module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Abelian.Injective.Dimension | {
"line": 309,
"column": 14
} | {
"line": 309,
"column": 55
} | {
"line": 309,
"column": 55
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nhd : injectiveDimension X = ⊥\n⊢ HasInjectiveDimensionLE X 0",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"WithBot... | [] | by simp [← injectiveDimension_le_iff, hd] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.Monomorphisms | {
"line": 37,
"column": 4
} | {
"line": 38,
"column": 18
} | {
"line": 38,
"column": 18
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nx✝³ x✝² x✝¹ : C\nf : x✝³ ⟶ x✝¹\ng : x✝² ⟶ x✝¹\nx✝ : Limits.HasPullback f g\nhf : epimorphisms C g\n⊢ epimorphisms C (Limits.pullback.fst f g)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryT... | [] | simp only [epimorphisms.iff] at hf ⊢
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.Monomorphisms | {
"line": 37,
"column": 4
} | {
"line": 38,
"column": 18
} | {
"line": 38,
"column": 18
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nx✝³ x✝² x✝¹ : C\nf : x✝³ ⟶ x✝¹\ng : x✝² ⟶ x✝¹\nx✝ : Limits.HasPullback f g\nhf : epimorphisms C g\n⊢ epimorphisms C (Limits.pullback.fst f g)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryT... | [] | simp only [epimorphisms.iff] at hf ⊢
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.Pseudoelements | {
"line": 312,
"column": 2
} | {
"line": 322,
"column": 24
} | {
"line": 324,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\n⊢ Function.Surjective (pseudoApply f) → Epi f",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"CategoryTheory.Over",
"CategoryTheory.Ep... | [] | intro h
have ⟨pbar, hpbar⟩ := h (𝟙 Q)
have ⟨p, hp⟩ := Quotient.exists_rep pbar
have : (⟦(p.hom ≫ f : Over Q)⟧ : Quotient (setoid Q)) = ⟦↑(𝟙 Q)⟧ := by
rw [← hp] at hpbar
exact hpbar
have ⟨R, x, y, _, ey, comm⟩ := Quotient.exact this
apply @epi_of_epi_fac _ _ _ _ _ (x ≫ p.hom) f y ey
dsimp at comm
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.Pseudoelements | {
"line": 312,
"column": 2
} | {
"line": 322,
"column": 24
} | {
"line": 324,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\n⊢ Function.Surjective (pseudoApply f) → Epi f",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"CategoryTheory.Over",
"CategoryTheory.Ep... | [] | intro h
have ⟨pbar, hpbar⟩ := h (𝟙 Q)
have ⟨p, hp⟩ := Quotient.exists_rep pbar
have : (⟦(p.hom ≫ f : Over Q)⟧ : Quotient (setoid Q)) = ⟦↑(𝟙 Q)⟧ := by
rw [← hp] at hpbar
exact hpbar
have ⟨R, x, y, _, ey, comm⟩ := Quotient.exact this
apply @epi_of_epi_fac _ _ _ _ _ (x ≫ p.hom) f y ey
dsimp at comm
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 246,
"column": 8
} | {
"line": 246,
"column": 58
} | {
"line": 247,
"column": 8
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX Y : D\nf : X ⟶ Y\nthis✝¹ : L.PreservesMonomorphisms\nth... | [
"C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX Y : D\nf : X ⟶ Y\nthis✝¹ : L.PreservesMonomorphisms\nthis✝ : L.mapA... | let e := L.mapArrow.objObjPreimageIso (Arrow.mk f) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 270,
"column": 8
} | {
"line": 270,
"column": 58
} | {
"line": 271,
"column": 8
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX Y : D\nf : X ⟶ Y\nthis✝¹ : L.PreservesEpimorphisms\nthi... | [
