac\u2081' fac\u2082'\n    rw [comp_eq _ _ z\u2081 fac\u2081, comp_eq _ _ z\u2082 fac\u2082]\n    apply Quot.sound\n    refine' LeftFractionRel.trans _ ((_ : LeftFractionRel p\u2081 p\u2082).trans _)\n    \u00b7 have eq : a\u2081.s \u226b z\u2081.f \u226b w\u2081.s = a\u2081.s \u226b t\u2081 \u226b w\u2081.f := by rw [\u2190 fac\u2081', reassoc_of% fac\u2081]\n      obtain \u27e8Z, u, hu, fac\u2083\u27e9 := HasLeftCalculusOfFractions.ext _ _ _ a\u2081.hs eq\n      simp only [assoc] at fac\u2083\n      refine' \u27e8Z, w\u2081.s \u226b u, u, _, _, _\u27e9\n      \u00b7 dsimp\n        simp only [assoc]\n      \u00b7 dsimp\n        simp only [assoc, fac\u2083]\n      \u00b7 dsimp\n        simp only [assoc]\n        exact W.comp_mem _ _ b.hs (W.comp_mem _ _ z\u2081.hs (W.comp_mem _ _ w\u2081.hs hu))\n    \u00b7 obtain \u27e8q, fac\u2083\u27e9 := exists_leftFraction (RightFraction.mk (z\u2081.s \u226b w\u2081.s)\n        (W.comp_mem _ _ z\u2081.hs w\u2081.hs) (z\u2082.s \u226b w\u2082.s))\n      dsimp at fac\u2083\n      simp only [assoc] at fac\u2083\n      have eq : a\u2081.s \u226b t\u2081 \u226b w\u2081.f \u226b q.f = a\u2081.s \u226b t\u2081 \u226b w\u2082.f \u226b q.s := by\n        rw [\u2190 reassoc_of% fac\u2081', \u2190 fac\u2083, reassoc_of% hst, reassoc_of% fac\u2082']\n      obtain \u27e8Z, u, hu, fac\u2084\u27e9 := HasLeftCalculusOfFractions.ext _ _ _ a\u2081.hs eq\n      simp only [assoc] at fac\u2084\n      refine' \u27e8Z, q.f \u226b u, q.s \u226b u, _, _, _\u27e9\n      \u00b7 simp only [assoc, reassoc_of% fac\u2083]\n      \u00b7 rw [assoc, assoc, assoc, assoc, fac\u2084, reassoc_of% hft]\n      \u00b7 simp only [assoc, \u2190 reassoc_of% fac\u2083]\n        exact W.comp_mem _ _ b.hs (W.comp_mem _ _ z\u2082.hs\n          (W.comp_mem _ _ w\u2082.hs (W.comp_mem _ _ q.hs hu)))\n    \u00b7 have eq : a\u2082.s \u226b z\u2082.f \u226b w\u2082.s = a\u2082.s \u226b t\u2082 \u226b w\u2082.f := by\n        rw [\u2190 fac\u2082', reassoc_of% fac\u2082]\n      obtain \u27e8Z, u, hu, fac\u2084\u27e9 := HasLeftCalculusOfFractions.ext _ _ _ a\u2082.hs eq\n      simp only [assoc] at fac\u2084\n      refine' \u27e8Z, u, w\u2082.s \u226b u, _, _, _\u27e9\n      \u00b7 dsimp\n        simp only [assoc]\n      \u00b7 dsimp\n        simp only [assoc, fac\u2084]\n      \u00b7 dsimp\n        simp only [assoc]\n        exact W.comp_mem _ _ b.hs (W.comp_mem _ _ z\u2082.hs (W.comp_mem _ _ w\u2082.hs hu))\n\nlemma Hom.comp_eq {X Y Z : C} (z\u2081 : W.LeftFraction X Y) (z\u2082 : W.LeftFraction Y Z) :\n    Hom.comp (mk z\u2081) (mk z\u2082) = z\u2081.comp z\u2082 := rfl\n\nend Localization\n\n/-- The constructed localized category for a morphism property\nthat has left calculus of fractions. -/\n@[nolint unusedArguments]\ndef Localization (_ : MorphismProperty C) := C\n\nnamespace Localization\n\nnoncomputable instance : Category (Localization W) where\n  Hom X Y := Localization.Hom W X Y\n  id X := Localization.Hom.mk (ofHom W (\ud835\udfd9 _))\n  comp f g := f.comp g\n  comp_id := by\n    rintro (X Y : C) f\n    obtain \u27e8z, rfl\u27e9 := Hom.mk_surjective f\n    change (Hom.mk z).comp (Hom.mk (ofHom W (\ud835\udfd9 Y))) = Hom.mk z\n    rw [Hom.comp_eq, comp_eq z (ofHom W (\ud835\udfd9 Y)) (ofInv z.s z.hs) (by simp)]\n    dsimp [comp\u2080]\n    simp only [comp_id, id_comp]\n  id_comp := by\n    rintro (X Y : C) f\n    obtain \u27e8z, rfl\u27e9 := Hom.mk_surjective f\n    change (Hom.mk (ofHom W (\ud835\udfd9 X))).comp (Hom.mk z) = Hom.mk z\n    rw [Hom.comp_eq, comp_eq (ofHom W (\ud835\udfd9 X)) z (ofHom W z.f) (by simp)]\n    dsimp\n    simp only [comp\u2080, id_comp, comp_id]\n  assoc := by\n    rintro (X\u2081 X\u2082 X\u2083 X\u2084 : C) f\u2081 f\u2082 f\u2083\n    obtain \u27e8z\u2081, rfl\u27e9 := Hom.mk_surjective f\u2081\n    obtain \u27e8z\u2082, rfl\u27e9 := Hom.mk_surjective f\u2082\n    obtain \u27e8z\u2083, rfl\u27e9 := Hom.mk_surjective f\u2083\n    change ((Hom.mk z\u2081).comp (Hom.mk z\u2082)).comp (Hom.mk z\u2083) =\n      (Hom.mk z\u2081).comp ((Hom.mk z\u2082).comp (Hom.mk z\u2083))\n    rw [Hom.comp_eq z\u2081 z\u2082, Hom.comp_eq z\u2082 z\u2083]\n    obtain \u27e8z\u2081\u2082, fac\u2081\u2082\u27e9 := exists_leftFraction (RightFraction.mk z\u2081.s z\u2081.hs z\u2082.f)\n    obtain \u27e8z\u2082\u2083, fac\u2082\u2083\u27e9 := exists_leftFraction (RightFraction.mk z\u2082.s z\u2082.hs z\u2083.f)\n    obtain \u27e8z', fac\u27e9 := exists_leftFraction (RightFraction.mk z\u2081\u2082.s z\u2081\u2082.hs z\u2082\u2083.f)\n    dsimp at fac\u2081\u2082 fac\u2082\u2083 fac\n    rw [comp_eq z\u2081 z\u2082 z\u2081\u2082 fac\u2081\u2082, comp_eq z\u2082 z\u2083 z\u2082\u2083 fac\u2082\u2083, comp\u2080, comp\u2080,\n      Hom.comp_eq, Hom.comp_eq,\n      comp_eq _ z\u2083 (mk z'.f (z\u2082\u2083.s \u226b z'.s) (W.comp_mem _ _ z\u2082\u2083.hs z'.hs))\n        (by dsimp; rw [assoc, reassoc_of% fac\u2082\u2083, fac]),\n      comp_eq z\u2081 _ (mk (z\u2081\u2082.f \u226b z'.f) z'.s z'.hs)\n        (by dsimp; rw [assoc, \u2190 reassoc_of% fac\u2081\u2082, fac])]\n    simp\n\nvariable (W)\n\n/-- The localization functor to the constructed localized category for a morphism property\nthat has left calculus of fractions. -/\n@[simps obj]\ndef Q : C \u2964 Localization W where\n  obj X := X\n  map f := Hom.mk (ofHom W f)\n  map_id _ := rfl\n  map_comp {X Y Z} f g := by\n    change _ = Hom.comp _ _\n    rw [Hom.comp_eq, comp_eq (ofHom W f) (ofHom W g) (ofHom W g) (by simp)]\n    simp only [ofHom, comp\u2080, comp_id]\n\nvariable {W}\n\n/-- The morphism on `Localization W` that is induced by a left fraction. -/\nabbrev homMk {X Y : C} (f : W.LeftFraction X Y) : (Q W).obj X \u27f6 (Q W).obj Y := Hom.mk f\n\nlemma homMk_eq_hom_mk {X Y : C} (f : W.LeftFraction X Y) : homMk f = Hom.mk f := rfl\n\nvariable (W)\n\nlemma Q_map {X Y : C} (f : X \u27f6 Y) : (Q W).map f = homMk (ofHom W f) := rfl\n\nvariable {W}\n\nlemma homMk_comp_homMk {X Y Z : C} (z\u2081 : W.LeftFraction X Y) (z\u2082 : W.LeftFraction Y Z)\n    (z\u2083 : W.LeftFraction z\u2081.Y' z\u2082.Y') (h\u2083 : z\u2082.f \u226b z\u2083.s = z\u2081.s \u226b z\u2083.f) :\n    homMk z\u2081 \u226b homMk z\u2082 = homMk (z\u2081.comp\u2080 z\u2082 z\u2083) := by\n  change Hom.comp _ _ = _\n  erw [Hom.comp_eq, comp_eq z\u2081 z\u2082 z\u2083 h\u2083]\n\nlemma homMk_eq_of_leftFractionRel {X Y : C} (z\u2081 z\u2082 : W.LeftFraction X Y)\n    (h : LeftFractionRel z\u2081 z\u2082) :\n    homMk z\u2081 = homMk z\u2082 :=\n  Quot.sound h\n\nlemma homMk_eq_iff_leftFractionRel {X Y : C} (z\u2081 z\u2082 : W.LeftFraction X Y) :\n    homMk z\u2081 = homMk z\u2082 \u2194 LeftFractionRel z\u2081 z\u2082 :=\n  @Equivalence.quot_mk_eq_iff _ _ (equivalenceLeftFractionRel W X Y) _ _\n\n/-- The morphism in `Localization W` that is the formal inverse of a morphism\nwhich belongs to `W`. -/\ndef Qinv {X Y : C} (s : X \u27f6 Y) (hs : W s) : (Q W).obj Y \u27f6 (Q W).obj X := homMk (ofInv s hs)\n\nlemma Q_map_comp_Qinv {X Y Y' : C} (f : X \u27f6 Y') (s : Y \u27f6 Y') (hs : W s) :\n    (Q W).map f \u226b Qinv s hs = homMk (mk f s hs) := by\n  dsimp only [Q_map, Qinv]\n  rw [homMk_comp_homMk (ofHom W f) (ofInv s hs) (ofHom W (\ud835\udfd9 _)) (by simp)]\n  simp\n\n/-- The isomorphism in `Localization W` that is induced by a morphism in `W`. -/\n@[simps]\ndef Qiso {X Y : C} (s : X \u27f6 Y) (hs : W s) : (Q W).obj X \u2245 (Q W).obj Y where\n  hom := (Q W).map s\n  inv := Qinv s hs\n  hom_inv_id := by\n    rw [Q_map_comp_Qinv]\n    apply homMk_eq_of_leftFractionRel\n    exact \u27e8_, \ud835\udfd9 Y, s, by simp, by simp, by simpa using hs\u27e9\n  inv_hom_id := by\n    dsimp only [Qinv, Q_map]\n    rw [homMk_comp_homMk (ofInv s hs) (ofHom W s) (ofHom W (\ud835\udfd9 Y)) (by simp)]\n    apply homMk_eq_of_leftFractionRel\n    exact \u27e8_, \ud835\udfd9 Y, \ud835\udfd9 Y, by simp, by simp, by simpa using W.id_mem Y\u27e9\n\n@[reassoc (attr := simp)]\nlemma Qiso_hom_inv_id {X Y : C} (s : X \u27f6 Y) (hs : W s) :\n    (Q W).map s \u226b Qinv s hs = \ud835\udfd9 _ := (Qiso s hs).hom_inv_id\n\n@[reassoc (attr := simp)]\nlemma Qiso_inv_hom_id {X Y : C} (s : X \u27f6 Y) (hs : W s) :\n    Qinv s hs  \u226b (Q W).map s = \ud835\udfd9 _ := (Qiso s hs).inv_hom_id\n\ninstance {X Y : C} (s : X \u27f6 Y) (hs : W s) : IsIso (Qinv s hs) :=\n  (inferInstance : IsIso (Qiso s hs).inv)\n\nsection\n\nvariable {E : Type*} [Category E]\n\n/-- The image by a functor which inverts `W` of an equivalence class of left fractions. -/\nnoncomputable def Hom.map {X Y : C} (f : Hom W X Y) (F : C \u2964 E) (hF : W.IsInvertedBy F) :\n    F.obj X \u27f6 F.obj Y :=\n  Quot.lift (fun f => f.map F hF) (by\n    intro a\u2081 a\u2082 \u27e8Z, t\u2081, t\u2082, hst, hft, h\u27e9\n    dsimp\n    have := hF _ h\n    rw [\u2190 cancel_mono (F.map (a\u2081.s \u226b t\u2081)), F.map_comp, map_comp_map_s_assoc,\n      \u2190 F.map_comp, \u2190 F.map_comp, hst, hft, F.map_comp,\n      F.map_comp, map_comp_map_s_assoc]) f\n\n@[simp]\nlemma Hom.map_mk {X Y : C} (f : LeftFraction W X Y)\n    (F : C \u2964 E) (hF : W.IsInvertedBy F) :\n  Hom.map (Hom.mk f) F hF = f.map F hF := rfl\n\nnamespace StrictUniversalPropertyFixedTarget\n\nvariable (W)\n\nlemma inverts : W.IsInvertedBy (Q W) := fun _ _ s hs =>\n  (inferInstance : IsIso (Qiso s hs).hom)\n\nvariable {W}\n\n/-- The functor `Localization W \u2964 E` that is induced by a functor `C \u2964 E` which inverts `W`,\nwhen `W` has a left calculus of fractions. -/\nnoncomputable def lift (F : C \u2964 E) (hF : W.IsInvertedBy F) :\n    Localization W \u2964 E where\n  obj X := F.obj X\n  map {X Y : C} f := f.map F hF\n  map_id := by\n    intro (X : C)\n    dsimp\n    change (Hom.mk (ofHom W (\ud835\udfd9 X))).map F hF = _\n    rw [Hom.map_mk, map_ofHom, F.map_id]\n  map_comp := by\n    rintro (X Y Z : C) f g\n    obtain \u27e8f, rfl\u27e9 := Hom.mk_surjective f\n    obtain \u27e8g, rfl\u27e9 := Hom.mk_surjective g\n    dsimp\n    obtain \u27e8z, fac\u27e9 := HasLeftCalculusOfFractions.exists_leftFraction\n      (RightFraction.mk f.s f.hs g.f)\n    rw [homMk_comp_homMk f g z fac, Hom.map_mk]\n    dsimp at fac \u22a2\n    have := hF _ g.hs\n    have := hF _ z.hs\n    rw [\u2190 cancel_mono (F.map g.s), assoc, map_comp_map_s,\n      \u2190 cancel_mono (F.map z.s), assoc, assoc, \u2190 F.map_comp,\n      \u2190 F.map_comp, map_comp_map_s, fac]\n    dsimp\n    rw [F.map_comp, F.map_comp, map_comp_map_s_assoc]\n\nlemma fac (F : C \u2964 E) (hF : W.IsInvertedBy F) : Q W \u22d9 lift F hF = F :=\n  Functor.ext (fun X => rfl) (fun X Y f => by\n    dsimp [lift]\n    rw [Q_map, Hom.map_mk, id_comp, comp_id, map_ofHom])\n\nlemma uniq (F\u2081 F\u2082 : Localization W \u2964 E) (h : Q W \u22d9 F\u2081 = Q W \u22d9 F\u2082) : F\u2081 = F\u2082 :=\n  Functor.ext (fun X => Functor.congr_obj h X) (by\n    rintro (X Y : C) f\n    obtain \u27e8f, rfl\u27e9 := Hom.mk_surjective f\n    rw [show Hom.mk f = homMk (mk f.f f.s f.hs) by rfl,\n      \u2190 Q_map_comp_Qinv f.f f.s f.hs, F\u2081.map_comp, F\u2082.map_comp, assoc]\n    erw [Functor.congr_hom h f.f]\n    rw [assoc, assoc]\n    congr 2\n    have := inverts W _ f.hs\n    rw [\u2190 cancel_epi (F\u2082.map ((Q W).map f.s)), \u2190 F\u2082.map_comp_assoc,\n      Qiso_hom_inv_id, Functor.map_id, id_comp]\n    erw [Functor.congr_hom h.symm f.s]\n    dsimp\n    rw [assoc, assoc, eqToHom_trans_assoc, eqToHom_refl, id_comp, \u2190 F\u2081.map_comp,\n      Qiso_hom_inv_id]\n    dsimp\n    rw [F\u2081.map_id, comp_id])\n\nend StrictUniversalPropertyFixedTarget\n\nvariable (W)\n\nopen StrictUniversalPropertyFixedTarget in\n/-- The universal property of the localization for the constructed localized category\nwhen there is a left calculus of fractions. -/\nnoncomputable def strictUniversalPropertyFixedTarget (E : Type*) [Category E] :\n    Localization.StrictUniversalPropertyFixedTarget (Q W) W E where\n  inverts := inverts W\n  lift := lift\n  fac := fac\n  uniq := uniq\n\ninstance : (Q W).IsLocalization W :=\n  Functor.IsLocalization.mk' _ _\n    (strictUniversalPropertyFixedTarget W _)\n    (strictUniversalPropertyFixedTarget W _)\n\nend\n\nlemma homMk_eq {X Y : C} (f : LeftFraction W X Y) :\n    homMk f = f.map (Q W) (Localization.inverts _ W) := by\n  have := Localization.inverts (Q W) W f.s f.hs\n  rw [\u2190 Q_map_comp_Qinv f.f f.s f.hs, \u2190 cancel_mono ((Q W).map f.s),\n    assoc, Qiso_inv_hom_id, comp_id, map_comp_map_s]\n\nlemma map_eq_iff {X Y : C} (f g : LeftFraction W X Y) :\n    f.map (LeftFraction.Localization.Q W) (Localization.inverts _ _) =\n        g.map (LeftFraction.Localization.Q W) (Localization.inverts _ _) \u2194\n      LeftFractionRel f g := by\n  simp only [\u2190 Hom.map_mk _ (Q W)]\n  constructor\n  \u00b7 intro h\n    rw [\u2190 homMk_eq_iff_leftFractionRel, homMk_eq, homMk_eq]\n    exact h\n  \u00b7 intro h\n    congr 1\n    exact Quot.sound h\n\nend Localization\n\nsection\n\nlemma map_eq {X Y : C} (\u03c6 : W.LeftFraction X Y) (L : C \u2964 D) [L.IsLocalization W] :\n    \u03c6.map L (Localization.inverts L W) =\n      L.map \u03c6.f \u226b (Localization.isoOfHom L W \u03c6.s \u03c6.hs).inv := rfl\n\nlemma map_compatibility {X Y : C}\n    (\u03c6 : W.LeftFraction X Y) {E : Type*} [Category E]\n    (L\u2081 : C \u2964 D) (L\u2082 : C \u2964 E) [L\u2081.IsLocalization W] [L\u2082.IsLocalization W] :\n    (Localization.uniq L\u2081 L\u2082 W).functor.map (\u03c6.map L\u2081 (Localization.inverts L\u2081 W)) =\n      (Localization.compUniqFunctor L\u2081 L\u2082 W).hom.app X \u226b\n        \u03c6.map L\u2082 (Localization.inverts L\u2082 W) \u226b\n        (Localization.compUniqFunctor L\u2081 L\u2082 W).inv.app Y := by\n  let e := Localization.compUniqFunctor L\u2081 L\u2082 W\n  have := Localization.inverts L\u2082 W \u03c6.s \u03c6.hs\n  rw [\u2190 cancel_mono (e.hom.app Y), assoc, assoc, e.inv_hom_id_app, comp_id,\n    \u2190 cancel_mono (L\u2082.map \u03c6.s), assoc, assoc, map_comp_map_s, \u2190 e.hom.naturality]\n  simpa [\u2190 Functor.map_comp_assoc, map_comp_map_s] using e.hom.naturality \u03c6.f\n\nlemma map_eq_of_map_eq {X Y : C}\n    (\u03c6\u2081 \u03c6\u2082 : W.LeftFraction X Y) {E : Type*} [Category E]\n    (L\u2081 : C \u2964 D) (L\u2082 : C \u2964 E) [L\u2081.IsLocalization W] [L\u2082.IsLocalization W]\n    (h : \u03c6\u2081.map L\u2081 (Localization.inverts L\u2081 W) = \u03c6\u2082.map L\u2081 (Localization.inverts L\u2081 W)) :\n    \u03c6\u2081.map L\u2082 (Localization.inverts L\u2082 W) = \u03c6\u2082.map L\u2082 (Localization.inverts L\u2082 W) := by\n  apply (Localization.uniq L\u2082 L\u2081 W).functor.map_injective\n  rw [map_compatibility \u03c6\u2081 L\u2082 L\u2081, map_compatibility \u03c6\u2082 L\u2082 L\u2081, h]\n\nend\n\nend LeftFraction\n\nend\n\nend MorphismProperty\n\nvariable (L : C \u2964 D) (W : MorphismProperty C) [L.IsLocalization W]\n\nsection\n\nvariable [W.HasLeftCalculusOfFractions]\n\nlemma Localization.exists_leftFraction {X Y : C} (f : L.obj X \u27f6 L.obj Y) :\n    \u2203 (\u03c6 : W.LeftFraction X Y), f = \u03c6.map L (Localization.inverts L W) := by\n  let E := Localization.uniq (MorphismProperty.LeftFraction.Localization.Q W) L W\n  let e : _ \u22d9 E.functor \u2245 L := Localization.compUniqFunctor _ _ _\n  obtain \u27e8f', rfl\u27e9 : \u2203 (f' : E.functor.obj X \u27f6 E.functor.obj Y),\n      f = e.inv.app _ \u226b f' \u226b e.hom.app _ := \u27e8e.hom.app _ \u226b f \u226b e.inv.app _, by simp\u27e9\n  obtain \u27e8g, rfl\u27e9 := E.functor.map_surjective f'\n  obtain \u27e8g, rfl\u27e9 := MorphismProperty.LeftFraction.Localization.Hom.mk_surjective g\n  refine' \u27e8g, _\u27e9\n  rw [\u2190 MorphismProperty.LeftFraction.Localization.homMk_eq_hom_mk,\n    MorphismProperty.LeftFraction.Localization.homMk_eq g,\n    g.map_compatibility (MorphismProperty.LeftFraction.Localization.Q W) L,\n    assoc, assoc, Iso.inv_hom_id_app, comp_id, Iso.inv_hom_id_app_assoc]\n\nlemma MorphismProperty.LeftFraction.map_eq_iff\n    {X Y : C} (\u03c6 \u03c8 : W.LeftFraction X Y) :\n    \u03c6.map L (Localization.inverts _ _) = \u03c8.map L (Localization.inverts _ _) \u2194\n      LeftFractionRel \u03c6 \u03c8 := by\n  constructor\n  \u00b7 intro h\n    rw [\u2190 MorphismProperty.LeftFraction.Localization.map_eq_iff]\n    apply map_eq_of_map_eq _ _ _ _ h\n  \u00b7 intro h\n    simp only [\u2190 Localization.Hom.map_mk _ L (Localization.inverts _ _)]\n    congr 1\n    exact Quot.sound h\n\nlemma MorphismProperty.map_eq_iff_postcomp {X Y : C} (f\u2081 f\u2082 : X \u27f6 Y) :\n    L.map f\u2081 = L.map f\u2082 \u2194 \u2203 (Z : C) (s : Y \u27f6 Z) (_ : W s), f\u2081 \u226b s = f\u2082 \u226b s := by\n  constructor\n  \u00b7 intro h\n    rw [\u2190 LeftFraction.map_ofHom W _ L (Localization.inverts _ _),\n      \u2190 LeftFraction.map_ofHom W _ L (Localization.inverts _ _),\n      LeftFraction.map_eq_iff] at h\n    obtain \u27e8Z, t\u2081, t\u2082, hst, hft, ht\u27e9 := h\n    dsimp at t\u2081 t\u2082 hst hft ht\n    simp only [id_comp] at hst\n    exact \u27e8Z, t\u2081, by simpa using ht, by rw [hft, hst]\u27e9\n  \u00b7 rintro \u27e8Z, s, hs, fac\u27e9\n    simp only [\u2190 cancel_mono (Localization.isoOfHom L W s hs).hom,\n      Localization.isoOfHom_hom, \u2190 L.map_comp, fac]\n\nlemma Localization.essSurj_mapArrow_of_hasLeftCalculusofFractions :\n    L.mapArrow.EssSurj where\n  mem_essImage f := by\n    have := Localization.essSurj L W\n    obtain \u27e8X, \u27e8eX\u27e9\u27e9 : \u2203 (X : C), Nonempty (L.obj X \u2245 f.left) :=\n      \u27e8_, \u27e8L.objObjPreimageIso f.left\u27e9\u27e9\n    obtain \u27e8Y, \u27e8eY\u27e9\u27e9 : \u2203 (Y : C), Nonempty (L.obj Y \u2245 f.right) :=\n      \u27e8_, \u27e8L.objObjPreimageIso f.right\u27e9\u27e9\n    obtain \u27e8\u03c6, h\u03c6\u27e9 := Localization.exists_leftFraction L W (eX.hom \u226b f.hom \u226b eY.inv)\n    refine' \u27e8Arrow.mk \u03c6.f, \u27e8Iso.symm _\u27e9\u27e9\n    refine' Arrow.isoMk eX.symm (eY.symm \u226a\u226b Localization.isoOfHom L W \u03c6.s \u03c6.hs) _\n    dsimp\n    simp only [\u2190 cancel_epi eX.hom, Iso.hom_inv_id_assoc, reassoc_of% h\u03c6,\n      MorphismProperty.LeftFraction.map_comp_map_s]\n\nend\n\n\nnamespace MorphismProperty\n\nvariable {W}\n\n/-- The right fraction in the opposite category corresponding to a left fraction. -/\n@[simps]\ndef LeftFraction.op {X Y : C} (\u03c6 : W.LeftFraction X Y) :\n    W.op.RightFraction (Opposite.op Y) (Opposite.op X) where\n  X' := Opposite.op \u03c6.Y'\n  s := \u03c6.s.op\n  hs := \u03c6.hs\n  f := \u03c6.f.op\n\n/-- The left fraction in the opposite category corresponding to a right fraction. -/\n@[simps]\ndef RightFraction.op {X Y : C} (\u03c6 : W.RightFraction X Y) :\n    W.op.LeftFraction (Opposite.op Y) (Opposite.op X) where\n  Y' := Opposite.op \u03c6.X'\n  s := \u03c6.s.op\n  hs := \u03c6.hs\n  f := \u03c6.f.op\n\n/-- The right fraction corresponding to a left fraction in the opposite category. -/\n@[simps]\ndef LeftFraction.unop {W : MorphismProperty C\u1d52\u1d56}\n    {X Y : C\u1d52\u1d56} (\u03c6 : W.LeftFraction X Y) :\n    W.unop.RightFraction (Opposite.unop Y) (Opposite.unop X) where\n  X' := Opposite.unop \u03c6.Y'\n  s := \u03c6.s.unop\n  hs := \u03c6.hs\n  f := \u03c6.f.unop\n\n/-- The left fraction corresponding to a right fraction in the opposite category. -/\n@[simps]\ndef RightFraction.unop {W : MorphismProperty C\u1d52\u1d56}\n    {X Y : C\u1d52\u1d56} (\u03c6 : W.RightFraction X Y) :\n    W.unop.LeftFraction (Opposite.unop Y) (Opposite.unop X) where\n  Y' := Opposite.unop \u03c6.X'\n  s := \u03c6.s.unop\n  hs := \u03c6.hs\n  f := \u03c6.f.unop\n\nlemma RightFraction.op_map\n    {X Y : C} (\u03c6 : W.RightFraction X Y) (L : C \u2964 D) (hL : W.IsInvertedBy L) :\n    (\u03c6.map L hL).op = \u03c6.op.map L.op hL.op := by\n  dsimp [map, LeftFraction.map]\n  rw [op_inv]\n\n", "theoremStatement": "lemma LeftFraction.op_map\n    {X Y : C} (\u03c6 : W.LeftFraction X Y) (L : C \u2964 D) (hL : W.IsInvertedBy L) :\n    (\u03c6.map L hL).op = \u03c6.op.map L.op hL.op ", "theoremName": "CategoryTheory.MorphismProperty.LeftFraction.op_map", "fileCreated": {"commit": "5a69fafcfcd944ce332f8e3cd38451a52e303960", "date": "2023-10-30"}, "theoremCreated": {"commit": "1c3231190ef98736c0f1f37d0a3d92403c28de79", "date": "2024-03-29"}, "file": "mathlib/Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean", 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"Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Opposite", "Mathlib.Tactic.Cases", "Mathlib.Combinatorics.Quiver.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.Convert", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Spread", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.AssertExists", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.Data.Prod.Basic", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Order.Notation", "Mathlib.Data.Set.Defs", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.MinMax", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.CategoryTheory.FinCategory.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.CategoryTheory.FinCategory.AsType", "Mathlib.CategoryTheory.Limits.Shapes.Equalizers", "Mathlib.Data.Finset.Option", "Mathlib.Data.Fintype.Option", "Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits", "Mathlib.CategoryTheory.Filtered.Basic", "Mathlib.Data.TypeMax", "Mathlib.CategoryTheory.Limits.Shapes.Images", "Mathlib.CategoryTheory.Limits.Types", "Mathlib.CategoryTheory.Limits.Filtered", "Mathlib.CategoryTheory.Limits.Shapes.Products", "Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Terminal", "Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects", "Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero", "Mathlib.CategoryTheory.Limits.Shapes.Kernels", "Mathlib.CategoryTheory.Limits.Opposites", "Mathlib.CategoryTheory.Limits.Shapes.Biproducts", "Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts", "Mathlib.CategoryTheory.Limits.Constructions.ZeroObjects", "Mathlib.CategoryTheory.Limits.Shapes.CommSq", "Mathlib.CategoryTheory.Limits.Shapes.RegularMono", "Mathlib.CategoryTheory.Limits.Shapes.KernelPair", "Mathlib.CategoryTheory.Limits.Shapes.Diagonal", "Mathlib.CategoryTheory.MorphismProperty", "Mathlib.CategoryTheory.Quotient", "Mathlib.CategoryTheory.PathCategory", "Mathlib.CategoryTheory.Category.Quiv", "Mathlib.CategoryTheory.Localization.Construction", "Mathlib.CategoryTheory.Localization.Predicate", "Mathlib.CategoryTheory.Localization.Opposite"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  dsimp [map, RightFraction.map]\n  rw [op_inv]", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 52}}
+{"srcContext": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Mario Carneiro, Floris van Doorn\n-/\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Finsupp.Defs\nimport Mathlib.Data.Set.Countable\nimport Mathlib.Logic.Small.Set\nimport Mathlib.Order.ConditionallyCompleteLattice.Basic\nimport Mathlib.Order.SuccPred.CompleteLinearOrder\nimport Mathlib.SetTheory.Cardinal.SchroederBernstein\nimport Mathlib.Tactic.PPWithUniv\n\n#align_import set_theory.cardinal.basic from \"leanprover-community/mathlib\"@\"3ff3f2d6a3118b8711063de7111a0d77a53219a8\"\n\n/-!\n# Cardinal Numbers\n\nWe define cardinal numbers as a quotient of types under the equivalence relation of equinumerity.\n\n## Main definitions\n\n* `Cardinal` is the type of cardinal numbers (in a given universe).\n* `Cardinal.mk \u03b1` or `#\u03b1` is the cardinality of `\u03b1`. The notation `#` lives in the locale\n  `Cardinal`.\n* Addition `c\u2081 + c\u2082` is defined by `Cardinal.add_def \u03b1 \u03b2 : #\u03b1 + #\u03b2 = #(\u03b1 \u2295 \u03b2)`.\n* Multiplication `c\u2081 * c\u2082` is defined by `Cardinal.mul_def : #\u03b1 * #\u03b2 = #(\u03b1 \u00d7 \u03b2)`.\n* The order `c\u2081 \u2264 c\u2082` is defined by `Cardinal.le_def \u03b1 \u03b2 : #\u03b1 \u2264 #\u03b2 \u2194 Nonempty (\u03b1 \u21aa \u03b2)`.\n* Exponentiation `c\u2081 ^ c\u2082` is defined by `Cardinal.power_def \u03b1 \u03b2 : #\u03b1 ^ #\u03b2 = #(\u03b2 \u2192 \u03b1)`.\n* `Cardinal.isLimit c` means that `c` is a (weak) limit cardinal: `c \u2260 0 \u2227 \u2200 x < c, succ x < c`.\n* `Cardinal.aleph0` or `\u2135\u2080` is the cardinality of `\u2115`. This definition is universe polymorphic:\n  `Cardinal.aleph0.{u} : Cardinal.{u}` (contrast with `\u2115 : Type`, which lives in a specific\n  universe). In some cases the universe level has to be given explicitly.\n* `Cardinal.sum` is the sum of an indexed family of cardinals, i.e. the cardinality of the\n  corresponding sigma type.\n* `Cardinal.prod` is the product of an indexed family of cardinals, i.e. the cardinality of the\n  corresponding pi type.\n* `Cardinal.powerlt a b` or `a ^< b` is defined as the supremum of `a ^ c` for `c < b`.\n\n## Main instances\n\n* Cardinals form a `CanonicallyOrderedCommSemiring` with the aforementioned sum and product.\n* Cardinals form a `SuccOrder`. Use `Order.succ c` for the smallest cardinal greater than `c`.\n* The less than relation on cardinals forms a well-order.\n* Cardinals form a `ConditionallyCompleteLinearOrderBot`. Bounded sets for cardinals in universe\n  `u` are precisely the sets indexed by some type in universe `u`, see\n  `Cardinal.bddAbove_iff_small`. One can use `sSup` for the cardinal supremum, and `sInf` for the\n  minimum of a set of cardinals.\n\n## Main Statements\n\n* Cantor's theorem: `Cardinal.cantor c : c < 2 ^ c`.\n* K\u00f6nig's theorem: `Cardinal.sum_lt_prod`\n\n## Implementation notes\n\n* There is a type of cardinal numbers in every universe level:\n  `Cardinal.{u} : Type (u + 1)` is the quotient of types in `Type u`.\n  The operation `Cardinal.lift` lifts cardinal numbers to a higher level.\n* Cardinal arithmetic specifically for infinite cardinals (like `\u03ba * \u03ba = \u03ba`) is in the file\n  `Mathlib/SetTheory/Cardinal/Ordinal.lean`.\n* There is an instance `Pow Cardinal`, but this will only fire if Lean already knows that both\n  the base and the exponent live in the same universe. As a workaround, you can add\n  ```\n    local infixr:80 \" ^' \" => @HPow.hPow Cardinal Cardinal Cardinal _\n  ```\n  to a file. This notation will work even if Lean doesn't know yet that the base and the exponent\n  live in the same universe (but no exponents in other types can be used).\n  (Porting note: This last point might need to be updated.)\n\n## References\n\n* <https://en.wikipedia.org/wiki/Cardinal_number>\n\n## Tags\n\ncardinal number, cardinal arithmetic, cardinal exponentiation, aleph,\nCantor's theorem, K\u00f6nig's theorem, Konig's theorem\n-/\n\n\nopen scoped Classical\nopen Function Set Order BigOperators\n\nnoncomputable section\n\nuniverse u v w\n\nvariable {\u03b1 \u03b2 : Type u}\n\n/-- The equivalence relation on types given by equivalence (bijective correspondence) of types.\n  Quotienting by this equivalence relation gives the cardinal numbers.\n-/\ninstance Cardinal.isEquivalent : Setoid (Type u) where\n  r \u03b1 \u03b2 := Nonempty (\u03b1 \u2243 \u03b2)\n  iseqv := \u27e8\n    fun \u03b1 => \u27e8Equiv.refl \u03b1\u27e9,\n    fun \u27e8e\u27e9 => \u27e8e.symm\u27e9,\n    fun \u27e8e\u2081\u27e9 \u27e8e\u2082\u27e9 => \u27e8e\u2081.trans e\u2082\u27e9\u27e9\n#align cardinal.is_equivalent Cardinal.isEquivalent\n\n/-- `Cardinal.{u}` is the type of cardinal numbers in `Type u`,\n  defined as the quotient of `Type u` by existence of an equivalence\n  (a bijection with explicit inverse). -/\n@[pp_with_univ]\ndef Cardinal : Type (u + 1) :=\n  Quotient Cardinal.isEquivalent\n#align cardinal Cardinal\n\nnamespace Cardinal\n\n/-- The cardinal number of a type -/\ndef mk : Type u \u2192 Cardinal :=\n  Quotient.mk'\n#align cardinal.mk Cardinal.mk\n\n@[inherit_doc]\nscoped prefix:max \"#\" => Cardinal.mk\n\ninstance canLiftCardinalType : CanLift Cardinal.{u} (Type u) mk fun _ => True :=\n  \u27e8fun c _ => Quot.inductionOn c fun \u03b1 => \u27e8\u03b1, rfl\u27e9\u27e9\n#align cardinal.can_lift_cardinal_Type Cardinal.canLiftCardinalType\n\n@[elab_as_elim]\ntheorem inductionOn {p : Cardinal \u2192 Prop} (c : Cardinal) (h : \u2200 \u03b1, p #\u03b1) : p c :=\n  Quotient.inductionOn c h\n#align cardinal.induction_on Cardinal.inductionOn\n\n@[elab_as_elim]\ntheorem inductionOn\u2082 {p : Cardinal \u2192 Cardinal \u2192 Prop} (c\u2081 : Cardinal) (c\u2082 : Cardinal)\n    (h : \u2200 \u03b1 \u03b2, p #\u03b1 #\u03b2) : p c\u2081 c\u2082 :=\n  Quotient.inductionOn\u2082 c\u2081 c\u2082 h\n#align cardinal.induction_on\u2082 Cardinal.inductionOn\u2082\n\n@[elab_as_elim]\ntheorem inductionOn\u2083 {p : Cardinal \u2192 Cardinal \u2192 Cardinal \u2192 Prop} (c\u2081 : Cardinal) (c\u2082 : Cardinal)\n    (c\u2083 : Cardinal) (h : \u2200 \u03b1 \u03b2 \u03b3, p #\u03b1 #\u03b2 #\u03b3) : p c\u2081 c\u2082 c\u2083 :=\n  Quotient.inductionOn\u2083 c\u2081 c\u2082 c\u2083 h\n#align cardinal.induction_on\u2083 Cardinal.inductionOn\u2083\n\nprotected theorem eq : #\u03b1 = #\u03b2 \u2194 Nonempty (\u03b1 \u2243 \u03b2) :=\n  Quotient.eq'\n#align cardinal.eq Cardinal.eq\n\n@[simp]\ntheorem mk'_def (\u03b1 : Type u) : @Eq Cardinal \u27e6\u03b1\u27e7 #\u03b1 :=\n  rfl\n#align cardinal.mk_def Cardinal.mk'_def\n\n@[simp]\ntheorem mk_out (c : Cardinal) : #c.out = c :=\n  Quotient.out_eq _\n#align cardinal.mk_out Cardinal.mk_out\n\n/-- The representative of the cardinal of a type is equivalent to the original type. -/\ndef outMkEquiv {\u03b1 : Type v} : (#\u03b1).out \u2243 \u03b1 :=\n  Nonempty.some <| Cardinal.eq.mp (by simp)\n#align cardinal.out_mk_equiv Cardinal.outMkEquiv\n\ntheorem mk_congr (e : \u03b1 \u2243 \u03b2) : #\u03b1 = #\u03b2 :=\n  Quot.sound \u27e8e\u27e9\n#align cardinal.mk_congr Cardinal.mk_congr\n\nalias _root_.Equiv.cardinal_eq := mk_congr\n#align equiv.cardinal_eq Equiv.cardinal_eq\n\n/-- Lift a function between `Type*`s to a function between `Cardinal`s. -/\ndef map (f : Type u \u2192 Type v) (hf : \u2200 \u03b1 \u03b2, \u03b1 \u2243 \u03b2 \u2192 f \u03b1 \u2243 f \u03b2) : Cardinal.{u} \u2192 Cardinal.{v} :=\n  Quotient.map f fun \u03b1 \u03b2 \u27e8e\u27e9 => \u27e8hf \u03b1 \u03b2 e\u27e9\n#align cardinal.map Cardinal.map\n\n@[simp]\ntheorem map_mk (f : Type u \u2192 Type v) (hf : \u2200 \u03b1 \u03b2, \u03b1 \u2243 \u03b2 \u2192 f \u03b1 \u2243 f \u03b2) (\u03b1 : Type u) :\n    map f hf #\u03b1 = #(f \u03b1) :=\n  rfl\n#align cardinal.map_mk Cardinal.map_mk\n\n/-- Lift a binary operation `Type* \u2192 Type* \u2192 Type*` to a binary operation on `Cardinal`s. -/\ndef map\u2082 (f : Type u \u2192 Type v \u2192 Type w) (hf : \u2200 \u03b1 \u03b2 \u03b3 \u03b4, \u03b1 \u2243 \u03b2 \u2192 \u03b3 \u2243 \u03b4 \u2192 f \u03b1 \u03b3 \u2243 f \u03b2 \u03b4) :\n    Cardinal.{u} \u2192 Cardinal.{v} \u2192 Cardinal.{w} :=\n  Quotient.map\u2082 f fun \u03b1 \u03b2 \u27e8e\u2081\u27e9 \u03b3 \u03b4 \u27e8e\u2082\u27e9 => \u27e8hf \u03b1 \u03b2 \u03b3 \u03b4 e\u2081 e\u2082\u27e9\n#align cardinal.map\u2082 Cardinal.map\u2082\n\n/-- The universe lift operation on cardinals. You can specify the universes explicitly with\n  `lift.{u v} : Cardinal.{v} \u2192 Cardinal.{max v u}` -/\n@[pp_with_univ]\ndef lift (c : Cardinal.{v}) : Cardinal.{max v u} :=\n  map ULift.{u, v} (fun _ _ e => Equiv.ulift.trans <| e.trans Equiv.ulift.symm) c\n#align cardinal.lift Cardinal.lift\n\n@[simp]\ntheorem mk_uLift (\u03b1) : #(ULift.{v, u} \u03b1) = lift.{v} #\u03b1 :=\n  rfl\n#align cardinal.mk_ulift Cardinal.mk_uLift\n\n-- Porting note: simpNF is not happy with universe levels, but this is needed as simp lemma\n-- further down in this file\n/-- `lift.{max u v, u}` equals `lift.{v, u}`. -/\n@[simp, nolint simpNF]\ntheorem lift_umax : lift.{max u v, u} = lift.{v, u} :=\n  funext fun a => inductionOn a fun _ => (Equiv.ulift.trans Equiv.ulift.symm).cardinal_eq\n#align cardinal.lift_umax Cardinal.lift_umax\n\n-- Porting note: simpNF is not happy with universe levels, but this is needed as simp lemma\n-- further down in this file\n/-- `lift.{max v u, u}` equals `lift.{v, u}`. -/\n@[simp, nolint simpNF]\ntheorem lift_umax' : lift.{max v u, u} = lift.{v, u} :=\n  lift_umax\n#align cardinal.lift_umax' Cardinal.lift_umax'\n\n-- Porting note: simpNF is not happy with universe levels, but this is needed as simp lemma\n-- further down in this file\n/-- A cardinal lifted to a lower or equal universe equals itself. -/\n@[simp, nolint simpNF]\ntheorem lift_id' (a : Cardinal.{max u v}) : lift.{u} a = a :=\n  inductionOn a fun _ => mk_congr Equiv.ulift\n#align cardinal.lift_id' Cardinal.lift_id'\n\n/-- A cardinal lifted to the same universe equals itself. -/\n@[simp]\ntheorem lift_id (a : Cardinal) : lift.{u, u} a = a :=\n  lift_id'.{u, u} a\n#align cardinal.lift_id Cardinal.lift_id\n\n/-- A cardinal lifted to the zero universe equals itself. -/\n-- porting note (#10618): simp can prove this\n-- @[simp]\ntheorem lift_uzero (a : Cardinal.{u}) : lift.{0} a = a :=\n  lift_id'.{0, u} a\n#align cardinal.lift_uzero Cardinal.lift_uzero\n\n@[simp]\ntheorem lift_lift.{u_1} (a : Cardinal.{u_1}) : lift.{w} (lift.{v} a) = lift.{max v w} a :=\n  inductionOn a fun _ => (Equiv.ulift.trans <| Equiv.ulift.trans Equiv.ulift.symm).cardinal_eq\n#align cardinal.lift_lift Cardinal.lift_lift\n\n/-- We define the order on cardinal numbers by `#\u03b1 \u2264 #\u03b2` if and only if\n  there exists an embedding (injective function) from \u03b1 to \u03b2. -/\ninstance : LE Cardinal.{u} :=\n  \u27e8fun q\u2081 q\u2082 =>\n    Quotient.liftOn\u2082 q\u2081 q\u2082 (fun \u03b1 \u03b2 => Nonempty <| \u03b1 \u21aa \u03b2) fun _ _ _ _ \u27e8e\u2081\u27e9 \u27e8e\u2082\u27e9 =>\n      propext \u27e8fun \u27e8e\u27e9 => \u27e8e.congr e\u2081 e\u2082\u27e9, fun \u27e8e\u27e9 => \u27e8e.congr e\u2081.symm e\u2082.symm\u27e9\u27e9\u27e9\n\ninstance partialOrder : PartialOrder Cardinal.{u} where\n  le := (\u00b7 \u2264 \u00b7)\n  le_refl := by\n    rintro \u27e8\u03b1\u27e9\n    exact \u27e8Embedding.refl _\u27e9\n  le_trans := by\n    rintro \u27e8\u03b1\u27e9 \u27e8\u03b2\u27e9 \u27e8\u03b3\u27e9 \u27e8e\u2081\u27e9 \u27e8e\u2082\u27e9\n    exact \u27e8e\u2081.trans e\u2082\u27e9\n  le_antisymm := by\n    rintro \u27e8\u03b1\u27e9 \u27e8\u03b2\u27e9 \u27e8e\u2081\u27e9 \u27e8e\u2082\u27e9\n    exact Quotient.sound (e\u2081.antisymm e\u2082)\n\ninstance linearOrder : LinearOrder Cardinal.{u} :=\n  { Cardinal.partialOrder with\n    le_total := by\n      rintro \u27e8\u03b1\u27e9 \u27e8\u03b2\u27e9\n      apply Embedding.total\n    decidableLE := Classical.decRel _ }\n\ntheorem le_def (\u03b1 \u03b2 : Type u) : #\u03b1 \u2264 #\u03b2 \u2194 Nonempty (\u03b1 \u21aa \u03b2) :=\n  Iff.rfl\n#align cardinal.le_def Cardinal.le_def\n\ntheorem mk_le_of_injective {\u03b1 \u03b2 : Type u} {f : \u03b1 \u2192 \u03b2} (hf : Injective f) : #\u03b1 \u2264 #\u03b2 :=\n  \u27e8\u27e8f, hf\u27e9\u27e9\n#align cardinal.mk_le_of_injective Cardinal.mk_le_of_injective\n\ntheorem _root_.Function.Embedding.cardinal_le {\u03b1 \u03b2 : Type u} (f : \u03b1 \u21aa \u03b2) : #\u03b1 \u2264 #\u03b2 :=\n  \u27e8f\u27e9\n#align function.embedding.cardinal_le Function.Embedding.cardinal_le\n\ntheorem mk_le_of_surjective {\u03b1 \u03b2 : Type u} {f : \u03b1 \u2192 \u03b2} (hf : Surjective f) : #\u03b2 \u2264 #\u03b1 :=\n  \u27e8Embedding.ofSurjective f hf\u27e9\n#align cardinal.mk_le_of_surjective Cardinal.mk_le_of_surjective\n\ntheorem le_mk_iff_exists_set {c : Cardinal} {\u03b1 : Type u} : c \u2264 #\u03b1 \u2194 \u2203 p : Set \u03b1, #p = c :=\n  \u27e8inductionOn c fun _ \u27e8\u27e8f, hf\u27e9\u27e9 => \u27e8Set.range f, (Equiv.ofInjective f hf).cardinal_eq.symm\u27e9,\n    fun \u27e8_, e\u27e9 => e \u25b8 \u27e8\u27e8Subtype.val, fun _ _ => Subtype.eq\u27e9\u27e9\u27e9\n#align cardinal.le_mk_iff_exists_set Cardinal.le_mk_iff_exists_set\n\ntheorem mk_subtype_le {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) : #(Subtype p) \u2264 #\u03b1 :=\n  \u27e8Embedding.subtype p\u27e9\n#align cardinal.mk_subtype_le Cardinal.mk_subtype_le\n\ntheorem mk_set_le (s : Set \u03b1) : #s \u2264 #\u03b1 :=\n  mk_subtype_le s\n#align cardinal.mk_set_le Cardinal.mk_set_le\n\n@[simp]\nlemma mk_preimage_down {s : Set \u03b1} : #(ULift.down.{v} \u207b\u00b9' s) = lift.{v} (#s) := by\n  rw [\u2190 mk_uLift, Cardinal.eq]\n  constructor\n  let f : ULift.down \u207b\u00b9' s \u2192 ULift s := fun x \u21a6 ULift.up (restrictPreimage s ULift.down x)\n  have : Function.Bijective f :=\n    ULift.up_bijective.comp (restrictPreimage_bijective _ (ULift.down_bijective))\n  exact Equiv.ofBijective f this\n\ntheorem out_embedding {c c' : Cardinal} : c \u2264 c' \u2194 Nonempty (c.out \u21aa c'.out) := by\n  trans\n  \u00b7 rw [\u2190 Quotient.out_eq c, \u2190 Quotient.out_eq c']\n  \u00b7 rw [mk'_def, mk'_def, le_def]\n#align cardinal.out_embedding Cardinal.out_embedding\n\ntheorem lift_mk_le {\u03b1 : Type v} {\u03b2 : Type w} :\n    lift.{max u w} #\u03b1 \u2264 lift.{max u v} #\u03b2 \u2194 Nonempty (\u03b1 \u21aa \u03b2) :=\n  \u27e8fun \u27e8f\u27e9 => \u27e8Embedding.congr Equiv.ulift Equiv.ulift f\u27e9, fun \u27e8f\u27e9 =>\n    \u27e8Embedding.congr Equiv.ulift.symm Equiv.ulift.symm f\u27e9\u27e9\n#align cardinal.lift_mk_le Cardinal.lift_mk_le\n\n/-- A variant of `Cardinal.lift_mk_le` with specialized universes.\nBecause Lean often can not realize it should use this specialization itself,\nwe provide this statement separately so you don't have to solve the specialization problem either.\n-/\ntheorem lift_mk_le' {\u03b1 : Type u} {\u03b2 : Type v} : lift.{v} #\u03b1 \u2264 lift.{u} #\u03b2 \u2194 Nonempty (\u03b1 \u21aa \u03b2) :=\n  lift_mk_le.{0}\n#align cardinal.lift_mk_le' Cardinal.lift_mk_le'\n\ntheorem lift_mk_eq {\u03b1 : Type u} {\u03b2 : Type v} :\n    lift.{max v w} #\u03b1 = lift.{max u w} #\u03b2 \u2194 Nonempty (\u03b1 \u2243 \u03b2) :=\n  Quotient.eq'.trans\n    \u27e8fun \u27e8f\u27e9 => \u27e8Equiv.ulift.symm.trans <| f.trans Equiv.ulift\u27e9, fun \u27e8f\u27e9 =>\n      \u27e8Equiv.ulift.trans <| f.trans Equiv.ulift.symm\u27e9\u27e9\n#align cardinal.lift_mk_eq Cardinal.lift_mk_eq\n\n/-- A variant of `Cardinal.lift_mk_eq` with specialized universes.\nBecause Lean often can not realize it should use this specialization itself,\nwe provide this statement separately so you don't have to solve the specialization problem either.\n-/\ntheorem lift_mk_eq' {\u03b1 : Type u} {\u03b2 : Type v} : lift.{v} #\u03b1 = lift.{u} #\u03b2 \u2194 Nonempty (\u03b1 \u2243 \u03b2) :=\n  lift_mk_eq.{u, v, 0}\n#align cardinal.lift_mk_eq' Cardinal.lift_mk_eq'\n\n@[simp]\ntheorem lift_le {a b : Cardinal.{v}} : lift.{u, v} a \u2264 lift.{u, v} b \u2194 a \u2264 b :=\n  inductionOn\u2082 a b fun \u03b1 \u03b2 => by\n    rw [\u2190 lift_umax]\n    exact lift_mk_le.{u}\n#align cardinal.lift_le Cardinal.lift_le\n\n-- Porting note: changed `simps` to `simps!` because the linter told to do so.\n/-- `Cardinal.lift` as an `OrderEmbedding`. -/\n@[simps! (config := .asFn)]\ndef liftOrderEmbedding : Cardinal.{v} \u21aao Cardinal.{max v u} :=\n  OrderEmbedding.ofMapLEIff lift.{u, v} fun _ _ => lift_le\n#align cardinal.lift_order_embedding Cardinal.liftOrderEmbedding\n\ntheorem lift_injective : Injective lift.{u, v} :=\n  liftOrderEmbedding.injective\n#align cardinal.lift_injective Cardinal.lift_injective\n\n@[simp]\ntheorem lift_inj {a b : Cardinal.{u}} : lift.{v, u} a = lift.{v, u} b \u2194 a = b :=\n  lift_injective.eq_iff\n#align cardinal.lift_inj Cardinal.lift_inj\n\n@[simp]\ntheorem lift_lt {a b : Cardinal.{u}} : lift.{v, u} a < lift.{v, u} b \u2194 a < b :=\n  liftOrderEmbedding.lt_iff_lt\n#align cardinal.lift_lt Cardinal.lift_lt\n\ntheorem lift_strictMono : StrictMono lift := fun _ _ => lift_lt.2\n#align cardinal.lift_strict_mono Cardinal.lift_strictMono\n\ntheorem lift_monotone : Monotone lift :=\n  lift_strictMono.monotone\n#align cardinal.lift_monotone Cardinal.lift_monotone\n\ninstance : Zero Cardinal.{u} :=\n  -- `PEmpty` might be more canonical, but this is convenient for defeq with natCast\n  \u27e8lift #(Fin 0)\u27e9\n\ninstance : Inhabited Cardinal.{u} :=\n  \u27e80\u27e9\n\n@[simp]\ntheorem mk_eq_zero (\u03b1 : Type u) [IsEmpty \u03b1] : #\u03b1 = 0 :=\n  (Equiv.equivOfIsEmpty \u03b1 (ULift (Fin 0))).cardinal_eq\n#align cardinal.mk_eq_zero Cardinal.mk_eq_zero\n\n@[simp]\ntheorem lift_zero : lift 0 = 0 := mk_eq_zero _\n#align cardinal.lift_zero Cardinal.lift_zero\n\n@[simp]\ntheorem lift_eq_zero {a : Cardinal.{v}} : lift.{u} a = 0 \u2194 a = 0 :=\n  lift_injective.eq_iff' lift_zero\n#align cardinal.lift_eq_zero Cardinal.lift_eq_zero\n\ntheorem mk_eq_zero_iff {\u03b1 : Type u} : #\u03b1 = 0 \u2194 IsEmpty \u03b1 :=\n  \u27e8fun e =>\n    let \u27e8h\u27e9 := Quotient.exact e\n    h.isEmpty,\n    @mk_eq_zero \u03b1\u27e9\n#align cardinal.mk_eq_zero_iff Cardinal.mk_eq_zero_iff\n\ntheorem mk_ne_zero_iff {\u03b1 : Type u} : #\u03b1 \u2260 0 \u2194 Nonempty \u03b1 :=\n  (not_iff_not.2 mk_eq_zero_iff).trans not_isEmpty_iff\n#align cardinal.mk_ne_zero_iff Cardinal.mk_ne_zero_iff\n\n@[simp]\ntheorem mk_ne_zero (\u03b1 : Type u) [Nonempty \u03b1] : #\u03b1 \u2260 0 :=\n  mk_ne_zero_iff.2 \u2039_\u203a\n#align cardinal.mk_ne_zero Cardinal.mk_ne_zero\n\ninstance : One Cardinal.{u} :=\n  -- `PUnit` might be more canonical, but this is convenient for defeq with natCast\n  \u27e8lift #(Fin 1)\u27e9\n\ninstance : Nontrivial Cardinal.{u} :=\n  \u27e8\u27e81, 0, mk_ne_zero _\u27e9\u27e9\n\ntheorem mk_eq_one (\u03b1 : Type u) [Unique \u03b1] : #\u03b1 = 1 :=\n  (Equiv.equivOfUnique \u03b1 (ULift (Fin 1))).cardinal_eq\n#align cardinal.mk_eq_one Cardinal.mk_eq_one\n\ntheorem le_one_iff_subsingleton {\u03b1 : Type u} : #\u03b1 \u2264 1 \u2194 Subsingleton \u03b1 :=\n  \u27e8fun \u27e8f\u27e9 => \u27e8fun _ _ => f.injective (Subsingleton.elim _ _)\u27e9, fun \u27e8h\u27e9 =>\n    \u27e8fun _ => ULift.up 0, fun _ _ _ => h _ _\u27e9\u27e9\n#align cardinal.le_one_iff_subsingleton Cardinal.le_one_iff_subsingleton\n\n@[simp]\ntheorem mk_le_one_iff_set_subsingleton {s : Set \u03b1} : #s \u2264 1 \u2194 s.Subsingleton :=\n  le_one_iff_subsingleton.trans s.subsingleton_coe\n#align cardinal.mk_le_one_iff_set_subsingleton Cardinal.mk_le_one_iff_set_subsingleton\n\nalias \u27e8_, _root_.Set.Subsingleton.cardinal_mk_le_one\u27e9 := mk_le_one_iff_set_subsingleton\n#align set.subsingleton.cardinal_mk_le_one Set.Subsingleton.cardinal_mk_le_one\n\ninstance : Add Cardinal.{u} :=\n  \u27e8map\u2082 Sum fun _ _ _ _ => Equiv.sumCongr\u27e9\n\ntheorem add_def (\u03b1 \u03b2 : Type u) : #\u03b1 + #\u03b2 = #(Sum \u03b1 \u03b2) :=\n  rfl\n#align cardinal.add_def Cardinal.add_def\n\ninstance : NatCast Cardinal.{u} :=\n  \u27e8fun n => lift #(Fin n)\u27e9\n\n@[simp]\ntheorem mk_sum (\u03b1 : Type u) (\u03b2 : Type v) : #(\u03b1 \u2295 \u03b2) = lift.{v, u} #\u03b1 + lift.{u, v} #\u03b2 :=\n  mk_congr (Equiv.ulift.symm.sumCongr Equiv.ulift.symm)\n#align cardinal.mk_sum Cardinal.mk_sum\n\n@[simp]\ntheorem mk_option {\u03b1 : Type u} : #(Option \u03b1) = #\u03b1 + 1 := by\n  rw [(Equiv.optionEquivSumPUnit.{u, u} \u03b1).cardinal_eq, mk_sum, mk_eq_one PUnit, lift_id, lift_id]\n#align cardinal.mk_option Cardinal.mk_option\n\n@[simp]\ntheorem mk_psum (\u03b1 : Type u) (\u03b2 : Type v) : #(PSum \u03b1 \u03b2) = lift.{v} #\u03b1 + lift.{u} #\u03b2 :=\n  (mk_congr (Equiv.psumEquivSum \u03b1 \u03b2)).trans (mk_sum \u03b1 \u03b2)\n#align cardinal.mk_psum Cardinal.mk_psum\n\n@[simp]\ntheorem mk_fintype (\u03b1 : Type u) [h : Fintype \u03b1] : #\u03b1 = Fintype.card \u03b1 :=\n  mk_congr (Fintype.equivOfCardEq (by simp))\n\nprotected theorem cast_succ (n : \u2115) : ((n + 1 : \u2115) : Cardinal.{u}) = n + 1 := by\n  change #(ULift.{u} (Fin (n+1))) = # (ULift.{u} (Fin n)) + 1\n  rw [\u2190 mk_option, mk_fintype, mk_fintype]\n  simp only [Fintype.card_ulift, Fintype.card_fin, Fintype.card_option]\n\ninstance : Mul Cardinal.{u} :=\n  \u27e8map\u2082 Prod fun _ _ _ _ => Equiv.prodCongr\u27e9\n\ntheorem mul_def (\u03b1 \u03b2 : Type u) : #\u03b1 * #\u03b2 = #(\u03b1 \u00d7 \u03b2) :=\n  rfl\n#align cardinal.mul_def Cardinal.mul_def\n\n@[simp]\ntheorem mk_prod (\u03b1 : Type u) (\u03b2 : Type v) : #(\u03b1 \u00d7 \u03b2) = lift.{v, u} #\u03b1 * lift.{u, v} #\u03b2 :=\n  mk_congr (Equiv.ulift.symm.prodCongr Equiv.ulift.symm)\n#align cardinal.mk_prod Cardinal.mk_prod\n\nprivate theorem mul_comm' (a b : Cardinal.{u}) : a * b = b * a :=\n  inductionOn\u2082 a b fun \u03b1 \u03b2 => mk_congr <| Equiv.prodComm \u03b1 \u03b2\n\n/-- The cardinal exponential. `#\u03b1 ^ #\u03b2` is the cardinal of `\u03b2 \u2192 \u03b1`. -/\ninstance instPowCardinal : Pow Cardinal.{u} Cardinal.{u} :=\n  \u27e8map\u2082 (fun \u03b1 \u03b2 => \u03b2 \u2192 \u03b1) fun _ _ _ _ e\u2081 e\u2082 => e\u2082.arrowCongr e\u2081\u27e9\n\ntheorem power_def (\u03b1 \u03b2 : Type u) : #\u03b1 ^ #\u03b2 = #(\u03b2 \u2192 \u03b1) :=\n  rfl\n#align cardinal.power_def Cardinal.power_def\n\ntheorem mk_arrow (\u03b1 : Type u) (\u03b2 : Type v) : #(\u03b1 \u2192 \u03b2) = (lift.{u} #\u03b2^lift.{v} #\u03b1) :=\n  mk_congr (Equiv.ulift.symm.arrowCongr Equiv.ulift.symm)\n#align cardinal.mk_arrow Cardinal.mk_arrow\n\n@[simp]\ntheorem lift_power (a b : Cardinal.{u}) : lift.{v} (a ^ b) = lift.{v} a ^ lift.{v} b :=\n  inductionOn\u2082 a b fun _ _ =>\n    mk_congr <| Equiv.ulift.trans (Equiv.ulift.arrowCongr Equiv.ulift).symm\n#align cardinal.lift_power Cardinal.lift_power\n\n@[simp]\ntheorem power_zero {a : Cardinal} : a ^ (0 : Cardinal) = 1 :=\n  inductionOn a fun _ => mk_eq_one _\n#align cardinal.power_zero Cardinal.power_zero\n\n@[simp]\ntheorem power_one {a : Cardinal.{u}} : a ^ (1 : Cardinal) = a :=\n  inductionOn a fun \u03b1 => mk_congr (Equiv.funUnique (ULift.{u} (Fin 1)) \u03b1)\n#align cardinal.power_one Cardinal.power_one\n\ntheorem power_add {a b c : Cardinal} : a ^ (b + c) = a ^ b * a ^ c :=\n  inductionOn\u2083 a b c fun \u03b1 \u03b2 \u03b3 => mk_congr <| Equiv.sumArrowEquivProdArrow \u03b2 \u03b3 \u03b1\n#align cardinal.power_add Cardinal.power_add\n\ninstance commSemiring : CommSemiring Cardinal.{u} where\n  zero := 0\n  one := 1\n  add := (\u00b7 + \u00b7)\n  mul := (\u00b7 * \u00b7)\n  zero_add a := inductionOn a fun \u03b1 => mk_congr <| Equiv.emptySum (ULift (Fin 0)) \u03b1\n  add_zero a := inductionOn a fun \u03b1 => mk_congr <| Equiv.sumEmpty \u03b1 (ULift (Fin 0))\n  add_assoc a b c := inductionOn\u2083 a b c fun \u03b1 \u03b2 \u03b3 => mk_congr <| Equiv.sumAssoc \u03b1 \u03b2 \u03b3\n  add_comm a b := inductionOn\u2082 a b fun \u03b1 \u03b2 => mk_congr <| Equiv.sumComm \u03b1 \u03b2\n  zero_mul a := inductionOn a fun \u03b1 => mk_eq_zero _\n  mul_zero a := inductionOn a fun \u03b1 => mk_eq_zero _\n  one_mul a := inductionOn a fun \u03b1 => mk_congr <| Equiv.uniqueProd \u03b1 (ULift (Fin 1))\n  mul_one a := inductionOn a fun \u03b1 => mk_congr <| Equiv.prodUnique \u03b1 (ULift (Fin 1))\n  mul_assoc a b c := inductionOn\u2083 a b c fun \u03b1 \u03b2 \u03b3 => mk_congr <| Equiv.prodAssoc \u03b1 \u03b2 \u03b3\n  mul_comm := mul_comm'\n  left_distrib a b c := inductionOn\u2083 a b c fun \u03b1 \u03b2 \u03b3 => mk_congr <| Equiv.prodSumDistrib \u03b1 \u03b2 \u03b3\n  right_distrib a b c := inductionOn\u2083 a b c fun \u03b1 \u03b2 \u03b3 => mk_congr <| Equiv.sumProdDistrib \u03b1 \u03b2 \u03b3\n  nsmul := nsmulRec\n  npow n c := c ^ (n : Cardinal)\n  npow_zero := @power_zero\n  npow_succ n c := show c ^ (\u2191(n + 1) : Cardinal) = c ^ (\u2191n : Cardinal) * c\n    by rw [Cardinal.cast_succ, power_add, power_one, mul_comm']\n  natCast := (fun n => lift.{u} #(Fin n) : \u2115 \u2192 Cardinal.{u})\n  natCast_zero := rfl\n  natCast_succ := Cardinal.cast_succ\n\n/-! Porting note (#11229): Deprecated section. Remove. -/\nsection deprecated\nset_option linter.deprecated false\n\n@[deprecated]\ntheorem power_bit0 (a b : Cardinal) : a ^ bit0 b = a ^ b * a ^ b :=\n  power_add\n#align cardinal.power_bit0 Cardinal.power_bit0\n\n@[deprecated]\ntheorem power_bit1 (a b : Cardinal) : a ^ bit1 b = a ^ b * a ^ b * a := by\n  rw [bit1, \u2190 power_bit0, power_add, power_one]\n#align cardinal.power_bit1 Cardinal.power_bit1\n\nend deprecated\n\n@[simp]\ntheorem one_power {a : Cardinal} : (1 : Cardinal) ^ a = 1 :=\n  inductionOn a fun _ => mk_eq_one _\n#align cardinal.one_power Cardinal.one_power\n\n-- porting note (#10618): simp can prove this\n-- @[simp]\ntheorem mk_bool : #Bool = 2 := by simp\n#align cardinal.mk_bool Cardinal.mk_bool\n\n-- porting note (#10618): simp can prove this\n-- @[simp]\ntheorem mk_Prop : #Prop = 2 := by simp\n#align cardinal.mk_Prop Cardinal.mk_Prop\n\n@[simp]\ntheorem zero_power {a : Cardinal} : a \u2260 0 \u2192 (0 : Cardinal) ^ a = 0 :=\n  inductionOn a fun _ heq =>\n    mk_eq_zero_iff.2 <|\n      isEmpty_pi.2 <|\n        let \u27e8a\u27e9 := mk_ne_zero_iff.1 heq\n        \u27e8a, inferInstance\u27e9\n#align cardinal.zero_power Cardinal.zero_power\n\ntheorem power_ne_zero {a : Cardinal} (b : Cardinal) : a \u2260 0 \u2192 a ^ b \u2260 0 :=\n  inductionOn\u2082 a b fun _ _ h =>\n    let \u27e8a\u27e9 := mk_ne_zero_iff.1 h\n    mk_ne_zero_iff.2 \u27e8fun _ => a\u27e9\n#align cardinal.power_ne_zero Cardinal.power_ne_zero\n\ntheorem mul_power {a b c : Cardinal} : (a * b) ^ c = a ^ c * b ^ c :=\n  inductionOn\u2083 a b c fun \u03b1 \u03b2 \u03b3 => mk_congr <| Equiv.arrowProdEquivProdArrow \u03b1 \u03b2 \u03b3\n#align cardinal.mul_power Cardinal.mul_power\n\ntheorem power_mul {a b c : Cardinal} : a ^ (b * c) = (a ^ b) ^ c := by\n  rw [mul_comm b c]\n  exact inductionOn\u2083 a b c fun \u03b1 \u03b2 \u03b3 => mk_congr <| Equiv.curry \u03b3 \u03b2 \u03b1\n#align cardinal.power_mul Cardinal.power_mul\n\n@[simp]\ntheorem pow_cast_right (a : Cardinal.{u}) (n : \u2115) : a ^ (\u2191n : Cardinal.{u}) = a ^ n :=\n  rfl\n#align cardinal.pow_cast_right Cardinal.pow_cast_right\n\n@[simp]\ntheorem lift_one : lift 1 = 1 := mk_eq_one _\n#align cardinal.lift_one Cardinal.lift_one\n\n@[simp]\ntheorem lift_eq_one {a : Cardinal.{v}} : lift.{u} a = 1 \u2194 a = 1 :=\n  lift_injective.eq_iff' lift_one\n\n@[simp]\ntheorem lift_add (a b : Cardinal.{u}) : lift.{v} (a + b) = lift.{v} a + lift.{v} b :=\n  inductionOn\u2082 a b fun _ _ =>\n    mk_congr <| Equiv.ulift.trans (Equiv.sumCongr Equiv.ulift Equiv.ulift).symm\n#align cardinal.lift_add Cardinal.lift_add\n\n@[simp]\ntheorem lift_mul (a b : Cardinal.{u}) : lift.{v} (a * b) = lift.{v} a * lift.{v} b :=\n  inductionOn\u2082 a b fun _ _ =>\n    mk_congr <| Equiv.ulift.trans (Equiv.prodCongr Equiv.ulift Equiv.ulift).symm\n#align cardinal.lift_mul Cardinal.lift_mul\n\n/-! Porting note (#11229): Deprecated section. Remove. -/\nsection deprecated\nset_option linter.deprecated false\n\n@[simp, deprecated]\ntheorem lift_bit0 (a : Cardinal) : lift.{v} (bit0 a) = bit0 (lift.{v} a) :=\n  lift_add a a\n#align cardinal.lift_bit0 Cardinal.lift_bit0\n\n@[simp, deprecated]\ntheorem lift_bit1 (a : Cardinal) : lift.{v} (bit1 a) = bit1 (lift.{v} a) := by simp [bit1]\n#align cardinal.lift_bit1 Cardinal.lift_bit1\n\nend deprecated\n\n-- Porting note: Proof used to be simp, needed to remind simp that 1 + 1 = 2\ntheorem lift_two : lift.{u, v} 2 = 2 := by simp [\u2190 one_add_one_eq_two]\n#align cardinal.lift_two Cardinal.lift_two\n\n@[simp]\ntheorem mk_set {\u03b1 : Type u} : #(Set \u03b1) = 2 ^ #\u03b1 := by simp [\u2190 one_add_one_eq_two, Set, mk_arrow]\n#align cardinal.mk_set Cardinal.mk_set\n\n/-- A variant of `Cardinal.mk_set` expressed in terms of a `Set` instead of a `Type`. -/\n@[simp]\ntheorem mk_powerset {\u03b1 : Type u} (s : Set \u03b1) : #(\u21a5(\ud835\udcab s)) = 2 ^ #(\u21a5s) :=\n  (mk_congr (Equiv.Set.powerset s)).trans mk_set\n#align cardinal.mk_powerset Cardinal.mk_powerset\n\ntheorem lift_two_power (a : Cardinal) : lift.{v} (2 ^ a) = 2 ^ lift.{v} a := by\n  simp [\u2190 one_add_one_eq_two]\n#align cardinal.lift_two_power Cardinal.lift_two_power\n\nsection OrderProperties\n\nopen Sum\n\nprotected theorem zero_le : \u2200 a : Cardinal, 0 \u2264 a := by\n  rintro \u27e8\u03b1\u27e9\n  exact \u27e8Embedding.ofIsEmpty\u27e9\n#align cardinal.zero_le Cardinal.zero_le\n\nprivate theorem add_le_add' : \u2200 {a b c d : Cardinal}, a \u2264 b \u2192 c \u2264 d \u2192 a + c \u2264 b + d := by\n  rintro \u27e8\u03b1\u27e9 \u27e8\u03b2\u27e9 \u27e8\u03b3\u27e9 \u27e8\u03b4\u27e9 \u27e8e\u2081\u27e9 \u27e8e\u2082\u27e9; exact \u27e8e\u2081.sumMap e\u2082\u27e9\n-- #align cardinal.add_le_add' Cardinal.add_le_add'\n\ninstance add_covariantClass : CovariantClass Cardinal Cardinal (\u00b7 + \u00b7) (\u00b7 \u2264 \u00b7) :=\n  \u27e8fun _ _ _ => add_le_add' le_rfl\u27e9\n#align cardinal.add_covariant_class Cardinal.add_covariantClass\n\ninstance add_swap_covariantClass : CovariantClass Cardinal Cardinal (swap (\u00b7 + \u00b7)) (\u00b7 \u2264 \u00b7) :=\n  \u27e8fun _ _ _ h => add_le_add' h le_rfl\u27e9\n#align cardinal.add_swap_covariant_class Cardinal.add_swap_covariantClass\n\ninstance canonicallyOrderedCommSemiring : CanonicallyOrderedCommSemiring Cardinal.{u} :=\n  { Cardinal.commSemiring,\n    Cardinal.partialOrder with\n    bot := 0\n    bot_le := Cardinal.zero_le\n    add_le_add_left := fun a b => add_le_add_left\n    exists_add_of_le := fun {a b} =>\n      inductionOn\u2082 a b fun \u03b1 \u03b2 \u27e8\u27e8f, hf\u27e9\u27e9 =>\n        have : Sum \u03b1 ((range f)\u1d9c : Set \u03b2) \u2243 \u03b2 :=\n          (Equiv.sumCongr (Equiv.ofInjective f hf) (Equiv.refl _)).trans <|\n            Equiv.Set.sumCompl (range f)\n        \u27e8#(\u21a5(range f)\u1d9c), mk_congr this.symm\u27e9\n    le_self_add := fun a b => (add_zero a).ge.trans <| add_le_add_left (Cardinal.zero_le _) _\n    eq_zero_or_eq_zero_of_mul_eq_zero := fun {a b} =>\n      inductionOn\u2082 a b fun \u03b1 \u03b2 => by\n        simpa only [mul_def, mk_eq_zero_iff, isEmpty_prod] using id }\n\ninstance : CanonicallyLinearOrderedAddCommMonoid Cardinal.{u} :=\n  { Cardinal.canonicallyOrderedCommSemiring, Cardinal.linearOrder with }\n\n-- Computable instance to prevent a non-computable one being found via the one above\ninstance : CanonicallyOrderedAddCommMonoid Cardinal.{u} :=\n  { Cardinal.canonicallyOrderedCommSemiring with }\n\ninstance : LinearOrderedCommMonoidWithZero Cardinal.{u} :=\n  { Cardinal.commSemiring,\n    Cardinal.linearOrder with\n    mul_le_mul_left := @mul_le_mul_left' _ _ _ _\n    zero_le_one := zero_le _ }\n\n-- Computable instance to prevent a non-computable one being found via the one above\ninstance : CommMonoidWithZero Cardinal.{u} :=\n  { Cardinal.canonicallyOrderedCommSemiring with }\n\n-- Porting note: new\n-- Computable instance to prevent a non-computable one being found via the one above\ninstance : CommMonoid Cardinal.{u} :=\n  { Cardinal.canonicallyOrderedCommSemiring with }\n\ntheorem zero_power_le (c : Cardinal.{u}) : (0 : Cardinal.{u}) ^ c \u2264 1 := by\n  by_cases h : c = 0\n  \u00b7 rw [h, power_zero]\n  \u00b7 rw [zero_power h]\n    apply zero_le\n#align cardinal.zero_power_le Cardinal.zero_power_le\n\ntheorem power_le_power_left : \u2200 {a b c : Cardinal}, a \u2260 0 \u2192 b \u2264 c \u2192 a ^ b \u2264 a ^ c := by\n  rintro \u27e8\u03b1\u27e9 \u27e8\u03b2\u27e9 \u27e8\u03b3\u27e9 h\u03b1 \u27e8e\u27e9\n  let \u27e8a\u27e9 := mk_ne_zero_iff.1 h\u03b1\n  exact \u27e8@Function.Embedding.arrowCongrLeft _ _ _ \u27e8a\u27e9 e\u27e9\n#align cardinal.power_le_power_left Cardinal.power_le_power_left\n\ntheorem self_le_power (a : Cardinal) {b : Cardinal} (hb : 1 \u2264 b) : a \u2264 a ^ b := by\n  rcases eq_or_ne a 0 with (rfl | ha)\n  \u00b7 exact zero_le _\n  \u00b7 convert power_le_power_left ha hb\n    exact power_one.symm\n#align cardinal.self_le_power Cardinal.self_le_power\n\n/-- **Cantor's theorem** -/\ntheorem cantor (a : Cardinal.{u}) : a < 2 ^ a := by\n  induction' a using Cardinal.inductionOn with \u03b1\n  rw [\u2190 mk_set]\n  refine' \u27e8\u27e8\u27e8singleton, fun a b => singleton_eq_singleton_iff.1\u27e9\u27e9, _\u27e9\n  rintro \u27e8\u27e8f, hf\u27e9\u27e9\n  exact cantor_injective f hf\n#align cardinal.cantor Cardinal.cantor\n\ninstance : NoMaxOrder Cardinal.{u} where exists_gt a := \u27e8_, cantor a\u27e9\n\n-- short-circuit type class inference\ninstance : DistribLattice Cardinal.{u} := inferInstance\n\ntheorem one_lt_iff_nontrivial {\u03b1 : Type u} : 1 < #\u03b1 \u2194 Nontrivial \u03b1 := by\n  rw [\u2190 not_le, le_one_iff_subsingleton, \u2190 not_nontrivial_iff_subsingleton, Classical.not_not]\n#align cardinal.one_lt_iff_nontrivial Cardinal.one_lt_iff_nontrivial\n\ntheorem power_le_max_power_one {a b c : Cardinal} (h : b \u2264 c) : a ^ b \u2264 max (a ^ c) 1 := by\n  by_cases ha : a = 0\n  \u00b7 simp [ha, zero_power_le]\n  \u00b7 exact (power_le_power_left ha h).trans (le_max_left _ _)\n#align cardinal.power_le_max_power_one Cardinal.power_le_max_power_one\n\ntheorem power_le_power_right {a b c : Cardinal} : a \u2264 b \u2192 a ^ c \u2264 b ^ c :=\n  inductionOn\u2083 a b c fun _ _ _ \u27e8e\u27e9 => \u27e8Embedding.arrowCongrRight e\u27e9\n#align cardinal.power_le_power_right Cardinal.power_le_power_right\n\ntheorem power_pos {a : Cardinal} (b : Cardinal) (ha : 0 < a) : 0 < a ^ b :=\n  (power_ne_zero _ ha.ne').bot_lt\n#align cardinal.power_pos Cardinal.power_pos\n\nend OrderProperties\n\nprotected theorem lt_wf : @WellFounded Cardinal.{u} (\u00b7 < \u00b7) :=\n  \u27e8fun a =>\n    by_contradiction fun h => by\n      let \u03b9 := { c : Cardinal // \u00acAcc (\u00b7 < \u00b7) c }\n      let f : \u03b9 \u2192 Cardinal := Subtype.val\n      haveI h\u03b9 : Nonempty \u03b9 := \u27e8\u27e8_, h\u27e9\u27e9\n      obtain \u27e8\u27e8c : Cardinal, hc : \u00acAcc (\u00b7 < \u00b7) c\u27e9, \u27e8h_1 : \u2200 j, (f \u27e8c, hc\u27e9).out \u21aa (f j).out\u27e9\u27e9 :=\n        Embedding.min_injective fun i => (f i).out\n      refine hc (Acc.intro _ fun j h' => by_contradiction fun hj => h'.2 ?_)\n      have : #_ \u2264 #_ := \u27e8h_1 \u27e8j, hj\u27e9\u27e9\n      simpa only [mk_out] using this\u27e9\n#align cardinal.lt_wf Cardinal.lt_wf\n\ninstance : WellFoundedRelation Cardinal.{u} :=\n  \u27e8(\u00b7 < \u00b7), Cardinal.lt_wf\u27e9\n\n-- Porting note: this no longer is automatically inferred.\ninstance : WellFoundedLT Cardinal.{u} :=\n  \u27e8Cardinal.lt_wf\u27e9\n\ninstance wo : @IsWellOrder Cardinal.{u} (\u00b7 < \u00b7) where\n#align cardinal.wo Cardinal.wo\n\ninstance : ConditionallyCompleteLinearOrderBot Cardinal :=\n  IsWellOrder.conditionallyCompleteLinearOrderBot _\n\n@[simp]\ntheorem sInf_empty : sInf (\u2205 : Set Cardinal.{u}) = 0 :=\n  dif_neg Set.not_nonempty_empty\n#align cardinal.Inf_empty Cardinal.sInf_empty\n\nlemma sInf_eq_zero_iff {s : Set Cardinal} : sInf s = 0 \u2194 s = \u2205 \u2228 \u2203 a \u2208 s, a = 0 := by\n  refine \u27e8fun h \u21a6 ?_, fun h \u21a6 ?_\u27e9\n  \u00b7 rcases s.eq_empty_or_nonempty with rfl | hne\n    \u00b7 exact Or.inl rfl\n    \u00b7 exact Or.inr \u27e8sInf s, csInf_mem hne, h\u27e9\n  \u00b7 rcases h with rfl | \u27e8a, ha, rfl\u27e9\n    \u00b7 exact Cardinal.sInf_empty\n    \u00b7 exact eq_bot_iff.2 (csInf_le' ha)\n\nlemma iInf_eq_zero_iff {\u03b9 : Sort*} {f : \u03b9 \u2192 Cardinal} :\n    (\u2a05 i, f i) = 0 \u2194 IsEmpty \u03b9 \u2228 \u2203 i, f i = 0 := by\n  simp [iInf, sInf_eq_zero_iff]\n\n/-- Note that the successor of `c` is not the same as `c + 1` except in the case of finite `c`. -/\ninstance : SuccOrder Cardinal :=\n  SuccOrder.ofSuccLeIff (fun c => sInf { c' | c < c' })\n    -- Porting note: Needed to insert `by apply` in the next line\n    \u27e8by apply lt_of_lt_of_le <| csInf_mem <| exists_gt _,\n    -- Porting note used to be just `csInf_le'`\n    fun h \u21a6 csInf_le' h\u27e9\n\ntheorem succ_def (c : Cardinal) : succ c = sInf { c' | c < c' } :=\n  rfl\n#align cardinal.succ_def Cardinal.succ_def\n\ntheorem succ_pos : \u2200 c : Cardinal, 0 < succ c :=\n  bot_lt_succ\n#align cardinal.succ_pos Cardinal.succ_pos\n\ntheorem succ_ne_zero (c : Cardinal) : succ c \u2260 0 :=\n  (succ_pos _).ne'\n#align cardinal.succ_ne_zero Cardinal.succ_ne_zero\n\ntheorem add_one_le_succ (c : Cardinal.{u}) : c + 1 \u2264 succ c := by\n  -- Porting note: rewrote the next three lines to avoid defeq abuse.\n  have : Set.Nonempty { c' | c < c' } := exists_gt c\n  simp_rw [succ_def, le_csInf_iff'' this, mem_setOf]\n  intro b hlt\n  rcases b, c with \u27e8\u27e8\u03b2\u27e9, \u27e8\u03b3\u27e9\u27e9\n  cases' le_of_lt hlt with f\n  have : \u00acSurjective f := fun hn => (not_le_of_lt hlt) (mk_le_of_surjective hn)\n  simp only [Surjective, not_forall] at this\n  rcases this with \u27e8b, hb\u27e9\n  calc\n    #\u03b3 + 1 = #(Option \u03b3) := mk_option.symm\n    _ \u2264 #\u03b2 := (f.optionElim b hb).cardinal_le\n#align cardinal.add_one_le_succ Cardinal.add_one_le_succ\n\n/-- A cardinal is a limit if it is not zero or a successor cardinal. Note that `\u2135\u2080` is a limit\n  cardinal by this definition, but `0` isn't.\n\n  Use `IsSuccLimit` if you want to include the `c = 0` case. -/\ndef IsLimit (c : Cardinal) : Prop :=\n  c \u2260 0 \u2227 IsSuccLimit c\n#align cardinal.is_limit Cardinal.IsLimit\n\nprotected theorem IsLimit.ne_zero {c} (h : IsLimit c) : c \u2260 0 :=\n  h.1\n#align cardinal.is_limit.ne_zero Cardinal.IsLimit.ne_zero\n\nprotected theorem IsLimit.isSuccLimit {c} (h : IsLimit c) : IsSuccLimit c :=\n  h.2\n#align cardinal.is_limit.is_succ_limit Cardinal.IsLimit.isSuccLimit\n\ntheorem IsLimit.succ_lt {x c} (h : IsLimit c) : x < c \u2192 succ x < c :=\n  h.isSuccLimit.succ_lt\n#align cardinal.is_limit.succ_lt Cardinal.IsLimit.succ_lt\n\ntheorem isSuccLimit_zero : IsSuccLimit (0 : Cardinal) :=\n  isSuccLimit_bot\n#align cardinal.is_succ_limit_zero Cardinal.isSuccLimit_zero\n\n/-- The indexed sum of cardinals is the cardinality of the\n  indexed disjoint union, i.e. sigma type. -/\ndef sum {\u03b9} (f : \u03b9 \u2192 Cardinal) : Cardinal :=\n  mk (\u03a3i, (f i).out)\n#align cardinal.sum Cardinal.sum\n\ntheorem le_sum {\u03b9} (f : \u03b9 \u2192 Cardinal) (i) : f i \u2264 sum f := by\n  rw [\u2190 Quotient.out_eq (f i)]\n  exact \u27e8\u27e8fun a => \u27e8i, a\u27e9, fun a b h => by injection h\u27e9\u27e9\n#align cardinal.le_sum Cardinal.le_sum\n\n@[simp]\ntheorem mk_sigma {\u03b9} (f : \u03b9 \u2192 Type*) : #(\u03a3 i, f i) = sum fun i => #(f i) :=\n  mk_congr <| Equiv.sigmaCongrRight fun _ => outMkEquiv.symm\n#align cardinal.mk_sigma Cardinal.mk_sigma\n\n@[simp]\ntheorem sum_const (\u03b9 : Type u) (a : Cardinal.{v}) :\n    (sum fun _ : \u03b9 => a) = lift.{v} #\u03b9 * lift.{u} a :=\n  inductionOn a fun \u03b1 =>\n    mk_congr <|\n      calc\n        (\u03a3 _ : \u03b9, Quotient.out #\u03b1) \u2243 \u03b9 \u00d7 Quotient.out #\u03b1 := Equiv.sigmaEquivProd _ _\n        _ \u2243 ULift \u03b9 \u00d7 ULift \u03b1 := Equiv.ulift.symm.prodCongr (outMkEquiv.trans Equiv.ulift.symm)\n#align cardinal.sum_const Cardinal.sum_const\n\ntheorem sum_const' (\u03b9 : Type u) (a : Cardinal.{u}) : (sum fun _ : \u03b9 => a) = #\u03b9 * a := by simp\n#align cardinal.sum_const' Cardinal.sum_const'\n\n@[simp]\ntheorem sum_add_distrib {\u03b9} (f g : \u03b9 \u2192 Cardinal) : sum (f + g) = sum f + sum g := by\n  have := mk_congr (Equiv.sigmaSumDistrib (Quotient.out \u2218 f) (Quotient.out \u2218 g))\n  simp only [comp_apply, mk_sigma, mk_sum, mk_out, lift_id] at this\n  exact this\n#align cardinal.sum_add_distrib Cardinal.sum_add_distrib\n\n@[simp]\ntheorem sum_add_distrib' {\u03b9} (f g : \u03b9 \u2192 Cardinal) :\n    (Cardinal.sum fun i => f i + g i) = sum f + sum g :=\n  sum_add_distrib f g\n#align cardinal.sum_add_distrib' Cardinal.sum_add_distrib'\n\n@[simp]\ntheorem lift_sum {\u03b9 : Type u} (f : \u03b9 \u2192 Cardinal.{v}) :\n    Cardinal.lift.{w} (Cardinal.sum f) = Cardinal.sum fun i => Cardinal.lift.{w} (f i) :=\n  Equiv.cardinal_eq <|\n    Equiv.ulift.trans <|\n      Equiv.sigmaCongrRight fun a =>\n    -- Porting note: Inserted universe hint .{_,_,v} below\n        Nonempty.some <| by rw [\u2190 lift_mk_eq.{_,_,v}, mk_out, mk_out, lift_lift]\n#align cardinal.lift_sum Cardinal.lift_sum\n\ntheorem sum_le_sum {\u03b9} (f g : \u03b9 \u2192 Cardinal) (H : \u2200 i, f i \u2264 g i) : sum f \u2264 sum g :=\n  \u27e8(Embedding.refl _).sigmaMap fun i =>\n      Classical.choice <| by have := H i; rwa [\u2190 Quot.out_eq (f i), \u2190 Quot.out_eq (g i)] at this\u27e9\n#align cardinal.sum_le_sum Cardinal.sum_le_sum\n\ntheorem mk_le_mk_mul_of_mk_preimage_le {c : Cardinal} (f : \u03b1 \u2192 \u03b2) (hf : \u2200 b : \u03b2, #(f \u207b\u00b9' {b}) \u2264 c) :\n    #\u03b1 \u2264 #\u03b2 * c := by\n  simpa only [\u2190 mk_congr (@Equiv.sigmaFiberEquiv \u03b1 \u03b2 f), mk_sigma, \u2190 sum_const'] using\n    sum_le_sum _ _ hf\n#align cardinal.mk_le_mk_mul_of_mk_preimage_le Cardinal.mk_le_mk_mul_of_mk_preimage_le\n\ntheorem lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le {\u03b1 : Type u} {\u03b2 : Type v} {c : Cardinal}\n    (f : \u03b1 \u2192 \u03b2) (hf : \u2200 b : \u03b2, lift.{v} #(f \u207b\u00b9' {b}) \u2264 c) : lift.{v} #\u03b1 \u2264 lift.{u} #\u03b2 * c :=\n  (mk_le_mk_mul_of_mk_preimage_le fun x : ULift.{v} \u03b1 => ULift.up.{u} (f x.1)) <|\n    ULift.forall.2 fun b =>\n      (mk_congr <|\n            (Equiv.ulift.image _).trans\n              (Equiv.trans\n                (by\n                  rw [Equiv.image_eq_preimage]\n                  /- Porting note: Need to insert the following `have` b/c bad fun coercion\n                   behaviour for Equivs -/\n                  have : DFunLike.coe (Equiv.symm (Equiv.ulift (\u03b1 := \u03b1))) = ULift.up (\u03b1 := \u03b1) := rfl\n                  rw [this]\n                  simp only [preimage, mem_singleton_iff, ULift.up_inj, mem_setOf_eq, coe_setOf]\n                  exact Equiv.refl _)\n                Equiv.ulift.symm)).trans_le\n        (hf b)\n#align cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le Cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le\n\n/-- The range of an indexed cardinal function, whose outputs live in a higher universe than the\n    inputs, is always bounded above. -/\ntheorem bddAbove_range {\u03b9 : Type u} (f : \u03b9 \u2192 Cardinal.{max u v}) : BddAbove (Set.range f) :=\n  \u27e8_, by\n    rintro a \u27e8i, rfl\u27e9\n    -- Porting note: Added universe reference below\n    exact le_sum.{v,u} f i\u27e9\n#align cardinal.bdd_above_range Cardinal.bddAbove_range\n\ninstance (a : Cardinal.{u}) : Small.{u} (Set.Iic a) := by\n  rw [\u2190 mk_out a]\n  apply @small_of_surjective (Set a.out) (Iic #a.out) _ fun x => \u27e8#x, mk_set_le x\u27e9\n  rintro \u27e8x, hx\u27e9\n  simpa using le_mk_iff_exists_set.1 hx\n\ninstance (a : Cardinal.{u}) : Small.{u} (Set.Iio a) :=\n  small_subset Iio_subset_Iic_self\n\n/-- A set of cardinals is bounded above iff it's small, i.e. it corresponds to a usual ZFC set. -/\ntheorem bddAbove_iff_small {s : Set Cardinal.{u}} : BddAbove s \u2194 Small.{u} s :=\n  \u27e8fun \u27e8a, ha\u27e9 => @small_subset _ (Iic a) s (fun x h => ha h) _, by\n    rintro \u27e8\u03b9, \u27e8e\u27e9\u27e9\n    suffices (range fun x : \u03b9 => (e.symm x).1) = s by\n      rw [\u2190 this]\n      apply bddAbove_range.{u, u}\n    ext x\n    refine' \u27e8_, fun hx => \u27e8e \u27e8x, hx\u27e9, _\u27e9\u27e9\n    \u00b7 rintro \u27e8a, rfl\u27e9\n      exact (e.symm a).2\n    \u00b7 simp_rw [Equiv.symm_apply_apply]\u27e9\n#align cardinal.bdd_above_iff_small Cardinal.bddAbove_iff_small\n\ntheorem bddAbove_of_small (s : Set Cardinal.{u}) [h : Small.{u} s] : BddAbove s :=\n  bddAbove_iff_small.2 h\n#align cardinal.bdd_above_of_small Cardinal.bddAbove_of_small\n\ntheorem bddAbove_image (f : Cardinal.{u} \u2192 Cardinal.{max u v}) {s : Set Cardinal.{u}}\n    (hs : BddAbove s) : BddAbove (f '' s) := by\n  rw [bddAbove_iff_small] at hs \u22a2\n  -- Porting note: added universes below\n  exact small_lift.{_,v,_} _\n#align cardinal.bdd_above_image Cardinal.bddAbove_image\n\ntheorem bddAbove_range_comp {\u03b9 : Type u} {f : \u03b9 \u2192 Cardinal.{v}} (hf : BddAbove (range f))\n    (g : Cardinal.{v} \u2192 Cardinal.{max v w}) : BddAbove (range (g \u2218 f)) := by\n  rw [range_comp]\n  exact bddAbove_image.{v,w} g hf\n#align cardinal.bdd_above_range_comp Cardinal.bddAbove_range_comp\n\ntheorem iSup_le_sum {\u03b9} (f : \u03b9 \u2192 Cardinal) : iSup f \u2264 sum f :=\n  ciSup_le' <| le_sum.{u_2,u_1} _\n#align cardinal.supr_le_sum Cardinal.iSup_le_sum\n\n-- Porting note: Added universe hint .{v,_} below\ntheorem sum_le_iSup_lift {\u03b9 : Type u}\n    (f : \u03b9 \u2192 Cardinal.{max u v}) : sum f \u2264 Cardinal.lift.{v,_} #\u03b9 * iSup f := by\n  rw [\u2190 (iSup f).lift_id, \u2190 lift_umax, lift_umax.{max u v, u}, \u2190 sum_const]\n  exact sum_le_sum _ _ (le_ciSup <| bddAbove_range.{u, v} f)\n#align cardinal.sum_le_supr_lift Cardinal.sum_le_iSup_lift\n\ntheorem sum_le_iSup {\u03b9 : Type u} (f : \u03b9 \u2192 Cardinal.{u}) : sum f \u2264 #\u03b9 * iSup f := by\n  rw [\u2190 lift_id #\u03b9]\n  exact sum_le_iSup_lift f\n#align cardinal.sum_le_supr Cardinal.sum_le_iSup\n\ntheorem sum_nat_eq_add_sum_succ (f : \u2115 \u2192 Cardinal.{u}) :\n    Cardinal.sum f = f 0 + Cardinal.sum fun i => f (i + 1) := by\n  refine' (Equiv.sigmaNatSucc fun i => Quotient.out (f i)).cardinal_eq.trans _\n  simp only [mk_sum, mk_out, lift_id, mk_sigma]\n#align cardinal.sum_nat_eq_add_sum_succ Cardinal.sum_nat_eq_add_sum_succ\n\n-- Porting note: LFS is not in normal form.\n-- @[simp]\n/-- A variant of `ciSup_of_empty` but with `0` on the RHS for convenience -/\nprotected theorem iSup_of_empty {\u03b9} (f : \u03b9 \u2192 Cardinal) [IsEmpty \u03b9] : iSup f = 0 :=\n  ciSup_of_empty f\n#align cardinal.supr_of_empty Cardinal.iSup_of_empty\n\nlemma exists_eq_of_iSup_eq_of_not_isSuccLimit\n    {\u03b9 : Type u} (f : \u03b9 \u2192 Cardinal.{v}) (\u03c9 : Cardinal.{v})\n    (h\u03c9 : \u00ac Order.IsSuccLimit \u03c9)\n    (h : \u2a06 i : \u03b9, f i = \u03c9) : \u2203 i, f i = \u03c9 := by\n  subst h\n  refine (isLUB_csSup' ?_).exists_of_not_isSuccLimit h\u03c9\n  contrapose! h\u03c9 with hf\n  rw [iSup, csSup_of_not_bddAbove hf, csSup_empty]\n  exact Order.isSuccLimit_bot\n\nlemma exists_eq_of_iSup_eq_of_not_isLimit\n    {\u03b9 : Type u} [h\u03b9 : Nonempty \u03b9] (f : \u03b9 \u2192 Cardinal.{v}) (hf : BddAbove (range f))\n    (\u03c9 : Cardinal.{v}) (h\u03c9 : \u00ac \u03c9.IsLimit)\n    (h : \u2a06 i : \u03b9, f i = \u03c9) : \u2203 i, f i = \u03c9 := by\n  refine (not_and_or.mp h\u03c9).elim (fun e \u21a6 \u27e8h\u03b9.some, ?_\u27e9)\n    (Cardinal.exists_eq_of_iSup_eq_of_not_isSuccLimit.{u, v} f \u03c9 \u00b7 h)\n  cases not_not.mp e\n  rw [\u2190 le_zero_iff] at h \u22a2\n  exact (le_ciSup hf _).trans h\n\n-- Porting note: simpNF is not happy with universe levels.\n@[simp, nolint simpNF]\ntheorem lift_mk_shrink (\u03b1 : Type u) [Small.{v} \u03b1] :\n    Cardinal.lift.{max u w} #(Shrink.{v} \u03b1) = Cardinal.lift.{max v w} #\u03b1 :=\n-- Porting note: Added .{v,u,w} universe hint below\n  lift_mk_eq.{v,u,w}.2 \u27e8(equivShrink \u03b1).symm\u27e9\n#align cardinal.lift_mk_shrink Cardinal.lift_mk_shrink\n\n@[simp]\ntheorem lift_mk_shrink' (\u03b1 : Type u) [Small.{v} \u03b1] :\n    Cardinal.lift.{u} #(Shrink.{v} \u03b1) = Cardinal.lift.{v} #\u03b1 :=\n  lift_mk_shrink.{u, v, 0} \u03b1\n#align cardinal.lift_mk_shrink' Cardinal.lift_mk_shrink'\n\n@[simp]\ntheorem lift_mk_shrink'' (\u03b1 : Type max u v) [Small.{v} \u03b1] :\n    Cardinal.lift.{u} #(Shrink.{v} \u03b1) = #\u03b1 := by\n  rw [\u2190 lift_umax', lift_mk_shrink.{max u v, v, 0} \u03b1, \u2190 lift_umax, lift_id]\n#align cardinal.lift_mk_shrink'' Cardinal.lift_mk_shrink''\n\n/-- The indexed product of cardinals is the cardinality of the Pi type\n  (dependent product). -/\ndef prod {\u03b9 : Type u} (f : \u03b9 \u2192 Cardinal) : Cardinal :=\n  #(\u2200 i, (f i).out)\n#align cardinal.prod Cardinal.prod\n\n@[simp]\ntheorem mk_pi {\u03b9 : Type u} (\u03b1 : \u03b9 \u2192 Type v) : #(\u2200 i, \u03b1 i) = prod fun i => #(\u03b1 i) :=\n  mk_congr <| Equiv.piCongrRight fun _ => outMkEquiv.symm\n#align cardinal.mk_pi Cardinal.mk_pi\n\n@[simp]\ntheorem prod_const (\u03b9 : Type u) (a : Cardinal.{v}) :\n    (prod fun _ : \u03b9 => a) = lift.{u} a ^ lift.{v} #\u03b9 :=\n  inductionOn a fun _ =>\n    mk_congr <| Equiv.piCongr Equiv.ulift.symm fun _ => outMkEquiv.trans Equiv.ulift.symm\n#align cardinal.prod_const Cardinal.prod_const\n\ntheorem prod_const' (\u03b9 : Type u) (a : Cardinal.{u}) : (prod fun _ : \u03b9 => a) = a ^ #\u03b9 :=\n  inductionOn a fun _ => (mk_pi _).symm\n#align cardinal.prod_const' Cardinal.prod_const'\n\ntheorem prod_le_prod {\u03b9} (f g : \u03b9 \u2192 Cardinal) (H : \u2200 i, f i \u2264 g i) : prod f \u2264 prod g :=\n  \u27e8Embedding.piCongrRight fun i =>\n      Classical.choice <| by have := H i; rwa [\u2190 mk_out (f i), \u2190 mk_out (g i)] at this\u27e9\n#align cardinal.prod_le_prod Cardinal.prod_le_prod\n\n@[simp]\ntheorem prod_eq_zero {\u03b9} (f : \u03b9 \u2192 Cardinal.{u}) : prod f = 0 \u2194 \u2203 i, f i = 0 := by\n  lift f to \u03b9 \u2192 Type u using fun _ => trivial\n  simp only [mk_eq_zero_iff, \u2190 mk_pi, isEmpty_pi]\n#align cardinal.prod_eq_zero Cardinal.prod_eq_zero\n\ntheorem prod_ne_zero {\u03b9} (f : \u03b9 \u2192 Cardinal) : prod f \u2260 0 \u2194 \u2200 i, f i \u2260 0 := by simp [prod_eq_zero]\n#align cardinal.prod_ne_zero Cardinal.prod_ne_zero\n\n@[simp]\ntheorem lift_prod {\u03b9 : Type u} (c : \u03b9 \u2192 Cardinal.{v}) :\n    lift.{w} (prod c) = prod fun i => lift.{w} (c i) := by\n  lift c to \u03b9 \u2192 Type v using fun _ => trivial\n  simp only [\u2190 mk_pi, \u2190 mk_uLift]\n  exact mk_congr (Equiv.ulift.trans <| Equiv.piCongrRight fun i => Equiv.ulift.symm)\n#align cardinal.lift_prod Cardinal.lift_prod\n\ntheorem prod_eq_of_fintype {\u03b1 : Type u} [h : Fintype \u03b1] (f : \u03b1 \u2192 Cardinal.{v}) :\n    prod f = Cardinal.lift.{u} (\u220f i, f i) := by\n  revert f\n  refine' Fintype.induction_empty_option _ _ _ \u03b1 (h_fintype := h)\n  \u00b7 intro \u03b1 \u03b2 h\u03b2 e h f\n    letI := Fintype.ofEquiv \u03b2 e.symm\n    rw [\u2190 e.prod_comp f, \u2190 h]\n    exact mk_congr (e.piCongrLeft _).symm\n  \u00b7 intro f\n    rw [Fintype.univ_pempty, Finset.prod_empty, lift_one, Cardinal.prod, mk_eq_one]\n  \u00b7 intro \u03b1 h\u03b1 h f\n    rw [Cardinal.prod, mk_congr Equiv.piOptionEquivProd, mk_prod, lift_umax'.{v, u}, mk_out, \u2190\n        Cardinal.prod, lift_prod, Fintype.prod_option, lift_mul, \u2190 h fun a => f (some a)]\n    simp only [lift_id]\n#align cardinal.prod_eq_of_fintype Cardinal.prod_eq_of_fintype\n\n-- Porting note: Inserted .{u,v} below\n@[simp]\ntheorem lift_sInf (s : Set Cardinal) : lift.{u,v} (sInf s) = sInf (lift.{u,v} '' s) := by\n  rcases eq_empty_or_nonempty s with (rfl | hs)\n  \u00b7 simp\n  \u00b7 exact lift_monotone.map_csInf hs\n#align cardinal.lift_Inf Cardinal.lift_sInf\n\n-- Porting note: Inserted .{u,v} below\n@[simp]\ntheorem lift_iInf {\u03b9} (f : \u03b9 \u2192 Cardinal) : lift.{u,v} (iInf f) = \u2a05 i, lift.{u,v} (f i) := by\n  unfold iInf\n  convert lift_sInf (range f)\n  simp_rw [\u2190 comp_apply (f := lift), range_comp]\n#align cardinal.lift_infi Cardinal.lift_iInf\n\ntheorem lift_down {a : Cardinal.{u}} {b : Cardinal.{max u v}} :\n    b \u2264 lift.{v,u} a \u2192 \u2203 a', lift.{v,u} a' = b :=\n  inductionOn\u2082 a b fun \u03b1 \u03b2 => by\n    rw [\u2190 lift_id #\u03b2, \u2190 lift_umax, \u2190 lift_umax.{u, v}, lift_mk_le.{v}]\n    exact fun \u27e8f\u27e9 =>\n      \u27e8#(Set.range f),\n        Eq.symm <| lift_mk_eq.{_, _, v}.2\n          \u27e8Function.Embedding.equivOfSurjective (Embedding.codRestrict _ f Set.mem_range_self)\n              fun \u27e8a, \u27e8b, e\u27e9\u27e9 => \u27e8b, Subtype.eq e\u27e9\u27e9\u27e9\n#align cardinal.lift_down Cardinal.lift_down\n\n-- Porting note: Inserted .{u,v} below\ntheorem le_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :\n    b \u2264 lift.{v,u} a \u2194 \u2203 a', lift.{v,u} a' = b \u2227 a' \u2264 a :=\n  \u27e8fun h =>\n    let \u27e8a', e\u27e9 := lift_down h\n    \u27e8a', e, lift_le.1 <| e.symm \u25b8 h\u27e9,\n    fun \u27e8_, e, h\u27e9 => e \u25b8 lift_le.2 h\u27e9\n#align cardinal.le_lift_iff Cardinal.le_lift_iff\n\n-- Porting note: Inserted .{u,v} below\ntheorem lt_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :\n    b < lift.{v,u} a \u2194 \u2203 a', lift.{v,u} a' = b \u2227 a' < a :=\n  \u27e8fun h =>\n    let \u27e8a', e\u27e9 := lift_down h.le\n    \u27e8a', e, lift_lt.1 <| e.symm \u25b8 h\u27e9,\n    fun \u27e8_, e, h\u27e9 => e \u25b8 lift_lt.2 h\u27e9\n#align cardinal.lt_lift_iff Cardinal.lt_lift_iff\n\n-- Porting note: Inserted .{u,v} below\n@[simp]\ntheorem lift_succ (a) : lift.{v,u} (succ a) = succ (lift.{v,u} a) :=\n  le_antisymm\n    (le_of_not_gt fun h => by\n      rcases lt_lift_iff.1 h with \u27e8b, e, h\u27e9\n      rw [lt_succ_iff, \u2190 lift_le, e] at h\n      exact h.not_lt (lt_succ _))\n    (succ_le_of_lt <| lift_lt.2 <| lt_succ a)\n#align cardinal.lift_succ Cardinal.lift_succ\n\n-- Porting note: simpNF is not happy with universe levels.\n-- Porting note: Inserted .{u,v} below\n@[simp, nolint simpNF]\ntheorem lift_umax_eq {a : Cardinal.{u}} {b : Cardinal.{v}} :\n    lift.{max v w} a = lift.{max u w} b \u2194 lift.{v} a = lift.{u} b := by\n  rw [\u2190 lift_lift.{v, w, u}, \u2190 lift_lift.{u, w, v}, lift_inj]\n#align cardinal.lift_umax_eq Cardinal.lift_umax_eq\n\n-- Porting note: Inserted .{u,v} below\n@[simp]\ntheorem lift_min {a b : Cardinal} : lift.{u,v} (min a b) = min (lift.{u,v} a) (lift.{u,v} b) :=\n  lift_monotone.map_min\n#align cardinal.lift_min Cardinal.lift_min\n\n-- Porting note: Inserted .{u,v} below\n@[simp]\ntheorem lift_max {a b : Cardinal} : lift.{u,v} (max a b) = max (lift.{u,v} a) (lift.{u,v} b) :=\n  lift_monotone.map_max\n#align cardinal.lift_max Cardinal.lift_max\n\n/-- The lift of a supremum is the supremum of the lifts. -/\ntheorem lift_sSup {s : Set Cardinal} (hs : BddAbove s) :\n    lift.{u} (sSup s) = sSup (lift.{u} '' s) := by\n  apply ((le_csSup_iff' (bddAbove_image.{_,u} _ hs)).2 fun c hc => _).antisymm (csSup_le' _)\n  \u00b7 intro c hc\n    by_contra h\n    obtain \u27e8d, rfl\u27e9 := Cardinal.lift_down (not_le.1 h).le\n    simp_rw [lift_le] at h hc\n    rw [csSup_le_iff' hs] at h\n    exact h fun a ha => lift_le.1 <| hc (mem_image_of_mem _ ha)\n  \u00b7 rintro i \u27e8j, hj, rfl\u27e9\n    exact lift_le.2 (le_csSup hs hj)\n#align cardinal.lift_Sup Cardinal.lift_sSup\n\n/-- The lift of a supremum is the supremum of the lifts. -/\ntheorem lift_iSup {\u03b9 : Type v} {f : \u03b9 \u2192 Cardinal.{w}} (hf : BddAbove (range f)) :\n    lift.{u} (iSup f) = \u2a06 i, lift.{u} (f i) := by\n  rw [iSup, iSup, lift_sSup hf, \u2190 range_comp]\n  simp [Function.comp]\n#align cardinal.lift_supr Cardinal.lift_iSup\n\n/-- To prove that the lift of a supremum is bounded by some cardinal `t`,\nit suffices to show that the lift of each cardinal is bounded by `t`. -/\ntheorem lift_iSup_le {\u03b9 : Type v} {f : \u03b9 \u2192 Cardinal.{w}} {t : Cardinal} (hf : BddAbove (range f))\n    (w : \u2200 i, lift.{u} (f i) \u2264 t) : lift.{u} (iSup f) \u2264 t := by\n  rw [lift_iSup hf]\n  exact ciSup_le' w\n#align cardinal.lift_supr_le Cardinal.lift_iSup_le\n\n@[simp]\ntheorem lift_iSup_le_iff {\u03b9 : Type v} {f : \u03b9 \u2192 Cardinal.{w}} (hf : BddAbove (range f))\n    {t : Cardinal} : lift.{u} (iSup f) \u2264 t \u2194 \u2200 i, lift.{u} (f i) \u2264 t := by\n  rw [lift_iSup hf]\n  exact ciSup_le_iff' (bddAbove_range_comp.{_,_,u} hf _)\n#align cardinal.lift_supr_le_iff Cardinal.lift_iSup_le_iff\n\nuniverse v' w'\n\n/-- To prove an inequality between the lifts to a common universe of two different supremums,\nit suffices to show that the lift of each cardinal from the smaller supremum\nif bounded by the lift of some cardinal from the larger supremum.\n-/\ntheorem lift_iSup_le_lift_iSup {\u03b9 : Type v} {\u03b9' : Type v'} {f : \u03b9 \u2192 Cardinal.{w}}\n    {f' : \u03b9' \u2192 Cardinal.{w'}} (hf : BddAbove (range f)) (hf' : BddAbove (range f')) {g : \u03b9 \u2192 \u03b9'}\n    (h : \u2200 i, lift.{w'} (f i) \u2264 lift.{w} (f' (g i))) : lift.{w'} (iSup f) \u2264 lift.{w} (iSup f') := by\n  rw [lift_iSup hf, lift_iSup hf']\n  exact ciSup_mono' (bddAbove_range_comp.{_,_,w} hf' _) fun i => \u27e8_, h i\u27e9\n#align cardinal.lift_supr_le_lift_supr Cardinal.lift_iSup_le_lift_iSup\n\n/-- A variant of `lift_iSup_le_lift_iSup` with universes specialized via `w = v` and `w' = v'`.\nThis is sometimes necessary to avoid universe unification issues. -/\ntheorem lift_iSup_le_lift_iSup' {\u03b9 : Type v} {\u03b9' : Type v'} {f : \u03b9 \u2192 Cardinal.{v}}\n    {f' : \u03b9' \u2192 Cardinal.{v'}} (hf : BddAbove (range f)) (hf' : BddAbove (range f')) (g : \u03b9 \u2192 \u03b9')\n    (h : \u2200 i, lift.{v'} (f i) \u2264 lift.{v} (f' (g i))) : lift.{v'} (iSup f) \u2264 lift.{v} (iSup f') :=\n  lift_iSup_le_lift_iSup hf hf' h\n#align cardinal.lift_supr_le_lift_supr' Cardinal.lift_iSup_le_lift_iSup'\n\n/-- `\u2135\u2080` is the smallest infinite cardinal. -/\ndef aleph0 : Cardinal.{u} :=\n  lift #\u2115\n#align cardinal.aleph_0 Cardinal.aleph0\n\n@[inherit_doc]\nscoped notation \"\u2135\u2080\" => Cardinal.aleph0\n\ntheorem mk_nat : #\u2115 = \u2135\u2080 :=\n  (lift_id _).symm\n#align cardinal.mk_nat Cardinal.mk_nat\n\ntheorem aleph0_ne_zero : \u2135\u2080 \u2260 0 :=\n  mk_ne_zero _\n#align cardinal.aleph_0_ne_zero Cardinal.aleph0_ne_zero\n\ntheorem aleph0_pos : 0 < \u2135\u2080 :=\n  pos_iff_ne_zero.2 aleph0_ne_zero\n#align cardinal.aleph_0_pos Cardinal.aleph0_pos\n\n@[simp]\ntheorem lift_aleph0 : lift \u2135\u2080 = \u2135\u2080 :=\n  lift_lift _\n#align cardinal.lift_aleph_0 Cardinal.lift_aleph0\n\n@[simp]\ntheorem aleph0_le_lift {c : Cardinal.{u}} : \u2135\u2080 \u2264 lift.{v} c \u2194 \u2135\u2080 \u2264 c := by\n  rw [\u2190 lift_aleph0.{u,v}, lift_le]\n#align cardinal.aleph_0_le_lift Cardinal.aleph0_le_lift\n\n@[simp]\ntheorem lift_le_aleph0 {c : Cardinal.{u}} : lift.{v} c \u2264 \u2135\u2080 \u2194 c \u2264 \u2135\u2080 := by\n  rw [\u2190 lift_aleph0.{u,v}, lift_le]\n#align cardinal.lift_le_aleph_0 Cardinal.lift_le_aleph0\n\n@[simp]\ntheorem aleph0_lt_lift {c : Cardinal.{u}} : \u2135\u2080 < lift.{v} c \u2194 \u2135\u2080 < c := by\n  rw [\u2190 lift_aleph0.{u,v}, lift_lt]\n#align cardinal.aleph_0_lt_lift Cardinal.aleph0_lt_lift\n\n@[simp]\ntheorem lift_lt_aleph0 {c : Cardinal.{u}} : lift.{v} c < \u2135\u2080 \u2194 c < \u2135\u2080 := by\n  rw [\u2190 lift_aleph0.{u,v}, lift_lt]\n#align cardinal.lift_lt_aleph_0 Cardinal.lift_lt_aleph0\n\n/-! ### Properties about the cast from `\u2115` -/\nsection castFromN\n\n-- porting note (#10618): simp can prove this\n-- @[simp]\ntheorem mk_fin (n : \u2115) : #(Fin n) = n := by simp\n#align cardinal.mk_fin Cardinal.mk_fin\n\n@[simp]\ntheorem lift_natCast (n : \u2115) : lift.{u} (n : Cardinal.{v}) = n := by induction n <;> simp [*]\n#align cardinal.lift_nat_cast Cardinal.lift_natCast\n\n-- See note [no_index around OfNat.ofNat]\n@[simp]\ntheorem lift_ofNat (n : \u2115) [n.AtLeastTwo] :\n    lift.{u} (no_index (OfNat.ofNat n : Cardinal.{v})) = OfNat.ofNat n :=\n  lift_natCast n\n\n@[simp]\ntheorem lift_eq_nat_iff {a : Cardinal.{u}} {n : \u2115} : lift.{v} a = n \u2194 a = n :=\n  lift_injective.eq_iff' (lift_natCast n)\n#align cardinal.lift_eq_nat_iff Cardinal.lift_eq_nat_iff\n\n@[simp]\ntheorem lift_eq_ofNat_iff {a : Cardinal.{u}} {n : \u2115} [n.AtLeastTwo] :\n    lift.{v} a = (no_index (OfNat.ofNat n)) \u2194 a = OfNat.ofNat n :=\n  lift_eq_nat_iff\n\n@[simp]\ntheorem nat_eq_lift_iff {n : \u2115} {a : Cardinal.{u}} :\n    (n : Cardinal) = lift.{v} a \u2194 (n : Cardinal) = a := by\n  rw [\u2190 lift_natCast.{v,u} n, lift_inj]\n#align cardinal.nat_eq_lift_iff Cardinal.nat_eq_lift_iff\n\n@[simp]\ntheorem zero_eq_lift_iff {a : Cardinal.{u}} :\n    (0 : Cardinal) = lift.{v} a \u2194 0 = a := by\n  simpa using nat_eq_lift_iff (n := 0)\n\n@[simp]\ntheorem one_eq_lift_iff {a : Cardinal.{u}} :\n    (1 : Cardinal) = lift.{v} a \u2194 1 = a := by\n  simpa using nat_eq_lift_iff (n := 1)\n\n-- See note [no_index around OfNat.ofNat]\n@[simp]\ntheorem ofNat_eq_lift_iff {a : Cardinal.{u}} {n : \u2115} [n.AtLeastTwo] :\n    (no_index (OfNat.ofNat n : Cardinal)) = lift.{v} a \u2194 (OfNat.ofNat n : Cardinal) = a :=\n  nat_eq_lift_iff\n\n@[simp]\ntheorem lift_le_nat_iff {a : Cardinal.{u}} {n : \u2115} : lift.{v} a \u2264 n \u2194 a \u2264 n := by\n  rw [\u2190 lift_natCast.{v,u}, lift_le]\n#align cardinal.lift_le_nat_iff Cardinal.lift_le_nat_iff\n\n@[simp]\ntheorem lift_le_one_iff {a : Cardinal.{u}} :\n    lift.{v} a \u2264 1 \u2194 a \u2264 1 := by\n  simpa using lift_le_nat_iff (n := 1)\n\n-- See note [no_index around OfNat.ofNat]\n@[simp]\ntheorem lift_le_ofNat_iff {a : Cardinal.{u}} {n : \u2115} [n.AtLeastTwo] :\n    lift.{v} a \u2264 (no_index (OfNat.ofNat n)) \u2194 a \u2264 OfNat.ofNat n :=\n  lift_le_nat_iff\n\n@[simp]\ntheorem nat_le_lift_iff {n : \u2115} {a : Cardinal.{u}} : n \u2264 lift.{v} a \u2194 n \u2264 a := by\n  rw [\u2190 lift_natCast.{v,u}, lift_le]\n#align cardinal.nat_le_lift_iff Cardinal.nat_le_lift_iff\n\n@[simp]\ntheorem one_le_lift_iff {a : Cardinal.{u}} :\n    (1 : Cardinal) \u2264 lift.{v} a \u2194 1 \u2264 a := by\n  simpa using nat_le_lift_iff (n := 1)\n\n-- See note [no_index around OfNat.ofNat]\n@[simp]\ntheorem ofNat_le_lift_iff {a : Cardinal.{u}} {n : \u2115} [n.AtLeastTwo] :\n    (no_index (OfNat.ofNat n : Cardinal)) \u2264 lift.{v} a \u2194 (OfNat.ofNat n : Cardinal) \u2264 a :=\n  nat_le_lift_iff\n\n@[simp]\ntheorem lift_lt_nat_iff {a : Cardinal.{u}} {n : \u2115} : lift.{v} a < n \u2194 a < n := by\n  rw [\u2190 lift_natCast.{v,u}, lift_lt]\n#align cardinal.lift_lt_nat_iff Cardinal.lift_lt_nat_iff\n\n-- See note [no_index around OfNat.ofNat]\n@[simp]\ntheorem lift_lt_ofNat_iff {a : Cardinal.{u}} {n : \u2115} [n.AtLeastTwo] :\n    lift.{v} a < (no_index (OfNat.ofNat n)) \u2194 a < OfNat.ofNat n :=\n  lift_lt_nat_iff\n\n@[simp]\ntheorem nat_lt_lift_iff {n : \u2115} {a : Cardinal.{u}} : n < lift.{v} a \u2194 n < a := by\n  rw [\u2190 lift_natCast.{v,u}, lift_lt]\n#align cardinal.nat_lt_lift_iff Cardinal.nat_lt_lift_iff\n\n-- See note [no_index around OfNat.ofNat]\n@[simp]\ntheorem zero_lt_lift_iff {a : Cardinal.{u}} :\n    (0 : Cardinal) < lift.{v} a \u2194 0 < a := by\n  simpa using nat_lt_lift_iff (n := 0)\n\n@[simp]\ntheorem one_lt_lift_iff {a : Cardinal.{u}} :\n    (1 : Cardinal) < lift.{v} a \u2194 1 < a := by\n  simpa using nat_lt_lift_iff (n := 1)\n\n-- See note [no_index around OfNat.ofNat]\n@[simp]\ntheorem ofNat_lt_lift_iff {a : Cardinal.{u}} {n : \u2115} [n.AtLeastTwo] :\n    (no_index (OfNat.ofNat n : Cardinal)) < lift.{v} a \u2194 (OfNat.ofNat n : Cardinal) < a :=\n  nat_lt_lift_iff\n\ntheorem lift_mk_fin (n : \u2115) : lift #(Fin n) = n := rfl\n#align cardinal.lift_mk_fin Cardinal.lift_mk_fin\n\ntheorem mk_coe_finset {\u03b1 : Type u} {s : Finset \u03b1} : #s = \u2191(Finset.card s) := by simp\n#align cardinal.mk_coe_finset Cardinal.mk_coe_finset\n\ntheorem mk_finset_of_fintype [Fintype \u03b1] : #(Finset \u03b1) = 2 ^ Fintype.card \u03b1 := by\n  simp [Pow.pow]\n#align cardinal.mk_finset_of_fintype Cardinal.mk_finset_of_fintype\n\n@[simp]\ntheorem mk_finsupp_lift_of_fintype (\u03b1 : Type u) (\u03b2 : Type v) [Fintype \u03b1] [Zero \u03b2] :\n    #(\u03b1 \u2192\u2080 \u03b2) = lift.{u} #\u03b2 ^ Fintype.card \u03b1 := by\n  simpa using (@Finsupp.equivFunOnFinite \u03b1 \u03b2 _ _).cardinal_eq\n#align cardinal.mk_finsupp_lift_of_fintype Cardinal.mk_finsupp_lift_of_fintype\n\ntheorem mk_finsupp_of_fintype (\u03b1 \u03b2 : Type u) [Fintype \u03b1] [Zero \u03b2] :\n    #(\u03b1 \u2192\u2080 \u03b2) = #\u03b2 ^ Fintype.card \u03b1 := by simp\n#align cardinal.mk_finsupp_of_fintype Cardinal.mk_finsupp_of_fintype\n\ntheorem card_le_of_finset {\u03b1} (s : Finset \u03b1) : (s.card : Cardinal) \u2264 #\u03b1 :=\n  @mk_coe_finset _ s \u25b8 mk_set_le _\n#align cardinal.card_le_of_finset Cardinal.card_le_of_finset\n\n-- Porting note: was `simp`. LHS is not normal form.\n-- @[simp, norm_cast]\n@[norm_cast]\ntheorem natCast_pow {m n : \u2115} : (\u2191(m ^ n) : Cardinal) = (\u2191m : Cardinal) ^ (\u2191n : Cardinal) := by\n  induction n <;> simp [pow_succ, power_add, *, Pow.pow]\n#align cardinal.nat_cast_pow Cardinal.natCast_pow\n\n-- porting note (#10618): simp can prove this\n-- @[simp, norm_cast]\n@[norm_cast]\ntheorem natCast_le {m n : \u2115} : (m : Cardinal) \u2264 n \u2194 m \u2264 n := by\n  rw [\u2190 lift_mk_fin, \u2190 lift_mk_fin, lift_le, le_def, Function.Embedding.nonempty_iff_card_le,\n    Fintype.card_fin, Fintype.card_fin]\n#align cardinal.nat_cast_le Cardinal.natCast_le\n\n-- porting note (#10618): simp can prove this\n-- @[simp, norm_cast]\n@[norm_cast]\ntheorem natCast_lt {m n : \u2115} : (m : Cardinal) < n \u2194 m < n := by\n  rw [lt_iff_le_not_le, \u2190 not_le]\n  simp only [natCast_le, not_le, and_iff_right_iff_imp]\n  exact fun h \u21a6 le_of_lt h\n#align cardinal.nat_cast_lt Cardinal.natCast_lt\n\ninstance : CharZero Cardinal :=\n  \u27e8StrictMono.injective fun _ _ => natCast_lt.2\u27e9\n\ntheorem natCast_inj {m n : \u2115} : (m : Cardinal) = n \u2194 m = n :=\n  Nat.cast_inj\n#align cardinal.nat_cast_inj Cardinal.natCast_inj\n\ntheorem natCast_injective : Injective ((\u2191) : \u2115 \u2192 Cardinal) :=\n  Nat.cast_injective\n#align cardinal.nat_cast_injective Cardinal.natCast_injective\n\n@[simp, norm_cast]\ntheorem nat_succ (n : \u2115) : (n.succ : Cardinal) = succ \u2191n := by\n  rw [Nat.cast_succ]\n  refine (add_one_le_succ _).antisymm (succ_le_of_lt ?_)\n  rw [\u2190 Nat.cast_succ]\n  exact natCast_lt.2 (Nat.lt_succ_self _)\n\nlemma succ_natCast (n : \u2115) : Order.succ (n : Cardinal) = n + 1 := by\n  rw [\u2190 Cardinal.nat_succ]\n  norm_cast\n\nlemma natCast_add_one_le_iff {n : \u2115} {c : Cardinal} : n + 1 \u2264 c \u2194 n < c := by\n  rw [\u2190 Order.succ_le_iff, Cardinal.succ_natCast]\n\nlemma two_le_iff_one_lt {c : Cardinal} : 2 \u2264 c \u2194 1 < c := by\n  convert natCast_add_one_le_iff\n  norm_cast\n\n@[simp]\ntheorem succ_zero : succ (0 : Cardinal) = 1 := by norm_cast\n#align cardinal.succ_zero Cardinal.succ_zero\n\ntheorem exists_finset_le_card (\u03b1 : Type*) (n : \u2115) (h : n \u2264 #\u03b1) :\n    \u2203 s : Finset \u03b1, n \u2264 s.card := by\n  obtain h\u03b1|h\u03b1 := finite_or_infinite \u03b1\n  \u00b7 let h\u03b1 := Fintype.ofFinite \u03b1\n    use Finset.univ\n    simpa only [mk_fintype, Nat.cast_le] using h\n  \u00b7 obtain \u27e8s, hs\u27e9 := Infinite.exists_subset_card_eq \u03b1 n\n    exact \u27e8s, hs.ge\u27e9\n\ntheorem card_le_of {\u03b1 : Type u} {n : \u2115} (H : \u2200 s : Finset \u03b1, s.card \u2264 n) : #\u03b1 \u2264 n := by\n  contrapose! H\n  apply exists_finset_le_card \u03b1 (n+1)\n  simpa only [nat_succ, succ_le_iff] using H\n#align cardinal.card_le_of Cardinal.card_le_of\n\ntheorem cantor' (a) {b : Cardinal} (hb : 1 < b) : a < b ^ a := by\n  rw [\u2190 succ_le_iff, (by norm_cast : succ (1 : Cardinal) = 2)] at hb\n  exact (cantor a).trans_le (power_le_power_right hb)\n#align cardinal.cantor' Cardinal.cantor'\n\ntheorem one_le_iff_pos {c : Cardinal} : 1 \u2264 c \u2194 0 < c := by\n  rw [\u2190 succ_zero, succ_le_iff]\n#align cardinal.one_le_iff_pos Cardinal.one_le_iff_pos\n\ntheorem one_le_iff_ne_zero {c : Cardinal} : 1 \u2264 c \u2194 c \u2260 0 := by\n  rw [one_le_iff_pos, pos_iff_ne_zero]\n#align cardinal.one_le_iff_ne_zero Cardinal.one_le_iff_ne_zero\n\n@[simp]\ntheorem lt_one_iff_zero {c : Cardinal} : c < 1 \u2194 c = 0 := by\n  simpa using lt_succ_bot_iff (a := c)\n\ntheorem nat_lt_aleph0 (n : \u2115) : (n : Cardinal.{u}) < \u2135\u2080 :=\n  succ_le_iff.1\n    (by\n      rw [\u2190 nat_succ, \u2190 lift_mk_fin, aleph0, lift_mk_le.{u}]\n      exact \u27e8\u27e8(\u2191), fun a b => Fin.ext\u27e9\u27e9)\n#align cardinal.nat_lt_aleph_0 Cardinal.nat_lt_aleph0\n\n@[simp]\ntheorem one_lt_aleph0 : 1 < \u2135\u2080 := by simpa using nat_lt_aleph0 1\n#align cardinal.one_lt_aleph_0 Cardinal.one_lt_aleph0\n\ntheorem one_le_aleph0 : 1 \u2264 \u2135\u2080 :=\n  one_lt_aleph0.le\n#align cardinal.one_le_aleph_0 Cardinal.one_le_aleph0\n\ntheorem lt_aleph0 {c : Cardinal} : c < \u2135\u2080 \u2194 \u2203 n : \u2115, c = n :=\n  \u27e8fun h => by\n    rcases lt_lift_iff.1 h with \u27e8c, rfl, h'\u27e9\n    rcases le_mk_iff_exists_set.1 h'.1 with \u27e8S, rfl\u27e9\n    suffices S.Finite by\n      lift S to Finset \u2115 using this\n      simp\n    contrapose! h'\n    haveI := Infinite.to_subtype h'\n    exact \u27e8Infinite.natEmbedding S\u27e9, fun \u27e8n, e\u27e9 => e.symm \u25b8 nat_lt_aleph0 _\u27e9\n#align cardinal.lt_aleph_0 Cardinal.lt_aleph0\n\n", "theoremStatement": "lemma 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"Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Fintype.Option", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  obtain \u27e8n, hn\u27e9 := Cardinal.lt_aleph0.mp h\n  rw [hn, succ_natCast]", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 73}}
