module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 181,
"column": 21
} | {
"line": 181,
"column": 23
} | [
{
"pp": "J : Type v'\ninst✝² : Category.{u', v'} J\nC : Type u\ninst✝¹ : Category.{v, u} C\nK : Type u_3\ninst✝ : Category.{v_3, u_3} K\ne : J ≌ K\nF : K ⥤ C\nc : Cocone F\nhc : IsUniversalColimit c\nF' : J ⥤ C\nc' : Cocone F'\nα : F' ⟶ e.functor ⋙ F\nf : c'.pt ⟶ (Cocone.whisker e.functor c).pt\ne' : α ≫ (Cocon... | hα | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 204,
"column": 21
} | {
"line": 204,
"column": 23
} | [
{
"pp": "J : Type v'\ninst✝² : Category.{u', v'} J\nC : Type u\ninst✝¹ : Category.{v, u} C\nK : Type u_3\ninst✝ : Category.{v_3, u_3} K\ne : J ≌ K\nF : K ⥤ C\nc : Cocone F\nhc : IsVanKampenColimit c\nF' : J ⥤ C\nc' : Cocone F'\nα : F' ⟶ e.functor ⋙ F\nf : c'.pt ⟶ (Cocone.whisker e.functor c).pt\ne' : α ≫ (Cocon... | hα | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 236,
"column": 20
} | {
"line": 236,
"column": 22
} | [
{
"pp": "J : Type v'\ninst✝⁴ : Category.{u', v'} J\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u_2\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : HasPullbacks D\ninst✝ : HasColimitsOfShape J D\nF : J ⥤ C ⥤ D\nc : Cocone F\nhc : ∀ (x : C), IsVanKampenColimit (((evaluation C D).obj x).mapCocone c)\nF' : J ⥤ C ⥤ ... | hα | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 266,
"column": 20
} | {
"line": 266,
"column": 22
} | [
{
"pp": "J : Type v'\ninst✝⁶ : Category.{u', v'} J\nC : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u_2\ninst✝⁴ : Category.{v_2, u_2} D\nGl : C ⥤ D\nGr : D ⥤ C\nadj : Gl ⊣ Gr\ninst✝³ : Gr.Full\ninst✝² : Gr.Faithful\nF : J ⥤ D\nc : Cocone (F ⋙ Gr)\nH : IsUniversalColimit c\ninst✝¹ : ∀ (X : D) (f : X ⟶ Gl.obj c.... | hα | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.LeftExact | {
"line": 172,
"column": 38
} | {
"line": 172,
"column": 56
} | [
{
"pp": "case e_a\nC : Type u\ninst✝¹² : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝¹¹ : Category.{t, w} D\ninst✝¹⁰ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\ninst✝⁹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nFD : D → D → Type u_1\nCD : D → Type t\ninst✝... | ← Iso.eq_inv_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 363,
"column": 20
} | {
"line": 363,
"column": 22
} | [
{
"pp": "J : Type v'\ninst✝⁸ : Category.{u', v'} J\nC : Type u\ninst✝⁷ : Category.{v, u} C\nD : Type u_2\ninst✝⁶ : Category.{v_2, u_2} D\ninst✝⁵ : HasColimitsOfShape J C\nGl : C ⥤ D\nGr : D ⥤ C\nadj : Gl ⊣ Gr\ninst✝⁴ : Gr.Full\ninst✝³ : Gr.Faithful\nF : J ⥤ D\nc : Cocone (F ⋙ Gr)\nH : IsVanKampenColimit c\ninst... | hα | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Localization.Adjunction | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 26
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} D₁\ninst✝⁴ : Category.{v_4, u_4} D₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\nL₁ : C₁ ⥤ D₁\nW₁ : MorphismProperty C₁\ninst✝³ : L₁.IsLocalization W₁\nL... | apply Localization.η_app | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Localization.Adjunction | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 26
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} D₁\ninst✝⁴ : Category.{v_4, u_4} D₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\nL₁ : C₁ ⥤ D₁\nW₁ : MorphismProperty C₁\ninst✝³ : L₁.IsLocalization W₁\nL... | apply Localization.η_app | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Localization.Adjunction | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 26
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nD₁ : Type u_3\nD₂ : Type u_4\ninst✝⁷ : Category.{v_1, u_1} C₁\ninst✝⁶ : Category.{v_2, u_2} C₂\ninst✝⁵ : Category.{v_3, u_3} D₁\ninst✝⁴ : Category.{v_4, u_4} D₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\nL₁ : C₁ ⥤ D₁\nW₁ : MorphismProperty C₁\ninst✝³ : L₁.IsLocalization W₁\nL... | apply Localization.η_app | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Localization.Adjunction | {
"line": 160,
"column": 2
} | {
"line": 162,
"column": 29
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\ninst✝³ : Category.{v_1, u_1} C₁\ninst✝² : Category.{v_2, u_2} C₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\ninst✝¹ : G.Full\ninst✝ : G.Faithful\n⊢ F.IsLocalization ((MorphismProperty.isomorphisms C₁).inverseImage F)",
"usedConstants": [
"CategoryTheory.Functor.op",
... | rw [← Functor.IsLocalization.op_iff, MorphismProperty.op_inverseImage,
MorphismProperty.op_isomorphisms]
exact adj.op.isLocalization | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Localization.Adjunction | {
"line": 160,
"column": 2
} | {
"line": 162,
"column": 29
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\ninst✝³ : Category.{v_1, u_1} C₁\ninst✝² : Category.{v_2, u_2} C₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\nadj : G ⊣ F\ninst✝¹ : G.Full\ninst✝ : G.Faithful\n⊢ F.IsLocalization ((MorphismProperty.isomorphisms C₁).inverseImage F)",
"usedConstants": [
"CategoryTheory.Functor.op",
... | rw [← Functor.IsLocalization.op_iff, MorphismProperty.op_inverseImage,
MorphismProperty.op_isomorphisms]
exact adj.op.isLocalization | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 424,
"column": 65
} | {
"line": 424,
"column": 82
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasInitial C\nH : IsUniversalColimit (BinaryCofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥_ C)))\nA : C\nf : A ⟶ ⊥_ C\n⊢ mapPair f f ≫ (BinaryCofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥_ C))).ι =\n (BinaryCofan.mk (𝟙 A) (𝟙 A)).ι ≫ (const (Discrete WalkingPair)).map f",
"usedConsta... | ext ⟨⟨⟩⟩ <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 424,
"column": 65
} | {
"line": 424,
"column": 82
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasInitial C\nH : IsUniversalColimit (BinaryCofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥_ C)))\nA : C\nf : A ⟶ ⊥_ C\n⊢ mapPair f f ≫ (BinaryCofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥_ C))).ι =\n (BinaryCofan.mk (𝟙 A) (𝟙 A)).ι ≫ (const (Discrete WalkingPair)).map f",
"usedConsta... | ext ⟨⟨⟩⟩ <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 424,
"column": 65
} | {
"line": 424,
"column": 82