"C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX Y : D\nf : X ⟶ Y\nthis✝¹ : L.PreservesEpimorphisms\nthis✝ : L.mapAr... | let e := L.mapArrow.objObjPreimageIso (Arrow.mk f) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Action.Limits | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 34
} | {
"line": 196,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝² : Category.{v_1, u_1} V\nG : Type u_2\ninst✝¹ : Monoid G\ninst✝ : HasFiniteLimits V\nthis : PreservesFiniteLimits ((evaluation (SingleObj G) V).obj (SingleObj.star G))\n⊢ PreservesFiniteLimits ((functorCategoryEquivalence V G).functor ⋙ (evaluation (SingleObj G) V).obj (SingleObj.s... | [] | apply comp_preservesFiniteLimits | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Adjunction.Quadruple | {
"line": 93,
"column": 11
} | {
"line": 93,
"column": 55
} | {
"line": 93,
"column": 56
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\nL : C ⥤ D\nF : D ⥤ C\nG : C ⥤ D\nR : D ⥤ C\nq : Quadruple L F G R\ninst✝¹ : F.Full\ninst✝ : F.Faithful\n⊢ (∀ (X : C), Epi (q.leftTriple.rightToLeft.app X)) ↔ ∀ (X : D), Mono (q.rightTriple.leftToRight.app X)",
"pp... | [
"C : Type u₁\nD : Type u₂\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\nL : C ⥤ D\nF : D ⥤ C\nG : C ⥤ D\nR : D ⥤ C\nq : Quadruple L F G R\ninst✝¹ : F.Full\ninst✝ : F.Faithful\n⊢ (∀ (X : C), Epi (q.leftTriple.rightToLeft.app X)) ↔ ∀ (X : D), Mono (q.rightTriple.adj₂.unit.app (F.obj X))"
] | mono_leftToRight_app_iff_mono_adj₂_unit_app, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.CategoryTheory.Action.Monoidal | {
"line": 335,
"column": 8
} | {
"line": 335,
"column": 55
} | {
"line": 335,
"column": 56
} | [
{
"pp": "V : Type u_1\ninst✝⁵ : Category.{v_1, u_1} V\nG : Type u_2\ninst✝⁴ : Monoid G\nW : Type u_3\ninst✝³ : Category.{v_2, u_3} W\ninst✝² : MonoidalCategory V\ninst✝¹ : MonoidalCategory W\nF : V ⥤ W\ninst✝ : F.OplaxMonoidal\ng : G\n⊢ F.map (𝟙 (𝟙_ V)) ≫ η F = η F ≫ 𝟙 (𝟙_ W)",
"ppTerm": "?m.115",
"... | [] | rw [map_id, Category.id_comp, Category.comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Bicategory.FunctorBicategory.Lax | {
"line": 43,
"column": 8
} | {
"line": 52,
"column": 20
} | {
"line": 52,
"column": 21
} | [] | [] | _ = 𝟙 _ ⊗≫ η.app a ◁ ((Γ.as.app a ▷ H.map f ≫ ι.naturality f)) ⊗≫
η.naturality f ▷ (ι.app b) ⊗≫ 𝟙 _ := by
bicategory
_ = 𝟙 _ ⊗≫ η.app a ◁ θ.naturality f ⊗≫
((η.app a ≫ G.map f) ◁ Γ.as.app b ≫ η.naturality f ▷ ι.app b) ⊗≫ 𝟙 _ := by
rw [Γ.as.naturality]
... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.CategoryTheory.Core | {
"line": 75,
"column": 2
} | {
"line": 76,
"column": 17
} | {
"line": 78,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y : Core C\nf g : X ⟶ Y\nh : f.iso.hom = g.iso.hom\n⊢ f = g",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"CategoryTheory.CoreHom.iso",
"CategoryTheory.Core.of",
"CategoryTheory.CoreHom.ext",
"CategoryTheory.Iso... | [] | apply CoreHom.ext
exact Iso.ext h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Core | {
"line": 75,
"column": 2
} | {
"line": 76,
"column": 17
} | {
"line": 78,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y : Core C\nf g : X ⟶ Y\nh : f.iso.hom = g.iso.hom\n⊢ f = g",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"CategoryTheory.CoreHom.iso",
"CategoryTheory.Core.of",
"CategoryTheory.CoreHom.ext",
"CategoryTheory.Iso... | [] | apply CoreHom.ext