+{"srcContext": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Mario Carneiro, Floris van Doorn\n-/\nimport Mathlib.Algebra.Module.Basic\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Data.Finsupp.Defs\nimport Mathlib.Data.Set.Countable\nimport Mathlib.Logic.Small.Set\nimport Mathlib.Order.ConditionallyCompleteLattice.Basic\nimport Mathlib.Order.SuccPred.CompleteLinearOrder\nimport Mathlib.SetTheory.Cardinal.SchroederBernstein\nimport Mathlib.Tactic.PPWithUniv\n\n#align_import set_theory.cardinal.basic from \"leanprover-community/mathlib\"@\"3ff3f2d6a3118b8711063de7111a0d77a53219a8\"\n\n/-!\n# Cardinal Numbers\n\nWe define cardinal numbers as a quotient of types under the equivalence relation of equinumerity.\n\n## Main definitions\n\n* `Cardinal` is the type of cardinal numbers (in a given universe).\n* `Cardinal.mk \u03b1` or `#\u03b1` is the cardinality of `\u03b1`. The notation `#` lives in the locale\n  `Cardinal`.\n* Addition `c\u2081 + c\u2082` is defined by `Cardinal.add_def \u03b1 \u03b2 : #\u03b1 + #\u03b2 = #(\u03b1 \u2295 \u03b2)`.\n* Multiplication `c\u2081 * c\u2082` is defined by `Cardinal.mul_def : #\u03b1 * #\u03b2 = #(\u03b1 \u00d7 \u03b2)`.\n* The order `c\u2081 \u2264 c\u2082` is defined by `Cardinal.le_def \u03b1 \u03b2 : #\u03b1 \u2264 #\u03b2 \u2194 Nonempty (\u03b1 \u21aa \u03b2)`.\n* Exponentiation `c\u2081 ^ c\u2082` is defined by `Cardinal.power_def \u03b1 \u03b2 : #\u03b1 ^ #\u03b2 = #(\u03b2 \u2192 \u03b1)`.\n* `Cardinal.isLimit c` means that `c` is a (weak) limit cardinal: `c \u2260 0 \u2227 \u2200 x < c, succ x < c`.\n* `Cardinal.aleph0` or `\u2135\u2080` is the cardinality of `\u2115`. This definition is universe polymorphic:\n  `Cardinal.aleph0.{u} : Cardinal.{u}` (contrast with `\u2115 : Type`, which lives in a specific\n  universe). In some cases the universe level has to be given explicitly.\n* `Cardinal.sum` is the sum of an indexed family of cardinals, i.e. the cardinality of the\n  corresponding sigma type.\n* `Cardinal.prod` is the product of an indexed family of cardinals, i.e. the cardinality of the\n  corresponding pi type.\n* `Cardinal.powerlt a b` or `a ^< b` is defined as the supremum of `a ^ c` for `c < b`.\n\n## Main instances\n\n* Cardinals form a `CanonicallyOrderedCommSemiring` with the aforementioned sum and product.\n* Cardinals form a `SuccOrder`. Use `Order.succ c` for the smallest cardinal greater than `c`.\n* The less than relation on cardinals forms a well-order.\n* Cardinals form a `ConditionallyCompleteLinearOrderBot`. Bounded sets for cardinals in universe\n  `u` are precisely the sets indexed by some type in universe `u`, see\n  `Cardinal.bddAbove_iff_small`. One can use `sSup` for the cardinal supremum, and `sInf` for the\n  minimum of a set of cardinals.\n\n## Main Statements\n\n* Cantor's theorem: `Cardinal.cantor c : c < 2 ^ c`.\n* K\u00f6nig's theorem: `Cardinal.sum_lt_prod`\n\n## Implementation notes\n\n* There is a type of cardinal numbers in every universe level:\n  `Cardinal.{u} : Type (u + 1)` is the quotient of types in `Type u`.\n  The operation `Cardinal.lift` lifts cardinal numbers to a higher level.\n* Cardinal arithmetic specifically for infinite cardinals (like `\u03ba * \u03ba = \u03ba`) is in the file\n  `Mathlib/SetTheory/Cardinal/Ordinal.lean`.\n* There is an instance `Pow Cardinal`, but this will only fire if Lean already knows that both\n  the base and the exponent live in the same universe. As a workaround, you can add\n  ```\n    local infixr:80 \" ^' \" => @HPow.hPow Cardinal Cardinal Cardinal _\n  ```\n  to a file. This notation will work even if Lean doesn't know yet that the base and the exponent\n  live in the same universe (but no exponents in other types can be used).\n  (Porting note: This last point might need to be updated.)\n\n## References\n\n* <https://en.wikipedia.org/wiki/Cardinal_number>\n\n## Tags\n\ncardinal number, cardinal arithmetic, cardinal exponentiation, aleph,\nCantor's theorem, K\u00f6nig's theorem, Konig's theorem\n-/\n\n\nopen scoped Classical\nopen Function Set Order BigOperators\n\nnoncomputable section\n\nuniverse u v w\n\nvariable {\u03b1 \u03b2 : Type u}\n\n/-- The equivalence relation on types given by equivalence (bijective correspondence) of types.\n  Quotienting by this equivalence relation gives the cardinal numbers.\n-/\ninstance Cardinal.isEquivalent : Setoid (Type u) where\n  r \u03b1 \u03b2 := Nonempty (\u03b1 \u2243 \u03b2)\n  iseqv := \u27e8\n    fun \u03b1 => \u27e8Equiv.refl \u03b1\u27e9,\n    fun \u27e8e\u27e9 => \u27e8e.symm\u27e9,\n    fun \u27e8e\u2081\u27e9 \u27e8e\u2082\u27e9 => \u27e8e\u2081.trans e\u2082\u27e9\u27e9\n#align cardinal.is_equivalent Cardinal.isEquivalent\n\n/-- `Cardinal.{u}` is the type of cardinal numbers in `Type u`,\n  defined as the quotient of `Type u` by existence of an equivalence\n  (a bijection with explicit inverse). -/\n@[pp_with_univ]\ndef Cardinal : Type (u + 1) :=\n  Quotient Cardinal.isEquivalent\n#align cardinal Cardinal\n\nnamespace Cardinal\n\n/-- The cardinal number of a type -/\ndef mk : Type u \u2192 Cardinal :=\n  Quotient.mk'\n#align cardinal.mk Cardinal.mk\n\n@[inherit_doc]\nscoped prefix:max \"#\" => Cardinal.mk\n\ninstance canLiftCardinalType : CanLift Cardinal.{u} (Type u) mk fun _ => True :=\n  \u27e8fun c _ => Quot.inductionOn c fun \u03b1 => \u27e8\u03b1, rfl\u27e9\u27e9\n#align cardinal.can_lift_cardinal_Type Cardinal.canLiftCardinalType\n\n@[elab_as_elim]\ntheorem inductionOn {p : Cardinal \u2192 Prop} (c : Cardinal) (h : \u2200 \u03b1, p #\u03b1) : p c :=\n  Quotient.inductionOn c h\n#align cardinal.induction_on Cardinal.inductionOn\n\n@[elab_as_elim]\ntheorem inductionOn\u2082 {p : Cardinal \u2192 Cardinal \u2192 Prop} (c\u2081 : Cardinal) (c\u2082 : Cardinal)\n    (h : \u2200 \u03b1 \u03b2, p #\u03b1 #\u03b2) : p c\u2081 c\u2082 :=\n  Quotient.inductionOn\u2082 c\u2081 c\u2082 h\n#align cardinal.induction_on\u2082 Cardinal.inductionOn\u2082\n\n@[elab_as_elim]\ntheorem inductionOn\u2083 {p : Cardinal \u2192 Cardinal \u2192 Cardinal \u2192 Prop} (c\u2081 : Cardinal) (c\u2082 : Cardinal)\n    (c\u2083 : Cardinal) (h : \u2200 \u03b1 \u03b2 \u03b3, p #\u03b1 #\u03b2 #\u03b3) : p c\u2081 c\u2082 c\u2083 :=\n  Quotient.inductionOn\u2083 c\u2081 c\u2082 c\u2083 h\n#align cardinal.induction_on\u2083 Cardinal.inductionOn\u2083\n\nprotected theorem eq : #\u03b1 = #\u03b2 \u2194 Nonempty (\u03b1 \u2243 \u03b2) :=\n  Quotient.eq'\n#align cardinal.eq Cardinal.eq\n\n@[simp]\ntheorem mk'_def (\u03b1 : Type u) : @Eq Cardinal \u27e6\u03b1\u27e7 #\u03b1 :=\n  rfl\n#align cardinal.mk_def Cardinal.mk'_def\n\n@[simp]\ntheorem mk_out (c : Cardinal) : #c.out = c :=\n  Quotient.out_eq _\n#align cardinal.mk_out Cardinal.mk_out\n\n/-- The representative of the cardinal of a type is equivalent to the original type. -/\ndef outMkEquiv {\u03b1 : Type v} : (#\u03b1).out \u2243 \u03b1 :=\n  Nonempty.some <| Cardinal.eq.mp (by simp)\n#align cardinal.out_mk_equiv Cardinal.outMkEquiv\n\ntheorem mk_congr (e : \u03b1 \u2243 \u03b2) : #\u03b1 = #\u03b2 :=\n  Quot.sound \u27e8e\u27e9\n#align cardinal.mk_congr Cardinal.mk_congr\n\nalias _root_.Equiv.cardinal_eq := mk_congr\n#align equiv.cardinal_eq Equiv.cardinal_eq\n\n/-- Lift a function between `Type*`s to a function between `Cardinal`s. -/\ndef map (f : Type u \u2192 Type v) (hf : \u2200 \u03b1 \u03b2, \u03b1 \u2243 \u03b2 \u2192 f \u03b1 \u2243 f \u03b2) : Cardinal.{u} \u2192 Cardinal.{v} :=\n  Quotient.map f fun \u03b1 \u03b2 \u27e8e\u27e9 => \u27e8hf \u03b1 \u03b2 e\u27e9\n#align cardinal.map Cardinal.map\n\n@[simp]\ntheorem map_mk (f : Type u \u2192 Type v) (hf : \u2200 \u03b1 \u03b2, \u03b1 \u2243 \u03b2 \u2192 f \u03b1 \u2243 f \u03b2) (\u03b1 : Type u) :\n    map f hf #\u03b1 = #(f \u03b1) :=\n  rfl\n#align cardinal.map_mk Cardinal.map_mk\n\n/-- Lift a binary operation `Type* \u2192 Type* \u2192 Type*` to a binary operation on `Cardinal`s. -/\ndef map\u2082 (f : Type u \u2192 Type v \u2192 Type w) (hf : \u2200 \u03b1 \u03b2 \u03b3 \u03b4, \u03b1 \u2243 \u03b2 \u2192 \u03b3 \u2243 \u03b4 \u2192 f \u03b1 \u03b3 \u2243 f \u03b2 \u03b4) :\n    Cardinal.{u} \u2192 Cardinal.{v} \u2192 Cardinal.{w} :=\n  Quotient.map\u2082 f fun \u03b1 \u03b2 \u27e8e\u2081\u27e9 \u03b3 \u03b4 \u27e8e\u2082\u27e9 => \u27e8hf \u03b1 \u03b2 \u03b3 \u03b4 e\u2081 e\u2082\u27e9\n#align cardinal.map\u2082 Cardinal.map\u2082\n\n/-- The universe lift operation on cardinals. You can specify the universes explicitly with\n  `lift.{u v} : Cardinal.{v} \u2192 Cardinal.{max v u}` -/\n@[pp_with_univ]\ndef lift (c : Cardinal.{v}) : Cardinal.{max v u} :=\n  map ULift.{u, v} (fun _ _ e => Equiv.ulift.trans <| e.trans Equiv.ulift.symm) c\n#align cardinal.lift Cardinal.lift\n\n@[simp]\ntheorem mk_uLift (\u03b1) : #(ULift.{v, u} \u03b1) = lift.{v} #\u03b1 :=\n  rfl\n#align cardinal.mk_ulift Cardinal.mk_uLift\n\n-- Porting note: simpNF is not happy with universe levels, but this is needed as simp lemma\n-- further down in this file\n/-- `lift.{max u v, u}` equals `lift.{v, u}`. -/\n@[simp, nolint simpNF]\ntheorem lift_umax : lift.{max u v, u} = lift.{v, u} :=\n  funext fun a => inductionOn a fun _ => (Equiv.ulift.trans Equiv.ulift.symm).cardinal_eq\n#align cardinal.lift_umax Cardinal.lift_umax\n\n-- Porting note: simpNF is not happy with universe levels, but this is needed as simp lemma\n-- further down in this file\n/-- `lift.{max v u, u}` equals `lift.{v, u}`. -/\n@[simp, nolint simpNF]\ntheorem lift_umax' : lift.{max v u, u} = lift.{v, u} :=\n  lift_umax\n#align cardinal.lift_umax' Cardinal.lift_umax'\n\n-- Porting note: simpNF is not happy with universe levels, but this is needed as simp lemma\n-- further down in this file\n/-- A cardinal lifted to a lower or equal universe equals itself. -/\n@[simp, nolint simpNF]\ntheorem lift_id' (a : Cardinal.{max u v}) : lift.{u} a = a :=\n  inductionOn a fun _ => mk_congr Equiv.ulift\n#align cardinal.lift_id' Cardinal.lift_id'\n\n/-- A cardinal lifted to the same universe equals itself. -/\n@[simp]\ntheorem lift_id (a : Cardinal) : lift.{u, u} a = a :=\n  lift_id'.{u, u} a\n#align cardinal.lift_id Cardinal.lift_id\n\n/-- A cardinal lifted to the zero universe equals itself. -/\n-- porting note (#10618): simp can prove this\n-- @[simp]\ntheorem lift_uzero (a : Cardinal.{u}) : lift.{0} a = a :=\n  lift_id'.{0, u} a\n#align cardinal.lift_uzero Cardinal.lift_uzero\n\n@[simp]\ntheorem lift_lift.{u_1} (a : Cardinal.{u_1}) : lift.{w} (lift.{v} a) = lift.{max v w} a :=\n  inductionOn a fun _ => (Equiv.ulift.trans <| Equiv.ulift.trans Equiv.ulift.symm).cardinal_eq\n#align cardinal.lift_lift Cardinal.lift_lift\n\n/-- We define the order on cardinal numbers by `#\u03b1 \u2264 #\u03b2` if and only if\n  there exists an embedding (injective function) from \u03b1 to \u03b2. -/\ninstance : LE Cardinal.{u} :=\n  \u27e8fun q\u2081 q\u2082 =>\n    Quotient.liftOn\u2082 q\u2081 q\u2082 (fun \u03b1 \u03b2 => Nonempty <| \u03b1 \u21aa \u03b2) fun _ _ _ _ \u27e8e\u2081\u27e9 \u27e8e\u2082\u27e9 =>\n      propext \u27e8fun \u27e8e\u27e9 => \u27e8e.congr e\u2081 e\u2082\u27e9, fun \u27e8e\u27e9 => \u27e8e.congr e\u2081.symm e\u2082.symm\u27e9\u27e9\u27e9\n\ninstance partialOrder : PartialOrder Cardinal.{u} where\n  le := (\u00b7 \u2264 \u00b7)\n  le_refl := by\n    rintro \u27e8\u03b1\u27e9\n    exact \u27e8Embedding.refl _\u27e9\n  le_trans := by\n    rintro \u27e8\u03b1\u27e9 \u27e8\u03b2\u27e9 \u27e8\u03b3\u27e9 \u27e8e\u2081\u27e9 \u27e8e\u2082\u27e9\n    exact \u27e8e\u2081.trans e\u2082\u27e9\n  le_antisymm := by\n    rintro \u27e8\u03b1\u27e9 \u27e8\u03b2\u27e9 \u27e8e\u2081\u27e9 \u27e8e\u2082\u27e9\n    exact Quotient.sound (e\u2081.antisymm e\u2082)\n\ninstance linearOrder : LinearOrder Cardinal.{u} :=\n  { Cardinal.partialOrder with\n    le_total := by\n      rintro \u27e8\u03b1\u27e9 \u27e8\u03b2\u27e9\n      apply Embedding.total\n    decidableLE := Classical.decRel _ }\n\ntheorem le_def (\u03b1 \u03b2 : Type u) : #\u03b1 \u2264 #\u03b2 \u2194 Nonempty (\u03b1 \u21aa \u03b2) :=\n  Iff.rfl\n#align cardinal.le_def Cardinal.le_def\n\ntheorem mk_le_of_injective {\u03b1 \u03b2 : Type u} {f : \u03b1 \u2192 \u03b2} (hf : Injective f) : #\u03b1 \u2264 #\u03b2 :=\n  \u27e8\u27e8f, hf\u27e9\u27e9\n#align cardinal.mk_le_of_injective Cardinal.mk_le_of_injective\n\ntheorem _root_.Function.Embedding.cardinal_le {\u03b1 \u03b2 : Type u} (f : \u03b1 \u21aa \u03b2) : #\u03b1 \u2264 #\u03b2 :=\n  \u27e8f\u27e9\n#align function.embedding.cardinal_le Function.Embedding.cardinal_le\n\ntheorem mk_le_of_surjective {\u03b1 \u03b2 : Type u} {f : \u03b1 \u2192 \u03b2} (hf : Surjective f) : #\u03b2 \u2264 #\u03b1 :=\n  \u27e8Embedding.ofSurjective f hf\u27e9\n#align cardinal.mk_le_of_surjective Cardinal.mk_le_of_surjective\n\ntheorem le_mk_iff_exists_set {c : Cardinal} {\u03b1 : Type u} : c \u2264 #\u03b1 \u2194 \u2203 p : Set \u03b1, #p = c :=\n  \u27e8inductionOn c fun _ \u27e8\u27e8f, hf\u27e9\u27e9 => \u27e8Set.range f, (Equiv.ofInjective f hf).cardinal_eq.symm\u27e9,\n    fun \u27e8_, e\u27e9 => e \u25b8 \u27e8\u27e8Subtype.val, fun _ _ => Subtype.eq\u27e9\u27e9\u27e9\n#align cardinal.le_mk_iff_exists_set Cardinal.le_mk_iff_exists_set\n\ntheorem mk_subtype_le {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) : #(Subtype p) \u2264 #\u03b1 :=\n  \u27e8Embedding.subtype p\u27e9\n#align cardinal.mk_subtype_le Cardinal.mk_subtype_le\n\ntheorem mk_set_le (s : Set \u03b1) : #s \u2264 #\u03b1 :=\n  mk_subtype_le s\n#align cardinal.mk_set_le Cardinal.mk_set_le\n\n@[simp]\nlemma mk_preimage_down {s : Set \u03b1} : #(ULift.down.{v} \u207b\u00b9' s) = lift.{v} (#s) := by\n  rw [\u2190 mk_uLift, Cardinal.eq]\n  constructor\n  let f : ULift.down \u207b\u00b9' s \u2192 ULift s := fun x \u21a6 ULift.up (restrictPreimage s ULift.down x)\n  have : Function.Bijective f :=\n    ULift.up_bijective.comp (restrictPreimage_bijective _ (ULift.down_bijective))\n  exact Equiv.ofBijective f this\n\ntheorem out_embedding {c c' : Cardinal} : c \u2264 c' \u2194 Nonempty (c.out \u21aa c'.out) := by\n  trans\n  \u00b7 rw [\u2190 Quotient.out_eq c, \u2190 Quotient.out_eq c']\n  \u00b7 rw [mk'_def, mk'_def, le_def]\n#align cardinal.out_embedding Cardinal.out_embedding\n\ntheorem lift_mk_le {\u03b1 : Type v} {\u03b2 : Type w} :\n    lift.{max u w} #\u03b1 \u2264 lift.{max u v} #\u03b2 \u2194 Nonempty (\u03b1 \u21aa \u03b2) :=\n  \u27e8fun \u27e8f\u27e9 => \u27e8Embedding.congr Equiv.ulift Equiv.ulift f\u27e9, fun \u27e8f\u27e9 =>\n    \u27e8Embedding.congr Equiv.ulift.symm Equiv.ulift.symm f\u27e9\u27e9\n#align cardinal.lift_mk_le Cardinal.lift_mk_le\n\n/-- A variant of `Cardinal.lift_mk_le` with specialized universes.\nBecause Lean often can not realize it should use this specialization itself,\nwe provide this statement separately so you don't have to solve the specialization problem either.\n-/\ntheorem lift_mk_le' {\u03b1 : Type u} {\u03b2 : Type v} : lift.{v} #\u03b1 \u2264 lift.{u} #\u03b2 \u2194 Nonempty (\u03b1 \u21aa \u03b2) :=\n  lift_mk_le.{0}\n#align cardinal.lift_mk_le' Cardinal.lift_mk_le'\n\ntheorem lift_mk_eq {\u03b1 : Type u} {\u03b2 : Type v} :\n    lift.{max v w} #\u03b1 = lift.{max u w} #\u03b2 \u2194 Nonempty (\u03b1 \u2243 \u03b2) :=\n  Quotient.eq'.trans\n    \u27e8fun \u27e8f\u27e9 => \u27e8Equiv.ulift.symm.trans <| f.trans Equiv.ulift\u27e9, fun \u27e8f\u27e9 =>\n      \u27e8Equiv.ulift.trans <| f.trans Equiv.ulift.symm\u27e9\u27e9\n#align cardinal.lift_mk_eq Cardinal.lift_mk_eq\n\n/-- A variant of `Cardinal.lift_mk_eq` with specialized universes.\nBecause Lean often can not realize it should use this specialization itself,\nwe provide this statement separately so you don't have to solve the specialization problem either.\n-/\ntheorem lift_mk_eq' {\u03b1 : Type u} {\u03b2 : Type v} : lift.{v} #\u03b1 = lift.{u} #\u03b2 \u2194 Nonempty (\u03b1 \u2243 \u03b2) :=\n  lift_mk_eq.{u, v, 0}\n#align cardinal.lift_mk_eq' Cardinal.lift_mk_eq'\n\n@[simp]\ntheorem lift_le {a b : Cardinal.{v}} : lift.{u, v} a \u2264 lift.{u, v} b \u2194 a \u2264 b :=\n  inductionOn\u2082 a b fun \u03b1 \u03b2 => by\n    rw [\u2190 lift_umax]\n    exact lift_mk_le.{u}\n#align cardinal.lift_le Cardinal.lift_le\n\n-- Porting note: changed `simps` to `simps!` because the linter told to do so.\n/-- `Cardinal.lift` as an `OrderEmbedding`. -/\n@[simps! (config := .asFn)]\ndef liftOrderEmbedding : Cardinal.{v} \u21aao Cardinal.{max v u} :=\n  OrderEmbedding.ofMapLEIff lift.{u, v} fun _ _ => lift_le\n#align cardinal.lift_order_embedding Cardinal.liftOrderEmbedding\n\ntheorem lift_injective : Injective lift.{u, v} :=\n  liftOrderEmbedding.injective\n#align cardinal.lift_injective Cardinal.lift_injective\n\n@[simp]\ntheorem lift_inj {a b : Cardinal.{u}} : lift.{v, u} a = lift.{v, u} b \u2194 a = b :=\n  lift_injective.eq_iff\n#align cardinal.lift_inj Cardinal.lift_inj\n\n@[simp]\ntheorem lift_lt {a b : Cardinal.{u}} : lift.{v, u} a < lift.{v, u} b \u2194 a < b :=\n  liftOrderEmbedding.lt_iff_lt\n#align cardinal.lift_lt Cardinal.lift_lt\n\ntheorem lift_strictMono : StrictMono lift := fun _ _ => lift_lt.2\n#align cardinal.lift_strict_mono Cardinal.lift_strictMono\n\ntheorem lift_monotone : Monotone lift :=\n  lift_strictMono.monotone\n#align cardinal.lift_monotone Cardinal.lift_monotone\n\ninstance : Zero Cardinal.{u} :=\n  -- `PEmpty` might be more canonical, but this is convenient for defeq with natCast\n  \u27e8lift #(Fin 0)\u27e9\n\ninstance : Inhabited Cardinal.{u} :=\n  \u27e80\u27e9\n\n@[simp]\ntheorem mk_eq_zero (\u03b1 : Type u) [IsEmpty \u03b1] : #\u03b1 = 0 :=\n  (Equiv.equivOfIsEmpty \u03b1 (ULift (Fin 0))).cardinal_eq\n#align cardinal.mk_eq_zero Cardinal.mk_eq_zero\n\n@[simp]\ntheorem lift_zero : lift 0 = 0 := mk_eq_zero _\n#align cardinal.lift_zero Cardinal.lift_zero\n\n@[simp]\ntheorem lift_eq_zero {a : Cardinal.{v}} : lift.{u} a = 0 \u2194 a = 0 :=\n  lift_injective.eq_iff' lift_zero\n#align cardinal.lift_eq_zero Cardinal.lift_eq_zero\n\ntheorem mk_eq_zero_iff {\u03b1 : Type u} : #\u03b1 = 0 \u2194 IsEmpty \u03b1 :=\n  \u27e8fun e =>\n    let \u27e8h\u27e9 := Quotient.exact e\n    h.isEmpty,\n    @mk_eq_zero \u03b1\u27e9\n#align cardinal.mk_eq_zero_iff Cardinal.mk_eq_zero_iff\n\ntheorem mk_ne_zero_iff {\u03b1 : Type u} : #\u03b1 \u2260 0 \u2194 Nonempty \u03b1 :=\n  (not_iff_not.2 mk_eq_zero_iff).trans not_isEmpty_iff\n#align cardinal.mk_ne_zero_iff Cardinal.mk_ne_zero_iff\n\n@[simp]\ntheorem mk_ne_zero (\u03b1 : Type u) [Nonempty \u03b1] : #\u03b1 \u2260 0 :=\n  mk_ne_zero_iff.2 \u2039_\u203a\n#align cardinal.mk_ne_zero Cardinal.mk_ne_zero\n\ninstance : One Cardinal.{u} :=\n  -- `PUnit` might be more canonical, but this is convenient for defeq with natCast\n  \u27e8lift #(Fin 1)\u27e9\n\ninstance : Nontrivial Cardinal.{u} :=\n  \u27e8\u27e81, 0, mk_ne_zero _\u27e9\u27e9\n\ntheorem mk_eq_one (\u03b1 : Type u) [Unique \u03b1] : #\u03b1 = 1 :=\n  (Equiv.equivOfUnique \u03b1 (ULift (Fin 1))).cardinal_eq\n#align cardinal.mk_eq_one Cardinal.mk_eq_one\n\ntheorem le_one_iff_subsingleton {\u03b1 : Type u} : #\u03b1 \u2264 1 \u2194 Subsingleton \u03b1 :=\n  \u27e8fun \u27e8f\u27e9 => \u27e8fun _ _ => f.injective (Subsingleton.elim _ _)\u27e9, fun \u27e8h\u27e9 =>\n    \u27e8fun _ => ULift.up 0, fun _ _ _ => h _ _\u27e9\u27e9\n#align cardinal.le_one_iff_subsingleton Cardinal.le_one_iff_subsingleton\n\n@[simp]\ntheorem mk_le_one_iff_set_subsingleton {s : Set \u03b1} : #s \u2264 1 \u2194 s.Subsingleton :=\n  le_one_iff_subsingleton.trans s.subsingleton_coe\n#align cardinal.mk_le_one_iff_set_subsingleton Cardinal.mk_le_one_iff_set_subsingleton\n\nalias \u27e8_, _root_.Set.Subsingleton.cardinal_mk_le_one\u27e9 := mk_le_one_iff_set_subsingleton\n#align set.subsingleton.cardinal_mk_le_one Set.Subsingleton.cardinal_mk_le_one\n\ninstance : Add Cardinal.{u} :=\n  \u27e8map\u2082 Sum fun _ _ _ _ => Equiv.sumCongr\u27e9\n\ntheorem add_def (\u03b1 \u03b2 : Type u) : #\u03b1 + #\u03b2 = #(Sum \u03b1 \u03b2) :=\n  rfl\n#align cardinal.add_def Cardinal.add_def\n\ninstance : NatCast Cardinal.{u} :=\n  \u27e8fun n => lift #(Fin n)\u27e9\n\n@[simp]\ntheorem mk_sum (\u03b1 : Type u) (\u03b2 : Type v) : #(\u03b1 \u2295 \u03b2) = lift.{v, u} #\u03b1 + lift.{u, v} #\u03b2 :=\n  mk_congr (Equiv.ulift.symm.sumCongr Equiv.ulift.symm)\n#align cardinal.mk_sum Cardinal.mk_sum\n\n@[simp]\ntheorem mk_option {\u03b1 : Type u} : #(Option \u03b1) = #\u03b1 + 1 := by\n  rw [(Equiv.optionEquivSumPUnit.{u, u} \u03b1).cardinal_eq, mk_sum, mk_eq_one PUnit, lift_id, lift_id]\n#align cardinal.mk_option Cardinal.mk_option\n\n@[simp]\ntheorem mk_psum (\u03b1 : Type u) (\u03b2 : Type v) : #(PSum \u03b1 \u03b2) = lift.{v} #\u03b1 + lift.{u} #\u03b2 :=\n  (mk_congr (Equiv.psumEquivSum \u03b1 \u03b2)).trans (mk_sum \u03b1 \u03b2)\n#align cardinal.mk_psum Cardinal.mk_psum\n\n@[simp]\ntheorem mk_fintype (\u03b1 : Type u) [h : Fintype \u03b1] : #\u03b1 = Fintype.card \u03b1 :=\n  mk_congr (Fintype.equivOfCardEq (by simp))\n\nprotected theorem cast_succ (n : \u2115) : ((n + 1 : \u2115) : Cardinal.{u}) = n + 1 := by\n  change #(ULift.{u} (Fin (n+1))) = # (ULift.{u} (Fin n)) + 1\n  rw [\u2190 mk_option, mk_fintype, mk_fintype]\n  simp only [Fintype.card_ulift, Fintype.card_fin, Fintype.card_option]\n\ninstance : Mul Cardinal.{u} :=\n  \u27e8map\u2082 Prod fun _ _ _ _ => Equiv.prodCongr\u27e9\n\ntheorem mul_def (\u03b1 \u03b2 : Type u) : #\u03b1 * #\u03b2 = #(\u03b1 \u00d7 \u03b2) :=\n  rfl\n#align cardinal.mul_def Cardinal.mul_def\n\n@[simp]\ntheorem mk_prod (\u03b1 : Type u) (\u03b2 : Type v) : #(\u03b1 \u00d7 \u03b2) = lift.{v, u} #\u03b1 * lift.{u, v} #\u03b2 :=\n  mk_congr (Equiv.ulift.symm.prodCongr Equiv.ulift.symm)\n#align cardinal.mk_prod Cardinal.mk_prod\n\nprivate theorem mul_comm' (a b : Cardinal.{u}) : a * b = b * a :=\n  inductionOn\u2082 a b fun \u03b1 \u03b2 => mk_congr <| Equiv.prodComm \u03b1 \u03b2\n\n/-- The cardinal exponential. `#\u03b1 ^ #\u03b2` is the cardinal of `\u03b2 \u2192 \u03b1`. -/\ninstance instPowCardinal : Pow Cardinal.{u} Cardinal.{u} :=\n  \u27e8map\u2082 (fun \u03b1 \u03b2 => \u03b2 \u2192 \u03b1) fun _ _ _ _ e\u2081 e\u2082 => e\u2082.arrowCongr e\u2081\u27e9\n\ntheorem power_def (\u03b1 \u03b2 : Type u) : #\u03b1 ^ #\u03b2 = #(\u03b2 \u2192 \u03b1) :=\n  rfl\n#align cardinal.power_def Cardinal.power_def\n\ntheorem mk_arrow (\u03b1 : Type u) (\u03b2 : Type v) : #(\u03b1 \u2192 \u03b2) = (lift.{u} #\u03b2^lift.{v} #\u03b1) :=\n  mk_congr (Equiv.ulift.symm.arrowCongr Equiv.ulift.symm)\n#align cardinal.mk_arrow Cardinal.mk_arrow\n\n@[simp]\ntheorem lift_power (a b : Cardinal.{u}) : lift.{v} (a ^ b) = lift.{v} a ^ lift.{v} b :=\n  inductionOn\u2082 a b fun _ _ =>\n    mk_congr <| Equiv.ulift.trans (Equiv.ulift.arrowCongr Equiv.ulift).symm\n#align cardinal.lift_power Cardinal.lift_power\n\n@[simp]\ntheorem power_zero {a : Cardinal} : a ^ (0 : Cardinal) = 1 :=\n  inductionOn a fun _ => mk_eq_one _\n#align cardinal.power_zero Cardinal.power_zero\n\n@[simp]\ntheorem power_one {a : Cardinal.{u}} : a ^ (1 : Cardinal) = a :=\n  inductionOn a fun \u03b1 => mk_congr (Equiv.funUnique (ULift.{u} (Fin 1)) \u03b1)\n#align cardinal.power_one Cardinal.power_one\n\ntheorem power_add {a b c : Cardinal} : a ^ (b + c) = a ^ b * a ^ c :=\n  inductionOn\u2083 a b c fun \u03b1 \u03b2 \u03b3 => mk_congr <| Equiv.sumArrowEquivProdArrow \u03b2 \u03b3 \u03b1\n#align cardinal.power_add Cardinal.power_add\n\ninstance commSemiring : CommSemiring Cardinal.{u} where\n  zero := 0\n  one := 1\n  add := (\u00b7 + \u00b7)\n  mul := (\u00b7 * \u00b7)\n  zero_add a := inductionOn a fun \u03b1 => mk_congr <| Equiv.emptySum (ULift (Fin 0)) \u03b1\n  add_zero a := inductionOn a fun \u03b1 => mk_congr <| Equiv.sumEmpty \u03b1 (ULift (Fin 0))\n  add_assoc a b c := inductionOn\u2083 a b c fun \u03b1 \u03b2 \u03b3 => mk_congr <| Equiv.sumAssoc \u03b1 \u03b2 \u03b3\n  add_comm a b := inductionOn\u2082 a b fun \u03b1 \u03b2 => mk_congr <| Equiv.sumComm \u03b1 \u03b2\n  zero_mul a := inductionOn a fun \u03b1 => mk_eq_zero _\n  mul_zero a := inductionOn a fun \u03b1 => mk_eq_zero _\n  one_mul a := inductionOn a fun \u03b1 => mk_congr <| Equiv.uniqueProd \u03b1 (ULift (Fin 1))\n  mul_one a := inductionOn a fun \u03b1 => mk_congr <| Equiv.prodUnique \u03b1 (ULift (Fin 1))\n  mul_assoc a b c := inductionOn\u2083 a b c fun \u03b1 \u03b2 \u03b3 => mk_congr <| Equiv.prodAssoc \u03b1 \u03b2 \u03b3\n  mul_comm := mul_comm'\n  left_distrib a b c := inductionOn\u2083 a b c fun \u03b1 \u03b2 \u03b3 => mk_congr <| Equiv.prodSumDistrib \u03b1 \u03b2 \u03b3\n  right_distrib a b c := inductionOn\u2083 a b c fun \u03b1 \u03b2 \u03b3 => mk_congr <| Equiv.sumProdDistrib \u03b1 \u03b2 \u03b3\n  nsmul := nsmulRec\n  npow n c := c ^ (n : Cardinal)\n  npow_zero := @power_zero\n  npow_succ n c := show c ^ (\u2191(n + 1) : Cardinal) = c ^ (\u2191n : Cardinal) * c\n    by rw [Cardinal.cast_succ, power_add, power_one, mul_comm']\n  natCast := (fun n => lift.{u} #(Fin n) : \u2115 \u2192 Cardinal.{u})\n  natCast_zero := rfl\n  natCast_succ := Cardinal.cast_succ\n\n/-! Porting note (#11229): Deprecated section. Remove. -/\nsection deprecated\nset_option linter.deprecated false\n\n@[deprecated]\ntheorem power_bit0 (a b : Cardinal) : a ^ bit0 b = a ^ b * a ^ b :=\n  power_add\n#align cardinal.power_bit0 Cardinal.power_bit0\n\n@[deprecated]\ntheorem power_bit1 (a b : Cardinal) : a ^ bit1 b = a ^ b * a ^ b * a := by\n  rw [bit1, \u2190 power_bit0, power_add, power_one]\n#align cardinal.power_bit1 Cardinal.power_bit1\n\nend deprecated\n\n@[simp]\ntheorem one_power {a : Cardinal} : (1 : Cardinal) ^ a = 1 :=\n  inductionOn a fun _ => mk_eq_one _\n#align cardinal.one_power Cardinal.one_power\n\n-- porting note (#10618): simp can prove this\n-- @[simp]\ntheorem mk_bool : #Bool = 2 := by simp\n#align cardinal.mk_bool Cardinal.mk_bool\n\n-- porting note (#10618): simp can prove this\n-- @[simp]\ntheorem mk_Prop : #Prop = 2 := by simp\n#align cardinal.mk_Prop Cardinal.mk_Prop\n\n@[simp]\ntheorem zero_power {a : Cardinal} : a \u2260 0 \u2192 (0 : Cardinal) ^ a = 0 :=\n  inductionOn a fun _ heq =>\n    mk_eq_zero_iff.2 <|\n      isEmpty_pi.2 <|\n        let \u27e8a\u27e9 := mk_ne_zero_iff.1 heq\n        \u27e8a, inferInstance\u27e9\n#align cardinal.zero_power Cardinal.zero_power\n\ntheorem power_ne_zero {a : Cardinal} (b : Cardinal) : a \u2260 0 \u2192 a ^ b \u2260 0 :=\n  inductionOn\u2082 a b fun _ _ h =>\n    let \u27e8a\u27e9 := mk_ne_zero_iff.1 h\n    mk_ne_zero_iff.2 \u27e8fun _ => a\u27e9\n#align cardinal.power_ne_zero Cardinal.power_ne_zero\n\ntheorem mul_power {a b c : Cardinal} : (a * b) ^ c = a ^ c * b ^ c :=\n  inductionOn\u2083 a b c fun \u03b1 \u03b2 \u03b3 => mk_congr <| Equiv.arrowProdEquivProdArrow \u03b1 \u03b2 \u03b3\n#align cardinal.mul_power Cardinal.mul_power\n\ntheorem power_mul {a b c : Cardinal} : a ^ (b * c) = (a ^ b) ^ c := by\n  rw [mul_comm b c]\n  exact inductionOn\u2083 a b c fun \u03b1 \u03b2 \u03b3 => mk_congr <| Equiv.curry \u03b3 \u03b2 \u03b1\n#align cardinal.power_mul Cardinal.power_mul\n\n@[simp]\ntheorem pow_cast_right (a : Cardinal.{u}) (n : \u2115) : a ^ (\u2191n : Cardinal.{u}) = a ^ n :=\n  rfl\n#align cardinal.pow_cast_right Cardinal.pow_cast_right\n\n@[simp]\ntheorem lift_one : lift 1 = 1 := mk_eq_one _\n#align cardinal.lift_one Cardinal.lift_one\n\n@[simp]\ntheorem lift_eq_one {a : Cardinal.{v}} : lift.{u} a = 1 \u2194 a = 1 :=\n  lift_injective.eq_iff' lift_one\n\n@[simp]\ntheorem lift_add (a b : Cardinal.{u}) : lift.{v} (a + b) = lift.{v} a + lift.{v} b :=\n  inductionOn\u2082 a b fun _ _ =>\n    mk_congr <| Equiv.ulift.trans (Equiv.sumCongr Equiv.ulift Equiv.ulift).symm\n#align cardinal.lift_add Cardinal.lift_add\n\n@[simp]\ntheorem lift_mul (a b : Cardinal.{u}) : lift.{v} (a * b) = lift.{v} a * lift.{v} b :=\n  inductionOn\u2082 a b fun _ _ =>\n    mk_congr <| Equiv.ulift.trans (Equiv.prodCongr Equiv.ulift Equiv.ulift).symm\n#align cardinal.lift_mul Cardinal.lift_mul\n\n/-! Porting note (#11229): Deprecated section. Remove. -/\nsection deprecated\nset_option linter.deprecated false\n\n@[simp, deprecated]\ntheorem lift_bit0 (a : Cardinal) : lift.{v} (bit0 a) = bit0 (lift.{v} a) :=\n  lift_add a a\n#align cardinal.lift_bit0 Cardinal.lift_bit0\n\n@[simp, deprecated]\ntheorem lift_bit1 (a : Cardinal) : lift.{v} (bit1 a) = bit1 (lift.{v} a) := by simp [bit1]\n#align cardinal.lift_bit1 Cardinal.lift_bit1\n\nend deprecated\n\n-- Porting note: Proof used to be simp, needed to remind simp that 1 + 1 = 2\ntheorem lift_two : lift.{u, v} 2 = 2 := by simp [\u2190 one_add_one_eq_two]\n#align cardinal.lift_two Cardinal.lift_two\n\n@[simp]\ntheorem mk_set {\u03b1 : Type u} : #(Set \u03b1) = 2 ^ #\u03b1 := by simp [\u2190 one_add_one_eq_two, Set, mk_arrow]\n#align cardinal.mk_set Cardinal.mk_set\n\n/-- A variant of `Cardinal.mk_set` expressed in terms of a `Set` instead of a `Type`. -/\n@[simp]\ntheorem mk_powerset {\u03b1 : Type u} (s : Set \u03b1) : #(\u21a5(\ud835\udcab s)) = 2 ^ #(\u21a5s) :=\n  (mk_congr (Equiv.Set.powerset s)).trans mk_set\n#align cardinal.mk_powerset Cardinal.mk_powerset\n\ntheorem lift_two_power (a : Cardinal) : lift.{v} (2 ^ a) = 2 ^ lift.{v} a := by\n  simp [\u2190 one_add_one_eq_two]\n#align cardinal.lift_two_power Cardinal.lift_two_power\n\nsection OrderProperties\n\nopen Sum\n\nprotected theorem zero_le : \u2200 a : Cardinal, 0 \u2264 a := by\n  rintro \u27e8\u03b1\u27e9\n  exact \u27e8Embedding.ofIsEmpty\u27e9\n#align cardinal.zero_le Cardinal.zero_le\n\nprivate theorem add_le_add' : \u2200 {a b c d : Cardinal}, a \u2264 b \u2192 c \u2264 d \u2192 a + c \u2264 b + d := by\n  rintro \u27e8\u03b1\u27e9 \u27e8\u03b2\u27e9 \u27e8\u03b3\u27e9 \u27e8\u03b4\u27e9 \u27e8e\u2081\u27e9 \u27e8e\u2082\u27e9; exact \u27e8e\u2081.sumMap e\u2082\u27e9\n-- #align cardinal.add_le_add' Cardinal.add_le_add'\n\ninstance add_covariantClass : CovariantClass Cardinal Cardinal (\u00b7 + \u00b7) (\u00b7 \u2264 \u00b7) :=\n  \u27e8fun _ _ _ => add_le_add' le_rfl\u27e9\n#align cardinal.add_covariant_class Cardinal.add_covariantClass\n\ninstance add_swap_covariantClass : CovariantClass Cardinal Cardinal (swap (\u00b7 + \u00b7)) (\u00b7 \u2264 \u00b7) :=\n  \u27e8fun _ _ _ h => add_le_add' h le_rfl\u27e9\n#align cardinal.add_swap_covariant_class Cardinal.add_swap_covariantClass\n\ninstance canonicallyOrderedCommSemiring : CanonicallyOrderedCommSemiring Cardinal.{u} :=\n  { Cardinal.commSemiring,\n    Cardinal.partialOrder with\n    bot := 0\n    bot_le := Cardinal.zero_le\n    add_le_add_left := fun a b => add_le_add_left\n    exists_add_of_le := fun {a b} =>\n      inductionOn\u2082 a b fun \u03b1 \u03b2 \u27e8\u27e8f, hf\u27e9\u27e9 =>\n        have : Sum \u03b1 ((range f)\u1d9c : Set \u03b2) \u2243 \u03b2 :=\n          (Equiv.sumCongr (Equiv.ofInjective f hf) (Equiv.refl _)).trans <|\n            Equiv.Set.sumCompl (range f)\n        \u27e8#(\u21a5(range f)\u1d9c), mk_congr this.symm\u27e9\n    le_self_add := fun a b => (add_zero a).ge.trans <| add_le_add_left (Cardinal.zero_le _) _\n    eq_zero_or_eq_zero_of_mul_eq_zero := fun {a b} =>\n      inductionOn\u2082 a b fun \u03b1 \u03b2 => by\n        simpa only [mul_def, mk_eq_zero_iff, isEmpty_prod] using id }\n\ninstance : CanonicallyLinearOrderedAddCommMonoid Cardinal.{u} :=\n  { Cardinal.canonicallyOrderedCommSemiring, Cardinal.linearOrder with }\n\n-- Computable instance to prevent a non-computable one being found via the one above\ninstance : CanonicallyOrderedAddCommMonoid Cardinal.{u} :=\n  { Cardinal.canonicallyOrderedCommSemiring with }\n\ninstance : LinearOrderedCommMonoidWithZero Cardinal.{u} :=\n  { Cardinal.commSemiring,\n    Cardinal.linearOrder with\n    mul_le_mul_left := @mul_le_mul_left' _ _ _ _\n    zero_le_one := zero_le _ }\n\n-- Computable instance to prevent a non-computable one being found via the one above\ninstance : CommMonoidWithZero Cardinal.{u} :=\n  { Cardinal.canonicallyOrderedCommSemiring with }\n\n-- Porting note: new\n-- Computable instance to prevent a non-computable one being found via the one above\ninstance : CommMonoid Cardinal.{u} :=\n  { Cardinal.canonicallyOrderedCommSemiring with }\n\ntheorem zero_power_le (c : Cardinal.{u}) : (0 : Cardinal.{u}) ^ c \u2264 1 := by\n  by_cases h : c = 0\n  \u00b7 rw [h, power_zero]\n  \u00b7 rw [zero_power h]\n    apply zero_le\n#align cardinal.zero_power_le Cardinal.zero_power_le\n\ntheorem power_le_power_left : \u2200 {a b c : Cardinal}, a \u2260 0 \u2192 b \u2264 c \u2192 a ^ b \u2264 a ^ c := by\n  rintro \u27e8\u03b1\u27e9 \u27e8\u03b2\u27e9 \u27e8\u03b3\u27e9 h\u03b1 \u27e8e\u27e9\n  let \u27e8a\u27e9 := mk_ne_zero_iff.1 h\u03b1\n  exact \u27e8@Function.Embedding.arrowCongrLeft _ _ _ \u27e8a\u27e9 e\u27e9\n#align cardinal.power_le_power_left Cardinal.power_le_power_left\n\ntheorem self_le_power (a : Cardinal) {b : Cardinal} (hb : 1 \u2264 b) : a \u2264 a ^ b := by\n  rcases eq_or_ne a 0 with (rfl | ha)\n  \u00b7 exact zero_le _\n  \u00b7 convert power_le_power_left ha hb\n    exact power_one.symm\n#align cardinal.self_le_power Cardinal.self_le_power\n\n/-- **Cantor's theorem** -/\ntheorem cantor (a : Cardinal.{u}) : a < 2 ^ a := by\n  induction' a using Cardinal.inductionOn with \u03b1\n  rw [\u2190 mk_set]\n  refine' \u27e8\u27e8\u27e8singleton, fun a b => singleton_eq_singleton_iff.1\u27e9\u27e9, _\u27e9\n  rintro \u27e8\u27e8f, hf\u27e9\u27e9\n  exact cantor_injective f hf\n#align cardinal.cantor Cardinal.cantor\n\ninstance : NoMaxOrder Cardinal.{u} where exists_gt a := \u27e8_, cantor a\u27e9\n\n-- short-circuit type class inference\ninstance : DistribLattice Cardinal.{u} := inferInstance\n\ntheorem one_lt_iff_nontrivial {\u03b1 : Type u} : 1 < #\u03b1 \u2194 Nontrivial \u03b1 := by\n  rw [\u2190 not_le, le_one_iff_subsingleton, \u2190 not_nontrivial_iff_subsingleton, Classical.not_not]\n#align cardinal.one_lt_iff_nontrivial Cardinal.one_lt_iff_nontrivial\n\ntheorem power_le_max_power_one {a b c : Cardinal} (h : b \u2264 c) : a ^ b \u2264 max (a ^ c) 1 := by\n  by_cases ha : a = 0\n  \u00b7 simp [ha, zero_power_le]\n  \u00b7 exact (power_le_power_left ha h).trans (le_max_left _ _)\n#align cardinal.power_le_max_power_one Cardinal.power_le_max_power_one\n\ntheorem power_le_power_right {a b c : Cardinal} : a \u2264 b \u2192 a ^ c \u2264 b ^ c :=\n  inductionOn\u2083 a b c fun _ _ _ \u27e8e\u27e9 => \u27e8Embedding.arrowCongrRight e\u27e9\n#align cardinal.power_le_power_right Cardinal.power_le_power_right\n\ntheorem power_pos {a : Cardinal} (b : Cardinal) (ha : 0 < a) : 0 < a ^ b :=\n  (power_ne_zero _ ha.ne').bot_lt\n#align cardinal.power_pos Cardinal.power_pos\n\nend OrderProperties\n\nprotected theorem lt_wf : @WellFounded Cardinal.{u} (\u00b7 < \u00b7) :=\n  \u27e8fun a =>\n    by_contradiction fun h => by\n      let \u03b9 := { c : Cardinal // \u00acAcc (\u00b7 < \u00b7) c }\n      let f : \u03b9 \u2192 Cardinal := Subtype.val\n      haveI h\u03b9 : Nonempty \u03b9 := \u27e8\u27e8_, h\u27e9\u27e9\n      obtain \u27e8\u27e8c : Cardinal, hc : \u00acAcc (\u00b7 < \u00b7) c\u27e9, \u27e8h_1 : \u2200 j, (f \u27e8c, hc\u27e9).out \u21aa (f j).out\u27e9\u27e9 :=\n        Embedding.min_injective fun i => (f i).out\n      refine hc (Acc.intro _ fun j h' => by_contradiction fun hj => h'.2 ?_)\n      have : #_ \u2264 #_ := \u27e8h_1 \u27e8j, hj\u27e9\u27e9\n      simpa only [mk_out] using this\u27e9\n#align cardinal.lt_wf Cardinal.lt_wf\n\ninstance : WellFoundedRelation Cardinal.{u} :=\n  \u27e8(\u00b7 < \u00b7), Cardinal.lt_wf\u27e9\n\n-- Porting note: this no longer is automatically inferred.\ninstance : WellFoundedLT Cardinal.{u} :=\n  \u27e8Cardinal.lt_wf\u27e9\n\ninstance wo : @IsWellOrder Cardinal.{u} (\u00b7 < \u00b7) where\n#align cardinal.wo Cardinal.wo\n\ninstance : ConditionallyCompleteLinearOrderBot Cardinal :=\n  IsWellOrder.conditionallyCompleteLinearOrderBot _\n\n@[simp]\ntheorem sInf_empty : sInf (\u2205 : Set Cardinal.{u}) = 0 :=\n  dif_neg Set.not_nonempty_empty\n#align cardinal.Inf_empty Cardinal.sInf_empty\n\nlemma sInf_eq_zero_iff {s : Set Cardinal} : sInf s = 0 \u2194 s = \u2205 \u2228 \u2203 a \u2208 s, a = 0 := by\n  refine \u27e8fun h \u21a6 ?_, fun h \u21a6 ?_\u27e9\n  \u00b7 rcases s.eq_empty_or_nonempty with rfl | hne\n    \u00b7 exact Or.inl rfl\n    \u00b7 exact Or.inr \u27e8sInf s, csInf_mem hne, h\u27e9\n  \u00b7 rcases h with rfl | \u27e8a, ha, rfl\u27e9\n    \u00b7 exact Cardinal.sInf_empty\n    \u00b7 exact eq_bot_iff.2 (csInf_le' ha)\n\nlemma iInf_eq_zero_iff {\u03b9 : Sort*} {f : \u03b9 \u2192 Cardinal} :\n    (\u2a05 i, f i) = 0 \u2194 IsEmpty \u03b9 \u2228 \u2203 i, f i = 0 := by\n  simp [iInf, sInf_eq_zero_iff]\n\n/-- Note that the successor of `c` is not the same as `c + 1` except in the case of finite `c`. -/\ninstance : SuccOrder Cardinal :=\n  SuccOrder.ofSuccLeIff (fun c => sInf { c' | c < c' })\n    -- Porting note: Needed to insert `by apply` in the next line\n    \u27e8by apply lt_of_lt_of_le <| csInf_mem <| exists_gt _,\n    -- Porting note used to be just `csInf_le'`\n    fun h \u21a6 csInf_le' h\u27e9\n\ntheorem succ_def (c : Cardinal) : succ c = sInf { c' | c < c' } :=\n  rfl\n#align cardinal.succ_def Cardinal.succ_def\n\ntheorem succ_pos : \u2200 c : Cardinal, 0 < succ c :=\n  bot_lt_succ\n#align cardinal.succ_pos Cardinal.succ_pos\n\ntheorem succ_ne_zero (c : Cardinal) : succ c \u2260 0 :=\n  (succ_pos _).ne'\n#align cardinal.succ_ne_zero Cardinal.succ_ne_zero\n\ntheorem add_one_le_succ (c : Cardinal.{u}) : c + 1 \u2264 succ c := by\n  -- Porting note: rewrote the next three lines to avoid defeq abuse.\n  have : Set.Nonempty { c' | c < c' } := exists_gt c\n  simp_rw [succ_def, le_csInf_iff'' this, mem_setOf]\n  intro b hlt\n  rcases b, c with \u27e8\u27e8\u03b2\u27e9, \u27e8\u03b3\u27e9\u27e9\n  cases' le_of_lt hlt with f\n  have : \u00acSurjective f := fun hn => (not_le_of_lt hlt) (mk_le_of_surjective hn)\n  simp only [Surjective, not_forall] at this\n  rcases this with \u27e8b, hb\u27e9\n  calc\n    #\u03b3 + 1 = #(Option \u03b3) := mk_option.symm\n    _ \u2264 #\u03b2 := (f.optionElim b hb).cardinal_le\n#align cardinal.add_one_le_succ Cardinal.add_one_le_succ\n\n/-- A cardinal is a limit if it is not zero or a successor cardinal. Note that `\u2135\u2080` is a limit\n  cardinal by this definition, but `0` isn't.\n\n  Use `IsSuccLimit` if you want to include the `c = 0` case. -/\ndef IsLimit (c : Cardinal) : Prop :=\n  c \u2260 0 \u2227 IsSuccLimit c\n#align cardinal.is_limit Cardinal.IsLimit\n\nprotected theorem IsLimit.ne_zero {c} (h : IsLimit c) : c \u2260 0 :=\n  h.1\n#align cardinal.is_limit.ne_zero Cardinal.IsLimit.ne_zero\n\nprotected theorem IsLimit.isSuccLimit {c} (h : IsLimit c) : IsSuccLimit c :=\n  h.2\n#align cardinal.is_limit.is_succ_limit Cardinal.IsLimit.isSuccLimit\n\ntheorem IsLimit.succ_lt {x c} (h : IsLimit c) : x < c \u2192 succ x < c :=\n  h.isSuccLimit.succ_lt\n#align cardinal.is_limit.succ_lt Cardinal.IsLimit.succ_lt\n\ntheorem isSuccLimit_zero : IsSuccLimit (0 : Cardinal) :=\n  isSuccLimit_bot\n#align cardinal.is_succ_limit_zero Cardinal.isSuccLimit_zero\n\n/-- The indexed sum of cardinals is the cardinality of the\n  indexed disjoint union, i.e. sigma type. -/\ndef sum {\u03b9} (f : \u03b9 \u2192 Cardinal) : Cardinal :=\n  mk (\u03a3i, (f i).out)\n#align cardinal.sum Cardinal.sum\n\ntheorem le_sum {\u03b9} (f : \u03b9 \u2192 Cardinal) (i) : f i \u2264 sum f := by\n  rw [\u2190 Quotient.out_eq (f i)]\n  exact \u27e8\u27e8fun a => \u27e8i, a\u27e9, fun a b h => by injection h\u27e9\u27e9\n#align cardinal.le_sum Cardinal.le_sum\n\n@[simp]\ntheorem mk_sigma {\u03b9} (f : \u03b9 \u2192 Type*) : #(\u03a3 i, f i) = sum fun i => #(f i) :=\n  mk_congr <| Equiv.sigmaCongrRight fun _ => outMkEquiv.symm\n#align cardinal.mk_sigma Cardinal.mk_sigma\n\n@[simp]\ntheorem sum_const (\u03b9 : Type u) (a : Cardinal.{v}) :\n    (sum fun _ : \u03b9 => a) = lift.{v} #\u03b9 * lift.{u} a :=\n  inductionOn a fun \u03b1 =>\n    mk_congr <|\n      calc\n        (\u03a3 _ : \u03b9, Quotient.out #\u03b1) \u2243 \u03b9 \u00d7 Quotient.out #\u03b1 := Equiv.sigmaEquivProd _ _\n        _ \u2243 ULift \u03b9 \u00d7 ULift \u03b1 := Equiv.ulift.symm.prodCongr (outMkEquiv.trans Equiv.ulift.symm)\n#align cardinal.sum_const Cardinal.sum_const\n\ntheorem sum_const' (\u03b9 : Type u) (a : Cardinal.{u}) : (sum fun _ : \u03b9 => a) = #\u03b9 * a := by simp\n#align cardinal.sum_const' Cardinal.sum_const'\n\n@[simp]\ntheorem sum_add_distrib {\u03b9} (f g : \u03b9 \u2192 Cardinal) : sum (f + g) = sum f + sum g := by\n  have := mk_congr (Equiv.sigmaSumDistrib (Quotient.out \u2218 f) (Quotient.out \u2218 g))\n  simp only [comp_apply, mk_sigma, mk_sum, mk_out, lift_id] at this\n  exact this\n#align cardinal.sum_add_distrib Cardinal.sum_add_distrib\n\n@[simp]\ntheorem sum_add_distrib' {\u03b9} (f g : \u03b9 \u2192 Cardinal) :\n    (Cardinal.sum fun i => f i + g i) = sum f + sum g :=\n  sum_add_distrib f g\n#align cardinal.sum_add_distrib' Cardinal.sum_add_distrib'\n\n@[simp]\ntheorem lift_sum {\u03b9 : Type u} (f : \u03b9 \u2192 Cardinal.{v}) :\n    Cardinal.lift.{w} (Cardinal.sum f) = Cardinal.sum fun i => Cardinal.lift.{w} (f i) :=\n  Equiv.cardinal_eq <|\n    Equiv.ulift.trans <|\n      Equiv.sigmaCongrRight fun a =>\n    -- Porting note: Inserted universe hint .{_,_,v} below\n        Nonempty.some <| by rw [\u2190 lift_mk_eq.{_,_,v}, mk_out, mk_out, lift_lift]\n#align cardinal.lift_sum Cardinal.lift_sum\n\ntheorem sum_le_sum {\u03b9} (f g : \u03b9 \u2192 Cardinal) (H : \u2200 i, f i \u2264 g i) : sum f \u2264 sum g :=\n  \u27e8(Embedding.refl _).sigmaMap fun i =>\n      Classical.choice <| by have := H i; rwa [\u2190 Quot.out_eq (f i), \u2190 Quot.out_eq (g i)] at this\u27e9\n#align cardinal.sum_le_sum Cardinal.sum_le_sum\n\ntheorem mk_le_mk_mul_of_mk_preimage_le {c : Cardinal} (f : \u03b1 \u2192 \u03b2) (hf : \u2200 b : \u03b2, #(f \u207b\u00b9' {b}) \u2264 c) :\n    #\u03b1 \u2264 #\u03b2 * c := by\n  simpa only [\u2190 mk_congr (@Equiv.sigmaFiberEquiv \u03b1 \u03b2 f), mk_sigma, \u2190 sum_const'] using\n    sum_le_sum _ _ hf\n#align cardinal.mk_le_mk_mul_of_mk_preimage_le Cardinal.mk_le_mk_mul_of_mk_preimage_le\n\ntheorem lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le {\u03b1 : Type u} {\u03b2 : Type v} {c : Cardinal}\n    (f : \u03b1 \u2192 \u03b2) (hf : \u2200 b : \u03b2, lift.{v} #(f \u207b\u00b9' {b}) \u2264 c) : lift.{v} #\u03b1 \u2264 lift.{u} #\u03b2 * c :=\n  (mk_le_mk_mul_of_mk_preimage_le fun x : ULift.{v} \u03b1 => ULift.up.{u} (f x.1)) <|\n    ULift.forall.2 fun b =>\n      (mk_congr <|\n            (Equiv.ulift.image _).trans\n              (Equiv.trans\n                (by\n                  rw [Equiv.image_eq_preimage]\n                  /- Porting note: Need to insert the following `have` b/c bad fun coercion\n                   behaviour for Equivs -/\n                  have : DFunLike.coe (Equiv.symm (Equiv.ulift (\u03b1 := \u03b1))) = ULift.up (\u03b1 := \u03b1) := rfl\n                  rw [this]\n                  simp only [preimage, mem_singleton_iff, ULift.up_inj, mem_setOf_eq, coe_setOf]\n                  exact Equiv.refl _)\n                Equiv.ulift.symm)).trans_le\n        (hf b)\n#align cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le Cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le\n\n/-- The range of an indexed cardinal function, whose outputs live in a higher universe than the\n    inputs, is always bounded above. -/\ntheorem bddAbove_range {\u03b9 : Type u} (f : \u03b9 \u2192 Cardinal.{max u v}) : BddAbove (Set.range f) :=\n  \u27e8_, by\n    rintro a \u27e8i, rfl\u27e9\n    -- Porting note: Added universe reference below\n    exact le_sum.{v,u} f i\u27e9\n#align cardinal.bdd_above_range Cardinal.bddAbove_range\n\ninstance (a : Cardinal.{u}) : Small.{u} (Set.Iic a) := by\n  rw [\u2190 mk_out a]\n  apply @small_of_surjective (Set a.out) (Iic #a.out) _ fun x => \u27e8#x, mk_set_le x\u27e9\n  rintro \u27e8x, hx\u27e9\n  simpa using le_mk_iff_exists_set.1 hx\n\ninstance (a : Cardinal.{u}) : Small.{u} (Set.Iio a) :=\n  small_subset Iio_subset_Iic_self\n\n/-- A set of cardinals is bounded above iff it's small, i.e. it corresponds to a usual ZFC set. -/\ntheorem bddAbove_iff_small {s : Set Cardinal.{u}} : BddAbove s \u2194 Small.{u} s :=\n  \u27e8fun \u27e8a, ha\u27e9 => @small_subset _ (Iic a) s (fun x h => ha h) _, by\n    rintro \u27e8\u03b9, \u27e8e\u27e9\u27e9\n    suffices (range fun x : \u03b9 => (e.symm x).1) = s by\n      rw [\u2190 this]\n      apply bddAbove_range.{u, u}\n    ext x\n    refine' \u27e8_, fun hx => \u27e8e \u27e8x, hx\u27e9, _\u27e9\u27e9\n    \u00b7 rintro \u27e8a, rfl\u27e9\n      exact (e.symm a).2\n    \u00b7 simp_rw [Equiv.symm_apply_apply]\u27e9\n#align cardinal.bdd_above_iff_small Cardinal.bddAbove_iff_small\n\ntheorem bddAbove_of_small (s : Set Cardinal.{u}) [h : Small.{u} s] : BddAbove s :=\n  bddAbove_iff_small.2 h\n#align cardinal.bdd_above_of_small Cardinal.bddAbove_of_small\n\ntheorem bddAbove_image (f : Cardinal.{u} \u2192 Cardinal.{max u v}) {s : Set Cardinal.{u}}\n    (hs : BddAbove s) : BddAbove (f '' s) := by\n  rw [bddAbove_iff_small] at hs \u22a2\n  -- Porting note: added universes below\n  exact small_lift.{_,v,_} _\n#align cardinal.bdd_above_image Cardinal.bddAbove_image\n\ntheorem bddAbove_range_comp {\u03b9 : Type u} {f : \u03b9 \u2192 Cardinal.{v}} (hf : BddAbove (range f))\n    (g : Cardinal.{v} \u2192 Cardinal.{max v w}) : BddAbove (range (g \u2218 f)) := by\n  rw [range_comp]\n  exact bddAbove_image.{v,w} g hf\n#align cardinal.bdd_above_range_comp Cardinal.bddAbove_range_comp\n\ntheorem iSup_le_sum {\u03b9} (f : \u03b9 \u2192 Cardinal) : iSup f \u2264 sum f :=\n  ciSup_le' <| le_sum.{u_2,u_1} _\n#align cardinal.supr_le_sum Cardinal.iSup_le_sum\n\n-- Porting note: Added universe hint .{v,_} below\ntheorem sum_le_iSup_lift {\u03b9 : Type u}\n    (f : \u03b9 \u2192 Cardinal.{max u v}) : sum f \u2264 Cardinal.lift.{v,_} #\u03b9 * iSup f := by\n  rw [\u2190 (iSup f).lift_id, \u2190 lift_umax, lift_umax.{max u v, u}, \u2190 sum_const]\n  exact sum_le_sum _ _ (le_ciSup <| bddAbove_range.{u, v} f)\n#align cardinal.sum_le_supr_lift Cardinal.sum_le_iSup_lift\n\ntheorem sum_le_iSup {\u03b9 : Type u} (f : \u03b9 \u2192 Cardinal.{u}) : sum f \u2264 #\u03b9 * iSup f := by\n  rw [\u2190 lift_id #\u03b9]\n  exact sum_le_iSup_lift f\n#align cardinal.sum_le_supr Cardinal.sum_le_iSup\n\ntheorem sum_nat_eq_add_sum_succ (f : \u2115 \u2192 Cardinal.{u}) :\n    Cardinal.sum f = f 0 + Cardinal.sum fun i => f (i + 1) := by\n  refine' (Equiv.sigmaNatSucc fun i => Quotient.out (f i)).cardinal_eq.trans _\n  simp only [mk_sum, mk_out, lift_id, mk_sigma]\n#align cardinal.sum_nat_eq_add_sum_succ Cardinal.sum_nat_eq_add_sum_succ\n\n-- Porting note: LFS is not in normal form.\n-- @[simp]\n/-- A variant of `ciSup_of_empty` but with `0` on the RHS for convenience -/\nprotected theorem iSup_of_empty {\u03b9} (f : \u03b9 \u2192 Cardinal) [IsEmpty \u03b9] : iSup f = 0 :=\n  ciSup_of_empty f\n#align cardinal.supr_of_empty Cardinal.iSup_of_empty\n\nlemma exists_eq_of_iSup_eq_of_not_isSuccLimit\n    {\u03b9 : Type u} (f : \u03b9 \u2192 Cardinal.{v}) (\u03c9 : Cardinal.{v})\n    (h\u03c9 : \u00ac Order.IsSuccLimit \u03c9)\n    (h : \u2a06 i : \u03b9, f i = \u03c9) : \u2203 i, f i = \u03c9 := by\n  subst h\n  refine (isLUB_csSup' ?_).exists_of_not_isSuccLimit h\u03c9\n  contrapose! h\u03c9 with hf\n  rw [iSup, csSup_of_not_bddAbove hf, csSup_empty]\n  exact Order.isSuccLimit_bot\n\nlemma exists_eq_of_iSup_eq_of_not_isLimit\n    {\u03b9 : Type u} [h\u03b9 : Nonempty \u03b9] (f : \u03b9 \u2192 Cardinal.{v}) (hf : BddAbove (range f))\n    (\u03c9 : Cardinal.{v}) (h\u03c9 : \u00ac \u03c9.IsLimit)\n    (h : \u2a06 i : \u03b9, f i = \u03c9) : \u2203 i, f i = \u03c9 := by\n  refine (not_and_or.mp h\u03c9).elim (fun e \u21a6 \u27e8h\u03b9.some, ?_\u27e9)\n    (Cardinal.exists_eq_of_iSup_eq_of_not_isSuccLimit.{u, v} f \u03c9 \u00b7 h)\n  cases not_not.mp e\n  rw [\u2190 le_zero_iff] at h \u22a2\n  exact (le_ciSup hf _).trans h\n\n-- Porting note: simpNF is not happy with universe levels.\n@[simp, nolint simpNF]\ntheorem lift_mk_shrink (\u03b1 : Type u) [Small.{v} \u03b1] :\n    Cardinal.lift.{max u w} #(Shrink.{v} \u03b1) = Cardinal.lift.{max v w} #\u03b1 :=\n-- Porting note: Added .{v,u,w} universe hint below\n  lift_mk_eq.{v,u,w}.2 \u27e8(equivShrink \u03b1).symm\u27e9\n#align cardinal.lift_mk_shrink Cardinal.lift_mk_shrink\n\n@[simp]\ntheorem lift_mk_shrink' (\u03b1 : Type u) [Small.{v} \u03b1] :\n    Cardinal.lift.{u} #(Shrink.{v} \u03b1) = Cardinal.lift.{v} #\u03b1 :=\n  lift_mk_shrink.{u, v, 0} \u03b1\n#align cardinal.lift_mk_shrink' Cardinal.lift_mk_shrink'\n\n@[simp]\ntheorem lift_mk_shrink'' (\u03b1 : Type max u v) [Small.{v} \u03b1] :\n    Cardinal.lift.{u} #(Shrink.{v} \u03b1) = #\u03b1 := by\n  rw [\u2190 lift_umax', lift_mk_shrink.{max u v, v, 0} \u03b1, \u2190 lift_umax, lift_id]\n#align cardinal.lift_mk_shrink'' Cardinal.lift_mk_shrink''\n\n/-- The indexed product of cardinals is the cardinality of the Pi type\n  (dependent product). -/\ndef prod {\u03b9 : Type u} (f : \u03b9 \u2192 Cardinal) : Cardinal :=\n  #(\u2200 i, (f i).out)\n#align cardinal.prod Cardinal.prod\n\n@[simp]\ntheorem mk_pi {\u03b9 : Type u} (\u03b1 : \u03b9 \u2192 Type v) : #(\u2200 i, \u03b1 i) = prod fun i => #(\u03b1 i) :=\n  mk_congr <| Equiv.piCongrRight fun _ => outMkEquiv.symm\n#align cardinal.mk_pi Cardinal.mk_pi\n\n@[simp]\ntheorem prod_const (\u03b9 : Type u) (a : Cardinal.{v}) :\n    (prod fun _ : \u03b9 => a) = lift.{u} a ^ lift.{v} #\u03b9 :=\n  inductionOn a fun _ =>\n    mk_congr <| Equiv.piCongr Equiv.ulift.symm fun _ => outMkEquiv.trans Equiv.ulift.symm\n#align cardinal.prod_const Cardinal.prod_const\n\ntheorem prod_const' (\u03b9 : Type u) (a : Cardinal.{u}) : (prod fun _ : \u03b9 => a) = a ^ #\u03b9 :=\n  inductionOn a fun _ => (mk_pi _).symm\n#align cardinal.prod_const' Cardinal.prod_const'\n\ntheorem prod_le_prod {\u03b9} (f g : \u03b9 \u2192 Cardinal) (H : \u2200 i, f i \u2264 g i) : prod f \u2264 prod g :=\n  \u27e8Embedding.piCongrRight fun i =>\n      Classical.choice <| by have := H i; rwa [\u2190 mk_out (f i), \u2190 mk_out (g i)] at this\u27e9\n#align cardinal.prod_le_prod Cardinal.prod_le_prod\n\n@[simp]\ntheorem prod_eq_zero {\u03b9} (f : \u03b9 \u2192 Cardinal.{u}) : prod f = 0 \u2194 \u2203 i, f i = 0 := by\n  lift f to \u03b9 \u2192 Type u using fun _ => trivial\n  simp only [mk_eq_zero_iff, \u2190 mk_pi, isEmpty_pi]\n#align cardinal.prod_eq_zero Cardinal.prod_eq_zero\n\ntheorem prod_ne_zero {\u03b9} (f : \u03b9 \u2192 Cardinal) : prod f \u2260 0 \u2194 \u2200 i, f i \u2260 0 := by simp [prod_eq_zero]\n#align cardinal.prod_ne_zero Cardinal.prod_ne_zero\n\n@[simp]\ntheorem lift_prod {\u03b9 : Type u} (c : \u03b9 \u2192 Cardinal.{v}) :\n    lift.{w} (prod c) = prod fun i => lift.{w} (c i) := by\n  lift c to \u03b9 \u2192 Type v using fun _ => trivial\n  simp only [\u2190 mk_pi, \u2190 mk_uLift]\n  exact mk_congr (Equiv.ulift.trans <| Equiv.piCongrRight fun i => Equiv.ulift.symm)\n#align cardinal.lift_prod Cardinal.lift_prod\n\ntheorem prod_eq_of_fintype {\u03b1 : Type u} [h : Fintype \u03b1] (f : \u03b1 \u2192 Cardinal.{v}) :\n    prod f = Cardinal.lift.{u} (\u220f i, f i) := by\n  revert f\n  refine' Fintype.induction_empty_option _ _ _ \u03b1 (h_fintype := h)\n  \u00b7 intro \u03b1 \u03b2 h\u03b2 e h f\n    letI := Fintype.ofEquiv \u03b2 e.symm\n    rw [\u2190 e.prod_comp f, \u2190 h]\n    exact mk_congr (e.piCongrLeft _).symm\n  \u00b7 intro f\n    rw [Fintype.univ_pempty, Finset.prod_empty, lift_one, Cardinal.prod, mk_eq_one]\n  \u00b7 intro \u03b1 h\u03b1 h f\n    rw [Cardinal.prod, mk_congr Equiv.piOptionEquivProd, mk_prod, lift_umax'.{v, u}, mk_out, \u2190\n        Cardinal.prod, lift_prod, Fintype.prod_option, lift_mul, \u2190 h fun a => f (some a)]\n    simp only [lift_id]\n#align cardinal.prod_eq_of_fintype Cardinal.prod_eq_of_fintype\n\n-- Porting note: Inserted .{u,v} below\n@[simp]\ntheorem lift_sInf (s : Set Cardinal) : lift.{u,v} (sInf s) = sInf (lift.{u,v} '' s) := by\n  rcases eq_empty_or_nonempty s with (rfl | hs)\n  \u00b7 simp\n  \u00b7 exact lift_monotone.map_csInf hs\n#align cardinal.lift_Inf Cardinal.lift_sInf\n\n-- Porting note: Inserted .{u,v} below\n@[simp]\ntheorem lift_iInf {\u03b9} (f : \u03b9 \u2192 Cardinal) : lift.{u,v} (iInf f) = \u2a05 i, lift.{u,v} (f i) := by\n  unfold iInf\n  convert lift_sInf (range f)\n  simp_rw [\u2190 comp_apply (f := lift), range_comp]\n#align cardinal.lift_infi Cardinal.lift_iInf\n\ntheorem lift_down {a : Cardinal.{u}} {b : Cardinal.{max u v}} :\n    b \u2264 lift.{v,u} a \u2192 \u2203 a', lift.{v,u} a' = b :=\n  inductionOn\u2082 a b fun \u03b1 \u03b2 => by\n    rw [\u2190 lift_id #\u03b2, \u2190 lift_umax, \u2190 lift_umax.{u, v}, lift_mk_le.{v}]\n    exact fun \u27e8f\u27e9 =>\n      \u27e8#(Set.range f),\n        Eq.symm <| lift_mk_eq.{_, _, v}.2\n          \u27e8Function.Embedding.equivOfSurjective (Embedding.codRestrict _ f Set.mem_range_self)\n              fun \u27e8a, \u27e8b, e\u27e9\u27e9 => \u27e8b, Subtype.eq e\u27e9\u27e9\u27e9\n#align cardinal.lift_down Cardinal.lift_down\n\n-- Porting note: Inserted .{u,v} below\ntheorem le_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :\n    b \u2264 lift.{v,u} a \u2194 \u2203 a', lift.{v,u} a' = b \u2227 a' \u2264 a :=\n  \u27e8fun h =>\n    let \u27e8a', e\u27e9 := lift_down h\n    \u27e8a', e, lift_le.1 <| e.symm \u25b8 h\u27e9,\n    fun \u27e8_, e, h\u27e9 => e \u25b8 lift_le.2 h\u27e9\n#align cardinal.le_lift_iff Cardinal.le_lift_iff\n\n-- Porting note: Inserted .{u,v} below\ntheorem lt_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :\n    b < lift.{v,u} a \u2194 \u2203 a', lift.{v,u} a' = b \u2227 a' < a :=\n  \u27e8fun h =>\n    let \u27e8a', e\u27e9 := lift_down h.le\n    \u27e8a', e, lift_lt.1 <| e.symm \u25b8 h\u27e9,\n    fun \u27e8_, e, h\u27e9 => e \u25b8 lift_lt.2 h\u27e9\n#align cardinal.lt_lift_iff Cardinal.lt_lift_iff\n\n-- Porting note: Inserted .{u,v} below\n@[simp]\ntheorem lift_succ (a) : lift.{v,u} (succ a) = succ (lift.{v,u} a) :=\n  le_antisymm\n    (le_of_not_gt fun h => by\n      rcases lt_lift_iff.1 h with \u27e8b, e, h\u27e9\n      rw [lt_succ_iff, \u2190 lift_le, e] at h\n      exact h.not_lt (lt_succ _))\n    (succ_le_of_lt <| lift_lt.2 <| lt_succ a)\n#align cardinal.lift_succ Cardinal.lift_succ\n\n-- Porting note: simpNF is not happy with universe levels.\n-- Porting note: Inserted .{u,v} below\n@[simp, nolint simpNF]\ntheorem lift_umax_eq {a : Cardinal.{u}} {b : Cardinal.{v}} :\n    lift.{max v w} a = lift.{max u w} b \u2194 lift.{v} a = lift.{u} b := by\n  rw [\u2190 lift_lift.{v, w, u}, \u2190 lift_lift.{u, w, v}, lift_inj]\n#align cardinal.lift_umax_eq Cardinal.lift_umax_eq\n\n-- Porting note: Inserted .{u,v} below\n@[simp]\ntheorem lift_min {a b : Cardinal} : lift.{u,v} (min a b) = min (lift.{u,v} a) (lift.{u,v} b) :=\n  lift_monotone.map_min\n#align cardinal.lift_min Cardinal.lift_min\n\n-- Porting note: Inserted .{u,v} below\n@[simp]\ntheorem lift_max {a b : Cardinal} : lift.{u,v} (max a b) = max (lift.{u,v} a) (lift.{u,v} b) :=\n  lift_monotone.map_max\n#align cardinal.lift_max Cardinal.lift_max\n\n/-- The lift of a supremum is the supremum of the lifts. -/\ntheorem lift_sSup {s : Set Cardinal} (hs : BddAbove s) :\n    lift.{u} (sSup s) = sSup (lift.{u} '' s) := by\n  apply ((le_csSup_iff' (bddAbove_image.{_,u} _ hs)).2 fun c hc => _).antisymm (csSup_le' _)\n  \u00b7 intro c hc\n    by_contra h\n    obtain \u27e8d, rfl\u27e9 := Cardinal.lift_down (not_le.1 h).le\n    simp_rw [lift_le] at h hc\n    rw [csSup_le_iff' hs] at h\n    exact h fun a ha => lift_le.1 <| hc (mem_image_of_mem _ ha)\n  \u00b7 rintro i \u27e8j, hj, rfl\u27e9\n    exact lift_le.2 (le_csSup hs hj)\n#align cardinal.lift_Sup Cardinal.lift_sSup\n\n/-- The lift of a supremum is the supremum of the lifts. -/\ntheorem lift_iSup {\u03b9 : Type v} {f : \u03b9 \u2192 Cardinal.{w}} (hf : BddAbove (range f)) :\n    lift.{u} (iSup f) = \u2a06 i, lift.{u} (f i) := by\n  rw [iSup, iSup, lift_sSup hf, \u2190 range_comp]\n  simp [Function.comp]\n#align cardinal.lift_supr Cardinal.lift_iSup\n\n/-- To prove that the lift of a supremum is bounded by some cardinal `t`,\nit suffices to show that the lift of each cardinal is bounded by `t`. -/\ntheorem lift_iSup_le {\u03b9 : Type v} {f : \u03b9 \u2192 Cardinal.{w}} {t : Cardinal} (hf : BddAbove (range f))\n    (w : \u2200 i, lift.{u} (f i) \u2264 t) : lift.{u} (iSup f) \u2264 t := by\n  rw [lift_iSup hf]\n  exact ciSup_le' w\n#align cardinal.lift_supr_le Cardinal.lift_iSup_le\n\n@[simp]\ntheorem lift_iSup_le_iff {\u03b9 : Type v} {f : \u03b9 \u2192 Cardinal.{w}} (hf : BddAbove (range f))\n    {t : Cardinal} : lift.{u} (iSup f) \u2264 t \u2194 \u2200 i, lift.{u} (f i) \u2264 t := by\n  rw [lift_iSup hf]\n  exact ciSup_le_iff' (bddAbove_range_comp.{_,_,u} hf _)\n#align cardinal.lift_supr_le_iff Cardinal.lift_iSup_le_iff\n\nuniverse v' w'\n\n/-- To prove an inequality between the lifts to a common universe of two different supremums,\nit suffices to show that the lift of each cardinal from the smaller supremum\nif bounded by the lift of some cardinal from the larger supremum.\n-/\ntheorem lift_iSup_le_lift_iSup {\u03b9 : Type v} {\u03b9' : Type v'} {f : \u03b9 \u2192 Cardinal.{w}}\n    {f' : \u03b9' \u2192 Cardinal.{w'}} (hf : BddAbove (range f)) (hf' : BddAbove (range f')) {g : \u03b9 \u2192 \u03b9'}\n    (h : \u2200 i, lift.{w'} (f i) \u2264 lift.{w} (f' (g i))) : lift.{w'} (iSup f) \u2264 lift.{w} (iSup f') := by\n  rw [lift_iSup hf, lift_iSup hf']\n  exact ciSup_mono' (bddAbove_range_comp.{_,_,w} hf' _) fun i => \u27e8_, h i\u27e9\n#align cardinal.lift_supr_le_lift_supr Cardinal.lift_iSup_le_lift_iSup\n\n/-- A variant of `lift_iSup_le_lift_iSup` with universes specialized via `w = v` and `w' = v'`.\nThis is sometimes necessary to avoid universe unification issues. -/\ntheorem lift_iSup_le_lift_iSup' {\u03b9 : Type v} {\u03b9' : Type v'} {f : \u03b9 \u2192 Cardinal.{v}}\n    {f' : \u03b9' \u2192 Cardinal.{v'}} (hf : BddAbove (range f)) (hf' : BddAbove (range f')) (g : \u03b9 \u2192 \u03b9')\n    (h : \u2200 i, lift.{v'} (f i) \u2264 lift.{v} (f' (g i))) : lift.{v'} (iSup f) \u2264 lift.{v} (iSup f') :=\n  lift_iSup_le_lift_iSup hf hf' h\n#align cardinal.lift_supr_le_lift_supr' Cardinal.lift_iSup_le_lift_iSup'\n\n/-- `\u2135\u2080` is the smallest infinite cardinal. -/\ndef aleph0 : Cardinal.{u} :=\n  lift #\u2115\n#align cardinal.aleph_0 Cardinal.aleph0\n\n@[inherit_doc]\nscoped notation \"\u2135\u2080\" => Cardinal.aleph0\n\ntheorem mk_nat : #\u2115 = \u2135\u2080 :=\n  (lift_id _).symm\n#align cardinal.mk_nat Cardinal.mk_nat\n\ntheorem aleph0_ne_zero : \u2135\u2080 \u2260 0 :=\n  mk_ne_zero _\n#align cardinal.aleph_0_ne_zero Cardinal.aleph0_ne_zero\n\ntheorem aleph0_pos : 0 < \u2135\u2080 :=\n  pos_iff_ne_zero.2 aleph0_ne_zero\n#align cardinal.aleph_0_pos Cardinal.aleph0_pos\n\n@[simp]\ntheorem lift_aleph0 : lift \u2135\u2080 = \u2135\u2080 :=\n  lift_lift _\n#align cardinal.lift_aleph_0 Cardinal.lift_aleph0\n\n@[simp]\ntheorem aleph0_le_lift {c : Cardinal.{u}} : \u2135\u2080 \u2264 lift.{v} c \u2194 \u2135\u2080 \u2264 c := by\n  rw [\u2190 lift_aleph0.{u,v}, lift_le]\n#align cardinal.aleph_0_le_lift Cardinal.aleph0_le_lift\n\n@[simp]\ntheorem lift_le_aleph0 {c : Cardinal.{u}} : lift.{v} c \u2264 \u2135\u2080 \u2194 c \u2264 \u2135\u2080 := by\n  rw [\u2190 lift_aleph0.{u,v}, lift_le]\n#align cardinal.lift_le_aleph_0 Cardinal.lift_le_aleph0\n\n@[simp]\ntheorem aleph0_lt_lift {c : Cardinal.{u}} : \u2135\u2080 < lift.{v} c \u2194 \u2135\u2080 < c := by\n  rw [\u2190 lift_aleph0.{u,v}, lift_lt]\n#align cardinal.aleph_0_lt_lift Cardinal.aleph0_lt_lift\n\n@[simp]\ntheorem lift_lt_aleph0 {c : Cardinal.{u}} : lift.{v} c < \u2135\u2080 \u2194 c < \u2135\u2080 := by\n  rw [\u2190 lift_aleph0.{u,v}, lift_lt]\n#align cardinal.lift_lt_aleph_0 Cardinal.lift_lt_aleph0\n\n/-! ### Properties about the cast from `\u2115` -/\nsection castFromN\n\n-- porting note (#10618): simp can prove this\n-- @[simp]\ntheorem mk_fin (n : \u2115) : #(Fin n) = n := by simp\n#align cardinal.mk_fin Cardinal.mk_fin\n\n@[simp]\ntheorem lift_natCast (n : \u2115) : lift.{u} (n : Cardinal.{v}) = n := by induction n <;> simp [*]\n#align cardinal.lift_nat_cast Cardinal.lift_natCast\n\n-- See note [no_index around OfNat.ofNat]\n@[simp]\ntheorem lift_ofNat (n : \u2115) [n.AtLeastTwo] :\n    lift.{u} (no_index (OfNat.ofNat n : Cardinal.{v})) = OfNat.ofNat n :=\n  lift_natCast n\n\n@[simp]\ntheorem lift_eq_nat_iff {a : Cardinal.{u}} {n : \u2115} : lift.{v} a = n \u2194 a = n :=\n  lift_injective.eq_iff' (lift_natCast n)\n#align cardinal.lift_eq_nat_iff Cardinal.lift_eq_nat_iff\n\n@[simp]\ntheorem lift_eq_ofNat_iff {a : Cardinal.{u}} {n : \u2115} [n.AtLeastTwo] :\n    lift.{v} a = (no_index (OfNat.ofNat n)) \u2194 a = OfNat.ofNat n :=\n  lift_eq_nat_iff\n\n@[simp]\ntheorem nat_eq_lift_iff {n : \u2115} {a : Cardinal.{u}} :\n    (n : Cardinal) = lift.{v} a \u2194 (n : Cardinal) = a := by\n  rw [\u2190 lift_natCast.{v,u} n, lift_inj]\n#align cardinal.nat_eq_lift_iff Cardinal.nat_eq_lift_iff\n\n@[simp]\ntheorem zero_eq_lift_iff {a : Cardinal.{u}} :\n    (0 : Cardinal) = lift.{v} a \u2194 0 = a := by\n  simpa using nat_eq_lift_iff (n := 0)\n\n@[simp]\ntheorem one_eq_lift_iff {a : Cardinal.{u}} :\n    (1 : Cardinal) = lift.{v} a \u2194 1 = a := by\n  simpa using nat_eq_lift_iff (n := 1)\n\n-- See note [no_index around OfNat.ofNat]\n@[simp]\ntheorem ofNat_eq_lift_iff {a : Cardinal.{u}} {n : \u2115} [n.AtLeastTwo] :\n    (no_index (OfNat.ofNat n : Cardinal)) = lift.{v} a \u2194 (OfNat.ofNat n : Cardinal) = a :=\n  nat_eq_lift_iff\n\n@[simp]\ntheorem lift_le_nat_iff {a : Cardinal.{u}} {n : \u2115} : lift.{v} a \u2264 n \u2194 a \u2264 n := by\n  rw [\u2190 lift_natCast.{v,u}, lift_le]\n#align cardinal.lift_le_nat_iff Cardinal.lift_le_nat_iff\n\n@[simp]\ntheorem lift_le_one_iff {a : Cardinal.{u}} :\n    lift.{v} a \u2264 1 \u2194 a \u2264 1 := by\n  simpa using lift_le_nat_iff (n := 1)\n\n-- See note [no_index around OfNat.ofNat]\n@[simp]\ntheorem lift_le_ofNat_iff {a : Cardinal.{u}} {n : \u2115} [n.AtLeastTwo] :\n    lift.{v} a \u2264 (no_index (OfNat.ofNat n)) \u2194 a \u2264 OfNat.ofNat n :=\n  lift_le_nat_iff\n\n@[simp]\ntheorem nat_le_lift_iff {n : \u2115} {a : Cardinal.{u}} : n \u2264 lift.{v} a \u2194 n \u2264 a := by\n  rw [\u2190 lift_natCast.{v,u}, lift_le]\n#align cardinal.nat_le_lift_iff Cardinal.nat_le_lift_iff\n\n@[simp]\ntheorem one_le_lift_iff {a : Cardinal.{u}} :\n    (1 : Cardinal) \u2264 lift.{v} a \u2194 1 \u2264 a := by\n  simpa using nat_le_lift_iff (n := 1)\n\n-- See note [no_index around OfNat.ofNat]\n@[simp]\ntheorem ofNat_le_lift_iff {a : Cardinal.{u}} {n : \u2115} [n.AtLeastTwo] :\n    (no_index (OfNat.ofNat n : Cardinal)) \u2264 lift.{v} a \u2194 (OfNat.ofNat n : Cardinal) \u2264 a :=\n  nat_le_lift_iff\n\n@[simp]\ntheorem lift_lt_nat_iff {a : Cardinal.{u}} {n : \u2115} : lift.{v} a < n \u2194 a < n := by\n  rw [\u2190 lift_natCast.{v,u}, lift_lt]\n#align cardinal.lift_lt_nat_iff Cardinal.lift_lt_nat_iff\n\n-- See note [no_index around OfNat.ofNat]\n@[simp]\ntheorem lift_lt_ofNat_iff {a : Cardinal.{u}} {n : \u2115} [n.AtLeastTwo] :\n    lift.{v} a < (no_index (OfNat.ofNat n)) \u2194 a < OfNat.ofNat n :=\n  lift_lt_nat_iff\n\n@[simp]\ntheorem nat_lt_lift_iff {n : \u2115} {a : Cardinal.{u}} : n < lift.{v} a \u2194 n < a := by\n  rw [\u2190 lift_natCast.{v,u}, lift_lt]\n#align cardinal.nat_lt_lift_iff Cardinal.nat_lt_lift_iff\n\n-- See note [no_index around OfNat.ofNat]\n@[simp]\ntheorem zero_lt_lift_iff {a : Cardinal.{u}} :\n    (0 : Cardinal) < lift.{v} a \u2194 0 < a := by\n  simpa using nat_lt_lift_iff (n := 0)\n\n@[simp]\ntheorem one_lt_lift_iff {a : Cardinal.{u}} :\n    (1 : Cardinal) < lift.{v} a \u2194 1 < a := by\n  simpa using nat_lt_lift_iff (n := 1)\n\n-- See note [no_index around OfNat.ofNat]\n@[simp]\ntheorem ofNat_lt_lift_iff {a : Cardinal.{u}} {n : \u2115} [n.AtLeastTwo] :\n    (no_index (OfNat.ofNat n : Cardinal)) < lift.{v} a \u2194 (OfNat.ofNat n : Cardinal) < a :=\n  nat_lt_lift_iff\n\ntheorem lift_mk_fin (n : \u2115) : lift #(Fin n) = n := rfl\n#align cardinal.lift_mk_fin Cardinal.lift_mk_fin\n\ntheorem mk_coe_finset {\u03b1 : Type u} {s : Finset \u03b1} : #s = \u2191(Finset.card s) := by simp\n#align cardinal.mk_coe_finset Cardinal.mk_coe_finset\n\ntheorem mk_finset_of_fintype [Fintype \u03b1] : #(Finset \u03b1) = 2 ^ Fintype.card \u03b1 := by\n  simp [Pow.pow]\n#align cardinal.mk_finset_of_fintype Cardinal.mk_finset_of_fintype\n\n@[simp]\ntheorem mk_finsupp_lift_of_fintype (\u03b1 : Type u) (\u03b2 : Type v) [Fintype \u03b1] [Zero \u03b2] :\n    #(\u03b1 \u2192\u2080 \u03b2) = lift.{u} #\u03b2 ^ Fintype.card \u03b1 := by\n  simpa using (@Finsupp.equivFunOnFinite \u03b1 \u03b2 _ _).cardinal_eq\n#align cardinal.mk_finsupp_lift_of_fintype Cardinal.mk_finsupp_lift_of_fintype\n\ntheorem mk_finsupp_of_fintype (\u03b1 \u03b2 : Type u) [Fintype \u03b1] [Zero \u03b2] :\n    #(\u03b1 \u2192\u2080 \u03b2) = #\u03b2 ^ Fintype.card \u03b1 := by simp\n#align cardinal.mk_finsupp_of_fintype Cardinal.mk_finsupp_of_fintype\n\ntheorem card_le_of_finset {\u03b1} (s : Finset \u03b1) : (s.card : Cardinal) \u2264 #\u03b1 :=\n  @mk_coe_finset _ s \u25b8 mk_set_le _\n#align cardinal.card_le_of_finset Cardinal.card_le_of_finset\n\n-- Porting note: was `simp`. LHS is not normal form.\n-- @[simp, norm_cast]\n@[norm_cast]\ntheorem natCast_pow {m n : \u2115} : (\u2191(m ^ n) : Cardinal) = (\u2191m : Cardinal) ^ (\u2191n : Cardinal) := by\n  induction n <;> simp [pow_succ, power_add, *, Pow.pow]\n#align cardinal.nat_cast_pow Cardinal.natCast_pow\n\n-- porting note (#10618): simp can prove this\n-- @[simp, norm_cast]\n@[norm_cast]\ntheorem natCast_le {m n : \u2115} : (m : Cardinal) \u2264 n \u2194 m \u2264 n := by\n  rw [\u2190 lift_mk_fin, \u2190 lift_mk_fin, lift_le, le_def, Function.Embedding.nonempty_iff_card_le,\n    Fintype.card_fin, Fintype.card_fin]\n#align cardinal.nat_cast_le Cardinal.natCast_le\n\n-- porting note (#10618): simp can prove this\n-- @[simp, norm_cast]\n@[norm_cast]\ntheorem natCast_lt {m n : \u2115} : (m : Cardinal) < n \u2194 m < n := by\n  rw [lt_iff_le_not_le, \u2190 not_le]\n  simp only [natCast_le, not_le, and_iff_right_iff_imp]\n  exact fun h \u21a6 le_of_lt h\n#align cardinal.nat_cast_lt Cardinal.natCast_lt\n\ninstance : CharZero Cardinal :=\n  \u27e8StrictMono.injective fun _ _ => natCast_lt.2\u27e9\n\ntheorem natCast_inj {m n : \u2115} : (m : Cardinal) = n \u2194 m = n :=\n  Nat.cast_inj\n#align cardinal.nat_cast_inj Cardinal.natCast_inj\n\ntheorem natCast_injective : Injective ((\u2191) : \u2115 \u2192 Cardinal) :=\n  Nat.cast_injective\n#align cardinal.nat_cast_injective Cardinal.natCast_injective\n\n@[simp, norm_cast]\ntheorem nat_succ (n : \u2115) : (n.succ : Cardinal) = succ \u2191n := by\n  rw [Nat.cast_succ]\n  refine (add_one_le_succ _).antisymm (succ_le_of_lt ?_)\n  rw [\u2190 Nat.cast_succ]\n  exact natCast_lt.2 (Nat.lt_succ_self _)\n\nlemma succ_natCast (n : \u2115) : Order.succ (n : Cardinal) = n + 1 := by\n  rw [\u2190 Cardinal.nat_succ]\n  norm_cast\n\nlemma natCast_add_one_le_iff {n : \u2115} {c : Cardinal} : n + 1 \u2264 c \u2194 n < c := by\n  rw [\u2190 Order.succ_le_iff, Cardinal.succ_natCast]\n\nlemma two_le_iff_one_lt {c : Cardinal} : 2 \u2264 c \u2194 1 < c := by\n  convert natCast_add_one_le_iff\n  norm_cast\n\n@[simp]\ntheorem succ_zero : succ (0 : Cardinal) = 1 := by norm_cast\n#align cardinal.succ_zero Cardinal.succ_zero\n\ntheorem exists_finset_le_card (\u03b1 : Type*) (n : \u2115) (h : n \u2264 #\u03b1) :\n    \u2203 s : Finset \u03b1, n \u2264 s.card := by\n  obtain h\u03b1|h\u03b1 := finite_or_infinite \u03b1\n  \u00b7 let h\u03b1 := Fintype.ofFinite \u03b1\n    use Finset.univ\n    simpa only [mk_fintype, Nat.cast_le] using h\n  \u00b7 obtain \u27e8s, hs\u27e9 := Infinite.exists_subset_card_eq \u03b1 n\n    exact \u27e8s, hs.ge\u27e9\n\ntheorem card_le_of {\u03b1 : Type u} {n : \u2115} (H : \u2200 s : Finset \u03b1, s.card \u2264 n) : #\u03b1 \u2264 n := by\n  contrapose! H\n  apply exists_finset_le_card \u03b1 (n+1)\n  simpa only [nat_succ, succ_le_iff] using H\n#align cardinal.card_le_of Cardinal.card_le_of\n\ntheorem cantor' (a) {b : Cardinal} (hb : 1 < b) : a < b ^ a := by\n  rw [\u2190 succ_le_iff, (by norm_cast : succ (1 : Cardinal) = 2)] at hb\n  exact (cantor a).trans_le (power_le_power_right hb)\n#align cardinal.cantor' Cardinal.cantor'\n\ntheorem one_le_iff_pos {c : Cardinal} : 1 \u2264 c \u2194 0 < c := by\n  rw [\u2190 succ_zero, succ_le_iff]\n#align cardinal.one_le_iff_pos Cardinal.one_le_iff_pos\n\ntheorem one_le_iff_ne_zero {c : Cardinal} : 1 \u2264 c \u2194 c \u2260 0 := by\n  rw [one_le_iff_pos, pos_iff_ne_zero]\n#align cardinal.one_le_iff_ne_zero Cardinal.one_le_iff_ne_zero\n\n@[simp]\ntheorem lt_one_iff_zero {c : Cardinal} : c < 1 \u2194 c = 0 := by\n  simpa using lt_succ_bot_iff (a := c)\n\ntheorem nat_lt_aleph0 (n : \u2115) : (n : Cardinal.{u}) < \u2135\u2080 :=\n  succ_le_iff.1\n    (by\n      rw [\u2190 nat_succ, \u2190 lift_mk_fin, aleph0, lift_mk_le.{u}]\n      exact \u27e8\u27e8(\u2191), fun a b => Fin.ext\u27e9\u27e9)\n#align cardinal.nat_lt_aleph_0 Cardinal.nat_lt_aleph0\n\n@[simp]\ntheorem one_lt_aleph0 : 1 < \u2135\u2080 := by simpa using nat_lt_aleph0 1\n#align cardinal.one_lt_aleph_0 Cardinal.one_lt_aleph0\n\ntheorem one_le_aleph0 : 1 \u2264 \u2135\u2080 :=\n  one_lt_aleph0.le\n#align cardinal.one_le_aleph_0 Cardinal.one_le_aleph0\n\ntheorem lt_aleph0 {c : Cardinal} : c < \u2135\u2080 \u2194 \u2203 n : \u2115, c = n :=\n  \u27e8fun h => by\n    rcases lt_lift_iff.1 h with \u27e8c, rfl, h'\u27e9\n    rcases le_mk_iff_exists_set.1 h'.1 with \u27e8S, rfl\u27e9\n    suffices S.Finite by\n      lift S to Finset \u2115 using this\n      simp\n    contrapose! h'\n    haveI := Infinite.to_subtype h'\n    exact \u27e8Infinite.natEmbedding S\u27e9, fun \u27e8n, e\u27e9 => e.symm \u25b8 nat_lt_aleph0 _\u27e9\n#align cardinal.lt_aleph_0 Cardinal.lt_aleph0\n\nlemma succ_eq_of_lt_aleph0 {c : Cardinal} (h : c < \u2135\u2080) : Order.succ c = c + 1 := by\n  obtain \u27e8n, hn\u27e9 := Cardinal.lt_aleph0.mp h\n  rw [hn, succ_natCast]\n\ntheorem aleph0_le {c : Cardinal} : \u2135\u2080 \u2264 c \u2194 \u2200 n : \u2115, \u2191n \u2264 c :=\n  \u27e8fun h n => (nat_lt_aleph0 _).le.trans h, fun h =>\n    le_of_not_lt fun hn => by\n      rcases lt_aleph0.1 hn with \u27e8n, rfl\u27e9\n      exact (Nat.lt_succ_self _).not_le (natCast_le.1 (h (n + 1)))\u27e9\n#align cardinal.aleph_0_le Cardinal.aleph0_le\n\ntheorem isSuccLimit_aleph0 : IsSuccLimit \u2135\u2080 :=\n  isSuccLimit_of_succ_lt fun a ha => by\n    rcases lt_aleph0.1 ha with \u27e8n, rfl\u27e9\n    rw [\u2190 nat_succ]\n    apply nat_lt_aleph0\n#align cardinal.is_succ_limit_aleph_0 Cardinal.isSuccLimit_aleph0\n\ntheorem isLimit_aleph0 : IsLimit \u2135\u2080 :=\n  \u27e8aleph0_ne_zero, isSuccLimit_aleph0\u27e9\n#align cardinal.is_limit_aleph_0 Cardinal.isLimit_aleph0\n\nlemma not_isLimit_natCast : (n : \u2115) \u2192 \u00ac IsLimit (n : Cardinal.{u})\n  | 0, e => e.1 rfl\n  | Nat.succ n, e => Order.not_isSuccLimit_succ _ (nat_succ n \u25b8 e.2)\n\ntheorem IsLimit.aleph0_le {c : Cardinal} (h : IsLimit c) : \u2135\u2080 \u2264 c := by\n  by_contra! h'\n  rcases lt_aleph0.1 h' with \u27e8n, rfl\u27e9\n  exact not_isLimit_natCast n h\n\nlemma exists_eq_natCast_of_iSup_eq {\u03b9 : Type u} [Nonempty \u03b9] (f : \u03b9 \u2192 Cardinal.{v})\n    (hf : BddAbove (range f)) (n : \u2115) (h : \u2a06 i, f i = n) : \u2203 i, f i = n :=\n  exists_eq_of_iSup_eq_of_not_isLimit.{u, v} f hf _ (not_isLimit_natCast n) h\n\n@[simp]\ntheorem range_natCast : range ((\u2191) : \u2115 \u2192 Cardinal) = Iio \u2135\u2080 :=\n  ext fun x => by simp only [mem_Iio, mem_range, eq_comm, lt_aleph0]\n#align cardinal.range_nat_cast Cardinal.range_natCast\n\ntheorem mk_eq_nat_iff {\u03b1 : Type u} {n : \u2115} : #\u03b1 = n \u2194 Nonempty (\u03b1 \u2243 Fin n) := by\n  rw [\u2190 lift_mk_fin, \u2190 lift_uzero #\u03b1, lift_mk_eq']\n#align cardinal.mk_eq_nat_iff Cardinal.mk_eq_nat_iff\n\ntheorem lt_aleph0_iff_finite {\u03b1 : Type u} : #\u03b1 < \u2135\u2080 \u2194 Finite \u03b1 := by\n  simp only [lt_aleph0, mk_eq_nat_iff, finite_iff_exists_equiv_fin]\n#align cardinal.lt_aleph_0_iff_finite Cardinal.lt_aleph0_iff_finite\n\ntheorem lt_aleph0_iff_fintype {\u03b1 : Type u} : #\u03b1 < \u2135\u2080 \u2194 Nonempty (Fintype \u03b1) :=\n  lt_aleph0_iff_finite.trans (finite_iff_nonempty_fintype _)\n#align cardinal.lt_aleph_0_iff_fintype Cardinal.lt_aleph0_iff_fintype\n\ntheorem lt_aleph0_of_finite (\u03b1 : Type u) [Finite \u03b1] : #\u03b1 < \u2135\u2080 :=\n  lt_aleph0_iff_finite.2 \u2039_\u203a\n#align cardinal.lt_aleph_0_of_finite Cardinal.lt_aleph0_of_finite\n\n-- porting note (#10618): simp can prove this\n-- @[simp]\ntheorem lt_aleph0_iff_set_finite {S : Set \u03b1} : #S < \u2135\u2080 \u2194 S.Finite :=\n  lt_aleph0_iff_finite.trans finite_coe_iff\n#align cardinal.lt_aleph_0_iff_set_finite Cardinal.lt_aleph0_iff_set_finite\n\nalias \u27e8_, _root_.Set.Finite.lt_aleph0\u27e9 := lt_aleph0_iff_set_finite\n#align set.finite.lt_aleph_0 Set.Finite.lt_aleph0\n\n@[simp]\ntheorem lt_aleph0_iff_subtype_finite {p : \u03b1 \u2192 Prop} : #{ x // p x } < \u2135\u2080 \u2194 { x | p x }.Finite :=\n  lt_aleph0_iff_set_finite\n#align cardinal.lt_aleph_0_iff_subtype_finite Cardinal.lt_aleph0_iff_subtype_finite\n\ntheorem mk_le_aleph0_iff : #\u03b1 \u2264 \u2135\u2080 \u2194 Countable \u03b1 := by\n  rw [countable_iff_nonempty_embedding, aleph0, \u2190 lift_uzero #\u03b1, lift_mk_le']\n#align cardinal.mk_le_aleph_0_iff Cardinal.mk_le_aleph0_iff\n\n@[simp]\ntheorem mk_le_aleph0 [Countable \u03b1] : #\u03b1 \u2264 \u2135\u2080 :=\n  mk_le_aleph0_iff.mpr \u2039_\u203a\n#align cardinal.mk_le_aleph_0 Cardinal.mk_le_aleph0\n\n-- porting note (#10618): simp can prove this\n-- @[simp]\ntheorem le_aleph0_iff_set_countable {s : Set \u03b1} : #s \u2264 \u2135\u2080 \u2194 s.Countable := mk_le_aleph0_iff\n#align cardinal.le_aleph_0_iff_set_countable Cardinal.le_aleph0_iff_set_countable\n\nalias \u27e8_, _root_.Set.Countable.le_aleph0\u27e9 := le_aleph0_iff_set_countable\n#align set.countable.le_aleph_0 Set.Countable.le_aleph0\n\n@[simp]\ntheorem le_aleph0_iff_subtype_countable {p : \u03b1 \u2192 Prop} :\n    #{ x // p x } \u2264 \u2135\u2080 \u2194 { x | p x }.Countable :=\n  le_aleph0_iff_set_countable\n#align cardinal.le_aleph_0_iff_subtype_countable Cardinal.le_aleph0_iff_subtype_countable\n\ninstance canLiftCardinalNat : CanLift Cardinal \u2115 (\u2191) fun x => x < \u2135\u2080 :=\n  \u27e8fun _ hx =>\n    let \u27e8n, hn\u27e9 := lt_aleph0.mp hx\n    \u27e8n, hn.symm\u27e9\u27e9\n#align cardinal.can_lift_cardinal_nat Cardinal.canLiftCardinalNat\n\ntheorem add_lt_aleph0 {a b : Cardinal} (ha : a < \u2135\u2080) (hb : b < \u2135\u2080) : a + b < \u2135\u2080 :=\n  match a, b, lt_aleph0.1 ha, lt_aleph0.1 hb with\n  | _, _, \u27e8m, rfl\u27e9, \u27e8n, rfl\u27e9 => by rw [\u2190 Nat.cast_add]; apply nat_lt_aleph0\n#align cardinal.add_lt_aleph_0 Cardinal.add_lt_aleph0\n\ntheorem add_lt_aleph0_iff {a b : Cardinal} : a + b < \u2135\u2080 \u2194 a < \u2135\u2080 \u2227 b < \u2135\u2080 :=\n  \u27e8fun h => \u27e8(self_le_add_right _ _).trans_lt h, (self_le_add_left _ _).trans_lt h\u27e9,\n   fun \u27e8h1, h2\u27e9 => add_lt_aleph0 h1 h2\u27e9\n#align cardinal.add_lt_aleph_0_iff Cardinal.add_lt_aleph0_iff\n\ntheorem aleph0_le_add_iff {a b : Cardinal} : \u2135\u2080 \u2264 a + b \u2194 \u2135\u2080 \u2264 a \u2228 \u2135\u2080 \u2264 b := by\n  simp only [\u2190 not_lt, add_lt_aleph0_iff, not_and_or]\n#align cardinal.aleph_0_le_add_iff Cardinal.aleph0_le_add_iff\n\n/-- See also `Cardinal.nsmul_lt_aleph0_iff_of_ne_zero` if you already have `n \u2260 0`. -/\ntheorem nsmul_lt_aleph0_iff {n : \u2115} {a : Cardinal} : n \u2022 a < \u2135\u2080 \u2194 n = 0 \u2228 a < \u2135\u2080 := by\n  cases n with\n  | zero => simpa using nat_lt_aleph0 0\n  | succ n =>\n      simp only [Nat.succ_ne_zero, false_or_iff]\n      induction' n with n ih\n      \u00b7 simp\n      rw [succ_nsmul, add_lt_aleph0_iff, ih, and_self_iff]\n#align cardinal.nsmul_lt_aleph_0_iff Cardinal.nsmul_lt_aleph0_iff\n\n/-- See also `Cardinal.nsmul_lt_aleph0_iff` for a hypothesis-free version. -/\ntheorem nsmul_lt_aleph0_iff_of_ne_zero {n : \u2115} {a : Cardinal} (h : n \u2260 0) : n \u2022 a < \u2135\u2080 \u2194 a < \u2135\u2080 :=\n  nsmul_lt_aleph0_iff.trans <| or_iff_right h\n#align cardinal.nsmul_lt_aleph_0_iff_of_ne_zero Cardinal.nsmul_lt_aleph0_iff_of_ne_zero\n\ntheorem mul_lt_aleph0 {a b : Cardinal} (ha : a < \u2135\u2080) (hb : b < \u2135\u2080) : a * b < \u2135\u2080 :=\n  match a, b, lt_aleph0.1 ha, lt_aleph0.1 hb with\n  | _, _, \u27e8m, rfl\u27e9, \u27e8n, rfl\u27e9 => by rw [\u2190 Nat.cast_mul]; apply nat_lt_aleph0\n#align cardinal.mul_lt_aleph_0 Cardinal.mul_lt_aleph0\n\ntheorem mul_lt_aleph0_iff {a b : Cardinal} : a * b < \u2135\u2080 \u2194 a = 0 \u2228 b = 0 \u2228 a < \u2135\u2080 \u2227 b < \u2135\u2080 := by\n  refine' \u27e8fun h => _, _\u27e9\n  \u00b7 by_cases ha : a = 0\n    \u00b7 exact Or.inl ha\n    right\n    by_cases hb : b = 0\n    \u00b7 exact Or.inl hb\n    right\n    rw [\u2190 Ne, \u2190 one_le_iff_ne_zero] at ha hb\n    constructor\n    \u00b7 rw [\u2190 mul_one a]\n      exact (mul_le_mul' le_rfl hb).trans_lt h\n    \u00b7 rw [\u2190 one_mul b]\n      exact (mul_le_mul' ha le_rfl).trans_lt h\n  rintro (rfl | rfl | \u27e8ha, hb\u27e9) <;> simp only [*, mul_lt_aleph0, aleph0_pos, zero_mul, mul_zero]\n#align cardinal.mul_lt_aleph_0_iff Cardinal.mul_lt_aleph0_iff\n\n/-- See also `Cardinal.aleph0_le_mul_iff`. -/\ntheorem aleph0_le_mul_iff {a b : Cardinal} : \u2135\u2080 \u2264 a * b \u2194 a \u2260 0 \u2227 b \u2260 0 \u2227 (\u2135\u2080 \u2264 a \u2228 \u2135\u2080 \u2264 b) := by\n  let h := (@mul_lt_aleph0_iff a b).not\n  rwa [not_lt, not_or, not_or, not_and_or, not_lt, not_lt] at h\n#align cardinal.aleph_0_le_mul_iff Cardinal.aleph0_le_mul_iff\n\n/-- See also `Cardinal.aleph0_le_mul_iff'`. -/\ntheorem aleph0_le_mul_iff' {a b : Cardinal.{u}} : \u2135\u2080 \u2264 a * b \u2194 a \u2260 0 \u2227 \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 a \u2227 b \u2260 0 := by\n  have : \u2200 {a : Cardinal.