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasInitial C\nH : IsUniversalColimit (BinaryCofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥_ C)))\nA : C\nf : A ⟶ ⊥_ C\n⊢ mapPair f f ≫ (BinaryCofan.mk (𝟙 (⊥_ C)) (𝟙 (⊥_ C))).ι =\n (BinaryCofan.mk (𝟙 A) (𝟙 A)).ι ≫ (const (Discrete WalkingPair)).map f",
"usedConsta... | ext ⟨⟨⟩⟩ <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.LocallySurjective | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 58
} | [
{
"pp": "case h\nC : Type u\ninst✝³ : Category.{v, u} C\nA : Type u'\ninst✝² : Category.{v', u'} A\nFA : A → A → Type u_1\nCA : A → Type w'\ninst✝¹ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)\ninst✝ : ConcreteCategory A FA\nF G : Cᵒᵖ ⥤ A\nf : F ⟶ G\nU : C\ns : ToType (F.obj (op U))\nV : C\ni : V ⟶ U\n⊢ ∃ t,\n ... | exact ⟨F.map i.op s, NatTrans.naturality_apply f i.op s⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 524,
"column": 4
} | {
"line": 524,
"column": 21
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasInitial C\nX Y : C\nc : BinaryCofan X Y\nh : IsVanKampenColimit c\n⊢ mapPair (initial.to X) (𝟙 Y) ≫ c.ι =\n (BinaryCofan.mk (initial.to Y) (𝟙 Y)).ι ≫ (const (Discrete WalkingPair)).map c.inr",
"usedConstants": [
"CategoryT... | ext ⟨⟨⟩⟩ <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 524,
"column": 4
} | {
"line": 524,
"column": 21
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasInitial C\nX Y : C\nc : BinaryCofan X Y\nh : IsVanKampenColimit c\n⊢ mapPair (initial.to X) (𝟙 Y) ≫ c.ι =\n (BinaryCofan.mk (initial.to Y) (𝟙 Y)).ι ≫ (const (Discrete WalkingPair)).map c.inr",
"usedConstants": [
"CategoryT... | ext ⟨⟨⟩⟩ <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 524,
"column": 4
} | {
"line": 524,
"column": 21
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasInitial C\nX Y : C\nc : BinaryCofan X Y\nh : IsVanKampenColimit c\n⊢ mapPair (initial.to X) (𝟙 Y) ≫ c.ι =\n (BinaryCofan.mk (initial.to Y) (𝟙 Y)).ι ≫ (const (Discrete WalkingPair)).map c.inr",
"usedConstants": [
"CategoryT... | ext ⟨⟨⟩⟩ <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 533,
"column": 4
} | {
"line": 533,
"column": 21
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\ninst✝ : HasInitial C\nc : BinaryCofan X Y\nh : IsVanKampenColimit c\n⊢ mapPair (initial.to X) (𝟙 Y) ≫ c.ι =\n (BinaryCofan.mk (initial.to Y) (𝟙 Y)).ι ≫ (const (Discrete WalkingPair)).map c.inr",
"usedConstants": [
"CategoryT... | ext ⟨⟨⟩⟩ <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 533,
"column": 4
} | {
"line": 533,
"column": 21
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\ninst✝ : HasInitial C\nc : BinaryCofan X Y\nh : IsVanKampenColimit c\n⊢ mapPair (initial.to X) (𝟙 Y) ≫ c.ι =\n (BinaryCofan.mk (initial.to Y) (𝟙 Y)).ι ≫ (const (Discrete WalkingPair)).map c.inr",
"usedConstants": [
"CategoryT... | ext ⟨⟨⟩⟩ <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 533,
"column": 4
} | {
"line": 533,
"column": 21
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\ninst✝ : HasInitial C\nc : BinaryCofan X Y\nh : IsVanKampenColimit c\n⊢ mapPair (initial.to X) (𝟙 Y) ≫ c.ι =\n (BinaryCofan.mk (initial.to Y) (𝟙 Y)).ι ≫ (const (Discrete WalkingPair)).map c.inr",
"usedConstants": [
"CategoryT... | ext ⟨⟨⟩⟩ <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 548,
"column": 18
} | {
"line": 548,
"column": 20
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nn : ℕ\nf : Fin (n + 1) → C\nc₁ : Cofan fun i ↦ f i.succ\nc₂ : BinaryCofan (f 0) c₁.pt\nt₁ : IsUniversalColimit c₁\nt₂ : IsUniversalColimit c₂\ninst✝ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i\nF : Discrete (Fin (n + 1)) ⥤ C\nc : Cocone F\nα : F ⟶ Discrete.... | hα | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 547,
"column": 46
} | {
"line": 599,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nn : ℕ\nf : Fin (n + 1) → C\nc₁ : Cofan fun i ↦ f i.succ\nc₂ : BinaryCofan (f 0) c₁.pt\nt₁ : IsUniversalColimit c₁\nt₂ : IsUniversalColimit c₂\ninst✝ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i\n⊢ IsUniversalColimit (extendCofan c₁ c₂)",
"usedConstants":... | by
intro F c α i e hα H
let F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
have : F = Discrete.functor F' := by
apply Functor.hext
· exact fun i ↦ rfl
· rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩
simp [F']
have t₁' := @t₁ (Discrete.functor (fun j ↦ F.obj ⟨j.succ⟩))
(Cofan.mk (pullback c₂.inr i) fun j ↦ p... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 608,
"column": 18
} | {
"line": 608,
"column": 20
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nn : ℕ\nf : Fin (n + 1) → C\nc₁ : Cofan fun i ↦ f i.succ\nc₂ : BinaryCofan (f 0) c₁.pt\nt₁ : IsVanKampenColimit c₁\nt₂ : IsVanKampenColimit c₂\ninst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i\ninst✝ : HasFiniteCoproducts C\nF : Discrete (Fin (n + 1)) ⥤ C\... | hα | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 695,
"column": 8
} | {
"line": 695,
"column": 48
} | [
{
"pp": "case refine_2.refine_1.isFalse\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasInitial C\nι : Type u_3\nX : ι → C\nc : Cofan X\nhc : IsVanKampenColimit c\ni j✝ : ι\ninst✝ : DecidableEq ι\nt : Cofan fun k ↦ if k = i then X i else ⊥_ C\nj : ι\nh✝ : ¬j = i\n⊢ (eqToHom ⋯ ≫ initial.to (X i)) ≫ eqToHom ... | rw [Category.assoc, ← IsIso.eq_inv_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 717,
"column": 2
} | {
"line": 728,
"column": 44
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasInitial C\nι : Type u_3\nF : Discrete ι ⥤ C\nc : Cocone F\nhc : IsVanKampenColimit c\ni : Discrete ι\n⊢ Mono (c.ι.app i)",
"usedConstants": [
"dite_cond_eq_true",
"Eq.mpr",
"CategoryTheory.Functor",
"CategoryTheory.Limits.Co... | let f : ι → C := F.obj ∘ Discrete.mk
have : F = Discrete.functor f :=
Functor.hext (fun i ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp [f])
clear_value f
subst this
refine PullbackCone.mono_of_isLimitMkIdId _ (IsPullback.isLimit ?_)
nth_rw 1 [← Category.id_comp (c.ι.app i)]
convert! IsPullback.paste_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.VanKampen | {
"line": 717,
"column": 2
} | {
"line": 728,
"column": 44
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasInitial C\nι : Type u_3\nF : Discrete ι ⥤ C\nc : Cocone F\nhc : IsVanKampenColimit c\ni : Discrete ι\n⊢ Mono (c.ι.app i)",
"usedConstants": [
"dite_cond_eq_true",