exact Iso.ext h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Bicategory.Strict.Pseudofunctor | {
"line": 188,
"column": 24
} | {
"line": 188,
"column": 42
} | {
"line": 188,
"column": 42
} | [
{
"pp": "B : Type u₁\nC : Type u₂\ninst✝² : Bicategory B\ninst✝¹ : Strict B\ninst✝ : Bicategory C\nF : B ⥤ᵖ C\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : B\nt : X₁ ⟶ Y₁\nl : X₁ ⟶ X₂\nr : Y₁ ⟶ Y₂\nb : X₂ ⟶ Y₂\nsq : CommSq t l r b\nφ : X₁ ⟶ Y₂\nhφ : t ≫ r = φ\n⊢ l ≫ b = φ",
"ppTerm": "?m.67",
"assigned": true,
"usedConstant... | [] | by rw [← hφ, sq.w] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.ConnectedComponents | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 38
} | {
"line": 126,
"column": 2
} | [
{
"pp": "J : Type u₁\ninst✝ : Category.{v₁, u₁} J\nj₁ j₂ : J\nhj₁ : ConnectedComponents.objectProperty (Quotient.mk'' j₂) j₁\nh₁₂ : Zigzag j₁ j₂\nl : List J\nhl₁ : List.IsChain Zag (j₁ :: l)\nhl₂ : (j₁ :: l).getLast ⋯ = j₂\nf : (x : J) → Zigzag x j₂ → (ConnectedComponents.mk j₂).Component := ⋯\ni : J\nhi : i ∈ ... | [] | · apply Relation.ReflTransGen.refl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.PFun | {
"line": 243,
"column": 4
} | {
"line": 252,
"column": 35
} | {
"line": 253,
"column": 4
} | [
{
"pp": "case inl\nα : Type u_1\nβ : Type u_2\nf : α →. β ⊕ α\na : α\nb : β\nh₁ : (f a).Dom\nh₂ : (f a).get h₁ = Sum.inl b\n⊢ ∃ h,\n b ∈\n WellFounded.fixF\n (fun a IH ↦\n Part.assert (f a).Dom fun hf ↦\n match e : (f a).get hf with\n | Sum.inl b => Part.some b\n ... | [
"case inr\nα : Type u_1\nβ : Type u_2\nf : α →. β ⊕ α\na : α\nb : β\na' : α\nh : Sum.inr a' ∈ f a\nh₃ : b ∈ f.fix a'\n⊢ ∃ h,\n b ∈\n WellFounded.fixF\n (fun a IH ↦\n Part.assert (f a).Dom fun hf ↦\n match e : (f a).get hf with\n | Sum.inl b => Part.some b\n |... | · refine ⟨⟨_, fun y h' => ?_⟩, ?_⟩
· injection Part.mem_unique ⟨h₁, h₂⟩ h'
· rw [WellFounded.fixF_eq]
-- Porting note: used to be simp [h₁, h₂]
apply Part.mem_assert h₁
split
next e =>
injection h₂.symm.trans e with h; simp [h]
next e =>
injection ... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Category.RelCat | {
"line": 108,
"column": 4
} | {
"line": 109,
"column": 32
} | {
"line": 111,
"column": 0
} | [
{
"pp": "case mpr\nX Y : RelCat\nr : X ⟶ Y\n⊢ (∃ f, graphFunctor.map f.hom = r) → IsIso r",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"CategoryTheory.IsIso",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"Exists",
"CategoryTheory.RelCat.instLargeC... | [] | rintro ⟨f, rfl⟩
apply graphFunctor.map_isIso | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Category.RelCat | {
"line": 108,
"column": 4
} | {
"line": 109,
"column": 32
} | {
"line": 111,
"column": 0
} | [
{
"pp": "case mpr\nX Y : RelCat\nr : X ⟶ Y\n⊢ (∃ f, graphFunctor.map f.hom = r) → IsIso r",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"CategoryTheory.IsIso",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"Exists",
"CategoryTheory.RelCat.instLargeC... | [] | rintro ⟨f, rfl⟩
apply graphFunctor.map_isIso | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Localization.Prod | {
"line": 83,
"column": 2
} | {
"line": 89,
"column": 51
} | {
"line": 90,
"column": 2