{u}}, \u2135\u2080 \u2264 a \u2192 a \u2260 0 := fun a => ne_bot_of_le_ne_bot aleph0_ne_zero a\n  simp only [aleph0_le_mul_iff, and_or_left, and_iff_right_of_imp this, @and_left_comm (a \u2260 0)]\n  simp only [and_comm, or_comm]\n#align cardinal.aleph_0_le_mul_iff' Cardinal.aleph0_le_mul_iff'\n\ntheorem mul_lt_aleph0_iff_of_ne_zero {a b : Cardinal} (ha : a \u2260 0) (hb : b \u2260 0) :\n    a * b < \u2135\u2080 \u2194 a < \u2135\u2080 \u2227 b < \u2135\u2080 := by simp [mul_lt_aleph0_iff, ha, hb]\n#align cardinal.mul_lt_aleph_0_iff_of_ne_zero Cardinal.mul_lt_aleph0_iff_of_ne_zero\n\ntheorem power_lt_aleph0 {a b : Cardinal} (ha : a < \u2135\u2080) (hb : b < \u2135\u2080) : a ^ b < \u2135\u2080 :=\n  match a, b, lt_aleph0.1 ha, lt_aleph0.1 hb with\n  | _, _, \u27e8m, rfl\u27e9, \u27e8n, rfl\u27e9 => by rw [\u2190 natCast_pow]; apply nat_lt_aleph0\n#align cardinal.power_lt_aleph_0 Cardinal.power_lt_aleph0\n\ntheorem eq_one_iff_unique {\u03b1 : Type*} : #\u03b1 = 1 \u2194 Subsingleton \u03b1 \u2227 Nonempty \u03b1 :=\n  calc\n    #\u03b1 = 1 \u2194 #\u03b1 \u2264 1 \u2227 1 \u2264 #\u03b1 := le_antisymm_iff\n    _ \u2194 Subsingleton \u03b1 \u2227 Nonempty \u03b1 :=\n      le_one_iff_subsingleton.and (one_le_iff_ne_zero.trans mk_ne_zero_iff)\n#align cardinal.eq_one_iff_unique Cardinal.eq_one_iff_unique\n\ntheorem infinite_iff {\u03b1 : Type u} : Infinite \u03b1 \u2194 \u2135\u2080 \u2264 #\u03b1 := by\n  rw [\u2190 not_lt, lt_aleph0_iff_finite, not_finite_iff_infinite]\n#align cardinal.infinite_iff Cardinal.infinite_iff\n\nlemma aleph0_le_mk_iff : \u2135\u2080 \u2264 #\u03b1 \u2194 Infinite \u03b1 := infinite_iff.symm\nlemma mk_lt_aleph0_iff : #\u03b1 < \u2135\u2080 \u2194 Finite \u03b1 := by simp [\u2190 not_le, aleph0_le_mk_iff]\n\n@[simp]\ntheorem aleph0_le_mk (\u03b1 : Type u) [Infinite \u03b1] : \u2135\u2080 \u2264 #\u03b1 :=\n  infinite_iff.1 \u2039_\u203a\n#align cardinal.aleph_0_le_mk Cardinal.aleph0_le_mk\n\n@[simp]\ntheorem mk_eq_aleph0 (\u03b1 : Type*) [Countable \u03b1] [Infinite \u03b1] : #\u03b1 = \u2135\u2080 :=\n  mk_le_aleph0.antisymm <| aleph0_le_mk _\n#align cardinal.mk_eq_aleph_0 Cardinal.mk_eq_aleph0\n\ntheorem denumerable_iff {\u03b1 : Type u} : Nonempty (Denumerable \u03b1) \u2194 #\u03b1 = \u2135\u2080 :=\n  \u27e8fun \u27e8h\u27e9 => mk_congr ((@Denumerable.eqv \u03b1 h).trans Equiv.ulift.symm), fun h => by\n    cases' Quotient.exact h with f\n    exact \u27e8Denumerable.mk' <| f.trans Equiv.ulift\u27e9\u27e9\n#align cardinal.denumerable_iff Cardinal.denumerable_iff\n\n-- porting note (#10618): simp can prove this\n-- @[simp]\ntheorem mk_denumerable (\u03b1 : Type u) [Denumerable \u03b1] : #\u03b1 = \u2135\u2080 :=\n  denumerable_iff.1 \u27e8\u2039_\u203a\u27e9\n#align cardinal.mk_denumerable Cardinal.mk_denumerable\n\ntheorem _root_.Set.countable_infinite_iff_nonempty_denumerable {\u03b1 : Type*} {s : Set \u03b1} :\n    s.Countable \u2227 s.Infinite \u2194 Nonempty (Denumerable s) := by\n  rw [nonempty_denumerable_iff, \u2190 Set.infinite_coe_iff, countable_coe_iff]\n\n@[simp]\ntheorem aleph0_add_aleph0 : \u2135\u2080 + \u2135\u2080 = \u2135\u2080 :=\n  mk_denumerable _\n#align cardinal.aleph_0_add_aleph_0 Cardinal.aleph0_add_aleph0\n\ntheorem aleph0_mul_aleph0 : \u2135\u2080 * \u2135\u2080 = \u2135\u2080 :=\n  mk_denumerable _\n#align cardinal.aleph_0_mul_aleph_0 Cardinal.aleph0_mul_aleph0\n\n@[simp]\ntheorem nat_mul_aleph0 {n : \u2115} (hn : n \u2260 0) : \u2191n * \u2135\u2080 = \u2135\u2080 :=\n  le_antisymm (lift_mk_fin n \u25b8 mk_le_aleph0) <|\n    le_mul_of_one_le_left (zero_le _) <| by\n      rwa [\u2190 Nat.cast_one, natCast_le, Nat.one_le_iff_ne_zero]\n#align cardinal.nat_mul_aleph_0 Cardinal.nat_mul_aleph0\n\n@[simp]\ntheorem aleph0_mul_nat {n : \u2115} (hn : n \u2260 0) : \u2135\u2080 * n = \u2135\u2080 := by rw [mul_comm, nat_mul_aleph0 hn]\n#align cardinal.aleph_0_mul_nat Cardinal.aleph0_mul_nat\n\n-- See note [no_index around OfNat.ofNat]\n@[simp]\ntheorem ofNat_mul_aleph0 {n : \u2115} [Nat.AtLeastTwo n] : no_index (OfNat.ofNat n) * \u2135\u2080 = \u2135\u2080 :=\n  nat_mul_aleph0 (OfNat.ofNat_ne_zero n)\n\n-- See note [no_index around OfNat.ofNat]\n@[simp]\ntheorem aleph0_mul_ofNat {n : \u2115} [Nat.AtLeastTwo n] : \u2135\u2080 * no_index (OfNat.ofNat n) = \u2135\u2080 :=\n  aleph0_mul_nat (OfNat.ofNat_ne_zero n)\n\n@[simp]\ntheorem add_le_aleph0 {c\u2081 c\u2082 : Cardinal} : c\u2081 + c\u2082 \u2264 \u2135\u2080 \u2194 c\u2081 \u2264 \u2135\u2080 \u2227 c\u2082 \u2264 \u2135\u2080 :=\n  \u27e8fun h => \u27e8le_self_add.trans h, le_add_self.trans h\u27e9, fun h =>\n    aleph0_add_aleph0 \u25b8 add_le_add h.1 h.2\u27e9\n#align cardinal.add_le_aleph_0 Cardinal.add_le_aleph0\n\n@[simp]\ntheorem aleph0_add_nat (n : \u2115) : \u2135\u2080 + n = \u2135\u2080 :=\n  (add_le_aleph0.2 \u27e8le_rfl, (nat_lt_aleph0 n).le\u27e9).antisymm le_self_add\n#align cardinal.aleph_0_add_nat Cardinal.aleph0_add_nat\n\n@[simp]\ntheorem nat_add_aleph0 (n : \u2115) : \u2191n + \u2135\u2080 = \u2135\u2080 := by rw [add_comm, aleph0_add_nat]\n#align cardinal.nat_add_aleph_0 Cardinal.nat_add_aleph0\n\n-- See note [no_index around OfNat.ofNat]\n@[simp]\ntheorem ofNat_add_aleph0 {n : \u2115} [Nat.AtLeastTwo n] : no_index (OfNat.ofNat n) + \u2135\u2080 = \u2135\u2080 :=\n  nat_add_aleph0 n\n\n-- See note [no_index around OfNat.ofNat]\n@[simp]\ntheorem aleph0_add_ofNat {n : \u2115} [Nat.AtLeastTwo n] : \u2135\u2080 + no_index (OfNat.ofNat n) = \u2135\u2080 :=\n  aleph0_add_nat n\n\ntheorem exists_nat_eq_of_le_nat {c : Cardinal} {n : \u2115} (h : c \u2264 n) : \u2203 m, m \u2264 n \u2227 c = m := by\n  lift c to \u2115 using h.trans_lt (nat_lt_aleph0 _)\n  exact \u27e8c, mod_cast h, rfl\u27e9\n#align cardinal.exists_nat_eq_of_le_nat Cardinal.exists_nat_eq_of_le_nat\n\ntheorem mk_int : #\u2124 = \u2135\u2080 :=\n  mk_denumerable \u2124\n#align cardinal.mk_int Cardinal.mk_int\n\ntheorem mk_pNat : #\u2115+ = \u2135\u2080 :=\n  mk_denumerable \u2115+\n#align cardinal.mk_pnat Cardinal.mk_pNat\n\nend castFromN\n\nvariable {c : Cardinal}\n\n/-- **K\u00f6nig's theorem** -/\ntheorem sum_lt_prod {\u03b9} (f g : \u03b9 \u2192 Cardinal) (H : \u2200 i, f i < g i) : sum f < prod g :=\n  lt_of_not_ge fun \u27e8F\u27e9 => by\n    have : Inhabited (\u2200 i : \u03b9, (g i).out) := by\n      refine' \u27e8fun i => Classical.choice <| mk_ne_zero_iff.1 _\u27e9\n      rw [mk_out]\n      exact (H i).ne_bot\n    let G := invFun F\n    have sG : Surjective G := invFun_surjective F.2\n    choose C hc using\n      show \u2200 i, \u2203 b, \u2200 a, G \u27e8i, a\u27e9 i \u2260 b by\n        intro i\n        simp only [not_exists.symm, not_forall.symm]\n        refine' fun h => (H i).not_le _\n        rw [\u2190 mk_out (f i), \u2190 mk_out (g i)]\n        exact \u27e8Embedding.ofSurjective _ h\u27e9\n    let \u27e8\u27e8i, a\u27e9, h\u27e9 := sG C\n    exact hc i a (congr_fun h _)\n#align cardinal.sum_lt_prod Cardinal.sum_lt_prod\n\n/-! Cardinalities of sets: cardinality of empty, finite sets, unions, subsets etc. -/\nsection sets\n\n-- porting note (#10618): simp can prove this\n-- @[simp]\ntheorem mk_empty : #Empty = 0 :=\n  mk_eq_zero _\n#align cardinal.mk_empty Cardinal.mk_empty\n\n-- porting note (#10618): simp can prove this\n-- @[simp]\ntheorem mk_pempty : #PEmpty = 0 :=\n  mk_eq_zero _\n#align cardinal.mk_pempty Cardinal.mk_pempty\n\n-- porting note (#10618): simp can prove this\n-- @[simp]\ntheorem mk_punit : #PUnit = 1 :=\n  mk_eq_one PUnit\n#align cardinal.mk_punit Cardinal.mk_punit\n\ntheorem mk_unit : #Unit = 1 :=\n  mk_punit\n#align cardinal.mk_unit Cardinal.mk_unit\n\n-- porting note (#10618): simp can prove this\n-- @[simp]\ntheorem mk_singleton {\u03b1 : Type u} (x : \u03b1) : #({x} : Set \u03b1) = 1 :=\n  mk_eq_one _\n#align cardinal.mk_singleton Cardinal.mk_singleton\n\n-- porting note (#10618): simp can prove this\n-- @[simp]\ntheorem mk_plift_true : #(PLift True) = 1 :=\n  mk_eq_one _\n#align cardinal.mk_plift_true Cardinal.mk_plift_true\n\n-- porting note (#10618): simp can prove this\n-- @[simp]\ntheorem mk_plift_false : #(PLift False) = 0 :=\n  mk_eq_zero _\n#align cardinal.mk_plift_false Cardinal.mk_plift_false\n\n@[simp]\ntheorem mk_vector (\u03b1 : Type u) (n : \u2115) : #(Vector \u03b1 n) = #\u03b1 ^ n :=\n  (mk_congr (Equiv.vectorEquivFin \u03b1 n)).trans <| by simp\n#align cardinal.mk_vector Cardinal.mk_vector\n\ntheorem mk_list_eq_sum_pow (\u03b1 : Type u) : #(List \u03b1) = sum fun n : \u2115 => #\u03b1 ^ n :=\n  calc\n    #(List \u03b1) = #(\u03a3n, Vector \u03b1 n) := mk_congr (Equiv.sigmaFiberEquiv List.length).symm\n    _ = sum fun n : \u2115 => #\u03b1 ^ n := by simp\n#align cardinal.mk_list_eq_sum_pow Cardinal.mk_list_eq_sum_pow\n\ntheorem mk_quot_le {\u03b1 : Type u} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} : #(Quot r) \u2264 #\u03b1 :=\n  mk_le_of_surjective Quot.exists_rep\n#align cardinal.mk_quot_le Cardinal.mk_quot_le\n\ntheorem mk_quotient_le {\u03b1 : Type u} {s : Setoid \u03b1} : #(Quotient s) \u2264 #\u03b1 :=\n  mk_quot_le\n#align cardinal.mk_quotient_le Cardinal.mk_quotient_le\n\ntheorem mk_subtype_le_of_subset {\u03b1 : Type u} {p q : \u03b1 \u2192 Prop} (h : \u2200 \u2983x\u2984, p x \u2192 q x) :\n    #(Subtype p) \u2264 #(Subtype q) :=\n  \u27e8Embedding.subtypeMap (Embedding.refl \u03b1) h\u27e9\n#align cardinal.mk_subtype_le_of_subset Cardinal.mk_subtype_le_of_subset\n\n-- porting note (#10618): simp can prove this\n-- @[simp]\ntheorem mk_emptyCollection (\u03b1 : Type u) : #(\u2205 : Set \u03b1) = 0 :=\n  mk_eq_zero _\n#align cardinal.mk_emptyc Cardinal.mk_emptyCollection\n\ntheorem mk_emptyCollection_iff {\u03b1 : Type u} {s : Set \u03b1} : #s = 0 \u2194 s = \u2205 := by\n  constructor\n  \u00b7 intro h\n    rw [mk_eq_zero_iff] at h\n    exact eq_empty_iff_forall_not_mem.2 fun x hx => h.elim' \u27e8x, hx\u27e9\n  \u00b7 rintro rfl\n    exact mk_emptyCollection _\n#align cardinal.mk_emptyc_iff Cardinal.mk_emptyCollection_iff\n\n@[simp]\ntheorem mk_univ {\u03b1 : Type u} : #(@univ \u03b1) = #\u03b1 :=\n  mk_congr (Equiv.Set.univ \u03b1)\n#align cardinal.mk_univ Cardinal.mk_univ\n\ntheorem mk_image_le {\u03b1 \u03b2 : Type u} {f : \u03b1 \u2192 \u03b2} {s : Set \u03b1} : #(f '' s) \u2264 #s :=\n  mk_le_of_surjective surjective_onto_image\n#align cardinal.mk_image_le Cardinal.mk_image_le\n\ntheorem mk_image_le_lift {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} {s : Set \u03b1} :\n    lift.{u} #(f '' s) \u2264 lift.{v} #s :=\n  lift_mk_le.{0}.mpr \u27e8Embedding.ofSurjective _ surjective_onto_image\u27e9\n#align cardinal.mk_image_le_lift Cardinal.mk_image_le_lift\n\ntheorem mk_range_le {\u03b1 \u03b2 : Type u} {f : \u03b1 \u2192 \u03b2} : #(range f) \u2264 #\u03b1 :=\n  mk_le_of_surjective surjective_onto_range\n#align cardinal.mk_range_le Cardinal.mk_range_le\n\ntheorem mk_range_le_lift {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} :\n    lift.{u} #(range f) \u2264 lift.{v} #\u03b1 :=\n  lift_mk_le.{0}.mpr \u27e8Embedding.ofSurjective _ surjective_onto_range\u27e9\n#align cardinal.mk_range_le_lift Cardinal.mk_range_le_lift\n\ntheorem mk_range_eq (f : \u03b1 \u2192 \u03b2) (h : Injective f) : #(range f) = #\u03b1 :=\n  mk_congr (Equiv.ofInjective f h).symm\n#align cardinal.mk_range_eq Cardinal.mk_range_eq\n\ntheorem mk_range_eq_lift {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} (hf : Injective f) :\n    lift.{max u w} #(range f) = lift.{max v w} #\u03b1 :=\n  lift_mk_eq.{v,u,w}.mpr \u27e8(Equiv.ofInjective f hf).symm\u27e9\n#align cardinal.mk_range_eq_lift Cardinal.mk_range_eq_lift\n\ntheorem mk_range_eq_of_injective {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} (hf : Injective f) :\n    lift.{u} #(range f) = lift.{v} #\u03b1 :=\n  lift_mk_eq'.mpr \u27e8(Equiv.ofInjective f hf).symm\u27e9\n#align cardinal.mk_range_eq_of_injective Cardinal.mk_range_eq_of_injective\n\nlemma lift_mk_le_lift_mk_of_injective {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} (hf : Injective f) :\n    Cardinal.lift.{v} (#\u03b1) \u2264 Cardinal.lift.{u} (#\u03b2) := by\n  rw [\u2190 Cardinal.mk_range_eq_of_injective hf]\n  exact Cardinal.lift_le.2 (Cardinal.mk_set_le _)\n\ntheorem mk_image_eq_of_injOn {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (s : Set \u03b1) (h : InjOn f s) :\n    #(f '' s) = #s :=\n  mk_congr (Equiv.Set.imageOfInjOn f s h).symm\n#align cardinal.mk_image_eq_of_inj_on Cardinal.mk_image_eq_of_injOn\n\ntheorem mk_image_eq_of_injOn_lift {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : Set \u03b1)\n    (h : InjOn f s) : lift.{u} #(f '' s) = lift.{v} #s :=\n  lift_mk_eq.{v, u, 0}.mpr \u27e8(Equiv.Set.imageOfInjOn f s h).symm\u27e9\n#align cardinal.mk_image_eq_of_inj_on_lift Cardinal.mk_image_eq_of_injOn_lift\n\ntheorem mk_image_eq {\u03b1 \u03b2 : Type u} {f : \u03b1 \u2192 \u03b2} {s : Set \u03b1} (hf : Injective f) : #(f '' s) = #s :=\n  mk_image_eq_of_injOn _ _ <| hf.injOn _\n#align cardinal.mk_image_eq Cardinal.mk_image_eq\n\ntheorem mk_image_eq_lift {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : Set \u03b1) (h : Injective f) :\n    lift.{u} #(f '' s) = lift.{v} #s :=\n  mk_image_eq_of_injOn_lift _ _ <| h.injOn _\n#align cardinal.mk_image_eq_lift Cardinal.mk_image_eq_lift\n\ntheorem mk_iUnion_le_sum_mk {\u03b1 \u03b9 : Type u} {f : \u03b9 \u2192 Set \u03b1} : #(\u22c3 i, f i) \u2264 sum fun i => #(f i) :=\n  calc\n    #(\u22c3 i, f i) \u2264 #(\u03a3i, f i) := mk_le_of_surjective (Set.sigmaToiUnion_surjective f)\n    _ = sum fun i => #(f i) := mk_sigma _\n#align cardinal.mk_Union_le_sum_mk Cardinal.mk_iUnion_le_sum_mk\n\ntheorem mk_iUnion_le_sum_mk_lift {\u03b1 : Type u} {\u03b9 : Type v} {f : \u03b9 \u2192 Set \u03b1} :\n    lift.{v} #(\u22c3 i, f i) \u2264 sum fun i => #(f i) :=\n  calc\n    lift.{v} #(\u22c3 i, f i) \u2264 #(\u03a3i, f i) :=\n      mk_le_of_surjective <| ULift.up_surjective.comp (Set.sigmaToiUnion_surjective f)\n    _ = sum fun i => #(f i) := mk_sigma _\n\ntheorem mk_iUnion_eq_sum_mk {\u03b1 \u03b9 : Type u} {f : \u03b9 \u2192 Set \u03b1}\n    (h : Pairwise fun i j => Disjoint (f i) (f j)) : #(\u22c3 i, f i) = sum fun i => #(f i) :=\n  calc\n    #(\u22c3 i, f i) = #(\u03a3i, f i) := mk_congr (Set.unionEqSigmaOfDisjoint h)\n    _ = sum fun i => #(f i) := mk_sigma _\n#align cardinal.mk_Union_eq_sum_mk Cardinal.mk_iUnion_eq_sum_mk\n\ntheorem mk_iUnion_eq_sum_mk_lift {\u03b1 : Type u} {\u03b9 : Type v} {f : \u03b9 \u2192 Set \u03b1}\n    (h : Pairwise fun i j => Disjoint (f i) (f j)) :\n    lift.{v} #(\u22c3 i, f i) = sum fun i => #(f i) :=\n  calc\n    lift.{v} #(\u22c3 i, f i) = #(\u03a3i, f i) :=\n      mk_congr <| .trans Equiv.ulift (Set.unionEqSigmaOfDisjoint h)\n    _ = sum fun i => #(f i) := mk_sigma _\n\ntheorem mk_iUnion_le {\u03b1 \u03b9 : Type u} (f : \u03b9 \u2192 Set \u03b1) : #(\u22c3 i, f i) \u2264 #\u03b9 * \u2a06 i, #(f i) :=\n  mk_iUnion_le_sum_mk.trans (sum_le_iSup _)\n#align cardinal.mk_Union_le Cardinal.mk_iUnion_le\n\ntheorem mk_iUnion_le_lift {\u03b1 : Type u} {\u03b9 : Type v} (f : \u03b9 \u2192 Set \u03b1) :\n    lift.{v} #(\u22c3 i, f i) \u2264 lift.{u} #\u03b9 * \u2a06 i, lift.{v} #(f i) := by\n  refine mk_iUnion_le_sum_mk_lift.trans <| Eq.trans_le ?_ (sum_le_iSup_lift _)\n  rw [\u2190 lift_sum, lift_id'.{_,u}]\n\ntheorem mk_sUnion_le {\u03b1 : Type u} (A : Set (Set \u03b1)) : #(\u22c3\u2080 A) \u2264 #A * \u2a06 s : A, #s := by\n  rw [sUnion_eq_iUnion]\n  apply mk_iUnion_le\n#align cardinal.mk_sUnion_le Cardinal.mk_sUnion_le\n\ntheorem mk_biUnion_le {\u03b9 \u03b1 : Type u} (A : \u03b9 \u2192 Set \u03b1) (s : Set \u03b9) :\n    #(\u22c3 x \u2208 s, A x) \u2264 #s * \u2a06 x : s, #(A x.1) := by\n  rw [biUnion_eq_iUnion]\n  apply mk_iUnion_le\n#align cardinal.mk_bUnion_le Cardinal.mk_biUnion_le\n\ntheorem mk_biUnion_le_lift {\u03b1 : Type u} {\u03b9 : Type v} (A : \u03b9 \u2192 Set \u03b1) (s : Set \u03b9) :\n    lift.{v} #(\u22c3 x \u2208 s, A x) \u2264 lift.{u} #s * \u2a06 x : s, lift.{v} #(A x.1) := by\n  rw [biUnion_eq_iUnion]\n  apply mk_iUnion_le_lift\n\ntheorem finset_card_lt_aleph0 (s : Finset \u03b1) : #(\u2191s : Set \u03b1) < \u2135\u2080 :=\n  lt_aleph0_of_finite _\n#align cardinal.finset_card_lt_aleph_0 Cardinal.finset_card_lt_aleph0\n\ntheorem mk_set_eq_nat_iff_finset {\u03b1} {s : Set \u03b1} {n : \u2115} :\n    #s = n \u2194 \u2203 t : Finset \u03b1, (t : Set \u03b1) = s \u2227 t.card = n := by\n  constructor\n  \u00b7 intro h\n    lift s to Finset \u03b1 using lt_aleph0_iff_set_finite.1 (h.symm \u25b8 nat_lt_aleph0 n)\n    simpa using h\n  \u00b7 rintro \u27e8t, rfl, rfl\u27e9\n    exact mk_coe_finset\n#align cardinal.mk_set_eq_nat_iff_finset Cardinal.mk_set_eq_nat_iff_finset\n\ntheorem mk_eq_nat_iff_finset {n : \u2115} :\n    #\u03b1 = n \u2194 \u2203 t : Finset \u03b1, (t : Set \u03b1) = univ \u2227 t.card = n :=\n  by rw [\u2190 mk_univ, mk_set_eq_nat_iff_finset]\n#align cardinal.mk_eq_nat_iff_finset Cardinal.mk_eq_nat_iff_finset\n\ntheorem mk_eq_nat_iff_fintype {n : \u2115} : #\u03b1 = n \u2194 \u2203 h : Fintype \u03b1, @Fintype.card \u03b1 h = n := by\n  rw [mk_eq_nat_iff_finset]\n  constructor\n  \u00b7 rintro \u27e8t, ht, hn\u27e9\n    exact \u27e8\u27e8t, eq_univ_iff_forall.1 ht\u27e9, hn\u27e9\n  \u00b7 rintro \u27e8\u27e8t, ht\u27e9, hn\u27e9\n    exact \u27e8t, eq_univ_iff_forall.2 ht, hn\u27e9\n#align cardinal.mk_eq_nat_iff_fintype Cardinal.mk_eq_nat_iff_fintype\n\ntheorem mk_union_add_mk_inter {\u03b1 : Type u} {S T : Set \u03b1} :\n    #(S \u222a T : Set \u03b1) + #(S \u2229 T : Set \u03b1) = #S + #T :=\n  Quot.sound \u27e8Equiv.Set.unionSumInter S T\u27e9\n#align cardinal.mk_union_add_mk_inter Cardinal.mk_union_add_mk_inter\n\n/-- The cardinality of a union is at most the sum of the cardinalities\nof the two sets. -/\ntheorem mk_union_le {\u03b1 : Type u} (S T : Set \u03b1) : #(S \u222a T : Set \u03b1) \u2264 #S + #T :=\n  @mk_union_add_mk_inter \u03b1 S T \u25b8 self_le_add_right #(S \u222a T : Set \u03b1) #(S \u2229 T : Set \u03b1)\n#align cardinal.mk_union_le Cardinal.mk_union_le\n\ntheorem mk_union_of_disjoint {\u03b1 : Type u} {S T : Set \u03b1} (H : Disjoint S T) :\n    #(S \u222a T : Set \u03b1) = #S + #T :=\n  Quot.sound \u27e8Equiv.Set.union H.le_bot\u27e9\n#align cardinal.mk_union_of_disjoint Cardinal.mk_union_of_disjoint\n\ntheorem mk_insert {\u03b1 : Type u} {s : Set \u03b1} {a : \u03b1} (h : a \u2209 s) :\n    #(insert a s : Set \u03b1) = #s + 1 := by\n  rw [\u2190 union_singleton, mk_union_of_disjoint, mk_singleton]\n  simpa\n#align cardinal.mk_insert Cardinal.mk_insert\n\n", "theoremStatement": "theorem mk_insert_le {\u03b1 : Type u} {s : Set \u03b1} {a : \u03b1} : #(insert a s : Set \u03b1) \u2264 #s + 1 ", "theoremName": "Cardinal.mk_insert_le", "fileCreated": {"commit": "f805dd1a8f5243708d8b831bb207e63649ac1331", "date": "2023-02-11"}, "theoremCreated": {"commit": 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"Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", 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"Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", 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+{"srcContext": "/-\nCopyright (c) 2023 Felix Weilacher. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Felix Weilacher, Yury G. Kudryashov, R\u00e9my Degenne\n-/\nimport Mathlib.MeasureTheory.MeasurableSpace.Basic\nimport Mathlib.Data.Set.MemPartition\nimport Mathlib.Order.Filter.CountableSeparatingOn\n\n/-!\n# Countably generated measurable spaces\n\nWe say a measurable space is countably generated if it can be generated by a countable set of sets.\n\nIn such a space, we can also build a sequence of finer and finer finite measurable partitions of\nthe space such that the measurable space is generated by the union of all partitions.\n\n## Main definitions\n\n* `MeasurableSpace.CountablyGenerated`: class stating that a measurable space is countably\n  generated.\n* `MeasurableSpace.countableGeneratingSet`: a countable set of sets that generates the \u03c3-algebra.\n* `MeasurableSpace.countablePartition`: sequences of finer and finer partitions of\n  a countably generated space, defined by taking the `memPartion` of an enumeration of the sets in\n  `countableGeneratingSet`.\n* `MeasurableSpace.SeparatesPoints` : class stating that a measurable space separates points.\n\n## Main statements\n\n* `MeasurableSpace.measurable_equiv_nat_bool_of_countablyGenerated`: if a measurable space is\n  countably generated and separates points, it is measure equivalent to a subset of the Cantor Space\n  `\u2115 \u2192 Bool` (equipped with the product sigma algebra).\n* `MeasurableSpace.measurable_injection_nat_bool_of_countablyGenerated`: If a measurable space\n  admits a countable sequence of measurable sets separating points,\n  it admits a measurable injection into the Cantor space `\u2115 \u2192 Bool`\n  `\u2115 \u2192 Bool` (equipped with the product sigma algebra).\n\nThe file also contains measurability results about `memPartition`, from which the properties of\n`countablePartition` are deduced.\n\n-/\n\nopen Set MeasureTheory\n\nnamespace MeasurableSpace\n\nvariable {\u03b1 \u03b2 : Type*}\n\n/-- We say a measurable space is countably generated\nif it can be generated by a countable set of sets. -/\nclass CountablyGenerated (\u03b1 : Type*) [m : MeasurableSpace \u03b1] : Prop where\n  isCountablyGenerated : \u2203 b : Set (Set \u03b1), b.Countable \u2227 m = generateFrom b\n#align measurable_space.countably_generated MeasurableSpace.CountablyGenerated\n\n/-- A countable set of sets that generate the measurable space.\nWe insert `\u2205` to ensure it is nonempty. -/\ndef countableGeneratingSet (\u03b1 : Type*) [MeasurableSpace \u03b1] [h : CountablyGenerated \u03b1] :\n    Set (Set \u03b1) :=\n  insert \u2205 h.isCountablyGenerated.choose\n\nlemma countable_countableGeneratingSet [MeasurableSpace \u03b1] [h : CountablyGenerated \u03b1] :\n    Set.Countable (countableGeneratingSet \u03b1) :=\n  Countable.insert _ h.isCountablyGenerated.choose_spec.1\n\nlemma generateFrom_countableGeneratingSet [m : MeasurableSpace \u03b1] [h : CountablyGenerated \u03b1] :\n    generateFrom (countableGeneratingSet \u03b1) = m :=\n  (generateFrom_insert_empty _).trans <| h.isCountablyGenerated.choose_spec.2.symm\n\nlemma empty_mem_countableGeneratingSet [MeasurableSpace \u03b1] [CountablyGenerated \u03b1] :\n    \u2205 \u2208 countableGeneratingSet \u03b1 := mem_insert _ _\n\nlemma nonempty_countableGeneratingSet [MeasurableSpace \u03b1] [CountablyGenerated \u03b1] :\n    Set.Nonempty (countableGeneratingSet \u03b1) :=\n  \u27e8\u2205, mem_insert _ _\u27e9\n\nlemma measurableSet_countableGeneratingSet [MeasurableSpace \u03b1] [CountablyGenerated \u03b1]\n    {s : Set \u03b1} (hs : s \u2208 countableGeneratingSet \u03b1) :\n    MeasurableSet s := by\n  rw [\u2190 generateFrom_countableGeneratingSet (\u03b1 := \u03b1)]\n  exact measurableSet_generateFrom hs\n\n/-- A countable sequence of sets generating the measurable space. -/\ndef natGeneratingSequence (\u03b1 : Type*) [MeasurableSpace \u03b1] [CountablyGenerated \u03b1] : \u2115 \u2192 (Set \u03b1) :=\n  enumerateCountable (countable_countableGeneratingSet (\u03b1 := \u03b1)) \u2205\n\nlemma generateFrom_natGeneratingSequence (\u03b1 : Type*) [m : MeasurableSpace \u03b1]\n    [CountablyGenerated \u03b1] : generateFrom (range (natGeneratingSequence _)) = m := by\n  rw [natGeneratingSequence, range_enumerateCountable_of_mem _ empty_mem_countableGeneratingSet,\n    generateFrom_countableGeneratingSet]\n\n", "theoremStatement": "lemma measurableSet_natGeneratingSequence [MeasurableSpace \u03b1] [CountablyGenerated \u03b1] (n : \u2115) :\n    MeasurableSet (natGeneratingSequence \u03b1 n) ", "theoremName": "MeasurableSpace.measurableSet_natGeneratingSequence", "fileCreated": {"commit": "e82ee277c7cf0855c96740746702d13f609f3d3d", "date": "2024-03-13"}, "theoremCreated": {"commit": "ac9ad40ffc0c10776f8c8b56909fadd83a9d0848", "date": "2024-03-25"}, "file": "mathlib/Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean", "module": "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "jsonFile": "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated.jsonl", 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"Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Embedding.Set", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableInter", "Mathlib.Order.Filter.CountableSeparatingOn"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n  measurableSet_countableGeneratingSet $ Set.enumerateCountable_mem _\n    empty_mem_countableGeneratingSet n", "proofType": "term", "proofLengthLines": 2, "proofLengthTokens": 111}}
+{"srcContext": "/-\nCopyright (c) 2023 Felix Weilacher. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Felix Weilacher, Yury G. Kudryashov, R\u00e9my Degenne\n-/\nimport Mathlib.MeasureTheory.MeasurableSpace.Basic\nimport Mathlib.Data.Set.MemPartition\nimport Mathlib.Order.Filter.CountableSeparatingOn\n\n/-!\n# Countably generated measurable spaces\n\nWe say a measurable space is countably generated if it can be generated by a countable set of sets.\n\nIn such a space, we can also build a sequence of finer and finer finite measurable partitions of\nthe space such that the measurable space is generated by the union of all partitions.\n\n## Main definitions\n\n* `MeasurableSpace.CountablyGenerated`: class stating that a measurable space is countably\n  generated.\n* `MeasurableSpace.countableGeneratingSet`: a countable set of sets that generates the \u03c3-algebra.\n* `MeasurableSpace.countablePartition`: sequences of finer and finer partitions of\n  a countably generated space, defined by taking the `memPartion` of an enumeration of the sets in\n  `countableGeneratingSet`.\n* `MeasurableSpace.SeparatesPoints` : class stating that a measurable space separates points.\n\n## Main statements\n\n* `MeasurableSpace.measurable_equiv_nat_bool_of_countablyGenerated`: if a measurable space is\n  countably generated and separates points, it is measure equivalent to a subset of the Cantor Space\n  `\u2115 \u2192 Bool` (equipped with the product sigma algebra).\n* `MeasurableSpace.measurable_injection_nat_bool_of_countablyGenerated`: If a measurable space\n  admits a countable sequence of measurable sets separating points,\n  it admits a measurable injection into the Cantor space `\u2115 \u2192 Bool`\n  `\u2115 \u2192 Bool` (equipped with the product sigma algebra).\n\nThe file also contains measurability results about `memPartition`, from which the properties of\n`countablePartition` are deduced.\n\n-/\n\nopen Set MeasureTheory\n\nnamespace MeasurableSpace\n\nvariable {\u03b1 \u03b2 : Type*}\n\n/-- We say a measurable space is countably generated\nif it can be generated by a countable set of sets. -/\nclass CountablyGenerated (\u03b1 : Type*) [m : MeasurableSpace \u03b1] : Prop where\n  isCountablyGenerated : \u2203 b : Set (Set \u03b1), b.Countable \u2227 m = generateFrom b\n#align measurable_space.countably_generated MeasurableSpace.CountablyGenerated\n\n/-- A countable set of sets that generate the measurable space.\nWe insert `\u2205` to ensure it is nonempty. -/\ndef countableGeneratingSet (\u03b1 : Type*) [MeasurableSpace \u03b1] [h : CountablyGenerated \u03b1] :\n    Set (Set \u03b1) :=\n  insert \u2205 h.isCountablyGenerated.choose\n\nlemma countable_countableGeneratingSet [MeasurableSpace \u03b1] [h : CountablyGenerated \u03b1] :\n    Set.Countable (countableGeneratingSet \u03b1) :=\n  Countable.insert _ h.isCountablyGenerated.choose_spec.1\n\nlemma generateFrom_countableGeneratingSet [m : MeasurableSpace \u03b1] [h : CountablyGenerated \u03b1] :\n    generateFrom (countableGeneratingSet \u03b1) = m :=\n  (generateFrom_insert_empty _).trans <| h.isCountablyGenerated.choose_spec.2.symm\n\nlemma empty_mem_countableGeneratingSet [MeasurableSpace \u03b1] [CountablyGenerated \u03b1] :\n    \u2205 \u2208 countableGeneratingSet \u03b1 := mem_insert _ _\n\nlemma nonempty_countableGeneratingSet [MeasurableSpace \u03b1] [CountablyGenerated \u03b1] :\n    Set.Nonempty (countableGeneratingSet \u03b1) :=\n  \u27e8\u2205, mem_insert _ _\u27e9\n\nlemma measurableSet_countableGeneratingSet [MeasurableSpace \u03b1] [CountablyGenerated \u03b1]\n    {s : Set \u03b1} (hs : s \u2208 countableGeneratingSet \u03b1) :\n    MeasurableSet s := by\n  rw [\u2190 generateFrom_countableGeneratingSet (\u03b1 := \u03b1)]\n  exact measurableSet_generateFrom hs\n\n/-- A countable sequence of sets generating the measurable space. -/\ndef natGeneratingSequence (\u03b1 : Type*) [MeasurableSpace \u03b1] [CountablyGenerated \u03b1] : \u2115 \u2192 (Set \u03b1) :=\n  enumerateCountable (countable_countableGeneratingSet (\u03b1 := \u03b1)) \u2205\n\nlemma generateFrom_natGeneratingSequence (\u03b1 : Type*) [m : MeasurableSpace \u03b1]\n    [CountablyGenerated \u03b1] : generateFrom (range (natGeneratingSequence _)) = m := by\n  rw [natGeneratingSequence, range_enumerateCountable_of_mem _ empty_mem_countableGeneratingSet,\n    generateFrom_countableGeneratingSet]\n\nlemma measurableSet_natGeneratingSequence [MeasurableSpace \u03b1] [CountablyGenerated \u03b1] (n : \u2115) :\n    MeasurableSet (natGeneratingSequence \u03b1 n) :=\n  measurableSet_countableGeneratingSet $ Set.enumerateCountable_mem _\n    empty_mem_countableGeneratingSet n\n\ntheorem CountablyGenerated.comap [m : MeasurableSpace \u03b2] [h : CountablyGenerated \u03b2] (f : \u03b1 \u2192 \u03b2) :\n    @CountablyGenerated \u03b1 (.comap f m) := by\n  rcases h with \u27e8\u27e8b, hbc, rfl\u27e9\u27e9\n  rw [comap_generateFrom]\n  letI := generateFrom (preimage f '' b)\n  exact \u27e8_, hbc.image _, rfl\u27e9\n\ntheorem CountablyGenerated.sup {m\u2081 m\u2082 : MeasurableSpace \u03b2} (h\u2081 : @CountablyGenerated \u03b2 m\u2081)\n    (h\u2082 : @CountablyGenerated \u03b2 m\u2082) : @CountablyGenerated \u03b2 (m\u2081 \u2294 m\u2082) := by\n  rcases h\u2081 with \u27e8\u27e8b\u2081, hb\u2081c, rfl\u27e9\u27e9\n  rcases h\u2082 with \u27e8\u27e8b\u2082, hb\u2082c, rfl\u27e9\u27e9\n  exact @mk _ (_ \u2294 _) \u27e8_, hb\u2081c.union hb\u2082c, generateFrom_sup_generateFrom\u27e9\n\n/-- Any measurable space structure on a countable space is countably generated. -/\ninstance (priority := 100) [MeasurableSpace \u03b1] [Countable \u03b1] : CountablyGenerated \u03b1 where\n  isCountablyGenerated := by\n    refine \u27e8\u22c3 y, {measurableAtom y}, countable_iUnion (fun i \u21a6 countable_singleton _), ?_\u27e9\n    refine le_antisymm ?_ (generateFrom_le (by simp [MeasurableSet.measurableAtom_of_countable]))\n    intro s hs\n    have : s = \u22c3 y \u2208 s, measurableAtom y := by\n      apply Subset.antisymm\n      \u00b7 intro x hx\n        simpa using \u27e8x, hx, by simp\u27e9\n      \u00b7 simp only [iUnion_subset_iff]\n        intro x hx\n        exact measurableAtom_subset hs hx\n    rw [this]\n    apply MeasurableSet.biUnion (to_countable s) (fun x _hx \u21a6 ?_)\n    apply measurableSet_generateFrom\n    simp\n\ninstance [MeasurableSpace \u03b1] [CountablyGenerated \u03b1] {p : \u03b1 \u2192 Prop} :\n    CountablyGenerated { x // p x } := .comap _\n\ninstance [MeasurableSpace \u03b1] [CountablyGenerated \u03b1] [MeasurableSpace \u03b2] [CountablyGenerated \u03b2] :\n    CountablyGenerated (\u03b1 \u00d7 \u03b2) :=\n  .sup (.comap Prod.fst) (.comap Prod.snd)\n\nsection SeparatesPoints\n\n/-- We say that a measurable space separates points if for any two distinct points,\nthere is a measurable set containing one but not the other. -/\nclass SeparatesPoints (\u03b1 : Type*) [m : MeasurableSpace \u03b1] : Prop where\n  separates : \u2200 x y : \u03b1, (\u2200 s, MeasurableSet s \u2192 (x \u2208 s \u2192 y \u2208 s)) \u2192 x = y\n\ntheorem separatesPoints_def [MeasurableSpace \u03b1] [hs : SeparatesPoints \u03b1] {x y : \u03b1}\n    (h : \u2200 s, MeasurableSet s \u2192 (x \u2208 s \u2192 y \u2208 s)) : x = y := hs.separates _ _ h\n\ntheorem exists_measurableSet_of_ne [MeasurableSpace \u03b1] [SeparatesPoints \u03b1] {x y : \u03b1}\n    (h : x \u2260 y) : \u2203 s, MeasurableSet s \u2227 x \u2208 s \u2227 y \u2209 s := by\n  contrapose! h\n  exact separatesPoints_def h\n\ntheorem separatesPoints_iff [MeasurableSpace \u03b1] : SeparatesPoints \u03b1 \u2194\n    \u2200 x y : \u03b1, (\u2200 s, MeasurableSet s \u2192 (x \u2208 s \u2194 y \u2208 s)) \u2192 x = y :=\n  \u27e8fun h => fun _ _ hxy => h.separates _ _ fun _ hs xs => (hxy _ hs).mp xs,\n    fun h => \u27e8fun _ _ hxy => h _ _ fun _ hs =>\n    \u27e8fun xs => hxy _ hs xs, not_imp_not.mp fun xs => hxy _ hs.compl xs\u27e9\u27e9\u27e9\n\n/-- If the measurable space generated by `S` separates points,\nthen this is witnessed by sets in `S`. -/\ntheorem separating_of_generateFrom (S : Set (Set \u03b1))\n    [h : @SeparatesPoints \u03b1 (generateFrom S)] :\n    \u2200 x y : \u03b1, (\u2200 s \u2208 S, x \u2208 s \u2194 y \u2208 s) \u2192 x = y := by\n  letI := generateFrom S\n  intros x y hxy\n  rw [\u2190 forall_generateFrom_mem_iff_mem_iff] at hxy\n  exact separatesPoints_def $ fun _ hs => (hxy _ hs).mp\n\ntheorem SeparatesPoints.mono {m m' : MeasurableSpace \u03b1} [hsep : @SeparatesPoints _ m] (h : m \u2264 m') :\n    @SeparatesPoints _ m' := @SeparatesPoints.mk _ m' fun _ _ hxy \u21a6\n    @SeparatesPoints.separates _ m hsep _ _ fun _ hs \u21a6 hxy _ (h _ hs)\n\ninstance (priority := 100) Subtype.separatesPoints [MeasurableSpace \u03b1] [h : SeparatesPoints \u03b1]\n    {s : Set \u03b1} : SeparatesPoints s :=\n  \u27e8fun _ _ hxy \u21a6 Subtype.val_injective $ h.1 _ _ fun _ ht \u21a6 hxy _ $ measurable_subtype_coe ht\u27e9\n\ninstance (priority := 100) separatesPoints_of_measurableSingletonClass [MeasurableSpace \u03b1]\n    [MeasurableSingletonClass \u03b1] : SeparatesPoints \u03b1 := by\n  refine \u27e8fun x y h \u21a6 ?_\u27e9\n  specialize h _ (MeasurableSet.singleton x)\n  simp_rw [mem_singleton_iff, forall_true_left] at h\n  exact h.symm\n\ninstance (priority := 100) [MeasurableSpace \u03b1] {s : Set \u03b1}\n    [h : CountablyGenerated s] [SeparatesPoints s] :\n    HasCountableSeparatingOn \u03b1 MeasurableSet s := by\n  suffices HasCountableSeparatingOn s MeasurableSet univ from this.of_subtype fun _ \u21a6 id\n  rcases h.1 with \u27e8b, hbc, hb\u27e9\n  refine \u27e8\u27e8b, hbc, fun t ht \u21a6 hb.symm \u25b8 .basic t ht, ?_\u27e9\u27e9\n  rw [hb] at \u2039SeparatesPoints s\u203a\n  convert separating_of_generateFrom b\n  simp\n\nvariable (\u03b1)\n\n/-- If a measurable space admits a countable separating family of measurable sets,\nthere is a countably generated coarser space which still separates points. -/\ntheorem exists_countablyGenerated_le_of_hasCountableSeparatingOn [m : MeasurableSpace \u03b1]\n    [h : HasCountableSeparatingOn \u03b1 MeasurableSet univ] :\n    \u2203 m' : MeasurableSpace \u03b1, @CountablyGenerated _ m' \u2227 @SeparatesPoints _ m' \u2227 m' \u2264 m := by\n  rcases h.1 with \u27e8b, bct, hbm, hb\u27e9\n  refine \u27e8generateFrom b, ?_, ?_, generateFrom_le hbm\u27e9\n  \u00b7 use b\n  rw [@separatesPoints_iff]\n  exact fun x y hxy => hb _ trivial _ trivial fun _ hs => hxy _ $ measurableSet_generateFrom hs\n\nopen scoped Classical\n\nopen Function\n\n/-- A map from a measurable space to the Cantor space `\u2115 \u2192 Bool` induced by a countable\nsequence of sets generating the measurable space. -/\nnoncomputable\ndef mapNatBool [MeasurableSpace \u03b1] [CountablyGenerated \u03b1] (x : \u03b1) (n : \u2115) :\n    Bool := x \u2208 natGeneratingSequence \u03b1 n\n\ntheorem measurable_mapNatBool [MeasurableSpace \u03b1] [CountablyGenerated \u03b1] :\n    Measurable (mapNatBool \u03b1) := by\n  rw [measurable_pi_iff]\n  refine fun n \u21a6 measurable_to_bool ?_\n  simp only [preimage, mem_singleton_iff, mapNatBool,\n    Bool.decide_iff, setOf_mem_eq]\n  apply measurableSet_natGeneratingSequence\n\n", "theoremStatement": "theorem injective_mapNatBool [MeasurableSpace \u03b1] [CountablyGenerated \u03b1]\n    [SeparatesPoints \u03b1] : Injective (mapNatBool \u03b1) ", "theoremName": "MeasurableSpace.injective_mapNatBool", "fileCreated": {"commit": "e82ee277c7cf0855c96740746702d13f609f3d3d", "date": "2024-03-13"}, "theoremCreated": {"commit": "ac9ad40ffc0c10776f8c8b56909fadd83a9d0848", "date": "2024-03-25"}, "file": "mathlib/Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean", "module": "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "jsonFile": "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated.jsonl", "positionMetadata": {"lineInFile": 221, "tokenPositionInFile": 9821, "theoremPositionInFile": 1}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": 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"Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Embedding.Set", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableInter", "Mathlib.Order.Filter.CountableSeparatingOn"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  intro x y hxy\n  rw [\u2190 generateFrom_natGeneratingSequence \u03b1] at *\n  apply separating_of_generateFrom (range (natGeneratingSequence _))\n  rintro - \u27e8n, rfl\u27e9\n  rw [\u2190 decide_eq_decide]\n  exact congr_fun hxy n", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 211}}
+{"srcContext": "/-\nCopyright (c) 2024 Jujian Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jujian Zhang\n-/\n\nimport Mathlib.LinearAlgebra.PiTensorProduct\nimport Mathlib.Algebra.Algebra.Bilinear\nimport Mathlib.Algebra.Algebra.Equiv\nimport Mathlib.Data.Finset.NoncommProd\n\n/-!\n# Tensor product of `R`-algebras and rings\n\nIf `(A\u1d62)` is a family of `R`-algebras then the `R`-tensor product `\u2a02\u1d62 A\u1d62` is an `R`-algebra as well\nwith structure map defined by `r \u21a6 r \u2022 1`.\n\nIn particular if we take `R` to be `\u2124`, then this collapses into the tensor product of rings.\n-/\n\nopen TensorProduct Function\n\nvariable {\u03b9 R' R : Type*} {A : \u03b9 \u2192 Type*}\n\nnamespace PiTensorProduct\n\nnoncomputable section AddCommMonoidWithOne\n\nvariable [CommSemiring R] [\u2200 i, AddCommMonoidWithOne (A i)] [\u2200 i, Module R (A i)]\n\ninstance instOne : One (\u2a02[R] i, A i) where\n  one := tprod R 1\n\nlemma one_def : 1 = tprod R (1 : \u03a0 i, A i) := rfl\n\ninstance instAddCommMonoidWithOne : AddCommMonoidWithOne (\u2a02[R] i, A i) where\n  __ := inferInstanceAs (AddCommMonoid (\u2a02[R] i, A i))\n  __ := instOne\n\nend AddCommMonoidWithOne\n\nnoncomputable section NonUnitalNonAssocSemiring\n\nvariable [CommSemiring R] [\u2200 i, NonUnitalNonAssocSemiring (A i)]\nvariable [\u2200 i, Module R (A i)] [\u2200 i, SMulCommClass R (A i) (A i)] [\u2200 i, IsScalarTower R (A i) (A i)]\n\nattribute [aesop safe] mul_add mul_smul_comm smul_mul_assoc add_mul in\n/--\nThe multiplication in tensor product of rings is induced by `(x\u1d62) * (y\u1d62) = (x\u1d62 * y\u1d62)`\n-/\ndef mul : (\u2a02[R] i, A i) \u2192\u2097[R] (\u2a02[R] i, A i) \u2192\u2097[R] (\u2a02[R] i, A i) :=\n  PiTensorProduct.piTensorHomMap\u2082 <| tprod R fun _ \u21a6 LinearMap.mul _ _\n\n@[simp] lemma mul_tprod_tprod (x y : (i : \u03b9) \u2192 A i) :\n    mul (tprod R x) (tprod R y) = tprod R (x * y) := by\n  simp only [mul, piTensorHomMap\u2082_tprod_tprod_tprod, LinearMap.mul_apply']\n  rfl\n\ninstance instMul : Mul (\u2a02[R] i, A i) where\n  mul x y := mul x y\n\nlemma mul_def (x y : \u2a02[R] i, A i) : x * y = mul x y := rfl\n\n@[simp] lemma tprod_mul_tprod (x y : (i : \u03b9) \u2192 A i) :\n    tprod R x * tprod R y = tprod R (x * y) :=\n  mul_tprod_tprod x y\n\ntheorem _root_.SemiconjBy.tprod {a\u2081 a\u2082 a\u2083 : \u03a0 i, A i}\n    (ha : SemiconjBy a\u2081 a\u2082 a\u2083) :\n    SemiconjBy (tprod R a\u2081) (tprod R a\u2082) (tprod R a\u2083) := by\n  rw [SemiconjBy, tprod_mul_tprod, tprod_mul_tprod, ha]\n\nnonrec theorem _root_.Commute.tprod {a\u2081 a\u2082 : \u03a0 i, A i} (ha : Commute a\u2081 a\u2082) :\n    Commute (tprod R a\u2081) (tprod R a\u2082) :=\n  ha.tprod\n\nlemma smul_tprod_mul_smul_tprod (r s : R) (x y : \u03a0 i, A i) :\n    (r \u2022 tprod R x) * (s \u2022 tprod R y) = (r * s) \u2022 tprod R (x * y) := by\n  change mul _ _ = _\n  rw [map_smul, map_smul, mul_comm r s, mul_smul]\n  simp only [LinearMap.smul_apply, mul_tprod_tprod]\n\ninstance instNonUnitalNonAssocSemiring : NonUnitalNonAssocSemiring (\u2a02[R] i, A i) where\n  __ := instMul\n  __ := inferInstanceAs (AddCommMonoid (\u2a02[R] i, A i))\n  left_distrib _ _ _ := (mul _).map_add _ _\n  right_distrib _ _ _ := mul.map_add\u2082 _ _ _\n  zero_mul _ := mul.map_zero\u2082 _\n  mul_zero _ := map_zero (mul _)\n\nend NonUnitalNonAssocSemiring\n\nnoncomputable section NonAssocSemiring\n\nvariable [CommSemiring R] [\u2200 i, NonAssocSemiring (A i)]\nvariable [\u2200 i, Module R (A i)] [\u2200 i, SMulCommClass R (A i) (A i)] [\u2200 i, IsScalarTower R (A i) (A i)]\n\nprotected lemma one_mul (x : \u2a02[R] i, A i) : mul (tprod R 1) x = x := by\n  induction x using PiTensorProduct.induction_on with\n  | smul_tprod => simp\n  | add _ _ h1 h2 => simp [map_add, h1, h2]\n\nprotected lemma mul_one (x : \u2a02[R] i, A i) : mul x (tprod R 1) = x := by\n  induction x using PiTensorProduct.induction_on with\n  | smul_tprod => simp\n  | add _ _ h1 h2 => simp [h1, h2]\n\ninstance instNonAssocSemiring : NonAssocSemiring (\u2a02[R] i, A i) where\n  __ := instNonUnitalNonAssocSemiring\n  one_mul := PiTensorProduct.one_mul\n  mul_one := PiTensorProduct.mul_one\n\nvariable (R) in\n/-- `PiTensorProduct.tprod` as a `MonoidHom`. -/\n@[simps]\ndef tprodMonoidHom : (\u03a0 i, A i) \u2192* \u2a02[R] i, A i where\n  toFun := tprod R\n  map_one' := rfl\n  map_mul' x y := (tprod_mul_tprod x y).symm\n\nend NonAssocSemiring\n\nnoncomputable section NonUnitalSemiring\n\nvariable [CommSemiring R] [\u2200 i, NonUnitalSemiring (A i)]\nvariable [\u2200 i, Module R (A i)] [\u2200 i, SMulCommClass R (A i) (A i)] [\u2200 i, IsScalarTower R (A i) (A i)]\n\nprotected lemma mul_assoc (x y z : \u2a02[R] i, A i) : mul (mul x y) z = mul x (mul y z) := by\n  -- restate as an equality of morphisms so that we can use `ext`\n  suffices LinearMap.llcomp R _ _ _ mul \u2218\u2097 mul =\n      (LinearMap.llcomp R _ _ _ LinearMap.lflip <| LinearMap.llcomp R _ _ _ mul.flip \u2218\u2097 mul).flip by\n    exact DFunLike.congr_fun (DFunLike.congr_fun (DFunLike.congr_fun this x) y) z\n  ext x y z\n  dsimp [\u2190 mul_def]\n  simpa only [tprod_mul_tprod] using congr_arg (tprod R) (mul_assoc x y z)\n\ninstance instNonUnitalSemiring : NonUnitalSemiring (\u2a02[R] i, A i) where\n  __ := instNonUnitalNonAssocSemiring\n  mul_assoc := PiTensorProduct.mul_assoc\n\nend NonUnitalSemiring\n\nnoncomputable section Semiring\n\nvariable [CommSemiring R'] [CommSemiring R] [\u2200 i, Semiring (A i)]\nvariable [Algebra R' R] [\u2200 i, Algebra R (A i)] [\u2200 i, Algebra R' (A i)]\nvariable [\u2200 i, IsScalarTower R' R (A i)]\n\ninstance instSemiring : Semiring (\u2a02[R] i, A i) where\n  __ := instNonUnitalSemiring\n  __ := instNonAssocSemiring\n\ninstance instAlgebra : Algebra R' (\u2a02[R] i, A i) where\n  __ := hasSMul'\n  toFun := (\u00b7 \u2022 1)\n  map_one' := by simp\n  map_mul' r s := show (r * s) \u2022 1 = mul (r \u2022 1) (s \u2022 1)  by\n    rw [LinearMap.map_smul_of_tower, LinearMap.map_smul_of_tower, LinearMap.smul_apply, mul_comm,\n      mul_smul]\n    congr\n    show (1 : \u2a02[R] i, A i) = 1 * 1\n    rw [mul_one]\n  map_zero' := by simp\n  map_add' := by simp [add_smul]\n  commutes' r x := by\n    simp only [RingHom.coe_mk, MonoidHom.coe_mk, OneHom.coe_mk]\n    change mul _ _ = mul _ _\n    rw [LinearMap.map_smul_of_tower, LinearMap.map_smul_of_tower, LinearMap.smul_apply]\n    change r \u2022 (1 * x) = r \u2022 (x * 1)\n    rw [mul_one, one_mul]\n  smul_def' r x := by\n    simp only [RingHom.coe_mk, MonoidHom.coe_mk, OneHom.coe_mk]\n    change _ = mul _ _\n    rw [LinearMap.map_smul_of_tower, LinearMap.smul_apply]\n    change _ = r \u2022 (1 * x)\n    rw [one_mul]\n\nlemma algebraMap_apply (r : R') (i : \u03b9) [DecidableEq \u03b9] :\n    algebraMap R' (\u2a02[R] i, A i) r = tprod R (Pi.mulSingle i (algebraMap R' (A i) r)) := by\n  change r \u2022 tprod R 1 = _\n  have : Pi.mulSingle i (algebraMap R' (A i) r) = update (fun i \u21a6 1) i (r \u2022 1) := by\n    rw [Algebra.algebraMap_eq_smul_one]; rfl\n  rw [this, \u2190 smul_one_smul R r (1 : A i), MultilinearMap.map_smul, update_eq_self, smul_one_smul]\n  congr\n\n/--\nThe map `A\u1d62 \u27f6 \u2a02\u1d62 A\u1d62` given by `a \u21a6 1 \u2297 ... \u2297 a \u2297 1 \u2297 ...`\n-/\n@[simps]\ndef singleAlgHom [DecidableEq \u03b9] (i : \u03b9) : A i \u2192\u2090[R] \u2a02[R] i, A i where\n  toFun a := tprod R (MonoidHom.single _ i a)\n  map_one' := by simp only [_root_.map_one]; rfl\n  map_mul' a a' := by simp\n  map_zero' := MultilinearMap.map_update_zero _ _ _\n  map_add' _ _ := MultilinearMap.map_add _ _ _ _ _\n  commutes' r := show tprodCoeff R _ _ = r \u2022 tprodCoeff R _ _ by\n    rw [Algebra.algebraMap_eq_smul_one]\n    erw [smul_tprodCoeff]\n    rfl\n\n/--\nLifting a multilinear map to an algebra homomorphism from tensor product\n-/\n@[simps!]\ndef liftAlgHom {S : Type*} [Semiring S] [Algebra R S]\n    (f : MultilinearMap R A S)\n    (one : f 1 = 1) (mul : \u2200 x y, f (x * y) = f x * f y) : (\u2a02[R] i, A i) \u2192\u2090[R] S :=\n  AlgHom.ofLinearMap (lift f) (show lift f (tprod R 1) = 1 by simp [one]) <|\n    LinearMap.map_mul_iff _ |>.mpr <| by aesop\n\n@[simp] lemma tprod_noncommProd {\u03ba : Type*} (s : Finset \u03ba) (x : \u03ba \u2192 \u03a0 i, A i) (hx) :\n    tprod R (s.noncommProd x hx) = s.noncommProd (fun k => tprod R (x k))\n      (hx.imp fun _ _ => Commute.tprod) :=\n  Finset.noncommProd_map s x _ (tprodMonoidHom R)\n\n/-- To show two algebra morphisms from finite tensor products are equal, it suffices to show that\nthey agree on elements of the form $1 \u2297 \u22ef \u2297 a \u2297 1 \u2297 \u22ef$. -/\n@[ext high]\ntheorem algHom_ext {S : Type*} [Finite \u03b9] [DecidableEq \u03b9] [Semiring S] [Algebra R S]\n    \u2983f g : (\u2a02[R] i, A i) \u2192\u2090[R] S\u2984 (h : \u2200 i, f.comp (singleAlgHom i) = g.comp (singleAlgHom i)) :\n    f = g :=\n  AlgHom.toLinearMap_injective <| PiTensorProduct.ext <| MultilinearMap.ext fun x =>\n    suffices f.toMonoidHom.comp (tprodMonoidHom R) = g.toMonoidHom.comp (tprodMonoidHom R) from\n      DFunLike.congr_fun this x\n    MonoidHom.pi_ext fun i xi => DFunLike.congr_fun (h i) xi\n\nend Semiring\n\nnoncomputable section Ring\n\nvariable [CommRing R] [\u2200 i, Ring (A i)] [\u2200 i, Algebra R (A i)]\n\ninstance instRing : Ring (\u2a02[R] i, A i) where\n  __ := instSemiring\n  __ := inferInstanceAs <| AddCommGroup (\u2a02[R] i, A i)\n\nend Ring\n\nnoncomputable section CommSemiring\n\nvariable [CommSemiring R] [\u2200 i, CommSemiring (A i)] [\u2200 i, Algebra R (A i)]\n\nprotected lemma mul_comm (x y : \u2a02[R] i, A i) : mul x y = mul y x := by\n  suffices mul (R := R) (A := A) = mul.flip from\n    DFunLike.congr_fun (DFunLike.congr_fun this x) y\n  ext x y\n  dsimp\n  simp only [mul_tprod_tprod, mul_tprod_tprod, mul_comm x y]\n\ninstance instCommSemiring : CommSemiring (\u2a02[R] i, A i) where\n  __ := instSemiring\n  __ := inferInstanceAs <| AddCommMonoid (\u2a02[R] i, A i)\n  mul_comm := PiTensorProduct.mul_comm\n\nopen scoped BigOperators in\n@[simp] lemma tprod_prod {\u03ba : Type*} (s : Finset \u03ba) (x : \u03ba \u2192 \u03a0 i, A i) :\n    tprod R (\u220f k in s, x k) = \u220f k in s, tprod R (x k) :=\n  map_prod (tprodMonoidHom R) x s\n\nsection\n\nopen BigOperators Function\n\nvariable [Fintype \u03b9]\n\nvariable (R \u03b9)\n\n/--\nThe algebra equivalence from the tensor product of the constant family with\nvalue `R` to `R`, given by multiplication of the entries.\n-/\nnoncomputable def constantBaseRingEquiv : (\u2a02[R] _ : \u03b9, R) \u2243\u2090[R] R :=\n  letI toFun := lift (MultilinearMap.mkPiAlgebra R \u03b9 R)\n  AlgEquiv.ofAlgHom\n    (AlgHom.ofLinearMap\n      toFun\n      ((lift.tprod _).trans Finset.prod_const_one)\n      (by\n        rw [LinearMap.map_mul_iff]\n        ext x y\n        show toFun (tprod R x * tprod R y) = toFun (tprod R x) * toFun (tprod R y)\n        simp_rw [tprod_mul_tprod, toFun, lift.tprod, MultilinearMap.mkPiAlgebra_apply,\n          Pi.mul_apply, Finset.prod_mul_distrib]))\n    (Algebra.ofId _ _)\n    (by ext)\n    (by classical ext)\n\nvariable {R \u03b9}\n\n", "theoremStatement": "@[simp]\ntheorem constantBaseRingEquiv_tprod (x : \u03b9 \u2192 R) :\n    constantBaseRingEquiv \u03b9 R (tprod R x) = \u220f i, x i ", "theoremName": "PiTensorProduct.constantBaseRingEquiv_tprod", "fileCreated": {"commit": "fb2570a4660a748bc85728de1b95619043ea8b15", "date": "2024-03-02"}, "theoremCreated": {"commit": "f2f4854ba1c616fc736419ad1254f15ccfb27a6f", "date": "2024-03-23"}, "file": "mathlib/Mathlib/RingTheory/PiTensorProduct.lean", "module": "Mathlib.RingTheory.PiTensorProduct", "jsonFile": "Mathlib.RingTheory.PiTensorProduct.jsonl", "positionMetadata": {"lineInFile": 290, "tokenPositionInFile": 10058, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, 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"Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Relation", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Data.Set.Prod", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.MinMax", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Nat", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Logic.Pairwise", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Fintype.Sort", "Mathlib.Data.List.FinRange", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.LinearAlgebra.Multilinear.TensorProduct", "Mathlib.LinearAlgebra.PiTensorProduct", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  simp [constantBaseRingEquiv]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 36}}
+{"srcContext": "/-\nCopyright (c) 2021 Kyle Miller. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kyle Miller\n-/\nimport Mathlib.Combinatorics.SimpleGraph.Subgraph\nimport Mathlib.Data.List.Rotate\n\n#align_import combinatorics.simple_graph.connectivity from \"leanprover-community/mathlib\"@\"b99e2d58a5e6861833fa8de11e51a81144258db4\"\n\n/-!\n\n# Graph connectivity\n\nIn a simple graph,\n\n* A *walk* is a finite sequence of adjacent vertices, and can be\n  thought of equally well as a sequence of directed edges.\n\n* A *trail* is a walk whose edges each appear no more than once.\n\n* A *path* is a trail whose vertices appear no more than once.\n\n* A *cycle* is a nonempty trail whose first and last vertices are the\n  same and whose vertices except for the first appear no more than once.\n\n**Warning:** graph theorists mean something different by \"path\" than\ndo homotopy theorists.  A \"walk\" in graph theory is a \"path\" in\nhomotopy theory.  Another warning: some graph theorists use \"path\" and\n\"simple path\" for \"walk\" and \"path.\"\n\nSome definitions and theorems have inspiration from multigraph\ncounterparts in [Chou1994].\n\n## Main definitions\n\n* `SimpleGraph.Walk` (with accompanying pattern definitions\n  `SimpleGraph.Walk.nil'` and `SimpleGraph.Walk.cons'`)\n\n* `SimpleGraph.Walk.IsTrail`, `SimpleGraph.Walk.IsPath`, and `SimpleGraph.Walk.IsCycle`.\n\n* `SimpleGraph.Path`\n\n* `SimpleGraph.Walk.map` and `SimpleGraph.Path.map` for the induced map on walks,\n  given an (injective) graph homomorphism.\n\n* `SimpleGraph.Reachable` for the relation of whether there exists\n  a walk between a given pair of vertices\n\n* `SimpleGraph.Preconnected` and `SimpleGraph.Connected` are predicates\n  on simple graphs for whether every vertex can be reached from every other,\n  and in the latter case, whether the vertex type is nonempty.\n\n* `SimpleGraph.ConnectedComponent` is the type of connected components of\n  a given graph.\n\n* `SimpleGraph.IsBridge` for whether an edge is a bridge edge\n\n## Main statements\n\n* `SimpleGraph.isBridge_iff_mem_and_forall_cycle_not_mem` characterizes bridge edges in terms of\n  there being no cycle containing them.\n\n## Tags\nwalks, trails, paths, circuits, cycles, bridge edges\n\n-/\n\nopen Function\n\nuniverse u v w\n\nnamespace SimpleGraph\n\nvariable {V : Type u} {V' : Type v} {V'' : Type w}\nvariable (G : SimpleGraph V) (G' : SimpleGraph V') (G'' : SimpleGraph V'')\n\n/-- A walk is a sequence of adjacent vertices.  For vertices `u v : V`,\nthe type `walk u v` consists of all walks starting at `u` and ending at `v`.\n\nWe say that a walk *visits* the vertices it contains.  The set of vertices a\nwalk visits is `SimpleGraph.Walk.support`.\n\nSee `SimpleGraph.Walk.nil'` and `SimpleGraph.Walk.cons'` for patterns that\ncan be useful in definitions since they make the vertices explicit. -/\ninductive Walk : V \u2192 V \u2192 Type u\n  | nil {u : V} : Walk u u\n  | cons {u v w : V} (h : G.Adj u v) (p : Walk v w) : Walk u w\n  deriving DecidableEq\n#align simple_graph.walk SimpleGraph.Walk\n\nattribute [refl] Walk.nil\n\n@[simps]\ninstance Walk.instInhabited (v : V) : Inhabited (G.Walk v v) := \u27e8Walk.nil\u27e9\n#align simple_graph.walk.inhabited SimpleGraph.Walk.instInhabited\n\n/-- The one-edge walk associated to a pair of adjacent vertices. -/\n@[match_pattern, reducible]\ndef Adj.toWalk {G : SimpleGraph V} {u v : V} (h : G.Adj u v) : G.Walk u v :=\n  Walk.cons h Walk.nil\n#align simple_graph.adj.to_walk SimpleGraph.Adj.toWalk\n\nnamespace Walk\n\nvariable {G}\n\n/-- Pattern to get `Walk.nil` with the vertex as an explicit argument. -/\n@[match_pattern]\nabbrev nil' (u : V) : G.Walk u u := Walk.nil\n#align simple_graph.walk.nil' SimpleGraph.Walk.nil'\n\n/-- Pattern to get `Walk.cons` with the vertices as explicit arguments. -/\n@[match_pattern]\nabbrev cons' (u v w : V) (h : G.Adj u v) (p : G.Walk v w) : G.Walk u w := Walk.cons h p\n#align simple_graph.walk.cons' SimpleGraph.Walk.cons'\n\n/-- Change the endpoints of a walk using equalities. This is helpful for relaxing\ndefinitional equality constraints and to be able to state otherwise difficult-to-state\nlemmas. While this is a simple wrapper around `Eq.rec`, it gives a canonical way to write it.\n\nThe simp-normal form is for the `copy` to be pushed outward. That way calculations can\noccur within the \"copy context.\" -/\nprotected def copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') : G.Walk u' v' :=\n  hu \u25b8 hv \u25b8 p\n#align simple_graph.walk.copy SimpleGraph.Walk.copy\n\n@[simp]\ntheorem copy_rfl_rfl {u v} (p : G.Walk u v) : p.copy rfl rfl = p := rfl\n#align simple_graph.walk.copy_rfl_rfl SimpleGraph.Walk.copy_rfl_rfl\n\n@[simp]\ntheorem copy_copy {u v u' v' u'' v''} (p : G.Walk u v)\n    (hu : u = u') (hv : v = v') (hu' : u' = u'') (hv' : v' = v'') :\n    (p.copy hu hv).copy hu' hv' = p.copy (hu.trans hu') (hv.trans hv') := by\n  subst_vars\n  rfl\n#align simple_graph.walk.copy_copy SimpleGraph.Walk.copy_copy\n\n@[simp]\ntheorem copy_nil {u u'} (hu : u = u') : (Walk.nil : G.Walk u u).copy hu hu = Walk.nil := by\n  subst_vars\n  rfl\n#align simple_graph.walk.copy_nil SimpleGraph.Walk.copy_nil\n\ntheorem copy_cons {u v w u' w'} (h : G.Adj u v) (p : G.Walk v w) (hu : u = u') (hw : w = w') :\n    (Walk.cons h p).copy hu hw = Walk.cons (hu \u25b8 h) (p.copy rfl hw) := by\n  subst_vars\n  rfl\n#align simple_graph.walk.copy_cons SimpleGraph.Walk.copy_cons\n\n@[simp]\ntheorem cons_copy {u v w v' w'} (h : G.Adj u v) (p : G.Walk v' w') (hv : v' = v) (hw : w' = w) :\n    Walk.cons h (p.copy hv hw) = (Walk.cons (hv \u25b8 h) p).copy rfl hw := by\n  subst_vars\n  rfl\n#align simple_graph.walk.cons_copy SimpleGraph.Walk.cons_copy\n\ntheorem exists_eq_cons_of_ne {u v : V} (hne : u \u2260 v) :\n    \u2200 (p : G.Walk u v), \u2203 (w : V) (h : G.Adj u w) (p' : G.Walk w v), p = cons h p'\n  | nil => (hne rfl).elim\n  | cons h p' => \u27e8_, h, p', rfl\u27e9\n#align simple_graph.walk.exists_eq_cons_of_ne SimpleGraph.Walk.exists_eq_cons_of_ne\n\n/-- The length of a walk is the number of edges/darts along it. -/\ndef length {u v : V} : G.Walk u v \u2192 \u2115\n  | nil => 0\n  | cons _ q => q.length.succ\n#align simple_graph.walk.length SimpleGraph.Walk.length\n\n/-- The concatenation of two compatible walks. -/\n@[trans]\ndef append {u v w : V} : G.Walk u v \u2192 G.Walk v w \u2192 G.Walk u w\n  | nil, q => q\n  | cons h p, q => cons h (p.append q)\n#align simple_graph.walk.append SimpleGraph.Walk.append\n\n/-- The reversed version of `SimpleGraph.Walk.cons`, concatenating an edge to\nthe end of a walk. -/\ndef concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) : G.Walk u w := p.append (cons h nil)\n#align simple_graph.walk.concat SimpleGraph.Walk.concat\n\ntheorem concat_eq_append {u v w : V} (p : G.Walk u v) (h : G.Adj v w) :\n    p.concat h = p.append (cons h nil) := rfl\n#align simple_graph.walk.concat_eq_append SimpleGraph.Walk.concat_eq_append\n\n/-- The concatenation of the reverse of the first walk with the second walk. -/\nprotected def reverseAux {u v w : V} : G.Walk u v \u2192 G.Walk u w \u2192 G.Walk v w\n  | nil, q => q\n  | cons h p, q => Walk.reverseAux p (cons (G.symm h) q)\n#align simple_graph.walk.reverse_aux SimpleGraph.Walk.reverseAux\n\n/-- The walk in reverse. -/\n@[symm]\ndef reverse {u v : V} (w : G.Walk u v) : G.Walk v u := w.reverseAux nil\n#align simple_graph.walk.reverse SimpleGraph.Walk.reverse\n\n/-- Get the `n`th vertex from a walk, where `n` is generally expected to be\nbetween `0` and `p.length`, inclusive.\nIf `n` is greater than or equal to `p.length`, the result is the path's endpoint. -/\ndef getVert {u v : V} : G.Walk u v \u2192 \u2115 \u2192 V\n  | nil, _ => u\n  | cons _ _, 0 => u\n  | cons _ q, n + 1 => q.getVert n\n#align simple_graph.walk.get_vert SimpleGraph.Walk.getVert\n\n@[simp]\ntheorem getVert_zero {u v} (w : G.Walk u v) : w.getVert 0 = u := by cases w <;> rfl\n#align simple_graph.walk.get_vert_zero SimpleGraph.Walk.getVert_zero\n\ntheorem getVert_of_length_le {u v} (w : G.Walk u v) {i : \u2115} (hi : w.length \u2264 i) :\n    w.getVert i = v := by\n  induction w generalizing i with\n  | nil => rfl\n  | cons _ _ ih =>\n    cases i\n    \u00b7 cases hi\n    \u00b7 exact ih (Nat.succ_le_succ_iff.1 hi)\n#align simple_graph.walk.get_vert_of_length_le SimpleGraph.Walk.getVert_of_length_le\n\n@[simp]\ntheorem getVert_length {u v} (w : G.Walk u v) : w.getVert w.length = v :=\n  w.getVert_of_length_le rfl.le\n#align simple_graph.walk.get_vert_length SimpleGraph.Walk.getVert_length\n\ntheorem adj_getVert_succ {u v} (w : G.Walk u v) {i : \u2115} (hi : i < w.length) :\n    G.Adj (w.getVert i) (w.getVert (i + 1)) := by\n  induction w generalizing i with\n  | nil => cases hi\n  | cons hxy _ ih =>\n    cases i\n    \u00b7 simp [getVert, hxy]\n    \u00b7 exact ih (Nat.succ_lt_succ_iff.1 hi)\n#align simple_graph.walk.adj_get_vert_succ SimpleGraph.Walk.adj_getVert_succ\n\n@[simp]\ntheorem cons_append {u v w x : V} (h : G.Adj u v) (p : G.Walk v w) (q : G.Walk w x) :\n    (cons h p).append q = cons h (p.append q) := rfl\n#align simple_graph.walk.cons_append SimpleGraph.Walk.cons_append\n\n@[simp]\ntheorem cons_nil_append {u v w : V} (h : G.Adj u v) (p : G.Walk v w) :\n    (cons h nil).append p = cons h p := rfl\n#align simple_graph.walk.cons_nil_append SimpleGraph.Walk.cons_nil_append\n\n@[simp]\ntheorem append_nil {u v : V} (p : G.Walk u v) : p.append nil = p := by\n  induction p with\n  | nil => rfl\n  | cons _ _ ih => rw [cons_append, ih]\n#align simple_graph.walk.append_nil SimpleGraph.Walk.append_nil\n\n@[simp]\ntheorem nil_append {u v : V} (p : G.Walk u v) : nil.append p = p :=\n  rfl\n#align simple_graph.walk.nil_append SimpleGraph.Walk.nil_append\n\ntheorem append_assoc {u v w x : V} (p : G.Walk u v) (q : G.Walk v w) (r : G.Walk w x) :\n    p.append (q.append r) = (p.append q).append r := by\n  induction p with\n  | nil => rfl\n  | cons h p' ih =>\n    dsimp only [append]\n    rw [ih]\n#align simple_graph.walk.append_assoc SimpleGraph.Walk.append_assoc\n\n@[simp]\ntheorem append_copy_copy {u v w u' v' w'} (p : G.Walk u v) (q : G.Walk v w)\n    (hu : u = u') (hv : v = v') (hw : w = w') :\n    (p.copy hu hv).append (q.copy hv hw) = (p.append q).copy hu hw := by\n  subst_vars\n  rfl\n#align simple_graph.walk.append_copy_copy SimpleGraph.Walk.append_copy_copy\n\ntheorem concat_nil {u v : V} (h : G.Adj u v) : nil.concat h = cons h nil := rfl\n#align simple_graph.walk.concat_nil SimpleGraph.Walk.concat_nil\n\n@[simp]\ntheorem concat_cons {u v w x : V} (h : G.Adj u v) (p : G.Walk v w) (h' : G.Adj w x) :\n    (cons h p).concat h' = cons h (p.concat h') := rfl\n#align simple_graph.walk.concat_cons SimpleGraph.Walk.concat_cons\n\ntheorem append_concat {u v w x : V} (p : G.Walk u v) (q : G.Walk v w) (h : G.Adj w x) :\n    p.append (q.concat h) = (p.append q).concat h := append_assoc _ _ _\n#align simple_graph.walk.append_concat SimpleGraph.Walk.append_concat\n\ntheorem concat_append {u v w x : V} (p : G.Walk u v) (h : G.Adj v w) (q : G.Walk w x) :\n    (p.concat h).append q = p.append (cons h q) := by\n  rw [concat_eq_append, \u2190 append_assoc, cons_nil_append]\n#align simple_graph.walk.concat_append SimpleGraph.Walk.concat_append\n\n/-- A non-trivial `cons` walk is representable as a `concat` walk. -/\ntheorem exists_cons_eq_concat {u v w : V} (h : G.Adj u v) (p : G.Walk v w) :\n    \u2203 (x : V) (q : G.Walk u x) (h' : G.Adj x w), cons h p = q.concat h' := by\n  induction p generalizing u with\n  | nil => exact \u27e8_, nil, h, rfl\u27e9\n  | cons h' p ih =>\n    obtain \u27e8y, q, h'', hc\u27e9 := ih h'\n    refine' \u27e8y, cons h q, h'', _\u27e9\n    rw [concat_cons, hc]\n#align simple_graph.walk.exists_cons_eq_concat SimpleGraph.Walk.exists_cons_eq_concat\n\n/-- A non-trivial `concat` walk is representable as a `cons` walk. -/\ntheorem exists_concat_eq_cons {u v w : V} :\n    \u2200 (p : G.Walk u v) (h : G.Adj v w),\n      \u2203 (x : V) (h' : G.Adj u x) (q : G.Walk x w), p.concat h = cons h' q\n  | nil, h => \u27e8_, h, nil, rfl\u27e9\n  | cons h' p, h => \u27e8_, h', Walk.concat p h, concat_cons _ _ _\u27e9\n#align simple_graph.walk.exists_concat_eq_cons SimpleGraph.Walk.exists_concat_eq_cons\n\n@[simp]\ntheorem reverse_nil {u : V} : (nil : G.Walk u u).reverse = nil := rfl\n#align simple_graph.walk.reverse_nil SimpleGraph.Walk.reverse_nil\n\ntheorem reverse_singleton {u v : V} (h : G.Adj u v) : (cons h nil).reverse = cons (G.symm h) nil :=\n  rfl\n#align simple_graph.walk.reverse_singleton SimpleGraph.Walk.reverse_singleton\n\n@[simp]\ntheorem cons_reverseAux {u v w x : V} (p : G.Walk u v) (q : G.Walk w x) (h : G.Adj w u) :\n    (cons h p).reverseAux q = p.reverseAux (cons (G.symm h) q) := rfl\n#align simple_graph.walk.cons_reverse_aux SimpleGraph.Walk.cons_reverseAux\n\n@[simp]\nprotected theorem append_reverseAux {u v w x : V}\n    (p : G.Walk u v) (q : G.Walk v w) (r : G.Walk u x) :\n    (p.append q).reverseAux r = q.reverseAux (p.reverseAux r) := by\n  induction p with\n  | nil => rfl\n  | cons h _ ih => exact ih q (cons (G.symm h) r)\n#align simple_graph.walk.append_reverse_aux SimpleGraph.Walk.append_reverseAux\n\n@[simp]\nprotected theorem reverseAux_append {u v w x : V}\n    (p : G.Walk u v) (q : G.Walk u w) (r : G.Walk w x) :\n    (p.reverseAux q).append r = p.reverseAux (q.append r) := by\n  induction p with\n  | nil => rfl\n  | cons h _ ih => simp [ih (cons (G.symm h) q)]\n#align simple_graph.walk.reverse_aux_append SimpleGraph.Walk.reverseAux_append\n\nprotected theorem reverseAux_eq_reverse_append {u v w : V} (p : G.Walk u v) (q : G.Walk u w) :\n    p.reverseAux q = p.reverse.append q := by simp [reverse]\n#align simple_graph.walk.reverse_aux_eq_reverse_append SimpleGraph.Walk.reverseAux_eq_reverse_append\n\n@[simp]\ntheorem reverse_cons {u v w : V} (h : G.Adj u v) (p : G.Walk v w) :\n    (cons h p).reverse = p.reverse.append (cons (G.symm h) nil) := by simp [reverse]\n#align simple_graph.walk.reverse_cons SimpleGraph.Walk.reverse_cons\n\n@[simp]\ntheorem reverse_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') :\n    (p.copy hu hv).reverse = p.reverse.copy hv hu := by\n  subst_vars\n  rfl\n#align simple_graph.walk.reverse_copy SimpleGraph.Walk.reverse_copy\n\n@[simp]\ntheorem reverse_append {u v w : V} (p : G.Walk u v) (q : G.Walk v w) :\n    (p.append q).reverse = q.reverse.append p.reverse := by simp [reverse]\n#align simple_graph.walk.reverse_append SimpleGraph.Walk.reverse_append\n\n@[simp]\ntheorem reverse_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) :\n    (p.concat h).reverse = cons (G.symm h) p.reverse := by simp [concat_eq_append]\n#align simple_graph.walk.reverse_concat SimpleGraph.Walk.reverse_concat\n\n@[simp]\ntheorem reverse_reverse {u v : V} (p : G.Walk u v) : p.reverse.reverse = p := by\n  induction p with\n  | nil => rfl\n  | cons _ _ ih => simp [ih]\n#align simple_graph.walk.reverse_reverse SimpleGraph.Walk.reverse_reverse\n\n@[simp]\ntheorem length_nil {u : V} : (nil : G.Walk u u).length = 0 := rfl\n#align simple_graph.walk.length_nil SimpleGraph.Walk.length_nil\n\n@[simp]\ntheorem length_cons {u v w : V} (h : G.Adj u v) (p : G.Walk v w) :\n    (cons h p).length = p.length + 1 := rfl\n#align simple_graph.walk.length_cons SimpleGraph.Walk.length_cons\n\n@[simp]\ntheorem length_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') :\n    (p.copy hu hv).length = p.length := by\n  subst_vars\n  rfl\n#align simple_graph.walk.length_copy SimpleGraph.Walk.length_copy\n\n@[simp]\ntheorem length_append {u v w : V} (p : G.Walk u v) (q : G.Walk v w) :\n    (p.append q).length = p.length + q.length := by\n  induction p with\n  | nil => simp\n  | cons _ _ ih => simp [ih, add_comm, add_left_comm, add_assoc]\n#align simple_graph.walk.length_append SimpleGraph.Walk.length_append\n\n@[simp]\ntheorem length_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) :\n    (p.concat h).length = p.length + 1 := length_append _ _\n#align simple_graph.walk.length_concat SimpleGraph.Walk.length_concat\n\n@[simp]\nprotected theorem length_reverseAux {u v w : V} (p : G.Walk u v) (q : G.Walk u w) :\n    (p.reverseAux q).length = p.length + q.length := by\n  induction p with\n  | nil => simp!\n  | cons _ _ ih => simp [ih, Nat.add_succ, Nat.succ_add]\n#align simple_graph.walk.length_reverse_aux SimpleGraph.Walk.length_reverseAux\n\n@[simp]\ntheorem length_reverse {u v : V} (p : G.Walk u v) : p.reverse.length = p.length := by simp [reverse]\n#align simple_graph.walk.length_reverse SimpleGraph.Walk.length_reverse\n\ntheorem eq_of_length_eq_zero {u v : V} : \u2200 {p : G.Walk u v}, p.length = 0 \u2192 u = v\n  | nil, _ => rfl\n#align simple_graph.walk.eq_of_length_eq_zero SimpleGraph.Walk.eq_of_length_eq_zero\n\n@[simp]\ntheorem exists_length_eq_zero_iff {u v : V} : (\u2203 p : G.Walk u v, p.length = 0) \u2194 u = v := by\n  constructor\n  \u00b7 rintro \u27e8p, hp\u27e9\n    exact eq_of_length_eq_zero hp\n  \u00b7 rintro rfl\n    exact \u27e8nil, rfl\u27e9\n#align simple_graph.walk.exists_length_eq_zero_iff SimpleGraph.Walk.exists_length_eq_zero_iff\n\n@[simp]\ntheorem length_eq_zero_iff {u : V} {p : G.Walk u u} : p.length = 0 \u2194 p = nil := by cases p <;> simp\n#align simple_graph.walk.length_eq_zero_iff SimpleGraph.Walk.length_eq_zero_iff\n\nsection ConcatRec\n\nvariable {motive : \u2200 u v : V, G.Walk u v \u2192 Sort*} (Hnil : \u2200 {u : V}, motive u u nil)\n  (Hconcat : \u2200 {u v w : V} (p : G.Walk u v) (h : G.Adj v w), motive u v p \u2192 motive u w (p.concat h))\n\n/-- Auxiliary definition for `SimpleGraph.Walk.concatRec` -/\ndef concatRecAux {u v : V} : (p : G.Walk u v) \u2192 motive v u p.reverse\n  | nil => Hnil\n  | cons h p => reverse_cons h p \u25b8 Hconcat p.reverse h.symm (concatRecAux p)\n#align simple_graph.walk.concat_rec_aux SimpleGraph.Walk.concatRecAux\n\n/-- Recursor on walks by inducting on `SimpleGraph.Walk.concat`.\n\nThis is inducting from the opposite end of the walk compared\nto `SimpleGraph.Walk.rec`, which inducts on `SimpleGraph.Walk.cons`. -/\n@[elab_as_elim]\ndef concatRec {u v : V} (p : G.Walk u v) : motive u v p :=\n  reverse_reverse p \u25b8 concatRecAux @Hnil @Hconcat p.reverse\n#align simple_graph.walk.concat_rec SimpleGraph.Walk.concatRec\n\n@[simp]\ntheorem concatRec_nil (u : V) :\n    @concatRec _ _ motive @Hnil @Hconcat _ _ (nil : G.Walk u u) = Hnil := rfl\n#align simple_graph.walk.concat_rec_nil SimpleGraph.Walk.concatRec_nil\n\n@[simp]\ntheorem concatRec_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) :\n    @concatRec _ _ motive @Hnil @Hconcat _ _ (p.concat h) =\n      Hconcat p h (concatRec @Hnil @Hconcat p) := by\n  simp only [concatRec]\n  apply eq_of_heq\n  apply rec_heq_of_heq\n  trans concatRecAux @Hnil @Hconcat (cons h.symm p.reverse)\n  \u00b7 congr\n    simp\n  \u00b7 rw [concatRecAux, rec_heq_iff_heq]\n    congr <;> simp [heq_rec_iff_heq]\n#align simple_graph.walk.concat_rec_concat SimpleGraph.Walk.concatRec_concat\n\nend ConcatRec\n\ntheorem concat_ne_nil {u v : V} (p : G.Walk u v) (h : G.Adj v u) : p.concat h \u2260 nil := by\n  cases p <;> simp [concat]\n#align simple_graph.walk.concat_ne_nil SimpleGraph.Walk.concat_ne_nil\n\ntheorem concat_inj {u v v' w : V} {p : G.Walk u v} {h : G.Adj v w} {p' : G.Walk u v'}\n    {h' : G.Adj v' w} (he : p.concat h = p'.concat h') : \u2203 hv : v = v', p.copy rfl hv = p' := by\n  induction p with\n  | nil =>\n    cases p'\n    \u00b7 exact \u27e8rfl, rfl\u27e9\n    \u00b7 exfalso\n      simp only [concat_nil, concat_cons, cons.injEq] at he\n      obtain \u27e8rfl, he\u27e9 := he\n      simp only [heq_iff_eq] at he\n      exact concat_ne_nil _ _ he.symm\n  | cons _ _ ih =>\n    rw [concat_cons] at he\n    cases p'\n    \u00b7 exfalso\n      simp only [concat_nil, cons.injEq] at he\n      obtain \u27e8rfl, he\u27e9 := he\n      rw [heq_iff_eq] at he\n      exact concat_ne_nil _ _ he\n    \u00b7 rw [concat_cons, cons.injEq] at he\n      obtain \u27e8rfl, he\u27e9 := he\n      rw [heq_iff_eq] at he\n      obtain \u27e8rfl, rfl\u27e9 := ih he\n      exact \u27e8rfl, rfl\u27e9\n#align simple_graph.walk.concat_inj SimpleGraph.Walk.concat_inj\n\n/-- The `support` of a walk is the list of vertices it visits in order. -/\ndef support {u v : V} : G.Walk u v \u2192 List V\n  | nil => [u]\n  | cons _ p => u :: p.support\n#align simple_graph.walk.support SimpleGraph.Walk.support\n\n/-- The `darts` of a walk is the list of darts it visits in order. -/\ndef darts {u v : V} : G.Walk u v \u2192 List G.Dart\n  | nil => []\n  | cons h p => \u27e8(u, _), h\u27e9 :: p.darts\n#align simple_graph.walk.darts SimpleGraph.Walk.darts\n\n/-- The `edges` of a walk is the list of edges it visits in order.\nThis is defined to be the list of edges underlying `SimpleGraph.Walk.darts`. -/\ndef edges {u v : V} (p : G.Walk u v) : List (Sym2 V) := p.darts.map Dart.edge\n#align simple_graph.walk.edges SimpleGraph.Walk.edges\n\n@[simp]\ntheorem support_nil {u : V} : (nil : G.Walk u u).support = [u] := rfl\n#align simple_graph.walk.support_nil SimpleGraph.Walk.support_nil\n\n@[simp]\ntheorem support_cons {u v w : V} (h : G.Adj u v) (p : G.Walk v w) :\n    (cons h p).support = u :: p.support := rfl\n#align simple_graph.walk.support_cons SimpleGraph.Walk.support_cons\n\n@[simp]\ntheorem support_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) :\n    (p.concat h).support = p.support.concat w := by\n  induction p <;> simp [*, concat_nil]\n#align simple_graph.walk.support_concat SimpleGraph.Walk.support_concat\n\n@[simp]\ntheorem support_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') :\n    (p.copy hu hv).support = p.support := by\n  subst_vars\n  rfl\n#align simple_graph.walk.support_copy SimpleGraph.Walk.support_copy\n\ntheorem support_append {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :\n    (p.append p').support = p.support ++ p'.support.tail := by\n  induction p <;> cases p' <;> simp [*]\n#align simple_graph.walk.support_append SimpleGraph.Walk.support_append\n\n@[simp]\ntheorem support_reverse {u v : V} (p : G.Walk u v) : p.reverse.support = p.support.reverse := by\n  induction p <;> simp [support_append, *]\n#align simple_graph.walk.support_reverse SimpleGraph.Walk.support_reverse\n\n@[simp]\ntheorem support_ne_nil {u v : V} (p : G.Walk u v) : p.support \u2260 [] := by cases p <;> simp\n#align simple_graph.walk.support_ne_nil SimpleGraph.Walk.support_ne_nil\n\ntheorem tail_support_append {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :\n    (p.append p').support.tail = p.support.tail ++ p'.support.tail := by\n  rw [support_append, List.tail_append_of_ne_nil _ _ (support_ne_nil _)]\n#align simple_graph.walk.tail_support_append SimpleGraph.Walk.tail_support_append\n\ntheorem support_eq_cons {u v : V} (p : G.Walk u v) : p.support = u :: p.support.tail := by\n  cases p <;> simp\n#align simple_graph.walk.support_eq_cons SimpleGraph.Walk.support_eq_cons\n\n@[simp]\ntheorem start_mem_support {u v : V} (p : G.Walk u v) : u \u2208 p.support := by cases p <;> simp\n#align simple_graph.walk.start_mem_support SimpleGraph.Walk.start_mem_support\n\n@[simp]\ntheorem end_mem_support {u v : V} (p : G.Walk u v) : v \u2208 p.support := by induction p <;> simp [*]\n#align simple_graph.walk.end_mem_support SimpleGraph.Walk.end_mem_support\n\n@[simp]\ntheorem support_nonempty {u v : V} (p : G.Walk u v) : { w | w \u2208 p.support }.Nonempty :=\n  \u27e8u, by simp\u27e9\n#align simple_graph.walk.support_nonempty SimpleGraph.Walk.support_nonempty\n\ntheorem mem_support_iff {u v w : V} (p : G.Walk u v) : w \u2208 p.support \u2194 w = u \u2228 w \u2208 p.support.tail :=\n  by cases p <;> simp\n#align simple_graph.walk.mem_support_iff SimpleGraph.Walk.mem_support_iff\n\ntheorem mem_support_nil_iff {u v : V} : u \u2208 (nil : G.Walk v v).support \u2194 u = v := by simp\n#align simple_graph.walk.mem_support_nil_iff SimpleGraph.Walk.mem_support_nil_iff\n\n@[simp]\ntheorem mem_tail_support_append_iff {t u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :\n    t \u2208 (p.append p').support.tail \u2194 t \u2208 p.support.tail \u2228 t \u2208 p'.support.tail := by\n  rw [tail_support_append, List.mem_append]\n#align simple_graph.walk.mem_tail_support_append_iff SimpleGraph.Walk.mem_tail_support_append_iff\n\n@[simp]\ntheorem end_mem_tail_support_of_ne {u v : V} (h : u \u2260 v) (p : G.Walk u v) : v \u2208 p.support.tail := by\n  obtain \u27e8_, _, _, rfl\u27e9 := exists_eq_cons_of_ne h p\n  simp\n#align simple_graph.walk.end_mem_tail_support_of_ne SimpleGraph.Walk.end_mem_tail_support_of_ne\n\n@[simp, nolint unusedHavesSuffices]\ntheorem mem_support_append_iff {t u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :\n    t \u2208 (p.append p').support \u2194 t \u2208 p.support \u2228 t \u2208 p'.support := by\n  simp only [mem_support_iff, mem_tail_support_append_iff]\n  obtain rfl | h := eq_or_ne t v <;> obtain rfl | h' := eq_or_ne t u <;>\n    -- this `have` triggers the unusedHavesSuffices linter:\n    (try have := h'.symm) <;> simp [*]\n#align simple_graph.walk.mem_support_append_iff SimpleGraph.Walk.mem_support_append_iff\n\n@[simp]\ntheorem subset_support_append_left {V : Type u} {G : SimpleGraph V} {u v w : V}\n    (p : G.Walk u v) (q : G.Walk v w) : p.support \u2286 (p.append q).support := by\n  simp only [Walk.support_append, List.subset_append_left]\n#align simple_graph.walk.subset_support_append_left SimpleGraph.Walk.subset_support_append_left\n\n@[simp]\ntheorem subset_support_append_right {V : Type u} {G : SimpleGraph V} {u v w : V}\n    (p : G.Walk u v) (q : G.Walk v w) : q.support \u2286 (p.append q).support := by\n  intro h\n  simp (config := { contextual := true }) only [mem_support_append_iff, or_true_iff, imp_true_iff]\n#align simple_graph.walk.subset_support_append_right SimpleGraph.Walk.subset_support_append_right\n\ntheorem coe_support {u v : V} (p : G.Walk u v) : (p.support : Multiset V) = {u} + p.support.tail :=\n  by cases p <;> rfl\n#align simple_graph.walk.coe_support SimpleGraph.Walk.coe_support\n\ntheorem coe_support_append {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :\n    ((p.append p').support : Multiset V) = {u} + p.support.tail + p'.support.tail := by\n  rw [support_append, \u2190 Multiset.coe_add, coe_support]\n#align simple_graph.walk.coe_support_append SimpleGraph.Walk.coe_support_append\n\ntheorem coe_support_append' [DecidableEq V] {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :\n    ((p.append p').support : Multiset V) = p.support + p'.support - {v} := by\n  rw [support_append, \u2190 Multiset.coe_add]\n  simp only [coe_support]\n  rw [add_comm ({v} : Multiset V)]\n  simp only [\u2190 add_assoc, add_tsub_cancel_right]\n#align simple_graph.walk.coe_support_append' SimpleGraph.Walk.coe_support_append'\n\ntheorem chain_adj_support {u v w : V} (h : G.Adj u v) :\n    \u2200 (p : G.Walk v w), List.Chain G.Adj u p.support\n  | nil => List.Chain.cons h List.Chain.nil\n  | cons h' p => List.Chain.cons h (chain_adj_support h' p)\n#align simple_graph.walk.chain_adj_support SimpleGraph.Walk.chain_adj_support\n\ntheorem chain'_adj_support {u v : V} : \u2200 (p : G.Walk u v), List.Chain' G.Adj p.support\n  | nil => List.Chain.nil\n  | cons h p => chain_adj_support h p\n#align simple_graph.walk.chain'_adj_support SimpleGraph.Walk.chain'_adj_support\n\ntheorem chain_dartAdj_darts {d : G.Dart} {v w : V} (h : d.snd = v) (p : G.Walk v w) :\n    List.Chain G.DartAdj d p.darts := by\n  induction p generalizing d with\n  | nil => exact List.Chain.nil\n  -- Porting note: needed to defer `h` and `rfl` to help elaboration\n  | cons h' p ih => exact List.Chain.cons (by exact h) (ih (by rfl))\n#align simple_graph.walk.chain_dart_adj_darts SimpleGraph.Walk.chain_dartAdj_darts\n\ntheorem chain'_dartAdj_darts {u v : V} : \u2200 (p : G.Walk u v), List.Chain' G.DartAdj p.darts\n  | nil => trivial\n  -- Porting note: needed to defer `rfl` to help elaboration\n  | cons h p => chain_dartAdj_darts (by rfl) p\n#align simple_graph.walk.chain'_dart_adj_darts SimpleGraph.Walk.chain'_dartAdj_darts\n\n/-- Every edge in a walk's edge list is an edge of the graph.\nIt is written in this form (rather than using `\u2286`) to avoid unsightly coercions. -/\ntheorem edges_subset_edgeSet {u v : V} :\n    \u2200 (p : G.Walk u v) \u2983e : Sym2 V\u2984, e \u2208 p.edges \u2192 e \u2208 G.edgeSet\n  | cons h' p', e, h => by\n    cases h\n    \u00b7 exact h'\n    next h' => exact edges_subset_edgeSet p' h'\n#align simple_graph.walk.edges_subset_edge_set SimpleGraph.Walk.edges_subset_edgeSet\n\ntheorem adj_of_mem_edges {u v x y : V} (p : G.Walk u v) (h : s(x, y) \u2208 p.edges) : G.Adj x y :=\n  edges_subset_edgeSet p h\n#align simple_graph.walk.adj_of_mem_edges SimpleGraph.Walk.adj_of_mem_edges\n\n@[simp]\ntheorem darts_nil {u : V} : (nil : G.Walk u u).darts = [] := rfl\n#align simple_graph.walk.darts_nil SimpleGraph.Walk.darts_nil\n\n@[simp]\ntheorem darts_cons {u v w : V} (h : G.Adj u v) (p : G.Walk v w) :\n    (cons h p).darts = \u27e8(u, v), h\u27e9 :: p.darts := rfl\n#align simple_graph.walk.darts_cons SimpleGraph.Walk.darts_cons\n\n@[simp]\ntheorem darts_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) :\n    (p.concat h).darts = p.darts.concat \u27e8(v, w), h\u27e9 := by\n  induction p <;> simp [*, concat_nil]\n#align simple_graph.walk.darts_concat SimpleGraph.Walk.darts_concat\n\n@[simp]\ntheorem darts_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') :\n    (p.copy hu hv).darts = p.darts := by\n  subst_vars\n  rfl\n#align simple_graph.walk.darts_copy SimpleGraph.Walk.darts_copy\n\n@[simp]\ntheorem darts_append {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :\n    (p.append p').darts = p.darts ++ p'.darts := by\n  induction p <;> simp [*]\n#align simple_graph.walk.darts_append SimpleGraph.Walk.darts_append\n\n@[simp]\ntheorem darts_reverse {u v : V} (p : G.Walk u v) :\n    p.reverse.darts = (p.darts.map Dart.symm).reverse := by\n  induction p <;> simp [*, Sym2.eq_swap]\n#align simple_graph.walk.darts_reverse SimpleGraph.Walk.darts_reverse\n\ntheorem mem_darts_reverse {u v : V} {d : G.Dart} {p : G.Walk u v} :\n    d \u2208 p.reverse.darts \u2194 d.symm \u2208 p.darts := by simp\n#align simple_graph.walk.mem_darts_reverse SimpleGraph.Walk.mem_darts_reverse\n\ntheorem cons_map_snd_darts {u v : V} (p : G.Walk u v) : (u :: p.darts.map (\u00b7.snd)) = p.support := by\n  induction p <;> simp! [*]\n#align simple_graph.walk.cons_map_snd_darts SimpleGraph.Walk.cons_map_snd_darts\n\ntheorem map_snd_darts {u v : V} (p : G.Walk u v) : p.darts.map (\u00b7.snd) = p.support.tail := by\n  simpa using congr_arg List.tail (cons_map_snd_darts p)\n#align simple_graph.walk.map_snd_darts SimpleGraph.Walk.map_snd_darts\n\ntheorem map_fst_darts_append {u v : V} (p : G.Walk u v) :\n    p.darts.map (\u00b7.fst) ++ [v] = p.support := by\n  induction p <;> simp! [*]\n#align simple_graph.walk.map_fst_darts_append SimpleGraph.Walk.map_fst_darts_append\n\ntheorem map_fst_darts {u v : V} (p : G.Walk u v) : p.darts.map (\u00b7.fst) = p.support.dropLast := by\n  simpa! using congr_arg List.dropLast (map_fst_darts_append p)\n#align simple_graph.walk.map_fst_darts SimpleGraph.Walk.map_fst_darts\n\n@[simp]\ntheorem edges_nil {u : V} : (nil : G.Walk u u).edges = [] := rfl\n#align simple_graph.walk.edges_nil SimpleGraph.Walk.edges_nil\n\n@[simp]\ntheorem edges_cons {u v w : V} (h : G.Adj u v) (p : G.Walk v w) :\n    (cons h p).edges = s(u, v) :: p.edges := rfl\n#align simple_graph.walk.edges_cons SimpleGraph.Walk.edges_cons\n\n@[simp]\ntheorem edges_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) :\n    (p.concat h).edges = p.edges.concat s(v, w) := by simp [edges]\n#align simple_graph.walk.edges_concat SimpleGraph.Walk.edges_concat\n\n@[simp]\ntheorem edges_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') :\n    (p.copy hu hv).edges = p.edges := by\n  subst_vars\n  rfl\n#align simple_graph.walk.edges_copy SimpleGraph.Walk.edges_copy\n\n@[simp]\ntheorem edges_append {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :\n    (p.append p').edges = p.edges ++ p'.edges := by simp [edges]\n#align simple_graph.walk.edges_append SimpleGraph.Walk.edges_append\n\n@[simp]\ntheorem edges_reverse {u v : V} (p : G.Walk u v) : p.reverse.edges = p.edges.reverse := by\n  simp [edges, List.map_reverse]\n#align simple_graph.walk.edges_reverse SimpleGraph.Walk.edges_reverse\n\n@[simp]\ntheorem length_support {u v : V} (p : G.Walk u v) : p.support.length = p.length + 1 := by\n  induction p <;> simp [*]\n#align simple_graph.walk.length_support SimpleGraph.Walk.length_support\n\n@[simp]\ntheorem length_darts {u v : V} (p : G.Walk u v) : p.darts.length = p.length := by\n  induction p <;> simp [*]\n#align simple_graph.walk.length_darts SimpleGraph.Walk.length_darts\n\n@[simp]\ntheorem length_edges {u v : V} (p : G.Walk u v) : p.edges.length = p.length := by simp [edges]\n#align simple_graph.walk.length_edges SimpleGraph.Walk.length_edges\n\ntheorem dart_fst_mem_support_of_mem_darts {u v : V} :\n    \u2200 (p : G.Walk u v) {d : G.Dart}, d \u2208 p.darts \u2192 d.fst \u2208 p.support\n  | cons h p', d, hd => by\n    simp only [support_cons, darts_cons, List.mem_cons] at hd \u22a2\n    rcases hd with (rfl | hd)\n    \u00b7 exact Or.inl rfl\n    \u00b7 exact Or.inr (dart_fst_mem_support_of_mem_darts _ hd)\n#align simple_graph.walk.dart_fst_mem_support_of_mem_darts SimpleGraph.Walk.dart_fst_mem_support_of_mem_darts\n\ntheorem dart_snd_mem_support_of_mem_darts {u v : V} (p : G.Walk u v) {d : G.Dart}\n    (h : d \u2208 p.darts) : d.snd \u2208 p.support := by\n  simpa using p.reverse.dart_fst_mem_support_of_mem_darts (by simp [h] : d.symm \u2208 p.reverse.darts)\n#align simple_graph.walk.dart_snd_mem_support_of_mem_darts SimpleGraph.Walk.dart_snd_mem_support_of_mem_darts\n\ntheorem fst_mem_support_of_mem_edges {t u v w : V} (p : G.Walk v w) (he : s(t, u) \u2208 p.edges) :\n    t \u2208 p.support := by\n  obtain \u27e8d, hd, he\u27e9 := List.mem_map.mp he\n  rw [dart_edge_eq_mk'_iff'] at he\n  rcases he with (\u27e8rfl, rfl\u27e9 | \u27e8rfl, rfl\u27e9)\n  \u00b7 exact dart_fst_mem_support_of_mem_darts _ hd\n  \u00b7 exact dart_snd_mem_support_of_mem_darts _ hd\n#align simple_graph.walk.fst_mem_support_of_mem_edges SimpleGraph.Walk.fst_mem_support_of_mem_edges\n\ntheorem snd_mem_support_of_mem_edges {t u v w : V} (p : G.Walk v w) (he : s(t, u) \u2208 p.edges) :\n    u \u2208 p.support := by\n  rw [Sym2.eq_swap] at he\n  exact p.fst_mem_support_of_mem_edges he\n#align simple_graph.walk.snd_mem_support_of_mem_edges SimpleGraph.Walk.snd_mem_support_of_mem_edges\n\ntheorem darts_nodup_of_support_nodup {u v : V} {p : G.Walk u v} (h : p.support.Nodup) :\n    p.darts.Nodup := by\n  induction p with\n  | nil => simp\n  | cons _ p' ih =>\n    simp only [darts_cons, support_cons, List.nodup_cons] at h \u22a2\n    exact \u27e8fun h' => h.1 (dart_fst_mem_support_of_mem_darts p' h'), ih h.2\u27e9\n#align simple_graph.walk.darts_nodup_of_support_nodup SimpleGraph.Walk.darts_nodup_of_support_nodup\n\ntheorem edges_nodup_of_support_nodup {u v : V} {p : G.Walk u v} (h : p.support.Nodup) :\n    p.edges.Nodup := by\n  induction p with\n  | nil => simp\n  | cons _ p' ih =>\n    simp only [edges_cons, support_cons, List.nodup_cons] at h \u22a2\n    exact \u27e8fun h' => h.1 (fst_mem_support_of_mem_edges p' h'), ih h.2\u27e9\n#align simple_graph.walk.edges_nodup_of_support_nodup SimpleGraph.Walk.edges_nodup_of_support_nodup\n\n/-- Predicate for the empty walk.\n\nSolves the dependent type problem where `p = G.Walk.nil` typechecks\nonly if `p` has defeq endpoints. -/\ninductive Nil : {v w : V} \u2192 G.Walk v w \u2192 Prop\n  | nil {u : V} : Nil (nil : G.Walk u u)\n\nvariable {u v w : V}\n\n@[simp] lemma nil_nil : (nil : G.Walk u u).Nil := Nil.nil\n\n@[simp] lemma not_nil_cons {h : G.Adj u v} {p : G.Walk v w} : \u00ac (cons h p).Nil := nofun\n\ninstance (p : G.Walk v w) : Decidable p.Nil :=\n  match p with\n  | nil => isTrue .nil\n  | cons _ _ => isFalse nofun\n\nprotected lemma Nil.eq {p : G.Walk v w} : p.Nil \u2192 v = w | .nil => rfl\n\nlemma not_nil_of_ne {p : G.Walk v w} : v \u2260 w \u2192 \u00ac p.Nil := mt Nil.eq\n\nlemma nil_iff_support_eq {p : G.Walk v w} : p.Nil \u2194 p.support = [v] := by\n  cases p <;> simp\n\nlemma nil_iff_length_eq {p : G.Walk v w} : p.Nil \u2194 p.length = 0 := by\n  cases p <;> simp\n\nlemma not_nil_iff {p : G.Walk v w} :\n    \u00ac p.Nil \u2194 \u2203 (u : V) (h : G.Adj v u) (q : G.Walk u w), p = cons h q := by\n  cases p <;> simp [*]\n\n@[elab_as_elim]\ndef notNilRec {motive : {u w : V} \u2192 (p : G.Walk u w) \u2192 (h : \u00ac p.Nil) \u2192 Sort*}\n    (cons : {u v w : V} \u2192 (h : G.Adj u v) \u2192 (q : G.Walk v w) \u2192 motive (cons h q) not_nil_cons)\n    (p : G.Walk u w) : (hp : \u00ac p.Nil) \u2192 motive p hp :=\n  match p with\n  | nil => fun hp => absurd .nil hp\n  | .cons h q => fun _ => cons h q\n\n/-- The second vertex along a non-nil walk. -/\ndef sndOfNotNil (p : G.Walk v w) (hp : \u00ac p.Nil) : V :=\n  p.notNilRec (@fun _ u _ _ _ => u) hp\n\n@[simp] lemma adj_sndOfNotNil {p : G.Walk v w} (hp : \u00ac p.Nil) :\n    G.Adj v (p.sndOfNotNil hp) :=\n  p.notNilRec (fun h _ => h) hp\n\n/-- The walk obtained by removing the first dart of a non-nil walk. -/\ndef tail (p : G.Walk u v) (hp : \u00ac p.Nil) : G.Walk (p.sndOfNotNil hp) v :=\n  p.notNilRec (fun _ q => q) hp\n\n/-- The first dart of a walk. -/\n@[simps]\ndef firstDart (p : G.Walk v w) (hp : \u00ac p.Nil) : G.Dart where\n  fst := v\n  snd := p.sndOfNotNil hp\n  is_adj := p.adj_sndOfNotNil hp\n\nlemma edge_firstDart (p : G.Walk v w) (hp : \u00ac p.Nil) :\n    (p.firstDart hp).edge = s(v, p.sndOfNotNil hp) := rfl\n\nvariable {x y : V} -- TODO: rename to u, v, w instead?\n\n@[simp] lemma cons_tail_eq (p : G.Walk x y) (hp : \u00ac p.Nil) :\n    cons (p.adj_sndOfNotNil hp) (p.tail hp) = p :=\n  p.notNilRec (fun _ _ => rfl) hp\n\n@[simp] lemma cons_support_tail (p : G.Walk x y) (hp : \u00ac p.Nil) :\n    x :: (p.tail hp).support = p.support := by\n  rw [\u2190 support_cons, cons_tail_eq]\n\n@[simp] lemma length_tail_add_one {p : G.Walk x y} (hp : \u00ac p.Nil) :\n    (p.tail hp).length + 1 = p.length := by\n  rw [\u2190 length_cons, cons_tail_eq]\n\n@[simp] lemma nil_copy {x' y' : V} {p : G.Walk x y} (hx : x = x') (hy : y = y') :\n    (p.copy hx hy).Nil = p.Nil := by\n  subst_vars; rfl\n\n/-! ### Trails, paths, circuits, cycles -/\n\n/-- A *trail* is a walk with no repeating edges. -/\n@[mk_iff isTrail_def]\nstructure IsTrail {u v : V} (p : G.Walk u v) : Prop where\n  edges_nodup : p.edges.Nodup\n#align simple_graph.walk.is_trail SimpleGraph.Walk.IsTrail\n#align simple_graph.walk.is_trail_def SimpleGraph.Walk.isTrail_def\n\n/-- A *path* is a walk with no repeating vertices.\nUse `SimpleGraph.Walk.IsPath.mk'` for a simpler constructor. -/\nstructure IsPath {u v : V} (p : G.Walk u v) extends IsTrail p : Prop where\n  support_nodup : p.support.Nodup\n#align simple_graph.walk.is_path SimpleGraph.Walk.IsPath\n\n-- Porting note: used to use `extends to_trail : is_trail p` in structure\nprotected lemma IsPath.isTrail {p : Walk G u v}(h : IsPath p) : IsTrail p := h.toIsTrail\n#align simple_graph.walk.is_path.to_trail SimpleGraph.Walk.IsPath.isTrail\n\n/-- A *circuit* at `u : V` is a nonempty trail beginning and ending at `u`. -/\n@[mk_iff isCircuit_def]\nstructure IsCircuit {u : V} (p : G.Walk u u) extends IsTrail p : Prop where\n  ne_nil : p \u2260 nil\n#align simple_graph.walk.is_circuit SimpleGraph.Walk.IsCircuit\n#align simple_graph.walk.is_circuit_def SimpleGraph.Walk.isCircuit_def\n\n-- Porting note: used to use `extends to_trail : is_trail p` in structure\nprotected lemma IsCircuit.isTrail {p : Walk G u u} (h : IsCircuit p) : IsTrail p := h.toIsTrail\n#align simple_graph.walk.is_circuit.to_trail SimpleGraph.Walk.IsCircuit.isTrail\n\n/-- A *cycle* at `u : V` is a circuit at `u` whose only repeating vertex\nis `u` (which appears exactly twice). -/\nstructure IsCycle {u : V} (p : G.Walk u u) extends IsCircuit p : Prop where\n  support_nodup : p.support.tail.Nodup\n#align simple_graph.walk.is_cycle SimpleGraph.Walk.IsCycle\n\n-- Porting note: used to use `extends to_circuit : is_circuit p` in structure\nprotected lemma IsCycle.isCircuit {p : Walk G u u} (h : IsCycle p) : IsCircuit p := h.toIsCircuit\n#align simple_graph.walk.is_cycle.to_circuit SimpleGraph.Walk.IsCycle.isCircuit\n\n@[simp]\ntheorem isTrail_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') :\n    (p.copy hu hv).IsTrail \u2194 p.IsTrail := by\n  subst_vars\n  rfl\n#align simple_graph.walk.is_trail_copy SimpleGraph.Walk.isTrail_copy\n\ntheorem IsPath.mk' {u v : V} {p : G.Walk u v} (h : p.support.Nodup) : p.IsPath :=\n  \u27e8\u27e8edges_nodup_of_support_nodup h\u27e9, h\u27e9\n#align simple_graph.walk.is_path.mk' SimpleGraph.Walk.IsPath.mk'\n\ntheorem isPath_def {u v : V} (p : G.Walk u v) : p.IsPath \u2194 p.support.Nodup :=\n  \u27e8IsPath.support_nodup, IsPath.mk'\u27e9\n#align simple_graph.walk.is_path_def SimpleGraph.Walk.isPath_def\n\n@[simp]\ntheorem isPath_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') :\n    (p.copy hu hv).IsPath \u2194 p.IsPath := by\n  subst_vars\n  rfl\n#align simple_graph.walk.is_path_copy SimpleGraph.Walk.isPath_copy\n\n@[simp]\ntheorem isCircuit_copy {u u'} (p : G.Walk u u) (hu : u = u') :\n    (p.copy hu hu).IsCircuit \u2194 p.IsCircuit := by\n  subst_vars\n  rfl\n#align simple_graph.walk.is_circuit_copy SimpleGraph.Walk.isCircuit_copy\n\ntheorem isCycle_def {u : V} (p : G.Walk u u) :\n    p.IsCycle \u2194 p.IsTrail \u2227 p \u2260 nil \u2227 p.support.tail.Nodup :=\n  Iff.intro (fun h => \u27e8h.1.1, h.1.2, h.2\u27e9) fun h => \u27e8\u27e8h.1, h.2.1\u27e9, h.2.2\u27e9\n#align simple_graph.walk.is_cycle_def SimpleGraph.Walk.isCycle_def\n\n@[simp]\ntheorem isCycle_copy {u u'} (p : G.Walk u u) (hu : u = u') :\n    (p.copy hu hu).IsCycle \u2194 p.IsCycle := by\n  subst_vars\n  rfl\n#align simple_graph.walk.is_cycle_copy SimpleGraph.Walk.isCycle_copy\n\n@[simp]\ntheorem IsTrail.nil {u : V} : (nil : G.Walk u u).IsTrail :=\n  \u27e8by simp [edges]\u27e9\n#align simple_graph.walk.is_trail.nil SimpleGraph.Walk.IsTrail.nil\n\ntheorem IsTrail.of_cons {u v w : V} {h : G.Adj u v} {p : G.Walk v w} :\n    (cons h p).IsTrail \u2192 p.IsTrail := by simp [isTrail_def]\n#align simple_graph.walk.is_trail.of_cons SimpleGraph.Walk.IsTrail.of_cons\n\n@[simp]\ntheorem cons_isTrail_iff {u v w : V} (h : G.Adj u v) (p : G.Walk v w) :\n    (cons h p).IsTrail \u2194 p.IsTrail \u2227 s(u, v) \u2209 p.edges := by simp [isTrail_def, and_comm]\n#align simple_graph.walk.cons_is_trail_iff SimpleGraph.Walk.cons_isTrail_iff\n\ntheorem IsTrail.reverse {u v : V} (p : G.Walk u v) (h : p.IsTrail) : p.reverse.IsTrail := by\n  simpa [isTrail_def] using h\n#align simple_graph.walk.is_trail.reverse SimpleGraph.Walk.IsTrail.reverse\n\n@[simp]\ntheorem reverse_isTrail_iff {u v : V} (p : G.Walk u v) : p.reverse.IsTrail \u2194 p.IsTrail := by\n  constructor <;>\n    \u00b7 intro h\n      convert h.reverse _\n      try rw [reverse_reverse]\n#align simple_graph.walk.reverse_is_trail_iff SimpleGraph.Walk.reverse_isTrail_iff\n\ntheorem IsTrail.of_append_left {u v w : V} {p : G.Walk u v} {q : G.Walk v w}\n    (h : (p.append q).IsTrail) : p.IsTrail := by\n  rw [isTrail_def, edges_append, List.nodup_append] at h\n  exact \u27e8h.1\u27e9\n#align simple_graph.walk.is_trail.of_append_left SimpleGraph.Walk.IsTrail.of_append_left\n\ntheorem IsTrail.of_append_right {u v w : V} {p : G.Walk u v} {q : G.Walk v w}\n    (h : (p.append q).IsTrail) : q.IsTrail := by\n  rw [isTrail_def, edges_append, List.nodup_append] at h\n  exact \u27e8h.2.1\u27e9\n#align simple_graph.walk.is_trail.of_append_right SimpleGraph.Walk.IsTrail.of_append_right\n\ntheorem IsTrail.count_edges_le_one [DecidableEq V] {u v : V} {p : G.Walk u v} (h : p.IsTrail)\n    (e : Sym2 V) : p.edges.count e \u2264 1 :=\n  List.nodup_iff_count_le_one.mp h.edges_nodup e\n#align simple_graph.walk.is_trail.count_edges_le_one SimpleGraph.Walk.IsTrail.count_edges_le_one\n\ntheorem IsTrail.count_edges_eq_one [DecidableEq V] {u v : V} {p : G.Walk u v} (h : p.IsTrail)\n    {e : Sym2 V} (he : e \u2208 p.edges) : p.edges.count e = 1 :=\n  List.count_eq_one_of_mem h.edges_nodup he\n#align simple_graph.walk.is_trail.count_edges_eq_one SimpleGraph.Walk.IsTrail.count_edges_eq_one\n\ntheorem IsPath.nil {u : V} : (nil : G.Walk u u).IsPath := by constructor <;> simp\n#align simple_graph.walk.is_path.nil SimpleGraph.Walk.IsPath.nil\n\ntheorem IsPath.of_cons {u v w : V} {h : G.Adj u v} {p : G.Walk v w} :\n    (cons h p).IsPath \u2192 p.IsPath := by simp [isPath_def]\n#align simple_graph.walk.is_path.of_cons SimpleGraph.Walk.IsPath.of_cons\n\n@[simp]\ntheorem cons_isPath_iff {u v w : V} (h : G.Adj u v) (p : G.Walk v w) :\n    (cons h p).IsPath \u2194 p.IsPath \u2227 u \u2209 p.support := by\n  constructor <;> simp (config := { contextual := true }) [isPath_def]\n#align simple_graph.walk.cons_is_path_iff SimpleGraph.Walk.cons_isPath_iff\n\nprotected lemma IsPath.cons {p : Walk G v w} (hp : p.IsPath) (hu : u \u2209 p.support) {h : G.Adj u v} :\n    (cons h p).IsPath :=\n  (cons_isPath_iff _ _).2 \u27e8hp, hu\u27e9\n\n@[simp]\ntheorem isPath_iff_eq_nil {u : V} (p : G.Walk u u) : p.IsPath \u2194 p = nil := by\n  cases p <;> simp [IsPath.nil]\n#align simple_graph.walk.is_path_iff_eq_nil SimpleGraph.Walk.isPath_iff_eq_nil\n\ntheorem IsPath.reverse {u v : V} {p : G.Walk u v} (h : p.IsPath) : p.reverse.IsPath := by\n  simpa [isPath_def] using h\n#align simple_graph.walk.is_path.reverse SimpleGraph.Walk.IsPath.reverse\n\n@[simp]\ntheorem isPath_reverse_iff {u v : V} (p : G.Walk u v) : p.reverse.IsPath \u2194 p.IsPath := by\n  constructor <;> intro h <;> convert h.reverse; simp\n#align simple_graph.walk.is_path_reverse_iff SimpleGraph.Walk.isPath_reverse_iff\n\ntheorem IsPath.of_append_left {u v w : V} {p : G.Walk u v} {q : G.Walk v w} :\n    (p.append q).IsPath \u2192 p.IsPath := by\n  simp only [isPath_def, support_append]\n  exact List.Nodup.of_append_left\n#align simple_graph.walk.is_path.of_append_left SimpleGraph.Walk.IsPath.of_append_left\n\ntheorem IsPath.of_append_right {u v w : V} {p : G.Walk u v} {q : G.Walk v w}\n    (h : (p.append q).IsPath) : q.IsPath := by\n  rw [\u2190 isPath_reverse_iff] at h \u22a2\n  rw [reverse_append] at h\n  apply h.of_append_left\n#align simple_graph.walk.is_path.of_append_right SimpleGraph.Walk.IsPath.of_append_right\n\n@[simp]\ntheorem IsCycle.not_of_nil {u : V} : \u00ac(nil : G.Walk u u).IsCycle := fun h => h.ne_nil rfl\n#align simple_graph.walk.is_cycle.not_of_nil SimpleGraph.Walk.IsCycle.not_of_nil\n\nlemma IsCycle.ne_bot : \u2200 {p : G.Walk u u}, p.IsCycle \u2192 G \u2260 \u22a5\n  | nil, hp => by cases hp.ne_nil rfl\n  | cons h _, hp => by rintro rfl; exact h\n\n", "theoremStatement": "lemma IsCycle.three_le_length {v : V} {p : G.Walk v v} (hp : p.IsCycle) : 3 \u2264 p.length ", "theoremName": "SimpleGraph.Walk.IsCycle.three_le_length", "fileCreated": {"commit": "67ba3174dc3eb7ad27c2265bb8444d627a04c72a", "date": "2023-03-01"}, "theoremCreated": {"commit": "af01418e44cde15e3685543a2a91b9aadbeb3458", "date": "2024-04-03"}, "file": "mathlib/Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "module": "Mathlib.Combinatorics.SimpleGraph.Connectivity", "jsonFile": "Mathlib.Combinatorics.SimpleGraph.Connectivity.jsonl", "positionMetadata": {"lineInFile": 1084, "tokenPositionInFile": 44518, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 110, "importedModules": ["Init.Prelude", 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"Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Data.Subtype", "Mathlib.Order.Notation", "Mathlib.Data.Set.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Cases", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Basic", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Tactic.PushNeg", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Tactic.CasesM", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.ByContra", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Data.Set.Prod", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.MinMax", "Mathlib.Tactic.Says", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Util.AssertExists", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Logic.Pairwise", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Data.Rel", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Nat.Defs", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Data.Sym.Sym2", "Mathlib.Combinatorics.SimpleGraph.Basic", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Sym.Card", "Mathlib.Combinatorics.SimpleGraph.Finite", "Mathlib.Combinatorics.SimpleGraph.Dart", "Mathlib.Data.FunLike.Fintype", "Mathlib.Combinatorics.SimpleGraph.Maps", "Mathlib.Combinatorics.SimpleGraph.Subgraph"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have \u27e8\u27e8hp, hp'\u27e9, _\u27e9 := hp\n  match p with\n  | .nil => simp at hp'\n  | .cons h .nil => simp at h\n  | .cons _ (.cons _ .nil) => simp at hp\n  | .cons _ (.cons _ (.cons _ _)) => simp_rw [SimpleGraph.Walk.length_cons]; omega", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 226}}