"Eq.mpr",
"CategoryTheory.Functor",
"CategoryTheory.Limits.Co... | let f : ι → C := F.obj ∘ Discrete.mk
have : F = Discrete.functor f :=
Functor.hext (fun i ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp [f])
clear_value f
subst this
refine PullbackCone.mono_of_isLimitMkIdId _ (IsPullback.isLimit ?_)
nth_rw 1 [← Category.id_comp (c.ι.app i)]
convert! IsPullback.paste_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Bicategory.LocallyDiscrete | {
"line": 163,
"column": 6
} | {
"line": 163,
"column": 46
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{u_2, u_1} C\nX₁ X₂ X₃ X₄ : C\nt : X₁ ⟶ X₂\nl : X₁ ⟶ X₃\nr : X₂ ⟶ X₄\nb : X₃ ⟶ X₄\nh : CommSq t l r b\n⊢ t.toLoc ≫ r.toLoc = l.toLoc ≫ b.toLoc",
"usedConstants": [
"CategoryTheory.LocallyDiscrete.categoryStruct",
"CategoryTheory.LocallyDiscrete.mk",
... | simp only [← Quiver.Hom.comp_toLoc, h.w] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Bicategory.LocallyDiscrete | {
"line": 163,
"column": 6
} | {
"line": 163,
"column": 46
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{u_2, u_1} C\nX₁ X₂ X₃ X₄ : C\nt : X₁ ⟶ X₂\nl : X₁ ⟶ X₃\nr : X₂ ⟶ X₄\nb : X₃ ⟶ X₄\nh : CommSq t l r b\n⊢ t.toLoc ≫ r.toLoc = l.toLoc ≫ b.toLoc",
"usedConstants": [
"CategoryTheory.LocallyDiscrete.categoryStruct",
"CategoryTheory.LocallyDiscrete.mk",
... | simp only [← Quiver.Hom.comp_toLoc, h.w] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Bicategory.LocallyDiscrete | {
"line": 163,
"column": 6
} | {
"line": 163,
"column": 46
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{u_2, u_1} C\nX₁ X₂ X₃ X₄ : C\nt : X₁ ⟶ X₂\nl : X₁ ⟶ X₃\nr : X₂ ⟶ X₄\nb : X₃ ⟶ X₄\nh : CommSq t l r b\n⊢ t.toLoc ≫ r.toLoc = l.toLoc ≫ b.toLoc",
"usedConstants": [
"CategoryTheory.LocallyDiscrete.categoryStruct",
"CategoryTheory.LocallyDiscrete.mk",
... | simp only [← Quiver.Hom.comp_toLoc, h.w] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Closed | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 30
} | [
{
"pp": "case a\nC : Type u\ninst✝ : Category.{v, u} C\nJ₁ : GrothendieckTopology C\nX Y : C\nf : Y ⟶ X\nS : Sieve X\nZ : C\ng : Z ⟶ Y\nhg : (Sieve.pullback f (J₁.close S)).arrows g\n⊢ Sieve.pullback g (Sieve.pullback f S) ∈ J₁ Z",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",... | rw [← Sieve.pullback_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Functor.Flat | {
"line": 133,
"column": 4
} | {
"line": 134,
"column": 79
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nx✝ : RepresentablyFlat F.op\nX : D\n⊢ IsFiltered (CostructuredArrow F X)",
"usedConstants": [
"CategoryTheory.Functor.op",
"Opposite",
"CategoryTheory.IsCofiltered",
... | suffices IsCofiltered (CostructuredArrow F X)ᵒᵖ from isFiltered_of_isCofiltered_op _
apply IsCofiltered.of_equivalence (costructuredArrowOpEquivalence _ _).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Functor.Flat | {
"line": 133,
"column": 4
} | {
"line": 134,
"column": 79
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nx✝ : RepresentablyFlat F.op\nX : D\n⊢ IsFiltered (CostructuredArrow F X)",
"usedConstants": [
"CategoryTheory.Functor.op",
"Opposite",
"CategoryTheory.IsCofiltered",
... | suffices IsCofiltered (CostructuredArrow F X)ᵒᵖ from isFiltered_of_isCofiltered_op _
apply IsCofiltered.of_equivalence (costructuredArrowOpEquivalence _ _).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Closed | {
"line": 198,
"column": 8
} | {
"line": 198,
"column": 47
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝ : Category.{v, u} C\nJ₁ : GrothendieckTopology C\nX : C\nS : Sieve X\nhS : S ∈ J₁ X\nx : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows\nhx : x.Compatible\n⊢ ∃ t, x.IsAmalgamation t",
"usedConstants": [
"Opposite",
"CategoryTheory... | Presieve.compatible_iff_sieveCompatible | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Hypercover.Zero | {
"line": 235,
"column": 56
} | {
"line": 236,
"column": 46
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nS : C\nE : PreZeroHypercover S\nT : C\nf : T ⟶ S\n⊢ (E.add f).presieve₀ = E.presieve₀ ⊔ Presieve.singleton f",
"usedConstants": [
"Lattice.toSemilatticeSup",
"CategoryTheory.Presieve",
"congrArg",
"CategoryTheory.PreZeroHypercover.presi... | by
simp [add, presieve₀_reindex, presieve₀_sum] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Coverage | {
"line": 236,
"column": 4
} | {
"line": 236,
"column": 30
} | [
{
"pp": "case transitive\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nK : Coverage C\nX✝ Y : C\nS✝ : Sieve X✝\nX : C\nR S : Sieve X\na✝ : K.Saturate X R\nhS : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → K.Saturate Y (Sieve.pullback f S)\nH1 : ∀ (f : Y ⟶ X), K.Saturate Y (Sieve.pullback f R)\na_ih✝ :\n ∀ ⦃Y_1 : C⦄ ⦃f ... | rw [← Sieve.pullback_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.Hypercover.IsSheaf | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 39
} | [
{
"pp": "case mpr\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\nH : J.OneHypercoverFamily\nP : Cᵒᵖ ⥤ A\ninst✝ : H.IsGenerating\nhP : ∀ ⦃X : C⦄ (E : J.OneHypercover X), H E → Nonempty (IsLimit (E.multifork P))\nX : C\nS : Sieve X\nhS : S ∈ J X\nE ... | exact ⟨IsSheafIff.isLimit hP hE le⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck | {
"line": 288,
"column": 4
} | {
"line": 289,
"column": 32
} | [
{
"pp": "C : Type u_3\ninst✝¹ : Category.{u_2, u_3} C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\nJ : GrothendieckTopology C\nι : Type (max u_3 u_2)\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nhf : Presieve.ofArrows X f ∈ J.toPrecoverage.coverings S\nY : C\ng : Y ⟶ S\nP : ι → C\np₁ : (i : ι) → P i ⟶ Y\np₂ : (i ... | rw [mem_toPrecoverage_iff, ← Sieve.ofArrows, Sieve.ofArrows_eq_pullback_of_isPullback _ h]
exact J.pullback_stable _ hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck | {
"line": 288,
"column": 4
} | {
"line": 289,
"column": 32
} | [
{
"pp": "C : Type u_3\ninst✝¹ : Category.{u_2, u_3} C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\nJ : GrothendieckTopology C\nι : Type (max u_3 u_2)\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nhf : Presieve.ofArrows X f ∈ J.toPrecoverage.coverings S\nY : C\ng : Y ⟶ S\nP : ι → C\np₁ : (i : ι) → P i ⟶ Y\np₂ : (i ... | rw [mem_toPrecoverage_iff, ← Sieve.ofArrows, Sieve.ofArrows_eq_pullback_of_isPullback _ h]