+{"srcContext": "/-\nCopyright (c) 2020 Damiano Testa. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Damiano Testa\n-/\nimport Mathlib.Algebra.Polynomial.Degree.Definitions\nimport Mathlib.Data.ENat.Basic\n\n#align_import data.polynomial.degree.trailing_degree from \"leanprover-community/mathlib\"@\"302eab4f46abb63de520828de78c04cb0f9b5836\"\n\n/-!\n# Trailing degree of univariate polynomials\n\n## Main definitions\n\n* `trailingDegree p`: the multiplicity of `X` in the polynomial `p`\n* `natTrailingDegree`: a variant of `trailingDegree` that takes values in the natural numbers\n* `trailingCoeff`: the coefficient at index `natTrailingDegree p`\n\nConverts most results about `degree`, `natDegree` and `leadingCoeff` to results about the bottom\nend of a polynomial\n-/\n\n\nnoncomputable section\n\nopen Function Polynomial Finsupp Finset\n\nopen scoped BigOperators Polynomial\n\nnamespace Polynomial\n\nuniverse u v\n\nvariable {R : Type u} {S : Type v} {a b : R} {n m : \u2115}\n\nsection Semiring\n\nvariable [Semiring R] {p q r : R[X]}\n\n/-- `trailingDegree p` is the multiplicity of `x` in the polynomial `p`, i.e. the smallest\n`X`-exponent in `p`.\n`trailingDegree p = some n` when `p \u2260 0` and `n` is the smallest power of `X` that appears\nin `p`, otherwise\n`trailingDegree 0 = \u22a4`. -/\ndef trailingDegree (p : R[X]) : \u2115\u221e :=\n  p.support.min\n#align polynomial.trailing_degree Polynomial.trailingDegree\n\ntheorem trailingDegree_lt_wf : WellFounded fun p q : R[X] => trailingDegree p < trailingDegree q :=\n  InvImage.wf trailingDegree wellFounded_lt\n#align polynomial.trailing_degree_lt_wf Polynomial.trailingDegree_lt_wf\n\n/-- `natTrailingDegree p` forces `trailingDegree p` to `\u2115`, by defining\n`natTrailingDegree \u22a4 = 0`. -/\ndef natTrailingDegree (p : R[X]) : \u2115 :=\n  (trailingDegree p).getD 0\n#align polynomial.nat_trailing_degree Polynomial.natTrailingDegree\n\n/-- `trailingCoeff p` gives the coefficient of the smallest power of `X` in `p`-/\ndef trailingCoeff (p : R[X]) : R :=\n  coeff p (natTrailingDegree p)\n#align polynomial.trailing_coeff Polynomial.trailingCoeff\n\n/-- a polynomial is `monic_at` if its trailing coefficient is 1 -/\ndef TrailingMonic (p : R[X]) :=\n  trailingCoeff p = (1 : R)\n#align polynomial.trailing_monic Polynomial.TrailingMonic\n\ntheorem TrailingMonic.def : TrailingMonic p \u2194 trailingCoeff p = 1 :=\n  Iff.rfl\n#align polynomial.trailing_monic.def Polynomial.TrailingMonic.def\n\ninstance TrailingMonic.decidable [DecidableEq R] : Decidable (TrailingMonic p) :=\n  inferInstanceAs <| Decidable (trailingCoeff p = (1 : R))\n#align polynomial.trailing_monic.decidable Polynomial.TrailingMonic.decidable\n\n@[simp]\ntheorem TrailingMonic.trailingCoeff {p : R[X]} (hp : p.TrailingMonic) : trailingCoeff p = 1 :=\n  hp\n#align polynomial.trailing_monic.trailing_coeff Polynomial.TrailingMonic.trailingCoeff\n\n@[simp]\ntheorem trailingDegree_zero : trailingDegree (0 : R[X]) = \u22a4 :=\n  rfl\n#align polynomial.trailing_degree_zero Polynomial.trailingDegree_zero\n\n@[simp]\ntheorem trailingCoeff_zero : trailingCoeff (0 : R[X]) = 0 :=\n  rfl\n#align polynomial.trailing_coeff_zero Polynomial.trailingCoeff_zero\n\n@[simp]\ntheorem natTrailingDegree_zero : natTrailingDegree (0 : R[X]) = 0 :=\n  rfl\n#align polynomial.nat_trailing_degree_zero Polynomial.natTrailingDegree_zero\n\ntheorem trailingDegree_eq_top : trailingDegree p = \u22a4 \u2194 p = 0 :=\n  \u27e8fun h => support_eq_empty.1 (Finset.min_eq_top.1 h), fun h => by simp [h]\u27e9\n#align polynomial.trailing_degree_eq_top Polynomial.trailingDegree_eq_top\n\ntheorem trailingDegree_eq_natTrailingDegree (hp : p \u2260 0) :\n    trailingDegree p = (natTrailingDegree p : \u2115\u221e) := by\n  let \u27e8n, hn\u27e9 :=\n    not_forall.1 (mt Option.eq_none_iff_forall_not_mem.2 (mt trailingDegree_eq_top.1 hp))\n  have hn : trailingDegree p = n := Classical.not_not.1 hn\n  rw [natTrailingDegree, hn]\n  rfl\n#align polynomial.trailing_degree_eq_nat_trailing_degree Polynomial.trailingDegree_eq_natTrailingDegree\n\ntheorem trailingDegree_eq_iff_natTrailingDegree_eq {p : R[X]} {n : \u2115} (hp : p \u2260 0) :\n    p.trailingDegree = n \u2194 p.natTrailingDegree = n := by\n  rw [trailingDegree_eq_natTrailingDegree hp]\n  exact WithTop.coe_eq_coe\n#align polynomial.trailing_degree_eq_iff_nat_trailing_degree_eq Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq\n\ntheorem trailingDegree_eq_iff_natTrailingDegree_eq_of_pos {p : R[X]} {n : \u2115} (hn : 0 < n) :\n    p.trailingDegree = n \u2194 p.natTrailingDegree = n := by\n  constructor\n  \u00b7 intro H\n    rwa [\u2190 trailingDegree_eq_iff_natTrailingDegree_eq]\n    rintro rfl\n    rw [trailingDegree_zero] at H\n    exact Option.noConfusion H\n  \u00b7 intro H\n    rwa [trailingDegree_eq_iff_natTrailingDegree_eq]\n    rintro rfl\n    rw [natTrailingDegree_zero] at H\n    rw [H] at hn\n    exact lt_irrefl _ hn\n#align polynomial.trailing_degree_eq_iff_nat_trailing_degree_eq_of_pos Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq_of_pos\n\ntheorem natTrailingDegree_eq_of_trailingDegree_eq_some {p : R[X]} {n : \u2115}\n    (h : trailingDegree p = n) : natTrailingDegree p = n :=\n  have hp0 : p \u2260 0 := fun hp0 => by rw [hp0] at h; exact Option.noConfusion h\n  Option.some_inj.1 <|\n    show (natTrailingDegree p : \u2115\u221e) = n by rwa [\u2190 trailingDegree_eq_natTrailingDegree hp0]\n#align polynomial.nat_trailing_degree_eq_of_trailing_degree_eq_some Polynomial.natTrailingDegree_eq_of_trailingDegree_eq_some\n\n@[simp]\ntheorem natTrailingDegree_le_trailingDegree : \u2191(natTrailingDegree p) \u2264 trailingDegree p := by\n  by_cases hp : p = 0;\n  \u00b7 rw [hp, trailingDegree_zero]\n    exact le_top\n  rw [trailingDegree_eq_natTrailingDegree hp]\n#align polynomial.nat_trailing_degree_le_trailing_degree Polynomial.natTrailingDegree_le_trailingDegree\n\ntheorem natTrailingDegree_eq_of_trailingDegree_eq [Semiring S] {q : S[X]}\n    (h : trailingDegree p = trailingDegree q) : natTrailingDegree p = natTrailingDegree q := by\n  unfold natTrailingDegree\n  rw [h]\n#align polynomial.nat_trailing_degree_eq_of_trailing_degree_eq Polynomial.natTrailingDegree_eq_of_trailingDegree_eq\n\ntheorem trailingDegree_le_of_ne_zero (h : coeff p n \u2260 0) : trailingDegree p \u2264 n :=\n  show @LE.le \u2115\u221e _ p.support.min n from min_le (mem_support_iff.2 h)\n#align polynomial.le_trailing_degree_of_ne_zero Polynomial.trailingDegree_le_of_ne_zero\n\ntheorem natTrailingDegree_le_of_ne_zero (h : coeff p n \u2260 0) : natTrailingDegree p \u2264 n := by\n  have : WithTop.some (natTrailingDegree p) = Nat.cast (natTrailingDegree p) := rfl\n  rw [\u2190 WithTop.coe_le_coe, this, \u2190 trailingDegree_eq_natTrailingDegree]\n  \u00b7 exact trailingDegree_le_of_ne_zero h\n  \u00b7 intro h\n    subst h\n    exact h rfl\n#align polynomial.nat_trailing_degree_le_of_ne_zero Polynomial.natTrailingDegree_le_of_ne_zero\n\n@[simp] lemma coeff_natTrailingDegree_eq_zero : coeff p p.natTrailingDegree = 0 \u2194 p = 0 := by\n  constructor\n  \u00b7 rintro h\n    by_contra hp\n    obtain \u27e8n, hpn, hn\u27e9 := by simpa using min_mem_image_coe $ support_nonempty.2 hp\n    obtain rfl := (trailingDegree_eq_iff_natTrailingDegree_eq hp).1 hn.symm\n    exact hpn h\n  \u00b7 rintro rfl\n    simp\n\nlemma coeff_natTrailingDegree_ne_zero : coeff p p.natTrailingDegree \u2260 0 \u2194 p \u2260 0 :=\n  coeff_natTrailingDegree_eq_zero.not\n\n@[simp] lemma natTrailingDegree_eq_zero : natTrailingDegree p = 0 \u2194 p = 0 \u2228 coeff p 0 \u2260 0 := by\n  constructor\n  \u00b7 rw [or_iff_not_imp_left]\n    rintro h hp\n    rwa [\u2190 h, coeff_natTrailingDegree_ne_zero]\n  \u00b7 rintro (rfl | h)\n    \u00b7 simp\n    \u00b7 exact nonpos_iff_eq_zero.1 $ natTrailingDegree_le_of_ne_zero h\n\n", "theoremStatement": "lemma trailingDegree_eq_zero : trailingDegree p = 0 \u2194 coeff p 0 \u2260 0 ", "theoremName": "Polynomial.trailingDegree_eq_zero", "fileCreated": {"commit": "56e439ec275a319deed4f81f6436af5f0d9b6f00", "date": "2024-04-05"}, "theoremCreated": {"commit": "3b9115ffc3cae6fbd8bd8d17a8c422e2e5870f9d", "date": "2024-03-29"}, "file": "mathlib/Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean", "module": "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "jsonFile": "Mathlib.Algebra.Polynomial.Degree.TrailingDegree.jsonl", "positionMetadata": {"lineInFile": 189, "tokenPositionInFile": 7376, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 48, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", 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"Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  obtain rfl | hp := eq_or_ne p 0\n  \u00b7 simp [WithTop.top_ne_zero (\u03b1 := \u2115)]\n  \u00b7 exact (trailingDegree_eq_iff_natTrailingDegree_eq hp).trans $\n      natTrailingDegree_eq_zero.trans $ or_iff_right hp", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 201}}
+{"srcContext": "/-\nCopyright (c) 2018 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl\n-/\nimport Mathlib.Algebra.Order.BigOperators.Ring.Finset\nimport Mathlib.Data.Nat.Totient\nimport Mathlib.GroupTheory.OrderOfElement\nimport Mathlib.GroupTheory.Subgroup.Simple\nimport Mathlib.Tactic.Group\nimport Mathlib.GroupTheory.Exponent\n\n#align_import group_theory.specific_groups.cyclic from \"leanprover-community/mathlib\"@\"0f6670b8af2dff699de1c0b4b49039b31bc13c46\"\n\n/-!\n# Cyclic groups\n\nA group `G` is called cyclic if there exists an element `g : G` such that every element of `G` is of\nthe form `g ^ n` for some `n : \u2115`. This file only deals with the predicate on a group to be cyclic.\nFor the concrete cyclic group of order `n`, see `Data.ZMod.Basic`.\n\n## Main definitions\n\n* `IsCyclic` is a predicate on a group stating that the group is cyclic.\n\n## Main statements\n\n* `isCyclic_of_prime_card` proves that a finite group of prime order is cyclic.\n* `isSimpleGroup_of_prime_card`, `IsSimpleGroup.isCyclic`,\n  and `IsSimpleGroup.prime_card` classify finite simple abelian groups.\n* `IsCyclic.exponent_eq_card`: For a finite cyclic group `G`, the exponent is equal to\n  the group's cardinality.\n* `IsCyclic.exponent_eq_zero_of_infinite`: Infinite cyclic groups have exponent zero.\n* `IsCyclic.iff_exponent_eq_card`: A finite commutative group is cyclic iff its exponent\n  is equal to its cardinality.\n\n## Tags\n\ncyclic group\n-/\n\n\nuniverse u\n\nvariable {\u03b1 : Type u} {a : \u03b1}\n\nsection Cyclic\n\nopen BigOperators\n\nattribute [local instance] setFintype\n\nopen Subgroup\n\n/-- A group is called *cyclic* if it is generated by a single element. -/\nclass IsAddCyclic (\u03b1 : Type u) [AddGroup \u03b1] : Prop where\n  exists_generator : \u2203 g : \u03b1, \u2200 x, x \u2208 AddSubgroup.zmultiples g\n#align is_add_cyclic IsAddCyclic\n\n/-- A group is called *cyclic* if it is generated by a single element. -/\n@[to_additive]\nclass IsCyclic (\u03b1 : Type u) [Group \u03b1] : Prop where\n  exists_generator : \u2203 g : \u03b1, \u2200 x, x \u2208 zpowers g\n#align is_cyclic IsCyclic\n\n@[to_additive]\ninstance (priority := 100) isCyclic_of_subsingleton [Group \u03b1] [Subsingleton \u03b1] : IsCyclic \u03b1 :=\n  \u27e8\u27e81, fun x => by\n      rw [Subsingleton.elim x 1]\n      exact mem_zpowers 1\u27e9\u27e9\n#align is_cyclic_of_subsingleton isCyclic_of_subsingleton\n#align is_add_cyclic_of_subsingleton isAddCyclic_of_subsingleton\n\n@[simp]\ntheorem isCyclic_multiplicative_iff [AddGroup \u03b1] : IsCyclic (Multiplicative \u03b1) \u2194 IsAddCyclic \u03b1 :=\n  \u27e8fun H \u21a6 \u27e8H.1\u27e9, fun H \u21a6 \u27e8H.1\u27e9\u27e9\n\ninstance isCyclic_multiplicative [AddGroup \u03b1] [IsAddCyclic \u03b1] : IsCyclic (Multiplicative \u03b1) :=\n  isCyclic_multiplicative_iff.mpr inferInstance\n\n@[simp]\ntheorem isAddCyclic_additive_iff [Group \u03b1] : IsAddCyclic (Additive \u03b1) \u2194 IsCyclic \u03b1 :=\n  \u27e8fun H \u21a6 \u27e8H.1\u27e9, fun H \u21a6 \u27e8H.1\u27e9\u27e9\n\ninstance isAddCyclic_additive [Group \u03b1] [IsCyclic \u03b1] : IsAddCyclic (Additive \u03b1) :=\n  isAddCyclic_additive_iff.mpr inferInstance\n\n/-- A cyclic group is always commutative. This is not an `instance` because often we have a better\nproof of `CommGroup`. -/\n@[to_additive\n      \"A cyclic group is always commutative. This is not an `instance` because often we have\n      a better proof of `AddCommGroup`.\"]\ndef IsCyclic.commGroup [hg : Group \u03b1] [IsCyclic \u03b1] : CommGroup \u03b1 :=\n  { hg with\n    mul_comm := fun x y =>\n      let \u27e8_, hg\u27e9 := IsCyclic.exists_generator (\u03b1 := \u03b1)\n      let \u27e8_, hn\u27e9 := hg x\n      let \u27e8_, hm\u27e9 := hg y\n      hm \u25b8 hn \u25b8 zpow_mul_comm _ _ _ }\n#align is_cyclic.comm_group IsCyclic.commGroup\n#align is_add_cyclic.add_comm_group IsAddCyclic.addCommGroup\n\nvariable [Group \u03b1]\n\n/-- A non-cyclic multiplicative group is non-trivial. -/\n@[to_additive \"A non-cyclic additive group is non-trivial.\"]\ntheorem Nontrivial.of_not_isCyclic (nc : \u00acIsCyclic \u03b1) : Nontrivial \u03b1 := by\n  contrapose! nc\n  exact @isCyclic_of_subsingleton _ _ (not_nontrivial_iff_subsingleton.mp nc)\n\n@[to_additive]\ntheorem MonoidHom.map_cyclic {G : Type*} [Group G] [h : IsCyclic G] (\u03c3 : G \u2192* G) :\n    \u2203 m : \u2124, \u2200 g : G, \u03c3 g = g ^ m := by\n  obtain \u27e8h, hG\u27e9 := IsCyclic.exists_generator (\u03b1 := G)\n  obtain \u27e8m, hm\u27e9 := hG (\u03c3 h)\n  refine' \u27e8m, fun g => _\u27e9\n  obtain \u27e8n, rfl\u27e9 := hG g\n  rw [MonoidHom.map_zpow, \u2190 hm, \u2190 zpow_mul, \u2190 zpow_mul']\n#align monoid_hom.map_cyclic MonoidHom.map_cyclic\n#align monoid_add_hom.map_add_cyclic AddMonoidHom.map_addCyclic\n/- 2024-02-21 -/ @[deprecated] alias MonoidAddHom.map_add_cyclic := AddMonoidHom.map_addCyclic\n\n@[to_additive]\ntheorem isCyclic_of_orderOf_eq_card [Fintype \u03b1] (x : \u03b1) (hx : orderOf x = Fintype.card \u03b1) :\n    IsCyclic \u03b1 := by\n  classical\n    use x\n    simp_rw [\u2190 SetLike.mem_coe, \u2190 Set.eq_univ_iff_forall]\n    rw [\u2190 Fintype.card_congr (Equiv.Set.univ \u03b1), \u2190 Fintype.card_zpowers] at hx\n    exact Set.eq_of_subset_of_card_le (Set.subset_univ _) (ge_of_eq hx)\n#align is_cyclic_of_order_of_eq_card isCyclic_of_orderOf_eq_card\n#align is_add_cyclic_of_order_of_eq_card isAddCyclic_of_addOrderOf_eq_card\n/- 2024-02-21 -/ @[deprecated] alias isAddCyclic_of_orderOf_eq_card :=\n  isAddCyclic_of_addOrderOf_eq_card\n\n/-- Any non-identity element of a finite group of prime order generates the group. -/\n@[to_additive \"Any non-identity element of a finite group of prime order generates the group.\"]\ntheorem zpowers_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : \u2115}\n    [hp : Fact p.Prime] (h : Fintype.card G = p) {g : G} (hg : g \u2260 1) : zpowers g = \u22a4 := by\n  have := card_subgroup_dvd_card (zpowers g)\n  simp_rw [h, Nat.dvd_prime hp.1, \u2190 eq_bot_iff_card, zpowers_eq_bot, hg, false_or, \u2190 h,\n    card_eq_iff_eq_top] at this\n  exact this\n\n@[to_additive]\ntheorem mem_zpowers_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : \u2115} [hp : Fact p.Prime]\n    (h : Fintype.card G = p) {g g' : G} (hg : g \u2260 1) : g' \u2208 zpowers g := by\n  simp_rw [zpowers_eq_top_of_prime_card h hg, Subgroup.mem_top]\n\n@[to_additive]\ntheorem mem_powers_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : \u2115} [hp : Fact p.Prime]\n    (h : Fintype.card G = p) {g g' : G} (hg : g \u2260 1) : g' \u2208 Submonoid.powers g := by\n  rw [mem_powers_iff_mem_zpowers]\n  exact mem_zpowers_of_prime_card h hg\n\n@[to_additive]\ntheorem powers_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G} {p : \u2115}\n    [hp : Fact p.Prime] (h : Fintype.card G = p) {g : G} (hg : g \u2260 1) : Submonoid.powers g = \u22a4 := by\n  ext x\n  simp [mem_powers_of_prime_card h hg]\n\n/-- A finite group of prime order is cyclic. -/\n@[to_additive \"A finite group of prime order is cyclic.\"]\ntheorem isCyclic_of_prime_card {\u03b1 : Type u} [Group \u03b1] [Fintype \u03b1] {p : \u2115} [hp : Fact p.Prime]\n    (h : Fintype.card \u03b1 = p) : IsCyclic \u03b1 := by\n  obtain \u27e8g, hg\u27e9 : \u2203 g, g \u2260 1 := Fintype.exists_ne_of_one_lt_card (h.symm \u25b8 hp.1.one_lt) 1\n  exact \u27e8g, fun g' \u21a6 mem_zpowers_of_prime_card h hg\u27e9\n#align is_cyclic_of_prime_card isCyclic_of_prime_card\n#align is_add_cyclic_of_prime_card isAddCyclic_of_prime_card\n\n@[to_additive]\ntheorem isCyclic_of_surjective {H G F : Type*} [Group H] [Group G] [hH : IsCyclic H]\n    [FunLike F H G] [MonoidHomClass F H G] (f : F) (hf : Function.Surjective f) :\n    IsCyclic G := by\n  obtain \u27e8x, hx\u27e9 := hH\n  refine \u27e8f x, fun a \u21a6 ?_\u27e9\n  obtain \u27e8a, rfl\u27e9 := hf a\n  obtain \u27e8n, rfl\u27e9 := hx a\n  exact \u27e8n, (map_zpow _ _ _).symm\u27e9\n\n@[to_additive]\ntheorem orderOf_eq_card_of_forall_mem_zpowers [Fintype \u03b1] {g : \u03b1} (hx : \u2200 x, x \u2208 zpowers g) :\n    orderOf g = Fintype.card \u03b1 := by\n  classical\n    rw [\u2190 Fintype.card_zpowers]\n    apply Fintype.card_of_finset'\n    simpa using hx\n#align order_of_eq_card_of_forall_mem_zpowers orderOf_eq_card_of_forall_mem_zpowers\n#align add_order_of_eq_card_of_forall_mem_zmultiples addOrderOf_eq_card_of_forall_mem_zmultiples\n\n@[to_additive]\ntheorem exists_pow_ne_one_of_isCyclic {G : Type*} [Group G] [Fintype G] [G_cyclic : IsCyclic G]\n    {k : \u2115} (k_pos : k \u2260 0) (k_lt_card_G : k < Fintype.card G) : \u2203 a : G, a ^ k \u2260 1 := by\n  rcases G_cyclic with \u27e8a, ha\u27e9\n  use a\n  contrapose! k_lt_card_G\n  convert orderOf_le_of_pow_eq_one k_pos.bot_lt k_lt_card_G\n  rw [\u2190 Fintype.card_zpowers, eq_comm, Subgroup.card_eq_iff_eq_top, eq_top_iff]\n  exact fun x _ \u21a6 ha x\n\n@[to_additive]\ntheorem Infinite.orderOf_eq_zero_of_forall_mem_zpowers [Infinite \u03b1] {g : \u03b1}\n    (h : \u2200 x, x \u2208 zpowers g) : orderOf g = 0 := by\n  classical\n    rw [orderOf_eq_zero_iff']\n    refine' fun n hn hgn => _\n    have ho := isOfFinOrder_iff_pow_eq_one.mpr \u27e8n, hn, hgn\u27e9\n    obtain \u27e8x, hx\u27e9 :=\n      Infinite.exists_not_mem_finset\n        (Finset.image (fun x => g ^ x) <| Finset.range <| orderOf g)\n    apply hx\n    rw [\u2190 ho.mem_powers_iff_mem_range_orderOf, Submonoid.mem_powers_iff]\n    obtain \u27e8k, hk\u27e9 := h x\n    dsimp at hk\n    obtain \u27e8k, rfl | rfl\u27e9 := k.eq_nat_or_neg\n    \u00b7 exact \u27e8k, mod_cast hk\u27e9\n    rw [\u2190 zpow_mod_orderOf] at hk\n    have : 0 \u2264 (-k % orderOf g : \u2124) := Int.emod_nonneg (-k) (mod_cast ho.orderOf_pos.ne')\n    refine' \u27e8(-k % orderOf g : \u2124).toNat, _\u27e9\n    rwa [\u2190 zpow_natCast, Int.toNat_of_nonneg this]\n#align infinite.order_of_eq_zero_of_forall_mem_zpowers Infinite.orderOf_eq_zero_of_forall_mem_zpowers\n#align infinite.add_order_of_eq_zero_of_forall_mem_zmultiples Infinite.addOrderOf_eq_zero_of_forall_mem_zmultiples\n\n@[to_additive]\ninstance Bot.isCyclic {\u03b1 : Type u} [Group \u03b1] : IsCyclic (\u22a5 : Subgroup \u03b1) :=\n  \u27e8\u27e81, fun x => \u27e80, Subtype.eq <| (zpow_zero (1 : \u03b1)).trans <| Eq.symm (Subgroup.mem_bot.1 x.2)\u27e9\u27e9\u27e9\n#align bot.is_cyclic Bot.isCyclic\n#align bot.is_add_cyclic Bot.isAddCyclic\n\n@[to_additive]\ninstance Subgroup.isCyclic {\u03b1 : Type u} [Group \u03b1] [IsCyclic \u03b1] (H : Subgroup \u03b1) : IsCyclic H :=\n  haveI := Classical.propDecidable\n  let \u27e8g, hg\u27e9 := IsCyclic.exists_generator (\u03b1 := \u03b1)\n  if hx : \u2203 x : \u03b1, x \u2208 H \u2227 x \u2260 (1 : \u03b1) then\n    let \u27e8x, hx\u2081, hx\u2082\u27e9 := hx\n    let \u27e8k, hk\u27e9 := hg x\n    have hk : g ^ k = x := hk\n    have hex : \u2203 n : \u2115, 0 < n \u2227 g ^ n \u2208 H :=\n      \u27e8k.natAbs,\n        Nat.pos_of_ne_zero fun h => hx\u2082 <| by\n          rw [\u2190 hk, Int.natAbs_eq_zero.mp h, zpow_zero], by\n            cases' k with k k\n            \u00b7 rw [Int.ofNat_eq_coe, Int.natAbs_cast k, \u2190 zpow_natCast, \u2190 Int.ofNat_eq_coe, hk]\n              exact hx\u2081\n            \u00b7 rw [Int.natAbs_negSucc, \u2190 Subgroup.inv_mem_iff H]; simp_all\u27e9\n    \u27e8\u27e8\u27e8g ^ Nat.find hex, (Nat.find_spec hex).2\u27e9, fun \u27e8x, hx\u27e9 =>\n        let \u27e8k, hk\u27e9 := hg x\n        have hk : g ^ k = x := hk\n        have hk\u2082 : g ^ ((Nat.find hex : \u2124) * (k / Nat.find hex : \u2124)) \u2208 H := by\n          rw [zpow_mul]\n          apply H.zpow_mem\n          exact mod_cast (Nat.find_spec hex).2\n        have hk\u2083 : g ^ (k % Nat.find hex : \u2124) \u2208 H :=\n          (Subgroup.mul_mem_cancel_right H hk\u2082).1 <| by\n            rw [\u2190 zpow_add, Int.emod_add_ediv, hk]; exact hx\n        have hk\u2084 : k % Nat.find hex = (k % Nat.find hex).natAbs := by\n          rw [Int.natAbs_of_nonneg\n              (Int.emod_nonneg _ (Int.natCast_ne_zero_iff_pos.2 (Nat.find_spec hex).1))]\n        have hk\u2085 : g ^ (k % Nat.find hex).natAbs \u2208 H := by rwa [\u2190 zpow_natCast, \u2190 hk\u2084]\n        have hk\u2086 : (k % (Nat.find hex : \u2124)).natAbs = 0 :=\n          by_contradiction fun h =>\n            Nat.find_min hex\n              (Int.ofNat_lt.1 <| by\n                rw [\u2190 hk\u2084]; exact Int.emod_lt_of_pos _ (Int.natCast_pos.2 (Nat.find_spec hex).1))\n              \u27e8Nat.pos_of_ne_zero h, hk\u2085\u27e9\n        \u27e8k / (Nat.find hex : \u2124),\n          Subtype.ext_iff_val.2\n            (by\n              suffices g ^ ((Nat.find hex : \u2124) * (k / Nat.find hex : \u2124)) = x by simpa [zpow_mul]\n              rw [Int.mul_ediv_cancel'\n                  (Int.dvd_of_emod_eq_zero (Int.natAbs_eq_zero.mp hk\u2086)),\n                hk])\u27e9\u27e9\u27e9\n  else by\n    have : H = (\u22a5 : Subgroup \u03b1) :=\n      Subgroup.ext fun x =>\n        \u27e8fun h => by simp at *; tauto, fun h => by rw [Subgroup.mem_bot.1 h]; exact H.one_mem\u27e9\n    subst this; infer_instance\n#align subgroup.is_cyclic Subgroup.isCyclic\n#align add_subgroup.is_add_cyclic AddSubgroup.isAddCyclic\n\nopen Finset Nat\n\nsection Classical\n\nopen scoped Classical\n\n@[to_additive IsAddCyclic.card_nsmul_eq_zero_le]\ntheorem IsCyclic.card_pow_eq_one_le [DecidableEq \u03b1] [Fintype \u03b1] [IsCyclic \u03b1] {n : \u2115} (hn0 : 0 < n) :\n    (univ.filter fun a : \u03b1 => a ^ n = 1).card \u2264 n :=\n  let \u27e8g, hg\u27e9 := IsCyclic.exists_generator (\u03b1 := \u03b1)\n  calc\n    (univ.filter fun a : \u03b1 => a ^ n = 1).card \u2264\n        (zpowers (g ^ (Fintype.card \u03b1 / Nat.gcd n (Fintype.card \u03b1))) : Set \u03b1).toFinset.card :=\n      card_le_card fun x hx =>\n        let \u27e8m, hm\u27e9 := show x \u2208 Submonoid.powers g from mem_powers_iff_mem_zpowers.2 <| hg x\n        Set.mem_toFinset.2\n          \u27e8(m / (Fintype.card \u03b1 / Nat.gcd n (Fintype.card \u03b1)) : \u2115), by\n            dsimp at hm\n            have hgmn : g ^ (m * Nat.gcd n (Fintype.card \u03b1)) = 1 := by\n              rw [pow_mul, hm, \u2190 pow_gcd_card_eq_one_iff]; exact (mem_filter.1 hx).2\n            dsimp only\n            rw [zpow_natCast, \u2190 pow_mul, Nat.mul_div_cancel_left', hm]\n            refine' Nat.dvd_of_mul_dvd_mul_right (gcd_pos_of_pos_left (Fintype.card \u03b1) hn0) _\n            conv_lhs =>\n              rw [Nat.div_mul_cancel (Nat.gcd_dvd_right _ _), \u2190\n                orderOf_eq_card_of_forall_mem_zpowers hg]\n            exact orderOf_dvd_of_pow_eq_one hgmn\u27e9\n    _ \u2264 n := by\n      let \u27e8m, hm\u27e9 := Nat.gcd_dvd_right n (Fintype.card \u03b1)\n      have hm0 : 0 < m :=\n        Nat.pos_of_ne_zero fun hm0 => by\n          rw [hm0, mul_zero, Fintype.card_eq_zero_iff] at hm\n          exact hm.elim' 1\n      simp only [Set.toFinset_card, SetLike.coe_sort_coe]\n      rw [Fintype.card_zpowers, orderOf_pow g, orderOf_eq_card_of_forall_mem_zpowers hg]\n      nth_rw 2 [hm]; nth_rw 3 [hm]\n      rw [Nat.mul_div_cancel_left _ (gcd_pos_of_pos_left _ hn0), gcd_mul_left_left, hm,\n        Nat.mul_div_cancel _ hm0]\n      exact le_of_dvd hn0 (Nat.gcd_dvd_left _ _)\n#align is_cyclic.card_pow_eq_one_le IsCyclic.card_pow_eq_one_le\n#align is_add_cyclic.card_pow_eq_one_le IsAddCyclic.card_nsmul_eq_zero_le\n/- 2024-02-21 -/\n@[deprecated] alias IsAddCyclic.card_pow_eq_one_le := IsAddCyclic.card_nsmul_eq_zero_le\n\nend Classical\n\n@[to_additive]\ntheorem IsCyclic.exists_monoid_generator [Finite \u03b1] [IsCyclic \u03b1] :\n    \u2203 x : \u03b1, \u2200 y : \u03b1, y \u2208 Submonoid.powers x := by\n  simp_rw [mem_powers_iff_mem_zpowers]\n  exact IsCyclic.exists_generator\n#align is_cyclic.exists_monoid_generator IsCyclic.exists_monoid_generator\n#align is_add_cyclic.exists_add_monoid_generator IsAddCyclic.exists_addMonoid_generator\n\n@[to_additive]\nlemma IsCyclic.exists_ofOrder_eq_natCard [h : IsCyclic \u03b1] : \u2203 g : \u03b1, orderOf g = Nat.card \u03b1 := by\n  obtain \u27e8g, hg\u27e9 := h.exists_generator\n  use g\n  rw [\u2190 card_zpowers g, (eq_top_iff' (zpowers g)).mpr hg]\n  exact Nat.card_congr (Equiv.Set.univ \u03b1)\n\n", "theoremStatement": "@[to_additive]\nlemma IsCyclic.iff_exists_ofOrder_eq_natCard_of_Fintype [Fintype \u03b1] :\n    IsCyclic \u03b1 \u2194 \u2203 g : \u03b1, orderOf g = Nat.card \u03b1 ", "theoremName": "IsCyclic.iff_exists_ofOrder_eq_natCard_of_Fintype", "fileCreated": {"commit": "f8480be7d847cea8df877e4d6503b5292a5b6724", "date": "2023-05-25"}, "theoremCreated": {"commit": "e18e9225ee93a51b683ef93d9e12b4300aabf5d4", "date": "2024-03-28"}, "file": "mathlib/Mathlib/GroupTheory/SpecificGroups/Cyclic.lean", "module": "Mathlib.GroupTheory.SpecificGroups.Cyclic", "jsonFile": "Mathlib.GroupTheory.SpecificGroups.Cyclic.jsonl", "positionMetadata": {"lineInFile": 342, "tokenPositionInFile": 14436, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 22, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", 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"Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Int.ModEq", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  refine \u27e8fun h \u21a6 h.exists_ofOrder_eq_natCard, fun h \u21a6 ?_\u27e9\n  obtain \u27e8g, hg\u27e9 := h\n  refine isCyclic_of_orderOf_eq_card g ?_\n  simp [hg]", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 140}}
+{"srcContext": "/-\nCopyright (c) 2024 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport Mathlib.Analysis.LocallyConvex.Bounded\nimport Mathlib.Topology.Algebra.Module.Multilinear.Basic\n\n/-!\n# Images of (von Neumann) bounded sets under continuous multilinear maps\n\nIn this file we prove that continuous multilinear maps\nsend von Neumann bounded sets to von Neumann bounded sets.\n\nWe prove 2 versions of the theorem:\none assumes that the index type is nonempty,\nand the other assumes that the codomain is a topological vector space.\n\n## Implementation notes\n\nWe do not assume the index type `\u03b9` to be finite.\nWhile for a nonzero continuous multilinear map\nthe family `\u2200 i, E i` has to be essentially finite\n(more precisely, all but finitely many `E i` has to be trivial),\nproving theorems without a `[Finite \u03b9]` assumption saves us some typeclass searches here and there.\n-/\n\nopen Bornology Filter Set Function\nopen scoped Topology BigOperators\n\nnamespace Bornology.IsVonNBounded\n\nvariable {\u03b9 \ud835\udd5c F : Type*} {E : \u03b9 \u2192 Type*} [NormedField \ud835\udd5c]\n  [\u2200 i, AddCommGroup (E i)] [\u2200 i, Module \ud835\udd5c (E i)] [\u2200 i, TopologicalSpace (E i)]\n  [AddCommGroup F] [Module \ud835\udd5c F] [TopologicalSpace F]\n\n/-- The image of a von Neumann bounded set under a continuous multilinear map\nis von Neumann bounded.\n\nThis version does not assume that the topologies on the domain and on the codomain\nagree with the vector space structure in any way\nbut it assumes that `\u03b9` is nonempty.\n-/\ntheorem image_multilinear' [Nonempty \u03b9] {s : Set (\u2200 i, E i)} (hs : IsVonNBounded \ud835\udd5c s)\n    (f : ContinuousMultilinearMap \ud835\udd5c E F) : IsVonNBounded \ud835\udd5c (f '' s) := fun V hV \u21a6 by\n  classical\n  if h\u2081 : \u2200 c : \ud835\udd5c, \u2016c\u2016 \u2264 1 then\n    exact absorbs_iff_norm.2 \u27e82, fun c hc \u21a6 by linarith [h\u2081 c]\u27e9\n  else\n    let _ : NontriviallyNormedField \ud835\udd5c := \u27e8by simpa using h\u2081\u27e9\n    obtain \u27e8I, t, ht\u2080, hft\u27e9 :\n        \u2203 (I : Finset \u03b9) (t : \u2200 i, Set (E i)), (\u2200 i, t i \u2208 \ud835\udcdd 0) \u2227 Set.pi I t \u2286 f \u207b\u00b9' V := by\n      have hfV : f \u207b\u00b9' V \u2208 \ud835\udcdd 0 := (map_continuous f).tendsto' _ _ f.map_zero hV\n      rwa [nhds_pi, Filter.mem_pi, exists_finite_iff_finset] at hfV\n    have : \u2200 i, \u2203 c : \ud835\udd5c, c \u2260 0 \u2227 \u2200 c' : \ud835\udd5c, \u2016c'\u2016 \u2264 \u2016c\u2016 \u2192 \u2200 x \u2208 s, c' \u2022 x i \u2208 t i := fun i \u21a6 by\n      rw [isVonNBounded_pi_iff] at hs\n      have := (hs i).tendsto_smallSets_nhds.eventually (mem_lift' (ht\u2080 i))\n      rcases NormedAddCommGroup.nhds_zero_basis_norm_lt.eventually_iff.1 this with \u27e8r, hr\u2080, hr\u27e9\n      rcases NormedField.exists_norm_lt \ud835\udd5c hr\u2080 with \u27e8c, hc\u2080, hc\u27e9\n      refine \u27e8c, norm_pos_iff.1 hc\u2080, fun c' hle x hx \u21a6 ?_\u27e9\n      exact hr (hle.trans_lt hc) \u27e8_, \u27e8x, hx, rfl\u27e9, rfl\u27e9\n    choose c hc\u2080 hc using this\n    rw [absorbs_iff_eventually_nhds_zero (mem_of_mem_nhds hV),\n      NormedAddCommGroup.nhds_zero_basis_norm_lt.eventually_iff]\n    have hc\u2080' : \u220f i in I, c i \u2260 0 := Finset.prod_ne_zero_iff.2 fun i _ \u21a6 hc\u2080 i\n    refine \u27e8\u2016\u220f i in I, c i\u2016, norm_pos_iff.2 hc\u2080', fun a ha \u21a6 mapsTo_image_iff.2 fun x hx \u21a6 ?_\u27e9\n    let \u27e8i\u2080\u27e9 := \u2039Nonempty \u03b9\u203a\n    set y := I.piecewise (fun i \u21a6 c i \u2022 x i) x\n    calc\n      a \u2022 f x = f (update y i\u2080 ((a / \u220f i in I, c i) \u2022 y i\u2080)) := by\n        rw [f.map_smul, update_eq_self, f.map_piecewise_smul, div_eq_mul_inv, mul_smul,\n          inv_smul_smul\u2080 hc\u2080']\n      _ \u2208 V := hft fun i hi \u21a6 by\n        rcases eq_or_ne i i\u2080 with rfl | hne\n        \u00b7 simp_rw [update_same, y, I.piecewise_eq_of_mem _ _ hi, smul_smul]\n          refine hc _ _ ?_ _ hx\n          calc\n            \u2016(a / \u220f i in I, c i) * c i\u2016 \u2264 (\u2016\u220f i in I, c i\u2016 / \u2016\u220f i in I, c i\u2016) * \u2016c i\u2016 := by\n              rw [norm_mul, norm_div]; gcongr; exact ha.out.le\n            _ \u2264 1 * \u2016c i\u2016 := by gcongr; apply div_self_le_one\n            _ = \u2016c i\u2016 := one_mul _\n        \u00b7 simp_rw [update_noteq hne, y, I.piecewise_eq_of_mem _ _ hi]\n          exact hc _ _ le_rfl _ hx\n\n", "theoremStatement": "/-- The image of a von Neumann bounded set under a continuous multilinear map\nis von Neumann bounded.\n\nThis version assumes that the codomain is a topological vector space.\n-/\ntheorem image_multilinear [ContinuousSMul \ud835\udd5c F] {s : Set (\u2200 i, E i)} (hs : IsVonNBounded \ud835\udd5c s)\n    (f : ContinuousMultilinearMap \ud835\udd5c E F) : IsVonNBounded \ud835\udd5c (f '' s) ", "theoremName": "Bornology.IsVonNBounded.image_multilinear", "fileCreated": {"commit": "0523bd89e63fc853eb95a502edafe4ce828b88ae", "date": "2024-03-24"}, "theoremCreated": {"commit": "0523bd89e63fc853eb95a502edafe4ce828b88ae", "date": "2024-03-24"}, "file": "mathlib/Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean", "module": "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "jsonFile": "Mathlib.Topology.Algebra.Module.Multilinear.Bounded.jsonl", "positionMetadata": {"lineInFile": 85, "tokenPositionInFile": 3781, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 61, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", 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"Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Data.Set.UnionLift", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.Convex.Function", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Order", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  cases isEmpty_or_nonempty \u03b9 with\n  | inl h =>\n    exact (isBounded_iff_isVonNBounded _).1 <|\n      @Set.Finite.isBounded _ (vonNBornology \ud835\udd5c F) _ (s.toFinite.image _)\n  | inr h => exact hs.image_multilinear' f", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 216}}
+{"srcContext": "/-\nCopyright (c) 2018 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\nimport Mathlib.Algebra.Algebra.Basic\nimport Mathlib.Algebra.Order.BigOperators.Ring.Finset\nimport Mathlib.Algebra.Order.Field.Canonical.Basic\nimport Mathlib.Algebra.Order.Nonneg.Field\nimport Mathlib.Algebra.Order.Nonneg.Floor\nimport Mathlib.Data.Real.Pointwise\nimport Mathlib.Order.ConditionallyCompleteLattice.Group\nimport Mathlib.Tactic.GCongr.Core\n\n#align_import data.real.nnreal from \"leanprover-community/mathlib\"@\"b1abe23ae96fef89ad30d9f4362c307f72a55010\"\n\n/-!\n# Nonnegative real numbers\n\nIn this file we define `NNReal` (notation: `\u211d\u22650`) to be the type of non-negative real numbers,\na.k.a. the interval `[0, \u221e)`. We also define the following operations and structures on `\u211d\u22650`:\n\n* the order on `\u211d\u22650` is the restriction of the order on `\u211d`; these relations define a conditionally\n  complete linear order with a bottom element, `ConditionallyCompleteLinearOrderBot`;\n\n* `a + b` and `a * b` are the restrictions of addition and multiplication of real numbers to `\u211d\u22650`;\n  these operations together with `0 = \u27e80, _\u27e9` and `1 = \u27e81, _\u27e9` turn `\u211d\u22650` into a conditionally\n  complete linear ordered archimedean commutative semifield; we have no typeclass for this in\n  `mathlib` yet, so we define the following instances instead:\n\n  - `LinearOrderedSemiring \u211d\u22650`;\n  - `OrderedCommSemiring \u211d\u22650`;\n  - `CanonicallyOrderedCommSemiring \u211d\u22650`;\n  - `LinearOrderedCommGroupWithZero \u211d\u22650`;\n  - `CanonicallyLinearOrderedAddCommMonoid \u211d\u22650`;\n  - `Archimedean \u211d\u22650`;\n  - `ConditionallyCompleteLinearOrderBot \u211d\u22650`.\n\n  These instances are derived from corresponding instances about the type `{x : \u03b1 // 0 \u2264 x}` in an\n  appropriate ordered field/ring/group/monoid `\u03b1`, see `Mathlib.Algebra.Order.Nonneg.Ring`.\n\n* `Real.toNNReal x` is defined as `\u27e8max x 0, _\u27e9`, i.e. `\u2191(Real.toNNReal x) = x` when `0 \u2264 x` and\n  `\u2191(Real.toNNReal x) = 0` otherwise.\n\nWe also define an instance `CanLift \u211d \u211d\u22650`. This instance can be used by the `lift` tactic to\nreplace `x : \u211d` and `hx : 0 \u2264 x` in the proof context with `x : \u211d\u22650` while replacing all occurrences\nof `x` with `\u2191x`. This tactic also works for a function `f : \u03b1 \u2192 \u211d` with a hypothesis\n`hf : \u2200 x, 0 \u2264 f x`.\n\n## Notations\n\nThis file defines `\u211d\u22650` as a localized notation for `NNReal`.\n-/\n\nopen BigOperators Function\n\n-- to ensure these instances are computable\n/-- Nonnegative real numbers. -/\ndef NNReal := { r : \u211d // 0 \u2264 r } deriving\n  Zero, One, Semiring, StrictOrderedSemiring, CommMonoidWithZero, CommSemiring,\n  SemilatticeInf, SemilatticeSup, DistribLattice, OrderedCommSemiring,\n  CanonicallyOrderedCommSemiring, Inhabited\n#align nnreal NNReal\n\nnamespace NNReal\n\nscoped notation \"\u211d\u22650\" => NNReal\n\nnoncomputable instance : FloorSemiring \u211d\u22650 := Nonneg.floorSemiring\ninstance instDenselyOrdered : DenselyOrdered \u211d\u22650 := Nonneg.instDenselyOrdered\ninstance : OrderBot \u211d\u22650 := inferInstance\ninstance : Archimedean \u211d\u22650 := Nonneg.archimedean\nnoncomputable instance : Sub \u211d\u22650 := Nonneg.sub\nnoncomputable instance : OrderedSub \u211d\u22650 := Nonneg.orderedSub\n\nnoncomputable instance : CanonicallyLinearOrderedSemifield \u211d\u22650 :=\n  Nonneg.canonicallyLinearOrderedSemifield\n\n/-- Coercion `\u211d\u22650 \u2192 \u211d`. -/\n@[coe] def toReal : \u211d\u22650 \u2192 \u211d := Subtype.val\n\ninstance : Coe \u211d\u22650 \u211d := \u27e8toReal\u27e9\n\n-- Simp lemma to put back `n.val` into the normal form given by the coercion.\n@[simp]\ntheorem val_eq_coe (n : \u211d\u22650) : n.val = n :=\n  rfl\n#align nnreal.val_eq_coe NNReal.val_eq_coe\n\ninstance canLift : CanLift \u211d \u211d\u22650 toReal fun r => 0 \u2264 r :=\n  Subtype.canLift _\n#align nnreal.can_lift NNReal.canLift\n\n@[ext] protected theorem eq {n m : \u211d\u22650} : (n : \u211d) = (m : \u211d) \u2192 n = m :=\n  Subtype.eq\n#align nnreal.eq NNReal.eq\n\nprotected theorem eq_iff {n m : \u211d\u22650} : (n : \u211d) = (m : \u211d) \u2194 n = m :=\n  Subtype.ext_iff.symm\n#align nnreal.eq_iff NNReal.eq_iff\n\ntheorem ne_iff {x y : \u211d\u22650} : (x : \u211d) \u2260 (y : \u211d) \u2194 x \u2260 y :=\n  not_congr <| NNReal.eq_iff\n#align nnreal.ne_iff NNReal.ne_iff\n\nprotected theorem \u00abforall\u00bb {p : \u211d\u22650 \u2192 Prop} :\n    (\u2200 x : \u211d\u22650, p x) \u2194 \u2200 (x : \u211d) (hx : 0 \u2264 x), p \u27e8x, hx\u27e9 :=\n  Subtype.forall\n#align nnreal.forall NNReal.forall\n\nprotected theorem \u00abexists\u00bb {p : \u211d\u22650 \u2192 Prop} :\n    (\u2203 x : \u211d\u22650, p x) \u2194 \u2203 (x : \u211d) (hx : 0 \u2264 x), p \u27e8x, hx\u27e9 :=\n  Subtype.exists\n#align nnreal.exists NNReal.exists\n\n/-- Reinterpret a real number `r` as a non-negative real number. Returns `0` if `r < 0`. -/\nnoncomputable def _root_.Real.toNNReal (r : \u211d) : \u211d\u22650 :=\n  \u27e8max r 0, le_max_right _ _\u27e9\n#align real.to_nnreal Real.toNNReal\n\ntheorem _root_.Real.coe_toNNReal (r : \u211d) (hr : 0 \u2264 r) : (Real.toNNReal r : \u211d) = r :=\n  max_eq_left hr\n#align real.coe_to_nnreal Real.coe_toNNReal\n\ntheorem _root_.Real.toNNReal_of_nonneg {r : \u211d} (hr : 0 \u2264 r) : r.toNNReal = \u27e8r, hr\u27e9 := by\n  simp_rw [Real.toNNReal, max_eq_left hr]\n#align real.to_nnreal_of_nonneg Real.toNNReal_of_nonneg\n\ntheorem _root_.Real.le_coe_toNNReal (r : \u211d) : r \u2264 Real.toNNReal r :=\n  le_max_left r 0\n#align real.le_coe_to_nnreal Real.le_coe_toNNReal\n\ntheorem coe_nonneg (r : \u211d\u22650) : (0 : \u211d) \u2264 r := r.2\n#align nnreal.coe_nonneg NNReal.coe_nonneg\n\n@[simp, norm_cast] theorem coe_mk (a : \u211d) (ha) : toReal \u27e8a, ha\u27e9 = a := rfl\n#align nnreal.coe_mk NNReal.coe_mk\n\nexample : Zero \u211d\u22650 := by infer_instance\n\nexample : One \u211d\u22650 := by infer_instance\n\nexample : Add \u211d\u22650 := by infer_instance\n\nnoncomputable example : Sub \u211d\u22650 := by infer_instance\n\nexample : Mul \u211d\u22650 := by infer_instance\n\nnoncomputable example : Inv \u211d\u22650 := by infer_instance\n\nnoncomputable example : Div \u211d\u22650 := by infer_instance\n\nexample : LE \u211d\u22650 := by infer_instance\n\nexample : Bot \u211d\u22650 := by infer_instance\n\nexample : Inhabited \u211d\u22650 := by infer_instance\n\nexample : Nontrivial \u211d\u22650 := by infer_instance\n\nprotected theorem coe_injective : Injective ((\u2191) : \u211d\u22650 \u2192 \u211d) := Subtype.coe_injective\n#align nnreal.coe_injective NNReal.coe_injective\n\n@[simp, norm_cast] lemma coe_inj {r\u2081 r\u2082 : \u211d\u22650} : (r\u2081 : \u211d) = r\u2082 \u2194 r\u2081 = r\u2082 :=\n  NNReal.coe_injective.eq_iff\n#align nnreal.coe_eq NNReal.coe_inj\n\n-- 2024-02-03\n@[deprecated] protected alias coe_eq := coe_inj\n\n@[simp, norm_cast] lemma coe_zero : ((0 : \u211d\u22650) : \u211d) = 0 := rfl\n#align nnreal.coe_zero NNReal.coe_zero\n\n@[simp, norm_cast] lemma coe_one : ((1 : \u211d\u22650) : \u211d) = 1 := rfl\n#align nnreal.coe_one NNReal.coe_one\n\n@[simp, norm_cast]\nprotected theorem coe_add (r\u2081 r\u2082 : \u211d\u22650) : ((r\u2081 + r\u2082 : \u211d\u22650) : \u211d) = r\u2081 + r\u2082 :=\n  rfl\n#align nnreal.coe_add NNReal.coe_add\n\n@[simp, norm_cast]\nprotected theorem coe_mul (r\u2081 r\u2082 : \u211d\u22650) : ((r\u2081 * r\u2082 : \u211d\u22650) : \u211d) = r\u2081 * r\u2082 :=\n  rfl\n#align nnreal.coe_mul NNReal.coe_mul\n\n@[simp, norm_cast]\nprotected theorem coe_inv (r : \u211d\u22650) : ((r\u207b\u00b9 : \u211d\u22650) : \u211d) = (r : \u211d)\u207b\u00b9 :=\n  rfl\n#align nnreal.coe_inv NNReal.coe_inv\n\n@[simp, norm_cast]\nprotected theorem coe_div (r\u2081 r\u2082 : \u211d\u22650) : ((r\u2081 / r\u2082 : \u211d\u22650) : \u211d) = (r\u2081 : \u211d) / r\u2082 :=\n  rfl\n#align nnreal.coe_div NNReal.coe_div\n\n#noalign nnreal.coe_bit0\n#noalign nnreal.coe_bit1\n\nprotected theorem coe_two : ((2 : \u211d\u22650) : \u211d) = 2 := rfl\n#align nnreal.coe_two NNReal.coe_two\n\n@[simp, norm_cast]\nprotected theorem coe_sub {r\u2081 r\u2082 : \u211d\u22650} (h : r\u2082 \u2264 r\u2081) : ((r\u2081 - r\u2082 : \u211d\u22650) : \u211d) = \u2191r\u2081 - \u2191r\u2082 :=\n  max_eq_left <| le_sub_comm.2 <| by simp [show (r\u2082 : \u211d) \u2264 r\u2081 from h]\n#align nnreal.coe_sub NNReal.coe_sub\n\nvariable {r r\u2081 r\u2082 : \u211d\u22650} {x y : \u211d}\n\n@[simp, norm_cast] lemma coe_eq_zero : (r : \u211d) = 0 \u2194 r = 0 := by rw [\u2190 coe_zero, coe_inj]\n#align coe_eq_zero NNReal.coe_eq_zero\n\n@[simp, norm_cast] lemma coe_eq_one : (r : \u211d) = 1 \u2194 r = 1 := by rw [\u2190 coe_one, coe_inj]\n#align coe_inj_one NNReal.coe_eq_one\n\n@[norm_cast] lemma coe_ne_zero : (r : \u211d) \u2260 0 \u2194 r \u2260 0 := coe_eq_zero.not\n#align nnreal.coe_ne_zero NNReal.coe_ne_zero\n\n@[norm_cast] lemma coe_ne_one : (r : \u211d) \u2260 1 \u2194 r \u2260 1 := coe_eq_one.not\n\nexample : CommSemiring \u211d\u22650 := by infer_instance\n\n/-- Coercion `\u211d\u22650 \u2192 \u211d` as a `RingHom`.\n\nPorting note (#11215): TODO: what if we define `Coe \u211d\u22650 \u211d` using this function? -/\ndef toRealHom : \u211d\u22650 \u2192+* \u211d where\n  toFun := (\u2191)\n  map_one' := NNReal.coe_one\n  map_mul' := NNReal.coe_mul\n  map_zero' := NNReal.coe_zero\n  map_add' := NNReal.coe_add\n#align nnreal.to_real_hom NNReal.toRealHom\n\n@[simp] theorem coe_toRealHom : \u21d1toRealHom = toReal := rfl\n#align nnreal.coe_to_real_hom NNReal.coe_toRealHom\n\nsection Actions\n\n/-- A `MulAction` over `\u211d` restricts to a `MulAction` over `\u211d\u22650`. -/\ninstance {M : Type*} [MulAction \u211d M] : MulAction \u211d\u22650 M :=\n  MulAction.compHom M toRealHom.toMonoidHom\n\ntheorem smul_def {M : Type*} [MulAction \u211d M] (c : \u211d\u22650) (x : M) : c \u2022 x = (c : \u211d) \u2022 x :=\n  rfl\n#align nnreal.smul_def NNReal.smul_def\n\ninstance {M N : Type*} [MulAction \u211d M] [MulAction \u211d N] [SMul M N] [IsScalarTower \u211d M N] :\n    IsScalarTower \u211d\u22650 M N where smul_assoc r := (smul_assoc (r : \u211d) : _)\n\ninstance smulCommClass_left {M N : Type*} [MulAction \u211d N] [SMul M N] [SMulCommClass \u211d M N] :\n    SMulCommClass \u211d\u22650 M N where smul_comm r := (smul_comm (r : \u211d) : _)\n#align nnreal.smul_comm_class_left NNReal.smulCommClass_left\n\ninstance smulCommClass_right {M N : Type*} [MulAction \u211d N] [SMul M N] [SMulCommClass M \u211d N] :\n    SMulCommClass M \u211d\u22650 N where smul_comm m r := (smul_comm m (r : \u211d) : _)\n#align nnreal.smul_comm_class_right NNReal.smulCommClass_right\n\n/-- A `DistribMulAction` over `\u211d` restricts to a `DistribMulAction` over `\u211d\u22650`. -/\ninstance {M : Type*} [AddMonoid M] [DistribMulAction \u211d M] : DistribMulAction \u211d\u22650 M :=\n  DistribMulAction.compHom M toRealHom.toMonoidHom\n\n/-- A `Module` over `\u211d` restricts to a `Module` over `\u211d\u22650`. -/\ninstance {M : Type*} [AddCommMonoid M] [Module \u211d M] : Module \u211d\u22650 M :=\n  Module.compHom M toRealHom\n\n-- Porting note (#11215): TODO: after this line, `\u2191` uses `Algebra.cast` instead of `toReal`\n/-- An `Algebra` over `\u211d` restricts to an `Algebra` over `\u211d\u22650`. -/\ninstance {A : Type*} [Semiring A] [Algebra \u211d A] : Algebra \u211d\u22650 A where\n  smul := (\u00b7 \u2022 \u00b7)\n  commutes' r x := by simp [Algebra.commutes]\n  smul_def' r x := by simp [\u2190 Algebra.smul_def (r : \u211d) x, smul_def]\n  toRingHom := (algebraMap \u211d A).comp (toRealHom : \u211d\u22650 \u2192+* \u211d)\n\ninstance : StarRing \u211d\u22650 := starRingOfComm\n\ninstance : TrivialStar \u211d\u22650 where\n  star_trivial _ := rfl\n\ninstance : StarModule \u211d\u22650 \u211d where\n  star_smul := by simp only [star_trivial, eq_self_iff_true, forall_const]\n\n-- verify that the above produces instances we might care about\nexample : Algebra \u211d\u22650 \u211d := by infer_instance\n\nexample : DistribMulAction \u211d\u22650\u02e3 \u211d := by infer_instance\n\nend Actions\n\nexample : MonoidWithZero \u211d\u22650 := by infer_instance\n\nexample : CommMonoidWithZero \u211d\u22650 := by infer_instance\n\nnoncomputable example : CommGroupWithZero \u211d\u22650 := by infer_instance\n\n@[simp, norm_cast]\ntheorem coe_indicator {\u03b1} (s : Set \u03b1) (f : \u03b1 \u2192 \u211d\u22650) (a : \u03b1) :\n    ((s.indicator f a : \u211d\u22650) : \u211d) = s.indicator (fun x => \u2191(f x)) a :=\n  (toRealHom : \u211d\u22650 \u2192+ \u211d).map_indicator _ _ _\n#align nnreal.coe_indicator NNReal.coe_indicator\n\n@[simp, norm_cast]\ntheorem coe_pow (r : \u211d\u22650) (n : \u2115) : ((r ^ n : \u211d\u22650) : \u211d) = (r : \u211d) ^ n := rfl\n#align nnreal.coe_pow NNReal.coe_pow\n\n@[simp, norm_cast]\ntheorem coe_zpow (r : \u211d\u22650) (n : \u2124) : ((r ^ n : \u211d\u22650) : \u211d) = (r : \u211d) ^ n := rfl\n#align nnreal.coe_zpow NNReal.coe_zpow\n\n@[norm_cast]\ntheorem coe_list_sum (l : List \u211d\u22650) : ((l.sum : \u211d\u22650) : \u211d) = (l.map (\u2191)).sum :=\n  map_list_sum toRealHom l\n#align nnreal.coe_list_sum NNReal.coe_list_sum\n\n@[norm_cast]\ntheorem coe_list_prod (l : List \u211d\u22650) : ((l.prod : \u211d\u22650) : \u211d) = (l.map (\u2191)).prod :=\n  map_list_prod toRealHom l\n#align nnreal.coe_list_prod NNReal.coe_list_prod\n\n@[norm_cast]\ntheorem coe_multiset_sum (s : Multiset \u211d\u22650) : ((s.sum : \u211d\u22650) : \u211d) = (s.map (\u2191)).sum :=\n  map_multiset_sum toRealHom s\n#align nnreal.coe_multiset_sum NNReal.coe_multiset_sum\n\n@[norm_cast]\ntheorem coe_multiset_prod (s : Multiset \u211d\u22650) : ((s.prod : \u211d\u22650) : \u211d) = (s.map (\u2191)).prod :=\n  map_multiset_prod toRealHom s\n#align nnreal.coe_multiset_prod NNReal.coe_multiset_prod\n\n@[norm_cast]\ntheorem coe_sum {\u03b1} {s : Finset \u03b1} {f : \u03b1 \u2192 \u211d\u22650} : \u2191(\u2211 a in s, f a) = \u2211 a in s, (f a : \u211d) :=\n  map_sum toRealHom _ _\n#align nnreal.coe_sum NNReal.coe_sum\n\ntheorem _root_.Real.toNNReal_sum_of_nonneg {\u03b1} {s : Finset \u03b1} {f : \u03b1 \u2192 \u211d}\n    (hf : \u2200 a, a \u2208 s \u2192 0 \u2264 f a) :\n    Real.toNNReal (\u2211 a in s, f a) = \u2211 a in s, Real.toNNReal (f a) := by\n  rw [\u2190 coe_inj, NNReal.coe_sum, Real.coe_toNNReal _ (Finset.sum_nonneg hf)]\n  exact Finset.sum_congr rfl fun x hxs => by rw [Real.coe_toNNReal _ (hf x hxs)]\n#align real.to_nnreal_sum_of_nonneg Real.toNNReal_sum_of_nonneg\n\n@[norm_cast]\ntheorem coe_prod {\u03b1} {s : Finset \u03b1} {f : \u03b1 \u2192 \u211d\u22650} : \u2191(\u220f a in s, f a) = \u220f a in s, (f a : \u211d) :=\n  map_prod toRealHom _ _\n#align nnreal.coe_prod NNReal.coe_prod\n\ntheorem _root_.Real.toNNReal_prod_of_nonneg {\u03b1} {s : Finset \u03b1} {f : \u03b1 \u2192 \u211d}\n    (hf : \u2200 a, a \u2208 s \u2192 0 \u2264 f a) :\n    Real.toNNReal (\u220f a in s, f a) = \u220f a in s, Real.toNNReal (f a) := by\n  rw [\u2190 coe_inj, NNReal.coe_prod, Real.coe_toNNReal _ (Finset.prod_nonneg hf)]\n  exact Finset.prod_congr rfl fun x hxs => by rw [Real.coe_toNNReal _ (hf x hxs)]\n#align real.to_nnreal_prod_of_nonneg Real.toNNReal_prod_of_nonneg\n\n-- Porting note (#11215): TODO: `simp`? `norm_cast`?\ntheorem coe_nsmul (r : \u211d\u22650) (n : \u2115) : \u2191(n \u2022 r) = n \u2022 (r : \u211d) := rfl\n#align nnreal.nsmul_coe NNReal.coe_nsmul\n\n@[simp, norm_cast]\nprotected theorem coe_natCast (n : \u2115) : (\u2191(\u2191n : \u211d\u22650) : \u211d) = n :=\n  map_natCast toRealHom n\n#align nnreal.coe_nat_cast NNReal.coe_natCast\n\n-- See note [no_index around OfNat.ofNat]\n@[simp, norm_cast]\nprotected theorem coe_ofNat (n : \u2115) [n.AtLeastTwo] :\n    (no_index (OfNat.ofNat n : \u211d\u22650) : \u211d) = OfNat.ofNat n :=\n  rfl\n\nnoncomputable example : LinearOrder \u211d\u22650 := by infer_instance\n\n@[simp, norm_cast] lemma coe_le_coe : (r\u2081 : \u211d) \u2264 r\u2082 \u2194 r\u2081 \u2264 r\u2082 := Iff.rfl\n#align nnreal.coe_le_coe NNReal.coe_le_coe\n\n@[simp, norm_cast] lemma coe_lt_coe : (r\u2081 : \u211d) < r\u2082 \u2194 r\u2081 < r\u2082 := Iff.rfl\n#align nnreal.coe_lt_coe NNReal.coe_lt_coe\n\n@[simp, norm_cast] lemma coe_pos : (0 : \u211d) < r \u2194 0 < r := Iff.rfl\n#align nnreal.coe_pos NNReal.coe_pos\n\n@[simp, norm_cast] lemma one_le_coe : 1 \u2264 (r : \u211d) \u2194 1 \u2264 r := by rw [\u2190 coe_le_coe, coe_one]\n@[simp, norm_cast] lemma one_lt_coe : 1 < (r : \u211d) \u2194 1 < r := by rw [\u2190 coe_lt_coe, coe_one]\n@[simp, norm_cast] lemma coe_le_one : (r : \u211d) \u2264 1 \u2194 r \u2264 1 := by rw [\u2190 coe_le_coe, coe_one]\n@[simp, norm_cast] lemma coe_lt_one : (r : \u211d) < 1 \u2194 r < 1 := by rw [\u2190 coe_lt_coe, coe_one]\n\n@[mono] lemma coe_mono : Monotone ((\u2191) : \u211d\u22650 \u2192 \u211d) := fun _ _ => NNReal.coe_le_coe.2\n#align nnreal.coe_mono NNReal.coe_mono\n\n/-- Alias for the use of `gcongr` -/\n@[gcongr] alias \u27e8_, GCongr.toReal_le_toReal\u27e9 := coe_le_coe\n\nprotected theorem _root_.Real.toNNReal_mono : Monotone Real.toNNReal := fun _ _ h =>\n  max_le_max h (le_refl 0)\n#align real.to_nnreal_mono Real.toNNReal_mono\n\n@[simp]\ntheorem _root_.Real.toNNReal_coe {r : \u211d\u22650} : Real.toNNReal r = r :=\n  NNReal.eq <| max_eq_left r.2\n#align real.to_nnreal_coe Real.toNNReal_coe\n\n@[simp]\ntheorem mk_natCast (n : \u2115) : @Eq \u211d\u22650 (\u27e8(n : \u211d), n.cast_nonneg\u27e9 : \u211d\u22650) n :=\n  NNReal.eq (NNReal.coe_natCast n).symm\n#align nnreal.mk_coe_nat NNReal.mk_natCast\n\n-- 2024-04-05\n@[deprecated] alias mk_coe_nat := mk_natCast\n\n-- Porting note: place this in the `Real` namespace\n@[simp]\ntheorem toNNReal_coe_nat (n : \u2115) : Real.toNNReal n = n :=\n  NNReal.eq <| by simp [Real.coe_toNNReal]\n#align nnreal.to_nnreal_coe_nat NNReal.toNNReal_coe_nat\n\n-- See note [no_index around OfNat.ofNat]\n@[simp]\ntheorem _root_.Real.toNNReal_ofNat (n : \u2115) [n.AtLeastTwo] :\n    Real.toNNReal (no_index (OfNat.ofNat n)) = OfNat.ofNat n :=\n  toNNReal_coe_nat n\n\n/-- `Real.toNNReal` and `NNReal.toReal : \u211d\u22650 \u2192 \u211d` form a Galois insertion. -/\nnoncomputable def gi : GaloisInsertion Real.toNNReal (\u2191) :=\n  GaloisInsertion.monotoneIntro NNReal.coe_mono Real.toNNReal_mono Real.le_coe_toNNReal fun _ =>\n    Real.toNNReal_coe\n#align nnreal.gi NNReal.gi\n\n-- note that anything involving the (decidability of the) linear order,\n-- will be noncomputable, everything else should not be.\nexample : OrderBot \u211d\u22650 := by infer_instance\n\nexample : PartialOrder \u211d\u22650 := by infer_instance\n\nnoncomputable example : CanonicallyLinearOrderedAddCommMonoid \u211d\u22650 := by infer_instance\n\nnoncomputable example : LinearOrderedAddCommMonoid \u211d\u22650 := by infer_instance\n\nexample : DistribLattice \u211d\u22650 := by infer_instance\n\nexample : SemilatticeInf \u211d\u22650 := by infer_instance\n\nexample : SemilatticeSup \u211d\u22650 := by infer_instance\n\nnoncomputable example : LinearOrderedSemiring \u211d\u22650 := by infer_instance\n\nexample : OrderedCommSemiring \u211d\u22650 := by infer_instance\n\nnoncomputable example : LinearOrderedCommMonoid \u211d\u22650 := by infer_instance\n\nnoncomputable example : LinearOrderedCommMonoidWithZero \u211d\u22650 := by infer_instance\n\nnoncomputable example : LinearOrderedCommGroupWithZero \u211d\u22650 := by infer_instance\n\nexample : CanonicallyOrderedCommSemiring \u211d\u22650 := by infer_instance\n\nexample : DenselyOrdered \u211d\u22650 := by infer_instance\n\nexample : NoMaxOrder \u211d\u22650 := by infer_instance\n\ninstance instPosSMulStrictMono {\u03b1} [Preorder \u03b1] [MulAction \u211d \u03b1] [PosSMulStrictMono \u211d \u03b1] :\n    PosSMulStrictMono \u211d\u22650 \u03b1 where\n  elim _r hr _a\u2081 _a\u2082 ha := (smul_lt_smul_of_pos_left ha (coe_pos.2 hr):)\n\ninstance instSMulPosStrictMono {\u03b1} [Zero \u03b1] [Preorder \u03b1] [MulAction \u211d \u03b1] [SMulPosStrictMono \u211d \u03b1] :\n    SMulPosStrictMono \u211d\u22650 \u03b1 where\n  elim _a ha _r\u2081 _r\u2082 hr := (smul_lt_smul_of_pos_right (coe_lt_coe.2 hr) ha:)\n\n/-- If `a` is a nonnegative real number, then the closed interval `[0, a]` in `\u211d` is order\nisomorphic to the interval `Set.Iic a`. -/\n-- Porting note (#11215): TODO: restore once `simps` supports `\u211d\u22650` @[simps!? apply_coe_coe]\ndef orderIsoIccZeroCoe (a : \u211d\u22650) : Set.Icc (0 : \u211d) a \u2243o Set.Iic a where\n  toEquiv := Equiv.Set.sep (Set.Ici 0) fun x : \u211d => x \u2264 a\n  map_rel_iff' := Iff.rfl\n#align nnreal.order_iso_Icc_zero_coe NNReal.orderIsoIccZeroCoe\n\n@[simp]\ntheorem orderIsoIccZeroCoe_apply_coe_coe (a : \u211d\u22650) (b : Set.Icc (0 : \u211d) a) :\n    (orderIsoIccZeroCoe a b : \u211d) = b :=\n  rfl\n\n@[simp]\ntheorem orderIsoIccZeroCoe_symm_apply_coe (a : \u211d\u22650) (b : Set.Iic a) :\n    ((orderIsoIccZeroCoe a).symm b : \u211d) = b :=\n  rfl\n#align nnreal.order_iso_Icc_zero_coe_symm_apply_coe NNReal.orderIsoIccZeroCoe_symm_apply_coe\n\n-- note we need the `@` to make the `Membership.mem` have a sensible type\ntheorem coe_image {s : Set \u211d\u22650} :\n    (\u2191) '' s = { x : \u211d | \u2203 h : 0 \u2264 x, @Membership.mem \u211d\u22650 _ _ \u27e8x, h\u27e9 s } :=\n  Subtype.coe_image\n#align nnreal.coe_image NNReal.coe_image\n\ntheorem bddAbove_coe {s : Set \u211d\u22650} : BddAbove (((\u2191) : \u211d\u22650 \u2192 \u211d) '' s) \u2194 BddAbove s :=\n  Iff.intro\n    (fun \u27e8b, hb\u27e9 =>\n      \u27e8Real.toNNReal b, fun \u27e8y, _\u27e9 hys =>\n        show y \u2264 max b 0 from le_max_of_le_left <| hb <| Set.mem_image_of_mem _ hys\u27e9)\n    fun \u27e8b, hb\u27e9 => \u27e8b, fun _ \u27e8_, hx, eq\u27e9 => eq \u25b8 hb hx\u27e9\n#align nnreal.bdd_above_coe NNReal.bddAbove_coe\n\ntheorem bddBelow_coe (s : Set \u211d\u22650) : BddBelow (((\u2191) : \u211d\u22650 \u2192 \u211d) '' s) :=\n  \u27e80, fun _ \u27e8q, _, eq\u27e9 => eq \u25b8 q.2\u27e9\n#align nnreal.bdd_below_coe NNReal.bddBelow_coe\n\nnoncomputable instance : ConditionallyCompleteLinearOrderBot \u211d\u22650 :=\n  Nonneg.conditionallyCompleteLinearOrderBot 0\n\n@[norm_cast]\ntheorem coe_sSup (s : Set \u211d\u22650) : (\u2191(sSup s) : \u211d) = sSup (((\u2191) : \u211d\u22650 \u2192 \u211d) '' s) := by\n  rcases Set.eq_empty_or_nonempty s with rfl|hs\n  \u00b7 simp\n  by_cases H : BddAbove s\n  \u00b7 have A : sSup (Subtype.val '' s) \u2208 Set.Ici 0 := by\n      apply Real.sSup_nonneg\n      rintro - \u27e8y, -, rfl\u27e9\n      exact y.2\n    exact (@subset_sSup_of_within \u211d (Set.Ici (0 : \u211d)) _ _ (_) s hs H A).symm\n  \u00b7 simp only [csSup_of_not_bddAbove H, csSup_empty, bot_eq_zero', NNReal.coe_zero]\n    apply (Real.sSup_of_not_bddAbove ?_).symm\n    contrapose! H\n    exact bddAbove_coe.1 H\n#align nnreal.coe_Sup NNReal.coe_sSup\n\n@[simp, norm_cast] -- Porting note: add `simp`\ntheorem coe_iSup {\u03b9 : Sort*} (s : \u03b9 \u2192 \u211d\u22650) : (\u2191(\u2a06 i, s i) : \u211d) = \u2a06 i, \u2191(s i) := by\n  rw [iSup, iSup, coe_sSup, \u2190 Set.range_comp]; rfl\n#align nnreal.coe_supr NNReal.coe_iSup\n\n@[norm_cast]\ntheorem coe_sInf (s : Set \u211d\u22650) : (\u2191(sInf s) : \u211d) = sInf (((\u2191) : \u211d\u22650 \u2192 \u211d) '' s) := by\n  rcases Set.eq_empty_or_nonempty s with rfl|hs\n  \u00b7 simp only [Set.image_empty, Real.sInf_empty, coe_eq_zero]\n    exact @subset_sInf_emptyset \u211d (Set.Ici (0 : \u211d)) _ _ (_)\n  have A : sInf (Subtype.val '' s) \u2208 Set.Ici 0 := by\n    apply Real.sInf_nonneg\n    rintro - \u27e8y, -, rfl\u27e9\n    exact y.2\n  exact (@subset_sInf_of_within \u211d (Set.Ici (0 : \u211d)) _ _ (_) s hs (OrderBot.bddBelow s) A).symm\n#align nnreal.coe_Inf NNReal.coe_sInf\n\n@[simp]\ntheorem sInf_empty : sInf (\u2205 : Set \u211d\u22650) = 0 := by\n  rw [\u2190 coe_eq_zero, coe_sInf, Set.image_empty, Real.sInf_empty]\n#align nnreal.Inf_empty NNReal.sInf_empty\n\n@[norm_cast]\ntheorem coe_iInf {\u03b9 : Sort*} (s : \u03b9 \u2192 \u211d\u22650) : (\u2191(\u2a05 i, s i) : \u211d) = \u2a05 i, \u2191(s i) := by\n  rw [iInf, iInf, coe_sInf, \u2190 Set.range_comp]; rfl\n#align nnreal.coe_infi NNReal.coe_iInf\n\ntheorem le_iInf_add_iInf {\u03b9 \u03b9' : Sort*} [Nonempty \u03b9] [Nonempty \u03b9'] {f : \u03b9 \u2192 \u211d\u22650} {g : \u03b9' \u2192 \u211d\u22650}\n    {a : \u211d\u22650} (h : \u2200 i j, a \u2264 f i + g j) : a \u2264 (\u2a05 i, f i) + \u2a05 j, g j := by\n  rw [\u2190 NNReal.coe_le_coe, NNReal.coe_add, coe_iInf, coe_iInf]\n  exact le_ciInf_add_ciInf h\n#align nnreal.le_infi_add_infi NNReal.le_iInf_add_iInf\n\nexample : Archimedean \u211d\u22650 := by infer_instance\n\n-- Porting note (#11215): TODO: remove?\ninstance covariant_add : CovariantClass \u211d\u22650 \u211d\u22650 (\u00b7 + \u00b7) (\u00b7 \u2264 \u00b7) := inferInstance\n#align nnreal.covariant_add NNReal.covariant_add\n\ninstance contravariant_add : ContravariantClass \u211d\u22650 \u211d\u22650 (\u00b7 + \u00b7) (\u00b7 < \u00b7) := inferInstance\n#align nnreal.contravariant_add NNReal.contravariant_add\n\ninstance covariant_mul : CovariantClass \u211d\u22650 \u211d\u22650 (\u00b7 * \u00b7) (\u00b7 \u2264 \u00b7) := inferInstance\n#align nnreal.covariant_mul NNReal.covariant_mul\n\n-- Porting note (#11215): TODO: delete?\nnonrec theorem le_of_forall_pos_le_add {a b : \u211d\u22650} (h : \u2200 \u03b5, 0 < \u03b5 \u2192 a \u2264 b + \u03b5) : a \u2264 b :=\n  le_of_forall_pos_le_add h\n#align nnreal.le_of_forall_pos_le_add NNReal.le_of_forall_pos_le_add\n\ntheorem lt_iff_exists_rat_btwn (a b : \u211d\u22650) :\n    a < b \u2194 \u2203 q : \u211a, 0 \u2264 q \u2227 a < Real.toNNReal q \u2227 Real.toNNReal q < b :=\n  Iff.intro\n    (fun h : (\u2191a : \u211d) < (\u2191b : \u211d) =>\n      let \u27e8q, haq, hqb\u27e9 := exists_rat_btwn h\n      have : 0 \u2264 (q : \u211d) := le_trans a.2 <| le_of_lt haq\n      \u27e8q, Rat.cast_nonneg.1 this, by\n        simp [Real.coe_toNNReal _ this, NNReal.coe_lt_coe.symm, haq, hqb]\u27e9)\n    fun \u27e8q, _, haq, hqb\u27e9 => lt_trans haq hqb\n#align nnreal.lt_iff_exists_rat_btwn NNReal.lt_iff_exists_rat_btwn\n\ntheorem bot_eq_zero : (\u22a5 : \u211d\u22650) = 0 := rfl\n#align nnreal.bot_eq_zero NNReal.bot_eq_zero\n\ntheorem mul_sup (a b c : \u211d\u22650) : a * (b \u2294 c) = a * b \u2294 a * c :=\n  mul_max_of_nonneg _ _ <| zero_le a\n#align nnreal.mul_sup NNReal.mul_sup\n\ntheorem sup_mul (a b c : \u211d\u22650) : (a \u2294 b) * c = a * c \u2294 b * c :=\n  max_mul_of_nonneg _ _ <| zero_le c\n#align nnreal.sup_mul NNReal.sup_mul\n\ntheorem mul_finset_sup {\u03b1} (r : \u211d\u22650) (s : Finset \u03b1) (f : \u03b1 \u2192 \u211d\u22650) :\n    r * s.sup f = s.sup fun a => r * f a :=\n  Finset.comp_sup_eq_sup_comp _ (NNReal.mul_sup r) (mul_zero r)\n#align nnreal.mul_finset_sup NNReal.mul_finset_sup\n\ntheorem finset_sup_mul {\u03b1} (s : Finset \u03b1) (f : \u03b1 \u2192 \u211d\u22650) (r : \u211d\u22650) :\n    s.sup f * r = s.sup fun a => f a * r :=\n  Finset.comp_sup_eq_sup_comp (\u00b7 * r) (fun x y => NNReal.sup_mul x y r) (zero_mul r)\n#align nnreal.finset_sup_mul NNReal.finset_sup_mul\n\ntheorem finset_sup_div {\u03b1} {f : \u03b1 \u2192 \u211d\u22650} {s : Finset \u03b1} (r : \u211d\u22650) :\n    s.sup f / r = s.sup fun a => f a / r := by simp only [div_eq_inv_mul, mul_finset_sup]\n#align nnreal.finset_sup_div NNReal.finset_sup_div\n\n@[simp, norm_cast]\ntheorem coe_max (x y : \u211d\u22650) : ((max x y : \u211d\u22650) : \u211d) = max (x : \u211d) (y : \u211d) :=\n  NNReal.coe_mono.map_max\n#align nnreal.coe_max NNReal.coe_max\n\n@[simp, norm_cast]\ntheorem coe_min (x y : \u211d\u22650) : ((min x y : \u211d\u22650) : \u211d) = min (x : \u211d) (y : \u211d) :=\n  NNReal.coe_mono.map_min\n#align nnreal.coe_min NNReal.coe_min\n\n@[simp]\ntheorem zero_le_coe {q : \u211d\u22650} : 0 \u2264 (q : \u211d) :=\n  q.2\n#align nnreal.zero_le_coe NNReal.zero_le_coe\n\ninstance instOrderedSMul {M : Type*} [OrderedAddCommMonoid M] [Module \u211d M] [OrderedSMul \u211d M] :\n    OrderedSMul \u211d\u22650 M where\n  smul_lt_smul_of_pos hab hc := (smul_lt_smul_of_pos_left hab (NNReal.coe_pos.2 hc) : _)\n  lt_of_smul_lt_smul_of_pos {a b c} hab _ :=\n    lt_of_smul_lt_smul_of_nonneg_left (by exact hab) (NNReal.coe_nonneg c)\n\nend NNReal\n\nopen NNReal\n\nnamespace Real\n\nsection ToNNReal\n\n@[simp]\ntheorem coe_toNNReal' (r : \u211d) : (Real.toNNReal r : \u211d) = max r 0 :=\n  rfl\n#align real.coe_to_nnreal' Real.coe_toNNReal'\n\n@[simp]\ntheorem toNNReal_zero : Real.toNNReal 0 = 0 := NNReal.eq <| coe_toNNReal _ le_rfl\n#align real.to_nnreal_zero Real.toNNReal_zero\n\n@[simp]\ntheorem toNNReal_one : Real.toNNReal 1 = 1 := NNReal.eq <| coe_toNNReal _ zero_le_one\n#align real.to_nnreal_one Real.toNNReal_one\n\n@[simp]\ntheorem toNNReal_pos {r : \u211d} : 0 < Real.toNNReal r \u2194 0 < r := by\n  simp [\u2190 NNReal.coe_lt_coe, lt_irrefl]\n#align real.to_nnreal_pos Real.toNNReal_pos\n\n@[simp]\ntheorem toNNReal_eq_zero {r : \u211d} : Real.toNNReal r = 0 \u2194 r \u2264 0 := by\n  simpa [-toNNReal_pos] using not_iff_not.2 (@toNNReal_pos r)\n#align real.to_nnreal_eq_zero Real.toNNReal_eq_zero\n\ntheorem toNNReal_of_nonpos {r : \u211d} : r \u2264 0 \u2192 Real.toNNReal r = 0 :=\n  toNNReal_eq_zero.2\n#align real.to_nnreal_of_nonpos Real.toNNReal_of_nonpos\n\nlemma toNNReal_eq_iff_eq_coe {r : \u211d} {p : \u211d\u22650} (hp : p \u2260 0) : r.toNNReal = p \u2194 r = p :=\n  \u27e8fun h \u21a6 h \u25b8 (coe_toNNReal _ <| not_lt.1 fun hlt \u21a6 hp <| h \u25b8 toNNReal_of_nonpos hlt.le).symm,\n    fun h \u21a6 h.symm \u25b8 toNNReal_coe\u27e9\n\n@[simp]\nlemma toNNReal_eq_one {r : \u211d} : r.toNNReal = 1 \u2194 r = 1 := toNNReal_eq_iff_eq_coe one_ne_zero\n\n@[simp]\nlemma toNNReal_eq_natCast {r : \u211d} {n : \u2115} (hn : n \u2260 0) : r.toNNReal = n \u2194 r = n :=\n  mod_cast toNNReal_eq_iff_eq_coe <| Nat.cast_ne_zero.2 hn\n\n@[simp]\nlemma toNNReal_eq_ofNat {r : \u211d} {n : \u2115} [n.AtLeastTwo] :\n    r.toNNReal = no_index (OfNat.ofNat n) \u2194 r = OfNat.ofNat n :=\n  toNNReal_eq_natCast (NeZero.ne n)\n\n@[simp]\ntheorem toNNReal_le_toNNReal_iff {r p : \u211d} (hp : 0 \u2264 p) :\n    toNNReal r \u2264 toNNReal p \u2194 r \u2264 p := by simp [\u2190 NNReal.coe_le_coe, hp]\n#align real.to_nnreal_le_to_nnreal_iff Real.toNNReal_le_toNNReal_iff\n\n@[simp]\nlemma toNNReal_le_one {r : \u211d} : r.toNNReal \u2264 1 \u2194 r \u2264 1 := by\n  simpa using toNNReal_le_toNNReal_iff zero_le_one\n\n@[simp]\nlemma one_lt_toNNReal {r : \u211d} : 1 < r.toNNReal \u2194 1 < r := by\n  simpa only [not_le] using toNNReal_le_one.not\n\n@[simp]\nlemma toNNReal_le_natCast {r : \u211d} {n : \u2115} : r.toNNReal \u2264 n \u2194 r \u2264 n := by\n  simpa using toNNReal_le_toNNReal_iff n.cast_nonneg\n\n@[simp]\nlemma natCast_lt_toNNReal {r : \u211d} {n : \u2115} : n < r.toNNReal \u2194 n < r := by\n  simpa only [not_le] using toNNReal_le_natCast.not\n\n@[simp]\nlemma toNNReal_le_ofNat {r : \u211d} {n : \u2115} [n.AtLeastTwo] :\n    r.toNNReal \u2264 no_index (OfNat.ofNat n) \u2194 r \u2264 n :=\n  toNNReal_le_natCast\n\n@[simp]\nlemma ofNat_lt_toNNReal {r : \u211d} {n : \u2115} [n.AtLeastTwo] :\n    no_index (OfNat.ofNat n) < r.toNNReal \u2194 n < r :=\n  natCast_lt_toNNReal\n\n@[simp]\ntheorem toNNReal_eq_toNNReal_iff {r p : \u211d} (hr : 0 \u2264 r) (hp : 0 \u2264 p) :\n    toNNReal r = toNNReal p \u2194 r = p := by simp [\u2190 coe_inj, coe_toNNReal, hr, hp]\n#align real.to_nnreal_eq_to_nnreal_iff Real.toNNReal_eq_toNNReal_iff\n\n@[simp]\ntheorem toNNReal_lt_toNNReal_iff' {r p : \u211d} : Real.toNNReal r < Real.toNNReal p \u2194 r < p \u2227 0 < p :=\n  NNReal.coe_lt_coe.symm.trans max_lt_max_left_iff\n#align real.to_nnreal_lt_to_nnreal_iff' Real.toNNReal_lt_toNNReal_iff'\n\ntheorem toNNReal_lt_toNNReal_iff {r p : \u211d} (h : 0 < p) :\n    Real.toNNReal r < Real.toNNReal p \u2194 r < p :=\n  toNNReal_lt_toNNReal_iff'.trans (and_iff_left h)\n#align real.to_nnreal_lt_to_nnreal_iff Real.toNNReal_lt_toNNReal_iff\n\ntheorem toNNReal_lt_toNNReal_iff_of_nonneg {r p : \u211d} (hr : 0 \u2264 r) :\n    Real.toNNReal r < Real.toNNReal p \u2194 r < p :=\n  toNNReal_lt_toNNReal_iff'.trans \u27e8And.left, fun h => \u27e8h, lt_of_le_of_lt hr h\u27e9\u27e9\n#align real.to_nnreal_lt_to_nnreal_iff_of_nonneg Real.toNNReal_lt_toNNReal_iff_of_nonneg\n\nlemma toNNReal_le_toNNReal_iff' {r p : \u211d} : r.toNNReal \u2264 p.toNNReal \u2194 r \u2264 p \u2228 r \u2264 0 := by\n  simp_rw [\u2190 not_lt, toNNReal_lt_toNNReal_iff', not_and_or]\n\nlemma toNNReal_le_toNNReal_iff_of_pos {r p : \u211d} (hr : 0 < r) : r.toNNReal \u2264 p.toNNReal \u2194 r \u2264 p := by\n  simp [toNNReal_le_toNNReal_iff', hr.not_le]\n\n@[simp]\nlemma one_le_toNNReal {r : \u211d} : 1 \u2264 r.toNNReal \u2194 1 \u2264 r := by\n  simpa using toNNReal_le_toNNReal_iff_of_pos one_pos\n\n@[simp]\nlemma toNNReal_lt_one {r : \u211d} : r.toNNReal < 1 \u2194 r < 1 := by simp only [\u2190 not_le, one_le_toNNReal]\n\n@[simp]\nlemma natCastle_toNNReal' {n : \u2115} {r : \u211d} : \u2191n \u2264 r.toNNReal \u2194 n \u2264 r \u2228 n = 0 := by\n  simpa [n.cast_nonneg.le_iff_eq] using toNNReal_le_toNNReal_iff' (r := n)\n\n@[simp]\nlemma toNNReal_lt_natCast' {n : \u2115} {r : \u211d} : r.toNNReal < n \u2194 r < n \u2227 n \u2260 0 := by\n  simpa [pos_iff_ne_zero] using toNNReal_lt_toNNReal_iff' (r := r) (p := n)\n\nlemma natCast_le_toNNReal {n : \u2115} {r : \u211d} (hn : n \u2260 0) : \u2191n \u2264 r.toNNReal \u2194 n \u2264 r := by simp [hn]\n\nlemma toNNReal_lt_natCast {r : \u211d} {n : \u2115} (hn : n \u2260 0) : r.toNNReal < n \u2194 r < n := by simp [hn]\n\n@[simp]\nlemma toNNReal_lt_ofNat {r : \u211d} {n : \u2115} [n.AtLeastTwo] :\n    r.toNNReal < no_index (OfNat.ofNat n) \u2194 r < OfNat.ofNat n :=\n  toNNReal_lt_natCast (NeZero.ne n)\n\n@[simp]\nlemma ofNat_le_toNNReal {n : \u2115} {r : \u211d} [n.AtLeastTwo] :\n    no_index (OfNat.ofNat n) \u2264 r.toNNReal \u2194 OfNat.ofNat n \u2264 r :=\n  natCast_le_toNNReal (NeZero.ne n)\n\n@[simp]\ntheorem toNNReal_add {r p : \u211d} (hr : 0 \u2264 r) (hp : 0 \u2264 p) :\n    Real.toNNReal (r + p) = Real.toNNReal r + Real.toNNReal p :=\n  NNReal.eq <| by simp [hr, hp, add_nonneg]\n#align real.to_nnreal_add Real.toNNReal_add\n\ntheorem toNNReal_add_toNNReal {r p : \u211d} (hr : 0 \u2264 r) (hp : 0 \u2264 p) :\n    Real.toNNReal r + Real.toNNReal p = Real.toNNReal (r + p) :=\n  (Real.toNNReal_add hr hp).symm\n#align real.to_nnreal_add_to_nnreal Real.toNNReal_add_toNNReal\n\ntheorem toNNReal_le_toNNReal {r p : \u211d} (h : r \u2264 p) : Real.toNNReal r \u2264 Real.toNNReal p :=\n  Real.toNNReal_mono h\n#align real.to_nnreal_le_to_nnreal Real.toNNReal_le_toNNReal\n\ntheorem toNNReal_add_le {r p : \u211d} : Real.toNNReal (r + p) \u2264 Real.toNNReal r + Real.toNNReal p :=\n  NNReal.coe_le_coe.1 <| max_le (add_le_add (le_max_left _ _) (le_max_left _ _)) NNReal.zero_le_coe\n#align real.to_nnreal_add_le Real.toNNReal_add_le\n\ntheorem toNNReal_le_iff_le_coe {r : \u211d} {p : \u211d\u22650} : toNNReal r \u2264 p \u2194 r \u2264 \u2191p :=\n  NNReal.gi.gc r p\n#align real.to_nnreal_le_iff_le_coe Real.toNNReal_le_iff_le_coe\n\ntheorem le_toNNReal_iff_coe_le {r : \u211d\u22650} {p : \u211d} (hp : 0 \u2264 p) : r \u2264 Real.toNNReal p \u2194 \u2191r \u2264 p := by\n  rw [\u2190 NNReal.coe_le_coe, Real.coe_toNNReal p hp]\n#align real.le_to_nnreal_iff_coe_le Real.le_toNNReal_iff_coe_le\n\ntheorem le_toNNReal_iff_coe_le' {r : \u211d\u22650} {p : \u211d} (hr : 0 < r) : r \u2264 Real.toNNReal p \u2194 \u2191r \u2264 p :=\n  (le_or_lt 0 p).elim le_toNNReal_iff_coe_le fun hp => by\n    simp only [(hp.trans_le r.coe_nonneg).not_le, toNNReal_eq_zero.2 hp.le, hr.not_le]\n#align real.le_to_nnreal_iff_coe_le' Real.le_toNNReal_iff_coe_le'\n\ntheorem toNNReal_lt_iff_lt_coe {r : \u211d} {p : \u211d\u22650} (ha : 0 \u2264 r) : Real.toNNReal r < p \u2194 r < \u2191p := by\n  rw [\u2190 NNReal.coe_lt_coe, Real.coe_toNNReal r ha]\n#align real.to_nnreal_lt_iff_lt_coe Real.toNNReal_lt_iff_lt_coe\n\ntheorem lt_toNNReal_iff_coe_lt {r : \u211d\u22650} {p : \u211d} : r < Real.toNNReal p \u2194 \u2191r < p :=\n  lt_iff_lt_of_le_iff_le toNNReal_le_iff_le_coe\n#align real.lt_to_nnreal_iff_coe_lt Real.lt_toNNReal_iff_coe_lt\n\n#noalign real.to_nnreal_bit0\n#noalign real.to_nnreal_bit1\n\ntheorem toNNReal_pow {x : \u211d} (hx : 0 \u2264 x) (n : \u2115) : (x ^ n).toNNReal = x.toNNReal ^ n := by\n  rw [\u2190 coe_inj, NNReal.coe_pow, Real.coe_toNNReal _ (pow_nonneg hx _),\n    Real.coe_toNNReal x hx]\n#align real.to_nnreal_pow Real.toNNReal_pow\n\ntheorem toNNReal_mul {p q : \u211d} (hp : 0 \u2264 p) :\n    Real.toNNReal (p * q) = Real.toNNReal p * Real.toNNReal q :=\n  NNReal.eq <| by simp [mul_max_of_nonneg, hp]\n#align real.to_nnreal_mul Real.toNNReal_mul\n\nend ToNNReal\n\nend Real\n\nopen Real\n\nnamespace NNReal\n\nsection Mul\n\ntheorem mul_eq_mul_left {a b c : \u211d\u22650} (h : a \u2260 0) : a * b = a * c \u2194 b = c := by\n  rw [mul_eq_mul_left_iff, or_iff_left h]\n#align nnreal.mul_eq_mul_left NNReal.mul_eq_mul_left\n\nend Mul\n\nsection Pow\n\ntheorem pow_antitone_exp {a : \u211d\u22650} (m n : \u2115) (mn : m \u2264 n) (a1 : a \u2264 1) : a ^ n \u2264 a ^ m :=\n  pow_le_pow_of_le_one (zero_le a) a1 mn\n#align nnreal.pow_antitone_exp NNReal.pow_antitone_exp\n\nnonrec theorem exists_pow_lt_of_lt_one {a b : \u211d\u22650} (ha : 0 < a) (hb : b < 1) :\n    \u2203 n : \u2115, b ^ n < a := by\n  simpa only [\u2190 coe_pow, NNReal.coe_lt_coe] using\n    exists_pow_lt_of_lt_one (NNReal.coe_pos.2 ha) (NNReal.coe_lt_coe.2 hb)\n#align nnreal.exists_pow_lt_of_lt_one NNReal.exists_pow_lt_of_lt_one\n\nnonrec theorem exists_mem_Ico_zpow {x : \u211d\u22650} {y : \u211d\u22650} (hx : x \u2260 0) (hy : 1 < y) :\n    \u2203 n : \u2124, x \u2208 Set.Ico (y ^ n) (y ^ (n + 1)) :=\n  exists_mem_Ico_zpow (\u03b1 := \u211d) hx.bot_lt hy\n#align nnreal.exists_mem_Ico_zpow NNReal.exists_mem_Ico_zpow\n\nnonrec theorem exists_mem_Ioc_zpow {x : \u211d\u22650} {y : \u211d\u22650} (hx : x \u2260 0) (hy : 1 < y) :\n    \u2203 n : \u2124, x \u2208 Set.Ioc (y ^ n) (y ^ (n + 1)) :=\n  exists_mem_Ioc_zpow (\u03b1 := \u211d) hx.bot_lt hy\n#align nnreal.exists_mem_Ioc_zpow NNReal.exists_mem_Ioc_zpow\n\nend Pow\n\nsection Sub\n\n/-!\n### Lemmas about subtraction\n\nIn this section we provide a few lemmas about subtraction that do not fit well into any other\ntypeclass. For lemmas about subtraction and addition see lemmas about `OrderedSub` in the file\n`Mathlib.Algebra.Order.Sub.Basic`. See also `mul_tsub` and `tsub_mul`.\n-/\n\ntheorem sub_def {r p : \u211d\u22650} : r - p = Real.toNNReal (r - p) :=\n  rfl\n#align nnreal.sub_def NNReal.sub_def\n\ntheorem coe_sub_def {r p : \u211d\u22650} : \u2191(r - p) = max (r - p : \u211d) 0 :=\n  rfl\n#align nnreal.coe_sub_def NNReal.coe_sub_def\n\nexample : OrderedSub \u211d\u22650 := by infer_instance\n\ntheorem sub_div (a b c : \u211d\u22650) : (a - b) / c = a / c - b / c :=\n  tsub_div _ _ _\n#align nnreal.sub_div NNReal.sub_div\n\nend Sub\n\nsection Inv\n\n#align nnreal.sum_div Finset.sum_div\n\n@[simp]\ntheorem inv_le {r p : \u211d\u22650} (h : r \u2260 0) : r\u207b\u00b9 \u2264 p \u2194 1 \u2264 r * p := by\n  rw [\u2190 mul_le_mul_left (pos_iff_ne_zero.2 h), mul_inv_cancel h]\n#align nnreal.inv_le NNReal.inv_le\n\ntheorem inv_le_of_le_mul {r p : \u211d\u22650} (h : 1 \u2264 r * p) : r\u207b\u00b9 \u2264 p := by\n  by_cases r = 0 <;> simp [*, inv_le]\n#align nnreal.inv_le_of_le_mul NNReal.inv_le_of_le_mul\n\n@[simp]\ntheorem le_inv_iff_mul_le {r p : \u211d\u22650} (h : p \u2260 0) : r \u2264 p\u207b\u00b9 \u2194 r * p \u2264 1 := by\n  rw [\u2190 mul_le_mul_left (pos_iff_ne_zero.2 h), mul_inv_cancel h, mul_comm]\n#align nnreal.le_inv_iff_mul_le NNReal.le_inv_iff_mul_le\n\n@[simp]\ntheorem lt_inv_iff_mul_lt {r p : \u211d\u22650} (h : p \u2260 0) : r < p\u207b\u00b9 \u2194 r * p < 1 := by\n  rw [\u2190 mul_lt_mul_left (pos_iff_ne_zero.2 h), mul_inv_cancel h, mul_comm]\n#align nnreal.lt_inv_iff_mul_lt NNReal.lt_inv_iff_mul_lt\n\ntheorem mul_le_iff_le_inv {a b r : \u211d\u22650} (hr : r \u2260 0) : r * a \u2264 b \u2194 a \u2264 r\u207b\u00b9 * b := by\n  have : 0 < r := lt_of_le_of_ne (zero_le r) hr.symm\n  rw [\u2190 mul_le_mul_left (inv_pos.mpr this), \u2190 mul_assoc, inv_mul_cancel hr, one_mul]\n#align nnreal.mul_le_iff_le_inv NNReal.mul_le_iff_le_inv\n\ntheorem le_div_iff_mul_le {a b r : \u211d\u22650} (hr : r \u2260 0) : a \u2264 b / r \u2194 a * r \u2264 b :=\n  le_div_iff\u2080 hr\n#align nnreal.le_div_iff_mul_le NNReal.le_div_iff_mul_le\n\ntheorem div_le_iff {a b r : \u211d\u22650} (hr : r \u2260 0) : a / r \u2264 b \u2194 a \u2264 b * r :=\n  div_le_iff\u2080 hr\n#align nnreal.div_le_iff NNReal.div_le_iff\n\nnonrec theorem div_le_iff' {a b r : \u211d\u22650} (hr : r \u2260 0) : a / r \u2264 b \u2194 a \u2264 r * b :=\n  @div_le_iff' \u211d _ a r b <| pos_iff_ne_zero.2 hr\n#align nnreal.div_le_iff' NNReal.div_le_iff'\n\ntheorem div_le_of_le_mul {a b c : \u211d\u22650} (h : a \u2264 b * c) : a / c \u2264 b :=\n  if h0 : c = 0 then by simp [h0] else (div_le_iff h0).2 h\n#align nnreal.div_le_of_le_mul NNReal.div_le_of_le_mul\n\ntheorem div_le_of_le_mul' {a b c : \u211d\u22650} (h : a \u2264 b * c) : a / b \u2264 c :=\n  div_le_of_le_mul <| mul_comm b c \u25b8 h\n#align nnreal.div_le_of_le_mul' NNReal.div_le_of_le_mul'\n\nnonrec theorem le_div_iff {a b r : \u211d\u22650} (hr : r \u2260 0) : a \u2264 b / r \u2194 a * r \u2264 b :=\n  @le_div_iff \u211d _ a b r <| pos_iff_ne_zero.2 hr\n#align nnreal.le_div_iff NNReal.le_div_iff\n\nnonrec theorem le_div_iff' {a b r : \u211d\u22650} (hr : r \u2260 0) : a \u2264 b / r \u2194 r * a \u2264 b :=\n  @le_div_iff' \u211d _ a b r <| pos_iff_ne_zero.2 hr\n#align nnreal.le_div_iff' NNReal.le_div_iff'\n\ntheorem div_lt_iff {a b r : \u211d\u22650} (hr : r \u2260 0) : a / r < b \u2194 a < b * r :=\n  lt_iff_lt_of_le_iff_le (le_div_iff hr)\n#align nnreal.div_lt_iff NNReal.div_lt_iff\n\ntheorem div_lt_iff' {a b r : \u211d\u22650} (hr : r \u2260 0) : a / r < b \u2194 a < r * b :=\n  lt_iff_lt_of_le_iff_le (le_div_iff' hr)\n#align nnreal.div_lt_iff' NNReal.div_lt_iff'\n\ntheorem lt_div_iff {a b r : \u211d\u22650} (hr : r \u2260 0) : a < b / r \u2194 a * r < b :=\n  lt_iff_lt_of_le_iff_le (div_le_iff hr)\n#align nnreal.lt_div_iff NNReal.lt_div_iff\n\ntheorem lt_div_iff' {a b r : \u211d\u22650} (hr : r \u2260 0) : a < b / r \u2194 r * a < b :=\n  lt_iff_lt_of_le_iff_le (div_le_iff' hr)\n#align nnreal.lt_div_iff' NNReal.lt_div_iff'\n\ntheorem mul_lt_of_lt_div {a b r : \u211d\u22650} (h : a < b / r) : a * r < b :=\n  (lt_div_iff fun hr => False.elim <| by simp [hr] at h).1 h\n#align nnreal.mul_lt_of_lt_div NNReal.mul_lt_of_lt_div\n\ntheorem div_le_div_left_of_le {a b c : \u211d\u22650} (c0 : c \u2260 0) (cb : c \u2264 b) :\n    a / b \u2264 a / c :=\n  div_le_div_of_nonneg_left (zero_le _) c0.bot_lt cb\n#align nnreal.div_le_div_left_of_le NNReal.div_le_div_left_of_le\u2093\n\nnonrec theorem div_le_div_left {a b c : \u211d\u22650} (a0 : 0 < a) (b0 : 0 < b) (c0 : 0 < c) :\n    a / b \u2264 a / c \u2194 c \u2264 b :=\n  div_le_div_left a0 b0 c0\n#align nnreal.div_le_div_left NNReal.div_le_div_left\n\ntheorem le_of_forall_lt_one_mul_le {x y : \u211d\u22650} (h : \u2200 a < 1, a * x \u2264 y) : x \u2264 y :=\n  le_of_forall_ge_of_dense fun a ha => by\n    have hx : x \u2260 0 := pos_iff_ne_zero.1 (lt_of_le_of_lt (zero_le _) ha)\n    have hx' : x\u207b\u00b9 \u2260 0 := by rwa [Ne, inv_eq_zero]\n    have : a * x\u207b\u00b9 < 1 := by rwa [\u2190 lt_inv_iff_mul_lt hx', inv_inv]\n    have : a * x\u207b\u00b9 * x \u2264 y := h _ this\n    rwa [mul_assoc, inv_mul_cancel hx, mul_one] at this\n#align nnreal.le_of_forall_lt_one_mul_le NNReal.le_of_forall_lt_one_mul_le\n\nnonrec theorem half_le_self (a : \u211d\u22650) : a / 2 \u2264 a :=\n  half_le_self bot_le\n#align nnreal.half_le_self NNReal.half_le_self\n\nnonrec theorem half_lt_self {a : \u211d\u22650} (h : a \u2260 0) : a / 2 < a :=\n  half_lt_self h.bot_lt\n#align nnreal.half_lt_self NNReal.half_lt_self\n\ntheorem div_lt_one_of_lt {a b : \u211d\u22650} (h : a < b) : a / b < 1 := by\n  rwa [div_lt_iff, one_mul]\n  exact ne_of_gt (lt_of_le_of_lt (zero_le _) h)\n#align nnreal.div_lt_one_of_lt NNReal.div_lt_one_of_lt\n\ntheorem _root_.Real.toNNReal_inv {x : \u211d} : Real.toNNReal x\u207b\u00b9 = (Real.toNNReal x)\u207b\u00b9 := by\n  rcases le_total 0 x with hx | hx\n  \u00b7 nth_rw 1 [\u2190 Real.coe_toNNReal x hx]\n    rw [\u2190 NNReal.coe_inv, Real.toNNReal_coe]\n  \u00b7 rw [toNNReal_eq_zero.mpr hx, inv_zero, toNNReal_eq_zero.mpr (inv_nonpos.mpr hx)]\n#align real.to_nnreal_inv Real.toNNReal_inv\n\ntheorem _root_.Real.toNNReal_div {x y : \u211d} (hx : 0 \u2264 x) :\n    Real.toNNReal (x / y) = Real.toNNReal x / Real.toNNReal y := by\n  rw [div_eq_mul_inv, div_eq_mul_inv, \u2190 Real.toNNReal_inv, \u2190 Real.toNNReal_mul hx]\n#align real.to_nnreal_div Real.toNNReal_div\n\ntheorem _root_.Real.toNNReal_div' {x y : \u211d} (hy : 0 \u2264 y) :\n    Real.toNNReal (x / y) = Real.toNNReal x / Real.toNNReal y := by\n  rw [div_eq_inv_mul, div_eq_inv_mul, Real.toNNReal_mul (inv_nonneg.2 hy), Real.toNNReal_inv]\n#align real.to_nnreal_div' Real.toNNReal_div'\n\ntheorem inv_lt_one_iff {x : \u211d\u22650} (hx : x \u2260 0) : x\u207b\u00b9 < 1 \u2194 1 < x := by\n  rw [\u2190 one_div, div_lt_iff hx, one_mul]\n#align nnreal.inv_lt_one_iff NNReal.inv_lt_one_iff\n\ntheorem zpow_pos {x : \u211d\u22650} (hx : x \u2260 0) (n : \u2124) : 0 < x ^ n :=\n  zpow_pos_of_pos hx.bot_lt _\n#align nnreal.zpow_pos NNReal.zpow_pos\n\ntheorem inv_lt_inv {x y : \u211d\u22650} (hx : x \u2260 0) (h : x < y) : y\u207b\u00b9 < x\u207b\u00b9 :=\n  inv_lt_inv_of_lt hx.bot_lt h\n#align nnreal.inv_lt_inv NNReal.inv_lt_inv\n\nend Inv\n\n@[simp]\ntheorem abs_eq (x : \u211d\u22650) : |(x : \u211d)| = x :=\n  abs_of_nonneg x.property\n#align nnreal.abs_eq NNReal.abs_eq\n\nsection Csupr\n\nopen Set\n\nvariable {\u03b9 : Sort*} {f : \u03b9 \u2192 \u211d\u22650}\n\ntheorem le_toNNReal_of_coe_le {x : \u211d\u22650} {y : \u211d} (h : \u2191x \u2264 y) : x \u2264 y.toNNReal :=\n  (le_toNNReal_iff_coe_le <| x.2.trans h).2 h\n#align nnreal.le_to_nnreal_of_coe_le NNReal.le_toNNReal_of_coe_le\n\nnonrec theorem sSup_of_not_bddAbove {s : Set \u211d\u22650} (hs : \u00acBddAbove s) : SupSet.sSup s = 0 := by\n  rw [\u2190 bddAbove_coe] at hs\n  rw [\u2190 coe_inj, coe_sSup, NNReal.coe_zero]\n  exact sSup_of_not_bddAbove hs\n#align nnreal.Sup_of_not_bdd_above NNReal.sSup_of_not_bddAbove\n\ntheorem iSup_of_not_bddAbove (hf : \u00acBddAbove (range f)) : \u2a06 i, f i = 0 :=\n  sSup_of_not_bddAbove hf\n#align nnreal.supr_of_not_bdd_above NNReal.iSup_of_not_bddAbove\n\ntheorem iSup_empty [IsEmpty \u03b9] (f : \u03b9 \u2192 \u211d\u22650) : \u2a06 i, f i = 0 := ciSup_of_empty f\n\ntheorem iInf_empty [IsEmpty \u03b9] (f : \u03b9 \u2192 \u211d\u22650) : \u2a05 i, f i = 0 := by\n  rw [_root_.iInf_of_isEmpty, sInf_empty]\n#align nnreal.infi_empty NNReal.iInf_empty\n\n@[simp]\ntheorem iInf_const_zero {\u03b1 : Sort*} : \u2a05 _ : \u03b1, (0 : \u211d\u22650) = 0 := by\n  rw [\u2190 coe_inj, coe_iInf]\n  exact Real.ciInf_const_zero\n#align nnreal.infi_const_zero NNReal.iInf_const_zero\n\ntheorem iInf_mul (f : \u03b9 \u2192 \u211d\u22650) (a : \u211d\u22650) : iInf f * a = \u2a05 i, f i * a := by\n  rw [\u2190 coe_inj, NNReal.coe_mul, coe_iInf, coe_iInf]\n  exact Real.iInf_mul_of_nonneg (NNReal.coe_nonneg _) _\n#align nnreal.infi_mul NNReal.iInf_mul\n\ntheorem mul_iInf (f : \u03b9 \u2192 \u211d\u22650) (a : \u211d\u22650) : a * iInf f = \u2a05 i, a * f i := by\n  simpa only [mul_comm] using iInf_mul f a\n#align nnreal.mul_infi NNReal.mul_iInf\n\ntheorem mul_iSup (f : \u03b9 \u2192 \u211d\u22650) (a : \u211d\u22650) : (a * \u2a06 i, f i) = \u2a06 i, a * f i := by\n  rw [\u2190 coe_inj, NNReal.coe_mul, NNReal.coe_iSup, NNReal.coe_iSup]\n  exact Real.mul_iSup_of_nonneg (NNReal.coe_nonneg _) _\n#align nnreal.mul_supr NNReal.mul_iSup\n\ntheorem iSup_mul (f : \u03b9 \u2192 \u211d\u22650) (a : \u211d\u22650) : (\u2a06 i, f i) * a = \u2a06 i, f i * a := by\n  rw [mul_comm, mul_iSup]\n  simp_rw [mul_comm]\n#align nnreal.supr_mul NNReal.iSup_mul\n\ntheorem iSup_div (f : \u03b9 \u2192 \u211d\u22650) (a : \u211d\u22650) : (\u2a06 i, f i) / a = \u2a06 i, f i / a := by\n  simp only [div_eq_mul_inv, iSup_mul]\n#align nnreal.supr_div NNReal.iSup_div\n\n-- Porting note: generalized to allow empty `\u03b9`\ntheorem mul_iSup_le {a : \u211d\u22650} {g : \u211d\u22650} {h : \u03b9 \u2192 \u211d\u22650} (H : \u2200 j, g * h j \u2264 a) : g * iSup h \u2264 a := by\n  rw [mul_iSup]\n  exact ciSup_le' H\n#align nnreal.mul_supr_le NNReal.mul_iSup_le\n\n-- Porting note: generalized to allow empty `\u03b9`\ntheorem iSup_mul_le {a : \u211d\u22650} {g : \u03b9 \u2192 \u211d\u22650} {h : \u211d\u22650} (H : \u2200 i, g i * h \u2264 a) : iSup g * h \u2264 a := by\n  rw [iSup_mul]\n  exact ciSup_le' H\n#align nnreal.supr_mul_le NNReal.iSup_mul_le\n\n-- Porting note: generalized to allow empty `\u03b9`\ntheorem iSup_mul_iSup_le {a : \u211d\u22650} {g h : \u03b9 \u2192 \u211d\u22650} (H : \u2200 i j, g i * h j \u2264 a) :\n    iSup g * iSup h \u2264 a :=\n  iSup_mul_le fun _ => mul_iSup_le <| H _\n#align nnreal.supr_mul_supr_le NNReal.iSup_mul_iSup_le\n\nvariable [Nonempty \u03b9]\n\ntheorem le_mul_iInf {a : \u211d\u22650} {g : \u211d\u22650} {h : \u03b9 \u2192 \u211d\u22650} (H : \u2200 j, a \u2264 g * h j) : a \u2264 g * iInf h := by\n  rw [mul_iInf]\n  exact le_ciInf H\n#align nnreal.le_mul_infi NNReal.le_mul_iInf\n\ntheorem le_iInf_mul {a : \u211d\u22650} {g : \u03b9 \u2192 \u211d\u22650} {h : \u211d\u22650} (H : \u2200 i, a \u2264 g i * h) : a \u2264 iInf g * h := by\n  rw [iInf_mul]\n  exact le_ciInf H\n#align nnreal.le_infi_mul NNReal.le_iInf_mul\n\ntheorem le_iInf_mul_iInf {a : \u211d\u22650} {g h : \u03b9 \u2192 \u211d\u22650} (H : \u2200 i j, a \u2264 g i * h j) :\n    a \u2264 iInf g * iInf h :=\n  le_iInf_mul fun i => le_mul_iInf <| H i\n#align nnreal.le_infi_mul_infi NNReal.le_iInf_mul_iInf\n\nend Csupr\n\nend NNReal\n\nnamespace Set\n\nnamespace OrdConnected\n\nvariable {s : Set \u211d} {t : Set \u211d\u22650}\n\ntheorem preimage_coe_nnreal_real (h : s.OrdConnected) : ((\u2191) \u207b\u00b9' s : Set \u211d\u22650).OrdConnected :=\n  h.preimage_mono NNReal.coe_mono\n#align set.ord_connected.preimage_coe_nnreal_real Set.OrdConnected.preimage_coe_nnreal_real\n\ntheorem image_coe_nnreal_real (h : t.OrdConnected) : ((\u2191) '' t : Set \u211d).OrdConnected :=\n  \u27e8forall_mem_image.2 fun x hx =>\n      forall_mem_image.2 fun _y hy z hz => \u27e8\u27e8z, x.2.trans hz.1\u27e9, h.out hx hy hz, rfl\u27e9\u27e9\n#align set.ord_connected.image_coe_nnreal_real Set.OrdConnected.image_coe_nnreal_real\n\n-- Porting note (#11215): TODO: does it generalize to a `GaloisInsertion`?\ntheorem image_real_toNNReal (h : s.OrdConnected) : (Real.toNNReal '' s).OrdConnected := by\n  refine' \u27e8forall_mem_image.2 fun x hx => forall_mem_image.2 fun y hy z hz => _\u27e9\n  rcases le_total y 0 with hy\u2080 | hy\u2080\n  \u00b7 rw [mem_Icc, Real.toNNReal_of_nonpos hy\u2080, nonpos_iff_eq_zero] at hz\n    exact \u27e8y, hy, (toNNReal_of_nonpos hy\u2080).trans hz.2.symm\u27e9\n  \u00b7 lift y to \u211d\u22650 using hy\u2080\n    rw [toNNReal_coe] at hz\n    exact \u27e8z, h.out hx hy \u27e8toNNReal_le_iff_le_coe.1 hz.1, hz.2\u27e9, toNNReal_coe\u27e9\n#align set.ord_connected.image_real_to_nnreal Set.OrdConnected.image_real_toNNReal\n\ntheorem preimage_real_toNNReal (h : t.OrdConnected) : (Real.toNNReal \u207b\u00b9' t).OrdConnected :=\n  h.preimage_mono Real.toNNReal_mono\n#align set.ord_connected.preimage_real_to_nnreal Set.OrdConnected.preimage_real_toNNReal\n\nend OrdConnected\n\nend Set\n\nnamespace Real\n\n/-- The absolute value on `\u211d` as a map to `\u211d\u22650`. -/\n-- Porting note (#11180): removed @[pp_nodot]\ndef nnabs : \u211d \u2192*\u2080 \u211d\u22650 where\n  toFun x := \u27e8|x|, abs_nonneg x\u27e9\n  map_zero' := by ext; simp\n  map_one' := by ext; simp\n  map_mul' x y := by ext; simp [abs_mul]\n#align real.nnabs Real.nnabs\n\n@[norm_cast, simp]\ntheorem coe_nnabs (x : \u211d) : (nnabs x : \u211d) = |x| :=\n  rfl\n#align real.coe_nnabs Real.coe_nnabs\n\n@[simp]\ntheorem nnabs_of_nonneg {x : \u211d} (h : 0 \u2264 x) : nnabs x = toNNReal x := by\n  ext\n  rw [coe_toNNReal x h, coe_nnabs, abs_of_nonneg h]\n#align real.nnabs_of_nonneg Real.nnabs_of_nonneg\n\ntheorem nnabs_coe (x : \u211d\u22650) : nnabs x = x := by simp\n#align real.nnabs_coe Real.nnabs_coe\n\ntheorem coe_toNNReal_le (x : \u211d) : (toNNReal x : \u211d) \u2264 |x| :=\n  max_le (le_abs_self _) (abs_nonneg _)\n#align real.coe_to_nnreal_le Real.coe_toNNReal_le\n\n", "theoremStatement": "@[simp] lemma toNNReal_abs (x : \u211d) : |x|.toNNReal = nnabs x ", "theoremName": "Real.toNNReal_abs", "fileCreated": {"commit": "8c9ab0d16f1e2cafec6bd5ff0e9485e46c6f9f61", "date": "2023-02-20"}, "theoremCreated": {"commit": "a0e02a9399daf134f538b3f0e09cbf560875b33c", "date": "2024-03-23"}, "file": "mathlib/Mathlib/Data/Real/NNReal.lean", "module": "Mathlib.Data.Real.NNReal", "jsonFile": "Mathlib.Data.Real.NNReal.jsonl", "positionMetadata": {"lineInFile": 1185, "tokenPositionInFile": 44735, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": 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"Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group"]}, "proofMetadata": {"hasProof": true, "proof": ":= NNReal.coe_injective <| by simp", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 34}}
+{"srcContext": "/-\nCopyright (c) 2018 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Johannes H\u00f6lzl, Scott Morrison, Jens Wagemaker\n-/\nimport Mathlib.Algebra.Polynomial.Derivative\nimport Mathlib.Algebra.Polynomial.RingDivision\nimport Mathlib.RingTheory.EuclideanDomain\n\n#align_import data.polynomial.field_division from \"leanprover-community/mathlib\"@\"bbeb185db4ccee8ed07dc48449414ebfa39cb821\"\n\n/-!\n# Theory of univariate polynomials\n\nThis file starts looking like the ring theory of $R[X]$\n\n-/\n\n\nnoncomputable section\n\nopen BigOperators Polynomial\n\nnamespace Polynomial\n\nuniverse u v w y z\n\nvariable {R : Type u} {S : Type v} {k : Type y} {A : Type z} {a b : R} {n : \u2115}\n\nsection CommRing\n\nvariable [CommRing R]\n\ntheorem rootMultiplicity_sub_one_le_derivative_rootMultiplicity_of_ne_zero\n    (p : R[X]) (t : R) (hnezero : derivative p \u2260 0) :\n    p.rootMultiplicity t - 1 \u2264 p.derivative.rootMultiplicity t :=\n  (le_rootMultiplicity_iff hnezero).2 <|\n    pow_sub_one_dvd_derivative_of_pow_dvd (p.pow_rootMultiplicity_dvd t)\n\ntheorem derivative_rootMultiplicity_of_root_of_mem_nonZeroDivisors\n    {p : R[X]} {t : R} (hpt : Polynomial.IsRoot p t)\n    (hnzd : (p.rootMultiplicity t : R) \u2208 nonZeroDivisors R) :\n    (derivative p).rootMultiplicity t = p.rootMultiplicity t - 1 := by\n  by_cases h : p = 0\n  \u00b7 simp only [h, map_zero, rootMultiplicity_zero]\n  obtain \u27e8g, hp, hndvd\u27e9 := p.exists_eq_pow_rootMultiplicity_mul_and_not_dvd h t\n  set m := p.rootMultiplicity t\n  have hm : m - 1 + 1 = m := Nat.sub_add_cancel <| (rootMultiplicity_pos h).2 hpt\n  have hndvd : \u00ac(X - C t) ^ m \u2223 derivative p := by\n    rw [hp, derivative_mul, dvd_add_left (dvd_mul_right _ _),\n      derivative_X_sub_C_pow, \u2190 hm, pow_succ, hm, mul_comm (C _), mul_assoc,\n      dvd_cancel_left_mem_nonZeroDivisors (monic_X_sub_C t |>.pow _ |>.mem_nonZeroDivisors)]\n    rw [dvd_iff_isRoot, IsRoot] at hndvd \u22a2\n    rwa [eval_mul, eval_C, mul_left_mem_nonZeroDivisors_eq_zero_iff hnzd]\n  have hnezero : derivative p \u2260 0 := fun h \u21a6 hndvd (by rw [h]; exact dvd_zero _)\n  exact le_antisymm (by rwa [rootMultiplicity_le_iff hnezero, hm])\n    (rootMultiplicity_sub_one_le_derivative_rootMultiplicity_of_ne_zero _ t hnezero)\n\ntheorem isRoot_iterate_derivative_of_lt_rootMultiplicity {p : R[X]} {t : R} {n : \u2115}\n    (hn : n < p.rootMultiplicity t) : (derivative^[n] p).IsRoot t :=\n  dvd_iff_isRoot.mp <| (dvd_pow_self _ <| Nat.sub_ne_zero_of_lt hn).trans\n    (pow_sub_dvd_iterate_derivative_of_pow_dvd _ <| p.pow_rootMultiplicity_dvd t)\n\nopen Finset in\ntheorem eval_iterate_derivative_rootMultiplicity {p : R[X]} {t : R} :\n    (derivative^[p.rootMultiplicity t] p).eval t =\n      (p.rootMultiplicity t).factorial \u2022 (p /\u2098 (X - C t) ^ p.rootMultiplicity t).eval t := by\n  set m := p.rootMultiplicity t with hm\n  conv_lhs => rw [\u2190 p.pow_mul_divByMonic_rootMultiplicity_eq t, \u2190 hm]\n  rw [iterate_derivative_mul, eval_finset_sum, sum_eq_single_of_mem _ (mem_range.mpr m.succ_pos)]\n  \u00b7 rw [m.choose_zero_right, one_smul, eval_mul, m.sub_zero, iterate_derivative_X_sub_pow_self,\n      eval_natCast, nsmul_eq_mul]; rfl\n  \u00b7 intro b hb hb0\n    rw [iterate_derivative_X_sub_pow, eval_smul, eval_mul, eval_smul, eval_pow,\n      Nat.sub_sub_self (mem_range_succ_iff.mp hb), eval_sub, eval_X, eval_C, sub_self,\n      zero_pow hb0, smul_zero, zero_mul, smul_zero]\n\ntheorem lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors\n    {p : R[X]} {t : R} {n : \u2115} (h : p \u2260 0)\n    (hroot : \u2200 m \u2264 n, (derivative^[m] p).IsRoot t)\n    (hnzd : (n.factorial : R) \u2208 nonZeroDivisors R) :\n    n < p.rootMultiplicity t := by\n  by_contra! h'\n  replace hroot := hroot _ h'\n  simp only [IsRoot, eval_iterate_derivative_rootMultiplicity] at hroot\n  obtain \u27e8q, hq\u27e9 := Nat.cast_dvd_cast (\u03b1 := R) <| Nat.factorial_dvd_factorial h'\n  rw [hq, mul_mem_nonZeroDivisors] at hnzd\n  rw [nsmul_eq_mul, mul_left_mem_nonZeroDivisors_eq_zero_iff hnzd.1] at hroot\n  exact eval_divByMonic_pow_rootMultiplicity_ne_zero t h hroot\n\ntheorem lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors'\n    {p : R[X]} {t : R} {n : \u2115} (h : p \u2260 0)\n    (hroot : \u2200 m \u2264 n, (derivative^[m] p).IsRoot t)\n    (hnzd : \u2200 m \u2264 n, m \u2260 0 \u2192 (m : R) \u2208 nonZeroDivisors R) :\n    n < p.rootMultiplicity t := by\n  apply lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors h hroot\n  clear hroot\n  induction' n with n ih\n  \u00b7 simp only [Nat.zero_eq, Nat.factorial_zero, Nat.cast_one]\n    exact Submonoid.one_mem _\n  \u00b7 rw [Nat.factorial_succ, Nat.cast_mul, mul_mem_nonZeroDivisors]\n    exact \u27e8hnzd _ le_rfl n.succ_ne_zero, ih fun m h \u21a6 hnzd m (h.trans n.le_succ)\u27e9\n\ntheorem lt_rootMultiplicity_iff_isRoot_iterate_derivative_of_mem_nonZeroDivisors\n    {p : R[X]} {t : R} {n : \u2115} (h : p \u2260 0)\n    (hnzd : (n.factorial : R) \u2208 nonZeroDivisors R) :\n    n < p.rootMultiplicity t \u2194 \u2200 m \u2264 n, (derivative^[m] p).IsRoot t :=\n  \u27e8fun hn _ hm \u21a6 isRoot_iterate_derivative_of_lt_rootMultiplicity <| hm.trans_lt hn,\n    fun hr \u21a6 lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors h hr hnzd\u27e9\n\ntheorem lt_rootMultiplicity_iff_isRoot_iterate_derivative_of_mem_nonZeroDivisors'\n    {p : R[X]} {t : R} {n : \u2115} (h : p \u2260 0)\n    (hnzd : \u2200 m \u2264 n, m \u2260 0 \u2192 (m : R) \u2208 nonZeroDivisors R) :\n    n < p.rootMultiplicity t \u2194 \u2200 m \u2264 n, (derivative^[m] p).IsRoot t :=\n  \u27e8fun hn _ hm \u21a6 isRoot_iterate_derivative_of_lt_rootMultiplicity <| Nat.lt_of_le_of_lt hm hn,\n    fun hr \u21a6 lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors' h hr hnzd\u27e9\n\ntheorem one_lt_rootMultiplicity_iff_isRoot_iterate_derivative\n    {p : R[X]} {t : R} (h : p \u2260 0) :\n    1 < p.rootMultiplicity t \u2194 \u2200 m \u2264 1, (derivative^[m] p).IsRoot t :=\n  lt_rootMultiplicity_iff_isRoot_iterate_derivative_of_mem_nonZeroDivisors h\n    (by rw [Nat.factorial_one, Nat.cast_one]; exact Submonoid.one_mem _)\n\ntheorem one_lt_rootMultiplicity_iff_isRoot\n    {p : R[X]} {t : R} (h : p \u2260 0) :\n    1 < p.rootMultiplicity t \u2194 p.IsRoot t \u2227 (derivative p).IsRoot t := by\n  rw [one_lt_rootMultiplicity_iff_isRoot_iterate_derivative h]\n  refine \u27e8fun h \u21a6 \u27e8h 0 (by norm_num), h 1 (by norm_num)\u27e9, fun \u27e8h0, h1\u27e9 m hm \u21a6 ?_\u27e9\n  obtain (_|_|m) := m\n  exacts [h0, h1, by omega]\n\nend CommRing\n\nsection IsDomain\n\nvariable [CommRing R] [IsDomain R]\n\ntheorem one_lt_rootMultiplicity_iff_isRoot_gcd\n    [GCDMonoid R[X]] {p : R[X]} {t : R} (h : p \u2260 0) :\n    1 < p.rootMultiplicity t \u2194 (gcd p (derivative p)).IsRoot t := by\n  simp_rw [one_lt_rootMultiplicity_iff_isRoot h, \u2190 dvd_iff_isRoot, dvd_gcd_iff]\n\ntheorem derivative_rootMultiplicity_of_root [CharZero R] {p : R[X]} {t : R} (hpt : p.IsRoot t) :\n    p.derivative.rootMultiplicity t = p.rootMultiplicity t - 1 := by\n  by_cases h : p = 0\n  \u00b7 rw [h, map_zero, rootMultiplicity_zero]\n  exact derivative_rootMultiplicity_of_root_of_mem_nonZeroDivisors hpt <|\n    mem_nonZeroDivisors_of_ne_zero <| Nat.cast_ne_zero.2 ((rootMultiplicity_pos h).2 hpt).ne'\n#align polynomial.derivative_root_multiplicity_of_root Polynomial.derivative_rootMultiplicity_of_root\n\ntheorem rootMultiplicity_sub_one_le_derivative_rootMultiplicity [CharZero R] (p : R[X]) (t : R) :\n    p.rootMultiplicity t - 1 \u2264 p.derivative.rootMultiplicity t := by\n  by_cases h : p.IsRoot t\n  \u00b7 exact (derivative_rootMultiplicity_of_root h).symm.le\n  \u00b7 rw [rootMultiplicity_eq_zero h, zero_tsub]\n    exact zero_le _\n#align polynomial.root_multiplicity_sub_one_le_derivative_root_multiplicity Polynomial.rootMultiplicity_sub_one_le_derivative_rootMultiplicity\n\ntheorem lt_rootMultiplicity_of_isRoot_iterate_derivative\n    [CharZero R] {p : R[X]} {t : R} {n : \u2115} (h : p \u2260 0)\n    (hroot : \u2200 m \u2264 n, (derivative^[m] p).IsRoot t) :\n    n < p.rootMultiplicity t :=\n  lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors h hroot <|\n    mem_nonZeroDivisors_of_ne_zero <| Nat.cast_ne_zero.2 <| Nat.factorial_ne_zero n\n\ntheorem lt_rootMultiplicity_iff_isRoot_iterate_derivative\n    [CharZero R] {p : R[X]} {t : R} {n : \u2115} (h : p \u2260 0) :\n    n < p.rootMultiplicity t \u2194 \u2200 m \u2264 n, (derivative^[m] p).IsRoot t :=\n  \u27e8fun hn _ hm \u21a6 isRoot_iterate_derivative_of_lt_rootMultiplicity <| Nat.lt_of_le_of_lt hm hn,\n    fun hr \u21a6 lt_rootMultiplicity_of_isRoot_iterate_derivative h hr\u27e9\n\nsection NormalizationMonoid\n\nvariable [NormalizationMonoid R]\n\ninstance instNormalizationMonoid : NormalizationMonoid R[X] where\n  normUnit p :=\n    \u27e8C \u2191(normUnit p.leadingCoeff), C \u2191(normUnit p.leadingCoeff)\u207b\u00b9, by\n      rw [\u2190 RingHom.map_mul, Units.mul_inv, C_1], by rw [\u2190 RingHom.map_mul, Units.inv_mul, C_1]\u27e9\n  normUnit_zero := Units.ext (by simp)\n  normUnit_mul hp0 hq0 :=\n    Units.ext\n      (by\n        dsimp\n        rw [Ne, \u2190 leadingCoeff_eq_zero] at *\n        rw [leadingCoeff_mul, normUnit_mul hp0 hq0, Units.val_mul, C_mul])\n  normUnit_coe_units u :=\n    Units.ext\n      (by\n        dsimp\n        rw [\u2190 mul_one u\u207b\u00b9, Units.val_mul, Units.eq_inv_mul_iff_mul_eq]\n        rcases Polynomial.isUnit_iff.1 \u27e8u, rfl\u27e9 with \u27e8_, \u27e8w, rfl\u27e9, h2\u27e9\n        rw [\u2190 h2, leadingCoeff_C, normUnit_coe_units, \u2190 C_mul, Units.mul_inv, C_1]\n        rfl)\n\n@[simp]\ntheorem coe_normUnit {p : R[X]} : (normUnit p : R[X]) = C \u2191(normUnit p.leadingCoeff) := by\n  simp [normUnit]\n#align polynomial.coe_norm_unit Polynomial.coe_normUnit\n\ntheorem leadingCoeff_normalize (p : R[X]) :\n    leadingCoeff (normalize p) = normalize (leadingCoeff p) := by simp\n#align polynomial.leading_coeff_normalize Polynomial.leadingCoeff_normalize\n\ntheorem Monic.normalize_eq_self {p : R[X]} (hp : p.Monic) : normalize p = p := by\n  simp only [Polynomial.coe_normUnit, normalize_apply, hp.leadingCoeff, normUnit_one,\n    Units.val_one, Polynomial.C.map_one, mul_one]\n#align polynomial.monic.normalize_eq_self Polynomial.Monic.normalize_eq_self\n\ntheorem roots_normalize {p : R[X]} : (normalize p).roots = p.roots := by\n  rw [normalize_apply, mul_comm, coe_normUnit, roots_C_mul _ (normUnit (leadingCoeff p)).ne_zero]\n#align polynomial.roots_normalize Polynomial.roots_normalize\n\n", "theoremStatement": "theorem normUnit_X : normUnit (X : Polynomial R) = 1 ", "theoremName": "Polynomial.normUnit_X", "fileCreated": {"commit": 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"Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Cases", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Group.Nat", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Opposite", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Algebra.Pi", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Set.UnionLift", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", 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"Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.EuclideanDomain"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have := coe_normUnit (R := R) (p := X)\n  rwa [leadingCoeff_X, normUnit_one, Units.val_one, map_one, Units.val_eq_one] at this", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 133}}
+{"srcContext": "/-\nCopyright (c) 2021 Kexing Ying. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kexing Ying, Yury Kudryashov\n-/\nimport Mathlib.MeasureTheory.Measure.Restrict\n\n#align_import measure_theory.measure.mutually_singular from \"leanprover-community/mathlib\"@\"70a4f2197832bceab57d7f41379b2592d1110570\"\n\n/-! # Mutually singular measures\n\nTwo measures `\u03bc`, `\u03bd` are said to be mutually singular (`MeasureTheory.Measure.MutuallySingular`,\nlocalized notation `\u03bc \u27c2\u2098 \u03bd`) if there exists a measurable set `s` such that `\u03bc s = 0` and\n`\u03bd s\u1d9c = 0`. The measurability of `s` is an unnecessary assumption (see\n`MeasureTheory.Measure.MutuallySingular.mk`) but we keep it because this way `rcases (h : \u03bc \u27c2\u2098 \u03bd)`\ngives us a measurable set and usually it is easy to prove measurability.\n\nIn this file we define the predicate `MeasureTheory.Measure.MutuallySingular` and prove basic\nfacts about it.\n\n## Tags\n\nmeasure, mutually singular\n-/\n\n\nopen Set\n\nopen MeasureTheory NNReal ENNReal\n\nnamespace MeasureTheory\n\nnamespace Measure\n\nvariable {\u03b1 : Type*} {m0 : MeasurableSpace \u03b1} {\u03bc \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1}\n\n/-- Two measures `\u03bc`, `\u03bd` are said to be mutually singular if there exists a measurable set `s`\nsuch that `\u03bc s = 0` and `\u03bd s\u1d9c = 0`. -/\ndef MutuallySingular {_ : MeasurableSpace \u03b1} (\u03bc \u03bd : Measure \u03b1) : Prop :=\n  \u2203 s : Set \u03b1, MeasurableSet s \u2227 \u03bc s = 0 \u2227 \u03bd s\u1d9c = 0\n#align measure_theory.measure.mutually_singular MeasureTheory.Measure.MutuallySingular\n\n@[inherit_doc MeasureTheory.Measure.MutuallySingular]\nscoped[MeasureTheory] infixl:60 \" \u27c2\u2098 \" => MeasureTheory.Measure.MutuallySingular\n\nnamespace MutuallySingular\n\ntheorem mk {s t : Set \u03b1} (hs : \u03bc s = 0) (ht : \u03bd t = 0) (hst : univ \u2286 s \u222a t) :\n    MutuallySingular \u03bc \u03bd := by\n  use toMeasurable \u03bc s, measurableSet_toMeasurable _ _, (measure_toMeasurable _).trans hs\n  refine' measure_mono_null (fun x hx => (hst trivial).resolve_left fun hxs => hx _) ht\n  exact subset_toMeasurable _ _ hxs\n#align measure_theory.measure.mutually_singular.mk MeasureTheory.Measure.MutuallySingular.mk\n\n/-- A set such that `\u03bc h.nullSet = 0` and `\u03bd h.nullSet\u1d9c = 0`. -/\ndef nullSet (h : \u03bc \u27c2\u2098 \u03bd) : Set \u03b1 := h.choose\n\nlemma measurableSet_nullSet (h : \u03bc \u27c2\u2098 \u03bd) : MeasurableSet h.nullSet := h.choose_spec.1\n\n@[simp]\nlemma measure_nullSet (h : \u03bc \u27c2\u2098 \u03bd) : \u03bc h.nullSet = 0 := h.choose_spec.2.1\n\n@[simp]\nlemma measure_compl_nullSet (h : \u03bc \u27c2\u2098 \u03bd) : \u03bd h.nullSet\u1d9c = 0 := h.choose_spec.2.2\n\n-- TODO: this is proved by simp, but is not simplified in other contexts without the @[simp]\n-- attribute. Also, the linter does not complain about that attribute.\n@[simp]\nlemma restrict_nullSet (h : \u03bc \u27c2\u2098 \u03bd) : \u03bc.restrict h.nullSet = 0 := by simp\n\n-- TODO: this is proved by simp, but is not simplified in other contexts without the @[simp]\n-- attribute. Also, the linter does not complain about that attribute.\n@[simp]\nlemma restrict_compl_nullSet (h : \u03bc \u27c2\u2098 \u03bd) : \u03bd.restrict h.nullSet\u1d9c = 0 := by simp\n\n@[simp]\ntheorem zero_right : \u03bc \u27c2\u2098 0 :=\n  \u27e8\u2205, MeasurableSet.empty, measure_empty, rfl\u27e9\n#align measure_theory.measure.mutually_singular.zero_right MeasureTheory.Measure.MutuallySingular.zero_right\n\n@[symm]\ntheorem symm (h : \u03bd \u27c2\u2098 \u03bc) : \u03bc \u27c2\u2098 \u03bd :=\n  let \u27e8i, hi, his, hit\u27e9 := h\n  \u27e8i\u1d9c, hi.compl, hit, (compl_compl i).symm \u25b8 his\u27e9\n#align measure_theory.measure.mutually_singular.symm MeasureTheory.Measure.MutuallySingular.symm\n\ntheorem comm : \u03bc \u27c2\u2098 \u03bd \u2194 \u03bd \u27c2\u2098 \u03bc :=\n  \u27e8fun h => h.symm, fun h => h.symm\u27e9\n#align measure_theory.measure.mutually_singular.comm MeasureTheory.Measure.MutuallySingular.comm\n\n@[simp]\ntheorem zero_left : 0 \u27c2\u2098 \u03bc :=\n  zero_right.symm\n#align measure_theory.measure.mutually_singular.zero_left MeasureTheory.Measure.MutuallySingular.zero_left\n\ntheorem mono_ac (h : \u03bc\u2081 \u27c2\u2098 \u03bd\u2081) (h\u03bc : \u03bc\u2082 \u226a \u03bc\u2081) (h\u03bd : \u03bd\u2082 \u226a \u03bd\u2081) : \u03bc\u2082 \u27c2\u2098 \u03bd\u2082 :=\n  let \u27e8s, hs, h\u2081, h\u2082\u27e9 := h\n  \u27e8s, hs, h\u03bc h\u2081, h\u03bd h\u2082\u27e9\n#align measure_theory.measure.mutually_singular.mono_ac MeasureTheory.Measure.MutuallySingular.mono_ac\n\ntheorem mono (h : \u03bc\u2081 \u27c2\u2098 \u03bd\u2081) (h\u03bc : \u03bc\u2082 \u2264 \u03bc\u2081) (h\u03bd : \u03bd\u2082 \u2264 \u03bd\u2081) : \u03bc\u2082 \u27c2\u2098 \u03bd\u2082 :=\n  h.mono_ac h\u03bc.absolutelyContinuous h\u03bd.absolutelyContinuous\n#align measure_theory.measure.mutually_singular.mono MeasureTheory.Measure.MutuallySingular.mono\n\n@[simp]\nlemma self_iff (\u03bc : Measure \u03b1) : \u03bc \u27c2\u2098 \u03bc \u2194 \u03bc = 0 := by\n  refine \u27e8?_, fun h \u21a6 by (rw [h]; exact zero_left)\u27e9\n  rintro \u27e8s, hs, h\u03bcs, h\u03bcs_compl\u27e9\n  suffices \u03bc Set.univ = 0 by rwa [measure_univ_eq_zero] at this\n  rw [\u2190 Set.union_compl_self s, measure_union disjoint_compl_right hs.compl, h\u03bcs, h\u03bcs_compl,\n    add_zero]\n\n@[simp]\ntheorem sum_left {\u03b9 : Type*} [Countable \u03b9] {\u03bc : \u03b9 \u2192 Measure \u03b1} : sum \u03bc \u27c2\u2098 \u03bd \u2194 \u2200 i, \u03bc i \u27c2\u2098 \u03bd := by\n  refine' \u27e8fun h i => h.mono (le_sum _ _) le_rfl, fun H => _\u27e9\n  choose s hsm hs\u03bc hs\u03bd using H\n  refine' \u27e8\u22c2 i, s i, MeasurableSet.iInter hsm, _, _\u27e9\n  \u00b7 rw [sum_apply _ (MeasurableSet.iInter hsm), ENNReal.tsum_eq_zero]\n    exact fun i => measure_mono_null (iInter_subset _ _) (hs\u03bc i)\n  \u00b7 rwa [compl_iInter, measure_iUnion_null_iff]\n#align measure_theory.measure.mutually_singular.sum_left MeasureTheory.Measure.MutuallySingular.sum_left\n\n@[simp]\ntheorem sum_right {\u03b9 : Type*} [Countable \u03b9] {\u03bd : \u03b9 \u2192 Measure \u03b1} : \u03bc \u27c2\u2098 sum \u03bd \u2194 \u2200 i, \u03bc \u27c2\u2098 \u03bd i :=\n  comm.trans <| sum_left.trans <| forall_congr' fun _ => comm\n#align measure_theory.measure.mutually_singular.sum_right MeasureTheory.Measure.MutuallySingular.sum_right\n\n@[simp]\ntheorem add_left_iff : \u03bc\u2081 + \u03bc\u2082 \u27c2\u2098 \u03bd \u2194 \u03bc\u2081 \u27c2\u2098 \u03bd \u2227 \u03bc\u2082 \u27c2\u2098 \u03bd := by\n  rw [\u2190 sum_cond, sum_left, Bool.forall_bool, cond, cond, and_comm]\n#align measure_theory.measure.mutually_singular.add_left_iff MeasureTheory.Measure.MutuallySingular.add_left_iff\n\n@[simp]\ntheorem add_right_iff : \u03bc \u27c2\u2098 \u03bd\u2081 + \u03bd\u2082 \u2194 \u03bc \u27c2\u2098 \u03bd\u2081 \u2227 \u03bc \u27c2\u2098 \u03bd\u2082 :=\n  comm.trans <| add_left_iff.trans <| and_congr comm comm\n#align measure_theory.measure.mutually_singular.add_right_iff MeasureTheory.Measure.MutuallySingular.add_right_iff\n\ntheorem add_left (h\u2081 : \u03bd\u2081 \u27c2\u2098 \u03bc) (h\u2082 : \u03bd\u2082 \u27c2\u2098 \u03bc) : \u03bd\u2081 + \u03bd\u2082 \u27c2\u2098 \u03bc :=\n  add_left_iff.2 \u27e8h\u2081, h\u2082\u27e9\n#align measure_theory.measure.mutually_singular.add_left MeasureTheory.Measure.MutuallySingular.add_left\n\ntheorem add_right (h\u2081 : \u03bc \u27c2\u2098 \u03bd\u2081) (h\u2082 : \u03bc \u27c2\u2098 \u03bd\u2082) : \u03bc \u27c2\u2098 \u03bd\u2081 + \u03bd\u2082 :=\n  add_right_iff.2 \u27e8h\u2081, h\u2082\u27e9\n#align measure_theory.measure.mutually_singular.add_right MeasureTheory.Measure.MutuallySingular.add_right\n\ntheorem smul (r : \u211d\u22650\u221e) (h : \u03bd \u27c2\u2098 \u03bc) : r \u2022 \u03bd \u27c2\u2098 \u03bc :=\n  h.mono_ac (AbsolutelyContinuous.rfl.smul r) AbsolutelyContinuous.rfl\n#align measure_theory.measure.mutually_singular.smul MeasureTheory.Measure.MutuallySingular.smul\n\ntheorem smul_nnreal (r : \u211d\u22650) (h : \u03bd \u27c2\u2098 \u03bc) : r \u2022 \u03bd \u27c2\u2098 \u03bc :=\n  h.smul r\n#align measure_theory.measure.mutually_singular.smul_nnreal MeasureTheory.Measure.MutuallySingular.smul_nnreal\n\nlemma restrict (h : \u03bc \u27c2\u2098 \u03bd) (s : Set \u03b1) : \u03bc.restrict s \u27c2\u2098 \u03bd := by\n  refine \u27e8h.nullSet, h.measurableSet_nullSet, ?_, h.measure_compl_nullSet\u27e9\n  rw [Measure.restrict_apply h.measurableSet_nullSet]\n  exact measure_mono_null (Set.inter_subset_left _ _) h.measure_nullSet\n\nend MutuallySingular\n\n", "theoremStatement": "lemma eq_zero_of_absolutelyContinuous_of_mutuallySingular {\u03bc \u03bd : Measure \u03b1}\n    (h_ac : \u03bc \u226a \u03bd) (h_ms : \u03bc \u27c2\u2098 \u03bd) :\n    \u03bc = 0 ", "theoremName": "MeasureTheory.Measure.eq_zero_of_absolutelyContinuous_of_mutuallySingular", "fileCreated": {"commit": 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"Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  rw [\u2190 Measure.MutuallySingular.self_iff]\n  exact h_ms.mono_ac Measure.AbsolutelyContinuous.rfl h_ac", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 107}}
+{"srcContext": "/-\nCopyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Robert Y. Lewis, Johannes H\u00f6lzl, Mario Carneiro, S\u00e9bastien Gou\u00ebzel\n-/\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Data.ENNReal.Real\nimport Mathlib.Topology.UniformSpace.Pi\nimport Mathlib.Topology.UniformSpace.UniformConvergence\nimport Mathlib.Topology.UniformSpace.UniformEmbedding\n\n#align_import topology.metric_space.emetric_space from \"leanprover-community/mathlib\"@\"c8f305514e0d47dfaa710f5a52f0d21b588e6328\"\n\n/-!\n# Extended metric spaces\n\nThis file is devoted to the definition and study of `EMetricSpace`s, i.e., metric\nspaces in which the distance is allowed to take the value \u221e. This extended distance is\ncalled `edist`, and takes values in `\u211d\u22650\u221e`.\n\nMany definitions and theorems expected on emetric spaces are already introduced on uniform spaces\nand topological spaces. For example: open and closed sets, compactness, completeness, continuity and\nuniform continuity.\n\nThe class `EMetricSpace` therefore extends `UniformSpace` (and `TopologicalSpace`).\n\nSince a lot of elementary properties don't require `eq_of_edist_eq_zero` we start setting up the\ntheory of `PseudoEMetricSpace`, where we don't require `edist x y = 0 \u2192 x = y` and we specialize\nto `EMetricSpace` at the end.\n-/\n\n\nopen Set Filter Classical\n\nopen scoped Uniformity Topology BigOperators Filter NNReal ENNReal Pointwise\n\nuniverse u v w\n\nvariable {\u03b1 : Type u} {\u03b2 : Type v} {X : Type*}\n\n/-- Characterizing uniformities associated to a (generalized) distance function `D`\nin terms of the elements of the uniformity. -/\ntheorem uniformity_dist_of_mem_uniformity [LinearOrder \u03b2] {U : Filter (\u03b1 \u00d7 \u03b1)} (z : \u03b2)\n    (D : \u03b1 \u2192 \u03b1 \u2192 \u03b2) (H : \u2200 s, s \u2208 U \u2194 \u2203 \u03b5 > z, \u2200 {a b : \u03b1}, D a b < \u03b5 \u2192 (a, b) \u2208 s) :\n    U = \u2a05 \u03b5 > z, \ud835\udcdf { p : \u03b1 \u00d7 \u03b1 | D p.1 p.2 < \u03b5 } :=\n  HasBasis.eq_biInf \u27e8fun s => by simp only [H, subset_def, Prod.forall, mem_setOf]\u27e9\n#align uniformity_dist_of_mem_uniformity uniformity_dist_of_mem_uniformity\n\n/-- `EDist \u03b1` means that `\u03b1` is equipped with an extended distance. -/\n@[ext]\nclass EDist (\u03b1 : Type*) where\n  edist : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n#align has_edist EDist\n\nexport EDist (edist)\n\n/-- Creating a uniform space from an extended distance. -/\ndef uniformSpaceOfEDist (edist : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\u221e) (edist_self : \u2200 x : \u03b1, edist x x = 0)\n    (edist_comm : \u2200 x y : \u03b1, edist x y = edist y x)\n    (edist_triangle : \u2200 x y z : \u03b1, edist x z \u2264 edist x y + edist y z) : UniformSpace \u03b1 :=\n  .ofFun edist edist_self edist_comm edist_triangle fun \u03b5 \u03b50 =>\n    \u27e8\u03b5 / 2, ENNReal.half_pos \u03b50.ne', fun _ h\u2081 _ h\u2082 =>\n      (ENNReal.add_lt_add h\u2081 h\u2082).trans_eq (ENNReal.add_halves _)\u27e9\n#align uniform_space_of_edist uniformSpaceOfEDist\n\n-- the uniform structure is embedded in the emetric space structure\n-- to avoid instance diamond issues. See Note [forgetful inheritance].\n/-- Extended (pseudo) metric spaces, with an extended distance `edist` possibly taking the\nvalue \u221e\n\nEach pseudo_emetric space induces a canonical `UniformSpace` and hence a canonical\n`TopologicalSpace`.\nThis is enforced in the type class definition, by extending the `UniformSpace` structure. When\ninstantiating a `PseudoEMetricSpace` structure, the uniformity fields are not necessary, they\nwill be filled in by default. There is a default value for the uniformity, that can be substituted\nin cases of interest, for instance when instantiating a `PseudoEMetricSpace` structure\non a product.\n\nContinuity of `edist` is proved in `Topology.Instances.ENNReal`\n-/\nclass PseudoEMetricSpace (\u03b1 : Type u) extends EDist \u03b1 : Type u where\n  edist_self : \u2200 x : \u03b1, edist x x = 0\n  edist_comm : \u2200 x y : \u03b1, edist x y = edist y x\n  edist_triangle : \u2200 x y z : \u03b1, edist x z \u2264 edist x y + edist y z\n  toUniformSpace : UniformSpace \u03b1 := uniformSpaceOfEDist edist edist_self edist_comm edist_triangle\n  uniformity_edist : \ud835\udce4 \u03b1 = \u2a05 \u03b5 > 0, \ud835\udcdf { p : \u03b1 \u00d7 \u03b1 | edist p.1 p.2 < \u03b5 } := by rfl\n#align pseudo_emetric_space PseudoEMetricSpace\n\nattribute [instance] PseudoEMetricSpace.toUniformSpace\n\n/- Pseudoemetric spaces are less common than metric spaces. Therefore, we work in a dedicated\nnamespace, while notions associated to metric spaces are mostly in the root namespace. -/\n\n", "theoremStatement": "/-- Two pseudo emetric space structures with the same edistance function coincide. -/\n@[ext]\nprotected theorem PseudoEMetricSpace.ext {\u03b1 : Type*} {m m' : PseudoEMetricSpace \u03b1}\n    (h : m.toEDist = m'.toEDist) : m = m' ", "theoremName": "PseudoEMetricSpace.ext", "fileCreated": {"commit": "dd781b4101c2d151a556ea1323191a6e56f3c92e", "date": "2023-08-31"}, "theoremCreated": {"commit": "4347fcd1eb22abd58825b25eb0110ea8b8414c7f", "date": "2024-03-28"}, "file": "mathlib/Mathlib/Topology/EMetricSpace/Basic.lean", "module": "Mathlib.Topology.EMetricSpace.Basic", "jsonFile": "Mathlib.Topology.EMetricSpace.Basic.jsonl", "positionMetadata": {"lineInFile": 94, "tokenPositionInFile": 4225, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, 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"Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Tactic.TypeStar", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Basic", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Cases", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.PushNeg", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Tactic.CasesM", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.ByContra", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Tactic.Says", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.AssertExists", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Nat.Defs", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Order.Filter.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  cases' m with ed  _ _ _ U hU\n  cases' m' with ed' _ _ _ U' hU'\n  congr 1\n  exact UniformSpace.ext (((show ed = ed' from h) \u25b8 hU).trans hU'.symm)", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 152}}
+{"srcContext": "/-\nCopyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Robert Y. Lewis, Johannes H\u00f6lzl, Mario Carneiro, S\u00e9bastien Gou\u00ebzel\n-/\nimport Mathlib.Data.Nat.Interval\nimport Mathlib.Data.ENNReal.Real\nimport Mathlib.Topology.UniformSpace.Pi\nimport Mathlib.Topology.UniformSpace.UniformConvergence\nimport Mathlib.Topology.UniformSpace.UniformEmbedding\n\n#align_import topology.metric_space.emetric_space from \"leanprover-community/mathlib\"@\"c8f305514e0d47dfaa710f5a52f0d21b588e6328\"\n\n/-!\n# Extended metric spaces\n\nThis file is devoted to the definition and study of `EMetricSpace`s, i.e., metric\nspaces in which the distance is allowed to take the value \u221e. This extended distance is\ncalled `edist`, and takes values in `\u211d\u22650\u221e`.\n\nMany definitions and theorems expected on emetric spaces are already introduced on uniform spaces\nand topological spaces. For example: open and closed sets, compactness, completeness, continuity and\nuniform continuity.\n\nThe class `EMetricSpace` therefore extends `UniformSpace` (and `TopologicalSpace`).\n\nSince a lot of elementary properties don't require `eq_of_edist_eq_zero` we start setting up the\ntheory of `PseudoEMetricSpace`, where we don't require `edist x y = 0 \u2192 x = y` and we specialize\nto `EMetricSpace` at the end.\n-/\n\n\nopen Set Filter Classical\n\nopen scoped Uniformity Topology BigOperators Filter NNReal ENNReal Pointwise\n\nuniverse u v w\n\nvariable {\u03b1 : Type u} {\u03b2 : Type v} {X : Type*}\n\n/-- Characterizing uniformities associated to a (generalized) distance function `D`\nin terms of the elements of the uniformity. -/\ntheorem uniformity_dist_of_mem_uniformity [LinearOrder \u03b2] {U : Filter (\u03b1 \u00d7 \u03b1)} (z : \u03b2)\n    (D : \u03b1 \u2192 \u03b1 \u2192 \u03b2) (H : \u2200 s, s \u2208 U \u2194 \u2203 \u03b5 > z, \u2200 {a b : \u03b1}, D a b < \u03b5 \u2192 (a, b) \u2208 s) :\n    U = \u2a05 \u03b5 > z, \ud835\udcdf { p : \u03b1 \u00d7 \u03b1 | D p.1 p.2 < \u03b5 } :=\n  HasBasis.eq_biInf \u27e8fun s => by simp only [H, subset_def, Prod.forall, mem_setOf]\u27e9\n#align uniformity_dist_of_mem_uniformity uniformity_dist_of_mem_uniformity\n\n/-- `EDist \u03b1` means that `\u03b1` is equipped with an extended distance. -/\n@[ext]\nclass EDist (\u03b1 : Type*) where\n  edist : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n#align has_edist EDist\n\nexport EDist (edist)\n\n/-- Creating a uniform space from an extended distance. -/\ndef uniformSpaceOfEDist (edist : \u03b1 \u2192 \u03b1 \u2192 \u211d\u22650\u221e) (edist_self : \u2200 x : \u03b1, edist x x = 0)\n    (edist_comm : \u2200 x y : \u03b1, edist x y = edist y x)\n    (edist_triangle : \u2200 x y z : \u03b1, edist x z \u2264 edist x y + edist y z) : UniformSpace \u03b1 :=\n  .ofFun edist edist_self edist_comm edist_triangle fun \u03b5 \u03b50 =>\n    \u27e8\u03b5 / 2, ENNReal.half_pos \u03b50.ne', fun _ h\u2081 _ h\u2082 =>\n      (ENNReal.add_lt_add h\u2081 h\u2082).trans_eq (ENNReal.add_halves _)\u27e9\n#align uniform_space_of_edist uniformSpaceOfEDist\n\n-- the uniform structure is embedded in the emetric space structure\n-- to avoid instance diamond issues. See Note [forgetful inheritance].\n/-- Extended (pseudo) metric spaces, with an extended distance `edist` possibly taking the\nvalue \u221e\n\nEach pseudo_emetric space induces a canonical `UniformSpace` and hence a canonical\n`TopologicalSpace`.\nThis is enforced in the type class definition, by extending the `UniformSpace` structure. When\ninstantiating a `PseudoEMetricSpace` structure, the uniformity fields are not necessary, they\nwill be filled in by default. There is a default value for the uniformity, that can be substituted\nin cases of interest, for instance when instantiating a `PseudoEMetricSpace` structure\non a product.\n\nContinuity of `edist` is proved in `Topology.Instances.ENNReal`\n-/\nclass PseudoEMetricSpace (\u03b1 : Type u) extends EDist \u03b1 : Type u where\n  edist_self : \u2200 x : \u03b1, edist x x = 0\n  edist_comm : \u2200 x y : \u03b1, edist x y = edist y x\n  edist_triangle : \u2200 x y z : \u03b1, edist x z \u2264 edist x y + edist y z\n  toUniformSpace : UniformSpace \u03b1 := uniformSpaceOfEDist edist edist_self edist_comm edist_triangle\n  uniformity_edist : \ud835\udce4 \u03b1 = \u2a05 \u03b5 > 0, \ud835\udcdf { p : \u03b1 \u00d7 \u03b1 | edist p.1 p.2 < \u03b5 } := by rfl\n#align pseudo_emetric_space PseudoEMetricSpace\n\nattribute [instance] PseudoEMetricSpace.toUniformSpace\n\n/- Pseudoemetric spaces are less common than metric spaces. Therefore, we work in a dedicated\nnamespace, while notions associated to metric spaces are mostly in the root namespace. -/\n\n/-- Two pseudo emetric space structures with the same edistance function coincide. -/\n@[ext]\nprotected theorem PseudoEMetricSpace.ext {\u03b1 : Type*} {m m' : PseudoEMetricSpace \u03b1}\n    (h : m.toEDist = m'.toEDist) : m = m' := by\n  cases' m with ed  _ _ _ U hU\n  cases' m' with ed' _ _ _ U' hU'\n  congr 1\n  exact UniformSpace.ext (((show ed = ed' from h) \u25b8 hU).trans hU'.symm)\n\nvariable [PseudoEMetricSpace \u03b1]\n\nexport PseudoEMetricSpace (edist_self edist_comm edist_triangle)\n\nattribute [simp] edist_self\n\n/-- Triangle inequality for the extended distance -/\ntheorem edist_triangle_left (x y z : \u03b1) : edist x y \u2264 edist z x + edist z y := by\n  rw [edist_comm z]; apply edist_triangle\n#align edist_triangle_left edist_triangle_left\n\ntheorem edist_triangle_right (x y z : \u03b1) : edist x y \u2264 edist x z + edist y z := by\n  rw [edist_comm y]; apply edist_triangle\n#align edist_triangle_right edist_triangle_right\n\ntheorem edist_congr_right {x y z : \u03b1} (h : edist x y = 0) : edist x z = edist y z := by\n  apply le_antisymm\n  \u00b7 rw [\u2190 zero_add (edist y z), \u2190 h]\n    apply edist_triangle\n  \u00b7 rw [edist_comm] at h\n    rw [\u2190 zero_add (edist x z), \u2190 h]\n    apply edist_triangle\n#align edist_congr_right edist_congr_right\n\ntheorem edist_congr_left {x y z : \u03b1} (h : edist x y = 0) : edist z x = edist z y := by\n  rw [edist_comm z x, edist_comm z y]\n  apply edist_congr_right h\n#align edist_congr_left edist_congr_left\n\n-- new theorem\ntheorem edist_congr {w x y z : \u03b1} (hl : edist w x = 0) (hr : edist y z = 0) :\n    edist w y = edist x z :=\n  (edist_congr_right hl).trans (edist_congr_left hr)\n\ntheorem edist_triangle4 (x y z t : \u03b1) : edist x t \u2264 edist x y + edist y z + edist z t :=\n  calc\n    edist x t \u2264 edist x z + edist z t := edist_triangle x z t\n    _ \u2264 edist x y + edist y z + edist z t := add_le_add_right (edist_triangle x y z) _\n#align edist_triangle4 edist_triangle4\n\n/-- The triangle (polygon) inequality for sequences of points; `Finset.Ico` version. -/\ntheorem edist_le_Ico_sum_edist (f : \u2115 \u2192 \u03b1) {m n} (h : m \u2264 n) :\n    edist (f m) (f n) \u2264 \u2211 i in Finset.Ico m n, edist (f i) (f (i + 1)) := by\n  induction n, h using Nat.le_induction with\n  | base => rw [Finset.Ico_self, Finset.sum_empty, edist_self]\n  | succ n hle ihn =>\n    calc\n      edist (f m) (f (n + 1)) \u2264 edist (f m) (f n) + edist (f n) (f (n + 1)) := edist_triangle _ _ _\n      _ \u2264 (\u2211 i in Finset.Ico m n, _) + _ := add_le_add ihn le_rfl\n      _ = \u2211 i in Finset.Ico m (n + 1), _ := by\n      { rw [Nat.Ico_succ_right_eq_insert_Ico hle, Finset.sum_insert, add_comm]; simp }\n#align edist_le_Ico_sum_edist edist_le_Ico_sum_edist\n\n/-- The triangle (polygon) inequality for sequences of points; `Finset.range` version. -/\ntheorem edist_le_range_sum_edist (f : \u2115 \u2192 \u03b1) (n : \u2115) :\n    edist (f 0) (f n) \u2264 \u2211 i in Finset.range n, edist (f i) (f (i + 1)) :=\n  Nat.Ico_zero_eq_range \u25b8 edist_le_Ico_sum_edist f (Nat.zero_le n)\n#align edist_le_range_sum_edist edist_le_range_sum_edist\n\n/-- A version of `edist_le_Ico_sum_edist` with each intermediate distance replaced\nwith an upper estimate. -/\ntheorem edist_le_Ico_sum_of_edist_le {f : \u2115 \u2192 \u03b1} {m n} (hmn : m \u2264 n) {d : \u2115 \u2192 \u211d\u22650\u221e}\n    (hd : \u2200 {k}, m \u2264 k \u2192 k < n \u2192 edist (f k) (f (k + 1)) \u2264 d k) :\n    edist (f m) (f n) \u2264 \u2211 i in Finset.Ico m n, d i :=\n  le_trans (edist_le_Ico_sum_edist f hmn) <|\n    Finset.sum_le_sum fun _k hk => hd (Finset.mem_Ico.1 hk).1 (Finset.mem_Ico.1 hk).2\n#align edist_le_Ico_sum_of_edist_le edist_le_Ico_sum_of_edist_le\n\n/-- A version of `edist_le_range_sum_edist` with each intermediate distance replaced\nwith an upper estimate. -/\ntheorem edist_le_range_sum_of_edist_le {f : \u2115 \u2192 \u03b1} (n : \u2115) {d : \u2115 \u2192 \u211d\u22650\u221e}\n    (hd : \u2200 {k}, k < n \u2192 edist (f k) (f (k + 1)) \u2264 d k) :\n    edist (f 0) (f n) \u2264 \u2211 i in Finset.range n, d i :=\n  Nat.Ico_zero_eq_range \u25b8 edist_le_Ico_sum_of_edist_le (zero_le n) fun _ => hd\n#align edist_le_range_sum_of_edist_le edist_le_range_sum_of_edist_le\n\n/-- Reformulation of the uniform structure in terms of the extended distance -/\ntheorem uniformity_pseudoedist : \ud835\udce4 \u03b1 = \u2a05 \u03b5 > 0, \ud835\udcdf { p : \u03b1 \u00d7 \u03b1 | edist p.1 p.2 < \u03b5 } :=\n  PseudoEMetricSpace.uniformity_edist\n#align uniformity_pseudoedist uniformity_pseudoedist\n\ntheorem uniformSpace_edist :\n    \u2039PseudoEMetricSpace \u03b1\u203a.toUniformSpace =\n      uniformSpaceOfEDist edist edist_self edist_comm edist_triangle :=\n  UniformSpace.ext uniformity_pseudoedist\n#align uniform_space_edist uniformSpace_edist\n\ntheorem uniformity_basis_edist :\n    (\ud835\udce4 \u03b1).HasBasis (fun \u03b5 : \u211d\u22650\u221e => 0 < \u03b5) fun \u03b5 => { p : \u03b1 \u00d7 \u03b1 | edist p.1 p.2 < \u03b5 } :=\n  (@uniformSpace_edist \u03b1 _).symm \u25b8 UniformSpace.hasBasis_ofFun \u27e81, one_pos\u27e9 _ _ _ _ _\n#align uniformity_basis_edist uniformity_basis_edist\n\n/-- Characterization of the elements of the uniformity in terms of the extended distance -/\ntheorem mem_uniformity_edist {s : Set (\u03b1 \u00d7 \u03b1)} :\n    s \u2208 \ud835\udce4 \u03b1 \u2194 \u2203 \u03b5 > 0, \u2200 {a b : \u03b1}, edist a b < \u03b5 \u2192 (a, b) \u2208 s :=\n  uniformity_basis_edist.mem_uniformity_iff\n#align mem_uniformity_edist mem_uniformity_edist\n\n/-- Given `f : \u03b2 \u2192 \u211d\u22650\u221e`, if `f` sends `{i | p i}` to a set of positive numbers\naccumulating to zero, then `f i`-neighborhoods of the diagonal form a basis of `\ud835\udce4 \u03b1`.\n\nFor specific bases see `uniformity_basis_edist`, `uniformity_basis_edist'`,\n`uniformity_basis_edist_nnreal`, and `uniformity_basis_edist_inv_nat`. -/\nprotected theorem EMetric.mk_uniformity_basis {\u03b2 : Type*} {p : \u03b2 \u2192 Prop} {f : \u03b2 \u2192 \u211d\u22650\u221e}\n    (hf\u2080 : \u2200 x, p x \u2192 0 < f x) (hf : \u2200 \u03b5, 0 < \u03b5 \u2192 \u2203 x, p x \u2227 f x \u2264 \u03b5) :\n    (\ud835\udce4 \u03b1).HasBasis p fun x => { p : \u03b1 \u00d7 \u03b1 | edist p.1 p.2 < f x } := by\n  refine' \u27e8fun s => uniformity_basis_edist.mem_iff.trans _\u27e9\n  constructor\n  \u00b7 rintro \u27e8\u03b5, \u03b5\u2080, h\u03b5\u27e9\n    rcases hf \u03b5 \u03b5\u2080 with \u27e8i, hi, H\u27e9\n    exact \u27e8i, hi, fun x hx => h\u03b5 <| lt_of_lt_of_le hx.out H\u27e9\n  \u00b7 exact fun \u27e8i, hi, H\u27e9 => \u27e8f i, hf\u2080 i hi, H\u27e9\n#align emetric.mk_uniformity_basis EMetric.mk_uniformity_basis\n\n/-- Given `f : \u03b2 \u2192 \u211d\u22650\u221e`, if `f` sends `{i | p i}` to a set of positive numbers\naccumulating to zero, then closed `f i`-neighborhoods of the diagonal form a basis of `\ud835\udce4 \u03b1`.\n\nFor specific bases see `uniformity_basis_edist_le` and `uniformity_basis_edist_le'`. -/\nprotected theorem EMetric.mk_uniformity_basis_le {\u03b2 : Type*} {p : \u03b2 \u2192 Prop} {f : \u03b2 \u2192 \u211d\u22650\u221e}\n    (hf\u2080 : \u2200 x, p x \u2192 0 < f x) (hf : \u2200 \u03b5, 0 < \u03b5 \u2192 \u2203 x, p x \u2227 f x \u2264 \u03b5) :\n    (\ud835\udce4 \u03b1).HasBasis p fun x => { p : \u03b1 \u00d7 \u03b1 | edist p.1 p.2 \u2264 f x } := by\n  refine' \u27e8fun s => uniformity_basis_edist.mem_iff.trans _\u27e9\n  constructor\n  \u00b7 rintro \u27e8\u03b5, \u03b5\u2080, h\u03b5\u27e9\n    rcases exists_between \u03b5\u2080 with \u27e8\u03b5', h\u03b5'\u27e9\n    rcases hf \u03b5' h\u03b5'.1 with \u27e8i, hi, H\u27e9\n    exact \u27e8i, hi, fun x hx => h\u03b5 <| lt_of_le_of_lt (le_trans hx.out H) h\u03b5'.2\u27e9\n  \u00b7 exact fun \u27e8i, hi, H\u27e9 => \u27e8f i, hf\u2080 i hi, fun x hx => H (le_of_lt hx.out)\u27e9\n#align emetric.mk_uniformity_basis_le EMetric.mk_uniformity_basis_le\n\ntheorem uniformity_basis_edist_le :\n    (\ud835\udce4 \u03b1).HasBasis (fun \u03b5 : \u211d\u22650\u221e => 0 < \u03b5) fun \u03b5 => { p : \u03b1 \u00d7 \u03b1 | edist p.1 p.2 \u2264 \u03b5 } :=\n  EMetric.mk_uniformity_basis_le (fun _ => id) fun \u03b5 \u03b5\u2080 => \u27e8\u03b5, \u03b5\u2080, le_refl \u03b5\u27e9\n#align uniformity_basis_edist_le uniformity_basis_edist_le\n\ntheorem uniformity_basis_edist' (\u03b5' : \u211d\u22650\u221e) (h\u03b5' : 0 < \u03b5') :\n    (\ud835\udce4 \u03b1).HasBasis (fun \u03b5 : \u211d\u22650\u221e => \u03b5 \u2208 Ioo 0 \u03b5') fun \u03b5 => { p : \u03b1 \u00d7 \u03b1 | edist p.1 p.2 < \u03b5 } :=\n  EMetric.mk_uniformity_basis (fun _ => And.left) fun \u03b5 \u03b5\u2080 =>\n    let \u27e8\u03b4, h\u03b4\u27e9 := exists_between h\u03b5'\n    \u27e8min \u03b5 \u03b4, \u27e8lt_min \u03b5\u2080 h\u03b4.1, lt_of_le_of_lt (min_le_right _ _) h\u03b4.2\u27e9, min_le_left _ _\u27e9\n#align uniformity_basis_edist' uniformity_basis_edist'\n\ntheorem uniformity_basis_edist_le' (\u03b5' : \u211d\u22650\u221e) (h\u03b5' : 0 < \u03b5') :\n    (\ud835\udce4 \u03b1).HasBasis (fun \u03b5 : \u211d\u22650\u221e => \u03b5 \u2208 Ioo 0 \u03b5') fun \u03b5 => { p : \u03b1 \u00d7 \u03b1 | edist p.1 p.2 \u2264 \u03b5 } :=\n  EMetric.mk_uniformity_basis_le (fun _ => And.left) fun \u03b5 \u03b5\u2080 =>\n    let \u27e8\u03b4, h\u03b4\u27e9 := exists_between h\u03b5'\n    \u27e8min \u03b5 \u03b4, \u27e8lt_min \u03b5\u2080 h\u03b4.1, lt_of_le_of_lt (min_le_right _ _) h\u03b4.2\u27e9, min_le_left _ _\u27e9\n#align uniformity_basis_edist_le' uniformity_basis_edist_le'\n\ntheorem uniformity_basis_edist_nnreal :\n    (\ud835\udce4 \u03b1).HasBasis (fun \u03b5 : \u211d\u22650 => 0 < \u03b5) fun \u03b5 => { p : \u03b1 \u00d7 \u03b1 | edist p.1 p.2 < \u03b5 } :=\n  EMetric.mk_uniformity_basis (fun _ => ENNReal.coe_pos.2) fun _\u03b5 \u03b5\u2080 =>\n    let \u27e8\u03b4, h\u03b4\u27e9 := ENNReal.lt_iff_exists_nnreal_btwn.1 \u03b5\u2080\n    \u27e8\u03b4, ENNReal.coe_pos.1 h\u03b4.1, le_of_lt h\u03b4.2\u27e9\n#align uniformity_basis_edist_nnreal uniformity_basis_edist_nnreal\n\ntheorem uniformity_basis_edist_nnreal_le :\n    (\ud835\udce4 \u03b1).HasBasis (fun \u03b5 : \u211d\u22650 => 0 < \u03b5) fun \u03b5 => { p : \u03b1 \u00d7 \u03b1 | edist p.1 p.2 \u2264 \u03b5 } :=\n  EMetric.mk_uniformity_basis_le (fun _ => ENNReal.coe_pos.2) fun _\u03b5 \u03b5\u2080 =>\n    let \u27e8\u03b4, h\u03b4\u27e9 := ENNReal.lt_iff_exists_nnreal_btwn.1 \u03b5\u2080\n    \u27e8\u03b4, ENNReal.coe_pos.1 h\u03b4.1, le_of_lt h\u03b4.2\u27e9\n#align uniformity_basis_edist_nnreal_le uniformity_basis_edist_nnreal_le\n\ntheorem uniformity_basis_edist_inv_nat :\n    (\ud835\udce4 \u03b1).HasBasis (fun _ => True) fun n : \u2115 => { p : \u03b1 \u00d7 \u03b1 | edist p.1 p.2 < (\u2191n)\u207b\u00b9 } :=\n  EMetric.mk_uniformity_basis (fun n _ \u21a6 ENNReal.inv_pos.2 <| ENNReal.natCast_ne_top n) fun _\u03b5 \u03b5\u2080 \u21a6\n    let \u27e8n, hn\u27e9 := ENNReal.exists_inv_nat_lt (ne_of_gt \u03b5\u2080)\n    \u27e8n, trivial, le_of_lt hn\u27e9\n#align uniformity_basis_edist_inv_nat uniformity_basis_edist_inv_nat\n\ntheorem uniformity_basis_edist_inv_two_pow :\n    (\ud835\udce4 \u03b1).HasBasis (fun _ => True) fun n : \u2115 => { p : \u03b1 \u00d7 \u03b1 | edist p.1 p.2 < 2\u207b\u00b9 ^ n } :=\n  EMetric.mk_uniformity_basis (fun _ _ => ENNReal.pow_pos (ENNReal.inv_pos.2 ENNReal.two_ne_top) _)\n    fun _\u03b5 \u03b5\u2080 =>\n    let \u27e8n, hn\u27e9 := ENNReal.exists_inv_two_pow_lt (ne_of_gt \u03b5\u2080)\n    \u27e8n, trivial, le_of_lt hn\u27e9\n#align uniformity_basis_edist_inv_two_pow uniformity_basis_edist_inv_two_pow\n\n/-- Fixed size neighborhoods of the diagonal belong to the uniform structure -/\ntheorem edist_mem_uniformity {\u03b5 : \u211d\u22650\u221e} (\u03b50 : 0 < \u03b5) : { p : \u03b1 \u00d7 \u03b1 | edist p.1 p.2 < \u03b5 } \u2208 \ud835\udce4 \u03b1 :=\n  mem_uniformity_edist.2 \u27e8\u03b5, \u03b50, id\u27e9\n#align edist_mem_uniformity edist_mem_uniformity\n\nnamespace EMetric\n\ninstance (priority := 900) instIsCountablyGeneratedUniformity : IsCountablyGenerated (\ud835\udce4 \u03b1) :=\n  isCountablyGenerated_of_seq \u27e8_, uniformity_basis_edist_inv_nat.eq_iInf\u27e9\n\n-- Porting note: changed explicit/implicit\n/-- \u03b5-\u03b4 characterization of uniform continuity on a set for pseudoemetric spaces -/\ntheorem uniformContinuousOn_iff [PseudoEMetricSpace \u03b2] {f : \u03b1 \u2192 \u03b2} {s : Set \u03b1} :\n    UniformContinuousOn f s \u2194\n      \u2200 \u03b5 > 0, \u2203 \u03b4 > 0, \u2200 {a}, a \u2208 s \u2192 \u2200 {b}, b \u2208 s \u2192 edist a b < \u03b4 \u2192 edist (f a) (f b) < \u03b5 :=\n  uniformity_basis_edist.uniformContinuousOn_iff uniformity_basis_edist\n#align emetric.uniform_continuous_on_iff EMetric.uniformContinuousOn_iff\n\n/-- \u03b5-\u03b4 characterization of uniform continuity on pseudoemetric spaces -/\ntheorem uniformContinuous_iff [PseudoEMetricSpace \u03b2] {f : \u03b1 \u2192 \u03b2} :\n    UniformContinuous f \u2194 \u2200 \u03b5 > 0, \u2203 \u03b4 > 0, \u2200 {a b : \u03b1}, edist a b < \u03b4 \u2192 edist (f a) (f b) < \u03b5 :=\n  uniformity_basis_edist.uniformContinuous_iff uniformity_basis_edist\n#align emetric.uniform_continuous_iff EMetric.uniformContinuous_iff\n\n-- Porting note (#10756): new lemma\ntheorem uniformInducing_iff [PseudoEMetricSpace \u03b2] {f : \u03b1 \u2192 \u03b2} :\n    UniformInducing f \u2194 UniformContinuous f \u2227\n      \u2200 \u03b4 > 0, \u2203 \u03b5 > 0, \u2200 {a b : \u03b1}, edist (f a) (f b) < \u03b5 \u2192 edist a b < \u03b4 :=\n  uniformInducing_iff'.trans <| Iff.rfl.and <|\n    ((uniformity_basis_edist.comap _).le_basis_iff uniformity_basis_edist).trans <| by\n      simp only [subset_def, Prod.forall]; rfl\n\n/-- \u03b5-\u03b4 characterization of uniform embeddings on pseudoemetric spaces -/\nnonrec theorem uniformEmbedding_iff [PseudoEMetricSpace \u03b2] {f : \u03b1 \u2192 \u03b2} :\n    UniformEmbedding f \u2194 Function.Injective f \u2227 UniformContinuous f \u2227\n      \u2200 \u03b4 > 0, \u2203 \u03b5 > 0, \u2200 {a b : \u03b1}, edist (f a) (f b) < \u03b5 \u2192 edist a b < \u03b4 :=\n  (uniformEmbedding_iff _).trans <| and_comm.trans <| Iff.rfl.and uniformInducing_iff\n#align emetric.uniform_embedding_iff EMetric.uniformEmbedding_iff\n\n/-- If a map between pseudoemetric spaces is a uniform embedding then the edistance between `f x`\nand `f y` is controlled in terms of the distance between `x` and `y`.\n\nIn fact, this lemma holds for a `UniformInducing` map.\nTODO: generalize? -/\ntheorem controlled_of_uniformEmbedding [PseudoEMetricSpace \u03b2] {f : \u03b1 \u2192 \u03b2} (h : UniformEmbedding f) :\n    (\u2200 \u03b5 > 0, \u2203 \u03b4 > 0, \u2200 {a b : \u03b1}, edist a b < \u03b4 \u2192 edist (f a) (f b) < \u03b5) \u2227\n      \u2200 \u03b4 > 0, \u2203 \u03b5 > 0, \u2200 {a b : \u03b1}, edist (f a) (f b) < \u03b5 \u2192 edist a b < \u03b4 :=\n  \u27e8uniformContinuous_iff.1 h.uniformContinuous, (uniformEmbedding_iff.1 h).2.2\u27e9\n#align emetric.controlled_of_uniform_embedding EMetric.controlled_of_uniformEmbedding\n\n/-- \u03b5-\u03b4 characterization of Cauchy sequences on pseudoemetric spaces -/\nprotected theorem cauchy_iff {f : Filter \u03b1} :\n    Cauchy f \u2194 f \u2260 \u22a5 \u2227 \u2200 \u03b5 > 0, \u2203 t \u2208 f, \u2200 x, x \u2208 t \u2192 \u2200 y, y \u2208 t \u2192 edist x y < \u03b5 := by\n  rw [\u2190 neBot_iff]; exact uniformity_basis_edist.cauchy_iff\n#align emetric.cauchy_iff EMetric.cauchy_iff\n\n/-- A very useful criterion to show that a space is complete is to show that all sequences\nwhich satisfy a bound of the form `edist (u n) (u m) < B N` for all `n m \u2265 N` are\nconverging. This is often applied for `B N = 2^{-N}`, i.e., with a very fast convergence to\n`0`, which makes it possible to use arguments of converging series, while this is impossible\nto do in general for arbitrary Cauchy sequences. -/\ntheorem complete_of_convergent_controlled_sequences (B : \u2115 \u2192 \u211d\u22650\u221e) (hB : \u2200 n, 0 < B n)\n    (H : \u2200 u : \u2115 \u2192 \u03b1, (\u2200 N n m : \u2115, N \u2264 n \u2192 N \u2264 m \u2192 edist (u n) (u m) < B N) \u2192\n      \u2203 x, Tendsto u atTop (\ud835\udcdd x)) :\n    CompleteSpace \u03b1 :=\n  UniformSpace.complete_of_convergent_controlled_sequences\n    (fun n => { p : \u03b1 \u00d7 \u03b1 | edist p.1 p.2 < B n }) (fun n => edist_mem_uniformity <| hB n) H\n#align emetric.complete_of_convergent_controlled_sequences EMetric.complete_of_convergent_controlled_sequences\n\n/-- A sequentially complete pseudoemetric space is complete. -/\ntheorem complete_of_cauchySeq_tendsto :\n    (\u2200 u : \u2115 \u2192 \u03b1, CauchySeq u \u2192 \u2203 a, Tendsto u atTop (\ud835\udcdd a)) \u2192 CompleteSpace \u03b1 :=\n  UniformSpace.complete_of_cauchySeq_tendsto\n#align emetric.complete_of_cauchy_seq_tendsto EMetric.complete_of_cauchySeq_tendsto\n\n/-- Expressing locally uniform convergence on a set using `edist`. -/\ntheorem tendstoLocallyUniformlyOn_iff {\u03b9 : Type*} [TopologicalSpace \u03b2] {F : \u03b9 \u2192 \u03b2 \u2192 \u03b1} {f : \u03b2 \u2192 \u03b1}\n    {p : Filter \u03b9} {s : Set \u03b2} :\n    TendstoLocallyUniformlyOn F f p s \u2194\n      \u2200 \u03b5 > 0, \u2200 x \u2208 s, \u2203 t \u2208 \ud835\udcdd[s] x, \u2200\u1da0 n in p, \u2200 y \u2208 t, edist (f y) (F n y) < \u03b5 := by\n  refine' \u27e8fun H \u03b5 h\u03b5 => H _ (edist_mem_uniformity h\u03b5), fun H u hu x hx => _\u27e9\n  rcases mem_uniformity_edist.1 hu with \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9\n  rcases H \u03b5 \u03b5pos x hx with \u27e8t, ht, Ht\u27e9\n  exact \u27e8t, ht, Ht.mono fun n hs x hx => h\u03b5 (hs x hx)\u27e9\n#align emetric.tendsto_locally_uniformly_on_iff EMetric.tendstoLocallyUniformlyOn_iff\n\n/-- Expressing uniform convergence on a set using `edist`. -/\ntheorem tendstoUniformlyOn_iff {\u03b9 : Type*} {F : \u03b9 \u2192 \u03b2 \u2192 \u03b1} {f : \u03b2 \u2192 \u03b1} {p : Filter \u03b9} {s : Set \u03b2} :\n    TendstoUniformlyOn F f p s \u2194 \u2200 \u03b5 > 0, \u2200\u1da0 n in p, \u2200 x \u2208 s, edist (f x) (F n x) < \u03b5 := by\n  refine' \u27e8fun H \u03b5 h\u03b5 => H _ (edist_mem_uniformity h\u03b5), fun H u hu => _\u27e9\n  rcases mem_uniformity_edist.1 hu with \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9\n  exact (H \u03b5 \u03b5pos).mono fun n hs x hx => h\u03b5 (hs x hx)\n#align emetric.tendsto_uniformly_on_iff EMetric.tendstoUniformlyOn_iff\n\n/-- Expressing locally uniform convergence using `edist`. -/\ntheorem tendstoLocallyUniformly_iff {\u03b9 : Type*} [TopologicalSpace \u03b2] {F : \u03b9 \u2192 \u03b2 \u2192 \u03b1} {f : \u03b2 \u2192 \u03b1}\n    {p : Filter \u03b9} :\n    TendstoLocallyUniformly F f p \u2194\n      \u2200 \u03b5 > 0, \u2200 x : \u03b2, \u2203 t \u2208 \ud835\udcdd x, \u2200\u1da0 n in p, \u2200 y \u2208 t, edist (f y) (F n y) < \u03b5 := by\n  simp only [\u2190 tendstoLocallyUniformlyOn_univ, tendstoLocallyUniformlyOn_iff, mem_univ,\n    forall_const, exists_prop, nhdsWithin_univ]\n#align emetric.tendsto_locally_uniformly_iff EMetric.tendstoLocallyUniformly_iff\n\n/-- Expressing uniform convergence using `edist`. -/\ntheorem tendstoUniformly_iff {\u03b9 : Type*} {F : \u03b9 \u2192 \u03b2 \u2192 \u03b1} {f : \u03b2 \u2192 \u03b1} {p : Filter \u03b9} :\n    TendstoUniformly F f p \u2194 \u2200 \u03b5 > 0, \u2200\u1da0 n in p, \u2200 x, edist (f x) (F n x) < \u03b5 := by\n  simp only [\u2190 tendstoUniformlyOn_univ, tendstoUniformlyOn_iff, mem_univ, forall_const]\n#align emetric.tendsto_uniformly_iff EMetric.tendstoUniformly_iff\n\nend EMetric\n\nopen EMetric\n\n/-- Auxiliary function to replace the uniformity on a pseudoemetric space with\na uniformity which is equal to the original one, but maybe not defeq.\nThis is useful if one wants to construct a pseudoemetric space with a\nspecified uniformity. See Note [forgetful inheritance] explaining why having definitionally\nthe right uniformity is often important.\n-/\ndef PseudoEMetricSpace.replaceUniformity {\u03b1} [U : UniformSpace \u03b1] (m : PseudoEMetricSpace \u03b1)\n    (H : \ud835\udce4[U] = \ud835\udce4[PseudoEMetricSpace.toUniformSpace]) : PseudoEMetricSpace \u03b1 where\n  edist := @edist _ m.toEDist\n  edist_self := edist_self\n  edist_comm := edist_comm\n  edist_triangle := edist_triangle\n  toUniformSpace := U\n  uniformity_edist := H.trans (@PseudoEMetricSpace.uniformity_edist \u03b1 _)\n#align pseudo_emetric_space.replace_uniformity PseudoEMetricSpace.replaceUniformity\n\n/-- The extended pseudometric induced by a function taking values in a pseudoemetric space. -/\ndef PseudoEMetricSpace.induced {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) (m : PseudoEMetricSpace \u03b2) :\n    PseudoEMetricSpace \u03b1 where\n  edist x y := edist (f x) (f y)\n  edist_self _ := edist_self _\n  edist_comm _ _ := edist_comm _ _\n  edist_triangle _ _ _ := edist_triangle _ _ _\n  toUniformSpace := UniformSpace.comap f m.toUniformSpace\n  uniformity_edist := (uniformity_basis_edist.comap (Prod.map f f)).eq_biInf\n#align pseudo_emetric_space.induced PseudoEMetricSpace.induced\n\n/-- Pseudoemetric space instance on subsets of pseudoemetric spaces -/\ninstance {\u03b1 : Type*} {p : \u03b1 \u2192 Prop} [PseudoEMetricSpace \u03b1] : PseudoEMetricSpace (Subtype p) :=\n  PseudoEMetricSpace.induced Subtype.val \u2039_\u203a\n\n/-- The extended pseudodistance on a subset of a pseudoemetric space is the restriction of\nthe original pseudodistance, by definition -/\ntheorem Subtype.edist_eq {p : \u03b1 \u2192 Prop} (x y : Subtype p) : edist x y = edist (x : \u03b1) y := rfl\n#align subtype.edist_eq Subtype.edist_eq\n\nnamespace MulOpposite\n\n/-- Pseudoemetric space instance on the multiplicative opposite of a pseudoemetric space. -/\n@[to_additive \"Pseudoemetric space instance on the additive opposite of a pseudoemetric space.\"]\ninstance {\u03b1 : Type*} [PseudoEMetricSpace \u03b1] : PseudoEMetricSpace \u03b1\u1d50\u1d52\u1d56 :=\n  PseudoEMetricSpace.induced unop \u2039_\u203a\n\n@[to_additive]\ntheorem edist_unop (x y : \u03b1\u1d50\u1d52\u1d56) : edist (unop x) (unop y) = edist x y := rfl\n#align mul_opposite.edist_unop MulOpposite.edist_unop\n#align add_opposite.edist_unop AddOpposite.edist_unop\n\n@[to_additive]\ntheorem edist_op (x y : \u03b1) : edist (op x) (op y) = edist x y := rfl\n#align mul_opposite.edist_op MulOpposite.edist_op\n#align add_opposite.edist_op AddOpposite.edist_op\n\nend MulOpposite\n\nsection ULift\n\ninstance : PseudoEMetricSpace (ULift \u03b1) := PseudoEMetricSpace.induced ULift.down \u2039_\u203a\n\ntheorem ULift.edist_eq (x y : ULift \u03b1) : edist x y = edist x.down y.down := rfl\n#align ulift.edist_eq ULift.edist_eq\n\n@[simp]\ntheorem ULift.edist_up_up (x y : \u03b1) : edist (ULift.up x) (ULift.up y) = edist x y := rfl\n#align ulift.edist_up_up ULift.edist_up_up\n\nend ULift\n\n/-- The product of two pseudoemetric spaces, with the max distance, is an extended\npseudometric spaces. We make sure that the uniform structure thus constructed is the one\ncorresponding to the product of uniform spaces, to avoid diamond problems. -/\ninstance Prod.pseudoEMetricSpaceMax [PseudoEMetricSpace \u03b2] : PseudoEMetricSpace (\u03b1 \u00d7 \u03b2) where\n  edist x y := edist x.1 y.1 \u2294 edist x.2 y.2\n  edist_self x := by simp\n  edist_comm x y := by simp [edist_comm]\n  edist_triangle x y z :=\n    max_le (le_trans (edist_triangle _ _ _) (add_le_add (le_max_left _ _) (le_max_left _ _)))\n      (le_trans (edist_triangle _ _ _) (add_le_add (le_max_right _ _) (le_max_right _ _)))\n  uniformity_edist := uniformity_prod.trans <| by\n    simp [PseudoEMetricSpace.uniformity_edist, \u2190 iInf_inf_eq, setOf_and]\n  toUniformSpace := inferInstance\n#align prod.pseudo_emetric_space_max Prod.pseudoEMetricSpaceMax\n\ntheorem Prod.edist_eq [PseudoEMetricSpace \u03b2] (x y : \u03b1 \u00d7 \u03b2) :\n    edist x y = max (edist x.1 y.1) (edist x.2 y.2) :=\n  rfl\n#align prod.edist_eq Prod.edist_eq\n\nsection Pi\n\nopen Finset\n\nvariable {\u03c0 : \u03b2 \u2192 Type*} [Fintype \u03b2]\n\n-- Porting note: reordered instances\ninstance [\u2200 b, EDist (\u03c0 b)] : EDist (\u2200 b, \u03c0 b) where\n  edist f g := Finset.sup univ fun b => edist (f b) (g b)\n\ntheorem edist_pi_def [\u2200 b, EDist (\u03c0 b)] (f g : \u2200 b, \u03c0 b) :\n    edist f g = Finset.sup univ fun b => edist (f b) (g b) :=\n  rfl\n#align edist_pi_def edist_pi_def\n\ntheorem edist_le_pi_edist [\u2200 b, EDist (\u03c0 b)] (f g : \u2200 b, \u03c0 b) (b : \u03b2) :\n    edist (f b) (g b) \u2264 edist f g :=\n  le_sup (f := fun b => edist (f b) (g b)) (Finset.mem_univ b)\n#align edist_le_pi_edist edist_le_pi_edist\n\ntheorem edist_pi_le_iff [\u2200 b, EDist (\u03c0 b)] {f g : \u2200 b, \u03c0 b} {d : \u211d\u22650\u221e} :\n    edist f g \u2264 d \u2194 \u2200 b, edist (f b) (g b) \u2264 d :=\n  Finset.sup_le_iff.trans <| by simp only [Finset.mem_univ, forall_const]\n#align edist_pi_le_iff edist_pi_le_iff\n\ntheorem edist_pi_const_le (a b : \u03b1) : (edist (fun _ : \u03b2 => a) fun _ => b) \u2264 edist a b :=\n  edist_pi_le_iff.2 fun _ => le_rfl\n#align edist_pi_const_le edist_pi_const_le\n\n@[simp]\ntheorem edist_pi_const [Nonempty \u03b2] (a b : \u03b1) : (edist (fun _ : \u03b2 => a) fun _ => b) = edist a b :=\n  Finset.sup_const univ_nonempty (edist a b)\n#align edist_pi_const edist_pi_const\n\n/-- The product of a finite number of pseudoemetric spaces, with the max distance, is still\na pseudoemetric space.\nThis construction would also work for infinite products, but it would not give rise\nto the product topology. Hence, we only formalize it in the good situation of finitely many\nspaces. -/\ninstance pseudoEMetricSpacePi [\u2200 b, PseudoEMetricSpace (\u03c0 b)] : PseudoEMetricSpace (\u2200 b, \u03c0 b) where\n  edist_self f := bot_unique <| Finset.sup_le <| by simp\n  edist_comm f g := by simp [edist_pi_def, edist_comm]\n  edist_triangle f g h := edist_pi_le_iff.2 fun b => le_trans (edist_triangle _ (g b) _)\n    (add_le_add (edist_le_pi_edist _ _ _) (edist_le_pi_edist _ _ _))\n  toUniformSpace := Pi.uniformSpace _\n  uniformity_edist := by\n    simp only [Pi.uniformity, PseudoEMetricSpace.uniformity_edist, comap_iInf, gt_iff_lt,\n      preimage_setOf_eq, comap_principal, edist_pi_def]\n    rw [iInf_comm]; congr; funext \u03b5\n    rw [iInf_comm]; congr; funext \u03b5pos\n    simp [setOf_forall, \u03b5pos]\n#align pseudo_emetric_space_pi pseudoEMetricSpacePi\n\nend Pi\n\nnamespace EMetric\n\nvariable {x y z : \u03b1} {\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e} {s t : Set \u03b1}\n\n/-- `EMetric.ball x \u03b5` is the set of all points `y` with `edist y x < \u03b5` -/\ndef ball (x : \u03b1) (\u03b5 : \u211d\u22650\u221e) : Set \u03b1 :=\n  { y | edist y x < \u03b5 }\n#align emetric.ball EMetric.ball\n\n@[simp] theorem mem_ball : y \u2208 ball x \u03b5 \u2194 edist y x < \u03b5 := Iff.rfl\n#align emetric.mem_ball EMetric.mem_ball\n\ntheorem mem_ball' : y \u2208 ball x \u03b5 \u2194 edist x y < \u03b5 := by rw [edist_comm, mem_ball]\n#align emetric.mem_ball' EMetric.mem_ball'\n\n/-- `EMetric.closedBall x \u03b5` is the set of all points `y` with `edist y x \u2264 \u03b5` -/\ndef closedBall (x : \u03b1) (\u03b5 : \u211d\u22650\u221e) :=\n  { y | edist y x \u2264 \u03b5 }\n#align emetric.closed_ball EMetric.closedBall\n\n@[simp] theorem mem_closedBall : y \u2208 closedBall x \u03b5 \u2194 edist y x \u2264 \u03b5 := Iff.rfl\n#align emetric.mem_closed_ball EMetric.mem_closedBall\n\ntheorem mem_closedBall' : y \u2208 closedBall x \u03b5 \u2194 edist x y \u2264 \u03b5 := by rw [edist_comm, mem_closedBall]\n#align emetric.mem_closed_ball' EMetric.mem_closedBall'\n\n@[simp]\ntheorem closedBall_top (x : \u03b1) : closedBall x \u221e = univ :=\n  eq_univ_of_forall fun _ => mem_setOf.2 le_top\n#align emetric.closed_ball_top EMetric.closedBall_top\n\ntheorem ball_subset_closedBall : ball x \u03b5 \u2286 closedBall x \u03b5 := fun _ h => le_of_lt h.out\n#align emetric.ball_subset_closed_ball EMetric.ball_subset_closedBall\n\ntheorem pos_of_mem_ball (hy : y \u2208 ball x \u03b5) : 0 < \u03b5 :=\n  lt_of_le_of_lt (zero_le _) hy\n#align emetric.pos_of_mem_ball EMetric.pos_of_mem_ball\n\ntheorem mem_ball_self (h : 0 < \u03b5) : x \u2208 ball x \u03b5 := by\n  rwa [mem_ball, edist_self]\n#align emetric.mem_ball_self EMetric.mem_ball_self\n\ntheorem mem_closedBall_self : x \u2208 closedBall x \u03b5 := by\n  rw [mem_closedBall, edist_self]; apply zero_le\n#align emetric.mem_closed_ball_self EMetric.mem_closedBall_self\n\ntheorem mem_ball_comm : x \u2208 ball y \u03b5 \u2194 y \u2208 ball x \u03b5 := by rw [mem_ball', mem_ball]\n#align emetric.mem_ball_comm EMetric.mem_ball_comm\n\ntheorem mem_closedBall_comm : x \u2208 closedBall y \u03b5 \u2194 y \u2208 closedBall x \u03b5 := by\n  rw [mem_closedBall', mem_closedBall]\n#align emetric.mem_closed_ball_comm EMetric.mem_closedBall_comm\n\ntheorem ball_subset_ball (h : \u03b5\u2081 \u2264 \u03b5\u2082) : ball x \u03b5\u2081 \u2286 ball x \u03b5\u2082 := fun _y (yx : _ < \u03b5\u2081) =>\n  lt_of_lt_of_le yx h\n#align emetric.ball_subset_ball EMetric.ball_subset_ball\n\ntheorem closedBall_subset_closedBall (h : \u03b5\u2081 \u2264 \u03b5\u2082) : closedBall x \u03b5\u2081 \u2286 closedBall x \u03b5\u2082 :=\n  fun _y (yx : _ \u2264 \u03b5\u2081) => le_trans yx h\n#align emetric.closed_ball_subset_closed_ball EMetric.closedBall_subset_closedBall\n\ntheorem ball_disjoint (h : \u03b5\u2081 + \u03b5\u2082 \u2264 edist x y) : Disjoint (ball x \u03b5\u2081) (ball y \u03b5\u2082) :=\n  Set.disjoint_left.mpr fun z h\u2081 h\u2082 =>\n    (edist_triangle_left x y z).not_lt <| (ENNReal.add_lt_add h\u2081 h\u2082).trans_le h\n#align emetric.ball_disjoint EMetric.ball_disjoint\n\ntheorem ball_subset (h : edist x y + \u03b5\u2081 \u2264 \u03b5\u2082) (h' : edist x y \u2260 \u221e) : ball x \u03b5\u2081 \u2286 ball y \u03b5\u2082 :=\n  fun z zx =>\n  calc\n    edist z y \u2264 edist z x + edist x y := edist_triangle _ _ _\n    _ = edist x y + edist z x := add_comm _ _\n    _ < edist x y + \u03b5\u2081 := ENNReal.add_lt_add_left h' zx\n    _ \u2264 \u03b5\u2082 := h\n#align emetric.ball_subset EMetric.ball_subset\n\ntheorem exists_ball_subset_ball (h : y \u2208 ball x \u03b5) : \u2203 \u03b5' > 0, ball y \u03b5' \u2286 ball x \u03b5 := by\n  have : 0 < \u03b5 - edist y x := by simpa using h\n  refine' \u27e8\u03b5 - edist y x, this, ball_subset _ (ne_top_of_lt h)\u27e9\n  exact (add_tsub_cancel_of_le (mem_ball.mp h).le).le\n#align emetric.exists_ball_subset_ball EMetric.exists_ball_subset_ball\n\ntheorem ball_eq_empty_iff : ball x \u03b5 = \u2205 \u2194 \u03b5 = 0 :=\n  eq_empty_iff_forall_not_mem.trans\n    \u27e8fun h => le_bot_iff.1 (le_of_not_gt fun \u03b50 => h _ (mem_ball_self \u03b50)), fun \u03b50 _ h =>\n      not_lt_of_le (le_of_eq \u03b50) (pos_of_mem_ball h)\u27e9\n#align emetric.ball_eq_empty_iff EMetric.ball_eq_empty_iff\n\ntheorem ordConnected_setOf_closedBall_subset (x : \u03b1) (s : Set \u03b1) :\n    OrdConnected { r | closedBall x r \u2286 s } :=\n  \u27e8fun _ _ _ h\u2081 _ h\u2082 => (closedBall_subset_closedBall h\u2082.2).trans h\u2081\u27e9\n#align emetric.ord_connected_set_of_closed_ball_subset EMetric.ordConnected_setOf_closedBall_subset\n\ntheorem ordConnected_setOf_ball_subset (x : \u03b1) (s : Set \u03b1) : OrdConnected { r | ball x r \u2286 s } :=\n  \u27e8fun _ _ _ h\u2081 _ h\u2082 => (ball_subset_ball h\u2082.2).trans h\u2081\u27e9\n#align emetric.ord_connected_set_of_ball_subset EMetric.ordConnected_setOf_ball_subset\n\n/-- Relation \u201ctwo points are at a finite edistance\u201d is an equivalence relation. -/\ndef edistLtTopSetoid : Setoid \u03b1 where\n  r x y := edist x y < \u22a4\n  iseqv :=\n    \u27e8fun x => by rw [edist_self]; exact ENNReal.coe_lt_top,\n      fun h => by rwa [edist_comm], fun hxy hyz =>\n        lt_of_le_of_lt (edist_triangle _ _ _) (ENNReal.add_lt_top.2 \u27e8hxy, hyz\u27e9)\u27e9\n#align emetric.edist_lt_top_setoid EMetric.edistLtTopSetoid\n\n@[simp]\ntheorem ball_zero : ball x 0 = \u2205 := by rw [EMetric.ball_eq_empty_iff]\n#align emetric.ball_zero EMetric.ball_zero\n\ntheorem nhds_basis_eball : (\ud835\udcdd x).HasBasis (fun \u03b5 : \u211d\u22650\u221e => 0 < \u03b5) (ball x) :=\n  nhds_basis_uniformity uniformity_basis_edist\n#align emetric.nhds_basis_eball EMetric.nhds_basis_eball\n\ntheorem nhdsWithin_basis_eball : (\ud835\udcdd[s] x).HasBasis (fun \u03b5 : \u211d\u22650\u221e => 0 < \u03b5) fun \u03b5 => ball x \u03b5 \u2229 s :=\n  nhdsWithin_hasBasis nhds_basis_eball s\n#align emetric.nhds_within_basis_eball EMetric.nhdsWithin_basis_eball\n\ntheorem nhds_basis_closed_eball : (\ud835\udcdd x).HasBasis (fun \u03b5 : \u211d\u22650\u221e => 0 < \u03b5) (closedBall x) :=\n  nhds_basis_uniformity uniformity_basis_edist_le\n#align emetric.nhds_basis_closed_eball EMetric.nhds_basis_closed_eball\n\ntheorem nhdsWithin_basis_closed_eball :\n    (\ud835\udcdd[s] x).HasBasis (fun \u03b5 : \u211d\u22650\u221e => 0 < \u03b5) fun \u03b5 => closedBall x \u03b5 \u2229 s :=\n  nhdsWithin_hasBasis nhds_basis_closed_eball s\n#align emetric.nhds_within_basis_closed_eball EMetric.nhdsWithin_basis_closed_eball\n\ntheorem nhds_eq : \ud835\udcdd x = \u2a05 \u03b5 > 0, \ud835\udcdf (ball x \u03b5) :=\n  nhds_basis_eball.eq_biInf\n#align emetric.nhds_eq EMetric.nhds_eq\n\ntheorem mem_nhds_iff : s \u2208 \ud835\udcdd x \u2194 \u2203 \u03b5 > 0, ball x \u03b5 \u2286 s :=\n  nhds_basis_eball.mem_iff\n#align emetric.mem_nhds_iff EMetric.mem_nhds_iff\n\ntheorem mem_nhdsWithin_iff : s \u2208 \ud835\udcdd[t] x \u2194 \u2203 \u03b5 > 0, ball x \u03b5 \u2229 t \u2286 s :=\n  nhdsWithin_basis_eball.mem_iff\n#align emetric.mem_nhds_within_iff EMetric.mem_nhdsWithin_iff\n\nsection\n\nvariable [PseudoEMetricSpace \u03b2] {f : \u03b1 \u2192 \u03b2}\n\ntheorem tendsto_nhdsWithin_nhdsWithin {t : Set \u03b2} {a b} :\n    Tendsto f (\ud835\udcdd[s] a) (\ud835\udcdd[t] b) \u2194\n      \u2200 \u03b5 > 0, \u2203 \u03b4 > 0, \u2200 \u2983x\u2984, x \u2208 s \u2192 edist x a < \u03b4 \u2192 f x \u2208 t \u2227 edist (f x) b < \u03b5 :=\n  (nhdsWithin_basis_eball.tendsto_iff nhdsWithin_basis_eball).trans <|\n    forall\u2082_congr fun \u03b5 _ => exists_congr fun \u03b4 => and_congr_right fun _ =>\n      forall_congr' fun x => by simp; tauto\n#align emetric.tendsto_nhds_within_nhds_within EMetric.tendsto_nhdsWithin_nhdsWithin\n\ntheorem tendsto_nhdsWithin_nhds {a b} :\n    Tendsto f (\ud835\udcdd[s] a) (\ud835\udcdd b) \u2194\n      \u2200 \u03b5 > 0, \u2203 \u03b4 > 0, \u2200 {x : \u03b1}, x \u2208 s \u2192 edist x a < \u03b4 \u2192 edist (f x) b < \u03b5 := by\n  rw [\u2190 nhdsWithin_univ b, tendsto_nhdsWithin_nhdsWithin]\n  simp only [mem_univ, true_and_iff]\n#align emetric.tendsto_nhds_within_nhds EMetric.tendsto_nhdsWithin_nhds\n\ntheorem tendsto_nhds_nhds {a b} :\n    Tendsto f (\ud835\udcdd a) (\ud835\udcdd b) \u2194 \u2200 \u03b5 > 0, \u2203 \u03b4 > 0, \u2200 \u2983x\u2984, edist x a < \u03b4 \u2192 edist (f x) b < \u03b5 :=\n  nhds_basis_eball.tendsto_iff nhds_basis_eball\n#align emetric.tendsto_nhds_nhds EMetric.tendsto_nhds_nhds\n\nend\n\ntheorem isOpen_iff : IsOpen s \u2194 \u2200 x \u2208 s, \u2203 \u03b5 > 0, ball x \u03b5 \u2286 s := by\n  simp [isOpen_iff_nhds, mem_nhds_iff]\n#align emetric.is_open_iff EMetric.isOpen_iff\n\ntheorem isOpen_ball : IsOpen (ball x \u03b5) :=\n  isOpen_iff.2 fun _ => exists_ball_subset_ball\n#align emetric.is_open_ball EMetric.isOpen_ball\n\ntheorem isClosed_ball_top : IsClosed (ball x \u22a4) :=\n  isOpen_compl_iff.1 <| isOpen_iff.2 fun _y hy =>\n    \u27e8\u22a4, ENNReal.coe_lt_top, fun _z hzy hzx =>\n      hy (edistLtTopSetoid.trans (edistLtTopSetoid.symm hzy) hzx)\u27e9\n#align emetric.is_closed_ball_top EMetric.isClosed_ball_top\n\ntheorem ball_mem_nhds (x : \u03b1) {\u03b5 : \u211d\u22650\u221e} (\u03b50 : 0 < \u03b5) : ball x \u03b5 \u2208 \ud835\udcdd x :=\n  isOpen_ball.mem_nhds (mem_ball_self \u03b50)\n#align emetric.ball_mem_nhds EMetric.ball_mem_nhds\n\ntheorem closedBall_mem_nhds (x : \u03b1) {\u03b5 : \u211d\u22650\u221e} (\u03b50 : 0 < \u03b5) : closedBall x \u03b5 \u2208 \ud835\udcdd x :=\n  mem_of_superset (ball_mem_nhds x \u03b50) ball_subset_closedBall\n#align emetric.closed_ball_mem_nhds EMetric.closedBall_mem_nhds\n\ntheorem ball_prod_same [PseudoEMetricSpace \u03b2] (x : \u03b1) (y : \u03b2) (r : \u211d\u22650\u221e) :\n    ball x r \u00d7\u02e2 ball y r = ball (x, y) r :=\n  ext fun z => by simp [Prod.edist_eq]\n#align emetric.ball_prod_same EMetric.ball_prod_same\n\ntheorem closedBall_prod_same [PseudoEMetricSpace \u03b2] (x : \u03b1) (y : \u03b2) (r : \u211d\u22650\u221e) :\n    closedBall x r \u00d7\u02e2 closedBall y r = closedBall (x, y) r :=\n  ext fun z => by simp [Prod.edist_eq]\n#align emetric.closed_ball_prod_same EMetric.closedBall_prod_same\n\n/-- \u03b5-characterization of the closure in pseudoemetric spaces -/\ntheorem mem_closure_iff : x \u2208 closure s \u2194 \u2200 \u03b5 > 0, \u2203 y \u2208 s, edist x y < \u03b5 :=\n  (mem_closure_iff_nhds_basis nhds_basis_eball).trans <| by simp only [mem_ball, edist_comm x]\n#align emetric.mem_closure_iff EMetric.mem_closure_iff\n\ntheorem tendsto_nhds {f : Filter \u03b2} {u : \u03b2 \u2192 \u03b1} {a : \u03b1} :\n    Tendsto u f (\ud835\udcdd a) \u2194 \u2200 \u03b5 > 0, \u2200\u1da0 x in f, edist (u x) a < \u03b5 :=\n  nhds_basis_eball.tendsto_right_iff\n#align emetric.tendsto_nhds EMetric.tendsto_nhds\n\ntheorem tendsto_atTop [Nonempty \u03b2] [SemilatticeSup \u03b2] {u : \u03b2 \u2192 \u03b1} {a : \u03b1} :\n    Tendsto u atTop (\ud835\udcdd a) \u2194 \u2200 \u03b5 > 0, \u2203 N, \u2200 n \u2265 N, edist (u n) a < \u03b5 :=\n  (atTop_basis.tendsto_iff nhds_basis_eball).trans <| by\n    simp only [exists_prop, true_and_iff, mem_Ici, mem_ball]\n#align emetric.tendsto_at_top EMetric.tendsto_atTop\n\ntheorem inseparable_iff : Inseparable x y \u2194 edist x y = 0 := by\n  simp [inseparable_iff_mem_closure, mem_closure_iff, edist_comm, forall_lt_iff_le']\n#align emetric.inseparable_iff EMetric.inseparable_iff\n\n-- see Note [nolint_ge]\n/-- In a pseudoemetric space, Cauchy sequences are characterized by the fact that, eventually,\nthe pseudoedistance between its elements is arbitrarily small -/\ntheorem cauchySeq_iff [Nonempty \u03b2] [SemilatticeSup \u03b2] {u : \u03b2 \u2192 \u03b1} :\n    CauchySeq u \u2194 \u2200 \u03b5 > 0, \u2203 N, \u2200 m, N \u2264 m \u2192 \u2200 n, N \u2264 n \u2192 edist (u m) (u n) < \u03b5 :=\n  uniformity_basis_edist.cauchySeq_iff\n#align emetric.cauchy_seq_iff EMetric.cauchySeq_iff\n\n/-- A variation around the emetric characterization of Cauchy sequences -/\ntheorem cauchySeq_iff' [Nonempty \u03b2] [SemilatticeSup \u03b2] {u : \u03b2 \u2192 \u03b1} :\n    CauchySeq u \u2194 \u2200 \u03b5 > (0 : \u211d\u22650\u221e), \u2203 N, \u2200 n \u2265 N, edist (u n) (u N) < \u03b5 :=\n  uniformity_basis_edist.cauchySeq_iff'\n#align emetric.cauchy_seq_iff' EMetric.cauchySeq_iff'\n\n/-- A variation of the emetric characterization of Cauchy sequences that deals with\n`\u211d\u22650` upper bounds. -/\ntheorem cauchySeq_iff_NNReal [Nonempty \u03b2] [SemilatticeSup \u03b2] {u : \u03b2 \u2192 \u03b1} :\n    CauchySeq u \u2194 \u2200 \u03b5 : \u211d\u22650, 0 < \u03b5 \u2192 \u2203 N, \u2200 n, N \u2264 n \u2192 edist (u n) (u N) < \u03b5 :=\n  uniformity_basis_edist_nnreal.cauchySeq_iff'\n#align emetric.cauchy_seq_iff_nnreal EMetric.cauchySeq_iff_NNReal\n\ntheorem totallyBounded_iff {s : Set \u03b1} :\n    TotallyBounded s \u2194 \u2200 \u03b5 > 0, \u2203 t : Set \u03b1, t.Finite \u2227 s \u2286 \u22c3 y \u2208 t, ball y \u03b5 :=\n  \u27e8fun H _\u03b5 \u03b50 => H _ (edist_mem_uniformity \u03b50), fun H _r ru =>\n    let \u27e8\u03b5, \u03b50, h\u03b5\u27e9 := mem_uniformity_edist.1 ru\n    let \u27e8t, ft, h\u27e9 := H \u03b5 \u03b50\n    \u27e8t, ft, h.trans <| iUnion\u2082_mono fun _ _ _ => h\u03b5\u27e9\u27e9\n#align emetric.totally_bounded_iff EMetric.totallyBounded_iff\n\ntheorem totallyBounded_iff' {s : Set \u03b1} :\n    TotallyBounded s \u2194 \u2200 \u03b5 > 0, \u2203 t, t \u2286 s \u2227 Set.Finite t \u2227 s \u2286 \u22c3 y \u2208 t, ball y \u03b5 :=\n  \u27e8fun H _\u03b5 \u03b50 => (totallyBounded_iff_subset.1 H) _ (edist_mem_uniformity \u03b50), fun H _r ru =>\n    let \u27e8\u03b5, \u03b50, h\u03b5\u27e9 := mem_uniformity_edist.1 ru\n    let \u27e8t, _, ft, h\u27e9 := H \u03b5 \u03b50\n    \u27e8t, ft, h.trans <| iUnion\u2082_mono fun _ _ _ => h\u03b5\u27e9\u27e9\n#align emetric.totally_bounded_iff' EMetric.totallyBounded_iff'\n\nsection Compact\n\n-- Porting note (#11215): TODO: generalize to a uniform space with metrizable uniformity\n/-- For a set `s` in a pseudo emetric space, if for every `\u03b5 > 0` there exists a countable\nset that is `\u03b5`-dense in `s`, then there exists a countable subset `t \u2286 s` that is dense in `s`. -/\ntheorem subset_countable_closure_of_almost_dense_set (s : Set \u03b1)\n    (hs : \u2200 \u03b5 > 0, \u2203 t : Set \u03b1, t.Countable \u2227 s \u2286 \u22c3 x \u2208 t, closedBall x \u03b5) :\n    \u2203 t, t \u2286 s \u2227 t.Countable \u2227 s \u2286 closure t := by\n  rcases s.eq_empty_or_nonempty with (rfl | \u27e8x\u2080, hx\u2080\u27e9)\n  \u00b7 exact \u27e8\u2205, empty_subset _, countable_empty, empty_subset _\u27e9\n  choose! T hTc hsT using fun n : \u2115 => hs n\u207b\u00b9 (by simp)\n  have : \u2200 r x, \u2203 y \u2208 s, closedBall x r \u2229 s \u2286 closedBall y (r * 2) := fun r x => by\n    rcases (closedBall x r \u2229 s).eq_empty_or_nonempty with (he | \u27e8y, hxy, hys\u27e9)\n    \u00b7 refine' \u27e8x\u2080, hx\u2080, _\u27e9\n      rw [he]\n      exact empty_subset _\n    \u00b7 refine' \u27e8y, hys, fun z hz => _\u27e9\n      calc\n        edist z y \u2264 edist z x + edist y x := edist_triangle_right _ _ _\n        _ \u2264 r + r := add_le_add hz.1 hxy\n        _ = r * 2 := (mul_two r).symm\n  choose f hfs hf using this\n  refine'\n    \u27e8\u22c3 n : \u2115, f n\u207b\u00b9 '' T n, iUnion_subset fun n => image_subset_iff.2 fun z _ => hfs _ _,\n      countable_iUnion fun n => (hTc n).image _, _\u27e9\n  refine' fun x hx => mem_closure_iff.2 fun \u03b5 \u03b50 => _\n  rcases ENNReal.exists_inv_nat_lt (ENNReal.half_pos \u03b50.lt.ne').ne' with \u27e8n, hn\u27e9\n  rcases mem_iUnion\u2082.1 (hsT n hx) with \u27e8y, hyn, hyx\u27e9\n  refine' \u27e8f n\u207b\u00b9 y, mem_iUnion.2 \u27e8n, mem_image_of_mem _ hyn\u27e9, _\u27e9\n  calc\n    edist x (f n\u207b\u00b9 y) \u2264 (n : \u211d\u22650\u221e)\u207b\u00b9 * 2 := hf _ _ \u27e8hyx, hx\u27e9\n    _ < \u03b5 := ENNReal.mul_lt_of_lt_div hn\n#align emetric.subset_countable_closure_of_almost_dense_set EMetric.subset_countable_closure_of_almost_dense_set\n\nopen TopologicalSpace in\n/-- If a set `s` is separable in a (pseudo extended) metric space, then it admits a countable dense\nsubset. This is not obvious, as the countable set whose closure covers `s` given by the definition\nof separability does not need in general to be contained in `s`. -/\ntheorem _root_.TopologicalSpace.IsSeparable.exists_countable_dense_subset\n    {s : Set \u03b1} (hs : IsSeparable s) : \u2203 t, t \u2286 s \u2227 t.Countable \u2227 s \u2286 closure t := by\n  have : \u2200 \u03b5 > 0, \u2203 t : Set \u03b1, t.Countable \u2227 s \u2286 \u22c3 x \u2208 t, closedBall x \u03b5 := fun \u03b5 \u03b50 => by\n    rcases hs with \u27e8t, htc, hst\u27e9\n    refine \u27e8t, htc, hst.trans fun x hx => ?_\u27e9\n    rcases mem_closure_iff.1 hx \u03b5 \u03b50 with \u27e8y, hyt, hxy\u27e9\n    exact mem_iUnion\u2082.2 \u27e8y, hyt, mem_closedBall.2 hxy.le\u27e9\n  exact subset_countable_closure_of_almost_dense_set _ this\n\nopen TopologicalSpace in\n/-- If a set `s` is separable, then the corresponding subtype is separable in a (pseudo extended)\nmetric space.  This is not obvious, as the countable set whose closure covers `s` does not need in\ngeneral to be contained in `s`. -/\ntheorem _root_.TopologicalSpace.IsSeparable.separableSpace {s : Set \u03b1} (hs : IsSeparable s) :\n    SeparableSpace s := by\n  rcases hs.exists_countable_dense_subset with \u27e8t, hts, htc, hst\u27e9\n  lift t to Set s using hts\n  refine \u27e8\u27e8t, countable_of_injective_of_countable_image (Subtype.coe_injective.injOn _) htc, ?_\u27e9\u27e9\n  rwa [inducing_subtype_val.dense_iff, Subtype.forall]\n#align topological_space.is_separable.separable_space TopologicalSpace.IsSeparable.separableSpace\n\n-- Porting note (#11215): TODO: generalize to metrizable spaces\n/-- A compact set in a pseudo emetric space is separable, i.e., it is a subset of the closure of a\ncountable set.  -/\ntheorem subset_countable_closure_of_compact {s : Set \u03b1} (hs : IsCompact s) :\n    \u2203 t, t \u2286 s \u2227 t.Countable \u2227 s \u2286 closure t := by\n  refine' subset_countable_closure_of_almost_dense_set s fun \u03b5 h\u03b5 => _\n  rcases totallyBounded_iff'.1 hs.totallyBounded \u03b5 h\u03b5 with \u27e8t, -, htf, hst\u27e9\n  exact \u27e8t, htf.countable, hst.trans <| iUnion\u2082_mono fun _ _ => ball_subset_closedBall\u27e9\n#align emetric.subset_countable_closure_of_compact EMetric.subset_countable_closure_of_compact\n\nend Compact\n\nsection SecondCountable\n\nopen TopologicalSpace\n\nvariable (\u03b1)\n\n/-- A sigma compact pseudo emetric space has second countable topology. -/\ninstance (priority := 90) secondCountable_of_sigmaCompact [SigmaCompactSpace \u03b1] :\n    SecondCountableTopology \u03b1 := by\n  suffices SeparableSpace \u03b1 by exact UniformSpace.secondCountable_of_separable \u03b1\n  choose T _ hTc hsubT using fun n =>\n    subset_countable_closure_of_compact (isCompact_compactCovering \u03b1 n)\n  refine' \u27e8\u27e8\u22c3 n, T n, countable_iUnion hTc, fun x => _\u27e9\u27e9\n  rcases iUnion_eq_univ_iff.1 (iUnion_compactCovering \u03b1) x with \u27e8n, hn\u27e9\n  exact closure_mono (subset_iUnion _ n) (hsubT _ hn)\n#align emetric.second_countable_of_sigma_compact EMetric.secondCountable_of_sigmaCompact\n\nvariable {\u03b1}\n\ntheorem secondCountable_of_almost_dense_set\n    (hs : \u2200 \u03b5 > 0, \u2203 t : Set \u03b1, t.Countable \u2227 \u22c3 x \u2208 t, closedBall x \u03b5 = univ) :\n    SecondCountableTopology \u03b1 := by\n  suffices SeparableSpace \u03b1 from UniformSpace.secondCountable_of_separable \u03b1\n  have : \u2200 \u03b5 > 0, \u2203 t : Set \u03b1, Set.Countable t \u2227 univ \u2286 \u22c3 x \u2208 t, closedBall x \u03b5 := by\n    simpa only [univ_subset_iff] using hs\n  rcases subset_countable_closure_of_almost_dense_set (univ : Set \u03b1) this with \u27e8t, -, htc, ht\u27e9\n  exact \u27e8\u27e8t, htc, fun x => ht (mem_univ x)\u27e9\u27e9\n#align emetric.second_countable_of_almost_dense_set EMetric.secondCountable_of_almost_dense_set\n\nend SecondCountable\n\nsection Diam\n\n/-- The diameter of a set in a pseudoemetric space, named `EMetric.diam` -/\nnoncomputable def diam (s : Set \u03b1) :=\n  \u2a06 (x \u2208 s) (y \u2208 s), edist x y\n#align emetric.diam EMetric.diam\n\ntheorem diam_eq_sSup (s : Set \u03b1) : diam s = sSup (image2 edist s s) := sSup_image2.symm\n\ntheorem diam_le_iff {d : \u211d\u22650\u221e} : diam s \u2264 d \u2194 \u2200 x \u2208 s, \u2200 y \u2208 s, edist x y \u2264 d := by\n  simp only [diam, iSup_le_iff]\n#align emetric.diam_le_iff EMetric.diam_le_iff\n\ntheorem diam_image_le_iff {d : \u211d\u22650\u221e} {f : \u03b2 \u2192 \u03b1} {s : Set \u03b2} :\n    diam (f '' s) \u2264 d \u2194 \u2200 x \u2208 s, \u2200 y \u2208 s, edist (f x) (f y) \u2264 d := by\n  simp only [diam_le_iff, forall_mem_image]\n#align emetric.diam_image_le_iff EMetric.diam_image_le_iff\n\ntheorem edist_le_of_diam_le {d} (hx : x \u2208 s) (hy : y \u2208 s) (hd : diam s \u2264 d) : edist x y \u2264 d :=\n  diam_le_iff.1 hd x hx y hy\n#align emetric.edist_le_of_diam_le EMetric.edist_le_of_diam_le\n\n/-- If two points belong to some set, their edistance is bounded by the diameter of the set -/\ntheorem edist_le_diam_of_mem (hx : x \u2208 s) (hy : y \u2208 s) : edist x y \u2264 diam s :=\n  edist_le_of_diam_le hx hy le_rfl\n#align emetric.edist_le_diam_of_mem EMetric.edist_le_diam_of_mem\n\n/-- If the distance between any two points in a set is bounded by some constant, this constant\nbounds the diameter. -/\ntheorem diam_le {d : \u211d\u22650\u221e} (h : \u2200 x \u2208 s, \u2200 y \u2208 s, edist x y \u2264 d) : diam s \u2264 d :=\n  diam_le_iff.2 h\n#align emetric.diam_le EMetric.diam_le\n\n/-- The diameter of a subsingleton vanishes. -/\ntheorem diam_subsingleton (hs : s.Subsingleton) : diam s = 0 :=\n  nonpos_iff_eq_zero.1 <| diam_le fun _x hx y hy => (hs hx hy).symm \u25b8 edist_self y \u25b8 le_rfl\n#align emetric.diam_subsingleton EMetric.diam_subsingleton\n\n/-- The diameter of the empty set vanishes -/\n@[simp]\ntheorem diam_empty : diam (\u2205 : Set \u03b1) = 0 :=\n  diam_subsingleton subsingleton_empty\n#align emetric.diam_empty EMetric.diam_empty\n\n/-- The diameter of a singleton vanishes -/\n@[simp]\ntheorem diam_singleton : diam ({x} : Set \u03b1) = 0 :=\n  diam_subsingleton subsingleton_singleton\n#align emetric.diam_singleton EMetric.diam_singleton\n\n@[to_additive (attr := simp)]\ntheorem diam_one [One \u03b1] : diam (1 : Set \u03b1) = 0 :=\n  diam_singleton\n#align emetric.diam_one EMetric.diam_one\n#align emetric.diam_zero EMetric.diam_zero\n\ntheorem diam_iUnion_mem_option {\u03b9 : Type*} (o : Option \u03b9) (s : \u03b9 \u2192 Set \u03b1) :\n    diam (\u22c3 i \u2208 o, s i) = \u2a06 i \u2208 o, diam (s i) := by cases o <;> simp\n#align emetric.diam_Union_mem_option EMetric.diam_iUnion_mem_option\n\ntheorem diam_insert : diam (insert x s) = max (\u2a06 y \u2208 s, edist x y) (diam s) :=\n  eq_of_forall_ge_iff fun d => by\n    simp only [diam_le_iff, forall_mem_insert, edist_self, edist_comm x, max_le_iff, iSup_le_iff,\n      zero_le, true_and_iff, forall_and, and_self_iff, \u2190 and_assoc]\n#align emetric.diam_insert EMetric.diam_insert\n\ntheorem diam_pair : diam ({x, y} : Set \u03b1) = edist x y := by\n  simp only [iSup_singleton, diam_insert, diam_singleton, ENNReal.max_zero_right]\n#align emetric.diam_pair EMetric.diam_pair\n\ntheorem diam_triple : diam ({x, y, z} : Set \u03b1) = max (max (edist x y) (edist x z)) (edist y z) := by\n  simp only [diam_insert, iSup_insert, iSup_singleton, diam_singleton, ENNReal.max_zero_right,\n    ENNReal.sup_eq_max]\n#align emetric.diam_triple EMetric.diam_triple\n\n/-- The diameter is monotonous with respect to inclusion -/\ntheorem diam_mono {s t : Set \u03b1} (h : s \u2286 t) : diam s \u2264 diam t :=\n  diam_le fun _x hx _y hy => edist_le_diam_of_mem (h hx) (h hy)\n#align emetric.diam_mono EMetric.diam_mono\n\n/-- The diameter of a union is controlled by the diameter of the sets, and the edistance\nbetween two points in the sets. -/\ntheorem diam_union {t : Set \u03b1} (xs : x \u2208 s) (yt : y \u2208 t) :\n    diam (s \u222a t) \u2264 diam s + edist x y + diam t := by\n  have A : \u2200 a \u2208 s, \u2200 b \u2208 t, edist a b \u2264 diam s + edist x y + diam t := fun a ha b hb =>\n    calc\n      edist a b \u2264 edist a x + edist x y + edist y b := edist_triangle4 _ _ _ _\n      _ \u2264 diam s + edist x y + diam t :=\n        add_le_add (add_le_add (edist_le_diam_of_mem ha xs) le_rfl) (edist_le_diam_of_mem yt hb)\n  refine' diam_le fun a ha b hb => _\n  cases' (mem_union _ _ _).1 ha with h'a h'a <;> cases' (mem_union _ _ _).1 hb with h'b h'b\n  \u00b7 calc\n      edist a b \u2264 diam s := edist_le_diam_of_mem h'a h'b\n      _ \u2264 diam s + (edist x y + diam t) := le_self_add\n      _ = diam s + edist x y + diam t := (add_assoc _ _ _).symm\n  \u00b7 exact A a h'a b h'b\n  \u00b7 have Z := A b h'b a h'a\n    rwa [edist_comm] at Z\n  \u00b7 calc\n      edist a b \u2264 diam t := edist_le_diam_of_mem h'a h'b\n      _ \u2264 diam s + edist x y + diam t := le_add_self\n#align emetric.diam_union EMetric.diam_union\n\ntheorem diam_union' {t : Set \u03b1} (h : (s \u2229 t).Nonempty) : diam (s \u222a t) \u2264 diam s + diam t := by\n  let \u27e8x, \u27e8xs, xt\u27e9\u27e9 := h\n  simpa using diam_union xs xt\n#align emetric.diam_union' EMetric.diam_union'\n\ntheorem diam_closedBall {r : \u211d\u22650\u221e} : diam (closedBall x r) \u2264 2 * r :=\n  diam_le fun a ha b hb =>\n    calc\n      edist a b \u2264 edist a x + edist b x := edist_triangle_right _ _ _\n      _ \u2264 r + r := add_le_add ha hb\n      _ = 2 * r := (two_mul r).symm\n#align emetric.diam_closed_ball EMetric.diam_closedBall\n\ntheorem diam_ball {r : \u211d\u22650\u221e} : diam (ball x r) \u2264 2 * r :=\n  le_trans (diam_mono ball_subset_closedBall) diam_closedBall\n#align emetric.diam_ball EMetric.diam_ball\n\ntheorem diam_pi_le_of_le {\u03c0 : \u03b2 \u2192 Type*} [Fintype \u03b2] [\u2200 b, PseudoEMetricSpace (\u03c0 b)]\n    {s : \u2200 b : \u03b2, Set (\u03c0 b)} {c : \u211d\u22650\u221e} (h : \u2200 b, diam (s b) \u2264 c) : diam (Set.pi univ s) \u2264 c := by\n  refine diam_le fun x hx y hy => edist_pi_le_iff.mpr ?_\n  rw [mem_univ_pi] at hx hy\n  exact fun b => diam_le_iff.1 (h b) (x b) (hx b) (y b) (hy b)\n#align emetric.diam_pi_le_of_le EMetric.diam_pi_le_of_le\n\nend Diam\n\nend EMetric\n\n--namespace\n/-- We now define `EMetricSpace`, extending `PseudoEMetricSpace`. -/\nclass EMetricSpace (\u03b1 : Type u) extends PseudoEMetricSpace \u03b1 : Type u where\n  eq_of_edist_eq_zero : \u2200 {x y : \u03b1}, edist x y = 0 \u2192 x = y\n#align emetric_space EMetricSpace\n\n", "theoremStatement": "@[ext]\nprotected theorem EMetricSpace.ext\n    {\u03b1 : Type*} {m m' : EMetricSpace \u03b1} (h : m.toEDist = m'.toEDist) : m = m' ", "theoremName": "EMetricSpace.ext", "fileCreated": {"commit": "dd781b4101c2d151a556ea1323191a6e56f3c92e", "date": "2023-08-31"}, 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"Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.WF", "Mathlib.Init.Data.Nat.Notation", "Std.Data.List.Basic", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Lean.Compiler.CSimpAttr", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Tactic.Lemma", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Tactic.TypeStar", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Basic", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Cases", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.PushNeg", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Tactic.CasesM", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.ByContra", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Tactic.Says", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.AssertExists", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Nat.Defs", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", 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"Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", 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"Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", 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"Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", 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"Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  cases m\n  cases m'\n  congr\n  ext1\n  assumption", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 54}}