exact J.pullback_stable _ hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 217,
"column": 27
} | {
"line": 217,
"column": 62
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝² : Category.{v_2, u_2} D\nK : GrothendieckTopology D\nG : C ⥤ D\nℱ : Dᵒᵖ ⥤ Type v\nℱ' : Sheaf K (Type v)\nα : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.obj\ninst✝¹ : G.IsCoverDense K\ninst✝ : G.IsLocallyFull K\nX : D\nx : ℱ.obj (op X)\nthis :\n ∀ {Z : D} {W... | simpa only [Category.assoc] using e | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 217,
"column": 27
} | {
"line": 217,
"column": 62
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝² : Category.{v_2, u_2} D\nK : GrothendieckTopology D\nG : C ⥤ D\nℱ : Dᵒᵖ ⥤ Type v\nℱ' : Sheaf K (Type v)\nα : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.obj\ninst✝¹ : G.IsCoverDense K\ninst✝ : G.IsLocallyFull K\nX : D\nx : ℱ.obj (op X)\nthis :\n ∀ {Z : D} {W... | simpa only [Category.assoc] using e | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 217,
"column": 27
} | {
"line": 217,
"column": 62
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝² : Category.{v_2, u_2} D\nK : GrothendieckTopology D\nG : C ⥤ D\nℱ : Dᵒᵖ ⥤ Type v\nℱ' : Sheaf K (Type v)\nα : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.obj\ninst✝¹ : G.IsCoverDense K\ninst✝ : G.IsLocallyFull K\nX : D\nx : ℱ.obj (op X)\nthis :\n ∀ {Z : D} {W... | simpa only [Category.assoc] using e | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.DenseSubsite.InducedTopology | {
"line": 61,
"column": 4
} | {
"line": 62,
"column": 79
} | [
{
"pp": "case mp\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\nG : C ⥤ D\nK : GrothendieckTopology D\ninst✝² : G.LocallyCoverDense K\ninst✝¹ : G.Full\ninst✝ : G.Faithful\nX : C\nS : Sieve X\n⊢ Sieve.functorPushforward G S ∈ K (G.obj X) → ∃ T, Sieve.functorPullback ... | intro hS
exact ⟨⟨_, hS⟩, (Sieve.fullyFaithfulFunctorGaloisCoinsertion G X).u_l_eq S⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.DenseSubsite.InducedTopology | {
"line": 61,
"column": 4
} | {
"line": 62,
"column": 79
} | [
{
"pp": "case mp\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\nG : C ⥤ D\nK : GrothendieckTopology D\ninst✝² : G.LocallyCoverDense K\ninst✝¹ : G.Full\ninst✝ : G.Faithful\nX : C\nS : Sieve X\n⊢ Sieve.functorPushforward G S ∈ K (G.obj X) → ∃ T, Sieve.functorPullback ... | intro hS
exact ⟨⟨_, hS⟩, (Sieve.fullyFaithfulFunctorGaloisCoinsertion G X).u_l_eq S⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 512,
"column": 8
} | {
"line": 512,
"column": 30
} | [
{
"pp": "case h\nC : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝⁵ : Category.{v_2, u_2} D\nE : Type u_3\ninst✝⁴ : Category.{v_3, u_3} E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nA : Type u_4\ninst✝³ : Category.{v_4, u_4} A\nG : C ⥤ D\ninst✝² : G.IsCov... | ← sheafHom_eq G α.hom, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.DenseSubsite.Basic | {
"line": 507,
"column": 19
} | {
"line": 512,
"column": 56
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝⁵ : Category.{v_2, u_2} D\nE : Type u_3\ninst✝⁴ : Category.{v_3, u_3} E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nA : Type u_4\ninst✝³ : Category.{v_4, u_4} A\nG : C ⥤ D\ninst✝² : G.IsCoverDense ... | by
intro ℱ ℱ' α β e
ext1
apply_fun fun e => e.hom at e
dsimp [sheafPushforwardContinuous] at e
rw [← sheafHom_eq G α.hom, ← sheafHom_eq G β.hom, e] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Constructions.Over.Connected | {
"line": 57,
"column": 52
} | {
"line": 61,
"column": 83
} | [
{
"pp": "J : Type u'\ninst✝³ : Category.{v', u'} J\nC : Type u\ninst✝² : Category.{v, u} C\nD : Type u_1\ninst✝¹ : Category.{v_1, u_1} D\nK : C ⥤ D\nX : C\ninst✝ : IsConnected J\nB : D\nF : J ⥤ CostructuredArrow K B\nc : Cone (F ⋙ proj K B)\nj : J\n⊢ K.map (c.π.app j) ≫ (F.obj j).hom =\n (((Functor.const J).... | by
let z : (Functor.const J).obj (K.obj c.pt) ⟶ _ :=
(CategoryTheory.Functor.constComp J c.pt K).inv ≫ Functor.whiskerRight c.π K ≫
natTransInCostructuredArrow F
convert! (nat_trans_from_is_connected z j (Classical.arbitrary J)) <;> simp [z] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 327,
"column": 40
} | {
"line": 327,
"column": 64
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\ninst✝ : HasPullbacks C\ni j : E.I₀ × F.I₀\nW : C\np₁ : W ⟶ pullback (E.f i.1) (F.f i.2)\np₂ : W ⟶ pullback (E.f j.1) (F.f j.2)\nw : p₁ ≫ pullback.fst (E.f i.1) (F.f i.2) ≫ E.f i.1 = p₂ ≫ pullback.fst (E.f j.1... | by simp [reassoc_of% h₁] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Final.Type | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 42
} | [
{
"pp": "case refine_2\nC : Type u₁\nD : Type u₂\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nP : D ⥤ Type w\ninst✝ : F.Initial\nt : ↑(F ⋙ P).sections\nval : (Y : D) → P.obj Y\nhval : ∀ (Y : D) (j : CostructuredArrow F Y), (ConcreteCategory.hom (P.map j.hom)) (↑t j.left) = val Y\n⊢ ∃ ... | refine ⟨⟨val, fun {Y₁ Y₂} f ↦ ?_⟩, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Limits.Final.Type | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 56
} | [
{
"pp": "case refine_2\nC : Type u₁\nD : Type u₂\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nP : D ⥤ Type w\ninst✝ : F.Final\nX : D\nx : P.obj X\n⊢ ∃ a, F.colimitTypePrecomp P a = P.ιColimitType X x",
"usedConstants": [
"CategoryTheory.Functor.Final.instNonemptyStructuredAr... | let Y : StructuredArrow X F := Classical.arbitrary _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Topology.Sheaves.SheafCondition.Sites | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 32
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y : TopCat\nf : X ⟶ Y\nF : Presheaf C Y\nh : IsOpenEmbedding ⇑(ConcreteCategory.hom f)\nhF : F.IsSheaf\n⊢ IsSheaf (h.functor.op ⋙ F)",
"usedConstants": [
"Topology.IsOpenEmbedding.functor_isContinuous",
"TopologicalSpace.Opens.instPartialOrder",
... | have := h.functor_isContinuous | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing | {
"line": 98,
"column": 2
} | {
"line": 104,
"column": 76
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nFC : C → C → Type u_2\nCC : C → Type u_3\ninst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝ : ConcreteCategory C FC\nX : TopCat\nF : Presheaf (Type u_4) X\nι : Type u_5\nU : ι → Opens ↑X\nsf : (i : ι) → ToType (F.obj (op (U i)))\nh : F.IsCompatible... | refine ⟨objPairwiseOfFamily sf, ?_⟩