+{"srcContext": "/-\nCopyright (c) 2024 Joseph Myers. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Myers\n-/\nimport Mathlib.Data.FunLike.Equiv\nimport Mathlib.Logic.Pairwise\n\n/-!\n# Interaction of equivalences with `Pairwise`\n-/\n\n", "theoremStatement": "lemma EmbeddingLike.pairwise_comp {X : Type*} {Y : Type*} {F} [FunLike F Y X] [EmbeddingLike F Y X]\n    (f : F) {p : X \u2192 X \u2192 Prop} (h : Pairwise p) : Pairwise (p on f) ", "theoremName": "EmbeddingLike.pairwise_comp", "fileCreated": {"commit": "30850dba6554df47eeb9a8aafae445d585f49e73", "date": "2024-03-26"}, "theoremCreated": {"commit": "30850dba6554df47eeb9a8aafae445d585f49e73", "date": "2024-03-26"}, "file": "mathlib/Mathlib/Logic/Equiv/Pairwise.lean", "module": "Mathlib.Logic.Equiv.Pairwise", "jsonFile": "Mathlib.Logic.Equiv.Pairwise.jsonl", "positionMetadata": {"lineInFile": 13, "tokenPositionInFile": 269, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, 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"Mathlib.Tactic.WLOG", "Mathlib.Util.AssertExists", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Logic.Pairwise"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n  h.comp_of_injective <| EmbeddingLike.injective f", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 53}}
+{"srcContext": "/-\nCopyright (c) 2019 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\nimport Mathlib.Algebra.Group.Equiv.Basic\nimport Mathlib.Control.Applicative\nimport Mathlib.Control.Traversable.Basic\nimport Mathlib.Data.List.Basic\nimport Mathlib.Logic.Equiv.Defs\n\n#align_import algebra.free from \"leanprover-community/mathlib\"@\"6d0adfa76594f304b4650d098273d4366edeb61b\"\n\n/-!\n# Free constructions\n\n## Main definitions\n\n* `FreeMagma \u03b1`: free magma (structure with binary operation without any axioms) over alphabet `\u03b1`,\n  defined inductively, with traversable instance and decidable equality.\n* `MagmaAssocQuotient \u03b1`: quotient of a magma `\u03b1` by the associativity equivalence relation.\n* `FreeSemigroup \u03b1`: free semigroup over alphabet `\u03b1`, defined as a structure with two fields\n  `head : \u03b1` and `tail : List \u03b1` (i.e. nonempty lists), with traversable instance and decidable\n  equality.\n* `FreeMagmaAssocQuotientEquiv \u03b1`: isomorphism between `MagmaAssocQuotient (FreeMagma \u03b1)` and\n  `FreeSemigroup \u03b1`.\n* `FreeMagma.lift`: the universal property of the free magma, expressing its adjointness.\n-/\n\nuniverse u v l\n\n/-- Free nonabelian additive magma over a given alphabet. -/\ninductive FreeAddMagma (\u03b1 : Type u) : Type u\n  | of : \u03b1 \u2192 FreeAddMagma \u03b1\n  | add : FreeAddMagma \u03b1 \u2192 FreeAddMagma \u03b1 \u2192 FreeAddMagma \u03b1\n  deriving DecidableEq\n#align free_add_magma FreeAddMagma\ncompile_inductive% FreeAddMagma\n\n/-- Free magma over a given alphabet. -/\n@[to_additive]\ninductive FreeMagma (\u03b1 : Type u) : Type u\n  | of : \u03b1 \u2192 FreeMagma \u03b1\n  | mul : FreeMagma \u03b1 \u2192 FreeMagma \u03b1 \u2192 FreeMagma \u03b1\n  deriving DecidableEq\n#align free_magma FreeMagma\ncompile_inductive% FreeMagma\n\nnamespace FreeMagma\n\nvariable {\u03b1 : Type u}\n\n@[to_additive]\ninstance [Inhabited \u03b1] : Inhabited (FreeMagma \u03b1) := \u27e8of default\u27e9\n\n@[to_additive]\ninstance : Mul (FreeMagma \u03b1) := \u27e8FreeMagma.mul\u27e9\n\n-- Porting note: invalid attribute 'match_pattern', declaration is in an imported module\n-- attribute [match_pattern] Mul.mul\n\n@[to_additive (attr := simp)]\ntheorem mul_eq (x y : FreeMagma \u03b1) : mul x y = x * y := rfl\n#align free_magma.mul_eq FreeMagma.mul_eq\n\n/- Porting note: these lemmas are autogenerated by the inductive definition and due to\nthe existence of mul_eq not in simp normal form -/\nattribute [nolint simpNF] FreeAddMagma.add.sizeOf_spec\nattribute [nolint simpNF] FreeMagma.mul.sizeOf_spec\nattribute [nolint simpNF] FreeAddMagma.add.injEq\nattribute [nolint simpNF] FreeMagma.mul.injEq\n\n/-- Recursor for `FreeMagma` using `x * y` instead of `FreeMagma.mul x y`. -/\n@[to_additive (attr := elab_as_elim) \"Recursor for `FreeAddMagma` using `x + y` instead of\n`FreeAddMagma.add x y`.\"]\ndef recOnMul {C : FreeMagma \u03b1 \u2192 Sort l} (x) (ih1 : \u2200 x, C (of x))\n    (ih2 : \u2200 x y, C x \u2192 C y \u2192 C (x * y)) : C x :=\n  FreeMagma.recOn x ih1 ih2\n#align free_magma.rec_on_mul FreeMagma.recOnMul\n\n@[to_additive (attr := ext 1100)]\ntheorem hom_ext {\u03b2 : Type v} [Mul \u03b2] {f g : FreeMagma \u03b1 \u2192\u2099* \u03b2} (h : f \u2218 of = g \u2218 of) : f = g :=\n  (DFunLike.ext _ _) fun x \u21a6 recOnMul x (congr_fun h) <| by intros; simp only [map_mul, *]\n#align free_magma.hom_ext FreeMagma.hom_ext\n\nend FreeMagma\n\n/-- Lifts a function `\u03b1 \u2192 \u03b2` to a magma homomorphism `FreeMagma \u03b1 \u2192 \u03b2` given a magma `\u03b2`. -/\ndef FreeMagma.liftAux {\u03b1 : Type u} {\u03b2 : Type v} [Mul \u03b2] (f : \u03b1 \u2192 \u03b2) : FreeMagma \u03b1 \u2192 \u03b2\n  | FreeMagma.of x => f x\n  -- Adaptation note: around nightly-2024-02-25, we need to write `mul x y` in the pattern here,\n  -- instead of `x * y`.\n  | mul x y => liftAux f x * liftAux f y\n#align free_magma.lift_aux FreeMagma.liftAux\n\n/-- Lifts a function `\u03b1 \u2192 \u03b2` to an additive magma homomorphism `FreeAddMagma \u03b1 \u2192 \u03b2` given\nan additive magma `\u03b2`. -/\ndef FreeAddMagma.liftAux {\u03b1 : Type u} {\u03b2 : Type v} [Add \u03b2] (f : \u03b1 \u2192 \u03b2) : FreeAddMagma \u03b1 \u2192 \u03b2\n  | FreeAddMagma.of x => f x\n  | x + y => liftAux f x + liftAux f y\n#align free_add_magma.lift_aux FreeAddMagma.liftAux\n\nattribute [to_additive existing] FreeMagma.liftAux\n\nnamespace FreeMagma\n\nsection lift\n\nvariable {\u03b1 : Type u} {\u03b2 : Type v} [Mul \u03b2] (f : \u03b1 \u2192 \u03b2)\n\n/-- The universal property of the free magma expressing its adjointness. -/\n@[to_additive (attr := simps symm_apply)\n\"The universal property of the free additive magma expressing its adjointness.\"]\ndef lift : (\u03b1 \u2192 \u03b2) \u2243 (FreeMagma \u03b1 \u2192\u2099* \u03b2) where\n  toFun f :=\n  { toFun := liftAux f\n    map_mul' := fun x y \u21a6 rfl }\n  invFun F := F \u2218 of\n  left_inv f := rfl\n  right_inv F := by ext; rfl\n#align free_magma.lift FreeMagma.lift\n\n@[to_additive (attr := simp)]\ntheorem lift_of (x) : lift f (of x) = f x := rfl\n#align free_magma.lift_of FreeMagma.lift_of\n\n@[to_additive (attr := simp)]\ntheorem lift_comp_of : lift f \u2218 of = f := rfl\n#align free_magma.lift_comp_of FreeMagma.lift_comp_of\n\n@[to_additive (attr := simp)]\ntheorem lift_comp_of' (f : FreeMagma \u03b1 \u2192\u2099* \u03b2) : lift (f \u2218 of) = f := lift.apply_symm_apply f\n#align free_magma.lift_comp_of' FreeMagma.lift_comp_of'\n\nend lift\n\nsection Map\n\nvariable {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2)\n\n/-- The unique magma homomorphism `FreeMagma \u03b1 \u2192\u2099* FreeMagma \u03b2` that sends\neach `of x` to `of (f x)`. -/\n@[to_additive \"The unique additive magma homomorphism `FreeAddMagma \u03b1 \u2192 FreeAddMagma \u03b2` that sends\neach `of x` to `of (f x)`.\"]\ndef map (f : \u03b1 \u2192 \u03b2) : FreeMagma \u03b1 \u2192\u2099* FreeMagma \u03b2 := lift (of \u2218 f)\n#align free_magma.map FreeMagma.map\n\n@[to_additive (attr := simp)]\ntheorem map_of (x) : map f (of x) = of (f x) := rfl\n#align free_magma.map_of FreeMagma.map_of\n\nend Map\n\nsection Category\n\nvariable {\u03b1 \u03b2 : Type u}\n\n@[to_additive]\ninstance : Monad FreeMagma where\n  pure := of\n  bind x f := lift f x\n\n/-- Recursor on `FreeMagma` using `pure` instead of `of`. -/\n@[to_additive (attr := elab_as_elim) \"Recursor on `FreeAddMagma` using `pure` instead of `of`.\"]\nprotected def recOnPure {C : FreeMagma \u03b1 \u2192 Sort l} (x) (ih1 : \u2200 x, C (pure x))\n    (ih2 : \u2200 x y, C x \u2192 C y \u2192 C (x * y)) : C x :=\n  FreeMagma.recOnMul x ih1 ih2\n#align free_magma.rec_on_pure FreeMagma.recOnPure\n\n-- Porting note (#10675): dsimp can not prove this\n@[to_additive (attr := simp, nolint simpNF)]\ntheorem map_pure (f : \u03b1 \u2192 \u03b2) (x) : (f <$> pure x : FreeMagma \u03b2) = pure (f x) := rfl\n#align free_magma.map_pure FreeMagma.map_pure\n\n@[to_additive (attr := simp)]\ntheorem map_mul' (f : \u03b1 \u2192 \u03b2) (x y : FreeMagma \u03b1) : f <$> (x * y) = f <$> x * f <$> y := rfl\n#align free_magma.map_mul' FreeMagma.map_mul'\n\n-- Porting note (#10675): dsimp can not prove this\n@[to_additive (attr := simp, nolint simpNF)]\ntheorem pure_bind (f : \u03b1 \u2192 FreeMagma \u03b2) (x) : pure x >>= f = f x := rfl\n#align free_magma.pure_bind FreeMagma.pure_bind\n\n@[to_additive (attr := simp)]\ntheorem mul_bind (f : \u03b1 \u2192 FreeMagma \u03b2) (x y : FreeMagma \u03b1) : x * y >>= f = (x >>= f) * (y >>= f) :=\n  rfl\n#align free_magma.mul_bind FreeMagma.mul_bind\n\n@[to_additive (attr := simp)]\ntheorem pure_seq {\u03b1 \u03b2 : Type u} {f : \u03b1 \u2192 \u03b2} {x : FreeMagma \u03b1} : pure f <*> x = f <$> x := rfl\n#align free_magma.pure_seq FreeMagma.pure_seq\n\n@[to_additive (attr := simp)]\ntheorem mul_seq {\u03b1 \u03b2 : Type u} {f g : FreeMagma (\u03b1 \u2192 \u03b2)} {x : FreeMagma \u03b1} :\n    f * g <*> x = (f <*> x) * (g <*> x) := rfl\n#align free_magma.mul_seq FreeMagma.mul_seq\n\n@[to_additive]\ninstance instLawfulMonad : LawfulMonad FreeMagma.{u} := LawfulMonad.mk'\n  (pure_bind := fun f x \u21a6 rfl)\n  (bind_assoc := fun x f g \u21a6 FreeMagma.recOnPure x (fun x \u21a6 rfl) fun x y ih1 ih2 \u21a6 by\n    rw [mul_bind, mul_bind, mul_bind, ih1, ih2])\n  (id_map := fun x \u21a6 FreeMagma.recOnPure x (fun _ \u21a6 rfl) fun x y ih1 ih2 \u21a6 by\n    rw [map_mul', ih1, ih2])\n\nend Category\n\nend FreeMagma\n\n/-- `FreeMagma` is traversable. -/\nprotected def FreeMagma.traverse {m : Type u \u2192 Type u} [Applicative m] {\u03b1 \u03b2 : Type u}\n    (F : \u03b1 \u2192 m \u03b2) : FreeMagma \u03b1 \u2192 m (FreeMagma \u03b2)\n  | FreeMagma.of x => FreeMagma.of <$> F x\n  -- Adaptation note: around nightly-2024-02-25, we need to write `mul x y` in the pattern here,\n  -- instead of `x * y`.\n  | mul x y => (\u00b7 * \u00b7) <$> x.traverse F <*> y.traverse F\n#align free_magma.traverse FreeMagma.traverse\n\n/-- `FreeAddMagma` is traversable. -/\nprotected def FreeAddMagma.traverse {m : Type u \u2192 Type u} [Applicative m] {\u03b1 \u03b2 : Type u}\n    (F : \u03b1 \u2192 m \u03b2) : FreeAddMagma \u03b1 \u2192 m (FreeAddMagma \u03b2)\n  | FreeAddMagma.of x => FreeAddMagma.of <$> F x\n  | x + y => (\u00b7 + \u00b7) <$> x.traverse F <*> y.traverse F\n#align free_add_magma.traverse FreeAddMagma.traverse\n\nattribute [to_additive existing] FreeMagma.traverse\n\nnamespace FreeMagma\n\nvariable {\u03b1 : Type u}\n\nsection Category\n\nvariable {\u03b2 : Type u}\n\n@[to_additive]\ninstance : Traversable FreeMagma := \u27e8@FreeMagma.traverse\u27e9\n\nvariable {m : Type u \u2192 Type u} [Applicative m] (F : \u03b1 \u2192 m \u03b2)\n\n@[to_additive (attr := simp)]\ntheorem traverse_pure (x) : traverse F (pure x : FreeMagma \u03b1) = pure <$> F x := rfl\n#align free_magma.traverse_pure FreeMagma.traverse_pure\n\n@[to_additive (attr := simp)]\ntheorem traverse_pure' : traverse F \u2218 pure = fun x \u21a6 (pure <$> F x : m (FreeMagma \u03b2)) := rfl\n#align free_magma.traverse_pure' FreeMagma.traverse_pure'\n\n@[to_additive (attr := simp)]\ntheorem traverse_mul (x y : FreeMagma \u03b1) :\n    traverse F (x * y) = (\u00b7 * \u00b7) <$> traverse F x <*> traverse F y := rfl\n#align free_magma.traverse_mul FreeMagma.traverse_mul\n\n@[to_additive (attr := simp)]\ntheorem traverse_mul' :\n    Function.comp (traverse F) \u2218 (HMul.hMul : FreeMagma \u03b1 \u2192 FreeMagma \u03b1 \u2192 FreeMagma \u03b1) = fun x y \u21a6\n      (\u00b7 * \u00b7) <$> traverse F x <*> traverse F y := rfl\n#align free_magma.traverse_mul' FreeMagma.traverse_mul'\n\n@[to_additive (attr := simp)]\ntheorem traverse_eq (x) : FreeMagma.traverse F x = traverse F x := rfl\n#align free_magma.traverse_eq FreeMagma.traverse_eq\n\n-- Porting note (#10675): dsimp can not prove this\n@[to_additive (attr := simp, nolint simpNF)]\ntheorem mul_map_seq (x y : FreeMagma \u03b1) :\n    ((\u00b7 * \u00b7) <$> x <*> y : Id (FreeMagma \u03b1)) = (x * y : FreeMagma \u03b1) := rfl\n#align free_magma.mul_map_seq FreeMagma.mul_map_seq\n\n@[to_additive]\ninstance : LawfulTraversable FreeMagma.{u} :=\n  { instLawfulMonad with\n    id_traverse := fun x \u21a6\n      FreeMagma.recOnPure x (fun x \u21a6 rfl) fun x y ih1 ih2 \u21a6 by\n        rw [traverse_mul, ih1, ih2, mul_map_seq]\n    comp_traverse := fun f g x \u21a6\n      FreeMagma.recOnPure x\n        (fun x \u21a6 by simp only [(\u00b7 \u2218 \u00b7), traverse_pure, traverse_pure', functor_norm])\n        (fun x y ih1 ih2 \u21a6 by\n          rw [traverse_mul, ih1, ih2, traverse_mul];\n          simp [Functor.Comp.map_mk, Functor.map_map, (\u00b7 \u2218 \u00b7), Comp.seq_mk, seq_map_assoc,\n            map_seq, traverse_mul])\n    naturality := fun \u03b7 \u03b1 \u03b2 f x \u21a6\n      FreeMagma.recOnPure x\n        (fun x \u21a6 by simp only [traverse_pure, functor_norm, Function.comp_apply])\n        (fun x y ih1 ih2 \u21a6 by simp only [traverse_mul, functor_norm, ih1, ih2])\n    traverse_eq_map_id := fun f x \u21a6\n      FreeMagma.recOnPure x (fun _ \u21a6 rfl) fun x y ih1 ih2 \u21a6 by\n        rw [traverse_mul, ih1, ih2, map_mul', mul_map_seq]; rfl }\n\nend Category\n\nend FreeMagma\n\n-- Porting note: changed String to Lean.Format\n/-- Representation of an element of a free magma. -/\nprotected def FreeMagma.repr {\u03b1 : Type u} [Repr \u03b1] : FreeMagma \u03b1 \u2192 Lean.Format\n  | FreeMagma.of x => repr x\n  -- Adaptation note: around nightly-2024-02-25, we need to write `mul x y` in the pattern here,\n  -- instead of `x * y`.\n  | mul x y => \"( \" ++ x.repr ++ \" * \" ++ y.repr ++ \" )\"\n#align free_magma.repr FreeMagma.repr\n\n/-- Representation of an element of a free additive magma. -/\nprotected def FreeAddMagma.repr {\u03b1 : Type u} [Repr \u03b1] : FreeAddMagma \u03b1 \u2192 Lean.Format\n  | FreeAddMagma.of x => repr x\n  | x + y => \"( \" ++ x.repr ++ \" + \" ++ y.repr ++ \" )\"\n#align free_add_magma.repr FreeAddMagma.repr\n\nattribute [to_additive existing] FreeMagma.repr\n\n@[to_additive]\ninstance {\u03b1 : Type u} [Repr \u03b1] : Repr (FreeMagma \u03b1) := \u27e8fun o _ => FreeMagma.repr o\u27e9\n\n/-- Length of an element of a free magma. -/\ndef FreeMagma.length {\u03b1 : Type u} : FreeMagma \u03b1 \u2192 \u2115\n  | FreeMagma.of _x => 1\n  -- Adaptation note: around nightly-2024-02-25, we need to write `mul x y` in the pattern here,\n  -- instead of `x * y`.\n  | mul x y => x.length + y.length\n#align free_magma.length FreeMagma.length\n\n/-- Length of an element of a free additive magma. -/\ndef FreeAddMagma.length {\u03b1 : Type u} : FreeAddMagma \u03b1 \u2192 \u2115\n  | FreeAddMagma.of _x => 1\n  | x + y => x.length + y.length\n#align free_add_magma.length FreeAddMagma.length\n\nattribute [to_additive existing (attr := simp)] FreeMagma.length\n\n", "theoremStatement": "/-- The length of an element of a free magma is positive. -/\n@[to_additive \"The length of an element of a free additive magma is positive.\"]\nlemma FreeMagma.length_pos {\u03b1 : Type u} (x : FreeMagma \u03b1) : 0 < x.length ", "theoremName": "FreeMagma.length_pos", "fileCreated": {"commit": "59ded3f32295156dc8fbb8faa7b8f472717052ea", "date": "2023-01-29"}, "theoremCreated": {"commit": "a77f5537068e761bc67a11b112ef6f6f625417af", "date": "2024-03-30"}, "file": "mathlib/Mathlib/Algebra/Free.lean", "module": "Mathlib.Algebra.Free", "jsonFile": "Mathlib.Algebra.Free.jsonl", "positionMetadata": {"lineInFile": 331, "tokenPositionInFile": 12271, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 18, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", 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"Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Data.Nat.Defs", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n  match x with\n  | FreeMagma.of _ => Nat.succ_pos 0\n  | mul y z => Nat.add_pos_left (length_pos y) z.length", "proofType": "term", "proofLengthLines": 3, "proofLengthTokens": 110}}
+{"srcContext": "/-\nCopyright (c) 2022 Xavier Roblot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alex J. Best, Xavier Roblot\n-/\nimport Mathlib.Analysis.Complex.Polynomial\nimport Mathlib.NumberTheory.NumberField.Norm\nimport Mathlib.NumberTheory.NumberField.Basic\nimport Mathlib.RingTheory.Norm\nimport Mathlib.Topology.Instances.Complex\nimport Mathlib.RingTheory.RootsOfUnity.Basic\n\n#align_import number_theory.number_field.embeddings from \"leanprover-community/mathlib\"@\"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c\"\n\n/-!\n# Embeddings of number fields\nThis file defines the embeddings of a number field into an algebraic closed field.\n\n## Main Definitions and Results\n* `NumberField.Embeddings.range_eval_eq_rootSet_minpoly`: let `x \u2208 K` with `K` number field and\n  let `A` be an algebraic closed field of char. 0, then the images of `x` by the embeddings of `K`\n   in `A` are exactly the roots in `A` of the minimal polynomial of `x` over `\u211a`.\n* `NumberField.Embeddings.pow_eq_one_of_norm_eq_one`: an algebraic integer whose conjugates are\n  all of norm one is a root of unity.\n* `NumberField.InfinitePlace`: the type of infinite places of a number field `K`.\n* `NumberField.InfinitePlace.mk_eq_iff`: two complex embeddings define the same infinite place iff\nthey are equal or complex conjugates.\n* `NumberField.InfinitePlace.prod_eq_abs_norm`: the infinite part of the product formula, that is\nfor `x \u2208 K`, we have `\u03a0_w \u2016x\u2016_w = |norm(x)|` where the product is over the infinite place `w` and\n`\u2016\u00b7\u2016_w` is the normalized absolute value for `w`.\n\n## Tags\nnumber field, embeddings, places, infinite places\n-/\n\nopen scoped Classical\n\nnamespace NumberField.Embeddings\n\nsection Fintype\n\nopen FiniteDimensional\n\nvariable (K : Type*) [Field K] [NumberField K]\nvariable (A : Type*) [Field A] [CharZero A]\n\n/-- There are finitely many embeddings of a number field. -/\nnoncomputable instance : Fintype (K \u2192+* A) :=\n  Fintype.ofEquiv (K \u2192\u2090[\u211a] A) RingHom.equivRatAlgHom.symm\n\nvariable [IsAlgClosed A]\n\n/-- The number of embeddings of a number field is equal to its finrank. -/\ntheorem card : Fintype.card (K \u2192+* A) = finrank \u211a K := by\n  rw [Fintype.ofEquiv_card RingHom.equivRatAlgHom.symm, AlgHom.card]\n#align number_field.embeddings.card NumberField.Embeddings.card\n\ninstance : Nonempty (K \u2192+* A) := by\n  rw [\u2190 Fintype.card_pos_iff, NumberField.Embeddings.card K A]\n  exact FiniteDimensional.finrank_pos\n\nend Fintype\n\nsection Roots\n\nopen Set Polynomial\n\nvariable (K A : Type*) [Field K] [NumberField K] [Field A] [Algebra \u211a A] [IsAlgClosed A] (x : K)\n\n/-- Let `A` be an algebraically closed field and let `x \u2208 K`, with `K` a number field.\nThe images of `x` by the embeddings of `K` in `A` are exactly the roots in `A` of\nthe minimal polynomial of `x` over `\u211a`. -/\ntheorem range_eval_eq_rootSet_minpoly :\n    (range fun \u03c6 : K \u2192+* A => \u03c6 x) = (minpoly \u211a x).rootSet A := by\n  convert (NumberField.isAlgebraic K).range_eval_eq_rootSet_minpoly A x using 1\n  ext a\n  exact \u27e8fun \u27e8\u03c6, h\u03c6\u27e9 => \u27e8\u03c6.toRatAlgHom, h\u03c6\u27e9, fun \u27e8\u03c6, h\u03c6\u27e9 => \u27e8\u03c6.toRingHom, h\u03c6\u27e9\u27e9\n#align number_field.embeddings.range_eval_eq_root_set_minpoly NumberField.Embeddings.range_eval_eq_rootSet_minpoly\n\nend Roots\n\nsection Bounded\n\nopen FiniteDimensional Polynomial Set\n\nvariable {K : Type*} [Field K] [NumberField K]\nvariable {A : Type*} [NormedField A] [IsAlgClosed A] [NormedAlgebra \u211a A]\n\ntheorem coeff_bdd_of_norm_le {B : \u211d} {x : K} (h : \u2200 \u03c6 : K \u2192+* A, \u2016\u03c6 x\u2016 \u2264 B) (i : \u2115) :\n    \u2016(minpoly \u211a x).coeff i\u2016 \u2264 max B 1 ^ finrank \u211a K * (finrank \u211a K).choose (finrank \u211a K / 2) := by\n  have hx := IsSeparable.isIntegral \u211a x\n  rw [\u2190 norm_algebraMap' A, \u2190 coeff_map (algebraMap \u211a A)]\n  refine coeff_bdd_of_roots_le _ (minpoly.monic hx)\n      (IsAlgClosed.splits_codomain _) (minpoly.natDegree_le x) (fun z hz => ?_) i\n  classical\n  rw [\u2190 Multiset.mem_toFinset] at hz\n  obtain \u27e8\u03c6, rfl\u27e9 := (range_eval_eq_rootSet_minpoly K A x).symm.subset hz\n  exact h \u03c6\n#align number_field.embeddings.coeff_bdd_of_norm_le NumberField.Embeddings.coeff_bdd_of_norm_le\n\nvariable (K A)\n\n/-- Let `B` be a real number. The set of algebraic integers in `K` whose conjugates are all\nsmaller in norm than `B` is finite. -/\ntheorem finite_of_norm_le (B : \u211d) : {x : K | IsIntegral \u2124 x \u2227 \u2200 \u03c6 : K \u2192+* A, \u2016\u03c6 x\u2016 \u2264 B}.Finite := by\n  let C := Nat.ceil (max B 1 ^ finrank \u211a K * (finrank \u211a K).choose (finrank \u211a K / 2))\n  have := bUnion_roots_finite (algebraMap \u2124 K) (finrank \u211a K) (finite_Icc (-C : \u2124) C)\n  refine this.subset fun x hx => ?_; simp_rw [mem_iUnion]\n  have h_map_\u211a_minpoly := minpoly.isIntegrallyClosed_eq_field_fractions' \u211a hx.1\n  refine \u27e8_, \u27e8?_, fun i => ?_\u27e9, mem_rootSet.2 \u27e8minpoly.ne_zero hx.1, minpoly.aeval \u2124 x\u27e9\u27e9\n  \u00b7 rw [\u2190 (minpoly.monic hx.1).natDegree_map (algebraMap \u2124 \u211a), \u2190 h_map_\u211a_minpoly]\n    exact minpoly.natDegree_le x\n  rw [mem_Icc, \u2190 abs_le, \u2190 @Int.cast_le \u211d]\n  refine (Eq.trans_le ?_ <| coeff_bdd_of_norm_le hx.2 i).trans (Nat.le_ceil _)\n  rw [h_map_\u211a_minpoly, coeff_map, eq_intCast, Int.norm_cast_rat, Int.norm_eq_abs, Int.cast_abs]\n#align number_field.embeddings.finite_of_norm_le NumberField.Embeddings.finite_of_norm_le\n\n/-- An algebraic integer whose conjugates are all of norm one is a root of unity. -/\ntheorem pow_eq_one_of_norm_eq_one {x : K} (hxi : IsIntegral \u2124 x) (hx : \u2200 \u03c6 : K \u2192+* A, \u2016\u03c6 x\u2016 = 1) :\n    \u2203 (n : \u2115) (_ : 0 < n), x ^ n = 1 := by\n  obtain \u27e8a, -, b, -, habne, h\u27e9 :=\n    @Set.Infinite.exists_ne_map_eq_of_mapsTo _ _ _ _ (x ^ \u00b7 : \u2115 \u2192 K) Set.infinite_univ\n      (by exact fun a _ => \u27e8hxi.pow a, fun \u03c6 => by simp [hx \u03c6]\u27e9) (finite_of_norm_le K A (1 : \u211d))\n  \u00b7 wlog hlt : b < a\n    \u00b7 exact this K A hxi hx b a habne.symm h.symm (habne.lt_or_lt.resolve_right hlt)\n    refine \u27e8a - b, tsub_pos_of_lt hlt, ?_\u27e9\n    rw [\u2190 Nat.sub_add_cancel hlt.le, pow_add, mul_left_eq_self\u2080] at h\n    refine h.resolve_right fun hp => ?_\n    specialize hx (IsAlgClosed.lift (NumberField.isAlgebraic K)).toRingHom\n    rw [pow_eq_zero hp, map_zero, norm_zero] at hx; norm_num at hx\n#align number_field.embeddings.pow_eq_one_of_norm_eq_one NumberField.Embeddings.pow_eq_one_of_norm_eq_one\n\nend Bounded\n\nend NumberField.Embeddings\n\nsection Place\n\nvariable {K : Type*} [Field K] {A : Type*} [NormedDivisionRing A] [Nontrivial A] (\u03c6 : K \u2192+* A)\n\n/-- An embedding into a normed division ring defines a place of `K` -/\ndef NumberField.place : AbsoluteValue K \u211d :=\n  (IsAbsoluteValue.toAbsoluteValue (norm : A \u2192 \u211d)).comp \u03c6.injective\n#align number_field.place NumberField.place\n\n@[simp]\ntheorem NumberField.place_apply (x : K) : (NumberField.place \u03c6) x = norm (\u03c6 x) := rfl\n#align number_field.place_apply NumberField.place_apply\n\nend Place\n\nnamespace NumberField.ComplexEmbedding\n\nopen Complex NumberField\n\nopen scoped ComplexConjugate\n\nvariable {K : Type*} [Field K] {k : Type*} [Field k]\n\n/-- The conjugate of a complex embedding as a complex embedding. -/\n@[reducible]\ndef conjugate (\u03c6 : K \u2192+* \u2102) : K \u2192+* \u2102 := star \u03c6\n#align number_field.complex_embedding.conjugate NumberField.ComplexEmbedding.conjugate\n\n@[simp]\ntheorem conjugate_coe_eq (\u03c6 : K \u2192+* \u2102) (x : K) : (conjugate \u03c6) x = conj (\u03c6 x) := rfl\n#align number_field.complex_embedding.conjugate_coe_eq NumberField.ComplexEmbedding.conjugate_coe_eq\n\ntheorem place_conjugate (\u03c6 : K \u2192+* \u2102) : place (conjugate \u03c6) = place \u03c6 := by\n  ext; simp only [place_apply, norm_eq_abs, abs_conj, conjugate_coe_eq]\n#align number_field.complex_embedding.place_conjugate NumberField.ComplexEmbedding.place_conjugate\n\n/-- An embedding into `\u2102` is real if it is fixed by complex conjugation. -/\n@[reducible]\ndef IsReal (\u03c6 : K \u2192+* \u2102) : Prop := IsSelfAdjoint \u03c6\n#align number_field.complex_embedding.is_real NumberField.ComplexEmbedding.IsReal\n\ntheorem isReal_iff {\u03c6 : K \u2192+* \u2102} : IsReal \u03c6 \u2194 conjugate \u03c6 = \u03c6 := isSelfAdjoint_iff\n#align number_field.complex_embedding.is_real_iff NumberField.ComplexEmbedding.isReal_iff\n\ntheorem isReal_conjugate_iff {\u03c6 : K \u2192+* \u2102} : IsReal (conjugate \u03c6) \u2194 IsReal \u03c6 :=\n  IsSelfAdjoint.star_iff\n#align number_field.complex_embedding.is_real_conjugate_iff NumberField.ComplexEmbedding.isReal_conjugate_iff\n\n/-- A real embedding as a ring homomorphism from `K` to `\u211d` . -/\ndef IsReal.embedding {\u03c6 : K \u2192+* \u2102} (h\u03c6 : IsReal \u03c6) : K \u2192+* \u211d where\n  toFun x := (\u03c6 x).re\n  map_one' := by simp only [map_one, one_re]\n  map_mul' := by\n    simp only [Complex.conj_eq_iff_im.mp (RingHom.congr_fun h\u03c6 _), map_mul, mul_re,\n      mul_zero, tsub_zero, eq_self_iff_true, forall_const]\n  map_zero' := by simp only [map_zero, zero_re]\n  map_add' := by simp only [map_add, add_re, eq_self_iff_true, forall_const]\n#align number_field.complex_embedding.is_real.embedding NumberField.ComplexEmbedding.IsReal.embedding\n\n@[simp]\ntheorem IsReal.coe_embedding_apply {\u03c6 : K \u2192+* \u2102} (h\u03c6 : IsReal \u03c6) (x : K) :\n    (h\u03c6.embedding x : \u2102) = \u03c6 x := by\n  apply Complex.ext\n  \u00b7 rfl\n  \u00b7 rw [ofReal_im, eq_comm, \u2190 Complex.conj_eq_iff_im]\n    exact RingHom.congr_fun h\u03c6 x\n#align number_field.complex_embedding.is_real.coe_embedding_apply NumberField.ComplexEmbedding.IsReal.coe_embedding_apply\n\nlemma IsReal.comp (f : k \u2192+* K) {\u03c6 : K \u2192+* \u2102} (h\u03c6 : IsReal \u03c6) :\n    IsReal (\u03c6.comp f) := by ext1 x; simpa using RingHom.congr_fun h\u03c6 (f x)\n\nlemma isReal_comp_iff {f : k \u2243+* K} {\u03c6 : K \u2192+* \u2102} :\n    IsReal (\u03c6.comp (f : k \u2192+* K)) \u2194 IsReal \u03c6 :=\n  \u27e8fun H \u21a6 by convert H.comp f.symm.toRingHom; ext1; simp, IsReal.comp _\u27e9\n\nlemma exists_comp_symm_eq_of_comp_eq [Algebra k K] [IsGalois k K] (\u03c6 \u03c8 : K \u2192+* \u2102)\n    (h : \u03c6.comp (algebraMap k K) = \u03c8.comp (algebraMap k K)) :\n    \u2203 \u03c3 : K \u2243\u2090[k] K, \u03c6.comp \u03c3.symm = \u03c8 := by\n  letI := (\u03c6.comp (algebraMap k K)).toAlgebra\n  letI := \u03c6.toAlgebra\n  have : IsScalarTower k K \u2102 := IsScalarTower.of_algebraMap_eq' rfl\n  let \u03c8' : K \u2192\u2090[k] \u2102 := { \u03c8 with commutes' := fun r \u21a6 (RingHom.congr_fun h r).symm }\n  use (AlgHom.restrictNormal' \u03c8' K).symm\n  ext1 x\n  exact AlgHom.restrictNormal_commutes \u03c8' K x\n\nvariable [Algebra k K] (\u03c6 : K \u2192+* \u2102) (\u03c3 : K \u2243\u2090[k] K)\n\n/--\n`IsConj \u03c6 \u03c3` states that `\u03c3 : K \u2243\u2090[k] K` is the conjugation under the embedding `\u03c6 : K \u2192+* \u2102`.\n-/\ndef IsConj : Prop := conjugate \u03c6 = \u03c6.comp \u03c3\n\nvariable {\u03c6 \u03c3}\n\nlemma IsConj.eq (h : IsConj \u03c6 \u03c3) (x) : \u03c6 (\u03c3 x) = star (\u03c6 x) := RingHom.congr_fun h.symm x\n\nlemma IsConj.ext {\u03c3\u2081 \u03c3\u2082 : K \u2243\u2090[k] K} (h\u2081 : IsConj \u03c6 \u03c3\u2081) (h\u2082 : IsConj \u03c6 \u03c3\u2082) : \u03c3\u2081 = \u03c3\u2082 :=\n  AlgEquiv.ext fun x \u21a6 \u03c6.injective ((h\u2081.eq x).trans (h\u2082.eq x).symm)\n\nlemma IsConj.ext_iff {\u03c3\u2081 \u03c3\u2082 : K \u2243\u2090[k] K} (h\u2081 : IsConj \u03c6 \u03c3\u2081) : \u03c3\u2081 = \u03c3\u2082 \u2194 IsConj \u03c6 \u03c3\u2082 :=\n  \u27e8fun e \u21a6 e \u25b8 h\u2081, h\u2081.ext\u27e9\n\nlemma IsConj.isReal_comp (h : IsConj \u03c6 \u03c3) : IsReal (\u03c6.comp (algebraMap k K)) := by\n  ext1 x\n  simp only [conjugate_coe_eq, RingHom.coe_comp, Function.comp_apply, \u2190 h.eq,\n    starRingEnd_apply, AlgEquiv.commutes]\n\nlemma isConj_one_iff : IsConj \u03c6 (1 : K \u2243\u2090[k] K) \u2194 IsReal \u03c6 := Iff.rfl\n\nalias \u27e8_, IsReal.isConjGal_one\u27e9 := ComplexEmbedding.isConj_one_iff\n\nlemma IsConj.symm (h\u03c3 : IsConj \u03c6 \u03c3) :\n    IsConj \u03c6 \u03c3.symm := RingHom.ext fun x \u21a6 by simpa using congr_arg star (h\u03c3.eq (\u03c3.symm x))\n\nlemma isConj_symm : IsConj \u03c6 \u03c3.symm \u2194 IsConj \u03c6 \u03c3 :=\n  \u27e8IsConj.symm, IsConj.symm\u27e9\n\nend NumberField.ComplexEmbedding\n\nsection InfinitePlace\n\nopen NumberField\n\nvariable {k : Type*} [Field k] (K : Type*) [Field K] {F : Type*} [Field F]\n\n/-- An infinite place of a number field `K` is a place associated to a complex embedding. -/\ndef NumberField.InfinitePlace := { w : AbsoluteValue K \u211d // \u2203 \u03c6 : K \u2192+* \u2102, place \u03c6 = w }\n#align number_field.infinite_place NumberField.InfinitePlace\n\ninstance [NumberField K] : Nonempty (NumberField.InfinitePlace K) := Set.instNonemptyRange _\n\nvariable {K}\n\n/-- Return the infinite place defined by a complex embedding `\u03c6`. -/\nnoncomputable def NumberField.InfinitePlace.mk (\u03c6 : K \u2192+* \u2102) : NumberField.InfinitePlace K :=\n  \u27e8place \u03c6, \u27e8\u03c6, rfl\u27e9\u27e9\n#align number_field.infinite_place.mk NumberField.InfinitePlace.mk\n\nnamespace NumberField.InfinitePlace\n\nopen NumberField\n\ninstance {K : Type*} [Field K] : FunLike (InfinitePlace K) K \u211d where\n  coe w x := w.1 x\n  coe_injective' := fun _ _ h => Subtype.eq (AbsoluteValue.ext fun x => congr_fun h x)\n\ninstance : MonoidWithZeroHomClass (InfinitePlace K) K \u211d where\n  map_mul w _ _ := w.1.map_mul _ _\n  map_one w := w.1.map_one\n  map_zero w := w.1.map_zero\n\ninstance : NonnegHomClass (InfinitePlace K) K \u211d where\n  apply_nonneg w _ := w.1.nonneg _\n\n@[simp]\ntheorem apply (\u03c6 : K \u2192+* \u2102) (x : K) : (mk \u03c6) x = Complex.abs (\u03c6 x) := rfl\n#align number_field.infinite_place.apply NumberField.InfinitePlace.apply\n\n/-- For an infinite place `w`, return an embedding `\u03c6` such that `w = infinite_place \u03c6` . -/\nnoncomputable def embedding (w : InfinitePlace K) : K \u2192+* \u2102 := w.2.choose\n#align number_field.infinite_place.embedding NumberField.InfinitePlace.embedding\n\n@[simp]\ntheorem mk_embedding (w : InfinitePlace K) : mk (embedding w) = w := Subtype.ext w.2.choose_spec\n#align number_field.infinite_place.mk_embedding NumberField.InfinitePlace.mk_embedding\n\n@[simp]\ntheorem mk_conjugate_eq (\u03c6 : K \u2192+* \u2102) : mk (ComplexEmbedding.conjugate \u03c6) = mk \u03c6 := by\n  refine DFunLike.ext _ _ (fun x => ?_)\n  rw [apply, apply, ComplexEmbedding.conjugate_coe_eq, Complex.abs_conj]\n#align number_field.infinite_place.mk_conjugate_eq NumberField.InfinitePlace.mk_conjugate_eq\n\ntheorem norm_embedding_eq (w : InfinitePlace K) (x : K) :\n    \u2016(embedding w) x\u2016 = w x := by\n  nth_rewrite 2 [\u2190 mk_embedding w]\n  rfl\n\ntheorem eq_iff_eq (x : K) (r : \u211d) : (\u2200 w : InfinitePlace K, w x = r) \u2194 \u2200 \u03c6 : K \u2192+* \u2102, \u2016\u03c6 x\u2016 = r :=\n  \u27e8fun hw \u03c6 => hw (mk \u03c6), by rintro h\u03c6 \u27e8w, \u27e8\u03c6, rfl\u27e9\u27e9; exact h\u03c6 \u03c6\u27e9\n#align number_field.infinite_place.eq_iff_eq NumberField.InfinitePlace.eq_iff_eq\n\ntheorem le_iff_le (x : K) (r : \u211d) : (\u2200 w : InfinitePlace K, w x \u2264 r) \u2194 \u2200 \u03c6 : K \u2192+* \u2102, \u2016\u03c6 x\u2016 \u2264 r :=\n  \u27e8fun hw \u03c6 => hw (mk \u03c6), by rintro h\u03c6 \u27e8w, \u27e8\u03c6, rfl\u27e9\u27e9; exact h\u03c6 \u03c6\u27e9\n#align number_field.infinite_place.le_iff_le NumberField.InfinitePlace.le_iff_le\n\ntheorem pos_iff {w : InfinitePlace K} {x : K} : 0 < w x \u2194 x \u2260 0 := AbsoluteValue.pos_iff w.1\n#align number_field.infinite_place.pos_iff NumberField.InfinitePlace.pos_iff\n\n@[simp]\ntheorem mk_eq_iff {\u03c6 \u03c8 : K \u2192+* \u2102} : mk \u03c6 = mk \u03c8 \u2194 \u03c6 = \u03c8 \u2228 ComplexEmbedding.conjugate \u03c6 = \u03c8 := by\n  constructor\n  \u00b7 -- We prove that the map \u03c8 \u2218 \u03c6\u207b\u00b9 between \u03c6(K) and \u2102 is uniform continuous, thus it is either the\n    -- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`\n    intro h\u2080\n    obtain \u27e8j, hi\u03c6\u27e9 := (\u03c6.injective).hasLeftInverse\n    let \u03b9 := RingEquiv.ofLeftInverse hi\u03c6\n    have hlip : LipschitzWith 1 (RingHom.comp \u03c8 \u03b9.symm.toRingHom) := by\n      change LipschitzWith 1 (\u03c8 \u2218 \u03b9.symm)\n      apply LipschitzWith.of_dist_le_mul\n      intro x y\n      rw [NNReal.coe_one, one_mul, NormedField.dist_eq, Function.comp_apply, Function.comp_apply,\n        \u2190 map_sub, \u2190 map_sub]\n      apply le_of_eq\n      suffices \u2016\u03c6 (\u03b9.symm (x - y))\u2016 = \u2016\u03c8 (\u03b9.symm (x - y))\u2016 by\n        rw [\u2190 this, \u2190 RingEquiv.ofLeftInverse_apply hi\u03c6 _, RingEquiv.apply_symm_apply \u03b9 _]\n        rfl\n      exact congrFun (congrArg (\u2191) h\u2080) _\n    cases\n      Complex.uniformContinuous_ringHom_eq_id_or_conj \u03c6.fieldRange hlip.uniformContinuous with\n    | inl h =>\n        left; ext1 x\n        conv_rhs => rw [\u2190 hi\u03c6 x]\n        exact (congrFun h (\u03b9 x)).symm\n    | inr h =>\n        right; ext1 x\n        conv_rhs => rw [\u2190 hi\u03c6 x]\n        exact (congrFun h (\u03b9 x)).symm\n  \u00b7 rintro (\u27e8h\u27e9 | \u27e8h\u27e9)\n    \u00b7 exact congr_arg mk h\n    \u00b7 rw [\u2190 mk_conjugate_eq]\n      exact congr_arg mk h\n#align number_field.infinite_place.mk_eq_iff NumberField.InfinitePlace.mk_eq_iff\n\n/-- An infinite place is real if it is defined by a real embedding. -/\ndef IsReal (w : InfinitePlace K) : Prop := \u2203 \u03c6 : K \u2192+* \u2102, ComplexEmbedding.IsReal \u03c6 \u2227 mk \u03c6 = w\n#align number_field.infinite_place.is_real NumberField.InfinitePlace.IsReal\n\n/-- An infinite place is complex if it is defined by a complex (ie. not real) embedding. -/\ndef IsComplex (w : InfinitePlace K) : Prop := \u2203 \u03c6 : K \u2192+* \u2102, \u00acComplexEmbedding.IsReal \u03c6 \u2227 mk \u03c6 = w\n#align number_field.infinite_place.is_complex NumberField.InfinitePlace.IsComplex\n\ntheorem embedding_mk_eq (\u03c6 : K \u2192+* \u2102) :\n    embedding (mk \u03c6) = \u03c6 \u2228 embedding (mk \u03c6) = ComplexEmbedding.conjugate \u03c6 := by\n  rw [@eq_comm _ _ \u03c6, @eq_comm _ _ (ComplexEmbedding.conjugate \u03c6), \u2190 mk_eq_iff, mk_embedding]\n\n@[simp]\ntheorem embedding_mk_eq_of_isReal {\u03c6 : K \u2192+* \u2102} (h : ComplexEmbedding.IsReal \u03c6) :\n    embedding (mk \u03c6) = \u03c6 := by\n  have := embedding_mk_eq \u03c6\n  rwa [ComplexEmbedding.isReal_iff.mp h, or_self] at this\n#align number_field.complex_embeddings.is_real.embedding_mk NumberField.InfinitePlace.embedding_mk_eq_of_isReal\n\ntheorem isReal_iff {w : InfinitePlace K} :\n    IsReal w \u2194 ComplexEmbedding.IsReal (embedding w) := by\n  refine \u27e8?_, fun h => \u27e8embedding w, h, mk_embedding w\u27e9\u27e9\n  rintro \u27e8\u03c6, \u27e8h\u03c6, rfl\u27e9\u27e9\n  rwa [embedding_mk_eq_of_isReal h\u03c6]\n#align number_field.infinite_place.is_real_iff NumberField.InfinitePlace.isReal_iff\n\ntheorem isComplex_iff {w : InfinitePlace K} :\n    IsComplex w \u2194 \u00acComplexEmbedding.IsReal (embedding w) := by\n  refine \u27e8?_, fun h => \u27e8embedding w, h, mk_embedding w\u27e9\u27e9\n  rintro \u27e8\u03c6, \u27e8h\u03c6, rfl\u27e9\u27e9\n  contrapose! h\u03c6\n  cases mk_eq_iff.mp (mk_embedding (mk \u03c6)) with\n  | inl h => rwa [h] at h\u03c6\n  | inr h => rwa [\u2190 ComplexEmbedding.isReal_conjugate_iff, h] at h\u03c6\n#align number_field.infinite_place.is_complex_iff NumberField.InfinitePlace.isComplex_iff\n\n@[simp]\ntheorem conjugate_embedding_eq_of_isReal {w : InfinitePlace K} (h : IsReal w) :\n    ComplexEmbedding.conjugate (embedding w) = embedding w :=\n  ComplexEmbedding.isReal_iff.mpr (isReal_iff.mp h)\n\n@[simp]\ntheorem not_isReal_iff_isComplex {w : InfinitePlace K} : \u00acIsReal w \u2194 IsComplex w := by\n  rw [isComplex_iff, isReal_iff]\n#align number_field.infinite_place.not_is_real_iff_is_complex NumberField.InfinitePlace.not_isReal_iff_isComplex\n\n@[simp]\ntheorem not_isComplex_iff_isReal {w : InfinitePlace K} : \u00acIsComplex w \u2194 IsReal w := by\n  rw [isComplex_iff, isReal_iff, not_not]\n\ntheorem isReal_or_isComplex (w : InfinitePlace K) : IsReal w \u2228 IsComplex w := by\n  rw [\u2190 not_isReal_iff_isComplex]; exact em _\n#align number_field.infinite_place.is_real_or_is_complex NumberField.InfinitePlace.isReal_or_isComplex\n\ntheorem ne_of_isReal_isComplex {w w' : InfinitePlace K} (h : IsReal w) (h' : IsComplex w') :\n    w \u2260 w' := fun h_eq \u21a6 not_isReal_iff_isComplex.mpr h' (h_eq \u25b8 h)\n\nvariable (K) in\n", "theoremStatement": "theorem disjoint_isReal_isComplex :\n    Disjoint {(w : InfinitePlace K) | IsReal w} {(w : InfinitePlace K) | IsComplex w} ", "theoremName": 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"Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", 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+{"srcContext": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Floris van Doorn, S\u00e9bastien Gou\u00ebzel, Alex J. Best\n-/\nimport Mathlib.Algebra.Ring.Commute\nimport Mathlib.Algebra.Ring.Int\nimport Mathlib.Algebra.Ring.Nat\nimport Mathlib.Data.List.Dedup\nimport Mathlib.Data.List.ProdSigma\nimport Mathlib.Data.List.Join\nimport Mathlib.Data.List.Perm\nimport Mathlib.Data.List.Range\nimport Mathlib.Data.List.Rotate\n\n#align_import data.list.big_operators.basic from \"leanprover-community/mathlib\"@\"6c5f73fd6f6cc83122788a80a27cdd54663609f4\"\n\n/-!\n# Sums and products from lists\n\nThis file provides basic results about `List.prod`, `List.sum`, which calculate the product and sum\nof elements of a list and `List.alternatingProd`, `List.alternatingSum`, their alternating\ncounterparts.\n-/\n\n-- Make sure we haven't imported `Data.Nat.Order.Basic`\nassert_not_exists OrderedSub\n\n-- TODO\n-- assert_not_exists Ring\n\nvariable {\u03b9 \u03b1 \u03b2 \u03b3 M N P M\u2080 G R : Type*}\n\nnamespace List\nsection Defs\n\n/-- Product of a list.\n\n`List.prod [a, b, c] = ((1 * a) * b) * c` -/\n@[to_additive \"Sum of a list.\\n\\n`List.sum [a, b, c] = ((0 + a) + b) + c`\"]\ndef prod {\u03b1} [Mul \u03b1] [One \u03b1] : List \u03b1 \u2192 \u03b1 :=\n  foldl (\u00b7 * \u00b7) 1\n#align list.prod List.prod\n#align list.sum List.sum\n\n/-- The alternating sum of a list. -/\ndef alternatingSum {G : Type*} [Zero G] [Add G] [Neg G] : List G \u2192 G\n  | [] => 0\n  | g :: [] => g\n  | g :: h :: t => g + -h + alternatingSum t\n#align list.alternating_sum List.alternatingSum\n\n/-- The alternating product of a list. -/\n@[to_additive existing]\ndef alternatingProd {G : Type*} [One G] [Mul G] [Inv G] : List G \u2192 G\n  | [] => 1\n  | g :: [] => g\n  | g :: h :: t => g * h\u207b\u00b9 * alternatingProd t\n#align list.alternating_prod List.alternatingProd\n\nend Defs\n\nsection MulOneClass\n\nvariable [MulOneClass M] {l : List M} {a : M}\n\n@[to_additive (attr := simp)]\ntheorem prod_nil : ([] : List M).prod = 1 :=\n  rfl\n#align list.prod_nil List.prod_nil\n#align list.sum_nil List.sum_nil\n\n@[to_additive]\ntheorem prod_singleton : [a].prod = a :=\n  one_mul a\n#align list.prod_singleton List.prod_singleton\n#align list.sum_singleton List.sum_singleton\n\n", "theoremStatement": "@[to_additive (attr := simp)]\ntheorem prod_one_cons : (1 :: l).prod = l.prod ", "theoremName": "List.prod_one_cons", "fileCreated": {"commit": "f09ccf67dae82f9c6a8bafa478949ac9cb871ad7", "date": "2024-03-28"}, "theoremCreated": {"commit": "2d2e953b5b7ce492e8b3062178fdfee8c4123f86", "date": "2024-03-25"}, "file": "mathlib/Mathlib/Algebra/BigOperators/List/Basic.lean", "module": "Mathlib.Algebra.BigOperators.List.Basic", "jsonFile": "Mathlib.Algebra.BigOperators.List.Basic.jsonl", "positionMetadata": {"lineInFile": 79, "tokenPositionInFile": 2197, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 19, "importedModules": ["Init.Prelude", "Init.Coe", 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"Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", 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"Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Join", "Mathlib.Data.List.Count", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Rotate"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  rw [prod, foldl, mul_one]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 33}}
+{"srcContext": "/-\nCopyright (c) 2022 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johanes H\u00f6lzl, Patrick Massot, Yury Kudryashov, Kevin Wilson, Heather Macbeth\n-/\nimport Mathlib.Order.Filter.Basic\n\n#align_import order.filter.prod from \"leanprover-community/mathlib\"@\"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce\"\n\n/-!\n# Product and coproduct filters\n\nIn this file we define `Filter.prod f g` (notation: `f \u00d7\u02e2 g`) and `Filter.coprod f g`. The product\nof two filters is the largest filter `l` such that `Filter.Tendsto Prod.fst l f` and\n`Filter.Tendsto Prod.snd l g`.\n\n## Implementation details\n\nThe product filter cannot be defined using the monad structure on filters. For example:\n\n```lean\nF := do {x \u2190 seq, y \u2190 top, return (x, y)}\nG := do {y \u2190 top, x \u2190 seq, return (x, y)}\n```\nhence:\n```lean\ns \u2208 F  \u2194  \u2203 n, [n..\u221e] \u00d7 univ \u2286 s\ns \u2208 G  \u2194  \u2200 i:\u2115, \u2203 n, [n..\u221e] \u00d7 {i} \u2286 s\n```\nNow `\u22c3 i, [i..\u221e] \u00d7 {i}` is in `G` but not in `F`.\nAs product filter we want to have `F` as result.\n\n## Notations\n\n* `f \u00d7\u02e2 g` : `Filter.prod f g`, localized in `Filter`.\n\n-/\n\nopen Set\n\nopen Filter\n\nnamespace Filter\n\nvariable {\u03b1 \u03b2 \u03b3 \u03b4 : Type*} {\u03b9 : Sort*}\n\nsection Prod\n\nvariable {s : Set \u03b1} {t : Set \u03b2} {f : Filter \u03b1} {g : Filter \u03b2}\n\n/-- Product of filters. This is the filter generated by cartesian products\nof elements of the component filters. -/\nprotected def prod (f : Filter \u03b1) (g : Filter \u03b2) : Filter (\u03b1 \u00d7 \u03b2) :=\n  f.comap Prod.fst \u2293 g.comap Prod.snd\n#align filter.prod Filter.prod\n\ninstance instSProd : SProd (Filter \u03b1) (Filter \u03b2) (Filter (\u03b1 \u00d7 \u03b2)) where\n  sprod := Filter.prod\n\ntheorem prod_mem_prod (hs : s \u2208 f) (ht : t \u2208 g) : s \u00d7\u02e2 t \u2208 f \u00d7\u02e2 g :=\n  inter_mem_inf (preimage_mem_comap hs) (preimage_mem_comap ht)\n#align filter.prod_mem_prod Filter.prod_mem_prod\n\ntheorem mem_prod_iff {s : Set (\u03b1 \u00d7 \u03b2)} {f : Filter \u03b1} {g : Filter \u03b2} :\n    s \u2208 f \u00d7\u02e2 g \u2194 \u2203 t\u2081 \u2208 f, \u2203 t\u2082 \u2208 g, t\u2081 \u00d7\u02e2 t\u2082 \u2286 s := by\n  simp only [SProd.sprod, Filter.prod]\n  constructor\n  \u00b7 rintro \u27e8t\u2081, \u27e8s\u2081, hs\u2081, hts\u2081\u27e9, t\u2082, \u27e8s\u2082, hs\u2082, hts\u2082\u27e9, rfl\u27e9\n    exact \u27e8s\u2081, hs\u2081, s\u2082, hs\u2082, fun p \u27e8h, h'\u27e9 => \u27e8hts\u2081 h, hts\u2082 h'\u27e9\u27e9\n  \u00b7 rintro \u27e8t\u2081, ht\u2081, t\u2082, ht\u2082, h\u27e9\n    exact mem_inf_of_inter (preimage_mem_comap ht\u2081) (preimage_mem_comap ht\u2082) h\n#align filter.mem_prod_iff Filter.mem_prod_iff\n\n@[simp]\ntheorem prod_mem_prod_iff [f.NeBot] [g.NeBot] : s \u00d7\u02e2 t \u2208 f \u00d7\u02e2 g \u2194 s \u2208 f \u2227 t \u2208 g :=\n  \u27e8fun h =>\n    let \u27e8_s', hs', _t', ht', H\u27e9 := mem_prod_iff.1 h\n    (prod_subset_prod_iff.1 H).elim\n      (fun \u27e8hs's, ht't\u27e9 => \u27e8mem_of_superset hs' hs's, mem_of_superset ht' ht't\u27e9) fun h =>\n      h.elim (fun hs'e => absurd hs'e (nonempty_of_mem hs').ne_empty) fun ht'e =>\n        absurd ht'e (nonempty_of_mem ht').ne_empty,\n    fun h => prod_mem_prod h.1 h.2\u27e9\n#align filter.prod_mem_prod_iff Filter.prod_mem_prod_iff\n\ntheorem mem_prod_principal {s : Set (\u03b1 \u00d7 \u03b2)} :\n    s \u2208 f \u00d7\u02e2 \ud835\udcdf t \u2194 { a | \u2200 b \u2208 t, (a, b) \u2208 s } \u2208 f := by\n  rw [\u2190 @exists_mem_subset_iff _ f, mem_prod_iff]\n  refine' exists_congr fun u => Iff.rfl.and \u27e8_, fun h => \u27e8t, mem_principal_self t, _\u27e9\u27e9\n  \u00b7 rintro \u27e8v, v_in, hv\u27e9 a a_in b b_in\n    exact hv (mk_mem_prod a_in <| v_in b_in)\n  \u00b7 rintro \u27e8x, y\u27e9 \u27e8hx, hy\u27e9\n    exact h hx y hy\n#align filter.mem_prod_principal Filter.mem_prod_principal\n\ntheorem mem_prod_top {s : Set (\u03b1 \u00d7 \u03b2)} :\n    s \u2208 f \u00d7\u02e2 (\u22a4 : Filter \u03b2) \u2194 { a | \u2200 b, (a, b) \u2208 s } \u2208 f := by\n  rw [\u2190 principal_univ, mem_prod_principal]\n  simp only [mem_univ, forall_true_left]\n#align filter.mem_prod_top Filter.mem_prod_top\n\ntheorem eventually_prod_principal_iff {p : \u03b1 \u00d7 \u03b2 \u2192 Prop} {s : Set \u03b2} :\n    (\u2200\u1da0 x : \u03b1 \u00d7 \u03b2 in f \u00d7\u02e2 \ud835\udcdf s, p x) \u2194 \u2200\u1da0 x : \u03b1 in f, \u2200 y : \u03b2, y \u2208 s \u2192 p (x, y) := by\n  rw [eventually_iff, eventually_iff, mem_prod_principal]\n  simp only [mem_setOf_eq]\n#align filter.eventually_prod_principal_iff Filter.eventually_prod_principal_iff\n\ntheorem comap_prod (f : \u03b1 \u2192 \u03b2 \u00d7 \u03b3) (b : Filter \u03b2) (c : Filter \u03b3) :\n    comap f (b \u00d7\u02e2 c) = comap (Prod.fst \u2218 f) b \u2293 comap (Prod.snd \u2218 f) c := by\n  erw [comap_inf, Filter.comap_comap, Filter.comap_comap]\n#align filter.comap_prod Filter.comap_prod\n\ntheorem prod_top : f \u00d7\u02e2 (\u22a4 : Filter \u03b2) = f.comap Prod.fst := by\n  dsimp only [SProd.sprod]\n  rw [Filter.prod, comap_top, inf_top_eq]\n#align filter.prod_top Filter.prod_top\n\ntheorem sup_prod (f\u2081 f\u2082 : Filter \u03b1) (g : Filter \u03b2) : (f\u2081 \u2294 f\u2082) \u00d7\u02e2 g = (f\u2081 \u00d7\u02e2 g) \u2294 (f\u2082 \u00d7\u02e2 g) := by\n  dsimp only [SProd.sprod]\n  rw [Filter.prod, comap_sup, inf_sup_right, \u2190 Filter.prod, \u2190 Filter.prod]\n#align filter.sup_prod Filter.sup_prod\n\ntheorem prod_sup (f : Filter \u03b1) (g\u2081 g\u2082 : Filter \u03b2) : f \u00d7\u02e2 (g\u2081 \u2294 g\u2082) = (f \u00d7\u02e2 g\u2081) \u2294 (f \u00d7\u02e2 g\u2082) := by\n  dsimp only [SProd.sprod]\n  rw [Filter.prod, comap_sup, inf_sup_left, \u2190 Filter.prod, \u2190 Filter.prod]\n#align filter.prod_sup Filter.prod_sup\n\ntheorem eventually_prod_iff {p : \u03b1 \u00d7 \u03b2 \u2192 Prop} :\n    (\u2200\u1da0 x in f \u00d7\u02e2 g, p x) \u2194\n      \u2203 pa : \u03b1 \u2192 Prop, (\u2200\u1da0 x in f, pa x) \u2227 \u2203 pb : \u03b2 \u2192 Prop, (\u2200\u1da0 y in g, pb y) \u2227\n        \u2200 {x}, pa x \u2192 \u2200 {y}, pb y \u2192 p (x, y) := by\n  simpa only [Set.prod_subset_iff] using @mem_prod_iff \u03b1 \u03b2 p f g\n#align filter.eventually_prod_iff Filter.eventually_prod_iff\n\ntheorem tendsto_fst : Tendsto Prod.fst (f \u00d7\u02e2 g) f :=\n  tendsto_inf_left tendsto_comap\n#align filter.tendsto_fst Filter.tendsto_fst\n\ntheorem tendsto_snd : Tendsto Prod.snd (f \u00d7\u02e2 g) g :=\n  tendsto_inf_right tendsto_comap\n#align filter.tendsto_snd Filter.tendsto_snd\n\n", "theoremStatement": "/-- If a function tends to a product `g \u00d7\u02e2 h` of filters, then its first component tends to\n`g`. See also `Filter.Tendsto.fst_nhds` for the special case of converging to a point in a\nproduct of two topological spaces. -/\ntheorem Tendsto.fst {h : Filter \u03b3} {m : \u03b1 \u2192 \u03b2 \u00d7 \u03b3} (H : Tendsto m f (g \u00d7\u02e2 h)) :\n    Tendsto (fun a \u21a6 (m a).1) f g ", "theoremName": "Filter.Tendsto.fst", "fileCreated": {"commit": "5a32630dfdc1d17107bfeefe01bc860ede116bc4", "date": "2023-01-23"}, "theoremCreated": {"commit": "64528268b3c2cf578639bc479828882a9ecd3a82", "date": "2024-03-31"}, "file": "mathlib/Mathlib/Order/Filter/Prod.lean", "module": "Mathlib.Order.Filter.Prod", "jsonFile": "Mathlib.Order.Filter.Prod.jsonl", "positionMetadata": {"lineInFile": 142, "tokenPositionInFile": 5251, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 8, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", 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"Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Order.Filter.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n  tendsto_fst.comp H", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 23}}
+{"srcContext": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Johan Commelin, Mario Carneiro\n-/\nimport Mathlib.Algebra.MvPolynomial.Basic\n\n#align_import data.mv_polynomial.rename from \"leanprover-community/mathlib\"@\"2f5b500a507264de86d666a5f87ddb976e2d8de4\"\n\n/-!\n# Renaming variables of polynomials\n\nThis file establishes the `rename` operation on multivariate polynomials,\nwhich modifies the set of variables.\n\n## Main declarations\n\n* `MvPolynomial.rename`\n* `MvPolynomial.renameEquiv`\n\n## Notation\n\nAs in other polynomial files, we typically use the notation:\n\n+ `\u03c3 \u03c4 \u03b1 : Type*` (indexing the variables)\n\n+ `R S : Type*` `[CommSemiring R]` `[CommSemiring S]` (the coefficients)\n\n+ `s : \u03c3 \u2192\u2080 \u2115`, a function from `\u03c3` to `\u2115` which is zero away from a finite set.\nThis will give rise to a monomial in `MvPolynomial \u03c3 R` which mathematicians might call `X^s`\n\n+ `r : R` elements of the coefficient ring\n\n+ `i : \u03c3`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians\n\n+ `p : MvPolynomial \u03c3 \u03b1`\n\n-/\n\n\nnoncomputable section\n\nopen BigOperators\n\nopen Set Function Finsupp AddMonoidAlgebra\n\nopen BigOperators\n\nvariable {\u03c3 \u03c4 \u03b1 R S : Type*} [CommSemiring R] [CommSemiring S]\n\nnamespace MvPolynomial\n\nsection Rename\n\n/-- Rename all the variables in a multivariable polynomial. -/\ndef rename (f : \u03c3 \u2192 \u03c4) : MvPolynomial \u03c3 R \u2192\u2090[R] MvPolynomial \u03c4 R :=\n  aeval (X \u2218 f)\n#align mv_polynomial.rename MvPolynomial.rename\n\ntheorem rename_C (f : \u03c3 \u2192 \u03c4) (r : R) : rename f (C r) = C r :=\n  eval\u2082_C _ _ _\nset_option linter.uppercaseLean3 false in\n#align mv_polynomial.rename_C MvPolynomial.rename_C\n\n@[simp]\ntheorem rename_X (f : \u03c3 \u2192 \u03c4) (i : \u03c3) : rename f (X i : MvPolynomial \u03c3 R) = X (f i) :=\n  eval\u2082_X _ _ _\nset_option linter.uppercaseLean3 false in\n#align mv_polynomial.rename_X MvPolynomial.rename_X\n\ntheorem map_rename (f : R \u2192+* S) (g : \u03c3 \u2192 \u03c4) (p : MvPolynomial \u03c3 R) :\n    map f (rename g p) = rename g (map f p) := by\n  apply MvPolynomial.induction_on p\n    (fun a => by simp only [map_C, rename_C])\n    (fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by\n    simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]\n#align mv_polynomial.map_rename MvPolynomial.map_rename\n\n@[simp]\ntheorem rename_rename (f : \u03c3 \u2192 \u03c4) (g : \u03c4 \u2192 \u03b1) (p : MvPolynomial \u03c3 R) :\n    rename g (rename f p) = rename (g \u2218 f) p :=\n  show rename g (eval\u2082 C (X \u2218 f) p) = _ by\n    simp only [rename, aeval_eq_eval\u2082Hom]\n    -- Porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.\n    -- Hopefully this is less prone to breaking\n    rw [eval\u2082_comp_left (eval\u2082Hom (algebraMap R (MvPolynomial \u03b1 R)) (X \u2218 g)) C (X \u2218 f) p]\n    simp only [(\u00b7 \u2218 \u00b7), eval\u2082Hom_X']\n    refine' eval\u2082Hom_congr _ rfl rfl\n    ext1; simp only [comp_apply, RingHom.coe_comp, eval\u2082Hom_C]\n#align mv_polynomial.rename_rename MvPolynomial.rename_rename\n\n@[simp]\ntheorem rename_id (p : MvPolynomial \u03c3 R) : rename id p = p :=\n  eval\u2082_eta p\n#align mv_polynomial.rename_id MvPolynomial.rename_id\n\ntheorem rename_monomial (f : \u03c3 \u2192 \u03c4) (d : \u03c3 \u2192\u2080 \u2115) (r : R) :\n    rename f (monomial d r) = monomial (d.mapDomain f) r := by\n  rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),\n    Finsupp.prod_mapDomain_index]\n  \u00b7 rfl\n  \u00b7 exact fun n => pow_zero _\n  \u00b7 exact fun n i\u2081 i\u2082 => pow_add _ _ _\n#align mv_polynomial.rename_monomial MvPolynomial.rename_monomial\n\ntheorem rename_eq (f : \u03c3 \u2192 \u03c4) (p : MvPolynomial \u03c3 R) :\n    rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p := by\n  simp only [rename, aeval_def, eval\u2082, Finsupp.mapDomain, algebraMap_eq, comp_apply,\n    X_pow_eq_monomial, \u2190 monomial_finsupp_sum_index]\n  rfl\n#align mv_polynomial.rename_eq MvPolynomial.rename_eq\n\ntheorem rename_injective (f : \u03c3 \u2192 \u03c4) (hf : Function.Injective f) :\n    Function.Injective (rename f : MvPolynomial \u03c3 R \u2192 MvPolynomial \u03c4 R) := by\n  have :\n    (rename f : MvPolynomial \u03c3 R \u2192 MvPolynomial \u03c4 R) = Finsupp.mapDomain (Finsupp.mapDomain f) :=\n    funext (rename_eq f)\n  rw [this]\n  exact Finsupp.mapDomain_injective (Finsupp.mapDomain_injective hf)\n#align mv_polynomial.rename_injective MvPolynomial.rename_injective\n\nsection\n\nvariable {f : \u03c3 \u2192 \u03c4} (hf : Function.Injective f)\n\nopen scoped Classical\n\n/-- Given a function between sets of variables `f : \u03c3 \u2192 \u03c4` that is injective with proof `hf`,\n  `MvPolynomial.killCompl hf` is the `AlgHom` from `R[\u03c4]` to `R[\u03c3]` that is left inverse to\n  `rename f : R[\u03c3] \u2192 R[\u03c4]` and sends the variables in the complement of the range of `f` to `0`. -/\ndef killCompl : MvPolynomial \u03c4 R \u2192\u2090[R] MvPolynomial \u03c3 R :=\n  aeval fun i => if h : i \u2208 Set.range f then X <| (Equiv.ofInjective f hf).symm \u27e8i, h\u27e9 else 0\n#align mv_polynomial.kill_compl MvPolynomial.killCompl\n\n", "theoremStatement": "theorem killCompl_C (r : R) : killCompl hf (C r) = C r ", "theoremName": "MvPolynomial.killCompl_C", "fileCreated": {"commit": "56e439ec275a319deed4f81f6436af5f0d9b6f00", "date": "2024-04-05"}, "theoremCreated": {"commit": "555069bd76c4f9e4e056e59201acf33d8e3b4fa8", "date": "2024-03-24"}, "file": "mathlib/Mathlib/Algebra/MvPolynomial/Rename.lean", "module": "Mathlib.Algebra.MvPolynomial.Rename", "jsonFile": "Mathlib.Algebra.MvPolynomial.Rename.jsonl", "positionMetadata": {"lineInFile": 135, 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"Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Regular.Pow", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Tactic.FinCases", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Set.UnionLift", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.MvPolynomial.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":= algHom_C _ _", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 15}}
+{"srcContext": "/-\nCopyright (c) 2024 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\nimport Mathlib.Algebra.Homology.Bifunctor\nimport Mathlib.Algebra.Homology.Homotopy\n\n/-!\n# The action of a bifunctor on homological complexes factors through homotopies\n\nGiven a `TotalComplexShape c\u2081 c\u2082 c`, a functor `F : C\u2081 \u2964 C\u2082 \u2964 D`,\nwe shall show in this file that up to homotopy the morphism\n`mapBifunctorMap f\u2081 f\u2082 F c` only depends on the homotopy classes of\nthe morphism `f\u2081` in `HomologicalComplex C c\u2081` and of\nthe morphism `f\u2082` in `HomologicalComplex C c\u2082` (TODO).\n\n-/\n\nopen CategoryTheory Category Limits\n\nvariable {C\u2081 C\u2082 D I\u2081 I\u2082 J : Type*} [Category C\u2081] [Category C\u2082] [Category D]\n  [Preadditive C\u2081] [Preadditive C\u2082] [Preadditive D]\n  {c\u2081 : ComplexShape I\u2081} {c\u2082 : ComplexShape I\u2082}\n\nnamespace HomologicalComplex\n\nvariable {K\u2081 L\u2081 : HomologicalComplex C\u2081 c\u2081} {f\u2081 f\u2081' : K\u2081 \u27f6 L\u2081} (h\u2081 : Homotopy f\u2081 f\u2081')\n  {K\u2082 L\u2082 : HomologicalComplex C\u2082 c\u2082} (f\u2082 : K\u2082 \u27f6 L\u2082)\n  (F : C\u2081 \u2964 C\u2082 \u2964 D) [F.Additive] [\u2200 X\u2081, (F.obj X\u2081).Additive]\n  (c : ComplexShape J) [DecidableEq J] [TotalComplexShape c\u2081 c\u2082 c]\n  [HasMapBifunctor K\u2081 K\u2082 F c]\n  [HasMapBifunctor L\u2081 L\u2082 F c]\n\nnamespace mapBifunctorMapHomotopy\n\n/-- Auxiliary definition for `mapBifunctorMapHomotopy\u2081`. -/\nnoncomputable def hom\u2081 (j j' : J) :\n    (mapBifunctor K\u2081 K\u2082 F c).X j \u27f6 (mapBifunctor L\u2081 L\u2082 F c).X j' :=\n  HomologicalComplex\u2082.totalDesc _\n    (fun i\u2081 i\u2082 _ => ComplexShape.\u03b5\u2081 c\u2081 c\u2082 c (c\u2081.prev i\u2081, i\u2082) \u2022\n      (F.map (h\u2081.hom i\u2081 (c\u2081.prev i\u2081))).app (K\u2082.X i\u2082) \u226b\n      (F.obj (L\u2081.X (c\u2081.prev i\u2081))).map (f\u2082.f i\u2082) \u226b \u03b9MapBifunctorOrZero L\u2081 L\u2082 F c _ _ j')\n\n", "theoremStatement": "@[reassoc]\nlemma \u03b9MapBifunctor_hom\u2081 (i\u2081 i\u2081' : I\u2081) (i\u2082 : I\u2082) (j j' : J)\n    (h : ComplexShape.\u03c0 c\u2081 c\u2082 c (i\u2081', i\u2082) = j) (h' : c\u2081.prev i\u2081' = i\u2081) :\n    \u03b9MapBifunctor K\u2081 K\u2082 F c i\u2081' i\u2082 j h \u226b hom\u2081 h\u2081 f\u2082 F c j j' = ComplexShape.\u03b5\u2081 c\u2081 c\u2082 c (i\u2081, i\u2082) \u2022\n      (F.map (h\u2081.hom i\u2081' i\u2081)).app (K\u2082.X i\u2082) \u226b (F.obj (L\u2081.X i\u2081)).map (f\u2082.f i\u2082) \u226b\n        \u03b9MapBifunctorOrZero L\u2081 L\u2082 F c _ _ j'", "theoremName": "HomologicalComplex.mapBifunctorMapHomotopy.\u03b9MapBifunctor_hom\u2081", "fileCreated": {"commit": "ac1182835d2d1cb42c76a0f4b60c42183b5c8ed9", "date": "2024-03-24"}, "theoremCreated": {"commit": "ac1182835d2d1cb42c76a0f4b60c42183b5c8ed9", "date": 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"Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.Endomorphism", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Terminal", "Mathlib.CategoryTheory.Limits.Shapes.Products", "Mathlib.CategoryTheory.Limits.Shapes.Equalizers", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Shapes.Images", "Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects", "Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero", "Mathlib.CategoryTheory.Limits.Shapes.Kernels", "Mathlib.CategoryTheory.Preadditive.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.CategoryTheory.Linear.Basic", "Mathlib.Algebra.Homology.ComplexShape", "Mathlib.Data.Int.Sqrt", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Units", "Mathlib.Data.Int.Order.Units", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Homology.ComplexShapeSigns", "Mathlib.CategoryTheory.Adjunction.Opposites", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.Limits.Creates", "Mathlib.CategoryTheory.Limits.Unit", "Mathlib.CategoryTheory.Limits.Comma", "Mathlib.CategoryTheory.Adjunction.Comma", "Mathlib.CategoryTheory.Adjunction.Limits", "Mathlib.CategoryTheory.Limits.Shapes.Equivalence", "Mathlib.CategoryTheory.Limits.ConeCategory", "Mathlib.CategoryTheory.Limits.Over", "Mathlib.CategoryTheory.Conj", "Mathlib.CategoryTheory.Adjunction.FullyFaithful", "Mathlib.CategoryTheory.Adjunction.Reflective", "Mathlib.CategoryTheory.Subobject.MonoOver", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.CategoryTheory.Subobject.Basic", "Mathlib.CategoryTheory.Subobject.FactorThru", "Mathlib.CategoryTheory.Subobject.WellPowered", "Mathlib.CategoryTheory.Subobject.Lattice", "Mathlib.CategoryTheory.Subobject.Limits", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.End", "Mathlib.CategoryTheory.Monoidal.NaturalTransformation", "Mathlib.CategoryTheory.Monoidal.Discrete", "Mathlib.CategoryTheory.Shift.Basic", "Mathlib.CategoryTheory.GradedObject", "Mathlib.Algebra.Homology.ShortComplex.Basic", "Mathlib.Algebra.Homology.HomologicalComplex", "Mathlib.Algebra.Homology.HomologicalBicomplex", "Mathlib.Algebra.Homology.TotalComplex", "Mathlib.CategoryTheory.GradedObject.Bifunctor", "Mathlib.Algebra.Homology.Bifunctor", "Mathlib.Algebra.Homology.ImageToKernel", "Mathlib.Algebra.Homology.Homology", "Mathlib.Algebra.Homology.Single", "Mathlib.CategoryTheory.FinCategory.Basic", "Mathlib.CategoryTheory.FinCategory.AsType", "Mathlib.CategoryTheory.Limits.Preserves.Finite", "Mathlib.CategoryTheory.Limits.ExactFunctor", "Mathlib.Algebra.Group.Ext", "Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits", "Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts", "Mathlib.CategoryTheory.Limits.Shapes.Biproducts", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Biproducts", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products", "Mathlib.CategoryTheory.Preadditive.Biproducts", "Mathlib.CategoryTheory.Preadditive.FunctorCategory", "Mathlib.CategoryTheory.Preadditive.AdditiveFunctor", "Mathlib.Algebra.Homology.Additive", "Mathlib.CategoryTheory.Linear.LinearFunctor", "Mathlib.Algebra.Homology.Linear", "Mathlib.Algebra.Homology.ShortComplex.LeftHomology", "Mathlib.CategoryTheory.Filtered.Basic", "Mathlib.Data.TypeMax", "Mathlib.CategoryTheory.Limits.Types", "Mathlib.CategoryTheory.Limits.Filtered", "Mathlib.CategoryTheory.Limits.Opposites", "Mathlib.Algebra.Homology.ShortComplex.RightHomology", "Mathlib.Algebra.Homology.ShortComplex.Homology", "Mathlib.Algebra.Homology.ShortComplex.QuasiIso", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels", "Mathlib.Algebra.Homology.ShortComplex.PreservesHomology", "Mathlib.CategoryTheory.Limits.Constructions.Pullbacks", "Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers", "Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts", "Mathlib.CategoryTheory.Limits.Constructions.Equalizers", "Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts", "Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers", "Mathlib.CategoryTheory.Limits.Shapes.RegularMono", "Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Basic", "Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Equalizers", "Mathlib.CategoryTheory.Abelian.Images", "Mathlib.CategoryTheory.Abelian.NonPreadditive", "Mathlib.CategoryTheory.Abelian.Basic", 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30}}
+{"srcContext": "/-\nCopyright (c) 2024 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou, Jack McKoen\n-/\nimport Mathlib.AlgebraicTopology.SimplicialSet\nimport Mathlib.CategoryTheory.ChosenFiniteProducts.FunctorCategory\nimport Mathlib.CategoryTheory.Monoidal.Types.Basic\n\n/-!\n# The monoidal category structure on simplicial sets\n\nThis file defines an instance of chosen finite products\nfor the category `SSet`. It follows from the fact\nthe `SSet` if a category of functors to the category\nof types and that the category of types have chosen\nfinite products. As a result, we obtain a monoidal\ncategory structure on `SSet`.\n\n-/\n\nuniverse u\n\nopen Simplicial CategoryTheory MonoidalCategory\n\nnamespace SSet\n\nnoncomputable instance : ChosenFiniteProducts SSet.{u} :=\n  (inferInstance : ChosenFiniteProducts (SimplexCategory\u1d52\u1d56 \u2964 Type u))\n\n@[simp]\nlemma leftUnitor_hom_app_apply (K : SSet.{u}) {\u0394 : SimplexCategory\u1d52\u1d56} (x : (\ud835\udfd9_ _ \u2297 K).obj \u0394) :\n    (\u03bb_ K).hom.app \u0394 x = x.2 := rfl\n\n@[simp]\nlemma leftUnitor_inv_app_apply (K : SSet.{u}) {\u0394 : SimplexCategory\u1d52\u1d56} (x : K.obj \u0394) :\n    (\u03bb_ K).inv.app \u0394 x = \u27e8PUnit.unit, x\u27e9 := rfl\n\n@[simp]\nlemma rightUnitor_hom_app_apply (K : SSet.{u}) {\u0394 : SimplexCategory\u1d52\u1d56} (x : (K \u2297 \ud835\udfd9_ _).obj \u0394) :\n    (\u03c1_ K).hom.app \u0394 x = x.1 := rfl\n\n@[simp]\nlemma rightUnitor_inv_app_apply (K : SSet.{u}) {\u0394 : SimplexCategory\u1d52\u1d56} (x : K.obj \u0394) :\n    (\u03c1_ K).inv.app \u0394 x = \u27e8x, PUnit.unit\u27e9 := rfl\n\n@[simp]\nlemma tensorHom_app_apply {K K' L L' : SSet.{u}} (f : K \u27f6 K') (g : L \u27f6 L')\n    {\u0394 : SimplexCategory\u1d52\u1d56} (x : (K \u2297 L).obj \u0394) :\n    (f \u2297 g).app \u0394 x = \u27e8f.app \u0394 x.1, g.app \u0394 x.2\u27e9 := rfl\n\n@[simp]\nlemma whiskerLeft_app_apply (K : SSet.{u}) {L L' : SSet.{u}} (g : L \u27f6 L')\n    {\u0394 : SimplexCategory\u1d52\u1d56} (x : (K \u2297 L).obj \u0394) :\n    (K \u25c1 g).app \u0394 x = \u27e8x.1, g.app \u0394 x.2\u27e9 := rfl\n\n@[simp]\nlemma whiskerRight_app_apply {K K' : SSet.{u}} (f : K \u27f6 K') (L : SSet.{u})\n    {\u0394 : SimplexCategory\u1d52\u1d56} (x : (K \u2297 L).obj \u0394) :\n    (f \u25b7 L).app \u0394 x = \u27e8f.app \u0394 x.1, x.2\u27e9 := rfl\n\n", "theoremStatement": "@[simp]\nlemma associator_hom_app_apply (K L M : SSet.{u}) {\u0394 : SimplexCategory\u1d52\u1d56}\n    (x : ((K \u2297 L) \u2297 M).obj \u0394) :\n    (\u03b1_ K L 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"Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Util.LibraryNote", "Std.Classes.Cast", "Std.Classes.Order", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.Lean.Name", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Match", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Tactic.Relation.Rfl", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Basic", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Expr", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Tactic.OpenPrivate", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.NameMapAttribute", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Logic", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Tactic.Eqns", "Mathlib.Init.Function", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.HashMap", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Simps.NotationClass", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Embedding.Set", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finset.Attr", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Fintype.Sort", "Mathlib.Data.Finset.Order", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.CategoryTheory.FintypeCat", "Mathlib.CategoryTheory.ConcreteCategory.BundledHom", "Mathlib.Order.Category.Preord", "Mathlib.Order.Category.PartOrd", "Mathlib.Order.Category.FinPartOrd", "Mathlib.Order.Category.Lat", "Mathlib.Order.Category.LinOrd", "Mathlib.CategoryTheory.Limits.Shapes.Equalizers", "Mathlib.CategoryTheory.Limits.Shapes.Images", "Mathlib.CategoryTheory.Limits.Shapes.RegularMono", "Mathlib.Order.Category.NonemptyFinLinOrd", "Mathlib.AlgebraicTopology.SimplexCategory", "Mathlib.CategoryTheory.Limits.Preserves.Limits", "Mathlib.CategoryTheory.Limits.FunctorCategory", "Mathlib.AlgebraicTopology.SimplicialObject", "Mathlib.Data.TypeMax", "Mathlib.CategoryTheory.Limits.Types", "Mathlib.CategoryTheory.Limits.Shapes.Products", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.CategoryTheory.Limits.Shapes.Types", "Mathlib.Data.Fin.VecNotation", "Mathlib.Data.Nat.Units", "Mathlib.Tactic.FinCases", "Mathlib.AlgebraicTopology.SimplicialSet", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.CategoryTheory.Monoidal.Free.Coherence", "Mathlib.CategoryTheory.Monoidal.NaturalTransformation", "Mathlib.CategoryTheory.Monoidal.Discrete", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Monoidal.Opposite", "Mathlib.CategoryTheory.Monoidal.Braided.Basic", "Mathlib.CategoryTheory.Monoidal.OfChosenFiniteProducts.Basic", "Mathlib.CategoryTheory.Monoidal.OfChosenFiniteProducts.Symmetric", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products", "Mathlib.CategoryTheory.FinCategory.Basic", "Mathlib.CategoryTheory.FinCategory.AsType", "Mathlib.Data.Fintype.Option", "Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits", "Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts", "Mathlib.Logic.Equiv.Fin", "Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts", "Mathlib.CategoryTheory.ChosenFiniteProducts", "Mathlib.CategoryTheory.ChosenFiniteProducts.FunctorCategory", "Mathlib.CategoryTheory.Monoidal.Types.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":= rfl", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 6}}
+{"srcContext": "/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.Algebra.BigOperators.Ring\nimport Mathlib.Algebra.Module.BigOperators\nimport Mathlib.NumberTheory.Divisors\nimport Mathlib.Data.Nat.Squarefree\nimport Mathlib.Data.Nat.GCD.BigOperators\nimport Mathlib.Data.Nat.Factorization.Basic\nimport Mathlib.Tactic.ArithMult\n\n#align_import number_theory.arithmetic_function from \"leanprover-community/mathlib\"@\"e8638a0fcaf73e4500469f368ef9494e495099b3\"\n\n/-!\n# Arithmetic Functions and Dirichlet Convolution\n\nThis file defines arithmetic functions, which are functions from `\u2115` to a specified type that map 0\nto 0. In the literature, they are often instead defined as functions from `\u2115+`. These arithmetic\nfunctions are endowed with a multiplication, given by Dirichlet convolution, and pointwise addition,\nto form the Dirichlet ring.\n\n## Main Definitions\n\n * `ArithmeticFunction R` consists of functions `f : \u2115 \u2192 R` such that `f 0 = 0`.\n * An arithmetic function `f` `IsMultiplicative` when `x.coprime y \u2192 f (x * y) = f x * f y`.\n * The pointwise operations `pmul` and `ppow` differ from the multiplication\n  and power instances on `ArithmeticFunction R`, which use Dirichlet multiplication.\n * `\u03b6` is the arithmetic function such that `\u03b6 x = 1` for `0 < x`.\n * `\u03c3 k` is the arithmetic function such that `\u03c3 k x = \u2211 y in divisors x, y ^ k` for `0 < x`.\n * `pow k` is the arithmetic function such that `pow k x = x ^ k` for `0 < x`.\n * `id` is the identity arithmetic function on `\u2115`.\n * `\u03c9 n` is the number of distinct prime factors of `n`.\n * `\u03a9 n` is the number of prime factors of `n` counted with multiplicity.\n * `\u03bc` is the M\u00f6bius function (spelled `moebius` in code).\n\n## Main Results\n\n * Several forms of M\u00f6bius inversion:\n * `sum_eq_iff_sum_mul_moebius_eq` for functions to a `CommRing`\n * `sum_eq_iff_sum_smul_moebius_eq` for functions to an `AddCommGroup`\n * `prod_eq_iff_prod_pow_moebius_eq` for functions to a `CommGroup`\n * `prod_eq_iff_prod_pow_moebius_eq_of_nonzero` for functions to a `CommGroupWithZero`\n * And variants that apply when the equalities only hold on a set `S : Set \u2115` such that\n  `m \u2223 n \u2192 n \u2208 S \u2192 m \u2208 S`:\n * `sum_eq_iff_sum_mul_moebius_eq_on` for functions to a `CommRing`\n * `sum_eq_iff_sum_smul_moebius_eq_on` for functions to an `AddCommGroup`\n * `prod_eq_iff_prod_pow_moebius_eq_on` for functions to a `CommGroup`\n * `prod_eq_iff_prod_pow_moebius_eq_on_of_nonzero` for functions to a `CommGroupWithZero`\n\n## Notation\n\nAll notation is localized in the namespace `ArithmeticFunction`.\n\nThe arithmetic functions `\u03b6`, `\u03c3`, `\u03c9`, `\u03a9` and `\u03bc` have Greek letter names.\n\nIn addition, there are separate locales `ArithmeticFunction.zeta` for `\u03b6`,\n`ArithmeticFunction.sigma` for `\u03c3`, `ArithmeticFunction.omega` for `\u03c9`,\n`ArithmeticFunction.Omega` for `\u03a9`, and `ArithmeticFunction.Moebius` for `\u03bc`,\nto allow for selective access to these notations.\n\nThe arithmetic function $$n \\mapsto \\prod_{p \\mid n} f(p)$$ is given custom notation\n`\u220f\u1d56 p \u2223 n, f p` when applied to `n`.\n\n## Tags\n\narithmetic functions, dirichlet convolution, divisors\n\n-/\n\nopen Finset\n\nopen BigOperators\n\nopen Nat\n\nvariable (R : Type*)\n\n/-- An arithmetic function is a function from `\u2115` that maps 0 to 0. In the literature, they are\n  often instead defined as functions from `\u2115+`. Multiplication on `ArithmeticFunctions` is by\n  Dirichlet convolution. -/\ndef ArithmeticFunction [Zero R] :=\n  ZeroHom \u2115 R\n#align nat.arithmetic_function ArithmeticFunction\n\ninstance ArithmeticFunction.zero [Zero R] : Zero (ArithmeticFunction R) :=\n  inferInstanceAs (Zero (ZeroHom \u2115 R))\n\ninstance [Zero R] : Inhabited (ArithmeticFunction R) := inferInstanceAs (Inhabited (ZeroHom \u2115 R))\n\nvariable {R}\n\nnamespace ArithmeticFunction\n\nsection Zero\n\nvariable [Zero R]\n\n--  porting note: used to be `CoeFun`\ninstance : FunLike (ArithmeticFunction R) \u2115 R :=\n  inferInstanceAs (FunLike (ZeroHom \u2115 R) \u2115 R)\n\n@[simp]\ntheorem toFun_eq (f : ArithmeticFunction R) : f.toFun = f := rfl\n#align nat.arithmetic_function.to_fun_eq ArithmeticFunction.toFun_eq\n\n@[simp]\ntheorem coe_mk (f : \u2115 \u2192 R) (hf) : @DFunLike.coe (ArithmeticFunction R) _ _ _\n    (ZeroHom.mk f hf) = f := rfl\n\n@[simp]\ntheorem map_zero {f : ArithmeticFunction R} : f 0 = 0 :=\n  ZeroHom.map_zero' f\n#align nat.arithmetic_function.map_zero ArithmeticFunction.map_zero\n\ntheorem coe_inj {f g : ArithmeticFunction R} : (f : \u2115 \u2192 R) = g \u2194 f = g :=\n  DFunLike.coe_fn_eq\n#align nat.arithmetic_function.coe_inj ArithmeticFunction.coe_inj\n\n@[simp]\ntheorem zero_apply {x : \u2115} : (0 : ArithmeticFunction R) x = 0 :=\n  ZeroHom.zero_apply x\n#align nat.arithmetic_function.zero_apply ArithmeticFunction.zero_apply\n\n@[ext]\ntheorem ext \u2983f g : ArithmeticFunction R\u2984 (h : \u2200 x, f x = g x) : f = g :=\n  ZeroHom.ext h\n#align nat.arithmetic_function.ext ArithmeticFunction.ext\n\ntheorem ext_iff {f g : ArithmeticFunction R} : f = g \u2194 \u2200 x, f x = g x :=\n  DFunLike.ext_iff\n#align nat.arithmetic_function.ext_iff ArithmeticFunction.ext_iff\n\nsection One\n\nvariable [One R]\n\ninstance one : One (ArithmeticFunction R) :=\n  \u27e8\u27e8fun x => ite (x = 1) 1 0, rfl\u27e9\u27e9\n\ntheorem one_apply {x : \u2115} : (1 : ArithmeticFunction R) x = ite (x = 1) 1 0 :=\n  rfl\n#align nat.arithmetic_function.one_apply ArithmeticFunction.one_apply\n\n@[simp]\ntheorem one_one : (1 : ArithmeticFunction R) 1 = 1 :=\n  rfl\n#align nat.arithmetic_function.one_one ArithmeticFunction.one_one\n\n@[simp]\ntheorem one_apply_ne {x : \u2115} (h : x \u2260 1) : (1 : ArithmeticFunction R) x = 0 :=\n  if_neg h\n#align nat.arithmetic_function.one_apply_ne ArithmeticFunction.one_apply_ne\n\nend One\n\nend Zero\n\n/-- Coerce an arithmetic function with values in `\u2115` to one with values in `R`. We cannot inline\nthis in `natCoe` because it gets unfolded too much. -/\n@[coe]  -- Porting note: added `coe` tag.\ndef natToArithmeticFunction [AddMonoidWithOne R] :\n    (ArithmeticFunction \u2115) \u2192 (ArithmeticFunction R) :=\n  fun f => \u27e8fun n => \u2191(f n), by simp\u27e9\n\ninstance natCoe [AddMonoidWithOne R] : Coe (ArithmeticFunction \u2115) (ArithmeticFunction R) :=\n  \u27e8natToArithmeticFunction\u27e9\n#align nat.arithmetic_function.nat_coe ArithmeticFunction.natCoe\n\n@[simp]\ntheorem natCoe_nat (f : ArithmeticFunction \u2115) : natToArithmeticFunction f = f :=\n  ext fun _ => cast_id _\n#align nat.arithmetic_function.nat_coe_nat ArithmeticFunction.natCoe_nat\n\n@[simp]\ntheorem natCoe_apply [AddMonoidWithOne R] {f : ArithmeticFunction \u2115} {x : \u2115} :\n    (f : ArithmeticFunction R) x = f x :=\n  rfl\n#align nat.arithmetic_function.nat_coe_apply ArithmeticFunction.natCoe_apply\n\n/-- Coerce an arithmetic function with values in `\u2124` to one with values in `R`. We cannot inline\nthis in `intCoe` because it gets unfolded too much. -/\n@[coe]\ndef ofInt [AddGroupWithOne R] :\n    (ArithmeticFunction \u2124) \u2192 (ArithmeticFunction R) :=\n  fun f => \u27e8fun n => \u2191(f n), by simp\u27e9\n\ninstance intCoe [AddGroupWithOne R] : Coe (ArithmeticFunction \u2124) (ArithmeticFunction R) :=\n  \u27e8ofInt\u27e9\n#align nat.arithmetic_function.int_coe ArithmeticFunction.intCoe\n\n@[simp]\ntheorem intCoe_int (f : ArithmeticFunction \u2124) : ofInt f = f :=\n  ext fun _ => Int.cast_id\n#align nat.arithmetic_function.int_coe_int ArithmeticFunction.intCoe_int\n\n@[simp]\ntheorem intCoe_apply [AddGroupWithOne R] {f : ArithmeticFunction \u2124} {x : \u2115} :\n    (f : ArithmeticFunction R) x = f x := rfl\n#align nat.arithmetic_function.int_coe_apply ArithmeticFunction.intCoe_apply\n\n@[simp]\ntheorem coe_coe [AddGroupWithOne R] {f : ArithmeticFunction \u2115} :\n    ((f : ArithmeticFunction \u2124) : ArithmeticFunction R) = (f : ArithmeticFunction R) := by\n  ext\n  simp\n#align nat.arithmetic_function.coe_coe ArithmeticFunction.coe_coe\n\n@[simp]\ntheorem natCoe_one [AddMonoidWithOne R] :\n    ((1 : ArithmeticFunction \u2115) : ArithmeticFunction R) = 1 := by\n  ext n\n  simp [one_apply]\n#align nat.arithmetic_function.nat_coe_one ArithmeticFunction.natCoe_one\n\n@[simp]\ntheorem intCoe_one [AddGroupWithOne R] : ((1 : ArithmeticFunction \u2124) :\n    ArithmeticFunction R) = 1 := by\n  ext n\n  simp [one_apply]\n#align nat.arithmetic_function.int_coe_one ArithmeticFunction.intCoe_one\n\nsection AddMonoid\n\nvariable [AddMonoid R]\n\ninstance add : Add (ArithmeticFunction R) :=\n  \u27e8fun f g => \u27e8fun n => f n + g n, by simp\u27e9\u27e9\n\n@[simp]\ntheorem add_apply {f g : ArithmeticFunction R} {n : \u2115} : (f + g) n = f n + g n :=\n  rfl\n#align nat.arithmetic_function.add_apply ArithmeticFunction.add_apply\n\ninstance instAddMonoid : AddMonoid (ArithmeticFunction R) :=\n  { ArithmeticFunction.zero R,\n    ArithmeticFunction.add with\n    add_assoc := fun _ _ _ => ext fun _ => add_assoc _ _ _\n    zero_add := fun _ => ext fun _ => zero_add _\n    add_zero := fun _ => ext fun _ => add_zero _\n    nsmul := nsmulRec }\n#align nat.arithmetic_function.add_monoid ArithmeticFunction.instAddMonoid\n\nend AddMonoid\n\ninstance instAddMonoidWithOne [AddMonoidWithOne R] : AddMonoidWithOne (ArithmeticFunction R) :=\n  { ArithmeticFunction.instAddMonoid,\n    ArithmeticFunction.one with\n    natCast := fun n => \u27e8fun x => if x = 1 then (n : R) else 0, by simp\u27e9\n    natCast_zero := by ext; simp\n    natCast_succ := fun n => by ext x; by_cases h : x = 1 <;> simp [h] }\n#align nat.arithmetic_function.add_monoid_with_one ArithmeticFunction.instAddMonoidWithOne\n\ninstance instAddCommMonoid [AddCommMonoid R] : AddCommMonoid (ArithmeticFunction R) :=\n  { ArithmeticFunction.instAddMonoid with add_comm := fun _ _ => ext fun _ => add_comm _ _ }\n\ninstance [NegZeroClass R] : Neg (ArithmeticFunction R) where\n  neg f := \u27e8fun n => -f n, by simp\u27e9\n\ninstance [AddGroup R] : AddGroup (ArithmeticFunction R) :=\n  { ArithmeticFunction.instAddMonoid with\n    add_left_neg := fun _ => ext fun _ => add_left_neg _\n    zsmul := zsmulRec }\n\ninstance [AddCommGroup R] : AddCommGroup (ArithmeticFunction R) :=\n  { show AddGroup (ArithmeticFunction R) by infer_instance with\n    add_comm := fun _ _ \u21a6 add_comm _ _ }\n\nsection SMul\n\nvariable {M : Type*} [Zero R] [AddCommMonoid M] [SMul R M]\n\n/-- The Dirichlet convolution of two arithmetic functions `f` and `g` is another arithmetic function\n  such that `(f * g) n` is the sum of `f x * g y` over all `(x,y)` such that `x * y = n`. -/\ninstance : SMul (ArithmeticFunction R) (ArithmeticFunction M) :=\n  \u27e8fun f g => \u27e8fun n => \u2211 x in divisorsAntidiagonal n, f x.fst \u2022 g x.snd, by simp\u27e9\u27e9\n\n@[simp]\ntheorem smul_apply {f : ArithmeticFunction R} {g : ArithmeticFunction M} {n : \u2115} :\n    (f \u2022 g) n = \u2211 x in divisorsAntidiagonal n, f x.fst \u2022 g x.snd :=\n  rfl\n#align nat.arithmetic_function.smul_apply ArithmeticFunction.smul_apply\n\nend SMul\n\n/-- The Dirichlet convolution of two arithmetic functions `f` and `g` is another arithmetic function\n  such that `(f * g) n` is the sum of `f x * g y` over all `(x,y)` such that `x * y = n`. -/\ninstance [Semiring R] : Mul (ArithmeticFunction R) :=\n  \u27e8(\u00b7 \u2022 \u00b7)\u27e9\n\n@[simp]\ntheorem mul_apply [Semiring R] {f g : ArithmeticFunction R} {n : \u2115} :\n    (f * g) n = \u2211 x in divisorsAntidiagonal n, f x.fst * g x.snd :=\n  rfl\n#align nat.arithmetic_function.mul_apply ArithmeticFunction.mul_apply\n\ntheorem mul_apply_one [Semiring R] {f g : ArithmeticFunction R} : (f * g) 1 = f 1 * g 1 := by simp\n#align nat.arithmetic_function.mul_apply_one ArithmeticFunction.mul_apply_one\n\n@[simp, norm_cast]\ntheorem natCoe_mul [Semiring R] {f g : ArithmeticFunction \u2115} :\n    (\u2191(f * g) : ArithmeticFunction R) = f * g := by\n  ext n\n  simp\n#align nat.arithmetic_function.nat_coe_mul ArithmeticFunction.natCoe_mul\n\n@[simp, norm_cast]\ntheorem intCoe_mul [Ring R] {f g : ArithmeticFunction \u2124} :\n    (\u2191(f * g) : ArithmeticFunction R) = \u2191f * g := by\n  ext n\n  simp\n#align nat.arithmetic_function.int_coe_mul ArithmeticFunction.intCoe_mul\n\nsection Module\n\nvariable {M : Type*} [Semiring R] [AddCommMonoid M] [Module R M]\n\ntheorem mul_smul' (f g : ArithmeticFunction R) (h : ArithmeticFunction M) :\n    (f * g) \u2022 h = f \u2022 g \u2022 h := by\n  ext n\n  simp only [mul_apply, smul_apply, sum_smul, mul_smul, smul_sum, Finset.sum_sigma']\n  apply Finset.sum_nbij' (fun \u27e8\u27e8_i, j\u27e9, \u27e8k, l\u27e9\u27e9 \u21a6 \u27e8(k, l * j), (l, j)\u27e9)\n    (fun \u27e8\u27e8i, _j\u27e9, \u27e8k, l\u27e9\u27e9 \u21a6 \u27e8(i * k, l), (i, k)\u27e9) <;> aesop (add simp mul_assoc)\n#align nat.arithmetic_function.mul_smul' ArithmeticFunction.mul_smul'\n\ntheorem one_smul' (b : ArithmeticFunction M) : (1 : ArithmeticFunction R) \u2022 b = b := by\n  ext x\n  rw [smul_apply]\n  by_cases x0 : x = 0\n  \u00b7 simp [x0]\n  have h : {(1, x)} \u2286 divisorsAntidiagonal x := by simp [x0]\n  rw [\u2190 sum_subset h]\n  \u00b7 simp\n  intro y ymem ynmem\n  have y1ne : y.fst \u2260 1 := by\n    intro con\n    simp only [Con, mem_divisorsAntidiagonal, one_mul, Ne] at ymem\n    simp only [mem_singleton, Prod.ext_iff] at ynmem\n    -- Porting note: `tauto` worked from here.\n    cases y\n    subst con\n    simp only [true_and, one_mul, x0, not_false_eq_true, and_true] at ynmem ymem\n    tauto\n\n  simp [y1ne]\n#align nat.arithmetic_function.one_smul' ArithmeticFunction.one_smul'\n\nend Module\n\nsection Semiring\n\nvariable [Semiring R]\n\ninstance instMonoid : Monoid (ArithmeticFunction R) :=\n  { one := One.one\n    mul := Mul.mul\n    one_mul := one_smul'\n    mul_one := fun f => by\n      ext x\n      rw [mul_apply]\n      by_cases x0 : x = 0\n      \u00b7 simp [x0]\n      have h : {(x, 1)} \u2286 divisorsAntidiagonal x := by simp [x0]\n      rw [\u2190 sum_subset h]\n      \u00b7 simp\n      intro y ymem ynmem\n      have y2ne : y.snd \u2260 1 := by\n        intro con\n        cases y; subst con -- Porting note: added\n        simp only [Con, mem_divisorsAntidiagonal, mul_one, Ne] at ymem\n        simp only [mem_singleton, Prod.ext_iff] at ynmem\n        tauto\n      simp [y2ne]\n    mul_assoc := mul_smul' }\n#align nat.arithmetic_function.monoid ArithmeticFunction.instMonoid\n\ninstance instSemiring : Semiring (ArithmeticFunction R) :=\n  -- Porting note: I reorganized this instance\n  { ArithmeticFunction.instAddMonoidWithOne,\n    ArithmeticFunction.instMonoid,\n    ArithmeticFunction.instAddCommMonoid with\n    zero_mul := fun f => by\n      ext\n      simp only [mul_apply, zero_mul, sum_const_zero, zero_apply]\n    mul_zero := fun f => by\n      ext\n      simp only [mul_apply, sum_const_zero, mul_zero, zero_apply]\n    left_distrib := fun a b c => by\n      ext\n      simp only [\u2190 sum_add_distrib, mul_add, mul_apply, add_apply]\n    right_distrib := fun a b c => by\n      ext\n      simp only [\u2190 sum_add_distrib, add_mul, mul_apply, add_apply] }\n#align nat.arithmetic_function.semiring ArithmeticFunction.instSemiring\n\nend Semiring\n\ninstance [CommSemiring R] : CommSemiring (ArithmeticFunction R) :=\n  { ArithmeticFunction.instSemiring with\n    mul_comm := fun f g => by\n      ext\n      rw [mul_apply, \u2190 map_swap_divisorsAntidiagonal, sum_map]\n      simp [mul_comm] }\n\ninstance [CommRing R] : CommRing (ArithmeticFunction R) :=\n  { ArithmeticFunction.instSemiring with\n    add_left_neg := add_left_neg\n    mul_comm := mul_comm\n    zsmul := (\u00b7 \u2022 \u00b7) }\n\ninstance {M : Type*} [Semiring R] [AddCommMonoid M] [Module R M] :\n    Module (ArithmeticFunction R) (ArithmeticFunction M) where\n  one_smul := one_smul'\n  mul_smul := mul_smul'\n  smul_add r x y := by\n    ext\n    simp only [sum_add_distrib, smul_add, smul_apply, add_apply]\n  smul_zero r := by\n    ext\n    simp only [smul_apply, sum_const_zero, smul_zero, zero_apply]\n  add_smul r s x := by\n    ext\n    simp only [add_smul, sum_add_distrib, smul_apply, add_apply]\n  zero_smul r := by\n    ext\n    simp only [smul_apply, sum_const_zero, zero_smul, zero_apply]\n\nsection Zeta\n\n/-- `\u03b6 0 = 0`, otherwise `\u03b6 x = 1`. The Dirichlet Series is the Riemann `\u03b6`.  -/\ndef zeta : ArithmeticFunction \u2115 :=\n  \u27e8fun x => ite (x = 0) 0 1, rfl\u27e9\n#align nat.arithmetic_function.zeta ArithmeticFunction.zeta\n\n@[inherit_doc]\nscoped[ArithmeticFunction] notation \"\u03b6\" => ArithmeticFunction.zeta\n\n@[inherit_doc]\nscoped[ArithmeticFunction.zeta] notation \"\u03b6\" => ArithmeticFunction.zeta\n\n@[simp]\ntheorem zeta_apply {x : \u2115} : \u03b6 x = if x = 0 then 0 else 1 :=\n  rfl\n#align nat.arithmetic_function.zeta_apply ArithmeticFunction.zeta_apply\n\ntheorem zeta_apply_ne {x : \u2115} (h : x \u2260 0) : \u03b6 x = 1 :=\n  if_neg h\n#align nat.arithmetic_function.zeta_apply_ne ArithmeticFunction.zeta_apply_ne\n\n-- Porting note: removed `@[simp]`, LHS not in normal form\ntheorem coe_zeta_smul_apply {M} [Semiring R] [AddCommMonoid M] [Module R M]\n    {f : ArithmeticFunction M} {x : \u2115} :\n    ((\u2191\u03b6 : ArithmeticFunction R) \u2022 f) x = \u2211 i in divisors x, f i := by\n  rw [smul_apply]\n  trans \u2211 i in divisorsAntidiagonal x, f i.snd\n  \u00b7 refine' sum_congr rfl fun i hi => _\n    rcases mem_divisorsAntidiagonal.1 hi with \u27e8rfl, h\u27e9\n    rw [natCoe_apply, zeta_apply_ne (left_ne_zero_of_mul h), cast_one, one_smul]\n  \u00b7 rw [\u2190 map_div_left_divisors, sum_map, Function.Embedding.coeFn_mk]\n#align nat.arithmetic_function.coe_zeta_smul_apply ArithmeticFunction.coe_zeta_smul_apply\n\n-- Porting note: removed `@[simp]` to make the linter happy.\ntheorem coe_zeta_mul_apply [Semiring R] {f : ArithmeticFunction R} {x : \u2115} :\n    (\u2191\u03b6 * f) x = \u2211 i in divisors x, f i :=\n  coe_zeta_smul_apply\n#align nat.arithmetic_function.coe_zeta_mul_apply ArithmeticFunction.coe_zeta_mul_apply\n\n-- Porting note: removed `@[simp]` to make the linter happy.\ntheorem coe_mul_zeta_apply [Semiring R] {f : ArithmeticFunction R} {x : \u2115} :\n    (f * \u03b6) x = \u2211 i in divisors x, f i := by\n  rw [mul_apply]\n  trans \u2211 i in divisorsAntidiagonal x, f i.1\n  \u00b7 refine' sum_congr rfl fun i hi => _\n    rcases mem_divisorsAntidiagonal.1 hi with \u27e8rfl, h\u27e9\n    rw [natCoe_apply, zeta_apply_ne (right_ne_zero_of_mul h), cast_one, mul_one]\n  \u00b7 rw [\u2190 map_div_right_divisors, sum_map, Function.Embedding.coeFn_mk]\n#align nat.arithmetic_function.coe_mul_zeta_apply ArithmeticFunction.coe_mul_zeta_apply\n\ntheorem zeta_mul_apply {f : ArithmeticFunction \u2115} {x : \u2115} : (\u03b6 * f) x = \u2211 i in divisors x, f i :=\n  coe_zeta_mul_apply\n  -- Porting note: was `by rw [\u2190 nat_coe_nat \u03b6, coe_zeta_mul_apply]`.  Is this `theorem` obsolete?\n#align nat.arithmetic_function.zeta_mul_apply ArithmeticFunction.zeta_mul_apply\n\ntheorem mul_zeta_apply {f : ArithmeticFunction \u2115} {x : \u2115} : (f * \u03b6) x = \u2211 i in divisors x, f i :=\n  coe_mul_zeta_apply\n  -- Porting note: was `by rw [\u2190 natCoe_nat \u03b6, coe_mul_zeta_apply]`.  Is this `theorem` obsolete=\n#align nat.arithmetic_function.mul_zeta_apply ArithmeticFunction.mul_zeta_apply\n\nend Zeta\n\nopen ArithmeticFunction\n\nsection Pmul\n\n/-- This is the pointwise product of `ArithmeticFunction`s. -/\ndef pmul [MulZeroClass R] (f g : ArithmeticFunction R) : ArithmeticFunction R :=\n  \u27e8fun x => f x * g x, by simp\u27e9\n#align nat.arithmetic_function.pmul ArithmeticFunction.pmul\n\n@[simp]\ntheorem pmul_apply [MulZeroClass R] {f g : ArithmeticFunction R} {x : \u2115} : f.pmul g x = f x * g x :=\n  rfl\n#align nat.arithmetic_function.pmul_apply ArithmeticFunction.pmul_apply\n\ntheorem pmul_comm [CommMonoidWithZero R] (f g : ArithmeticFunction R) : f.pmul g = g.pmul f := by\n  ext\n  simp [mul_comm]\n#align nat.arithmetic_function.pmul_comm ArithmeticFunction.pmul_comm\n\nlemma pmul_assoc [CommMonoidWithZero R] (f\u2081 f\u2082 f\u2083 : ArithmeticFunction R) :\n    pmul (pmul f\u2081 f\u2082) f\u2083 = pmul f\u2081 (pmul f\u2082 f\u2083) := by\n  ext\n  simp only [pmul_apply, mul_assoc]\n\nsection NonAssocSemiring\n\nvariable [NonAssocSemiring R]\n\n@[simp]\ntheorem pmul_zeta (f : ArithmeticFunction R) : f.pmul \u2191\u03b6 = f := by\n  ext x\n  cases x <;> simp [Nat.succ_ne_zero]\n#align nat.arithmetic_function.pmul_zeta ArithmeticFunction.pmul_zeta\n\n@[simp]\ntheorem zeta_pmul (f : ArithmeticFunction R) : (\u03b6 : ArithmeticFunction R).pmul f = f := by\n  ext x\n  cases x <;> simp [Nat.succ_ne_zero]\n#align nat.arithmetic_function.zeta_pmul ArithmeticFunction.zeta_pmul\n\nend NonAssocSemiring\n\nvariable [Semiring R]\n\n/-- This is the pointwise power of `ArithmeticFunction`s. -/\ndef ppow (f : ArithmeticFunction R) (k : \u2115) : ArithmeticFunction R :=\n  if h0 : k = 0 then \u03b6 else \u27e8fun x \u21a6 f x ^ k, by simp_rw [map_zero, zero_pow h0]\u27e9\n#align nat.arithmetic_function.ppow ArithmeticFunction.ppow\n\n@[simp]\ntheorem ppow_zero {f : ArithmeticFunction R} : f.ppow 0 = \u03b6 := by rw [ppow, dif_pos rfl]\n#align nat.arithmetic_function.ppow_zero ArithmeticFunction.ppow_zero\n\n@[simp]\ntheorem ppow_apply {f : ArithmeticFunction R} {k x : \u2115} (kpos : 0 < k) : f.ppow k x = f x ^ k := by\n  rw [ppow, dif_neg (Nat.ne_of_gt kpos)]\n  rfl\n#align nat.arithmetic_function.ppow_apply ArithmeticFunction.ppow_apply\n\ntheorem ppow_succ' {f : ArithmeticFunction R} {k : \u2115} : f.ppow (k + 1) = f.pmul (f.ppow k) := by\n  ext x\n  rw [ppow_apply (Nat.succ_pos k), _root_.pow_succ']\n  induction k <;> simp\n#align nat.arithmetic_function.ppow_succ ArithmeticFunction.ppow_succ'\n\ntheorem ppow_succ {f : ArithmeticFunction R} {k : \u2115} {kpos : 0 < k} :\n    f.ppow (k + 1) = (f.ppow k).pmul f := by\n  ext x\n  rw [ppow_apply (Nat.succ_pos k), _root_.pow_succ]\n  induction k <;> simp\n#align nat.arithmetic_function.ppow_succ' ArithmeticFunction.ppow_succ\n\nend Pmul\n\nsection Pdiv\n\n/-- This is the pointwise division of `ArithmeticFunction`s. -/\ndef pdiv [GroupWithZero R] (f g : ArithmeticFunction R) : ArithmeticFunction R :=\n  \u27e8fun n => f n / g n, by simp only [map_zero, ne_eq, not_true, div_zero]\u27e9\n\n@[simp]\ntheorem pdiv_apply [GroupWithZero R] (f g : ArithmeticFunction R) (n : \u2115) :\n    pdiv f g n = f n / g n := rfl\n\n/-- This result only holds for `DivisionSemiring`s instead of `GroupWithZero`s because zeta takes\nvalues in \u2115, and hence the coercion requires an `AddMonoidWithOne`. TODO: Generalise zeta -/\n@[simp]\ntheorem pdiv_zeta [DivisionSemiring R] (f : ArithmeticFunction R) :\n    pdiv f zeta = f := by\n  ext n\n  cases n <;> simp [succ_ne_zero]\n\nend Pdiv\n\nsection ProdPrimeFactors\n\n/-- The map $n \\mapsto \\prod_{p \\mid n} f(p)$ as an arithmetic function -/\ndef prodPrimeFactors [CommMonoidWithZero R] (f : \u2115 \u2192 R) : ArithmeticFunction R where\n  toFun d := if d = 0 then 0 else \u220f p in d.primeFactors, f p\n  map_zero' := if_pos rfl\n\nopen Std.ExtendedBinder\n\n/-- `\u220f\u1d56 p \u2223 n, f p` is custom notation for `prodPrimeFactors f n` -/\nscoped syntax (name := bigproddvd) \"\u220f\u1d56 \" extBinder \" \u2223 \" term \", \" term:67 : term\nscoped macro_rules (kind := bigproddvd)\n  | `(\u220f\u1d56 $x:ident \u2223 $n, $r) => `(prodPrimeFactors (fun $x \u21a6 $r) $n)\n\n@[simp]\ntheorem prodPrimeFactors_apply [CommMonoidWithZero R] {f: \u2115 \u2192 R} {n : \u2115} (hn : n \u2260 0) :\n    \u220f\u1d56 p \u2223 n, f p = \u220f p in n.primeFactors, f p :=\n  if_neg hn\n\nend ProdPrimeFactors\n\n/-- Multiplicative functions -/\ndef IsMultiplicative [MonoidWithZero R] (f : ArithmeticFunction R) : Prop :=\n  f 1 = 1 \u2227 \u2200 {m n : \u2115}, m.Coprime n \u2192 f (m * n) = f m * f n\n#align nat.arithmetic_function.is_multiplicative ArithmeticFunction.IsMultiplicative\n\nnamespace IsMultiplicative\n\nsection MonoidWithZero\n\nvariable [MonoidWithZero R]\n\n@[simp, arith_mult]\ntheorem map_one {f : ArithmeticFunction R} (h : f.IsMultiplicative) : f 1 = 1 :=\n  h.1\n#align nat.arithmetic_function.is_multiplicative.map_one ArithmeticFunction.IsMultiplicative.map_one\n\n@[simp]\ntheorem map_mul_of_coprime {f : ArithmeticFunction R} (hf : f.IsMultiplicative) {m n : \u2115}\n    (h : m.Coprime n) : f (m * n) = f m * f n :=\n  hf.2 h\n#align nat.arithmetic_function.is_multiplicative.map_mul_of_coprime ArithmeticFunction.IsMultiplicative.map_mul_of_coprime\n\nend MonoidWithZero\n\ntheorem map_prod {\u03b9 : Type*} [CommMonoidWithZero R] (g : \u03b9 \u2192 \u2115) {f : ArithmeticFunction R}\n    (hf : f.IsMultiplicative) (s : Finset \u03b9) (hs : (s : Set \u03b9).Pairwise (Coprime on g)) :\n    f (\u220f i in s, g i) = \u220f i in s, f (g i) := by\n  classical\n    induction' s using Finset.induction_on with a s has ih hs\n    \u00b7 simp [hf]\n    rw [coe_insert, Set.pairwise_insert_of_symmetric (Coprime.symmetric.comap g)] at hs\n    rw [prod_insert has, prod_insert has, hf.map_mul_of_coprime, ih hs.1]\n    exact .prod_right fun i hi => hs.2 _ hi (hi.ne_of_not_mem has).symm\n#align nat.arithmetic_function.is_multiplicative.map_prod ArithmeticFunction.IsMultiplicative.map_prod\n\ntheorem map_prod_of_prime [CommSemiring R] {f : ArithmeticFunction R}\n    (h_mult : ArithmeticFunction.IsMultiplicative f)\n    (t : Finset \u2115) (ht : \u2200 p \u2208 t, p.Prime) :\n    f (\u220f a in t, a) = \u220f a : \u2115 in t, f a :=\n  map_prod _ h_mult t fun x hx y hy hxy => (coprime_primes (ht x hx) (ht y hy)).mpr hxy\n\ntheorem map_prod_of_subset_primeFactors [CommSemiring R] {f : ArithmeticFunction R}\n    (h_mult : ArithmeticFunction.IsMultiplicative f) (l : \u2115)\n    (t : Finset \u2115) (ht : t \u2286 l.primeFactors) :\n    f (\u220f a in t, a) = \u220f a : \u2115 in t, f a :=\n  map_prod_of_prime h_mult t fun _ a => prime_of_mem_primeFactors (ht a)\n\n@[arith_mult]\ntheorem natCast {f : ArithmeticFunction \u2115} [Semiring R] (h : f.IsMultiplicative) :\n    IsMultiplicative (f : ArithmeticFunction R) :=\n                                 -- Porting note: was `by simp [cop, h]`\n  \u27e8by simp [h], fun {m n} cop => by simp [h.2 cop]\u27e9\n#align nat.arithmetic_function.is_multiplicative.nat_cast ArithmeticFunction.IsMultiplicative.natCast\n\n@[arith_mult]\ntheorem intCast {f : ArithmeticFunction \u2124} [Ring R] (h : f.IsMultiplicative) :\n    IsMultiplicative (f : ArithmeticFunction R) :=\n                                 -- Porting note: was `by simp [cop, h]`\n  \u27e8by simp [h], fun {m n} cop => by simp [h.2 cop]\u27e9\n#align nat.arithmetic_function.is_multiplicative.int_cast ArithmeticFunction.IsMultiplicative.intCast\n\n@[arith_mult]\ntheorem mul [CommSemiring R] {f g : ArithmeticFunction R} (hf : f.IsMultiplicative)\n    (hg : g.IsMultiplicative) : IsMultiplicative (f * g) := by\n  refine \u27e8by simp [hf.1, hg.1], ?_\u27e9\n  simp only [mul_apply]\n  intro m n cop\n  rw [sum_mul_sum, \u2190 sum_product']\n  symm\n  apply sum_nbij fun ((i, j), k, l) \u21a6 (i * k, j * l)\n  \u00b7 rintro \u27e8\u27e8a1, a2\u27e9, \u27e8b1, b2\u27e9\u27e9 h\n    simp only [mem_divisorsAntidiagonal, Ne, mem_product] at h\n    rcases h with \u27e8\u27e8rfl, ha\u27e9, \u27e8rfl, hb\u27e9\u27e9\n    simp only [mem_divisorsAntidiagonal, Nat.mul_eq_zero, Ne]\n    constructor\n    \u00b7 ring\n    rw [Nat.mul_eq_zero] at *\n    apply not_or_of_not ha hb\n  \u00b7 simp only [Set.InjOn, mem_coe, mem_divisorsAntidiagonal, Ne, mem_product, Prod.mk.inj_iff]\n    rintro \u27e8\u27e8a1, a2\u27e9, \u27e8b1, b2\u27e9\u27e9 \u27e8\u27e8rfl, ha\u27e9, \u27e8rfl, hb\u27e9\u27e9 \u27e8\u27e8c1, c2\u27e9, \u27e8d1, d2\u27e9\u27e9 hcd h\n    simp only [Prod.mk.inj_iff] at h\n    ext <;> dsimp only\n    \u00b7 trans Nat.gcd (a1 * a2) (a1 * b1)\n      \u00b7 rw [Nat.gcd_mul_left, cop.coprime_mul_left.coprime_mul_right_right.gcd_eq_one, mul_one]\n      \u00b7 rw [\u2190 hcd.1.1, \u2190 hcd.2.1] at cop\n        rw [\u2190 hcd.1.1, h.1, Nat.gcd_mul_left,\n          cop.coprime_mul_left.coprime_mul_right_right.gcd_eq_one, mul_one]\n    \u00b7 trans Nat.gcd (a1 * a2) (a2 * b2)\n      \u00b7 rw [mul_comm, Nat.gcd_mul_left, cop.coprime_mul_right.coprime_mul_left_right.gcd_eq_one,\n          mul_one]\n      \u00b7 rw [\u2190 hcd.1.1, \u2190 hcd.2.1] at cop\n        rw [\u2190 hcd.1.1, h.2, mul_comm, Nat.gcd_mul_left,\n          cop.coprime_mul_right.coprime_mul_left_right.gcd_eq_one, mul_one]\n    \u00b7 trans Nat.gcd (b1 * b2) (a1 * b1)\n      \u00b7 rw [mul_comm, Nat.gcd_mul_right,\n          cop.coprime_mul_right.coprime_mul_left_right.symm.gcd_eq_one, one_mul]\n      \u00b7 rw [\u2190 hcd.1.1, \u2190 hcd.2.1] at cop\n        rw [\u2190 hcd.2.1, h.1, mul_comm c1 d1, Nat.gcd_mul_left,\n          cop.coprime_mul_right.coprime_mul_left_right.symm.gcd_eq_one, mul_one]\n    \u00b7 trans Nat.gcd (b1 * b2) (a2 * b2)\n      \u00b7 rw [Nat.gcd_mul_right, cop.coprime_mul_left.coprime_mul_right_right.symm.gcd_eq_one,\n          one_mul]\n      \u00b7 rw [\u2190 hcd.1.1, \u2190 hcd.2.1] at cop\n        rw [\u2190 hcd.2.1, h.2, Nat.gcd_mul_right,\n          cop.coprime_mul_left.coprime_mul_right_right.symm.gcd_eq_one, one_mul]\n  \u00b7 simp only [Set.SurjOn, Set.subset_def, mem_coe, mem_divisorsAntidiagonal, Ne, mem_product,\n      Set.mem_image, exists_prop, Prod.mk.inj_iff]\n    rintro \u27e8b1, b2\u27e9 h\n    dsimp at h\n    use ((b1.gcd m, b2.gcd m), (b1.gcd n, b2.gcd n))\n    rw [\u2190 cop.gcd_mul _, \u2190 cop.gcd_mul _, \u2190 h.1, Nat.gcd_mul_gcd_of_coprime_of_mul_eq_mul cop h.1,\n      Nat.gcd_mul_gcd_of_coprime_of_mul_eq_mul cop.symm _]\n    \u00b7 rw [Nat.mul_eq_zero, not_or] at h\n      simp [h.2.1, h.2.2]\n    rw [mul_comm n m, h.1]\n  \u00b7 simp only [mem_divisorsAntidiagonal, Ne, mem_product]\n    rintro \u27e8\u27e8a1, a2\u27e9, \u27e8b1, b2\u27e9\u27e9 \u27e8\u27e8rfl, ha\u27e9, \u27e8rfl, hb\u27e9\u27e9\n    dsimp only\n    rw [hf.map_mul_of_coprime cop.coprime_mul_right.coprime_mul_right_right,\n      hg.map_mul_of_coprime cop.coprime_mul_left.coprime_mul_left_right]\n    ring\n#align nat.arithmetic_function.is_multiplicative.mul ArithmeticFunction.IsMultiplicative.mul\n\n@[arith_mult]\ntheorem pmul [CommSemiring R] {f g : ArithmeticFunction R} (hf : f.IsMultiplicative)\n    (hg : g.IsMultiplicative) : IsMultiplicative (f.pmul g) :=\n  \u27e8by simp [hf, hg], fun {m n} cop => by\n    simp only [pmul_apply, hf.map_mul_of_coprime cop, hg.map_mul_of_coprime cop]\n    ring\u27e9\n#align nat.arithmetic_function.is_multiplicative.pmul ArithmeticFunction.IsMultiplicative.pmul\n\n@[arith_mult]\ntheorem pdiv [CommGroupWithZero R] {f g : ArithmeticFunction R} (hf : IsMultiplicative f)\n    (hg : IsMultiplicative g) : IsMultiplicative (pdiv f g) :=\n  \u27e8 by simp [hf, hg], fun {m n} cop => by\n    simp only [pdiv_apply, map_mul_of_coprime hf cop, map_mul_of_coprime hg cop,\n      div_eq_mul_inv, mul_inv]\n    apply mul_mul_mul_comm \u27e9\n\n/-- For any multiplicative function `f` and any `n > 0`,\nwe can evaluate `f n` by evaluating `f` at `p ^ k` over the factorization of `n` -/\nnonrec  -- Porting note: added\ntheorem multiplicative_factorization [CommMonoidWithZero R] (f : ArithmeticFunction R)\n    (hf : f.IsMultiplicative) {n : \u2115} (hn : n \u2260 0) :\n    f n = n.factorization.prod fun p k => f (p ^ k) :=\n  multiplicative_factorization f (fun _ _ => hf.2) hf.1 hn\n#align nat.arithmetic_function.is_multiplicative.multiplicative_factorization ArithmeticFunction.IsMultiplicative.multiplicative_factorization\n\n/-- A recapitulation of the definition of multiplicative that is simpler for proofs -/\ntheorem iff_ne_zero [MonoidWithZero R] {f : ArithmeticFunction R} :\n    IsMultiplicative f \u2194\n      f 1 = 1 \u2227 \u2200 {m n : \u2115}, m \u2260 0 \u2192 n \u2260 0 \u2192 m.Coprime n \u2192 f (m * n) = f m * f n := by\n  refine' and_congr_right' (forall\u2082_congr fun m n => \u27e8fun h _ _ => h, fun h hmn => _\u27e9)\n  rcases eq_or_ne m 0 with (rfl | hm)\n  \u00b7 simp\n  rcases eq_or_ne n 0 with (rfl | hn)\n  \u00b7 simp\n  exact h hm hn hmn\n#align nat.arithmetic_function.is_multiplicative.iff_ne_zero ArithmeticFunction.IsMultiplicative.iff_ne_zero\n\n/-- Two multiplicative functions `f` and `g` are equal if and only if\nthey agree on prime powers -/\ntheorem eq_iff_eq_on_prime_powers [CommMonoidWithZero R] (f : ArithmeticFunction R)\n    (hf : f.IsMultiplicative) (g : ArithmeticFunction R) (hg : g.IsMultiplicative) :\n    f = g \u2194 \u2200 p i : \u2115, Nat.Prime p \u2192 f (p ^ i) = g (p ^ i) := by\n  constructor\n  \u00b7 intro h p i _\n    rw [h]\n  intro h\n  ext n\n  by_cases hn : n = 0\n  \u00b7 rw [hn, ArithmeticFunction.map_zero, ArithmeticFunction.map_zero]\n  rw [multiplicative_factorization f hf hn, multiplicative_factorization g hg hn]\n  exact Finset.prod_congr rfl fun p hp \u21a6 h p _ (Nat.prime_of_mem_primeFactors hp)\n#align nat.arithmetic_function.is_multiplicative.eq_iff_eq_on_prime_powers ArithmeticFunction.IsMultiplicative.eq_iff_eq_on_prime_powers\n\n@[arith_mult]\ntheorem prodPrimeFactors [CommMonoidWithZero R] (f : \u2115 \u2192 R) :\n    IsMultiplicative (prodPrimeFactors f) := by\n  rw [iff_ne_zero]\n  refine \u27e8prodPrimeFactors_apply one_ne_zero, ?_\u27e9\n  intro x y hx hy hxy\n  have hxy\u2080 : x * y \u2260 0 := mul_ne_zero hx hy\n  rw [prodPrimeFactors_apply hxy\u2080, prodPrimeFactors_apply hx, prodPrimeFactors_apply hy,\n    Nat.primeFactors_mul hx hy, \u2190 Finset.prod_union hxy.disjoint_primeFactors]\n\ntheorem prodPrimeFactors_add_of_squarefree [CommSemiring R] {f g : ArithmeticFunction R}\n    (hf : IsMultiplicative f) (hg : IsMultiplicative g) {n : \u2115} (hn : Squarefree n) :\n    \u220f\u1d56 p \u2223 n, (f + g) p = (f * g) n := by\n  rw [prodPrimeFactors_apply hn.ne_zero]\n  simp_rw [add_apply (f:=f) (g:=g)]\n  rw [Finset.prod_add, mul_apply, sum_divisorsAntidiagonal (f \u00b7 * g \u00b7),\n    \u2190 divisors_filter_squarefree_of_squarefree hn, sum_divisors_filter_squarefree hn.ne_zero,\n    factors_eq]\n  apply Finset.sum_congr rfl\n  intro t ht\n  rw [t.prod_val, Function.id_def,\n    \u2190 prod_primeFactors_sdiff_of_squarefree hn (Finset.mem_powerset.mp ht),\n    hf.map_prod_of_subset_primeFactors n t (Finset.mem_powerset.mp ht),\n    \u2190 hg.map_prod_of_subset_primeFactors n (_ \\ t) (Finset.sdiff_subset _ t)]\n\ntheorem lcm_apply_mul_gcd_apply [CommMonoidWithZero R] {f : ArithmeticFunction R}\n    (hf : f.IsMultiplicative) {x y : \u2115} :\n    f (x.lcm y) * f (x.gcd y) = f x * f y := by\n  by_cases hx : x = 0\n  \u00b7 simp only [hx, f.map_zero, zero_mul, Nat.lcm_zero_left, Nat.gcd_zero_left]\n  by_cases hy : y = 0\n  \u00b7 simp only [hy, f.map_zero, mul_zero, Nat.lcm_zero_right, Nat.gcd_zero_right, zero_mul]\n  have hgcd_ne_zero : x.gcd y \u2260 0 := gcd_ne_zero_left hx\n  have hlcm_ne_zero : x.lcm y \u2260 0 := lcm_ne_zero hx hy\n  have hfi_zero : \u2200 {i}, f (i ^ 0) = 1 := by\n    intro i; rw [Nat.pow_zero, hf.1]\n  iterate 4 rw [hf.multiplicative_factorization f (by assumption),\n    Finsupp.prod_of_support_subset _ _ _ (fun _ _ => hfi_zero)\n      (s := (x.primeFactors \u2294 y.primeFactors))]\n  \u00b7 rw [\u2190 Finset.prod_mul_distrib, \u2190 Finset.prod_mul_distrib]\n    apply Finset.prod_congr rfl\n    intro p _\n    rcases Nat.le_or_le (x.factorization p) (y.factorization p) with h | h <;>\n      simp only [factorization_lcm hx hy, ge_iff_le, Finsupp.sup_apply, h, sup_of_le_right,\n        sup_of_le_left, inf_of_le_right, Nat.factorization_gcd hx hy, Finsupp.inf_apply,\n        inf_of_le_left, mul_comm]\n  \u00b7 apply Finset.subset_union_right\n  \u00b7 apply Finset.subset_union_left\n  \u00b7 rw [factorization_gcd hx hy, Finsupp.support_inf, Finset.sup_eq_union]\n    apply Finset.inter_subset_union\n  \u00b7 simp [factorization_lcm hx hy]\n\nend IsMultiplicative\n\nsection SpecialFunctions\n\n/-- The identity on `\u2115` as an `ArithmeticFunction`.  -/\nnonrec  -- Porting note (#11445): added\ndef id : ArithmeticFunction \u2115 :=\n  \u27e8id, rfl\u27e9\n#align nat.arithmetic_function.id ArithmeticFunction.id\n\n@[simp]\ntheorem id_apply {x : \u2115} : id x = x :=\n  rfl\n#align nat.arithmetic_function.id_apply ArithmeticFunction.id_apply\n\n/-- `pow k n = n ^ k`, except `pow 0 0 = 0`. -/\ndef pow (k : \u2115) : ArithmeticFunction \u2115 :=\n  id.ppow k\n#align nat.arithmetic_function.pow ArithmeticFunction.pow\n\n@[simp]\ntheorem pow_apply {k n : \u2115} : pow k n = if k = 0 \u2227 n = 0 then 0 else n ^ k := by\n  cases k\n  \u00b7 simp [pow]\n  rename_i k  -- Porting note: added\n  simp [pow, k.succ_pos.ne']\n#align nat.arithmetic_function.pow_apply ArithmeticFunction.pow_apply\n\ntheorem pow_zero_eq_zeta : pow 0 = \u03b6 := by\n  ext n\n  simp\n#align nat.arithmetic_function.pow_zero_eq_zeta ArithmeticFunction.pow_zero_eq_zeta\n\n/-- `\u03c3 k n` is the sum of the `k`th powers of the divisors of `n` -/\ndef sigma (k : \u2115) : ArithmeticFunction \u2115 :=\n  \u27e8fun n => \u2211 d in divisors n, d ^ k, by simp\u27e9\n#align nat.arithmetic_function.sigma ArithmeticFunction.sigma\n\n@[inherit_doc]\nscoped[ArithmeticFunction] notation \"\u03c3\" => ArithmeticFunction.sigma\n\n@[inherit_doc]\nscoped[ArithmeticFunction.sigma] notation \"\u03c3\" => ArithmeticFunction.sigma\n\ntheorem sigma_apply {k n : \u2115} : \u03c3 k n = \u2211 d in divisors n, d ^ k :=\n  rfl\n#align nat.arithmetic_function.sigma_apply ArithmeticFunction.sigma_apply\n\ntheorem sigma_one_apply (n : \u2115) : \u03c3 1 n = \u2211 d in divisors n, d := by simp [sigma_apply]\n#align nat.arithmetic_function.sigma_one_apply ArithmeticFunction.sigma_one_apply\n\ntheorem sigma_zero_apply (n : \u2115) : \u03c3 0 n = (divisors n).card := by simp [sigma_apply]\n#align nat.arithmetic_function.sigma_zero_apply ArithmeticFunction.sigma_zero_apply\n\ntheorem sigma_zero_apply_prime_pow {p i : \u2115} (hp : p.Prime) : \u03c3 0 (p ^ i) = i + 1 := by\n  rw [sigma_zero_apply, divisors_prime_pow hp, card_map, card_range]\n#align nat.arithmetic_function.sigma_zero_apply_prime_pow ArithmeticFunction.sigma_zero_apply_prime_pow\n\ntheorem zeta_mul_pow_eq_sigma {k : \u2115} : \u03b6 * pow k = \u03c3 k := by\n  ext\n  rw [sigma, zeta_mul_apply]\n  apply sum_congr rfl\n  intro x hx\n  rw [pow_apply, if_neg (not_and_of_not_right _ _)]\n  contrapose! hx\n  simp [hx]\n#align nat.arithmetic_function.zeta_mul_pow_eq_sigma ArithmeticFunction.zeta_mul_pow_eq_sigma\n\n@[arith_mult]\ntheorem isMultiplicative_one [MonoidWithZero R] : IsMultiplicative (1 : ArithmeticFunction R) :=\n  IsMultiplicative.iff_ne_zero.2\n    \u27e8by simp, by\n      intro m n hm _hn hmn\n      rcases eq_or_ne m 1 with (rfl | hm')\n      \u00b7 simp\n      rw [one_apply_ne, one_apply_ne hm', zero_mul]\n      rw [Ne, mul_eq_one, not_and_or]\n      exact Or.inl hm'\u27e9\n#align nat.arithmetic_function.is_multiplicative_one ArithmeticFunction.isMultiplicative_one\n\n@[arith_mult]\ntheorem isMultiplicative_zeta : IsMultiplicative \u03b6 :=\n  IsMultiplicative.iff_ne_zero.2 \u27e8by simp, by simp (config := { contextual := true })\u27e9\n#align nat.arithmetic_function.is_multiplicative_zeta ArithmeticFunction.isMultiplicative_zeta\n\n@[arith_mult]\ntheorem isMultiplicative_id : IsMultiplicative ArithmeticFunction.id :=\n  \u27e8rfl, fun {_ _} _ => rfl\u27e9\n#align nat.arithmetic_function.is_multiplicative_id ArithmeticFunction.isMultiplicative_id\n\n@[arith_mult]\ntheorem IsMultiplicative.ppow [CommSemiring R] {f : ArithmeticFunction R} (hf : f.IsMultiplicative)\n    {k : \u2115} : IsMultiplicative (f.ppow k) := by\n  induction' k with k hi\n  \u00b7 exact isMultiplicative_zeta.natCast\n  \u00b7 rw [ppow_succ']\n    apply hf.pmul hi\n#align nat.arithmetic_function.is_multiplicative.ppow ArithmeticFunction.IsMultiplicative.ppow\n\n@[arith_mult]\ntheorem isMultiplicative_pow {k : \u2115} : IsMultiplicative (pow k) :=\n  isMultiplicative_id.ppow\n#align nat.arithmetic_function.is_multiplicative_pow ArithmeticFunction.isMultiplicative_pow\n\n@[arith_mult]\ntheorem isMultiplicative_sigma {k : \u2115} : IsMultiplicative (\u03c3 k) := by\n  rw [\u2190 zeta_mul_pow_eq_sigma]\n  apply isMultiplicative_zeta.mul isMultiplicative_pow\n#align nat.arithmetic_function.is_multiplicative_sigma ArithmeticFunction.isMultiplicative_sigma\n\n/-- `\u03a9 n` is the number of prime factors of `n`. -/\ndef cardFactors : ArithmeticFunction \u2115 :=\n  \u27e8fun n => n.factors.length, by simp\u27e9\n#align nat.arithmetic_function.card_factors ArithmeticFunction.cardFactors\n\n@[inherit_doc]\nscoped[ArithmeticFunction] notation \"\u03a9\" => ArithmeticFunction.cardFactors\n\n@[inherit_doc]\nscoped[ArithmeticFunction.Omega] notation \"\u03a9\" => ArithmeticFunction.cardFactors\n\ntheorem cardFactors_apply {n : \u2115} : \u03a9 n = n.factors.length :=\n  rfl\n#align nat.arithmetic_function.card_factors_apply ArithmeticFunction.cardFactors_apply\n\n@[simp, nolint simpNF] -- this is a `dsimp` lemma\nlemma cardFactors_zero : \u03a9 0 = 0 := rfl\n\n@[simp] theorem cardFactors_one : \u03a9 1 = 0 := rfl\n#align nat.arithmetic_function.card_factors_one ArithmeticFunction.cardFactors_one\n\n@[simp]\ntheorem cardFactors_eq_one_iff_prime {n : \u2115} : \u03a9 n = 1 \u2194 n.Prime := by\n  refine' \u27e8fun h => _, fun h => List.length_eq_one.2 \u27e8n, factors_prime h\u27e9\u27e9\n  cases' n with n\n  \u00b7 simp at h\n  rcases List.length_eq_one.1 h with \u27e8x, hx\u27e9\n  rw [\u2190 prod_factors n.succ_ne_zero, hx, List.prod_singleton]\n  apply prime_of_mem_factors\n  rw [hx, List.mem_singleton]\n#align nat.arithmetic_function.card_factors_eq_one_iff_prime ArithmeticFunction.cardFactors_eq_one_iff_prime\n\ntheorem cardFactors_mul {m n : \u2115} (m0 : m \u2260 0) (n0 : n \u2260 0) : \u03a9 (m * n) = \u03a9 m + \u03a9 n := by\n  rw [cardFactors_apply, cardFactors_apply, cardFactors_apply, \u2190 Multiset.coe_card, \u2190 factors_eq,\n    UniqueFactorizationMonoid.normalizedFactors_mul m0 n0, factors_eq, factors_eq,\n    Multiset.card_add, Multiset.coe_card, Multiset.coe_card]\n#align nat.arithmetic_function.card_factors_mul ArithmeticFunction.cardFactors_mul\n\ntheorem cardFactors_multiset_prod {s : Multiset \u2115} (h0 : s.prod \u2260 0) :\n    \u03a9 s.prod = (Multiset.map \u03a9 s).sum := by\n  induction s using Multiset.induction_on with\n  | empty => simp\n  | cons ih => simp_all [cardFactors_mul, not_or]\n#align nat.arithmetic_function.card_factors_multiset_prod ArithmeticFunction.cardFactors_multiset_prod\n\n@[simp]\ntheorem cardFactors_apply_prime {p : \u2115} (hp : p.Prime) : \u03a9 p = 1 :=\n  cardFactors_eq_one_iff_prime.2 hp\n#align nat.arithmetic_function.card_factors_apply_prime ArithmeticFunction.cardFactors_apply_prime\n\n@[simp]\ntheorem cardFactors_apply_prime_pow {p k : \u2115} (hp : p.Prime) : \u03a9 (p ^ k) = k := by\n  rw [cardFactors_apply, hp.factors_pow, List.length_replicate]\n#align nat.arithmetic_function.card_factors_apply_prime_pow ArithmeticFunction.cardFactors_apply_prime_pow\n\n/-- `\u03c9 n` is the number of distinct prime factors of `n`. -/\ndef cardDistinctFactors : ArithmeticFunction \u2115 :=\n  \u27e8fun n => n.factors.dedup.length, by simp\u27e9\n#align nat.arithmetic_function.card_distinct_factors ArithmeticFunction.cardDistinctFactors\n\n@[inherit_doc]\nscoped[ArithmeticFunction] notation \"\u03c9\" => ArithmeticFunction.cardDistinctFactors\n\n@[inherit_doc]\nscoped[ArithmeticFunction.omega] notation \"\u03c9\" => ArithmeticFunction.cardDistinctFactors\n\ntheorem cardDistinctFactors_zero : \u03c9 0 = 0 := by simp\n#align nat.arithmetic_function.card_distinct_factors_zero ArithmeticFunction.cardDistinctFactors_zero\n\n@[simp]\ntheorem cardDistinctFactors_one : \u03c9 1 = 0 := by simp [cardDistinctFactors]\n#align nat.arithmetic_function.card_distinct_factors_one ArithmeticFunction.cardDistinctFactors_one\n\ntheorem cardDistinctFactors_apply {n : \u2115} : \u03c9 n = n.factors.dedup.length :=\n  rfl\n#align nat.arithmetic_function.card_distinct_factors_apply ArithmeticFunction.cardDistinctFactors_apply\n\ntheorem cardDistinctFactors_eq_cardFactors_iff_squarefree {n : \u2115} (h0 : n \u2260 0) :\n    \u03c9 n = \u03a9 n \u2194 Squarefree n := by\n  rw [squarefree_iff_nodup_factors h0, cardDistinctFactors_apply]\n  constructor <;> intro h\n  \u00b7 rw [\u2190 n.factors.dedup_sublist.eq_of_length h]\n    apply List.nodup_dedup\n  \u00b7 rw [h.dedup]\n    rfl\n#align nat.arithmetic_function.card_distinct_factors_eq_card_factors_iff_squarefree ArithmeticFunction.cardDistinctFactors_eq_cardFactors_iff_squarefree\n\n@[simp]\ntheorem cardDistinctFactors_apply_prime_pow {p k : \u2115} (hp : p.Prime) (hk : k \u2260 0) :\n    \u03c9 (p ^ k) = 1 := by\n  rw [cardDistinctFactors_apply, hp.factors_pow, List.replicate_dedup hk, List.length_singleton]\n#align nat.arithmetic_function.card_distinct_factors_apply_prime_pow ArithmeticFunction.cardDistinctFactors_apply_prime_pow\n\n@[simp]\ntheorem cardDistinctFactors_apply_prime {p : \u2115} (hp : p.Prime) : \u03c9 p = 1 := by\n  rw [\u2190 pow_one p, cardDistinctFactors_apply_prime_pow hp one_ne_zero]\n#align nat.arithmetic_function.card_distinct_factors_apply_prime ArithmeticFunction.cardDistinctFactors_apply_prime\n\n/-- `\u03bc` is the M\u00f6bius function. If `n` is squarefree with an even number of distinct prime factors,\n  `\u03bc n = 1`. If `n` is squarefree with an odd number of distinct prime factors, `\u03bc n = -1`.\n  If `n` is not squarefree, `\u03bc n = 0`. -/\ndef moebius : ArithmeticFunction \u2124 :=\n  \u27e8fun n => if Squarefree n then (-1) ^ cardFactors n else 0, by simp\u27e9\n#align nat.arithmetic_function.moebius ArithmeticFunction.moebius\n\n@[inherit_doc]\nscoped[ArithmeticFunction] notation \"\u03bc\" => ArithmeticFunction.moebius\n\n@[inherit_doc]\nscoped[ArithmeticFunction.Moebius] notation \"\u03bc\" => ArithmeticFunction.moebius\n\n@[simp]\ntheorem moebius_apply_of_squarefree {n : \u2115} (h : Squarefree n) : \u03bc n = (-1) ^ cardFactors n :=\n  if_pos h\n#align nat.arithmetic_function.moebius_apply_of_squarefree ArithmeticFunction.moebius_apply_of_squarefree\n\n@[simp]\ntheorem moebius_eq_zero_of_not_squarefree {n : \u2115} (h : \u00acSquarefree n) : \u03bc n = 0 :=\n  if_neg h\n#align nat.arithmetic_function.moebius_eq_zero_of_not_squarefree ArithmeticFunction.moebius_eq_zero_of_not_squarefree\n\ntheorem moebius_apply_one : \u03bc 1 = 1 := by simp\n#align nat.arithmetic_function.moebius_apply_one ArithmeticFunction.moebius_apply_one\n\ntheorem moebius_ne_zero_iff_squarefree {n : \u2115} : \u03bc n \u2260 0 \u2194 Squarefree n := by\n  constructor <;> intro h\n  \u00b7 contrapose! h\n    simp [h]\n  \u00b7 simp [h, pow_ne_zero]\n#align nat.arithmetic_function.moebius_ne_zero_iff_squarefree ArithmeticFunction.moebius_ne_zero_iff_squarefree\n\ntheorem moebius_eq_or (n : \u2115) : \u03bc n = 0 \u2228 \u03bc n = 1 \u2228 \u03bc n = -1 := by\n  simp only [moebius, coe_mk]\n  split_ifs\n  \u00b7 right\n    exact neg_one_pow_eq_or ..\n  \u00b7 left\n    rfl\n\ntheorem moebius_ne_zero_iff_eq_or {n : \u2115} : \u03bc n \u2260 0 \u2194 \u03bc n = 1 \u2228 \u03bc n = -1 := by\n  have := moebius_eq_or n\n  aesop\n#align nat.arithmetic_function.moebius_ne_zero_iff_eq_or ArithmeticFunction.moebius_ne_zero_iff_eq_or\n\ntheorem moebius_sq_eq_one_of_squarefree {l : \u2115} (hl : Squarefree l) : \u03bc l ^ 2 = 1 := by\n  rw [moebius_apply_of_squarefree hl, \u2190 pow_mul, mul_comm, pow_mul, neg_one_sq, one_pow]\n\ntheorem abs_moebius_eq_one_of_squarefree {l : \u2115} (hl : Squarefree l) : |\u03bc l| = 1 := by\n  simp only [moebius_apply_of_squarefree hl, abs_pow, abs_neg, abs_one, one_pow]\n\ntheorem moebius_sq {n : \u2115} :\n    \u03bc n ^ 2 = if Squarefree n then 1 else 0 := by\n  split_ifs with h\n  \u00b7 exact moebius_sq_eq_one_of_squarefree h\n  \u00b7 simp only [pow_eq_zero_iff, moebius_eq_zero_of_not_squarefree h,\n    zero_pow (show 2 \u2260 0 by norm_num)]\n\n", "theoremStatement": "theorem abs_moebius {n : \u2115} :\n    |\u03bc n| = if Squarefree n then 1 else 0 ", "theoremName": "ArithmeticFunction.abs_moebius", "fileCreated": {"commit": "5c0483b6d2d047cc4fb513eaa7cfefe1175fb6e5", "date": "2023-05-30"}, "theoremCreated": {"commit": "d2f16b1bef1031d71e1591e1bdb1350839926ad5", "date": "2024-03-28"}, 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"Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.NumberTheory.Divisors", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Data.Rat.BigOperators", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Data.Finsupp.Basic", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Set.UnionLift", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Tactic.FinCases", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.GroupTheory.Finiteness", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.Multiplicity", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.PrimeFin", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.Data.Nat.Multiplicity", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  split_ifs with h\n  \u00b7 exact abs_moebius_eq_one_of_squarefree h\n  \u00b7 simp only [moebius_eq_zero_of_not_squarefree h, abs_zero]", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 131}}
+{"srcContext": "/-\nCopyright (c) 2019 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn, Yury Kudryashov\n-/\nimport Mathlib.Data.Set.Lattice\n\n#align_import data.set.intervals.disjoint from \"leanprover-community/mathlib\"@\"207cfac9fcd06138865b5d04f7091e46d9320432\"\n\n/-!\n# Extra lemmas about intervals\n\nThis file contains lemmas about intervals that cannot be included into `Data.Set.Intervals.Basic`\nbecause this would create an `import` cycle. Namely, lemmas in this file can use definitions\nfrom `Data.Set.Lattice`, including `Disjoint`.\n\nWe consider various intersections and unions of half infinite intervals.\n-/\n\n\nuniverse u v w\n\nvariable {\u03b9 : Sort u} {\u03b1 : Type v} {\u03b2 : Type w}\n\nopen Set\n\nopen OrderDual (toDual)\n\nnamespace Set\n\nsection Preorder\n\nvariable [Preorder \u03b1] {a b c : \u03b1}\n\n@[simp]\ntheorem Iic_disjoint_Ioi (h : a \u2264 b) : Disjoint (Iic a) (Ioi b) :=\n  disjoint_left.mpr fun _ ha hb => (h.trans_lt hb).not_le ha\n#align set.Iic_disjoint_Ioi Set.Iic_disjoint_Ioi\n\n@[simp]\ntheorem Iio_disjoint_Ici (h : a \u2264 b) : Disjoint (Iio a) (Ici b) :=\n  disjoint_left.mpr fun _ ha hb => (h.trans_lt' ha).not_le hb\n\n@[simp]\ntheorem Iic_disjoint_Ioc (h : a \u2264 b) : Disjoint (Iic a) (Ioc b c) :=\n  (Iic_disjoint_Ioi h).mono le_rfl Ioc_subset_Ioi_self\n#align set.Iic_disjoint_Ioc Set.Iic_disjoint_Ioc\n\n@[simp]\ntheorem Ioc_disjoint_Ioc_same : Disjoint (Ioc a b) (Ioc b c) :=\n  (Iic_disjoint_Ioc le_rfl).mono Ioc_subset_Iic_self le_rfl\n#align set.Ioc_disjoint_Ioc_same Set.Ioc_disjoint_Ioc_same\n\n@[simp]\ntheorem Ico_disjoint_Ico_same : Disjoint (Ico a b) (Ico b c) :=\n  disjoint_left.mpr fun _ hab hbc => hab.2.not_le hbc.1\n#align set.Ico_disjoint_Ico_same Set.Ico_disjoint_Ico_same\n\n@[simp]\ntheorem Ici_disjoint_Iic : Disjoint (Ici a) (Iic b) \u2194 \u00aca \u2264 b := by\n  rw [Set.disjoint_iff_inter_eq_empty, Ici_inter_Iic, Icc_eq_empty_iff]\n#align set.Ici_disjoint_Iic Set.Ici_disjoint_Iic\n\n@[simp]\ntheorem Iic_disjoint_Ici : Disjoint (Iic a) (Ici b) \u2194 \u00acb \u2264 a :=\n  disjoint_comm.trans Ici_disjoint_Iic\n#align set.Iic_disjoint_Ici Set.Iic_disjoint_Ici\n\n@[simp]\ntheorem Ioc_disjoint_Ioi (h : b \u2264 c) : Disjoint (Ioc a b) (Ioi c) :=\n  disjoint_left.mpr (fun _ hx hy \u21a6 (hx.2.trans h).not_lt hy)\n\ntheorem Ioc_disjoint_Ioi_same : Disjoint (Ioc a b) (Ioi b) :=\n  Ioc_disjoint_Ioi le_rfl\n\n@[simp]\ntheorem iUnion_Iic : \u22c3 a : \u03b1, Iic a = univ :=\n  iUnion_eq_univ_iff.2 fun x => \u27e8x, right_mem_Iic\u27e9\n#align set.Union_Iic Set.iUnion_Iic\n\n@[simp]\ntheorem iUnion_Ici : \u22c3 a : \u03b1, Ici a = univ :=\n  iUnion_eq_univ_iff.2 fun x => \u27e8x, left_mem_Ici\u27e9\n#align set.Union_Ici Set.iUnion_Ici\n\n@[simp]\ntheorem iUnion_Icc_right (a : \u03b1) : \u22c3 b, Icc a b = Ici a := by\n  simp only [\u2190 Ici_inter_Iic, \u2190 inter_iUnion, iUnion_Iic, inter_univ]\n#align set.Union_Icc_right Set.iUnion_Icc_right\n\n@[simp]\ntheorem iUnion_Ioc_right (a : \u03b1) : \u22c3 b, Ioc a b = Ioi a := by\n  simp only [\u2190 Ioi_inter_Iic, \u2190 inter_iUnion, iUnion_Iic, inter_univ]\n#align set.Union_Ioc_right Set.iUnion_Ioc_right\n\n@[simp]\ntheorem iUnion_Icc_left (b : \u03b1) : \u22c3 a, Icc a b = Iic b := by\n  simp only [\u2190 Ici_inter_Iic, \u2190 iUnion_inter, iUnion_Ici, univ_inter]\n#align set.Union_Icc_left Set.iUnion_Icc_left\n\n@[simp]\ntheorem iUnion_Ico_left (b : \u03b1) : \u22c3 a, Ico a b = Iio b := by\n  simp only [\u2190 Ici_inter_Iio, \u2190 iUnion_inter, iUnion_Ici, univ_inter]\n#align set.Union_Ico_left Set.iUnion_Ico_left\n\n@[simp]\ntheorem iUnion_Iio [NoMaxOrder \u03b1] : \u22c3 a : \u03b1, Iio a = univ :=\n  iUnion_eq_univ_iff.2 exists_gt\n#align set.Union_Iio Set.iUnion_Iio\n\n@[simp]\ntheorem iUnion_Ioi [NoMinOrder \u03b1] : \u22c3 a : \u03b1, Ioi a = univ :=\n  iUnion_eq_univ_iff.2 exists_lt\n#align set.Union_Ioi Set.iUnion_Ioi\n\n@[simp]\ntheorem iUnion_Ico_right [NoMaxOrder \u03b1] (a : \u03b1) : \u22c3 b, Ico a b = Ici a := by\n  simp only [\u2190 Ici_inter_Iio, \u2190 inter_iUnion, iUnion_Iio, inter_univ]\n#align set.Union_Ico_right Set.iUnion_Ico_right\n\n@[simp]\ntheorem iUnion_Ioo_right [NoMaxOrder \u03b1] (a : \u03b1) : \u22c3 b, Ioo a b = Ioi a := by\n  simp only [\u2190 Ioi_inter_Iio, \u2190 inter_iUnion, iUnion_Iio, inter_univ]\n#align set.Union_Ioo_right Set.iUnion_Ioo_right\n\n@[simp]\ntheorem iUnion_Ioc_left [NoMinOrder \u03b1] (b : \u03b1) : \u22c3 a, Ioc a b = Iic b := by\n  simp only [\u2190 Ioi_inter_Iic, \u2190 iUnion_inter, iUnion_Ioi, univ_inter]\n#align set.Union_Ioc_left Set.iUnion_Ioc_left\n\n@[simp]\ntheorem iUnion_Ioo_left [NoMinOrder \u03b1] (b : \u03b1) : \u22c3 a, Ioo a b = Iio b := by\n  simp only [\u2190 Ioi_inter_Iio, \u2190 iUnion_inter, iUnion_Ioi, univ_inter]\n#align set.Union_Ioo_left Set.iUnion_Ioo_left\n\nend Preorder\n\nsection LinearOrder\n\nvariable [LinearOrder \u03b1] {a\u2081 a\u2082 b\u2081 b\u2082 : \u03b1}\n\n@[simp]\ntheorem Ico_disjoint_Ico : Disjoint (Ico a\u2081 a\u2082) (Ico b\u2081 b\u2082) \u2194 min a\u2082 b\u2082 \u2264 max a\u2081 b\u2081 := by\n  simp_rw [Set.disjoint_iff_inter_eq_empty, Ico_inter_Ico, Ico_eq_empty_iff, inf_eq_min, sup_eq_max,\n    not_lt]\n#align set.Ico_disjoint_Ico Set.Ico_disjoint_Ico\n\n@[simp]\ntheorem Ioc_disjoint_Ioc : Disjoint (Ioc a\u2081 a\u2082) (Ioc b\u2081 b\u2082) \u2194 min a\u2082 b\u2082 \u2264 max a\u2081 b\u2081 := by\n  have h : _ \u2194 min (toDual a\u2081) (toDual b\u2081) \u2264 max (toDual a\u2082) (toDual b\u2082) := Ico_disjoint_Ico\n  simpa only [dual_Ico] using h\n#align set.Ioc_disjoint_Ioc Set.Ioc_disjoint_Ioc\n\n@[simp]\ntheorem Ioo_disjoint_Ioo [DenselyOrdered \u03b1] :\n    Disjoint (Set.Ioo a\u2081 a\u2082) (Set.Ioo b\u2081 b\u2082) \u2194 min a\u2082 b\u2082 \u2264 max a\u2081 b\u2081 := by\n  simp_rw [Set.disjoint_iff_inter_eq_empty, Ioo_inter_Ioo, Ioo_eq_empty_iff, inf_eq_min, sup_eq_max,\n    not_lt]\n\n/-- If two half-open intervals are disjoint and the endpoint of one lies in the other,\n  then it must be equal to the endpoint of the other. -/\ntheorem eq_of_Ico_disjoint {x\u2081 x\u2082 y\u2081 y\u2082 : \u03b1} (h : Disjoint (Ico x\u2081 x\u2082) (Ico y\u2081 y\u2082)) (hx : x\u2081 < x\u2082)\n    (h2 : x\u2082 \u2208 Ico y\u2081 y\u2082) : y\u2081 = x\u2082 := by\n  rw [Ico_disjoint_Ico, min_eq_left (le_of_lt h2.2), le_max_iff] at h\n  apply le_antisymm h2.1\n  exact h.elim (fun h => absurd hx (not_lt_of_le h)) id\n#align set.eq_of_Ico_disjoint Set.eq_of_Ico_disjoint\n\n@[simp]\ntheorem iUnion_Ico_eq_Iio_self_iff {f : \u03b9 \u2192 \u03b1} {a : \u03b1} :\n    \u22c3 i, Ico (f i) a = Iio a \u2194 \u2200 x < a, \u2203 i, f i \u2264 x := by\n  simp [\u2190 Ici_inter_Iio, \u2190 iUnion_inter, subset_def]\n#align set.Union_Ico_eq_Iio_self_iff Set.iUnion_Ico_eq_Iio_self_iff\n\n@[simp]\ntheorem iUnion_Ioc_eq_Ioi_self_iff {f : \u03b9 \u2192 \u03b1} {a : \u03b1} :\n    \u22c3 i, Ioc a (f i) = Ioi a \u2194 \u2200 x, a < x \u2192 \u2203 i, x \u2264 f i := by\n  simp [\u2190 Ioi_inter_Iic, \u2190 inter_iUnion, subset_def]\n#align set.Union_Ioc_eq_Ioi_self_iff Set.iUnion_Ioc_eq_Ioi_self_iff\n\n@[simp]\ntheorem biUnion_Ico_eq_Iio_self_iff {p : \u03b9 \u2192 Prop} {f : \u2200 i, p i \u2192 \u03b1} {a : \u03b1} :\n    \u22c3 (i) (hi : p i), Ico (f i hi) a = Iio a \u2194 \u2200 x < a, \u2203 i hi, f i hi \u2264 x := by\n  simp [\u2190 Ici_inter_Iio, \u2190 iUnion_inter, subset_def]\n#align set.bUnion_Ico_eq_Iio_self_iff Set.biUnion_Ico_eq_Iio_self_iff\n\n@[simp]\ntheorem biUnion_Ioc_eq_Ioi_self_iff {p : \u03b9 \u2192 Prop} {f : \u2200 i, p i \u2192 \u03b1} {a : \u03b1} :\n    \u22c3 (i) (hi : p i), Ioc a (f i hi) = Ioi a \u2194 \u2200 x, a < x \u2192 \u2203 i hi, x \u2264 f i hi := by\n  simp [\u2190 Ioi_inter_Iic, \u2190 inter_iUnion, subset_def]\n#align set.bUnion_Ioc_eq_Ioi_self_iff Set.biUnion_Ioc_eq_Ioi_self_iff\n\nend LinearOrder\n\nend Set\n\nsection UnionIxx\n\nvariable [LinearOrder \u03b1] {s : Set \u03b1} {a : \u03b1} {f : \u03b9 \u2192 \u03b1}\n\ntheorem IsGLB.biUnion_Ioi_eq (h : IsGLB s a) : \u22c3 x \u2208 s, Ioi x = Ioi a := by\n  refine' (iUnion\u2082_subset fun x hx => _).antisymm fun x hx => _\n  \u00b7 exact Ioi_subset_Ioi (h.1 hx)\n  \u00b7 rcases h.exists_between hx with \u27e8y, hys, _, hyx\u27e9\n    exact mem_biUnion hys hyx\n#align is_glb.bUnion_Ioi_eq IsGLB.biUnion_Ioi_eq\n\ntheorem IsGLB.iUnion_Ioi_eq (h : IsGLB (range f) a) : \u22c3 x, Ioi (f x) = Ioi a :=\n  biUnion_range.symm.trans h.biUnion_Ioi_eq\n#align is_glb.Union_Ioi_eq IsGLB.iUnion_Ioi_eq\n\ntheorem IsLUB.biUnion_Iio_eq (h : IsLUB s a) : \u22c3 x \u2208 s, Iio x = Iio a :=\n  h.dual.biUnion_Ioi_eq\n#align is_lub.bUnion_Iio_eq IsLUB.biUnion_Iio_eq\n\ntheorem IsLUB.iUnion_Iio_eq (h : IsLUB (range f) a) : \u22c3 x, Iio (f x) = Iio a :=\n  h.dual.iUnion_Ioi_eq\n#align is_lub.Union_Iio_eq IsLUB.iUnion_Iio_eq\n\ntheorem IsGLB.biUnion_Ici_eq_Ioi (a_glb : IsGLB s a) (a_not_mem : a \u2209 s) :\n    \u22c3 x \u2208 s, Ici x = Ioi a := by\n  refine' (iUnion\u2082_subset fun x hx => _).antisymm fun x hx => _\n  \u00b7 exact Ici_subset_Ioi.mpr (lt_of_le_of_ne (a_glb.1 hx) fun h => (h \u25b8 a_not_mem) hx)\n  \u00b7 rcases a_glb.exists_between hx with \u27e8y, hys, _, hyx\u27e9\n    rw [mem_iUnion\u2082]\n    exact \u27e8y, hys, hyx.le\u27e9\n#align is_glb.bUnion_Ici_eq_Ioi IsGLB.biUnion_Ici_eq_Ioi\n\ntheorem IsGLB.biUnion_Ici_eq_Ici (a_glb : IsGLB s a) (a_mem : a \u2208 s) :\n    \u22c3 x \u2208 s, Ici x = Ici a := by\n  refine' (iUnion\u2082_subset fun x hx => _).antisymm fun x hx => _\n  \u00b7 exact Ici_subset_Ici.mpr (mem_lowerBounds.mp a_glb.1 x hx)\n  \u00b7 exact mem_iUnion\u2082.mpr \u27e8a, a_mem, hx\u27e9\n#align is_glb.bUnion_Ici_eq_Ici IsGLB.biUnion_Ici_eq_Ici\n\ntheorem IsLUB.biUnion_Iic_eq_Iio (a_lub : IsLUB s a) (a_not_mem : a \u2209 s) :\n    \u22c3 x \u2208 s, Iic x = Iio a :=\n  a_lub.dual.biUnion_Ici_eq_Ioi a_not_mem\n#align is_lub.bUnion_Iic_eq_Iio IsLUB.biUnion_Iic_eq_Iio\n\ntheorem IsLUB.biUnion_Iic_eq_Iic (a_lub : IsLUB s a) (a_mem : a \u2208 s) : \u22c3 x \u2208 s, Iic x = Iic a :=\n  a_lub.dual.biUnion_Ici_eq_Ici a_mem\n#align is_lub.bUnion_Iic_eq_Iic IsLUB.biUnion_Iic_eq_Iic\n\ntheorem iUnion_Ici_eq_Ioi_iInf {R : Type*} [CompleteLinearOrder R] {f : \u03b9 \u2192 R}\n    (no_least_elem : \u2a05 i, f i \u2209 range f) : \u22c3 i : \u03b9, Ici (f i) = Ioi (\u2a05 i, f i) := by\n  simp only [\u2190 IsGLB.biUnion_Ici_eq_Ioi (@isGLB_iInf _ _ _ f) no_least_elem, mem_range,\n    iUnion_exists, iUnion_iUnion_eq']\n#align Union_Ici_eq_Ioi_infi iUnion_Ici_eq_Ioi_iInf\n\ntheorem iUnion_Iic_eq_Iio_iSup {R : Type*} [CompleteLinearOrder R] {f : \u03b9 \u2192 R}\n    (no_greatest_elem : (\u2a06 i, f i) \u2209 range f) : \u22c3 i : \u03b9, Iic (f i) = Iio (\u2a06 i, f i) :=\n  @iUnion_Ici_eq_Ioi_iInf \u03b9 (OrderDual R) _ f no_greatest_elem\n#align Union_Iic_eq_Iio_supr iUnion_Iic_eq_Iio_iSup\n\ntheorem iUnion_Ici_eq_Ici_iInf {R : Type*} [CompleteLinearOrder R] {f : \u03b9 \u2192 R}\n    (has_least_elem : (\u2a05 i, f i) \u2208 range f) : \u22c3 i : \u03b9, Ici (f i) = Ici (\u2a05 i, f i) := by\n  simp only [\u2190 IsGLB.biUnion_Ici_eq_Ici (@isGLB_iInf _ _ _ f) has_least_elem, mem_range,\n    iUnion_exists, iUnion_iUnion_eq']\n#align Union_Ici_eq_Ici_infi iUnion_Ici_eq_Ici_iInf\n\ntheorem iUnion_Iic_eq_Iic_iSup {R : Type*} [CompleteLinearOrder R] {f : \u03b9 \u2192 R}\n    (has_greatest_elem : (\u2a06 i, f i) \u2208 range f) : \u22c3 i : \u03b9, Iic (f i) = Iic (\u2a06 i, f i) :=\n  @iUnion_Ici_eq_Ici_iInf \u03b9 (OrderDual R) _ f has_greatest_elem\n#align Union_Iic_eq_Iic_supr iUnion_Iic_eq_Iic_iSup\n\n", "theoremStatement": "theorem iUnion_Iio_eq_univ_iff : \u22c3 i, Iio (f i) = univ \u2194 (\u00ac BddAbove (range f)) ", "theoremName": "iUnion_Iio_eq_univ_iff", "fileCreated": {"commit": "74306d8176906b9de77391cba18e48349db8baf5", "date": "2022-12-26"}, "theoremCreated": {"commit": "78660691e262ccf3c22390bda656410f4c7e7356", "date": "2024-03-26"}, "file": "mathlib/Mathlib/Data/Set/Intervals/Disjoint.lean", "module": "Mathlib.Data.Set.Intervals.Disjoint", "jsonFile": "Mathlib.Data.Set.Intervals.Disjoint.jsonl", "positionMetadata": {"lineInFile": 267, "tokenPositionInFile": 10161, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 37, "importedModules": ["Init.Prelude", "Init.Coe", 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"Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", 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"Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.Conv", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Cases", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.Convert", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Spread", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.AssertExists", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Logic.Pairwise", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Order.Notation", "Mathlib.Data.Set.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Data.Set.Prod", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.MinMax", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  simp [not_bddAbove_iff, Set.eq_univ_iff_forall]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 55}}
+{"srcContext": "/-\nCopyright (c) 2018 Guy Leroy. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes H\u00f6lzl, Mario Carneiro\n-/\nimport Mathlib.Algebra.Group.Commute.Units\nimport Mathlib.Algebra.GroupWithZero.Power\nimport Mathlib.Algebra.Ring.Regular\nimport Mathlib.Data.Int.Dvd.Basic\nimport Mathlib.Data.Nat.GCD.Basic\nimport Mathlib.Order.Bounds.Basic\n\n#align_import data.int.gcd from \"leanprover-community/mathlib\"@\"47a1a73351de8dd6c8d3d32b569c8e434b03ca47\"\n\n/-!\n# Extended GCD and divisibility over \u2124\n\n## Main definitions\n\n* Given `x y : \u2115`, `xgcd x y` computes the pair of integers `(a, b)` such that\n  `gcd x y = x * a + y * b`. `gcdA x y` and `gcdB x y` are defined to be `a` and `b`,\n  respectively.\n\n## Main statements\n\n* `gcd_eq_gcd_ab`: B\u00e9zout's lemma, given `x y : \u2115`, `gcd x y = x * gcdA x y + y * gcdB x y`.\n\n## Tags\n\nB\u00e9zout's lemma, Bezout's lemma\n-/\n\n/-! ### Extended Euclidean algorithm -/\n\n\nnamespace Nat\n\n/-- Helper function for the extended GCD algorithm (`Nat.xgcd`). -/\ndef xgcdAux : \u2115 \u2192 \u2124 \u2192 \u2124 \u2192 \u2115 \u2192 \u2124 \u2192 \u2124 \u2192 \u2115 \u00d7 \u2124 \u00d7 \u2124\n  | 0, _, _, r', s', t' => (r', s', t')\n  | succ k, s, t, r', s', t' =>\n    have : r' % succ k < succ k := mod_lt _ <| (succ_pos _).gt\n    let q := r' / succ k\n    xgcdAux (r' % succ k) (s' - q * s) (t' - q * t) (succ k) s t\n#align nat.xgcd_aux Nat.xgcdAux\n\n-- Porting note: these are not in mathlib3; these equation lemmas are to fix\n-- complaints by the Lean 4 `unusedHavesSuffices` linter obtained when `simp [xgcdAux]` is used.\ntheorem xgcdAux_zero {s t : \u2124} {r' : \u2115} {s' t' : \u2124} : xgcdAux 0 s t r' s' t' = (r', s', t') := rfl\n\ntheorem xgcdAux_succ {k : \u2115} {s t : \u2124} {r' : \u2115} {s' t' : \u2124} : xgcdAux (succ k) s t r' s' t' =\n    xgcdAux (r' % succ k) (s' - (r' / succ k) * s) (t' - (r' / succ k) * t) (succ k) s t := rfl\n\n@[simp]\ntheorem xgcd_zero_left {s t r' s' t'} : xgcdAux 0 s t r' s' t' = (r', s', t') := by simp [xgcdAux]\n#align nat.xgcd_zero_left Nat.xgcd_zero_left\n\ntheorem xgcdAux_rec {r s t r' s' t'} (h : 0 < r) :\n    xgcdAux r s t r' s' t' = xgcdAux (r' % r) (s' - r' / r * s) (t' - r' / r * t) r s t := by\n  obtain \u27e8r, rfl\u27e9 := Nat.exists_eq_succ_of_ne_zero h.ne'\n  rfl\n#align nat.xgcd_aux_rec Nat.xgcdAux_rec\n\n/-- Use the extended GCD algorithm to generate the `a` and `b` values\n  satisfying `gcd x y = x * a + y * b`. -/\ndef xgcd (x y : \u2115) : \u2124 \u00d7 \u2124 :=\n  (xgcdAux x 1 0 y 0 1).2\n#align nat.xgcd Nat.xgcd\n\n/-- The extended GCD `a` value in the equation `gcd x y = x * a + y * b`. -/\ndef gcdA (x y : \u2115) : \u2124 :=\n  (xgcd x y).1\n#align nat.gcd_a Nat.gcdA\n\n/-- The extended GCD `b` value in the equation `gcd x y = x * a + y * b`. -/\ndef gcdB (x y : \u2115) : \u2124 :=\n  (xgcd x y).2\n#align nat.gcd_b Nat.gcdB\n\n@[simp]\ntheorem gcdA_zero_left {s : \u2115} : gcdA 0 s = 0 := by\n  unfold gcdA\n  rw [xgcd, xgcd_zero_left]\n#align nat.gcd_a_zero_left Nat.gcdA_zero_left\n\n@[simp]\ntheorem gcdB_zero_left {s : \u2115} : gcdB 0 s = 1 := by\n  unfold gcdB\n  rw [xgcd, xgcd_zero_left]\n#align nat.gcd_b_zero_left Nat.gcdB_zero_left\n\n@[simp]\ntheorem gcdA_zero_right {s : \u2115} (h : s \u2260 0) : gcdA s 0 = 1 := by\n  unfold gcdA xgcd\n  obtain \u27e8s, rfl\u27e9 := Nat.exists_eq_succ_of_ne_zero h\n  -- Porting note (https://github.com/leanprover/lean4/issues/2330):\n  -- `simp [xgcdAux_succ]` crashes Lean here\n  rw [xgcdAux_succ]\n  rfl\n#align nat.gcd_a_zero_right Nat.gcdA_zero_right\n\n@[simp]\ntheorem gcdB_zero_right {s : \u2115} (h : s \u2260 0) : gcdB s 0 = 0 := by\n  unfold gcdB xgcd\n  obtain \u27e8s, rfl\u27e9 := Nat.exists_eq_succ_of_ne_zero h\n  -- Porting note (https://github.com/leanprover/lean4/issues/2330):\n  -- `simp [xgcdAux_succ]` crashes Lean here\n  rw [xgcdAux_succ]\n  rfl\n#align nat.gcd_b_zero_right Nat.gcdB_zero_right\n\n@[simp]\ntheorem xgcdAux_fst (x y) : \u2200 s t s' t', (xgcdAux x s t y s' t').1 = gcd x y :=\n  gcd.induction x y (by simp) fun x y h IH s t s' t' => by\n    simp only [h, xgcdAux_rec, IH]\n    rw [\u2190 gcd_rec]\n#align nat.xgcd_aux_fst Nat.xgcdAux_fst\n\ntheorem xgcdAux_val (x y) : xgcdAux x 1 0 y 0 1 = (gcd x y, xgcd x y) := by\n  rw [xgcd, \u2190 xgcdAux_fst x y 1 0 0 1]\n#align nat.xgcd_aux_val Nat.xgcdAux_val\n\ntheorem xgcd_val (x y) : xgcd x y = (gcdA x y, gcdB x y) := by\n  unfold gcdA gcdB; cases xgcd x y; rfl\n#align nat.xgcd_val Nat.xgcd_val\n\nsection\n\nvariable (x y : \u2115)\n\nprivate def P : \u2115 \u00d7 \u2124 \u00d7 \u2124 \u2192 Prop\n  | (r, s, t) => (r : \u2124) = x * s + y * t\n\ntheorem xgcdAux_P {r r'} :\n    \u2200 {s t s' t'}, P x y (r, s, t) \u2192 P x y (r', s', t') \u2192 P x y (xgcdAux r s t r' s' t') := by\n  induction r, r' using gcd.induction with\n  | H0 => simp\n  | H1 a b h IH =>\n    intro s t s' t' p p'\n    rw [xgcdAux_rec h]; refine' IH _ p; dsimp [P] at *\n    rw [Int.emod_def]; generalize (b / a : \u2124) = k\n    rw [p, p', mul_sub, sub_add_eq_add_sub, mul_sub, add_mul, mul_comm k t, mul_comm k s,\n      \u2190 mul_assoc, \u2190 mul_assoc, add_comm (x * s * k), \u2190 add_sub_assoc, sub_sub]\nset_option linter.uppercaseLean3 false in\n#align nat.xgcd_aux_P Nat.xgcdAux_P\n\n/-- **B\u00e9zout's lemma**: given `x y : \u2115`, `gcd x y = x * a + y * b`, where `a = gcd_a x y` and\n`b = gcd_b x y` are computed by the extended Euclidean algorithm.\n-/\ntheorem gcd_eq_gcd_ab : (gcd x y : \u2124) = x * gcdA x y + y * gcdB x y := by\n  have := @xgcdAux_P x y x y 1 0 0 1 (by simp [P]) (by simp [P])\n  rwa [xgcdAux_val, xgcd_val] at this\n#align nat.gcd_eq_gcd_ab Nat.gcd_eq_gcd_ab\n\nend\n\ntheorem exists_mul_emod_eq_gcd {k n : \u2115} (hk : gcd n k < k) : \u2203 m, n * m % k = gcd n k := by\n  have hk' := Int.ofNat_ne_zero.2 (ne_of_gt (lt_of_le_of_lt (zero_le (gcd n k)) hk))\n  have key := congr_arg (fun (m : \u2124) => (m % k).toNat) (gcd_eq_gcd_ab n k)\n  simp only at key\n  rw [Int.add_mul_emod_self_left, \u2190 Int.natCast_mod, Int.toNat_natCast, mod_eq_of_lt hk] at key\n  refine' \u27e8(n.gcdA k % k).toNat, Eq.trans (Int.ofNat.inj _) key.symm\u27e9\n  rw [Int.ofNat_eq_coe, Int.natCast_mod, Int.ofNat_mul, Int.toNat_of_nonneg (Int.emod_nonneg _ hk'),\n    Int.ofNat_eq_coe, Int.toNat_of_nonneg (Int.emod_nonneg _ hk'), Int.mul_emod, Int.emod_emod,\n    \u2190 Int.mul_emod]\n#align nat.exists_mul_mod_eq_gcd Nat.exists_mul_emod_eq_gcd\n\ntheorem exists_mul_emod_eq_one_of_coprime {k n : \u2115} (hkn : Coprime n k) (hk : 1 < k) :\n    \u2203 m, n * m % k = 1 :=\n  Exists.recOn (exists_mul_emod_eq_gcd (lt_of_le_of_lt (le_of_eq hkn) hk)) fun m hm \u21a6\n    \u27e8m, hm.trans hkn\u27e9\n#align nat.exists_mul_mod_eq_one_of_coprime Nat.exists_mul_emod_eq_one_of_coprime\n\nend Nat\n\n/-! ### Divisibility over \u2124 -/\n\n\nnamespace Int\n\ntheorem gcd_def (i j : \u2124) : gcd i j = Nat.gcd i.natAbs j.natAbs := rfl\n\nprotected theorem coe_nat_gcd (m n : \u2115) : Int.gcd \u2191m \u2191n = Nat.gcd m n :=\n  rfl\n#align int.coe_nat_gcd Int.coe_nat_gcd\n\n/-- The extended GCD `a` value in the equation `gcd x y = x * a + y * b`. -/\ndef gcdA : \u2124 \u2192 \u2124 \u2192 \u2124\n  | ofNat m, n => m.gcdA n.natAbs\n  | -[m+1], n => -m.succ.gcdA n.natAbs\n#align int.gcd_a Int.gcdA\n\n/-- The extended GCD `b` value in the equation `gcd x y = x * a + y * b`. -/\ndef gcdB : \u2124 \u2192 \u2124 \u2192 \u2124\n  | m, ofNat n => m.natAbs.gcdB n\n  | m, -[n+1] => -m.natAbs.gcdB n.succ\n#align int.gcd_b Int.gcdB\n\n/-- **B\u00e9zout's lemma** -/\ntheorem gcd_eq_gcd_ab : \u2200 x y : \u2124, (gcd x y : \u2124) = x * gcdA x y + y * gcdB x y\n  | (m : \u2115), (n : \u2115) => Nat.gcd_eq_gcd_ab _ _\n  | (m : \u2115), -[n+1] =>\n    show (_ : \u2124) = _ + -(n + 1) * -_ by rw [neg_mul_neg]; apply Nat.gcd_eq_gcd_ab\n  | -[m+1], (n : \u2115) =>\n    show (_ : \u2124) = -(m + 1) * -_ + _ by rw [neg_mul_neg]; apply Nat.gcd_eq_gcd_ab\n  | -[m+1], -[n+1] =>\n    show (_ : \u2124) = -(m + 1) * -_ + -(n + 1) * -_ by\n      rw [neg_mul_neg, neg_mul_neg]\n      apply Nat.gcd_eq_gcd_ab\n#align int.gcd_eq_gcd_ab Int.gcd_eq_gcd_ab\n\ntheorem natAbs_ediv (a b : \u2124) (H : b \u2223 a) : natAbs (a / b) = natAbs a / natAbs b := by\n  rcases Nat.eq_zero_or_pos (natAbs b) with (h | h)\n  \u00b7 rw [natAbs_eq_zero.1 h]\n    simp [Int.ediv_zero]\n  calc\n    natAbs (a / b) = natAbs (a / b) * 1 := by rw [mul_one]\n    _ = natAbs (a / b) * (natAbs b / natAbs b) := by rw [Nat.div_self h]\n    _ = natAbs (a / b) * natAbs b / natAbs b := by rw [Nat.mul_div_assoc _ dvd_rfl]\n    _ = natAbs (a / b * b) / natAbs b := by rw [natAbs_mul (a / b) b]\n    _ = natAbs a / natAbs b := by rw [Int.ediv_mul_cancel H]\n#align int.nat_abs_div Int.natAbs_ediv\n\n/-- special case of `mul_dvd_mul_iff_right` for `\u2124`.\nDuplicated here to keep simple imports for this file. -/\ntheorem dvd_of_mul_dvd_mul_left {i j k : \u2124} (k_non_zero : k \u2260 0) (H : k * i \u2223 k * j) : i \u2223 j :=\n  Dvd.elim H fun l H1 => by rw [mul_assoc] at H1; exact \u27e8_, mul_left_cancel\u2080 k_non_zero H1\u27e9\n#align int.dvd_of_mul_dvd_mul_left Int.dvd_of_mul_dvd_mul_left\n\n", "theoremStatement": "/-- special case of `mul_dvd_mul_iff_right` for `\u2124`.\nDuplicated here to keep simple imports for this file. -/\ntheorem dvd_of_mul_dvd_mul_right {i j k : \u2124} (k_non_zero : k \u2260 0) (H : i * k \u2223 j * k) : i \u2223 j ", "theoremName": "Int.dvd_of_mul_dvd_mul_right", "fileCreated": {"commit": "71723d8bddec4ec20864b84941e36ebb4827ac47", "date": "2022-12-23"}, "theoremCreated": {"commit": "8aa6d64b1a16e31ddbfa2b69e619b3e108b8fcb9", "date": "2022-12-21"}, "file": "mathlib/Mathlib/Data/Int/GCD.lean", "module": "Mathlib.Data.Int.GCD", "jsonFile": "Mathlib.Data.Int.GCD.jsonl", "positionMetadata": {"lineInFile": 230, "tokenPositionInFile": 8449, "theoremPositionInFile": 0}, 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+{"srcContext": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.CategoryTheory.Comma.StructuredArrow\nimport Mathlib.CategoryTheory.Groupoid\nimport Mathlib.CategoryTheory.PUnit\n\n#align_import category_theory.elements from \"leanprover-community/mathlib\"@\"8a318021995877a44630c898d0b2bc376fceef3b\"\n\n/-!\n# The category of elements\n\nThis file defines the category of elements, also known as (a special case of) the Grothendieck\nconstruction.\n\nGiven a functor `F : C \u2964 Type`, an object of `F.Elements` is a pair `(X : C, x : F.obj X)`.\nA morphism `(X, x) \u27f6 (Y, y)` is a morphism `f : X \u27f6 Y` in `C`, so `F.map f` takes `x` to `y`.\n\n## Implementation notes\n\nThis construction is equivalent to a special case of a comma construction, so this is mostly just a\nmore convenient API. We prove the equivalence in\n`CategoryTheory.CategoryOfElements.structuredArrowEquivalence`.\n\n## References\n* [Emily Riehl, *Category Theory in Context*, Section 2.4][riehl2017]\n* <https://en.wikipedia.org/wiki/Category_of_elements>\n* <https://ncatlab.org/nlab/show/category+of+elements>\n\n## Tags\ncategory of elements, Grothendieck construction, comma category\n-/\n\n\nnamespace CategoryTheory\n\nuniverse w v u\n\nvariable {C : Type u} [Category.{v} C]\n\n/-- The type of objects for the category of elements of a functor `F : C \u2964 Type`\nis a pair `(X : C, x : F.obj X)`.\n-/\ndef Functor.Elements (F : C \u2964 Type w) :=\n  \u03a3c : C, F.obj c\n#align category_theory.functor.elements CategoryTheory.Functor.Elements\n\n-- Porting note: added because Sigma.ext would be triggered automatically\nlemma Functor.Elements.ext {F : C \u2964 Type w} (x y : F.Elements) (h\u2081 : x.fst = y.fst)\n    (h\u2082 : F.map (eqToHom h\u2081) x.snd = y.snd) : x = y := by\n  cases x\n  cases y\n  cases h\u2081\n  simp only [eqToHom_refl, FunctorToTypes.map_id_apply] at h\u2082\n  simp [h\u2082]\n\n/-- The category structure on `F.Elements`, for `F : C \u2964 Type`.\n    A morphism `(X, x) \u27f6 (Y, y)` is a morphism `f : X \u27f6 Y` in `C`, so `F.map f` takes `x` to `y`.\n -/\ninstance categoryOfElements (F : C \u2964 Type w) : Category.{v} F.Elements where\n  Hom p q := { f : p.1 \u27f6 q.1 // (F.map f) p.2 = q.2 }\n  id p := \u27e8\ud835\udfd9 p.1, by aesop_cat\u27e9\n  comp {X Y Z} f g := \u27e8f.val \u226b g.val, by simp [f.2, g.2]\u27e9\n#align category_theory.category_of_elements CategoryTheory.categoryOfElements\n\nnamespace CategoryOfElements\n\n@[ext]\ntheorem ext (F : C \u2964 Type w) {x y : F.Elements} (f g : x \u27f6 y) (w : f.val = g.val) : f = g :=\n  Subtype.ext_val w\n#align category_theory.category_of_elements.ext CategoryTheory.CategoryOfElements.ext\n\n@[simp]\ntheorem comp_val {F : C \u2964 Type w} {p q r : F.Elements} {f : p \u27f6 q} {g : q \u27f6 r} :\n    (f \u226b g).val = f.val \u226b g.val :=\n  rfl\n#align category_theory.category_of_elements.comp_val CategoryTheory.CategoryOfElements.comp_val\n\n@[simp]\ntheorem id_val {F : C \u2964 Type w} {p : F.Elements} : (\ud835\udfd9 p : p \u27f6 p).val = \ud835\udfd9 p.1 :=\n  rfl\n#align category_theory.category_of_elements.id_val CategoryTheory.CategoryOfElements.id_val\n\n", "theoremStatement": "@[simp]\ntheorem map_snd {F : C \u2964 Type w} {p q : F.Elements} (f : p \u27f6 q) : (F.map f.val) p.2 = q.2 ", "theoremName": "CategoryTheory.CategoryOfElements.map_snd", "fileCreated": {"commit": "de2321d01fed2b8dfa8db6d498ef79d0a9faadfc", "date": "2023-03-31"}, "theoremCreated": {"commit": "1a8b123383eb596a9dad0ca3f4c0081bd4d990a1", "date": "2021-08-03"}, "file": "mathlib/Mathlib/CategoryTheory/Elements.lean", "module": "Mathlib.CategoryTheory.Elements", "jsonFile": "Mathlib.CategoryTheory.Elements.jsonl", "positionMetadata": {"lineInFile": 86, "tokenPositionInFile": 3012, 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"Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Opposite", "Mathlib.Tactic.Cases", "Mathlib.Combinatorics.Quiver.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", 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"Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Spread", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.AssertExists", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.Data.Prod.Basic", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Control.ULift", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", 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+{"srcContext": "/-\nCopyright (c) 2023 Anatole Dedecker. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anatole Dedecker, Etienne Marion\n-/\n\nimport Mathlib.Topology.Homeomorph\nimport Mathlib.Topology.StoneCech\nimport Mathlib.Topology.Filter\nimport Mathlib.Order.Filter.Cofinite\n\n/-!\n# Proper maps between topological spaces\n\nThis file develops the basic theory of proper maps between topological spaces. A map `f : X \u2192 Y`\nbetween two topological spaces is said to be **proper** if it is continuous and satisfies\nthe following equivalent conditions:\n1. `f` is closed and has compact fibers.\n2. `f` is **universally closed**, in the sense that for any topological space `Z`, the map\n  `Prod.map f id : X \u00d7 Z \u2192 Y \u00d7 Z` is closed.\n3. For any `\u2131 : Filter X`, all cluster points of `map f \u2131` are images by `f` of some cluster point\n  of `\u2131`.\n\nWe take 3 as the definition in `IsProperMap`, and we show the equivalence with 1, 2, and some\nother variations. We also show the usual characterization of proper maps to a locally compact\nHausdorff space as continuous maps such that preimages of compact sets are compact.\n\n## Main statements\n\n* `isProperMap_iff_ultrafilter`: characterization of proper maps in terms of limits of ultrafilters\n  instead of cluster points of filters.\n* `IsProperMap.pi_map`: any product of proper maps is proper.\n* `isProperMap_iff_isClosedMap_and_compact_fibers`: a map is proper if and only if it is\n  continuous, closed, and has compact fibers\n* `isProperMap_iff_isCompact_preimage`: a map to a locally compact Hausdorff space is proper if\n  and only if it is continuous and preimages of compact sets are compact.\n* `isProperMap_iff_universally_closed`: a map is proper if and only if it is continuous and\n  universally closed, in the sense of condition 2. above.\n\n## Implementation notes\n\nIn algebraic geometry, it is common to also ask that proper maps are *separated*, in the sense of\n[Stacks: definition OCY1](https://stacks.math.columbia.edu/tag/0CY1). We don't follow this\nconvention because it is unclear whether it would give the right notion in all cases, and in\nparticular for the theory of proper group actions. That means that our terminology does **NOT**\nalign with that of [Stacks: Characterizing proper maps](https://stacks.math.columbia.edu/tag/005M),\ninstead our definition of `IsProperMap` coincides with what they call \"Bourbaki-proper\".\n\nRegarding the proofs, we don't really follow Bourbaki and go for more filter-heavy proofs,\nas usual. In particular, their arguments rely heavily on restriction of closed maps (see\n`IsClosedMap.restrictPreimage`), which makes them somehow annoying to formalize in type theory.\nIn contrast, the filter-based proofs work really well thanks to the existing API.\n\nIn fact, the filter proofs work so well that I thought this would be a great pedagogical resource\nabout how we use filters. For that reason, **all interesting proofs in this file are commented**,\nso don't hesitate to have a look!\n\n## TODO\n\n* prove the equivalence with condition 3 of\n  [Stacks: Theorem 005R](https://stacks.math.columbia.edu/tag/005R). Note that they mean something\n  different by \"universally closed\".\n\n## References\n\n* [N. Bourbaki, *General Topology*][bourbaki1966]\n* [Stacks: Characterizing proper maps](https://stacks.math.columbia.edu/tag/005M)\n-/\n\nopen Filter Topology Function Set\nopen Prod (fst snd)\n\nvariable {X Y Z W \u03b9 : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]\n  [TopologicalSpace W] {f : X \u2192 Y} {g : Y \u2192 Z}\n\nuniverse u v\n\n/-- A map `f : X \u2192 Y` between two topological spaces is said to be **proper** if it is continuous\nand, for all `\u2131 : Filter X`, any cluster point of `map f \u2131` is the image by `f` of a cluster point\nof `\u2131`. -/\n@[mk_iff isProperMap_iff_clusterPt]\nstructure IsProperMap (f : X \u2192 Y) extends Continuous f : Prop where\n  /-- By definition, if `f` is a proper map and `\u2131` is any filter on `X`, then any cluster point of\n  `map f \u2131` is the image by `f` of some cluster point of `\u2131`. -/\n  clusterPt_of_mapClusterPt :\n    \u2200 \u2983\u2131 : Filter X\u2984, \u2200 \u2983y : Y\u2984, MapClusterPt y \u2131 f \u2192 \u2203 x, f x = y \u2227 ClusterPt x \u2131\n\n/-- Definition of proper maps. See also `isClosedMap_iff_clusterPt` for a related criterion\nfor closed maps. -/\nadd_decl_doc isProperMap_iff_clusterPt\n\n/-- By definition, a proper map is continuous. -/\n@[continuity, fun_prop]\nlemma IsProperMap.continuous (h : IsProperMap f) : Continuous f := h.toContinuous\n\n/-- A proper map is closed. -/\nlemma IsProperMap.isClosedMap (h : IsProperMap f) : IsClosedMap f := by\n  rw [isClosedMap_iff_clusterPt]\n  exact fun s y \u21a6 h.clusterPt_of_mapClusterPt (\u2131 := \ud835\udcdf s) (y := y)\n\n/-- Characterization of proper maps by ultrafilters. -/\nlemma isProperMap_iff_ultrafilter : IsProperMap f \u2194 Continuous f \u2227\n    \u2200 \u2983\ud835\udcb0 : Ultrafilter X\u2984, \u2200 \u2983y : Y\u2984, Tendsto f \ud835\udcb0 (\ud835\udcdd y) \u2192 \u2203 x, f x = y \u2227 \ud835\udcb0 \u2264 \ud835\udcdd x := by\n  -- This is morally trivial since ultrafilters give all the information about cluster points.\n  rw [isProperMap_iff_clusterPt]\n  refine and_congr_right (fun _ \u21a6 ?_)\n  constructor <;> intro H\n  \u00b7 intro \ud835\udcb0 y (hY : (Ultrafilter.map f \ud835\udcb0 : Filter Y) \u2264 _)\n    simp_rw [\u2190 Ultrafilter.clusterPt_iff] at hY \u22a2\n    exact H hY\n  \u00b7 simp_rw [MapClusterPt, ClusterPt, \u2190 Filter.push_pull', map_neBot_iff, \u2190 exists_ultrafilter_iff,\n      forall_exists_index]\n    intro \u2131 y \ud835\udcb0 hy\n    rcases H (tendsto_iff_comap.mpr <| hy.trans inf_le_left) with \u27e8x, hxy, hx\u27e9\n    exact \u27e8x, hxy, \ud835\udcb0, le_inf hx (hy.trans inf_le_right)\u27e9\n\nlemma isProperMap_iff_ultrafilter_of_t2 [T2Space Y] : IsProperMap f \u2194 Continuous f \u2227\n    \u2200 \u2983\ud835\udcb0 : Ultrafilter X\u2984, \u2200 \u2983y : Y\u2984, Tendsto f \ud835\udcb0 (\ud835\udcdd y) \u2192 \u2203 x, \ud835\udcb0.1 \u2264 \ud835\udcdd x :=\n  isProperMap_iff_ultrafilter.trans <| and_congr_right fun hc \u21a6 forall\u2083_congr fun _\ud835\udcb0 _y hy \u21a6\n    exists_congr fun x \u21a6 and_iff_right_of_imp fun h \u21a6\n      tendsto_nhds_unique ((hc.tendsto x).mono_left h) hy\n\n/-- If `f` is proper and converges to `y` along some ultrafilter `\ud835\udcb0`, then `\ud835\udcb0` converges to some\n`x` such that `f x = y`. -/\nlemma IsProperMap.ultrafilter_le_nhds_of_tendsto (h : IsProperMap f) \u2983\ud835\udcb0 : Ultrafilter X\u2984 \u2983y : Y\u2984\n    (hy : Tendsto f \ud835\udcb0 (\ud835\udcdd y)) : \u2203 x, f x = y \u2227 \ud835\udcb0 \u2264 \ud835\udcdd x :=\n  (isProperMap_iff_ultrafilter.mp h).2 hy\n\n/-- The composition of two proper maps is proper. -/\nlemma IsProperMap.comp (hf : IsProperMap f) (hg : IsProperMap g) :\n    IsProperMap (g \u2218 f) := by\n  refine \u27e8by continuity, fun \u2131 z h \u21a6 ?_\u27e9\n  rw [mapClusterPt_comp] at h\n  rcases hg.clusterPt_of_mapClusterPt h with \u27e8y, rfl, hy\u27e9\n  rcases hf.clusterPt_of_mapClusterPt hy with \u27e8x, rfl, hx\u27e9\n  use x, rfl\n\n\n/-- If the composition of two continuous functions `g \u2218 f` is proper and `f` is surjective,\nthen `g` is proper. -/\nlemma isProperMap_of_comp_of_surj (hf : Continuous f)\n    (hg : Continuous g) (hgf : IsProperMap (g \u2218 f)) (f_surj : f.Surjective) : IsProperMap g := by\n  refine \u27e8hg, fun \u2131 z h \u21a6 ?_\u27e9\n  rw [\u2190 \u2131.map_comap_of_surjective f_surj, \u2190 mapClusterPt_comp] at h\n  rcases hgf.clusterPt_of_mapClusterPt h with \u27e8x, rfl, hx\u27e9\n  rw [\u2190 \u2131.map_comap_of_surjective f_surj]\n  exact \u27e8f x, rfl, hx.map hf.continuousAt tendsto_map\u27e9\n\n/-- If the composition of two continuous functions `g \u2218 f` is proper and `g` is injective,\nthen `f` is proper. -/\nlemma isProperMap_of_comp_of_inj {f : X \u2192 Y} {g : Y \u2192 Z} (hf : Continuous f) (hg : Continuous g)\n    (hgf : IsProperMap (g \u2218 f)) (g_inj : g.Injective) : IsProperMap f := by\n  refine \u27e8hf, fun \u2131 y h \u21a6 ?_\u27e9\n  rcases hgf.clusterPt_of_mapClusterPt (h.map hg.continuousAt tendsto_map) with \u27e8x, hx1, hx2\u27e9\n  exact \u27e8x, g_inj hx1, hx2\u27e9\n\n/-- If the composition of two continuous functions `f : X \u2192 Y` and `g : Y \u2192 Z` is proper\nand `Y` is T2, then `f` is proper. -/\nlemma isProperMap_of_comp_of_t2 [T2Space Y] (hf : Continuous f) (hg : Continuous g)\n    (hgf : IsProperMap (g \u2218 f)) : IsProperMap f := by\n  rw [isProperMap_iff_ultrafilter_of_t2]\n  refine \u27e8hf, fun \ud835\udcb0 y h \u21a6 ?_\u27e9\n  rw [isProperMap_iff_ultrafilter] at hgf\n  rcases hgf.2 ((hg.tendsto y).comp h) with \u27e8x, -, hx\u27e9\n  exact \u27e8x, hx\u27e9\n\n/-- A binary product of proper maps is proper. -/\nlemma IsProperMap.prod_map {g : Z \u2192 W} (hf : IsProperMap f) (hg : IsProperMap g) :\n    IsProperMap (Prod.map f g) := by\n  simp_rw [isProperMap_iff_ultrafilter] at hf hg \u22a2\n  constructor\n  -- Continuity is clear.\n  \u00b7 exact hf.1.prod_map hg.1\n  -- Let `\ud835\udcb0 : Ultrafilter (X \u00d7 Z)`, and assume that `f \u00d7 g` tends to some `(y, w) : Y \u00d7 W`\n  -- along `\ud835\udcb0`.\n  \u00b7 intro \ud835\udcb0 \u27e8y, w\u27e9 hyw\n  -- That means that `f` tends to `y` along `map fst \ud835\udcb0` and `g` tends to `w` along `map snd \ud835\udcb0`.\n    simp_rw [nhds_prod_eq, tendsto_prod_iff'] at hyw\n  -- Thus, by properness of `f` and `g`, we get some `x : X` and `z : Z` such that `f x = y`,\n  -- `g z = w`, `map fst \ud835\udcb0` tends to  `x`, and `map snd \ud835\udcb0` tends to `y`.\n    rcases hf.2 (show Tendsto f (Ultrafilter.map fst \ud835\udcb0) (\ud835\udcdd y) by simpa using hyw.1) with\n      \u27e8x, hxy, hx\u27e9\n    rcases hg.2 (show Tendsto g (Ultrafilter.map snd \ud835\udcb0) (\ud835\udcdd w) by simpa using hyw.2) with\n      \u27e8z, hzw, hz\u27e9\n  -- By the properties of the product topology, that means that `\ud835\udcb0` tends to `(x, z)`,\n  -- which completes the proof since `(f \u00d7 g)(x, z) = (y, w)`.\n    refine \u27e8\u27e8x, z\u27e9, Prod.ext hxy hzw, ?_\u27e9\n    rw [nhds_prod_eq, le_prod]\n    exact \u27e8hx, hz\u27e9\n\n/-- Any product of proper maps is proper. -/\nlemma IsProperMap.pi_map {X Y : \u03b9 \u2192 Type*} [\u2200 i, TopologicalSpace (X i)]\n    [\u2200 i, TopologicalSpace (Y i)] {f : (i : \u03b9) \u2192 X i \u2192 Y i} (h : \u2200 i, IsProperMap (f i)) :\n    IsProperMap (fun (x : \u2200 i, X i) i \u21a6 f i (x i)) := by\n  simp_rw [isProperMap_iff_ultrafilter] at h \u22a2\n  constructor\n  -- Continuity is clear.\n  \u00b7 exact continuous_pi fun i \u21a6 (h i).1.comp (continuous_apply i)\n  -- Let `\ud835\udcb0 : Ultrafilter (\u03a0 i, X i)`, and assume that `\u03a0 i, f i` tends to some `y : \u03a0 i, Y i`\n  -- along `\ud835\udcb0`.\n  \u00b7 intro \ud835\udcb0 y hy\n  -- That means that each `f i` tends to `y i` along `map (eval i) \ud835\udcb0`.\n    have : \u2200 i, Tendsto (f i) (Ultrafilter.map (eval i) \ud835\udcb0) (\ud835\udcdd (y i)) := by\n      simpa [tendsto_pi_nhds] using hy\n  -- Thus, by properness of all the `f i`s, we can choose some `x : \u03a0 i, X i` such that, for all\n  -- `i`, `f i (x i) = y i` and `map (eval i) \ud835\udcb0` tends to  `x i`.\n    choose x hxy hx using fun i \u21a6 (h i).2 (this i)\n  -- By the properties of the product topology, that means that `\ud835\udcb0` tends to `x`,\n  -- which completes the proof since `(\u03a0 i, f i) x = y`.\n    refine \u27e8x, funext hxy, ?_\u27e9\n    rwa [nhds_pi, le_pi]\n\n/-- The preimage of a compact set by a proper map is again compact. See also\n`isProperMap_iff_isCompact_preimage` which proves that this property completely characterizes\nproper map when the codomain is locally compact and Hausdorff. -/\nlemma IsProperMap.isCompact_preimage (h : IsProperMap f) {K : Set Y} (hK : IsCompact K) :\n    IsCompact (f \u207b\u00b9' K) := by\n  rw [isCompact_iff_ultrafilter_le_nhds]\n  -- Let `\ud835\udcb0 \u2264 \ud835\udcdf (f \u207b\u00b9' K)` an ultrafilter.\n  intro \ud835\udcb0 h\ud835\udcb0\n  -- In other words, we have `map f \ud835\udcb0 \u2264 \ud835\udcdf K`\n  rw [\u2190 comap_principal, \u2190 map_le_iff_le_comap, \u2190 Ultrafilter.coe_map] at h\ud835\udcb0\n  -- Thus, by compactness of `K`, the ultrafilter `map f \ud835\udcb0` tends to some `y \u2208 K`.\n  rcases hK.ultrafilter_le_nhds _ h\ud835\udcb0 with \u27e8y, hyK, hy\u27e9\n  -- Then, by properness of `f`, that means that `\ud835\udcb0` tends to some `x \u2208 f \u207b\u00b9' {y} \u2286 f \u207b\u00b9' K`,\n  -- which completes the proof.\n  rcases h.ultrafilter_le_nhds_of_tendsto hy with \u27e8x, rfl, hx\u27e9\n  exact \u27e8x, hyK, hx\u27e9\n\n/-- A map is proper if and only if it is closed and its fibers are compact. -/\ntheorem isProperMap_iff_isClosedMap_and_compact_fibers :\n    IsProperMap f \u2194 Continuous f \u2227 IsClosedMap f \u2227 \u2200 y, IsCompact (f \u207b\u00b9' {y}) := by\n  constructor <;> intro H\n  -- Note: In Bourbaki, the direct implication is proved by going through universally closed maps.\n  -- We could do the same (using a `TFAE` cycle) but proving it directly from\n  -- `IsProperMap.isCompact_preimage` is nice enough already so we don't bother with that.\n  \u00b7 exact \u27e8H.continuous, H.isClosedMap, fun y \u21a6 H.isCompact_preimage isCompact_singleton\u27e9\n  \u00b7 rw [isProperMap_iff_clusterPt]\n  -- Let `\u2131 : Filter X` and `y` some cluster point of `map f \u2131`.\n    refine \u27e8H.1, fun \u2131 y hy \u21a6 ?_\u27e9\n  -- That means that the singleton `pure y` meets the \"closure\" of `map f \u2131`, by which we mean\n  -- `Filter.lift' (map f \u2131) closure`. But `f` is closed, so\n  -- `closure (map f \u2131) = map f (closure \u2131)` (see `IsClosedMap.lift'_closure_map_eq`).\n  -- Thus `map f (closure \u2131 \u2293 \ud835\udcdf (f \u207b\u00b9' {y})) = map f (closure \u2131) \u2293 \ud835\udcdf {y} \u2260 \u22a5`, hence\n  -- `closure \u2131 \u2293 \ud835\udcdf (f \u207b\u00b9' {y}) \u2260 \u22a5`.\n    rw [H.2.1.mapClusterPt_iff_lift'_closure H.1] at hy\n  -- Now, applying the compactness of `f \u207b\u00b9' {y}` to the nontrivial filter\n  -- `closure \u2131 \u2293 \ud835\udcdf (f \u207b\u00b9' {y})`, we obtain that it has a cluster point `x \u2208 f \u207b\u00b9' {y}`.\n    rcases H.2.2 y (f := Filter.lift' \u2131 closure \u2293 \ud835\udcdf (f \u207b\u00b9' {y})) inf_le_right with \u27e8x, hxy, hx\u27e9\n    refine \u27e8x, hxy, ?_\u27e9\n  -- In particular `x` is a cluster point of `closure \u2131`. Since cluster points of `closure \u2131`\n  -- are exactly cluster points of `\u2131` (see `clusterPt_lift'_closure_iff`), this completes\n  -- the proof.\n    rw [\u2190 clusterPt_lift'_closure_iff]\n    exact hx.mono inf_le_left\n\n/-- An injective and continuous function is proper if and only if it is closed. -/\nlemma isProperMap_iff_isClosedMap_of_inj (f_cont : Continuous f) (f_inj : f.Injective) :\n    IsProperMap f \u2194 IsClosedMap f := by\n  refine \u27e8fun h \u21a6 h.isClosedMap, fun h \u21a6 ?_\u27e9\n  rw [isProperMap_iff_isClosedMap_and_compact_fibers]\n  exact \u27e8f_cont, h, fun y \u21a6 (subsingleton_singleton.preimage f_inj).isCompact\u27e9\n\n", "theoremStatement": "/-- A injective continuous and closed map is proper. -/\nlemma isProperMap_of_isClosedMap_of_inj (f_cont : Continuous f) (f_inj : f.Injective)\n    (f_closed : IsClosedMap f) : IsProperMap f ", "theoremName": "isProperMap_of_isClosedMap_of_inj", "fileCreated": {"commit": 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"Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Cases", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Function.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Algebra.Ring.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Fin.OrderHom", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Embedding.Set", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Finset.Attr", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", 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"Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Util.AtomM", "Mathlib.Data.List.TFAE", 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+{"srcContext": "/-\nCopyright (c) 2024 Thomas Browning. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Thomas Browning\n-/\nimport Mathlib.GroupTheory.Perm.Cycle.Type\nimport Mathlib.Algebra.GroupPower.IterateHom\nimport Mathlib.GroupTheory.OrderOfElement\n\n/-!\n# Fixed-point-free automorphisms\n\nThis file defines fixed-point-free automorphisms and proves some basic properties.\n\nAn automorphism `\u03c6` of a group `G` is fixed-point-free if `1 : G` is the only fixed point of `\u03c6`.\n-/\n\nnamespace MonoidHom\n\nvariable {G : Type*}\n\nsection Definitions\n\nvariable (\u03c6 : G \u2192 G)\n\n/-- A function `\u03c6 : G \u2192 G` is fixed-point-free if `1 : G` is the only fixed point of `\u03c6`. -/\ndef FixedPointFree [One G] := \u2200 g, \u03c6 g = g \u2192 g = 1\n\n/-- The commutator map `g \u21a6 g / \u03c6 g`. If `\u03c6 g = h * g * h\u207b\u00b9`, then `g / \u03c6 g` is exactly the\n  commutator `[g, h] = g * h * g\u207b\u00b9 * h\u207b\u00b9`. -/\ndef commutatorMap [Div G] (g : G) := g / \u03c6 g\n\n@[simp] theorem commutatorMap_apply [Div G] (g : G) : commutatorMap \u03c6 g = g / \u03c6 g := rfl\n\nend Definitions\n\nnamespace FixedPointFree\n\n-- todo: refactor Mathlib/Algebra/GroupPower/IterateHom to generalize \u03c6 to MonoidHomClass\nvariable [Group G] {\u03c6 : G \u2192* G} (h\u03c6 : FixedPointFree \u03c6)\n\ntheorem commutatorMap_injective : Function.Injective (commutatorMap \u03c6) := by\n  refine' fun x y h \u21a6 inv_mul_eq_one.mp <| h\u03c6 _ _\n  rwa [map_mul, map_inv, eq_inv_mul_iff_mul_eq, \u2190 mul_assoc, \u2190 eq_div_iff_mul_eq', \u2190 division_def]\n\nvariable [Finite G]\n\ntheorem commutatorMap_surjective : Function.Surjective (commutatorMap \u03c6) :=\n  Finite.surjective_of_injective h\u03c6.commutatorMap_injective\n\ntheorem prod_pow_eq_one {n : \u2115} (hn : \u03c6^[n] = _root_.id) (g : G) :\n    ((List.range n).map (fun k \u21a6 \u03c6^[k] g)).prod = 1 := by\n  obtain \u27e8g, rfl\u27e9 := commutatorMap_surjective h\u03c6 g\n  simp only [commutatorMap_apply, iterate_map_div, \u2190 Function.iterate_succ_apply]\n  rw [List.prod_range_div', Function.iterate_zero_apply, hn, Function.id_def, div_self']\n\ntheorem coe_eq_inv_of_sq_eq_one (h2 : \u03c6^[2] = _root_.id) : \u21d1\u03c6 = (\u00b7\u207b\u00b9) := by\n  ext g\n  have key : 1 * g * \u03c6 g = 1 := h\u03c6.prod_pow_eq_one h2 g\n  rwa [one_mul, \u2190 inv_eq_iff_mul_eq_one, eq_comm] at key\n\nsection Involutive\n\nvariable (h2 : Function.Involutive \u03c6)\n\n", "theoremStatement": "theorem coe_eq_inv_of_involutive : \u21d1\u03c6 = (\u00b7\u207b\u00b9) ", "theoremName": "MonoidHom.FixedPointFree.coe_eq_inv_of_involutive", "fileCreated": {"commit": "8b3db4c313a6f4de7128342a47e0815b1d0c4bb6", "date": "2024-03-24"}, "theoremCreated": {"commit": "8b3db4c313a6f4de7128342a47e0815b1d0c4bb6", 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"Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Multiset.Fold", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.Group.Embedding", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Option", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Fintype.Perm", "Mathlib.Data.Int.ModEq", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Regular.SMul", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", 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"Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.Multiplicity", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n  coe_eq_inv_of_sq_eq_one h\u03c6  (funext h2)", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 44}}