let G := (Pairwise.diagram U).op ⋙ F
rintro (i | ⟨i, j⟩) (i' | ⟨i', j'⟩) (_ | _ | _ | _)
· exact ConcreteCategory.congr_hom (G.map_id <| op <| Pairwise.single i) _
· rfl
· exact (h i' i).symm
· exact ConcreteCategory.congr_hom (G.map_id <| op <| Pairwise.pair i j) _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing | {
"line": 98,
"column": 2
} | {
"line": 104,
"column": 76
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nFC : C → C → Type u_2\nCC : C → Type u_3\ninst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝ : ConcreteCategory C FC\nX : TopCat\nF : Presheaf (Type u_4) X\nι : Type u_5\nU : ι → Opens ↑X\nsf : (i : ι) → ToType (F.obj (op (U i)))\nh : F.IsCompatible... | refine ⟨objPairwiseOfFamily sf, ?_⟩
let G := (Pairwise.diagram U).op ⋙ F
rintro (i | ⟨i, j⟩) (i' | ⟨i', j'⟩) (_ | _ | _ | _)
· exact ConcreteCategory.congr_hom (G.map_id <| op <| Pairwise.single i) _
· rfl
· exact (h i' i).symm
· exact ConcreteCategory.congr_hom (G.map_id <| op <| Pairwise.pair i j) _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections | {
"line": 410,
"column": 6
} | {
"line": 410,
"column": 17
} | [
{
"pp": "case property.refine_3.single.left\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nF : Sheaf C X\nU V : Opens ↑X\ns✝ : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)\nι : ULift.{w, 0} WalkingPair → Opens ↑X :=\n fun x ↦\n match x with\n | { down := j } => WalkingPai... | convert! h₁ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections | {
"line": 420,
"column": 6
} | {
"line": 420,
"column": 17
} | [
{
"pp": "case property.refine_3.pair.left.e_a\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nF : Sheaf C X\nU V : Opens ↑X\ns✝ : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)\nι : ULift.{w, 0} WalkingPair → Opens ↑X :=\n fun x ↦\n match x with\n | { down := j } => WalkingP... | convert! h₁ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.CategoryTheory.Limits.Preserves.Over | {
"line": 44,
"column": 4
} | {
"line": 47,
"column": 8
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : C\nJ : Type u_3\nhJ : Category.{u_2, u_3} J\nhJ' : IsFiltered J\nF : J ⥤ Under X\nc : Cocone F\nhc : IsColimit c\ns : Cocone (F ⋙ Under.forget X)\ni j : J\n⊢ (F.obj j).hom ≫ s.ι.app j = (((Functor.const J).obj (Under.mk ((F.obj i).hom ≫ s.ι.app i))).obj ... | obtain ⟨k, hik, hjk, -⟩ := IsFilteredOrEmpty.cocone_objs i j
simp only [Functor.const_obj_obj, Under.mk_right, Under.mk_hom,
← s.w hjk, ← s.w hik]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Preserves.Over | {
"line": 44,
"column": 4
} | {
"line": 47,
"column": 8
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : C\nJ : Type u_3\nhJ : Category.{u_2, u_3} J\nhJ' : IsFiltered J\nF : J ⥤ Under X\nc : Cocone F\nhc : IsColimit c\ns : Cocone (F ⋙ Under.forget X)\ni j : J\n⊢ (F.obj j).hom ≫ s.ι.app j = (((Functor.const J).obj (Under.mk ((F.obj i).hom ≫ s.ι.app i))).obj ... | obtain ⟨k, hik, hjk, -⟩ := IsFilteredOrEmpty.cocone_objs i j
simp only [Functor.const_obj_obj, Under.mk_right, Under.mk_hom,
← s.w hjk, ← s.w hik]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Constructions.EventuallyConstant | {
"line": 229,
"column": 4
} | {
"line": 230,
"column": 30
} | [
{
"pp": "J : Type u_1\nC : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} C\nF : J ⥤ C\ni₀ : J\nh : F.IsEventuallyConstantFrom i₀\ninst✝ : IsFiltered J\ns : Cocone F\nj : J\n⊢ (F.map (rightToMax i₀ j) ≫ (h.isoMap (leftToMax i₀ j) ⋯).inv) ≫ s.ι.app i₀ = s.ι.app j",
"usedConstants": [
... | rw [← s.w (IsFiltered.rightToMax i₀ j), ← s.w (IsFiltered.leftToMax i₀ j), assoc,
isoMap_inv_hom_id_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Category.Ring.Small | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 95
} | [
{
"pp": "P Q : ObjectProperty CommRingCat\ninst✝ : ObjectProperty.EssentiallySmall.{u, u, u + 1} Q\nhPQ : ∀ (S : CommRingCat), P S → ∃ s, Ideal.span s = ⊤ ∧ ∀ f ∈ s, Q (of (Localization.Away f))\nQ' : ObjectProperty CommRingCat\nw✝ : ObjectProperty.Small.{u, u, u + 1} Q'\nhQ'Q : Q' ≤ Q\nhQQ' : Q ≤ Q'.isoClosure... | refine (RingEquiv.injective _).comp (Localization.algebraMap_injective_of_span_eq_top _ hs) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Category.Ring.FinitePresentation | {
"line": 94,
"column": 4
} | {
"line": 110,
"column": 18
} | [
{
"pp": "J : Type uJ\ninst✝² : Category.{vJ, uJ} J\ninst✝¹ : IsFiltered J\nR : CommRingCat\nF : J ⥤ CommRingCat\nα : (Functor.const J).obj R ⟶ F\nS : CommRingCat\nc : Cocone F\nhc : IsColimit c\ninst✝ : PreservesColimit F (forget CommRingCat)\ng : S ⟶ c.pt\nhc' : IsColimit ((forget CommRingCat).mapCocone c)\nn ... | choose j x h using fun i ↦ Types.jointly_surjective_of_isColimit hc' ((π ≫ g) (.X i))
obtain ⟨i, ⟨hi⟩⟩ : ∃ i, Nonempty (∀ a, (j a ⟶ i)) := by
have : ∃ i, ∀ a, Nonempty (j a ⟶ i) := by
simpa using IsFiltered.sup_objs_exists (Finset.univ.image j)
simpa [← exists_true_iff_nonempty, Classical.skolem... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.Ring.FinitePresentation | {
"line": 94,
"column": 4
} | {
"line": 110,
"column": 18
} | [
{
"pp": "J : Type uJ\ninst✝² : Category.{vJ, uJ} J\ninst✝¹ : IsFiltered J\nR : CommRingCat\nF : J ⥤ CommRingCat\nα : (Functor.const J).obj R ⟶ F\nS : CommRingCat\nc : Cocone F\nhc : IsColimit c\ninst✝ : PreservesColimit F (forget CommRingCat)\ng : S ⟶ c.pt\nhc' : IsColimit ((forget CommRingCat).mapCocone c)\nn ... | choose j x h using fun i ↦ Types.jointly_surjective_of_isColimit hc' ((π ≫ g) (.X i))
obtain ⟨i, ⟨hi⟩⟩ : ∃ i, Nonempty (∀ a, (j a ⟶ i)) := by
have : ∃ i, ∀ a, Nonempty (j a ⟶ i) := by
simpa using IsFiltered.sup_objs_exists (Finset.univ.image j)
simpa [← exists_true_iff_nonempty, Classical.skolem... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.Ring.Under.Limits | {
"line": 173,
"column": 82
} | {
"line": 175,
"column": 36
} | [
{
"pp": "R S : CommRingCat\ninst✝¹ : Algebra ↑R ↑S\ninst✝ : Module.Flat ↑R ↑S\nA B : Under R\nf g : A ⟶ B\n⊢ (AlgHom.tensorEqualizerEquiv (↑S) (↑S) (toAlgHom f) (toAlgHom g)).toUnder.hom ≫\n (equalizerFork' (Algebra.TensorProduct.map (AlgHom.id ↑S ↑S) (toAlgHom f))\n (Algebra.TensorProduct.map (Al... | by
ext
apply AlgHom.coe_tensorEqualizer | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.Ring.Under.Property | {
"line": 187,
"column": 6
} | {
"line": 187,
"column": 75
} | [
{
"pp": "P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhPi : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P\nhPe : HasEqualizers fun {R S} [CommRing R] [CommRing S] ↦ P\nhPse : HasStableEqualizers fun {R S} [CommRing R] [CommRing S] ↦ P\ninst✝ : (toMorphismProperty fu... | ← preservesLimit_iff_of_iso_diagram _ (diagramIsoParallelPair _).symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.CharP.Invertible | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 19
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\np : ℕ\ninst✝ : CharP R p\nn : ℕ\nhp : Nat.Prime p\nh : IsUnit ↑n\nthis : Nontrivial R\n⊢ ¬↑n = 0",
"usedConstants": [
"AddGroupWithOne.toAddMonoidWithOne",
"AddMonoidWithOne.toNatCast",
"Nat.cast",
"IsUnit.ne_zero",
"Ring.toSemiring",
... | exact h.ne_zero | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.CharP.MixedCharZero | {
"line": 128,
"column": 6
} | {
"line": 128,
"column": 21
} | [
{
"pp": "case h.right.intro\nR : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I ≠ ⊤\nright✝ : CharP (R ⧸ I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I ≤ M\nr : ℕ\nhr : CharP (R ⧸ M) r\n⊢ CharP (R ⧸ M) p",
"usedConstants": [
"CharP.cast_eq_zero",
"Eq.mpr",... | | intro r hr => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.Algebra.CharP.MixedCharZero | {
"line": 230,
"column": 31
} | {
"line": 230,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nh : ∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)\nthis : Fact (∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I))\na b : ℚ\n⊢ ↑(a * b).num * (↑b.den * ↑a.den) = ↑(a * b).num * ↑↑a.den * ↑↑b.den",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
... | Int.cast_natCast | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.CharP.MixedCharZero | {
"line": 239,
"column": 31
} | {
"line": 239,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nh : ∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I)\nthis : Fact (∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I))\na b : ℚ\n⊢ ↑(a + b).num * (↑b.den * ↑a.den) = ↑(a + b).num * ↑↑a.den * ↑↑b.den",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NonAssocSemiring.toA... | Int.cast_natCast | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.CharZero.Quotient | {
"line": 35,
"column": 41
} | {
"line": 35,
"column": 51
} | [
{
"pp": "R : Type u_1\ninst✝¹ : DivisionRing R\ninst✝ : CharZero R\np r : R\nz : ℤ\nhz : z ≠ 0\nhz' : ↑z ≠ 0\n⊢ (∃ k, k • p = z • r) ↔ ∃ k k_1, (z * k_1) • p + ↑↑k • p = z • r",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"HMul.hMul",
"Ring.toNonAssocRing",
"Monoid.toMulOneClass"... | ← add_smul | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.CharZero.Quotient | {
"line": 52,
"column": 4
} | {
"line": 52,
"column": 20
} | [
{
"pp": "R : Type u_1\ninst✝¹ : DivisionRing R\ninst✝ : CharZero R\np r : R\nn : ℕ\nhn : n ≠ 0\n⊢ (∃ k, r - ↑k • (p / ↑↑n) ∈ zmultiples p) ↔ ∃ k, r - ↑k • (p / ↑n) ∈ zmultiples p",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"instHSMul",
"instHDiv",
"AddG... | Int.cast_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.CharZero.Quotient | {
"line": 70,
"column": 64
} | {
"line": 70,
"column": 80
} | [
{
"pp": "R : Type u_1\ninst✝¹ : DivisionRing R\ninst✝ : CharZero R\np : R\nψ θ : R ⧸ AddSubgroup.zmultiples p\nn : ℕ\nhz : n ≠ 0\n⊢ (∃ k, ψ = θ + ↑(↑k • (p / ↑↑n))) ↔ ∃ k, ψ = θ + ↑k • ↑(p / ↑n)",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"instHSMul",
"instHD... | Int.cast_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.FreeCommRing | {
"line": 279,
"column": 17
} | {
"line": 279,
"column": 32
} | [
{
"pp": "case refine_3\nα : Type u\nx : FreeCommRing α\ns : Set α\ninst✝ : DecidablePred fun x ↦ x ∈ s\nhxs : x.IsSupported s\np : α\nhps : p ∈ s\nn : FreeCommRing α\nih : (map Subtype.val) ((restriction s) n) = n\n⊢ (map Subtype.val) ((restriction s) (of p) * (restriction s) n) = of p * n",
"usedConstants"... | restriction_of, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.FreeCommRing | {
"line": 352,
"column": 2
} | {
"line": 360,
"column": 48
} | [
{
"pp": "α : Type u\nx : FreeCommRing α\n⊢ ∃ a, ↑a = x",
"usedConstants": [
"NegZeroClass.toNeg",
"Equiv.instEquivLike",
"HMul.hMul",
"FreeCommRing",
"FreeRing.of",
"CommSemiring.toSemiring",
"FreeRing.lift",
"AddGroupWithOne.toAddMonoidWithOne",
"instCo... | induction x with
| neg_one => use -1; rfl
| of b => exact ⟨FreeRing.of b, rfl⟩
| add _ _ hx hy =>
rcases hx with ⟨x, rfl⟩; rcases hy with ⟨y, rfl⟩
exact ⟨x + y, (FreeRing.lift _).map_add _ _⟩
| mul _ _ hx hy =>
rcases hx with ⟨x, rfl⟩; rcases hy with ⟨y, rfl⟩
exact ⟨x * y, (FreeRing.lift _).map_... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.FreeCommRing | {
"line": 352,
"column": 2
} | {
"line": 360,
"column": 48
} | [
{
"pp": "α : Type u\nx : FreeCommRing α\n⊢ ∃ a, ↑a = x",
"usedConstants": [
"NegZeroClass.toNeg",
"Equiv.instEquivLike",
"HMul.hMul",
"FreeCommRing",
"FreeRing.of",
"CommSemiring.toSemiring",
"FreeRing.lift",
"AddGroupWithOne.toAddMonoidWithOne",
"instCo... | induction x with
| neg_one => use -1; rfl
| of b => exact ⟨FreeRing.of b, rfl⟩
| add _ _ hx hy =>
rcases hx with ⟨x, rfl⟩; rcases hy with ⟨y, rfl⟩
exact ⟨x + y, (FreeRing.lift _).map_add _ _⟩
| mul _ _ hx hy =>
rcases hx with ⟨x, rfl⟩; rcases hy with ⟨y, rfl⟩
exact ⟨x * y, (FreeRing.lift _).map_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.FreeCommRing | {
"line": 352,
"column": 2
} | {
"line": 360,
"column": 48
} | [
{
"pp": "α : Type u\nx : FreeCommRing α\n⊢ ∃ a, ↑a = x",
"usedConstants": [
"NegZeroClass.toNeg",
"Equiv.instEquivLike",
"HMul.hMul",
"FreeCommRing",
"FreeRing.of",
"CommSemiring.toSemiring",
"FreeRing.lift",
"AddGroupWithOne.toAddMonoidWithOne",
"instCo... | induction x with
| neg_one => use -1; rfl
| of b => exact ⟨FreeRing.of b, rfl⟩
| add _ _ hx hy =>
rcases hx with ⟨x, rfl⟩; rcases hy with ⟨y, rfl⟩
exact ⟨x + y, (FreeRing.lift _).map_add _ _⟩
| mul _ _ hx hy =>
rcases hx with ⟨x, rfl⟩; rcases hy with ⟨y, rfl⟩
exact ⟨x * y, (FreeRing.lift _).map_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Stream.Init | {