+{"srcContext": "/-\nCopyright (c) 2024 Michael Stoll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Stoll\n-/\nimport Mathlib.Analysis.InnerProductSpace.Basic\nimport Mathlib.Analysis.Normed.Field.InfiniteSum\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.NumberTheory.LSeries.Convergence\n\n/-!\n# Dirichlet convolution of sequences and products of L-series\n\nWe define the *Dirichlet convolution* `f \u235f g` of two sequences `f g : \u2115 \u2192 R` with values in a\nsemiring `R` by `(f \u235f g) n = \u2211 (k * m = n), f k * g m` when `n \u2260 0` and `(f \u235f g) 0 = 0`.\nTechnically, this is done by transporting the existing definition for `ArithmeticFunction R`;\nsee `LSeries.convolution`. We show that these definitions agree (`LSeries.convolution_def`).\n\nWe then consider the case `R = \u2102` and show that `L (f \u235f g) = L f * L g` on the common domain\nof convergence of the L-series `L f`  and `L g` of `f` and `g`; see `LSeries_convolution`\nand `LSeries_convolution'`.\n-/\n\nopen scoped LSeries.notation\n\nopen Complex LSeries\n\n/-!\n### Dirichlet convolution of two functions\n-/\n\nopen BigOperators Nat\n\n/-- We turn any function `\u2115 \u2192 R` into an `ArithmeticFunction R` by setting its value at `0`\nto be zero. -/\ndef toArithmeticFunction {R : Type*} [Zero R] (f : \u2115 \u2192 R) : ArithmeticFunction R where\n  toFun n := if n = 0 then 0 else f n\n  map_zero' := rfl\n\nlemma toArithmeticFunction_congr {R : Type*} [Zero R] {f f' : \u2115 \u2192 R}\n    (h : \u2200 {n}, n \u2260 0 \u2192 f n = f' n) :\n    toArithmeticFunction f = toArithmeticFunction f' := by\n  ext \u27e8- | _\u27e9\n  \u00b7 simp only [zero_eq, ArithmeticFunction.map_zero]\n  \u00b7 simp only [toArithmeticFunction, ArithmeticFunction.coe_mk, succ_ne_zero, \u2193reduceIte,\n      ne_eq, not_false_eq_true, h]\n\n/-- If we consider an arithmetic function just as a function and turn it back into an\narithmetic function, it is the same as before. -/\n@[simp]\nlemma ArithmeticFunction.toArithmeticFunction_eq_self {R : Type*} [Zero R]\n    (f : ArithmeticFunction R) :\n    toArithmeticFunction f = f := by\n  ext n\n  simp (config := {contextual := true}) [toArithmeticFunction, ArithmeticFunction.map_zero]\n\n/-- Dirichlet convolution of two sequences.\n\nWe define this in terms of the already existing definition for arithmetic functions. -/\nnoncomputable def LSeries.convolution {R : Type*} [Semiring R] (f g : \u2115 \u2192 R) : \u2115 \u2192 R :=\n  \u21d1(toArithmeticFunction f * toArithmeticFunction g)\n\n@[inherit_doc]\nscoped[LSeries.notation] infixl:70 \" \u235f \" => LSeries.convolution\n\nlemma LSeries.convolution_congr {R : Type*} [Semiring R] {f f' g g' : \u2115 \u2192 R}\n    (hf : \u2200 {n}, n \u2260 0 \u2192 f n = f' n) (hg : \u2200 {n}, n \u2260 0 \u2192 g n = g' n) :\n    f \u235f g = f' \u235f g' := by\n  simp only [convolution, toArithmeticFunction_congr hf, toArithmeticFunction_congr hg]\n\n/-- The product of two arithmetic functions defines the same function as the Dirichlet convolution\nof the functions defined by them. -/\nlemma ArithmeticFunction.coe_mul {R : Type*} [Semiring R] (f g : ArithmeticFunction R) :\n    f \u235f g = \u21d1(f * g) := by\n  simp only [convolution, ArithmeticFunction.toArithmeticFunction_eq_self]\n\nnamespace LSeries\n\nlemma convolution_def {R : Type*} [Semiring R] (f g : \u2115 \u2192 R) :\n    f \u235f g = fun n \u21a6 \u2211 p in n.divisorsAntidiagonal, f p.1 * g p.2 := by\n  ext n\n  simp only [convolution, toArithmeticFunction, ArithmeticFunction.mul_apply,\n    ArithmeticFunction.coe_mk, mul_ite, mul_zero, ite_mul, zero_mul]\n  refine Finset.sum_congr rfl fun p hp \u21a6 ?_\n  obtain \u27e8h\u2081, h\u2082\u27e9 := ne_zero_of_mem_divisorsAntidiagonal hp\n  simp only [h\u2082, \u2193reduceIte, h\u2081]\n\n@[simp]\nlemma convolution_map_zero {R : Type*} [Semiring R] (f g : \u2115 \u2192 R) : (f \u235f g) 0 = 0 := by\n  simp only [convolution_def, divisorsAntidiagonal_zero, Finset.sum_empty]\n\n\n/-!\n### Multiplication of L-series\n-/\n\n/-- We give an expression of the `LSeries.term` of the convolution of two functions\nin terms of a sum over `Nat.divisorsAntidiagonal`. -/\nlemma term_convolution (f g : \u2115 \u2192 \u2102) (s : \u2102) (n : \u2115) :\n    term (f \u235f g) s n = \u2211 p in n.divisorsAntidiagonal, term f s p.1 * term g s p.2 := by\n  rcases eq_or_ne n 0 with rfl | hn\n  \u00b7 simp only [term_zero, divisorsAntidiagonal_zero, Finset.sum_empty]\n  -- now `n \u2260 0`\n  rw [term_of_ne_zero hn, convolution_def, Finset.sum_div]\n  refine Finset.sum_congr rfl fun p hp \u21a6 ?_\n  have \u27e8hp\u2081, hp\u2082\u27e9 := ne_zero_of_mem_divisorsAntidiagonal hp\n  rw [term_of_ne_zero hp\u2081 f s, term_of_ne_zero hp\u2082 g s, mul_comm_div, div_div, \u2190 mul_div_assoc,\n    \u2190 natCast_mul_natCast_cpow, \u2190 cast_mul, mul_comm p.2, (mem_divisorsAntidiagonal.mp hp).1]\n\nopen Set in\n/-- We give an expression of the `LSeries.term` of the convolution of two functions\nin terms of an a priori infinte sum over all pairs `(k, m)` with `k * m = n`\n(the set we sum over is infinite when `n = 0`). This is the version needed for the\nproof that `L (f \u235f g) = L f * L g`. -/\nlemma term_convolution' (f g : \u2115 \u2192 \u2102) (s : \u2102) :\n    term (f \u235f g) s = fun n \u21a6\n      \u2211' (b : (fun p : \u2115 \u00d7 \u2115 \u21a6 p.1 * p.2) \u207b\u00b9' {n}), term f s b.val.1 * term g s b.val.2 := by\n  ext n\n  rcases eq_or_ne n 0 with rfl | hn\n  \u00b7 -- show that both sides vanish when `n = 0`; this is the hardest part of the proof!\n    refine (term_zero ..).trans ?_\n    -- the right hand sum is over the union below, but in each term, one factor is always zero\n    have hS : (fun p \u21a6 p.1 * p.2) \u207b\u00b9' {0} = {0} \u00d7\u02e2 univ \u222a univ \u00d7\u02e2 {0} := by\n      ext\n      simp only [mem_preimage, mem_singleton_iff, Nat.mul_eq_zero, mem_union, mem_prod, mem_univ,\n        and_true, true_and]\n    have : \u2200 p : (fun p : \u2115 \u00d7 \u2115 \u21a6 p.1 * p.2) \u207b\u00b9' {0}, term f s p.val.1 * term g s p.val.2 = 0 := by\n      rintro \u27e8\u27e8p\u2081, p\u2082\u27e9, hp\u27e9\n      rcases hS \u25b8 hp with \u27e8rfl, -\u27e9 | \u27e8-, rfl\u27e9 <;> simp only [term_zero, zero_mul, mul_zero]\n    simp only [this, tsum_zero]\n  -- now `n \u2260 0`\n  rw [show (fun p : \u2115 \u00d7 \u2115 \u21a6 p.1 * p.2) \u207b\u00b9' {n} = n.divisorsAntidiagonal by ext; simp [hn],\n    Finset.tsum_subtype' n.divisorsAntidiagonal fun p \u21a6 term f s p.1 * term g s p.2,\n    term_convolution f g s n]\n\nend LSeries\n\nopen Set in\n/-- The L-series of the convolution product `f \u235f g` of two sequences `f` and `g`\nequals the product of their L-series, assuming both L-series converge. -/\nlemma LSeriesHasSum.convolution {f g : \u2115 \u2192 \u2102} {s a b : \u2102} (hf : LSeriesHasSum f s a)\n    (hg : LSeriesHasSum g s b) :\n    LSeriesHasSum (f \u235f g) s (a * b) := by\n  simp only [LSeriesHasSum, term_convolution']\n  have hsum := summable_mul_of_summable_norm hf.summable.norm hg.summable.norm\n  exact (HasSum.mul hf hg hsum).tsum_fiberwise (fun p \u21a6 p.1 * p.2)\n\n/-- The L-series of the convolution product `f \u235f g` of two sequences `f` and `g`\nequals the product of their L-series, assuming both L-series converge. -/\nlemma LSeries_convolution' {f g : \u2115 \u2192 \u2102} {s : \u2102} (hf : LSeriesSummable f s)\n    (hg : LSeriesSummable g s) :\n    LSeries (f \u235f g) s = LSeries f s * LSeries g s :=\n  (LSeriesHasSum.convolution hf.LSeriesHasSum hg.LSeriesHasSum).LSeries_eq\n\n/-- The L-series of the convolution product `f \u235f g` of two sequences `f` and `g`\nequals the product of their L-series in their common half-plane of absolute convergence. -/\nlemma LSeries_convolution {f g : \u2115 \u2192 \u2102} {s : \u2102}\n    (hf : abscissaOfAbsConv f < s.re) (hg : abscissaOfAbsConv g < s.re) :\n    LSeries (f \u235f g) s = LSeries f s * LSeries g s :=\n  LSeries_convolution' (LSeriesSummable_of_abscissaOfAbsConv_lt_re hf)\n    (LSeriesSummable_of_abscissaOfAbsConv_lt_re hg)\n\n/-- The L-series of the convolution product `f \u235f g` of two sequences `f` and `g`\nis summable when both L-series are summable. -/\nlemma LSeriesSummable.convolution {f g : \u2115 \u2192 \u2102} {s : \u2102} (hf : LSeriesSummable f s)\n    (hg : LSeriesSummable g s) :\n    LSeriesSummable (f \u235f g) s :=\n  (LSeriesHasSum.convolution hf.LSeriesHasSum hg.LSeriesHasSum).LSeriesSummable\n\nnamespace ArithmeticFunction\n\n/-!\n### Versions for arithmetic functions\n-/\n\n/-- The L-series of the (convolution) product of two `\u2102`-valued arithmetic functions `f` and `g`\nequals the product of their L-series, assuming both L-series converge. -/\nlemma LSeriesHasSum_mul {f g : ArithmeticFunction \u2102} {s a b : \u2102} (hf : LSeriesHasSum \u2197f s a)\n    (hg : LSeriesHasSum \u2197g s b) :\n    LSeriesHasSum \u2197(f * g) s (a * b) :=\n  coe_mul f g \u25b8 hf.convolution hg\n\n/-- The L-series of the (convolution) product of two `\u2102`-valued arithmetic functions `f` and `g`\nequals the product of their L-series, assuming both L-series converge. -/\nlemma LSeries_mul' {f g : ArithmeticFunction \u2102} {s : \u2102} (hf : LSeriesSummable \u2197f s)\n    (hg : LSeriesSummable \u2197g s) :\n    LSeries \u2197(f * g) s = LSeries \u2197f s * LSeries \u2197g s :=\n  coe_mul f g \u25b8 LSeries_convolution' hf hg\n\n/-- The L-series of the (convolution) product of two `\u2102`-valued arithmetic functions `f` and `g`\nequals the product of their L-series in their common half-plane of absolute convergence. -/\nlemma LSeries_mul {f g : ArithmeticFunction \u2102} {s : \u2102}\n    (hf : abscissaOfAbsConv \u2197f < s.re) (hg : abscissaOfAbsConv \u2197g < s.re) :\n    LSeries \u2197(f * g) s = LSeries \u2197f s * LSeries \u2197g s :=\n  coe_mul f g \u25b8 LSeries_convolution hf hg\n\n", "theoremStatement": "/-- The L-series of the (convolution) product of two `\u2102`-valued arithmetic functions `f` and `g`\nis summable when both L-series are summable. -/\nlemma LSeriesSummable_mul 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"Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n  coe_mul f g \u25b8 hf.convolution hg", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 36}}
+{"srcContext": "/-\nCopyright (c) 2021 R\u00e9my Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: R\u00e9my Degenne, S\u00e9bastien Gou\u00ebzel\n-/\nimport Mathlib.Analysis.NormedSpace.BoundedLinearMaps\nimport Mathlib.MeasureTheory.Measure.WithDensity\nimport Mathlib.MeasureTheory.Function.SimpleFuncDense\nimport Mathlib.Topology.Algebra.Module.FiniteDimension\n\n#align_import measure_theory.function.strongly_measurable.basic from \"leanprover-community/mathlib\"@\"3b52265189f3fb43aa631edffce5d060fafaf82f\"\n\n/-!\n# Strongly measurable and finitely strongly measurable functions\n\nA function `f` is said to be strongly measurable if `f` is the sequential limit of simple functions.\nIt is said to be finitely strongly measurable with respect to a measure `\u03bc` if the supports\nof those simple functions have finite measure. We also provide almost everywhere versions of\nthese notions.\n\nAlmost everywhere strongly measurable functions form the largest class of functions that can be\nintegrated using the Bochner integral.\n\nIf the target space has a second countable topology, strongly measurable and measurable are\nequivalent.\n\nIf the measure is sigma-finite, strongly measurable and finitely strongly measurable are equivalent.\n\nThe main property of finitely strongly measurable functions is\n`FinStronglyMeasurable.exists_set_sigmaFinite`: there exists a measurable set `t` such that the\nfunction is supported on `t` and `\u03bc.restrict t` is sigma-finite. As a consequence, we can prove some\nresults for those functions as if the measure was sigma-finite.\n\n## Main definitions\n\n* `StronglyMeasurable f`: `f : \u03b1 \u2192 \u03b2` is the limit of a sequence `fs : \u2115 \u2192 SimpleFunc \u03b1 \u03b2`.\n* `FinStronglyMeasurable f \u03bc`: `f : \u03b1 \u2192 \u03b2` is the limit of a sequence `fs : \u2115 \u2192 SimpleFunc \u03b1 \u03b2`\n  such that for all `n \u2208 \u2115`, the measure of the support of `fs n` is finite.\n* `AEStronglyMeasurable f \u03bc`: `f` is almost everywhere equal to a `StronglyMeasurable` function.\n* `AEFinStronglyMeasurable f \u03bc`: `f` is almost everywhere equal to a `FinStronglyMeasurable`\n  function.\n\n* `AEFinStronglyMeasurable.sigmaFiniteSet`: a measurable set `t` such that\n  `f =\u1d50[\u03bc.restrict t\u1d9c] 0` and `\u03bc.restrict t` is sigma-finite.\n\n## Main statements\n\n* `AEFinStronglyMeasurable.exists_set_sigmaFinite`: there exists a measurable set `t` such that\n  `f =\u1d50[\u03bc.restrict t\u1d9c] 0` and `\u03bc.restrict t` is sigma-finite.\n\nWe provide a solid API for strongly measurable functions, and for almost everywhere strongly\nmeasurable functions, as a basis for the Bochner integral.\n\n## References\n\n* Hyt\u00f6nen, Tuomas, Jan Van Neerven, Mark Veraar, and Lutz Weis. Analysis in Banach spaces.\n  Springer, 2016.\n\n-/\n\n\nopen MeasureTheory Filter TopologicalSpace Function Set MeasureTheory.Measure\n\nopen ENNReal Topology MeasureTheory NNReal BigOperators\n\nvariable {\u03b1 \u03b2 \u03b3 \u03b9 : Type*} [Countable \u03b9]\n\nnamespace MeasureTheory\n\nlocal infixr:25 \" \u2192\u209b \" => SimpleFunc\n\nsection Definitions\n\nvariable [TopologicalSpace \u03b2]\n\n/-- A function is `StronglyMeasurable` if it is the limit of simple functions. -/\ndef StronglyMeasurable [MeasurableSpace \u03b1] (f : \u03b1 \u2192 \u03b2) : Prop :=\n  \u2203 fs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2, \u2200 x, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\n#align measure_theory.strongly_measurable MeasureTheory.StronglyMeasurable\n\n/-- The notation for StronglyMeasurable giving the measurable space instance explicitly. -/\nscoped notation \"StronglyMeasurable[\" m \"]\" => @MeasureTheory.StronglyMeasurable _ _ _ m\n\n/-- A function is `FinStronglyMeasurable` with respect to a measure if it is the limit of simple\n  functions with support with finite measure. -/\ndef FinStronglyMeasurable [Zero \u03b2]\n    {_ : MeasurableSpace \u03b1} (f : \u03b1 \u2192 \u03b2) (\u03bc : Measure \u03b1 := by volume_tac) : Prop :=\n  \u2203 fs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2, (\u2200 n, \u03bc (support (fs n)) < \u221e) \u2227 \u2200 x, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))\n#align measure_theory.fin_strongly_measurable MeasureTheory.FinStronglyMeasurable\n\n/-- A function is `AEStronglyMeasurable` with respect to a measure `\u03bc` if it is almost everywhere\nequal to the limit of a sequence of simple functions. -/\ndef AEStronglyMeasurable\n    {_ : MeasurableSpace \u03b1} (f : \u03b1 \u2192 \u03b2) (\u03bc : Measure \u03b1 := by volume_tac) : Prop :=\n  \u2203 g, StronglyMeasurable g \u2227 f =\u1d50[\u03bc] g\n#align measure_theory.ae_strongly_measurable MeasureTheory.AEStronglyMeasurable\n\n/-- A function is `AEFinStronglyMeasurable` with respect to a measure if it is almost everywhere\nequal to the limit of a sequence of simple functions with support with finite measure. -/\ndef AEFinStronglyMeasurable\n    [Zero \u03b2] {_ : MeasurableSpace \u03b1} (f : \u03b1 \u2192 \u03b2) (\u03bc : Measure \u03b1 := by volume_tac) : Prop :=\n  \u2203 g, FinStronglyMeasurable g \u03bc \u2227 f =\u1d50[\u03bc] g\n#align measure_theory.ae_fin_strongly_measurable MeasureTheory.AEFinStronglyMeasurable\n\nend Definitions\n\nopen MeasureTheory\n\n/-! ## Strongly measurable functions -/\n\n@[aesop 30% apply (rule_sets := [Measurable])]\nprotected theorem StronglyMeasurable.aestronglyMeasurable {\u03b1 \u03b2} {_ : MeasurableSpace \u03b1}\n    [TopologicalSpace \u03b2] {f : \u03b1 \u2192 \u03b2} {\u03bc : Measure \u03b1} (hf : StronglyMeasurable f) :\n    AEStronglyMeasurable f \u03bc :=\n  \u27e8f, hf, EventuallyEq.refl _ _\u27e9\n#align measure_theory.strongly_measurable.ae_strongly_measurable MeasureTheory.StronglyMeasurable.aestronglyMeasurable\n\n@[simp]\ntheorem Subsingleton.stronglyMeasurable {\u03b1 \u03b2} [MeasurableSpace \u03b1] [TopologicalSpace \u03b2]\n    [Subsingleton \u03b2] (f : \u03b1 \u2192 \u03b2) : StronglyMeasurable f := by\n  let f_sf : \u03b1 \u2192\u209b \u03b2 := \u27e8f, fun x => ?_, Set.Subsingleton.finite Set.subsingleton_of_subsingleton\u27e9\n  \u00b7 exact \u27e8fun _ => f_sf, fun x => tendsto_const_nhds\u27e9\n  \u00b7 have h_univ : f \u207b\u00b9' {x} = Set.univ := by\n      ext1 y\n      simp [eq_iff_true_of_subsingleton]\n    rw [h_univ]\n    exact MeasurableSet.univ\n#align measure_theory.subsingleton.strongly_measurable MeasureTheory.Subsingleton.stronglyMeasurable\n\ntheorem SimpleFunc.stronglyMeasurable {\u03b1 \u03b2} {_ : MeasurableSpace \u03b1} [TopologicalSpace \u03b2]\n    (f : \u03b1 \u2192\u209b \u03b2) : StronglyMeasurable f :=\n  \u27e8fun _ => f, fun _ => tendsto_const_nhds\u27e9\n#align measure_theory.simple_func.strongly_measurable MeasureTheory.SimpleFunc.stronglyMeasurable\n\n@[nontriviality]\ntheorem StronglyMeasurable.of_finite [Finite \u03b1] {_ : MeasurableSpace \u03b1}\n    [MeasurableSingletonClass \u03b1] [TopologicalSpace \u03b2]\n    (f : \u03b1 \u2192 \u03b2) : StronglyMeasurable f :=\n  \u27e8fun _ => SimpleFunc.ofFinite f, fun _ => tendsto_const_nhds\u27e9\n\n@[deprecated] -- Since 2024/02/05\nalias stronglyMeasurable_of_fintype := StronglyMeasurable.of_finite\n\n@[deprecated StronglyMeasurable.of_finite]\ntheorem stronglyMeasurable_of_isEmpty [IsEmpty \u03b1] {_ : MeasurableSpace \u03b1} [TopologicalSpace \u03b2]\n    (f : \u03b1 \u2192 \u03b2) : StronglyMeasurable f :=\n  .of_finite f\n#align measure_theory.strongly_measurable_of_is_empty MeasureTheory.StronglyMeasurable.of_finite\n\ntheorem stronglyMeasurable_const {\u03b1 \u03b2} {_ : MeasurableSpace \u03b1} [TopologicalSpace \u03b2] {b : \u03b2} :\n    StronglyMeasurable fun _ : \u03b1 => b :=\n  \u27e8fun _ => SimpleFunc.const \u03b1 b, fun _ => tendsto_const_nhds\u27e9\n#align measure_theory.strongly_measurable_const MeasureTheory.stronglyMeasurable_const\n\n@[to_additive]\ntheorem stronglyMeasurable_one {\u03b1 \u03b2} {_ : MeasurableSpace \u03b1} [TopologicalSpace \u03b2] [One \u03b2] :\n    StronglyMeasurable (1 : \u03b1 \u2192 \u03b2) :=\n  stronglyMeasurable_const\n#align measure_theory.strongly_measurable_one MeasureTheory.stronglyMeasurable_one\n#align measure_theory.strongly_measurable_zero MeasureTheory.stronglyMeasurable_zero\n\n/-- A version of `stronglyMeasurable_const` that assumes `f x = f y` for all `x, y`.\nThis version works for functions between empty types. -/\ntheorem stronglyMeasurable_const' {\u03b1 \u03b2} {m : MeasurableSpace \u03b1} [TopologicalSpace \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (hf : \u2200 x y, f x = f y) : StronglyMeasurable f := by\n  nontriviality \u03b1\n  inhabit \u03b1\n  convert stronglyMeasurable_const (\u03b2 := \u03b2) using 1\n  exact funext fun x => hf x default\n#align measure_theory.strongly_measurable_const' MeasureTheory.stronglyMeasurable_const'\n\n-- Porting note: changed binding type of `MeasurableSpace \u03b1`.\n@[simp]\ntheorem Subsingleton.stronglyMeasurable' {\u03b1 \u03b2} [MeasurableSpace \u03b1] [TopologicalSpace \u03b2]\n    [Subsingleton \u03b1] (f : \u03b1 \u2192 \u03b2) : StronglyMeasurable f :=\n  stronglyMeasurable_const' fun x y => by rw [Subsingleton.elim x y]\n#align measure_theory.subsingleton.strongly_measurable' MeasureTheory.Subsingleton.stronglyMeasurable'\n\nnamespace StronglyMeasurable\n\nvariable {f g : \u03b1 \u2192 \u03b2}\n\nsection BasicPropertiesInAnyTopologicalSpace\n\nvariable [TopologicalSpace \u03b2]\n\n/-- A sequence of simple functions such that\n`\u2200 x, Tendsto (fun n => hf.approx n x) atTop (\ud835\udcdd (f x))`.\nThat property is given by `stronglyMeasurable.tendsto_approx`. -/\nprotected noncomputable def approx {_ : MeasurableSpace \u03b1} (hf : StronglyMeasurable f) :\n    \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 :=\n  hf.choose\n#align measure_theory.strongly_measurable.approx MeasureTheory.StronglyMeasurable.approx\n\nprotected theorem tendsto_approx {_ : MeasurableSpace \u03b1} (hf : StronglyMeasurable f) :\n    \u2200 x, Tendsto (fun n => hf.approx n x) atTop (\ud835\udcdd (f x)) :=\n  hf.choose_spec\n#align measure_theory.strongly_measurable.tendsto_approx MeasureTheory.StronglyMeasurable.tendsto_approx\n\n/-- Similar to `stronglyMeasurable.approx`, but enforces that the norm of every function in the\nsequence is less than `c` everywhere. If `\u2016f x\u2016 \u2264 c` this sequence of simple functions verifies\n`Tendsto (fun n => hf.approxBounded n x) atTop (\ud835\udcdd (f x))`. -/\nnoncomputable def approxBounded {_ : MeasurableSpace \u03b1} [Norm \u03b2] [SMul \u211d \u03b2]\n    (hf : StronglyMeasurable f) (c : \u211d) : \u2115 \u2192 SimpleFunc \u03b1 \u03b2 := fun n =>\n  (hf.approx n).map fun x => min 1 (c / \u2016x\u2016) \u2022 x\n#align measure_theory.strongly_measurable.approx_bounded MeasureTheory.StronglyMeasurable.approxBounded\n\ntheorem tendsto_approxBounded_of_norm_le {\u03b2} {f : \u03b1 \u2192 \u03b2} [NormedAddCommGroup \u03b2] [NormedSpace \u211d \u03b2]\n    {m : MeasurableSpace \u03b1} (hf : StronglyMeasurable[m] f) {c : \u211d} {x : \u03b1} (hfx : \u2016f x\u2016 \u2264 c) :\n    Tendsto (fun n => hf.approxBounded c n x) atTop (\ud835\udcdd (f x)) := by\n  have h_tendsto := hf.tendsto_approx x\n  simp only [StronglyMeasurable.approxBounded, SimpleFunc.coe_map, Function.comp_apply]\n  by_cases hfx0 : \u2016f x\u2016 = 0\n  \u00b7 rw [norm_eq_zero] at hfx0\n    rw [hfx0] at h_tendsto \u22a2\n    have h_tendsto_norm : Tendsto (fun n => \u2016hf.approx n x\u2016) atTop (\ud835\udcdd 0) := by\n      convert h_tendsto.norm\n      rw [norm_zero]\n    refine' squeeze_zero_norm (fun n => _) h_tendsto_norm\n    calc\n      \u2016min 1 (c / \u2016hf.approx n x\u2016) \u2022 hf.approx n x\u2016 =\n          \u2016min 1 (c / \u2016hf.approx n x\u2016)\u2016 * \u2016hf.approx n x\u2016 :=\n        norm_smul _ _\n      _ \u2264 \u2016(1 : \u211d)\u2016 * \u2016hf.approx n x\u2016 := by\n        refine' mul_le_mul_of_nonneg_right _ (norm_nonneg _)\n        rw [norm_one, Real.norm_of_nonneg]\n        \u00b7 exact min_le_left _ _\n        \u00b7 exact le_min zero_le_one (div_nonneg ((norm_nonneg _).trans hfx) (norm_nonneg _))\n      _ = \u2016hf.approx n x\u2016 := by rw [norm_one, one_mul]\n  rw [\u2190 one_smul \u211d (f x)]\n  refine' Tendsto.smul _ h_tendsto\n  have : min 1 (c / \u2016f x\u2016) = 1 := by\n    rw [min_eq_left_iff, one_le_div (lt_of_le_of_ne (norm_nonneg _) (Ne.symm hfx0))]\n    exact hfx\n  nth_rw 2 [this.symm]\n  refine' Tendsto.min tendsto_const_nhds _\n  exact Tendsto.div tendsto_const_nhds h_tendsto.norm hfx0\n#align measure_theory.strongly_measurable.tendsto_approx_bounded_of_norm_le MeasureTheory.StronglyMeasurable.tendsto_approxBounded_of_norm_le\n\ntheorem tendsto_approxBounded_ae {\u03b2} {f : \u03b1 \u2192 \u03b2} [NormedAddCommGroup \u03b2] [NormedSpace \u211d \u03b2]\n    {m m0 : MeasurableSpace \u03b1} {\u03bc : Measure \u03b1} (hf : StronglyMeasurable[m] f) {c : \u211d}\n    (hf_bound : \u2200\u1d50 x \u2202\u03bc, \u2016f x\u2016 \u2264 c) :\n    \u2200\u1d50 x \u2202\u03bc, Tendsto (fun n => hf.approxBounded c n x) atTop (\ud835\udcdd (f x)) := by\n  filter_upwards [hf_bound] with x hfx using tendsto_approxBounded_of_norm_le hf hfx\n#align measure_theory.strongly_measurable.tendsto_approx_bounded_ae MeasureTheory.StronglyMeasurable.tendsto_approxBounded_ae\n\ntheorem norm_approxBounded_le {\u03b2} {f : \u03b1 \u2192 \u03b2} [SeminormedAddCommGroup \u03b2] [NormedSpace \u211d \u03b2]\n    {m : MeasurableSpace \u03b1} {c : \u211d} (hf : StronglyMeasurable[m] f) (hc : 0 \u2264 c) (n : \u2115) (x : \u03b1) :\n    \u2016hf.approxBounded c n x\u2016 \u2264 c := by\n  simp only [StronglyMeasurable.approxBounded, SimpleFunc.coe_map, Function.comp_apply]\n  refine' (norm_smul_le _ _).trans _\n  by_cases h0 : \u2016hf.approx n x\u2016 = 0\n  \u00b7 simp only [h0, _root_.div_zero, min_eq_right, zero_le_one, norm_zero, mul_zero]\n    exact hc\n  rcases le_total \u2016hf.approx n x\u2016 c with h | h\n  \u00b7 rw [min_eq_left _]\n    \u00b7 simpa only [norm_one, one_mul] using h\n    \u00b7 rwa [one_le_div (lt_of_le_of_ne (norm_nonneg _) (Ne.symm h0))]\n  \u00b7 rw [min_eq_right _]\n    \u00b7 rw [norm_div, norm_norm, mul_comm, mul_div, div_eq_mul_inv, mul_comm, \u2190 mul_assoc,\n        inv_mul_cancel h0, one_mul, Real.norm_of_nonneg hc]\n    \u00b7 rwa [div_le_one (lt_of_le_of_ne (norm_nonneg _) (Ne.symm h0))]\n#align measure_theory.strongly_measurable.norm_approx_bounded_le MeasureTheory.StronglyMeasurable.norm_approxBounded_le\n\ntheorem _root_.stronglyMeasurable_bot_iff [Nonempty \u03b2] [T2Space \u03b2] :\n    StronglyMeasurable[\u22a5] f \u2194 \u2203 c, f = fun _ => c := by\n  cases' isEmpty_or_nonempty \u03b1 with h\u03b1 h\u03b1\n  \u00b7 simp only [@Subsingleton.stronglyMeasurable' _ _ \u22a5 _ _ f,\n      eq_iff_true_of_subsingleton, exists_const]\n  refine' \u27e8fun hf => _, fun hf_eq => _\u27e9\n  \u00b7 refine' \u27e8f h\u03b1.some, _\u27e9\n    let fs := hf.approx\n    have h_fs_tendsto : \u2200 x, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x)) := hf.tendsto_approx\n    have : \u2200 n, \u2203 c, \u2200 x, fs n x = c := fun n => SimpleFunc.simpleFunc_bot (fs n)\n    let cs n := (this n).choose\n    have h_cs_eq : \u2200 n, \u21d1(fs n) = fun _ => cs n := fun n => funext (this n).choose_spec\n    conv at h_fs_tendsto => enter [x, 1, n]; rw [h_cs_eq]\n    have h_tendsto : Tendsto cs atTop (\ud835\udcdd (f h\u03b1.some)) := h_fs_tendsto h\u03b1.some\n    ext1 x\n    exact tendsto_nhds_unique (h_fs_tendsto x) h_tendsto\n  \u00b7 obtain \u27e8c, rfl\u27e9 := hf_eq\n    exact stronglyMeasurable_const\n#align strongly_measurable_bot_iff stronglyMeasurable_bot_iff\n\nend BasicPropertiesInAnyTopologicalSpace\n\ntheorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace \u03b2] [Zero \u03b2]\n    {m : MeasurableSpace \u03b1} {\u03bc : Measure \u03b1} (hf_meas : StronglyMeasurable f) {t : Set \u03b1}\n    (ht : MeasurableSet t) (hft_zero : \u2200 x \u2208 t\u1d9c, f x = 0) (ht\u03bc : SigmaFinite (\u03bc.restrict t)) :\n    FinStronglyMeasurable f \u03bc := by\n  haveI : SigmaFinite (\u03bc.restrict t) := ht\u03bc\n  let S := spanningSets (\u03bc.restrict t)\n  have hS_meas : \u2200 n, MeasurableSet (S n) := measurable_spanningSets (\u03bc.restrict t)\n  let f_approx := hf_meas.approx\n  let fs n := SimpleFunc.restrict (f_approx n) (S n \u2229 t)\n  have h_fs_t_compl : \u2200 n, \u2200 x, x \u2209 t \u2192 fs n x = 0 := by\n    intro n x hxt\n    rw [SimpleFunc.restrict_apply _ ((hS_meas n).inter ht)]\n    refine' Set.indicator_of_not_mem _ _\n    simp [hxt]\n  refine' \u27e8fs, _, fun x => _\u27e9\n  \u00b7 simp_rw [SimpleFunc.support_eq]\n    refine' fun n => (measure_biUnion_finset_le _ _).trans_lt _\n    refine' ENNReal.sum_lt_top_iff.mpr fun y hy => _\n    rw [SimpleFunc.restrict_preimage_singleton _ ((hS_meas n).inter ht)]\n    swap\n    \u00b7 letI : (y : \u03b2) \u2192 Decidable (y = 0) := fun y => Classical.propDecidable _\n      rw [Finset.mem_filter] at hy\n      exact hy.2\n    refine' (measure_mono (Set.inter_subset_left _ _)).trans_lt _\n    have h_lt_top := measure_spanningSets_lt_top (\u03bc.restrict t) n\n    rwa [Measure.restrict_apply' ht] at h_lt_top\n  \u00b7 by_cases hxt : x \u2208 t\n    swap\n    \u00b7 rw [funext fun n => h_fs_t_compl n x hxt, hft_zero x hxt]\n      exact tendsto_const_nhds\n    have h : Tendsto (fun n => (f_approx n) x) atTop (\ud835\udcdd (f x)) := hf_meas.tendsto_approx x\n    obtain \u27e8n\u2081, hn\u2081\u27e9 : \u2203 n, \u2200 m, n \u2264 m \u2192 fs m x = f_approx m x := by\n      obtain \u27e8n, hn\u27e9 : \u2203 n, \u2200 m, n \u2264 m \u2192 x \u2208 S m \u2229 t := by\n        rsuffices \u27e8n, hn\u27e9 : \u2203 n, \u2200 m, n \u2264 m \u2192 x \u2208 S m\n        \u00b7 exact \u27e8n, fun m hnm => Set.mem_inter (hn m hnm) hxt\u27e9\n        rsuffices \u27e8n, hn\u27e9 : \u2203 n, x \u2208 S n\n        \u00b7 exact \u27e8n, fun m hnm => monotone_spanningSets (\u03bc.restrict t) hnm hn\u27e9\n        rw [\u2190 Set.mem_iUnion, iUnion_spanningSets (\u03bc.restrict t)]\n        trivial\n      refine' \u27e8n, fun m hnm => _\u27e9\n      simp_rw [fs, SimpleFunc.restrict_apply _ ((hS_meas m).inter ht),\n        Set.indicator_of_mem (hn m hnm)]\n    rw [tendsto_atTop'] at h \u22a2\n    intro s hs\n    obtain \u27e8n\u2082, hn\u2082\u27e9 := h s hs\n    refine' \u27e8max n\u2081 n\u2082, fun m hm => _\u27e9\n    rw [hn\u2081 m ((le_max_left _ _).trans hm.le)]\n    exact hn\u2082 m ((le_max_right _ _).trans hm.le)\n#align measure_theory.strongly_measurable.fin_strongly_measurable_of_set_sigma_finite MeasureTheory.StronglyMeasurable.finStronglyMeasurable_of_set_sigmaFinite\n\n/-- If the measure is sigma-finite, all strongly measurable functions are\n  `FinStronglyMeasurable`. -/\n@[aesop 5% apply (rule_sets := [Measurable])]\nprotected theorem finStronglyMeasurable [TopologicalSpace \u03b2] [Zero \u03b2] {m0 : MeasurableSpace \u03b1}\n    (hf : StronglyMeasurable f) (\u03bc : Measure \u03b1) [SigmaFinite \u03bc] : FinStronglyMeasurable f \u03bc :=\n  hf.finStronglyMeasurable_of_set_sigmaFinite MeasurableSet.univ (by simp)\n    (by rwa [Measure.restrict_univ])\n#align measure_theory.strongly_measurable.fin_strongly_measurable MeasureTheory.StronglyMeasurable.finStronglyMeasurable\n\n/-- A strongly measurable function is measurable. -/\n@[aesop 5% apply (rule_sets := [Measurable])]\nprotected theorem measurable {_ : MeasurableSpace \u03b1} [TopologicalSpace \u03b2] [PseudoMetrizableSpace \u03b2]\n    [MeasurableSpace \u03b2] [BorelSpace \u03b2] (hf : StronglyMeasurable f) : Measurable f :=\n  measurable_of_tendsto_metrizable (fun n => (hf.approx n).measurable)\n    (tendsto_pi_nhds.mpr hf.tendsto_approx)\n#align measure_theory.strongly_measurable.measurable MeasureTheory.StronglyMeasurable.measurable\n\n/-- A strongly measurable function is almost everywhere measurable. -/\n@[aesop 5% apply (rule_sets := [Measurable])]\nprotected theorem aemeasurable {_ : MeasurableSpace \u03b1} [TopologicalSpace \u03b2]\n    [PseudoMetrizableSpace \u03b2] [MeasurableSpace \u03b2] [BorelSpace \u03b2] {\u03bc : Measure \u03b1}\n    (hf : StronglyMeasurable f) : AEMeasurable f \u03bc :=\n  hf.measurable.aemeasurable\n#align measure_theory.strongly_measurable.ae_measurable MeasureTheory.StronglyMeasurable.aemeasurable\n\ntheorem _root_.Continuous.comp_stronglyMeasurable {_ : MeasurableSpace \u03b1} [TopologicalSpace \u03b2]\n    [TopologicalSpace \u03b3] {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : Continuous g) (hf : StronglyMeasurable f) :\n    StronglyMeasurable fun x => g (f x) :=\n  \u27e8fun n => SimpleFunc.map g (hf.approx n), fun x => (hg.tendsto _).comp (hf.tendsto_approx x)\u27e9\n#align continuous.comp_strongly_measurable Continuous.comp_stronglyMeasurable\n\n@[to_additive]\nnonrec theorem measurableSet_mulSupport {m : MeasurableSpace \u03b1} [One \u03b2] [TopologicalSpace \u03b2]\n    [MetrizableSpace \u03b2] (hf : StronglyMeasurable f) : MeasurableSet (mulSupport f) := by\n  borelize \u03b2\n  exact measurableSet_mulSupport hf.measurable\n#align measure_theory.strongly_measurable.measurable_set_mul_support MeasureTheory.StronglyMeasurable.measurableSet_mulSupport\n#align measure_theory.strongly_measurable.measurable_set_support MeasureTheory.StronglyMeasurable.measurableSet_support\n\nprotected theorem mono {m m' : MeasurableSpace \u03b1} [TopologicalSpace \u03b2]\n    (hf : StronglyMeasurable[m'] f) (h_mono : m' \u2264 m) : StronglyMeasurable[m] f := by\n  let f_approx : \u2115 \u2192 @SimpleFunc \u03b1 m \u03b2 := fun n =>\n    @SimpleFunc.mk \u03b1 m \u03b2\n      (hf.approx n)\n      (fun x => h_mono _ (SimpleFunc.measurableSet_fiber' _ x))\n      (SimpleFunc.finite_range (hf.approx n))\n  exact \u27e8f_approx, hf.tendsto_approx\u27e9\n#align measure_theory.strongly_measurable.mono MeasureTheory.StronglyMeasurable.mono\n\nprotected theorem prod_mk {m : MeasurableSpace \u03b1} [TopologicalSpace \u03b2] [TopologicalSpace \u03b3]\n    {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b3} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :\n    StronglyMeasurable fun x => (f x, g x) := by\n  refine' \u27e8fun n => SimpleFunc.pair (hf.approx n) (hg.approx n), fun x => _\u27e9\n  rw [nhds_prod_eq]\n  exact Tendsto.prod_mk (hf.tendsto_approx x) (hg.tendsto_approx x)\n#align measure_theory.strongly_measurable.prod_mk MeasureTheory.StronglyMeasurable.prod_mk\n\ntheorem comp_measurable [TopologicalSpace \u03b2] {_ : MeasurableSpace \u03b1} {_ : MeasurableSpace \u03b3}\n    {f : \u03b1 \u2192 \u03b2} {g : \u03b3 \u2192 \u03b1} (hf : StronglyMeasurable f) (hg : Measurable g) :\n    StronglyMeasurable (f \u2218 g) :=\n  \u27e8fun n => SimpleFunc.comp (hf.approx n) g hg, fun x => hf.tendsto_approx (g x)\u27e9\n#align measure_theory.strongly_measurable.comp_measurable MeasureTheory.StronglyMeasurable.comp_measurable\n\ntheorem of_uncurry_left [TopologicalSpace \u03b2] {_ : MeasurableSpace \u03b1} {_ : MeasurableSpace \u03b3}\n    {f : \u03b1 \u2192 \u03b3 \u2192 \u03b2} (hf : StronglyMeasurable (uncurry f)) {x : \u03b1} : StronglyMeasurable (f x) :=\n  hf.comp_measurable measurable_prod_mk_left\n#align measure_theory.strongly_measurable.of_uncurry_left MeasureTheory.StronglyMeasurable.of_uncurry_left\n\ntheorem of_uncurry_right [TopologicalSpace \u03b2] {_ : MeasurableSpace \u03b1} {_ : MeasurableSpace \u03b3}\n    {f : \u03b1 \u2192 \u03b3 \u2192 \u03b2} (hf : StronglyMeasurable (uncurry f)) {y : \u03b3} :\n    StronglyMeasurable fun x => f x y :=\n  hf.comp_measurable measurable_prod_mk_right\n#align measure_theory.strongly_measurable.of_uncurry_right MeasureTheory.StronglyMeasurable.of_uncurry_right\n\nsection Arithmetic\n\nvariable {m\u03b1 : MeasurableSpace \u03b1} [TopologicalSpace \u03b2]\n\n@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]\nprotected theorem mul [Mul \u03b2] [ContinuousMul \u03b2] (hf : StronglyMeasurable f)\n    (hg : StronglyMeasurable g) : StronglyMeasurable (f * g) :=\n  \u27e8fun n => hf.approx n * hg.approx n, fun x => (hf.tendsto_approx x).mul (hg.tendsto_approx x)\u27e9\n#align measure_theory.strongly_measurable.mul MeasureTheory.StronglyMeasurable.mul\n#align measure_theory.strongly_measurable.add MeasureTheory.StronglyMeasurable.add\n\n@[to_additive (attr := measurability)]\ntheorem mul_const [Mul \u03b2] [ContinuousMul \u03b2] (hf : StronglyMeasurable f) (c : \u03b2) :\n    StronglyMeasurable fun x => f x * c :=\n  hf.mul stronglyMeasurable_const\n#align measure_theory.strongly_measurable.mul_const MeasureTheory.StronglyMeasurable.mul_const\n#align measure_theory.strongly_measurable.add_const MeasureTheory.StronglyMeasurable.add_const\n\n@[to_additive (attr := measurability)]\ntheorem const_mul [Mul \u03b2] [ContinuousMul \u03b2] (hf : StronglyMeasurable f) (c : \u03b2) :\n    StronglyMeasurable fun x => c * f x :=\n  stronglyMeasurable_const.mul hf\n#align measure_theory.strongly_measurable.const_mul MeasureTheory.StronglyMeasurable.const_mul\n#align measure_theory.strongly_measurable.const_add MeasureTheory.StronglyMeasurable.const_add\n\n@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable])) const_nsmul]\nprotected theorem pow [Monoid \u03b2] [ContinuousMul \u03b2] (hf : StronglyMeasurable f) (n : \u2115) :\n    StronglyMeasurable (f ^ n) :=\n  \u27e8fun k => hf.approx k ^ n, fun x => (hf.tendsto_approx x).pow n\u27e9\n\n@[to_additive (attr := measurability)]\nprotected theorem inv [Inv \u03b2] [ContinuousInv \u03b2] (hf : StronglyMeasurable f) :\n    StronglyMeasurable f\u207b\u00b9 :=\n  \u27e8fun n => (hf.approx n)\u207b\u00b9, fun x => (hf.tendsto_approx x).inv\u27e9\n#align measure_theory.strongly_measurable.inv MeasureTheory.StronglyMeasurable.inv\n#align measure_theory.strongly_measurable.neg MeasureTheory.StronglyMeasurable.neg\n\n@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]\nprotected theorem div [Div \u03b2] [ContinuousDiv \u03b2] (hf : StronglyMeasurable f)\n    (hg : StronglyMeasurable g) : StronglyMeasurable (f / g) :=\n  \u27e8fun n => hf.approx n / hg.approx n, fun x => (hf.tendsto_approx x).div' (hg.tendsto_approx x)\u27e9\n#align measure_theory.strongly_measurable.div MeasureTheory.StronglyMeasurable.div\n#align measure_theory.strongly_measurable.sub MeasureTheory.StronglyMeasurable.sub\n\n@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]\nprotected theorem smul {\ud835\udd5c} [TopologicalSpace \ud835\udd5c] [SMul \ud835\udd5c \u03b2] [ContinuousSMul \ud835\udd5c \u03b2] {f : \u03b1 \u2192 \ud835\udd5c}\n    {g : \u03b1 \u2192 \u03b2} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :\n    StronglyMeasurable fun x => f x \u2022 g x :=\n  continuous_smul.comp_stronglyMeasurable (hf.prod_mk hg)\n#align measure_theory.strongly_measurable.smul MeasureTheory.StronglyMeasurable.smul\n#align measure_theory.strongly_measurable.vadd MeasureTheory.StronglyMeasurable.vadd\n\n@[to_additive (attr := measurability)]\nprotected theorem const_smul {\ud835\udd5c} [SMul \ud835\udd5c \u03b2] [ContinuousConstSMul \ud835\udd5c \u03b2] (hf : StronglyMeasurable f)\n    (c : \ud835\udd5c) : StronglyMeasurable (c \u2022 f) :=\n  \u27e8fun n => c \u2022 hf.approx n, fun x => (hf.tendsto_approx x).const_smul c\u27e9\n#align measure_theory.strongly_measurable.const_smul MeasureTheory.StronglyMeasurable.const_smul\n\n@[to_additive (attr := measurability)]\nprotected theorem const_smul' {\ud835\udd5c} [SMul \ud835\udd5c \u03b2] [ContinuousConstSMul \ud835\udd5c \u03b2] (hf : StronglyMeasurable f)\n    (c : \ud835\udd5c) : StronglyMeasurable fun x => c \u2022 f x :=\n  hf.const_smul c\n#align measure_theory.strongly_measurable.const_smul' MeasureTheory.StronglyMeasurable.const_smul'\n\n@[to_additive (attr := measurability)]\nprotected theorem smul_const {\ud835\udd5c} [TopologicalSpace \ud835\udd5c] [SMul \ud835\udd5c \u03b2] [ContinuousSMul \ud835\udd5c \u03b2] {f : \u03b1 \u2192 \ud835\udd5c}\n    (hf : StronglyMeasurable f) (c : \u03b2) : StronglyMeasurable fun x => f x \u2022 c :=\n  continuous_smul.comp_stronglyMeasurable (hf.prod_mk stronglyMeasurable_const)\n#align measure_theory.strongly_measurable.smul_const MeasureTheory.StronglyMeasurable.smul_const\n#align measure_theory.strongly_measurable.vadd_const MeasureTheory.StronglyMeasurable.vadd_const\n\n/-- In a normed vector space, the addition of a measurable function and a strongly measurable\nfunction is measurable. Note that this is not true without further second-countability assumptions\nfor the addition of two measurable functions. -/\ntheorem _root_.Measurable.add_stronglyMeasurable\n    {\u03b1 E : Type*} {_ : MeasurableSpace \u03b1} [AddGroup E] [TopologicalSpace E]\n    [MeasurableSpace E] [BorelSpace E] [ContinuousAdd E] [PseudoMetrizableSpace E]\n    {g f : \u03b1 \u2192 E} (hg : Measurable g) (hf : StronglyMeasurable f) :\n    Measurable (g + f) := by\n  rcases hf with \u27e8\u03c6, h\u03c6\u27e9\n  have : Tendsto (fun n x \u21a6 g x + \u03c6 n x) atTop (\ud835\udcdd (g + f)) :=\n    tendsto_pi_nhds.2 (fun x \u21a6 tendsto_const_nhds.add (h\u03c6 x))\n  apply measurable_of_tendsto_metrizable (fun n \u21a6 ?_) this\n  exact hg.add_simpleFunc _\n\n/-- In a normed vector space, the subtraction of a measurable function and a strongly measurable\nfunction is measurable. Note that this is not true without further second-countability assumptions\nfor the subtraction of two measurable functions. -/\ntheorem _root_.Measurable.sub_stronglyMeasurable\n    {\u03b1 E : Type*} {_ : MeasurableSpace \u03b1} [AddCommGroup E] [TopologicalSpace E]\n    [MeasurableSpace E] [BorelSpace E] [ContinuousAdd E] [ContinuousNeg E] [PseudoMetrizableSpace E]\n    {g f : \u03b1 \u2192 E} (hg : Measurable g) (hf : StronglyMeasurable f) :\n    Measurable (g - f) := by\n  rw [sub_eq_add_neg]\n  exact hg.add_stronglyMeasurable hf.neg\n\n/-- In a normed vector space, the addition of a strongly measurable function and a measurable\nfunction is measurable. Note that this is not true without further second-countability assumptions\nfor the addition of two measurable functions. -/\ntheorem _root_.Measurable.stronglyMeasurable_add\n    {\u03b1 E : Type*} {_ : MeasurableSpace \u03b1} [AddGroup E] [TopologicalSpace E]\n    [MeasurableSpace E] [BorelSpace E] [ContinuousAdd E] [PseudoMetrizableSpace E]\n    {g f : \u03b1 \u2192 E} (hg : Measurable g) (hf : StronglyMeasurable f) :\n    Measurable (f + g) := by\n  rcases hf with \u27e8\u03c6, h\u03c6\u27e9\n  have : Tendsto (fun n x \u21a6 \u03c6 n x + g x) atTop (\ud835\udcdd (f + g)) :=\n    tendsto_pi_nhds.2 (fun x \u21a6 (h\u03c6 x).add tendsto_const_nhds)\n  apply measurable_of_tendsto_metrizable (fun n \u21a6 ?_) this\n  exact hg.simpleFunc_add _\n\nend Arithmetic\n\nsection MulAction\n\nvariable {M G G\u2080 : Type*}\nvariable [TopologicalSpace \u03b2]\nvariable [Monoid M] [MulAction M \u03b2] [ContinuousConstSMul M \u03b2]\nvariable [Group G] [MulAction G \u03b2] [ContinuousConstSMul G \u03b2]\nvariable [GroupWithZero G\u2080] [MulAction G\u2080 \u03b2] [ContinuousConstSMul G\u2080 \u03b2]\n\ntheorem _root_.stronglyMeasurable_const_smul_iff {m : MeasurableSpace \u03b1} (c : G) :\n    (StronglyMeasurable fun x => c \u2022 f x) \u2194 StronglyMeasurable f :=\n  \u27e8fun h => by simpa only [inv_smul_smul] using h.const_smul' c\u207b\u00b9, fun h => h.const_smul c\u27e9\n#align strongly_measurable_const_smul_iff stronglyMeasurable_const_smul_iff\n\nnonrec theorem _root_.IsUnit.stronglyMeasurable_const_smul_iff {_ : MeasurableSpace \u03b1} {c : M}\n    (hc : IsUnit c) :\n    (StronglyMeasurable fun x => c \u2022 f x) \u2194 StronglyMeasurable f :=\n  let \u27e8u, hu\u27e9 := hc\n  hu \u25b8 stronglyMeasurable_const_smul_iff u\n#align is_unit.strongly_measurable_const_smul_iff IsUnit.stronglyMeasurable_const_smul_iff\n\ntheorem _root_.stronglyMeasurable_const_smul_iff\u2080 {_ : MeasurableSpace \u03b1} {c : G\u2080} (hc : c \u2260 0) :\n    (StronglyMeasurable fun x => c \u2022 f x) \u2194 StronglyMeasurable f :=\n  (IsUnit.mk0 _ hc).stronglyMeasurable_const_smul_iff\n#align strongly_measurable_const_smul_iff\u2080 stronglyMeasurable_const_smul_iff\u2080\n\nend MulAction\n\nsection Order\n\nvariable [MeasurableSpace \u03b1] [TopologicalSpace \u03b2]\n\nopen Filter\n\nopen Filter\n\n@[aesop safe 20 (rule_sets := [Measurable])]\nprotected theorem sup [Sup \u03b2] [ContinuousSup \u03b2] (hf : StronglyMeasurable f)\n    (hg : StronglyMeasurable g) : StronglyMeasurable (f \u2294 g) :=\n  \u27e8fun n => hf.approx n \u2294 hg.approx n, fun x =>\n    (hf.tendsto_approx x).sup_nhds (hg.tendsto_approx x)\u27e9\n#align measure_theory.strongly_measurable.sup MeasureTheory.StronglyMeasurable.sup\n\n@[aesop safe 20 (rule_sets := [Measurable])]\nprotected theorem inf [Inf \u03b2] [ContinuousInf \u03b2] (hf : StronglyMeasurable f)\n    (hg : StronglyMeasurable g) : StronglyMeasurable (f \u2293 g) :=\n  \u27e8fun n => hf.approx n \u2293 hg.approx n, fun x =>\n    (hf.tendsto_approx x).inf_nhds (hg.tendsto_approx x)\u27e9\n#align measure_theory.strongly_measurable.inf MeasureTheory.StronglyMeasurable.inf\n\nend Order\n\n/-!\n### Big operators: `\u220f` and `\u2211`\n-/\n\n\nsection Monoid\n\nvariable {M : Type*} [Monoid M] [TopologicalSpace M] [ContinuousMul M] {m : MeasurableSpace \u03b1}\n\n@[to_additive (attr := measurability)]\ntheorem _root_.List.stronglyMeasurable_prod' (l : List (\u03b1 \u2192 M))\n    (hl : \u2200 f \u2208 l, StronglyMeasurable f) : StronglyMeasurable l.prod := by\n  induction' l with f l ihl; \u00b7 exact stronglyMeasurable_one\n  rw [List.forall_mem_cons] at hl\n  rw [List.prod_cons]\n  exact hl.1.mul (ihl hl.2)\n#align list.strongly_measurable_prod' List.stronglyMeasurable_prod'\n#align list.strongly_measurable_sum' List.stronglyMeasurable_sum'\n\n@[to_additive (attr := measurability)]\ntheorem _root_.List.stronglyMeasurable_prod (l : List (\u03b1 \u2192 M))\n    (hl : \u2200 f \u2208 l, StronglyMeasurable f) :\n    StronglyMeasurable fun x => (l.map fun f : \u03b1 \u2192 M => f x).prod := by\n  simpa only [\u2190 Pi.list_prod_apply] using l.stronglyMeasurable_prod' hl\n#align list.strongly_measurable_prod List.stronglyMeasurable_prod\n#align list.strongly_measurable_sum List.stronglyMeasurable_sum\n\nend Monoid\n\nsection CommMonoid\n\nvariable {M : Type*} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M] {m : MeasurableSpace \u03b1}\n\n@[to_additive (attr := measurability)]\ntheorem _root_.Multiset.stronglyMeasurable_prod' (l : Multiset (\u03b1 \u2192 M))\n    (hl : \u2200 f \u2208 l, StronglyMeasurable f) : StronglyMeasurable l.prod := by\n  rcases l with \u27e8l\u27e9\n  simpa using l.stronglyMeasurable_prod' (by simpa using hl)\n#align multiset.strongly_measurable_prod' Multiset.stronglyMeasurable_prod'\n#align multiset.strongly_measurable_sum' Multiset.stronglyMeasurable_sum'\n\n@[to_additive (attr := measurability)]\ntheorem _root_.Multiset.stronglyMeasurable_prod (s : Multiset (\u03b1 \u2192 M))\n    (hs : \u2200 f \u2208 s, StronglyMeasurable f) :\n    StronglyMeasurable fun x => (s.map fun f : \u03b1 \u2192 M => f x).prod := by\n  simpa only [\u2190 Pi.multiset_prod_apply] using s.stronglyMeasurable_prod' hs\n#align multiset.strongly_measurable_prod Multiset.stronglyMeasurable_prod\n#align multiset.strongly_measurable_sum Multiset.stronglyMeasurable_sum\n\n@[to_additive (attr := measurability)]\ntheorem _root_.Finset.stronglyMeasurable_prod' {\u03b9 : Type*} {f : \u03b9 \u2192 \u03b1 \u2192 M} (s : Finset \u03b9)\n    (hf : \u2200 i \u2208 s, StronglyMeasurable (f i)) : StronglyMeasurable (\u220f i in s, f i) :=\n  Finset.prod_induction _ _ (fun _a _b ha hb => ha.mul hb) (@stronglyMeasurable_one \u03b1 M _ _ _) hf\n#align finset.strongly_measurable_prod' Finset.stronglyMeasurable_prod'\n#align finset.strongly_measurable_sum' Finset.stronglyMeasurable_sum'\n\n@[to_additive (attr := measurability)]\ntheorem _root_.Finset.stronglyMeasurable_prod {\u03b9 : Type*} {f : \u03b9 \u2192 \u03b1 \u2192 M} (s : Finset \u03b9)\n    (hf : \u2200 i \u2208 s, StronglyMeasurable (f i)) : StronglyMeasurable fun a => \u220f i in s, f i a := by\n  simpa only [\u2190 Finset.prod_apply] using s.stronglyMeasurable_prod' hf\n#align finset.strongly_measurable_prod Finset.stronglyMeasurable_prod\n#align finset.strongly_measurable_sum Finset.stronglyMeasurable_sum\n\nend CommMonoid\n\n/-- The range of a strongly measurable function is separable. -/\nprotected theorem isSeparable_range {m : MeasurableSpace \u03b1} [TopologicalSpace \u03b2]\n    (hf : StronglyMeasurable f) : TopologicalSpace.IsSeparable (range f) := by\n  have : IsSeparable (closure (\u22c3 n, range (hf.approx n))) :=\n    .closure <| .iUnion fun n => (hf.approx n).finite_range.isSeparable\n  apply this.mono\n  rintro _ \u27e8x, rfl\u27e9\n  apply mem_closure_of_tendsto (hf.tendsto_approx x)\n  filter_upwards with n\n  apply mem_iUnion_of_mem n\n  exact mem_range_self _\n#align measure_theory.strongly_measurable.is_separable_range MeasureTheory.StronglyMeasurable.isSeparable_range\n\ntheorem separableSpace_range_union_singleton {_ : MeasurableSpace \u03b1} [TopologicalSpace \u03b2]\n    [PseudoMetrizableSpace \u03b2] (hf : StronglyMeasurable f) {b : \u03b2} :\n    SeparableSpace (range f \u222a {b} : Set \u03b2) :=\n  letI := pseudoMetrizableSpacePseudoMetric \u03b2\n  (hf.isSeparable_range.union (finite_singleton _).isSeparable).separableSpace\n#align measure_theory.strongly_measurable.separable_space_range_union_singleton MeasureTheory.StronglyMeasurable.separableSpace_range_union_singleton\n\nsection SecondCountableStronglyMeasurable\n\nvariable {m\u03b1 : MeasurableSpace \u03b1} [MeasurableSpace \u03b2]\n\n/-- In a space with second countable topology, measurable implies strongly measurable. -/\n@[aesop 90% apply (rule_sets := [Measurable])]\ntheorem _root_.Measurable.stronglyMeasurable [TopologicalSpace \u03b2] [PseudoMetrizableSpace \u03b2]\n    [SecondCountableTopology \u03b2] [OpensMeasurableSpace \u03b2] (hf : Measurable f) :\n    StronglyMeasurable f := by\n  letI := pseudoMetrizableSpacePseudoMetric \u03b2\n  nontriviality \u03b2; inhabit \u03b2\n  exact \u27e8SimpleFunc.approxOn f hf Set.univ default (Set.mem_univ _), fun x \u21a6\n    SimpleFunc.tendsto_approxOn hf (Set.mem_univ _) (by rw [closure_univ]; simp)\u27e9\n#align measurable.strongly_measurable Measurable.stronglyMeasurable\n\n/-- In a space with second countable topology, strongly measurable and measurable are equivalent. -/\ntheorem _root_.stronglyMeasurable_iff_measurable [TopologicalSpace \u03b2] [MetrizableSpace \u03b2]\n    [BorelSpace \u03b2] [SecondCountableTopology \u03b2] : StronglyMeasurable f \u2194 Measurable f :=\n  \u27e8fun h => h.measurable, fun h => Measurable.stronglyMeasurable h\u27e9\n#align strongly_measurable_iff_measurable stronglyMeasurable_iff_measurable\n\n@[measurability]\ntheorem _root_.stronglyMeasurable_id [TopologicalSpace \u03b1] [PseudoMetrizableSpace \u03b1]\n    [OpensMeasurableSpace \u03b1] [SecondCountableTopology \u03b1] : StronglyMeasurable (id : \u03b1 \u2192 \u03b1) :=\n  measurable_id.stronglyMeasurable\n#align strongly_measurable_id stronglyMeasurable_id\n\nend SecondCountableStronglyMeasurable\n\n/-- A function is strongly measurable if and only if it is measurable and has separable\nrange. -/\ntheorem _root_.stronglyMeasurable_iff_measurable_separable {m : MeasurableSpace \u03b1}\n    [TopologicalSpace \u03b2] [PseudoMetrizableSpace \u03b2] [MeasurableSpace \u03b2] [BorelSpace \u03b2] :\n    StronglyMeasurable f \u2194 Measurable f \u2227 IsSeparable (range f) := by\n  refine \u27e8fun H \u21a6 \u27e8H.measurable, H.isSeparable_range\u27e9, fun \u27e8Hm, Hsep\u27e9  \u21a6 ?_\u27e9\n  have := Hsep.secondCountableTopology\n  have Hm' : StronglyMeasurable (rangeFactorization f) := Hm.subtype_mk.stronglyMeasurable\n  exact continuous_subtype_val.comp_stronglyMeasurable Hm'\n#align strongly_measurable_iff_measurable_separable stronglyMeasurable_iff_measurable_separable\n\n/-- A continuous function is strongly measurable when either the source space or the target space\nis second-countable. -/\ntheorem _root_.Continuous.stronglyMeasurable [MeasurableSpace \u03b1] [TopologicalSpace \u03b1]\n    [OpensMeasurableSpace \u03b1] [TopologicalSpace \u03b2] [PseudoMetrizableSpace \u03b2]\n    [h : SecondCountableTopologyEither \u03b1 \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : Continuous f) :\n    StronglyMeasurable f := by\n  borelize \u03b2\n  cases h.out\n  \u00b7 rw [stronglyMeasurable_iff_measurable_separable]\n    refine' \u27e8hf.measurable, _\u27e9\n    exact isSeparable_range hf\n  \u00b7 exact hf.measurable.stronglyMeasurable\n#align continuous.strongly_measurable Continuous.stronglyMeasurable\n\n/-- A continuous function whose support is contained in a compact set is strongly measurable. -/\n@[to_additive]\ntheorem _root_.Continuous.stronglyMeasurable_of_mulSupport_subset_isCompact\n    [MeasurableSpace \u03b1] [TopologicalSpace \u03b1] [OpensMeasurableSpace \u03b1] [MeasurableSpace \u03b2]\n    [TopologicalSpace \u03b2] [PseudoMetrizableSpace \u03b2] [BorelSpace \u03b2] [One \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (hf : Continuous f) {k : Set \u03b1} (hk : IsCompact k)\n    (h'f : mulSupport f \u2286 k) : StronglyMeasurable f := by\n  letI : PseudoMetricSpace \u03b2 := pseudoMetrizableSpacePseudoMetric \u03b2\n  rw [stronglyMeasurable_iff_measurable_separable]\n  exact \u27e8hf.measurable, (isCompact_range_of_mulSupport_subset_isCompact hf hk h'f).isSeparable\u27e9\n\n/-- A continuous function with compact support is strongly measurable. -/\n@[to_additive]\ntheorem _root_.Continuous.stronglyMeasurable_of_hasCompactMulSupport\n    [MeasurableSpace \u03b1] [TopologicalSpace \u03b1] [OpensMeasurableSpace \u03b1] [MeasurableSpace \u03b2]\n    [TopologicalSpace \u03b2] [PseudoMetrizableSpace \u03b2] [BorelSpace \u03b2] [One \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (hf : Continuous f) (h'f : HasCompactMulSupport f) : StronglyMeasurable f :=\n  hf.stronglyMeasurable_of_mulSupport_subset_isCompact h'f (subset_mulTSupport f)\n\n/-- A continuous function with compact support on a product space is strongly measurable for the\nproduct sigma-algebra. The subtlety is that we do not assume that the spaces are separable, so the\nproduct of the Borel sigma algebras might not contain all open sets, but still it contains enough\nof them to approximate compactly supported continuous functions. -/\nlemma _root_.HasCompactSupport.stronglyMeasurable_of_prod {X Y : Type*} [Zero \u03b1]\n    [TopologicalSpace X] [TopologicalSpace Y] [MeasurableSpace X] [MeasurableSpace Y]\n    [OpensMeasurableSpace X] [OpensMeasurableSpace Y] [TopologicalSpace \u03b1] [PseudoMetrizableSpace \u03b1]\n    {f : X \u00d7 Y \u2192 \u03b1} (hf : Continuous f) (h'f : HasCompactSupport f) :\n    StronglyMeasurable f := by\n  borelize \u03b1\n  apply stronglyMeasurable_iff_measurable_separable.2 \u27e8h'f.measurable_of_prod hf, ?_\u27e9\n  letI : PseudoMetricSpace \u03b1 := pseudoMetrizableSpacePseudoMetric \u03b1\n  exact IsCompact.isSeparable (s := range f) (h'f.isCompact_range hf)\n\n/-- If `g` is a topological embedding, then `f` is strongly measurable iff `g \u2218 f` is. -/\ntheorem _root_.Embedding.comp_stronglyMeasurable_iff {m : MeasurableSpace \u03b1} [TopologicalSpace \u03b2]\n    [PseudoMetrizableSpace \u03b2] [TopologicalSpace \u03b3] [PseudoMetrizableSpace \u03b3] {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2}\n    (hg : Embedding g) : (StronglyMeasurable fun x => g (f x)) \u2194 StronglyMeasurable f := by\n  letI := pseudoMetrizableSpacePseudoMetric \u03b3\n  borelize \u03b2 \u03b3\n  refine'\n    \u27e8fun H => stronglyMeasurable_iff_measurable_separable.2 \u27e8_, _\u27e9, fun H =>\n      hg.continuous.comp_stronglyMeasurable H\u27e9\n  \u00b7 let G : \u03b2 \u2192 range g := rangeFactorization g\n    have hG : ClosedEmbedding G :=\n      { hg.codRestrict _ _ with\n        isClosed_range := by\n          rw [surjective_onto_range.range_eq]\n          exact isClosed_univ }\n    have : Measurable (G \u2218 f) := Measurable.subtype_mk H.measurable\n    exact hG.measurableEmbedding.measurable_comp_iff.1 this\n  \u00b7 have : IsSeparable (g \u207b\u00b9' range (g \u2218 f)) := hg.isSeparable_preimage H.isSeparable_range\n    rwa [range_comp, hg.inj.preimage_image] at this\n#align embedding.comp_strongly_measurable_iff Embedding.comp_stronglyMeasurable_iff\n\n/-- A sequential limit of strongly measurable functions is strongly measurable. -/\ntheorem _root_.stronglyMeasurable_of_tendsto {\u03b9 : Type*} {m : MeasurableSpace \u03b1}\n    [TopologicalSpace \u03b2] [PseudoMetrizableSpace \u03b2] (u : Filter \u03b9) [NeBot u] [IsCountablyGenerated u]\n    {f : \u03b9 \u2192 \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2} (hf : \u2200 i, StronglyMeasurable (f i)) (lim : Tendsto f u (\ud835\udcdd g)) :\n    StronglyMeasurable g := by\n  borelize \u03b2\n  refine' stronglyMeasurable_iff_measurable_separable.2 \u27e8_, _\u27e9\n  \u00b7 exact measurable_of_tendsto_metrizable' u (fun i => (hf i).measurable) lim\n  \u00b7 rcases u.exists_seq_tendsto with \u27e8v, hv\u27e9\n    have : IsSeparable (closure (\u22c3 i, range (f (v i)))) :=\n      .closure <| .iUnion fun i => (hf (v i)).isSeparable_range\n    apply this.mono\n    rintro _ \u27e8x, rfl\u27e9\n    rw [tendsto_pi_nhds] at lim\n    apply mem_closure_of_tendsto ((lim x).comp hv)\n    filter_upwards with n\n    apply mem_iUnion_of_mem n\n    exact mem_range_self _\n#align strongly_measurable_of_tendsto stronglyMeasurable_of_tendsto\n\nprotected theorem piecewise {m : MeasurableSpace \u03b1} [TopologicalSpace \u03b2] {s : Set \u03b1}\n    {_ : DecidablePred (\u00b7 \u2208 s)} (hs : MeasurableSet s) (hf : StronglyMeasurable f)\n    (hg : StronglyMeasurable g) : StronglyMeasurable (Set.piecewise s f g) := by\n  refine' \u27e8fun n => SimpleFunc.piecewise s hs (hf.approx n) (hg.approx n), fun x => _\u27e9\n  by_cases hx : x \u2208 s\n  \u00b7 simpa [@Set.piecewise_eq_of_mem _ _ _ _ _ (fun _ => Classical.propDecidable _) _ hx,\n      hx] using hf.tendsto_approx x\n  \u00b7 simpa [@Set.piecewise_eq_of_not_mem _ _ _ _ _ (fun _ => Classical.propDecidable _) _ hx,\n      hx] using hg.tendsto_approx x\n#align measure_theory.strongly_measurable.piecewise MeasureTheory.StronglyMeasurable.piecewise\n\n/-- this is slightly different from `StronglyMeasurable.piecewise`. It can be used to show\n`StronglyMeasurable (ite (x=0) 0 1)` by\n`exact StronglyMeasurable.ite (measurableSet_singleton 0) stronglyMeasurable_const\nstronglyMeasurable_const`, but replacing `StronglyMeasurable.ite` by\n`StronglyMeasurable.piecewise` in that example proof does not work. -/\nprotected theorem ite {_ : MeasurableSpace \u03b1} [TopologicalSpace \u03b2] {p : \u03b1 \u2192 Prop}\n    {_ : DecidablePred p} (hp : MeasurableSet { a : \u03b1 | p a }) (hf : StronglyMeasurable f)\n    (hg : StronglyMeasurable g) : StronglyMeasurable fun x => ite (p x) (f x) (g x) :=\n  StronglyMeasurable.piecewise hp hf hg\n#align measure_theory.strongly_measurable.ite MeasureTheory.StronglyMeasurable.ite\n\n@[measurability]\ntheorem _root_.MeasurableEmbedding.stronglyMeasurable_extend {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b3} {g' : \u03b3 \u2192 \u03b2}\n    {m\u03b1 : MeasurableSpace \u03b1} {m\u03b3 : MeasurableSpace \u03b3} [TopologicalSpace \u03b2]\n    (hg : MeasurableEmbedding g) (hf : StronglyMeasurable f) (hg' : StronglyMeasurable g') :\n    StronglyMeasurable (Function.extend g f g') := by\n  refine' \u27e8fun n => SimpleFunc.extend (hf.approx n) g hg (hg'.approx n), _\u27e9\n  intro x\n  by_cases hx : \u2203 y, g y = x\n  \u00b7 rcases hx with \u27e8y, rfl\u27e9\n    simpa only [SimpleFunc.extend_apply, hg.injective, Injective.extend_apply] using\n      hf.tendsto_approx y\n  \u00b7 simpa only [hx, SimpleFunc.extend_apply', not_false_iff, extend_apply'] using\n      hg'.tendsto_approx x\n#align measurable_embedding.strongly_measurable_extend MeasurableEmbedding.stronglyMeasurable_extend\n\ntheorem _root_.MeasurableEmbedding.exists_stronglyMeasurable_extend {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b3}\n    {_ : MeasurableSpace \u03b1} {_ : MeasurableSpace \u03b3} [TopologicalSpace \u03b2]\n    (hg : MeasurableEmbedding g) (hf : StronglyMeasurable f) (hne : \u03b3 \u2192 Nonempty \u03b2) :\n    \u2203 f' : \u03b3 \u2192 \u03b2, StronglyMeasurable f' \u2227 f' \u2218 g = f :=\n  \u27e8Function.extend g f fun x => Classical.choice (hne x),\n    hg.stronglyMeasurable_extend hf (stronglyMeasurable_const' fun _ _ => rfl),\n    funext fun _ => hg.injective.extend_apply _ _ _\u27e9\n#align measurable_embedding.exists_strongly_measurable_extend MeasurableEmbedding.exists_stronglyMeasurable_extend\n\ntheorem _root_.stronglyMeasurable_of_stronglyMeasurable_union_cover {m : MeasurableSpace \u03b1}\n    [TopologicalSpace \u03b2] {f : \u03b1 \u2192 \u03b2} (s t : Set \u03b1) (hs : MeasurableSet s) (ht : MeasurableSet t)\n    (h : univ \u2286 s \u222a t) (hc : StronglyMeasurable fun a : s => f a)\n    (hd : StronglyMeasurable fun a : t => f a) : StronglyMeasurable f := by\n  nontriviality \u03b2; inhabit \u03b2\n  suffices Function.extend Subtype.val (fun x : s \u21a6 f x)\n      (Function.extend (\u2191) (fun x : t \u21a6 f x) fun _ \u21a6 default) = f from\n    this \u25b8 (MeasurableEmbedding.subtype_coe hs).stronglyMeasurable_extend hc <|\n      (MeasurableEmbedding.subtype_coe ht).stronglyMeasurable_extend hd stronglyMeasurable_const\n  ext x\n  by_cases hxs : x \u2208 s\n  \u00b7 lift x to s using hxs\n    simp [Subtype.coe_injective.extend_apply]\n  \u00b7 lift x to t using (h trivial).resolve_left hxs\n    rw [extend_apply', Subtype.coe_injective.extend_apply]\n    exact fun \u27e8y, hy\u27e9 \u21a6 hxs <| hy \u25b8 y.2\n#align strongly_measurable_of_strongly_measurable_union_cover stronglyMeasurable_of_stronglyMeasurable_union_cover\n\ntheorem _root_.stronglyMeasurable_of_restrict_of_restrict_compl {_ : MeasurableSpace \u03b1}\n    [TopologicalSpace \u03b2] {f : \u03b1 \u2192 \u03b2} {s : Set \u03b1} (hs : MeasurableSet s)\n    (h\u2081 : StronglyMeasurable (s.restrict f)) (h\u2082 : StronglyMeasurable (s\u1d9c.restrict f)) :\n    StronglyMeasurable f :=\n  stronglyMeasurable_of_stronglyMeasurable_union_cover s s\u1d9c hs hs.compl (union_compl_self s).ge h\u2081\n    h\u2082\n#align strongly_measurable_of_restrict_of_restrict_compl stronglyMeasurable_of_restrict_of_restrict_compl\n\n@[measurability]\nprotected theorem indicator {_ : MeasurableSpace \u03b1} [TopologicalSpace \u03b2] [Zero \u03b2]\n    (hf : StronglyMeasurable f) {s : Set \u03b1} (hs : MeasurableSet s) :\n    StronglyMeasurable (s.indicator f) :=\n  hf.piecewise hs stronglyMeasurable_const\n#align measure_theory.strongly_measurable.indicator MeasureTheory.StronglyMeasurable.indicator\n\n@[aesop safe 20 apply (rule_sets := [Measurable])]\nprotected theorem dist {_ : MeasurableSpace \u03b1} {\u03b2 : Type*} [PseudoMetricSpace \u03b2] {f g : \u03b1 \u2192 \u03b2}\n    (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :\n    StronglyMeasurable fun x => dist (f x) (g x) :=\n  continuous_dist.comp_stronglyMeasurable (hf.prod_mk hg)\n#align measure_theory.strongly_measurable.dist MeasureTheory.StronglyMeasurable.dist\n\n@[measurability]\nprotected theorem norm {_ : MeasurableSpace \u03b1} {\u03b2 : Type*} [SeminormedAddCommGroup \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (hf : StronglyMeasurable f) : StronglyMeasurable fun x => \u2016f x\u2016 :=\n  continuous_norm.comp_stronglyMeasurable hf\n#align measure_theory.strongly_measurable.norm MeasureTheory.StronglyMeasurable.norm\n\n@[measurability]\nprotected theorem nnnorm {_ : MeasurableSpace \u03b1} {\u03b2 : Type*} [SeminormedAddCommGroup \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (hf : StronglyMeasurable f) : StronglyMeasurable fun x => \u2016f x\u2016\u208a :=\n  continuous_nnnorm.comp_stronglyMeasurable hf\n#align measure_theory.strongly_measurable.nnnorm MeasureTheory.StronglyMeasurable.nnnorm\n\n@[measurability]\nprotected theorem ennnorm {_ : MeasurableSpace \u03b1} {\u03b2 : Type*} [SeminormedAddCommGroup \u03b2]\n    {f : \u03b1 \u2192 \u03b2} (hf : StronglyMeasurable f) : Measurable fun a => (\u2016f a\u2016\u208a : \u211d\u22650\u221e) :=\n  (ENNReal.continuous_coe.comp_stronglyMeasurable hf.nnnorm).measurable\n#align measure_theory.strongly_measurable.ennnorm MeasureTheory.StronglyMeasurable.ennnorm\n\n@[measurability]\nprotected theorem real_toNNReal {_ : MeasurableSpace \u03b1} {f : \u03b1 \u2192 \u211d} (hf : StronglyMeasurable f) :\n    StronglyMeasurable fun x => (f x).toNNReal :=\n  continuous_real_toNNReal.comp_stronglyMeasurable hf\n#align measure_theory.strongly_measurable.real_to_nnreal MeasureTheory.StronglyMeasurable.real_toNNReal\n\ntheorem measurableSet_eq_fun {m : MeasurableSpace \u03b1} {E} [TopologicalSpace E] [MetrizableSpace E]\n    {f g : \u03b1 \u2192 E} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :\n    MeasurableSet { x | f x = g x } := by\n  borelize (E \u00d7 E)\n  exact (hf.prod_mk hg).measurable isClosed_diagonal.measurableSet\n#align measure_theory.strongly_measurable.measurable_set_eq_fun MeasureTheory.StronglyMeasurable.measurableSet_eq_fun\n\ntheorem measurableSet_lt {m : MeasurableSpace \u03b1} [TopologicalSpace \u03b2] [LinearOrder \u03b2]\n    [OrderClosedTopology \u03b2] [PseudoMetrizableSpace \u03b2] {f g : \u03b1 \u2192 \u03b2} (hf : StronglyMeasurable f)\n    (hg : StronglyMeasurable g) : MeasurableSet { a | f a < g a } := by\n  borelize (\u03b2 \u00d7 \u03b2)\n  exact (hf.prod_mk hg).measurable isOpen_lt_prod.measurableSet\n#align measure_theory.strongly_measurable.measurable_set_lt MeasureTheory.StronglyMeasurable.measurableSet_lt\n\ntheorem measurableSet_le {m : MeasurableSpace \u03b1} [TopologicalSpace \u03b2] [Preorder \u03b2]\n    [OrderClosedTopology \u03b2] [PseudoMetrizableSpace \u03b2] {f g : \u03b1 \u2192 \u03b2} (hf : StronglyMeasurable f)\n    (hg : StronglyMeasurable g) : MeasurableSet { a | f a \u2264 g a } := by\n  borelize (\u03b2 \u00d7 \u03b2)\n  exact (hf.prod_mk hg).measurable isClosed_le_prod.measurableSet\n#align measure_theory.strongly_measurable.measurable_set_le MeasureTheory.StronglyMeasurable.measurableSet_le\n\ntheorem stronglyMeasurable_in_set {m : MeasurableSpace \u03b1} [TopologicalSpace \u03b2] [Zero \u03b2] {s : Set \u03b1}\n    {f : \u03b1 \u2192 \u03b2} (hs : MeasurableSet s) (hf : StronglyMeasurable f)\n    (hf_zero : \u2200 x, x \u2209 s \u2192 f x = 0) :\n    \u2203 fs : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2,\n      (\u2200 x, Tendsto (fun n => fs n x) atTop (\ud835\udcdd (f x))) \u2227 \u2200 x \u2209 s, \u2200 n, fs n x = 0 := by\n  let g_seq_s : \u2115 \u2192 @SimpleFunc \u03b1 m \u03b2 := fun n => (hf.approx n).restrict s\n  have hg_eq : \u2200 x \u2208 s, \u2200 n, g_seq_s n x = hf.approx n x := by\n    intro x hx n\n    rw [SimpleFunc.coe_restrict _ hs, Set.indicator_of_mem hx]\n  have hg_zero : \u2200 x \u2209 s, \u2200 n, g_seq_s n x = 0 := by\n    intro x hx n\n    rw [SimpleFunc.coe_restrict _ hs, Set.indicator_of_not_mem hx]\n  refine' \u27e8g_seq_s, fun x => _, hg_zero\u27e9\n  by_cases hx : x \u2208 s\n  \u00b7 simp_rw [hg_eq x hx]\n    exact hf.tendsto_approx x\n  \u00b7 simp_rw [hg_zero x hx, hf_zero x hx]\n    exact tendsto_const_nhds\n#align measure_theory.strongly_measurable.strongly_measurable_in_set MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_set\n\n/-- If the restriction to a set `s` of a \u03c3-algebra `m` is included in the restriction to `s` of\nanother \u03c3-algebra `m\u2082` (hypothesis `hs`), the set `s` is `m` measurable and a function `f` supported\non `s` is `m`-strongly-measurable, then `f` is also `m\u2082`-strongly-measurable. -/\ntheorem stronglyMeasurable_of_measurableSpace_le_on {\u03b1 E} {m m\u2082 : MeasurableSpace \u03b1}\n    [TopologicalSpace E] [Zero E] {s : Set \u03b1} {f : \u03b1 \u2192 E} (hs_m : MeasurableSet[m] s)\n    (hs : \u2200 t, MeasurableSet[m] (s \u2229 t) \u2192 MeasurableSet[m\u2082] (s \u2229 t))\n    (hf : StronglyMeasurable[m] f) (hf_zero : \u2200 x \u2209 s, f x = 0) :\n    StronglyMeasurable[m\u2082] f := by\n  have hs_m\u2082 : MeasurableSet[m\u2082] s := by\n    rw [\u2190 Set.inter_univ s]\n    refine' hs Set.univ _\n    rwa [Set.inter_univ]\n  obtain \u27e8g_seq_s, hg_seq_tendsto, hg_seq_zero\u27e9 := stronglyMeasurable_in_set hs_m hf hf_zero\n  let g_seq_s\u2082 : \u2115 \u2192 @SimpleFunc \u03b1 m\u2082 E := fun n =>\n    { toFun := g_seq_s n\n      measurableSet_fiber' := fun x => by\n        rw [\u2190 Set.inter_univ (g_seq_s n \u207b\u00b9' {x}), \u2190 Set.union_compl_self s,\n          Set.inter_union_distrib_left, Set.inter_comm (g_seq_s n \u207b\u00b9' {x})]\n        refine' MeasurableSet.union (hs _ (hs_m.inter _)) _\n        \u00b7 exact @SimpleFunc.measurableSet_fiber _ _ m _ _\n        by_cases hx : x = 0\n        \u00b7 suffices g_seq_s n \u207b\u00b9' {x} \u2229 s\u1d9c = s\u1d9c by\n            rw [this]\n            exact hs_m\u2082.compl\n          ext1 y\n          rw [hx, Set.mem_inter_iff, Set.mem_preimage, Set.mem_singleton_iff]\n          exact \u27e8fun h => h.2, fun h => \u27e8hg_seq_zero y h n, h\u27e9\u27e9\n        \u00b7 suffices g_seq_s n \u207b\u00b9' {x} \u2229 s\u1d9c = \u2205 by\n            rw [this]\n            exact MeasurableSet.empty\n          ext1 y\n          simp only [mem_inter_iff, mem_preimage, mem_singleton_iff, mem_compl_iff,\n            mem_empty_iff_false, iff_false_iff, not_and, not_not_mem]\n          refine' Function.mtr fun hys => _\n          rw [hg_seq_zero y hys n]\n          exact Ne.symm hx\n      finite_range' := @SimpleFunc.finite_range _ _ m (g_seq_s n) }\n  exact \u27e8g_seq_s\u2082, hg_seq_tendsto\u27e9\n#align measure_theory.strongly_measurable.strongly_measurable_of_measurable_space_le_on MeasureTheory.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on\n\n/-- If a function `f` is strongly measurable w.r.t. a sub-\u03c3-algebra `m` and the measure is \u03c3-finite\non `m`, then there exists spanning measurable sets with finite measure on which `f` has bounded\nnorm. In particular, `f` is integrable on each of those sets. -/\ntheorem exists_spanning_measurableSet_norm_le [SeminormedAddCommGroup \u03b2] {m m0 : MeasurableSpace \u03b1}\n    (hm : m \u2264 m0) (hf : StronglyMeasurable[m] f) (\u03bc : Measure \u03b1) [SigmaFinite (\u03bc.trim hm)] :\n    \u2203 s : \u2115 \u2192 Set \u03b1,\n      (\u2200 n, MeasurableSet[m] (s n) \u2227 \u03bc (s n) < \u221e \u2227 \u2200 x \u2208 s n, \u2016f x\u2016 \u2264 n) \u2227\n      \u22c3 i, s i = Set.univ := by\n  obtain \u27e8s, hs, hs_univ\u27e9 := exists_spanning_measurableSet_le hf.nnnorm.measurable (\u03bc.trim hm)\n  refine \u27e8s, fun n \u21a6 \u27e8(hs n).1, (le_trim hm).trans_lt (hs n).2.1, fun x hx \u21a6 ?_\u27e9, hs_univ\u27e9\n  have hx_nnnorm : \u2016f x\u2016\u208a \u2264 n := (hs n).2.2 x hx\n  rw [\u2190 coe_nnnorm]\n  norm_cast\n#align measure_theory.strongly_measurable.exists_spanning_measurable_set_norm_le MeasureTheory.StronglyMeasurable.exists_spanning_measurableSet_norm_le\n\nend StronglyMeasurable\n\n/-! ## Finitely strongly measurable functions -/\n\n\ntheorem finStronglyMeasurable_zero {\u03b1 \u03b2} {m : MeasurableSpace \u03b1} {\u03bc : Measure \u03b1} [Zero \u03b2]\n    [TopologicalSpace \u03b2] : FinStronglyMeasurable (0 : \u03b1 \u2192 \u03b2) \u03bc :=\n  \u27e80, by\n    simp only [Pi.zero_apply, SimpleFunc.coe_zero, support_zero', measure_empty,\n      zero_lt_top, forall_const],\n    fun _ => tendsto_const_nhds\u27e9\n#align measure_theory.fin_strongly_measurable_zero MeasureTheory.finStronglyMeasurable_zero\n\nnamespace FinStronglyMeasurable\n\nvariable {m0 : MeasurableSpace \u03b1} {\u03bc : Measure \u03b1} {f g : \u03b1 \u2192 \u03b2}\n\ntheorem aefinStronglyMeasurable [Zero \u03b2] [TopologicalSpace \u03b2] (hf : FinStronglyMeasurable f \u03bc) :\n    AEFinStronglyMeasurable f \u03bc :=\n  \u27e8f, hf, ae_eq_refl f\u27e9\n#align measure_theory.fin_strongly_measurable.ae_fin_strongly_measurable MeasureTheory.FinStronglyMeasurable.aefinStronglyMeasurable\n\nsection sequence\n\nvariable [Zero \u03b2] [TopologicalSpace \u03b2] (hf : FinStronglyMeasurable f \u03bc)\n\n/-- A sequence of simple functions such that `\u2200 x, Tendsto (fun n \u21a6 hf.approx n x) atTop (\ud835\udcdd (f x))`\nand `\u2200 n, \u03bc (support (hf.approx n)) < \u221e`. These properties are given by\n`FinStronglyMeasurable.tendsto_approx` and `FinStronglyMeasurable.fin_support_approx`. -/\nprotected noncomputable def approx : \u2115 \u2192 \u03b1 \u2192\u209b \u03b2 :=\n  hf.choose\n#align measure_theory.fin_strongly_measurable.approx MeasureTheory.FinStronglyMeasurable.approx\n\nprotected theorem fin_support_approx : \u2200 n, \u03bc (support (hf.approx n)) < \u221e :=\n  hf.choose_spec.1\n#align measure_theory.fin_strongly_measurable.fin_support_approx MeasureTheory.FinStronglyMeasurable.fin_support_approx\n\nprotected theorem tendsto_approx : \u2200 x, Tendsto (fun n => hf.approx n x) atTop (\ud835\udcdd (f x)) :=\n  hf.choose_spec.2\n#align measure_theory.fin_strongly_measurable.tendsto_approx MeasureTheory.FinStronglyMeasurable.tendsto_approx\n\nend sequence\n\n/-- A finitely strongly measurable function is strongly measurable. -/\n@[aesop 5% apply (rule_sets := [Measurable])]\nprotected theorem stronglyMeasurable [Zero \u03b2] [TopologicalSpace \u03b2]\n    (hf : FinStronglyMeasurable f \u03bc) : StronglyMeasurable f :=\n  \u27e8hf.approx, hf.tendsto_approx\u27e9\n#align measure_theory.fin_strongly_measurable.strongly_measurable MeasureTheory.FinStronglyMeasurable.stronglyMeasurable\n\ntheorem exists_set_sigmaFinite [Zero \u03b2] [TopologicalSpace \u03b2] [T2Space \u03b2]\n    (hf : FinStronglyMeasurable f \u03bc) :\n    \u2203 t, MeasurableSet t \u2227 (\u2200 x \u2208 t\u1d9c, f x = 0) \u2227 SigmaFinite (\u03bc.restrict t) := by\n  rcases hf with \u27e8fs, hT_lt_top, h_approx\u27e9\n  let T n := support (fs n)\n  have hT_meas : \u2200 n, MeasurableSet (T n) := fun n => SimpleFunc.measurableSet_support (fs n)\n  let t := \u22c3 n, T n\n  refine' \u27e8t, MeasurableSet.iUnion hT_meas, _, _\u27e9\n  \u00b7 have h_fs_zero : \u2200 n, \u2200 x \u2208 t\u1d9c, fs n x = 0 := by\n      intro n x hxt\n      rw [Set.mem_compl_iff, Set.mem_iUnion, not_exists] at hxt\n      simpa [T] using hxt n\n    refine' fun x hxt => tendsto_nhds_unique (h_approx x) _\n    rw [funext fun n => h_fs_zero n x hxt]\n    exact tendsto_const_nhds\n  \u00b7 refine' \u27e8\u27e8\u27e8fun n => t\u1d9c \u222a T n, fun _ => trivial, fun n => _, _\u27e9\u27e9\u27e9\n    \u00b7 rw [Measure.restrict_apply' (MeasurableSet.iUnion hT_meas), Set.union_inter_distrib_right,\n        Set.compl_inter_self t, Set.empty_union]\n      exact (measure_mono (Set.inter_subset_left _ _)).trans_lt (hT_lt_top n)\n    \u00b7 rw [\u2190 Set.union_iUnion t\u1d9c T]\n      exact Set.compl_union_self _\n#align measure_theory.fin_strongly_measurable.exists_set_sigma_finite MeasureTheory.FinStronglyMeasurable.exists_set_sigmaFinite\n\n/-- A finitely strongly measurable function is measurable. -/\nprotected theorem measurable [Zero \u03b2] [TopologicalSpace \u03b2] [PseudoMetrizableSpace \u03b2]\n    [MeasurableSpace \u03b2] [BorelSpace \u03b2] (hf : FinStronglyMeasurable f \u03bc) : Measurable f :=\n  hf.stronglyMeasurable.measurable\n#align measure_theory.fin_strongly_measurable.measurable MeasureTheory.FinStronglyMeasurable.measurable\n\nsection Arithmetic\n\nvariable [TopologicalSpace \u03b2]\n\n@[aesop safe 20 (rule_sets := [Measurable])]\nprotected theorem mul [MonoidWithZero \u03b2] [ContinuousMul \u03b2] (hf : FinStronglyMeasurable f \u03bc)\n    (hg : FinStronglyMeasurable g \u03bc) : FinStronglyMeasurable (f * g) \u03bc := by\n  refine'\n    \u27e8fun n => hf.approx n * hg.approx n, _, fun x =>\n      (hf.tendsto_approx x).mul (hg.tendsto_approx x)\u27e9\n  intro n\n  exact (measure_mono (support_mul_subset_left _ _)).trans_lt (hf.fin_support_approx n)\n#align measure_theory.fin_strongly_measurable.mul MeasureTheory.FinStronglyMeasurable.mul\n\n@[aesop safe 20 (rule_sets := [Measurable])]\nprotected theorem add [AddMonoid \u03b2] [ContinuousAdd \u03b2] (hf : FinStronglyMeasurable f \u03bc)\n    (hg : FinStronglyMeasurable g \u03bc) : FinStronglyMeasurable (f + g) \u03bc :=\n  \u27e8fun n => hf.approx n + hg.approx n, fun n =>\n    (measure_mono (Function.support_add _ _)).trans_lt\n      ((measure_union_le _ _).trans_lt\n        (ENNReal.add_lt_top.mpr \u27e8hf.fin_support_approx n, hg.fin_support_approx n\u27e9)),\n    fun x => (hf.tendsto_approx x).add (hg.tendsto_approx x)\u27e9\n#align measure_theory.fin_strongly_measurable.add MeasureTheory.FinStronglyMeasurable.add\n\n@[measurability]\nprotected theorem neg [AddGroup \u03b2] [TopologicalAddGroup \u03b2] (hf : FinStronglyMeasurable f \u03bc) :\n    FinStronglyMeasurable (-f) \u03bc := by\n  refine' \u27e8fun n => -hf.approx n, fun n => _, fun x => (hf.tendsto_approx x).neg\u27e9\n  suffices \u03bc (Function.support fun x => -(hf.approx n) x) < \u221e by convert this\n  rw [Function.support_neg (hf.approx n)]\n  exact hf.fin_support_approx n\n#align measure_theory.fin_strongly_measurable.neg MeasureTheory.FinStronglyMeasurable.neg\n\n@[measurability]\nprotected theorem sub [AddGroup \u03b2] [ContinuousSub \u03b2] (hf : FinStronglyMeasurable f \u03bc)\n    (hg : FinStronglyMeasurable g \u03bc) : FinStronglyMeasurable (f - g) \u03bc :=\n  \u27e8fun n => hf.approx n - hg.approx n, fun n =>\n    (measure_mono (Function.support_sub _ _)).trans_lt\n      ((measure_union_le _ _).trans_lt\n        (ENNReal.add_lt_top.mpr \u27e8hf.fin_support_approx n, hg.fin_support_approx n\u27e9)),\n    fun x => (hf.tendsto_approx x).sub (hg.tendsto_approx x)\u27e9\n#align measure_theory.fin_strongly_measurable.sub MeasureTheory.FinStronglyMeasurable.sub\n\n@[measurability]\nprotected theorem const_smul {\ud835\udd5c} [TopologicalSpace \ud835\udd5c] [AddMonoid \u03b2] [Monoid \ud835\udd5c]\n    [DistribMulAction \ud835\udd5c \u03b2] [ContinuousSMul \ud835\udd5c \u03b2] (hf : FinStronglyMeasurable f \u03bc) (c : \ud835\udd5c) :\n    FinStronglyMeasurable (c \u2022 f) \u03bc := by\n  refine' \u27e8fun n => c \u2022 hf.approx n, fun n => _, fun x => (hf.tendsto_approx x).const_smul c\u27e9\n  rw [SimpleFunc.coe_smul]\n  exact (measure_mono (support_const_smul_subset c _)).trans_lt (hf.fin_support_approx n)\n#align measure_theory.fin_strongly_measurable.const_smul MeasureTheory.FinStronglyMeasurable.const_smul\n\nend Arithmetic\n\nsection Order\n\nvariable [TopologicalSpace \u03b2] [Zero \u03b2]\n\n@[aesop safe 20 (rule_sets := [Measurable])]\nprotected theorem sup [SemilatticeSup \u03b2] [ContinuousSup \u03b2] (hf : FinStronglyMeasurable f \u03bc)\n    (hg : FinStronglyMeasurable g \u03bc) : FinStronglyMeasurable (f \u2294 g) \u03bc := by\n  refine'\n    \u27e8fun n => hf.approx n \u2294 hg.approx n, fun n => _, fun x =>\n      (hf.tendsto_approx x).sup_nhds (hg.tendsto_approx x)\u27e9\n  refine' (measure_mono (support_sup _ _)).trans_lt _\n  exact measure_union_lt_top_iff.mpr \u27e8hf.fin_support_approx n, hg.fin_support_approx n\u27e9\n#align measure_theory.fin_strongly_measurable.sup MeasureTheory.FinStronglyMeasurable.sup\n\n@[aesop safe 20 (rule_sets := [Measurable])]\nprotected theorem inf [SemilatticeInf \u03b2] [ContinuousInf \u03b2] (hf : FinStronglyMeasurable f \u03bc)\n    (hg : FinStronglyMeasurable g \u03bc) : FinStronglyMeasurable (f \u2293 g) \u03bc := by\n  refine'\n    \u27e8fun n => hf.approx n \u2293 hg.approx n, fun n => _, fun x =>\n      (hf.tendsto_approx x).inf_nhds (hg.tendsto_approx x)\u27e9\n  refine' (measure_mono (support_inf _ _)).trans_lt _\n  exact measure_union_lt_top_iff.mpr \u27e8hf.fin_support_approx n, hg.fin_support_approx n\u27e9\n#align measure_theory.fin_strongly_measurable.inf MeasureTheory.FinStronglyMeasurable.inf\n\nend Order\n\nend FinStronglyMeasurable\n\ntheorem finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite {\u03b1 \u03b2} {f : \u03b1 \u2192 \u03b2}\n    [TopologicalSpace \u03b2] [T2Space \u03b2] [Zero \u03b2] {_ : MeasurableSpace \u03b1} {\u03bc : Measure \u03b1} :\n    FinStronglyMeasurable f \u03bc \u2194\n      StronglyMeasurable f \u2227\n        \u2203 t, MeasurableSet t \u2227 (\u2200 x \u2208 t\u1d9c, f x = 0) \u2227 SigmaFinite (\u03bc.restrict t) :=\n  \u27e8fun hf => \u27e8hf.stronglyMeasurable, hf.exists_set_sigmaFinite\u27e9, fun hf =>\n    hf.1.finStronglyMeasurable_of_set_sigmaFinite hf.2.choose_spec.1 hf.2.choose_spec.2.1\n      hf.2.choose_spec.2.2\u27e9\n#align measure_theory.fin_strongly_measurable_iff_strongly_measurable_and_exists_set_sigma_finite MeasureTheory.finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite\n\ntheorem aefinStronglyMeasurable_zero {\u03b1 \u03b2} {_ : MeasurableSpace \u03b1} (\u03bc : Measure \u03b1) [Zero \u03b2]\n    [TopologicalSpace \u03b2] : AEFinStronglyMeasurable (0 : \u03b1 \u2192 \u03b2) \u03bc :=\n  \u27e80, finStronglyMeasurable_zero, EventuallyEq.rfl\u27e9\n#align measure_theory.ae_fin_strongly_measurable_zero MeasureTheory.aefinStronglyMeasurable_zero\n\n/-! ## Almost everywhere strongly measurable functions -/\n\n@[measurability]\ntheorem aestronglyMeasurable_const {\u03b1 \u03b2} {_ : MeasurableSpace \u03b1} {\u03bc : Measure \u03b1}\n    [TopologicalSpace \u03b2] {b : \u03b2} : AEStronglyMeasurable (fun _ : \u03b1 => b) \u03bc :=\n  stronglyMeasurable_const.aestronglyMeasurable\n#align measure_theory.ae_strongly_measurable_const MeasureTheory.aestronglyMeasurable_const\n\n@[to_additive (attr := measurability)]\ntheorem aestronglyMeasurable_one {\u03b1 \u03b2} {_ : MeasurableSpace \u03b1} {\u03bc : Measure \u03b1} [TopologicalSpace \u03b2]\n    [One \u03b2] : AEStronglyMeasurable (1 : \u03b1 \u2192 \u03b2) \u03bc :=\n  stronglyMeasurable_one.aestronglyMeasurable\n#align measure_theory.ae_strongly_measurable_one MeasureTheory.aestronglyMeasurable_one\n#align measure_theory.ae_strongly_measurable_zero MeasureTheory.aestronglyMeasurable_zero\n\n@[simp]\ntheorem Subsingleton.aestronglyMeasurable {_ : MeasurableSpace \u03b1} [TopologicalSpace \u03b2]\n    [Subsingleton \u03b2] {\u03bc : Measure \u03b1} (f : \u03b1 \u2192 \u03b2) : AEStronglyMeasurable f \u03bc :=\n  (Subsingleton.stronglyMeasurable f).aestronglyMeasurable\n#align measure_theory.subsingleton.ae_strongly_measurable MeasureTheory.Subsingleton.aestronglyMeasurable\n\n@[simp]\ntheorem Subsingleton.aestronglyMeasurable' {_ : MeasurableSpace \u03b1} [TopologicalSpace \u03b2]\n    [Subsingleton \u03b1] {\u03bc : Measure \u03b1} (f : \u03b1 \u2192 \u03b2) : AEStronglyMeasurable f \u03bc :=\n  (Subsingleton.stronglyMeasurable' f).aestronglyMeasurable\n#align measure_theory.subsingleton.ae_strongly_measurable' MeasureTheory.Subsingleton.aestronglyMeasurable'\n\n@[simp]\ntheorem aestronglyMeasurable_zero_measure [MeasurableSpace \u03b1] [TopologicalSpace \u03b2] (f : \u03b1 \u2192 \u03b2) :\n    AEStronglyMeasurable f (0 : Measure \u03b1) := by\n  nontriviality \u03b1\n  inhabit \u03b1\n  exact \u27e8fun _ => f default, stronglyMeasurable_const, rfl\u27e9\n#align measure_theory.ae_strongly_measurable_zero_measure MeasureTheory.aestronglyMeasurable_zero_measure\n\n@[measurability]\ntheorem SimpleFunc.aestronglyMeasurable {_ : MeasurableSpace \u03b1} {\u03bc : Measure \u03b1} [TopologicalSpace \u03b2]\n    (f : \u03b1 \u2192\u209b \u03b2) : AEStronglyMeasurable f \u03bc :=\n  f.stronglyMeasurable.aestronglyMeasurable\n#align measure_theory.simple_func.ae_strongly_measurable MeasureTheory.SimpleFunc.aestronglyMeasurable\n\nnamespace AEStronglyMeasurable\n\nvariable {m : MeasurableSpace \u03b1} {\u03bc \u03bd : Measure \u03b1} [TopologicalSpace \u03b2] [TopologicalSpace \u03b3]\n  {f g : \u03b1 \u2192 \u03b2}\n\nsection Mk\n\n/-- A `StronglyMeasurable` function such that `f =\u1d50[\u03bc] hf.mk f`. See lemmas\n`stronglyMeasurable_mk` and `ae_eq_mk`. -/\nprotected noncomputable def mk (f : \u03b1 \u2192 \u03b2) (hf : AEStronglyMeasurable f \u03bc) : \u03b1 \u2192 \u03b2 :=\n  hf.choose\n#align measure_theory.ae_strongly_measurable.mk MeasureTheory.AEStronglyMeasurable.mk\n\ntheorem stronglyMeasurable_mk (hf : AEStronglyMeasurable f \u03bc) : StronglyMeasurable (hf.mk f) :=\n  hf.choose_spec.1\n#align measure_theory.ae_strongly_measurable.strongly_measurable_mk MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk\n\ntheorem measurable_mk [PseudoMetrizableSpace \u03b2] [MeasurableSpace \u03b2] [BorelSpace \u03b2]\n    (hf : AEStronglyMeasurable f \u03bc) : Measurable (hf.mk f) :=\n  hf.stronglyMeasurable_mk.measurable\n#align measure_theory.ae_strongly_measurable.measurable_mk MeasureTheory.AEStronglyMeasurable.measurable_mk\n\ntheorem ae_eq_mk (hf : AEStronglyMeasurable f \u03bc) : f =\u1d50[\u03bc] hf.mk f :=\n  hf.choose_spec.2\n#align measure_theory.ae_strongly_measurable.ae_eq_mk MeasureTheory.AEStronglyMeasurable.ae_eq_mk\n\n@[aesop 5% apply (rule_sets := [Measurable])]\nprotected theorem aemeasurable {\u03b2} [MeasurableSpace \u03b2] [TopologicalSpace \u03b2]\n    [PseudoMetrizableSpace \u03b2] [BorelSpace \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : AEStronglyMeasurable f \u03bc) :\n    AEMeasurable f \u03bc :=\n  \u27e8hf.mk f, hf.stronglyMeasurable_mk.measurable, hf.ae_eq_mk\u27e9\n#align measure_theory.ae_strongly_measurable.ae_measurable MeasureTheory.AEStronglyMeasurable.aemeasurable\n\nend Mk\n\ntheorem congr (hf : AEStronglyMeasurable f \u03bc) (h : f =\u1d50[\u03bc] g) : AEStronglyMeasurable g \u03bc :=\n  \u27e8hf.mk f, hf.stronglyMeasurable_mk, h.symm.trans hf.ae_eq_mk\u27e9\n#align measure_theory.ae_strongly_measurable.congr MeasureTheory.AEStronglyMeasurable.congr\n\ntheorem _root_.aestronglyMeasurable_congr (h : f =\u1d50[\u03bc] g) :\n    AEStronglyMeasurable f \u03bc \u2194 AEStronglyMeasurable g \u03bc :=\n  \u27e8fun hf => hf.congr h, fun hg => hg.congr h.symm\u27e9\n#align ae_strongly_measurable_congr aestronglyMeasurable_congr\n\ntheorem mono_measure {\u03bd : Measure \u03b1} (hf : AEStronglyMeasurable f \u03bc) (h : \u03bd \u2264 \u03bc) :\n    AEStronglyMeasurable f \u03bd :=\n  \u27e8hf.mk f, hf.stronglyMeasurable_mk, Eventually.filter_mono (ae_mono h) hf.ae_eq_mk\u27e9\n#align measure_theory.ae_strongly_measurable.mono_measure MeasureTheory.AEStronglyMeasurable.mono_measure\n\nprotected lemma mono_ac (h : \u03bd \u226a \u03bc) (h\u03bc : AEStronglyMeasurable f \u03bc) : AEStronglyMeasurable f \u03bd :=\n  let \u27e8g, hg, hg'\u27e9 := h\u03bc; \u27e8g, hg, h.ae_eq hg'\u27e9\n#align measure_theory.ae_strongly_measurable.mono' MeasureTheory.AEStronglyMeasurable.mono_ac\n#align measure_theory.ae_strongly_measurable_of_absolutely_continuous MeasureTheory.AEStronglyMeasurable.mono_ac\n\n@[deprecated] protected alias mono' := AEStronglyMeasurable.mono_ac\n\ntheorem mono_set {s t} (h : s \u2286 t) (ht : AEStronglyMeasurable f (\u03bc.restrict t)) :\n    AEStronglyMeasurable f (\u03bc.restrict s) :=\n  ht.mono_measure (restrict_mono h le_rfl)\n#align measure_theory.ae_strongly_measurable.mono_set MeasureTheory.AEStronglyMeasurable.mono_set\n\nprotected theorem restrict (hfm : AEStronglyMeasurable f \u03bc) {s} :\n    AEStronglyMeasurable f (\u03bc.restrict s) :=\n  hfm.mono_measure Measure.restrict_le_self\n#align measure_theory.ae_strongly_measurable.restrict MeasureTheory.AEStronglyMeasurable.restrict\n\ntheorem ae_mem_imp_eq_mk {s} (h : AEStronglyMeasurable f (\u03bc.restrict s)) :\n    \u2200\u1d50 x \u2202\u03bc, x \u2208 s \u2192 f x = h.mk f x :=\n  ae_imp_of_ae_restrict h.ae_eq_mk\n#align measure_theory.ae_strongly_measurable.ae_mem_imp_eq_mk MeasureTheory.AEStronglyMeasurable.ae_mem_imp_eq_mk\n\n/-- The composition of a continuous function and an ae strongly measurable function is ae strongly\nmeasurable. -/\ntheorem _root_.Continuous.comp_aestronglyMeasurable {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : Continuous g)\n    (hf : AEStronglyMeasurable f \u03bc) : AEStronglyMeasurable (fun x => g (f x)) \u03bc :=\n  \u27e8_, hg.comp_stronglyMeasurable hf.stronglyMeasurable_mk, EventuallyEq.fun_comp hf.ae_eq_mk g\u27e9\n#align continuous.comp_ae_strongly_measurable Continuous.comp_aestronglyMeasurable\n\n/-- A continuous function from `\u03b1` to `\u03b2` is ae strongly measurable when one of the two spaces is\nsecond countable. -/\ntheorem _root_.Continuous.aestronglyMeasurable [TopologicalSpace \u03b1] [OpensMeasurableSpace \u03b1]\n    [PseudoMetrizableSpace \u03b2] [SecondCountableTopologyEither \u03b1 \u03b2] (hf : Continuous f) :\n    AEStronglyMeasurable f \u03bc :=\n  hf.stronglyMeasurable.aestronglyMeasurable\n#align continuous.ae_strongly_measurable Continuous.aestronglyMeasurable\n\nprotected theorem prod_mk {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b3} (hf : AEStronglyMeasurable f \u03bc)\n    (hg : AEStronglyMeasurable g \u03bc) : AEStronglyMeasurable (fun x => (f x, g x)) \u03bc :=\n  \u27e8fun x => (hf.mk f x, hg.mk g x), hf.stronglyMeasurable_mk.prod_mk hg.stronglyMeasurable_mk,\n    hf.ae_eq_mk.prod_mk hg.ae_eq_mk\u27e9\n#align measure_theory.ae_strongly_measurable.prod_mk MeasureTheory.AEStronglyMeasurable.prod_mk\n\n", "theoremStatement": "/-- The composition of a continuous function of two variables and two ae strongly measurable\nfunctions is ae strongly measurable. -/\ntheorem _root_.Continuous.comp_aestronglyMeasurable\u2082\n    {\u03b2' : Type*} [TopologicalSpace \u03b2']\n    {g : \u03b2 \u2192 \u03b2' \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} {f' : \u03b1 \u2192 \u03b2'} (hg : Continuous g.uncurry)\n    (hf : AEStronglyMeasurable f \u03bc) (h'f : AEStronglyMeasurable f' \u03bc) :\n    AEStronglyMeasurable (fun x => g (f x) (f' x)) \u03bc ", "theoremName": "Continuous.comp_aestronglyMeasurable\u2082", "fileCreated": 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"Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.List.Enum", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.LinearAlgebra.Pi", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Extr", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Data.Set.UnionLift", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", 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