"line": 212,
"column": 2
} | {
"line": 213,
"column": 17
} | [
{
"pp": "α : Type u\na : α\n⊢ const a = a :: const a",
"usedConstants": [
"instOfNatNat",
"Nat.casesAuxOn",
"instHAdd",
"Stream'.get",
"HAdd.hAdd",
"Stream'.const",
"Nat",
"Eq.ndrec",
"instAddNat",
"Eq.refl",
"Stream'.cons",
"OfNat.ofNa... | apply Stream'.ext; intro n
cases n <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Stream.Init | {
"line": 212,
"column": 2
} | {
"line": 213,
"column": 17
} | [
{
"pp": "α : Type u\na : α\n⊢ const a = a :: const a",
"usedConstants": [
"instOfNatNat",
"Nat.casesAuxOn",
"instHAdd",
"Stream'.get",
"HAdd.hAdd",
"Stream'.const",
"Nat",
"Eq.ndrec",
"instAddNat",
"Eq.refl",
"Stream'.cons",
"OfNat.ofNa... | apply Stream'.ext; intro n
cases n <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Stream.Init | {
"line": 528,
"column": 33
} | {
"line": 528,
"column": 48
} | [
{
"pp": "α : Type u\ns : Stream' α\nn : ℕ\n⊢ List.take 0 (take n s) = take 0 s",
"usedConstants": [
"Stream'.take",
"Eq.mpr",
"congrArg",
"id",
"List.take_zero",
"instOfNatNat",
"List",
"Nat",
"OfNat.ofNat",
"Eq",
"List.take",
"List.nil... | List.take_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.IntermediateField.Basic | {
"line": 273,
"column": 56
} | {
"line": 275,
"column": 5
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nS : Subalgebra K L\ninv_mem : ∀ x ∈ S, x⁻¹ ∈ S\n⊢ (S.toIntermediateField inv_mem).toSubalgebra = S",
"usedConstants": [
"Subalgebra.instSetLike",
"Iff.rfl",
"Membership.mem",
"Field.toSemifi... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.IntermediateField.Basic | {
"line": 279,
"column": 67
} | {
"line": 281,
"column": 5
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nS : IntermediateField K L\n⊢ S.toIntermediateField ⋯ = S",
"usedConstants": [
"IntermediateField",
"Iff.rfl",
"Membership.mem",
"Subalgebra.toIntermediateField",
"IntermediateField.ins... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Seq.Computation | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 67
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\nf : β → α ⊕ β\nb : β\n⊢ Computation α",
"usedConstants": [
"Option.some",
"Sum",
"instOfNatNat",
"Subtype.mk",
"Stream'",
"Stream'.corec'",
"instHAdd",
"HAdd.hAdd",
"Sum.inr",
"Nat",
"instAddNat",
... | refine ⟨Stream'.corec' (Corec.f f) (Sum.inr b), fun n a' h => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.FieldTheory.IntermediateField.Basic | {
"line": 296,
"column": 52
} | {
"line": 298,
"column": 5
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nS : Subalgebra K L\nhS : IsField ↥S\n⊢ (S.toIntermediateField' hS).toSubalgebra = S",
"usedConstants": [
"Subalgebra.instSetLike",
"Subalgebra.toIntermediateField'",
"Iff.rfl",
"Membership.m... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.IntermediateField.Basic | {
"line": 302,
"column": 67
} | {
"line": 304,
"column": 5
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nS : IntermediateField K L\n⊢ S.toIntermediateField' ⋯ = S",
"usedConstants": [
"Subalgebra.toIntermediateField'",
"IntermediateField",
"Iff.rfl",
"Membership.mem",
"Field.toIsField",
... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Seq.Computation | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 21
} | [
{
"pp": "case inr.h.a\nα : Type u\nβ : Type v\nf : β → α ⊕ β\nb b' : β\nh : f b = Sum.inr b'\n⊢ (tail ⟨Stream'.corec' (Corec.f f) (Sum.inr b), ⋯⟩).val = ⟨Stream'.corec' (Corec.f f) (Sum.inr b'), ⋯⟩.val",
"usedConstants": [
"Computation.corec._proof_1",
"Computation.tail",
"Option.some",
... | dsimp [corec, tail] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Data.Seq.Defs | {
"line": 320,
"column": 2
} | {
"line": 320,
"column": 21
} | [
{
"pp": "case some.a\nα : Type u\nβ : Type v\nf : β → Option (α × β)\nb : β\na : α\nb' : β\nh : f b = some (a, b')\n⊢ ↑(tail ⟨corec' (Corec.f f) (some b), ⋯⟩) = ↑⟨corec' (Corec.f f) (some b'), ⋯⟩",
"usedConstants": [
"Option.some",
"id",
"Subtype.mk",
"Stream'",
"Stream'.corec'... | dsimp [corec, tail] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Algebra.Colimit.Ring | {
"line": 86,
"column": 29
} | {
"line": 91,
"column": 82
} | [
{
"pp": "ι : Type u_1\ninst✝³ : Preorder ι\nG : ι → Type u_2\ninst✝² : (i : ι) → CommRing (G i)\nf : (i j : ι) → i ≤ j → G i → G j\ninst✝¹ : Nonempty ι\ninst✝ : IsDirectedOrder ι\nz : DirectLimit G f\n⊢ ∃ i x, (of G f i) x = z",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocS... | by
obtain ⟨z, rfl⟩ := Ideal.Quotient.mk_surjective z
refine z.induction_on ⟨Classical.arbitrary ι, -1, by simp; rfl⟩ (fun ⟨i, x⟩ ↦ ⟨i, x, rfl⟩) ?_ ?_
<;> rintro x' y' ⟨i, x, hx⟩ ⟨j, y, hy⟩ <;> have ⟨k, hik, hjk⟩ := exists_ge_ge i j
· exact ⟨k, f i k hik x + f j k hjk y, by rw [map_add, of_f, of_f, hx, hy]; rf... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Colimit.Ring | {
"line": 108,
"column": 53
} | {
"line": 110,
"column": 64
} | [
{
"pp": "ι : Type u_1\ninst✝³ : Preorder ι\nG : ι → Type u_2\ninst✝² : (i : ι) → CommRing (G i)\nf' : (i j : ι) → i ≤ j → G i →+* G j\ninst✝¹ : Nonempty ι\ninst✝ : IsDirectedOrder ι\nq : (DirectLimit G fun i j h ↦ ⇑(f' i j h))[X]\nq₁ q₂ : (DirectLimit G fun i j h ↦ ⇑(f' i j h))[X]\nx✝¹ : ∃ i p, Polynomial.map (... | by
rw [Polynomial.map_add, map_map, map_map, ← ih₁, ← ih₂]
congr 2 <;> ext x <;> simp_rw [RingHom.comp_apply, of_f] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.ContinuedFractions.Determinant | {
"line": 68,
"column": 2
} | {
"line": 71,
"column": 9
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝ : Field K\ng : GenContFract K\nn : ℕ\nterminatedAt_n : g.TerminatedAt n\n⊢ g.nums n * g.dens (n + 1) - g.dens n * g.nums (n + 1) = ∏ i ∈ Finset.range (n + 1), -(g.partNums.get? i).getD 0",
"usedConstants": [
"GenContFract.s",
"GenContFract.nums_stable_of_te... | · rw [dens_stable_of_terminated n.le_succ terminatedAt_n,
nums_stable_of_terminated n.le_succ terminatedAt_n, Finset.prod_range_succ,
partNum_none_iff_s_none.mpr terminatedAt_n]
grind | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.ContinuedFractions.Computation.Translations | {
"line": 109,
"column": 17
} | {
"line": 109,
"column": 79
} | [
{
"pp": "case succ\nK : Type u_1\ninst✝³ : DivisionRing K\ninst✝² : LinearOrder K\ninst✝¹ : FloorRing K\ninst✝ : IsStrictOrderedRing K\na : ℤ\nn : ℕ\nih : IntFractPair.stream (↑a) (n + 1) = none\n⊢ IntFractPair.stream (↑a) (n + 1 + 1) = none",
"usedConstants": [
"Iff.mpr",
"Int.cast",
"Gen... | exact IntFractPair.succ_nth_stream_eq_none_iff.mpr (Or.inl ih) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.ContinuedFractions.Computation.Translations | {
"line": 109,
"column": 17
} | {
"line": 109,
"column": 79
} | [
{
"pp": "case succ\nK : Type u_1\ninst✝³ : DivisionRing K\ninst✝² : LinearOrder K\ninst✝¹ : FloorRing K\ninst✝ : IsStrictOrderedRing K\na : ℤ\nn : ℕ\nih : IntFractPair.stream (↑a) (n + 1) = none\n⊢ IntFractPair.stream (↑a) (n + 1 + 1) = none",
"usedConstants": [
"Iff.mpr",
"Int.cast",
"Gen... | exact IntFractPair.succ_nth_stream_eq_none_iff.mpr (Or.inl ih) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.ContinuedFractions.Computation.Translations | {
"line": 109,
"column": 17
} | {
"line": 109,
"column": 79
} | [
{
"pp": "case succ\nK : Type u_1\ninst✝³ : DivisionRing K\ninst✝² : LinearOrder K\ninst✝¹ : FloorRing K\ninst✝ : IsStrictOrderedRing K\na : ℤ\nn : ℕ\nih : IntFractPair.stream (↑a) (n + 1) = none\n⊢ IntFractPair.stream (↑a) (n + 1 + 1) = none",
"usedConstants": [
"Iff.mpr",
"Int.cast",
"Gen... | exact IntFractPair.succ_nth_stream_eq_none_iff.mpr (Or.inl ih) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.ContinuedFractions.Computation.Translations | {
"line": 104,
"column": 50
} | {
"line": 109,
"column": 79
} | [
{
"pp": "K : Type u_1\ninst✝³ : DivisionRing K\ninst✝² : LinearOrder K\ninst✝¹ : FloorRing K\ninst✝ : IsStrictOrderedRing K\na : ℤ\nn : ℕ\n⊢ IntFractPair.stream (↑a) (n + 1) = none",
"usedConstants": [
"Iff.mpr",
"Int.cast",
"Nat.recAux",
"Int.floor",
"GenContFract.IntFractPair... | by
induction n with
| zero =>
refine IntFractPair.stream_eq_none_of_fr_eq_zero (IntFractPair.stream_zero (a : K)) ?_
simp only [IntFractPair.of, Int.fract_intCast]
| succ n ih => exact IntFractPair.succ_nth_stream_eq_none_iff.mpr (Or.inl ih) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Fib.Basic | {
"line": 220,
"column": 42
} | {
"line": 220,
"column": 67
} | [
{
"pp": "case inr\nm n : ℕ\nh : n.pred.succ = n\n⊢ (fib m).gcd (fib (n + m)) = (fib m).gcd (fib n)",
"usedConstants": [
"Nat.gcd",
"Eq.mpr",
"Nat.succ_eq_add_one",
"congrArg",
"id",
"instOfNatNat",
"instHAdd",
"HAdd.hAdd",
"Nat.fib",
"Nat",
"... | rw [← h, succ_eq_add_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 7
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\nv : K\ninst✝ : FloorRing K\nn : ℕ\nterminatedAt_n : (of v).TerminatedAt n\n⊢ ∃ a, ∀ b ≥ a, v = (of v).convs b",
"usedConstants": [
"GenContFract.convs",
"Preorder.toLE",
"Field.toDivisionRing",
"GE.ge",
"GenContFr... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Data.Seq.Basic | {
"line": 1009,
"column": 2
} | {
"line": 1009,
"column": 55
} | [
{
"pp": "α : Type u\ns : Seq α\n⊢ (Seq.map ret s).join = s",
"usedConstants": [
"Stream'.Seq",
"False",
"Stream'.Seq.coinduction2",
"congrArg",
"and_self",
"Stream'.Seq.map_cons",
"Option.some",
"Exists",
"Stream'.Seq1.ret",
"Eq.mp",
"Stream'... | apply coinduction2 s; intro s; cases s <;> simp [ret] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Seq.Basic | {
"line": 1009,
"column": 2
} | {
"line": 1009,
"column": 55
} | [
{
"pp": "α : Type u\ns : Seq α\n⊢ (Seq.map ret s).join = s",
"usedConstants": [
"Stream'.Seq",
"False",
"Stream'.Seq.coinduction2",
"congrArg",
"and_self",
"Stream'.Seq.map_cons",
"Option.some",
"Exists",
"Stream'.Seq1.ret",
"Eq.mp",
"Stream'... | apply coinduction2 s; intro s; cases s <;> simp [ret] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.ContinuedFractions.Computation.TerminatesIffRat | {
"line": 76,
"column": 6
} | {
"line": 78,
"column": 16
} | [
{
"pp": "case succ.zero\nK : Type u_1\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : FloorRing K\nv : K\ng : GenContFract K := of v\nIH : ∀ m < 0 + 1, ∃ conts, (of v).contsAux m = Pair.map Rat.cast conts\n⊢ ∃ conts, (of v).contsAux (0 + 1) = Pair.map Rat.cast conts",
"use... | · suffices ∃ conts : Pair ℚ, Pair.mk g.h 1 = conts.map (↑) by simpa [contsAux]
use Pair.mk ⌊v⌋ 1
simp [g] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.ContinuedFractions.Computation.TerminatesIffRat | {
"line": 219,
"column": 53
} | {
"line": 219,
"column": 58
} | [
{
"pp": "case mp\nK : Type u_1\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : FloorRing K\nv : K\nq : ℚ\nv_eq_q : v = ↑q\nn : ℕ\nh : (of v).s.TerminatedAt n\n⊢ (of q).Terminates",
"usedConstants": [
"GenContFract.s",
"Stream'.Seq.TerminatedAt",
"Rat",
... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.ContinuedFractions.Computation.TerminatesIffRat | {
"line": 219,
"column": 53
} | {
"line": 219,
"column": 58
} | [
{
"pp": "case mpr\nK : Type u_1\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : FloorRing K\nv : K\nq : ℚ\nv_eq_q : v = ↑q\nn : ℕ\nh : (of q).s.TerminatedAt n\n⊢ (of v).Terminates",
"usedConstants": [
"GenContFract.s",
"Stream'.Seq.TerminatedAt",
"Field.t... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries | {
"line": 113,
"column": 22
} | {
"line": 113,
"column": 87
} | [
{
"pp": "K : Type u_1\nv : K\ninst✝⁴ : Field K\ninst✝³ : LinearOrder K\ninst✝² : IsStrictOrderedRing K\ninst✝¹ : FloorRing K\ninst✝ : Archimedean K\nε : K\nε_pos : ε > 0\nN' : ℕ\none_div_ε_lt_N' : 1 / ε < ↑N'\nN : ℕ := max N' 5\nn : ℕ\nn_ge_N : n ≥ N\ng : GenContFract K := of v\nnot_terminatedAt_n : ¬g.Terminat... | exact_mod_cast le_fib_self <| le_trans (le_max_right N' 5) n_ge_N | Lean.Parser.Tactic._aux_Init_TacticsExtra___macroRules_Lean_Parser_Tactic_tacticExact_mod_cast__1 | Lean.Parser.Tactic.tacticExact_mod_cast_ |
Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries | {
"line": 113,
"column": 22
} | {
"line": 113,
"column": 87
} | [
{
"pp": "K : Type u_1\nv : K\ninst✝⁴ : Field K\ninst✝³ : LinearOrder K\ninst✝² : IsStrictOrderedRing K\ninst✝¹ : FloorRing K\ninst✝ : Archimedean K\nε : K\nε_pos : ε > 0\nN' : ℕ\none_div_ε_lt_N' : 1 / ε < ↑N'\nN : ℕ := max N' 5\nn : ℕ\nn_ge_N : n ≥ N\ng : GenContFract K := of v\nnot_terminatedAt_n : ¬g.Terminat... | exact_mod_cast le_fib_self <| le_trans (le_max_right N' 5) n_ge_N | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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