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nD₁ : Type u₃\nD₂ : Type u₄\ninst✝⁶ : Category.{v₁, u₁} C₁\ninst✝⁵ : Category.{v₂, u₂} C₂\ninst✝⁴ : Category.{v₃, u₃} D₁\ninst✝³ : Category.{v₄, u₄} D₂\nL₁ : C₁ ⥤ D₁\nW₁ : MorphismProperty C₁\nL₂ : C₂ ⥤ D₂\nW₂ : MorphismProperty C₂\nE : Type u₅\ninst✝² : Category.{v₅, u₅} E\n... | [
"C₁ : Type u₁\nC₂ : Type u₂\nD₁ : Type u₃\nD₂ : Type u₄\ninst✝⁶ : Category.{v₁, u₁} C₁\ninst✝⁵ : Category.{v₂, u₂} C₂\ninst✝⁴ : Category.{v₃, u₃} D₁\ninst✝³ : Category.{v₄, u₄} D₂\nL₁ : C₁ ⥤ D₁\nW₁ : MorphismProperty C₁\nL₂ : C₂ ⥤ D₂\nW₂ : MorphismProperty C₂\nE : Type u₅\ninst✝² : Category.{v₅, u₅} E\nF : C₁ × C₂ ... | haveI : ∀ (X₁ : W₁.Localization),
IsIso (((Functor.flip (prodLift₁ F hF)).map f₂).app X₁) := fun X₁ => by
obtain ⟨X₁, rfl⟩ := (Construction.objEquiv W₁).surjective X₁
exact ((MorphismProperty.isomorphisms E).arrow_mk_iso_iff
(((Functor.mapArrowFunctor _ _).mapIso
(eqToIso (Functor.congr_obj ... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.CategoryTheory.LocallyCartesianClosed.ChosenPullbacksAlong | {
"line": 303,
"column": 50
} | {
"line": 303,
"column": 90
} | {
"line": 303,
"column": 90
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nY Z X : C\nf : Y ⟶ X\ng : Z ⟶ X\ninst✝² : ChosenPullbacksAlong g\nY' Z' X' Y'' Z'' X'' : C\nf' : Y' ⟶ X'\ng' : Z' ⟶ X'\nf'' : Y'' ⟶ X''\ng'' : Z'' ⟶ X''\ninst✝¹ : ChosenPullbacksAlong g'\ninst✝ : ChosenPullbacksAlong g''\nγ₁ : Y' ⟶ Y\nγ₂ : Z' ⟶ Z\nγ₃ : X' ⟶ X\... | [] | by rw [reassoc_of% comm₂', comm₂, assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sums.Basic | {
"line": 271,
"column": 22
} | {
"line": 272,
"column": 20
} | {
"line": 274,
"column": 0
} | [
{
"pp": "A : Type u₁\ninst✝³ : Category.{v₁, u₁} A\nB : Type u₂\ninst✝² : Category.{v₂, u₂} B\nC : Type u₃\ninst✝¹ : Category.{v₃, u₃} C\nD : Type u₄\ninst✝ : Category.{v₄, u₄} D\nF G : A ⥤ C\nH I : B ⥤ C\nα : F ⟶ G\nβ : H ⟶ I\nX Y : A ⊕ B\nf : X ⟶ Y\n⊢ ((F.sum' H).map f ≫\n match Y with\n | inl X => ... | [] | by
cases f <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Enriched.FunctorCategory | {
"line": 148,
"column": 18
} | {
"line": 148,
"column": 49
} | {
"line": 149,
"column": 6
} | [
{
"pp": "V : Type u₁\ninst✝⁸ : Category.{v₁, u₁} V\ninst✝⁷ : MonoidalCategory V\nC : Type u₂\ninst✝⁶ : Category.{v₂, u₂} C\nJ : Type u₃\ninst✝⁵ : Category.{v₃, u₃} J\nK : Type u₄\ninst✝⁴ : Category.{v₄, u₄} K\ninst✝³ : EnrichedOrdinaryCategory V C\nF₁ F₂ F₃ F₄ : J ⥤ C\ninst✝² : HasEnrichedHom V F₁ F₂\ninst✝¹ : ... | [
"V : Type u₁\ninst✝⁸ : Category.{v₁, u₁} V\ninst✝⁷ : MonoidalCategory V\nC : Type u₂\ninst✝⁶ : Category.{v₂, u₂} C\nJ : Type u₃\ninst✝⁵ : Category.{v₃, u₃} J\nK : Type u₄\ninst✝⁴ : Category.{v₄, u₄} K\ninst✝³ : EnrichedOrdinaryCategory V C\nF₁ F₂ F₃ F₄ : J ⥤ C\ninst✝² : HasEnrichedHom V F₁ F₂\ninst✝¹ : HasEnrichedH... | rw [assoc, tensorHom_def_assoc] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.CategoryTheory.Enriched.FunctorCategory | {
"line": 148,
"column": 18
} | {
"line": 148,
"column": 49
} | {
"line": 149,
"column": 6
} | [
{
"pp": "V : Type u₁\ninst✝⁸ : Category.{v₁, u₁} V\ninst✝⁷ : MonoidalCategory V\nC : Type u₂\ninst✝⁶ : Category.{v₂, u₂} C\nJ : Type u₃\ninst✝⁵ : Category.{v₃, u₃} J\nK : Type u₄\ninst✝⁴ : Category.{v₄, u₄} K\ninst✝³ : EnrichedOrdinaryCategory V C\nF₁ F₂ F₃ F₄ : J ⥤ C\ninst✝² : HasEnrichedHom V F₁ F₂\ninst✝¹ : ... | [
"V : Type u₁\ninst✝⁸ : Category.{v₁, u₁} V\ninst✝⁷ : MonoidalCategory V\nC : Type u₂\ninst✝⁶ : Category.{v₂, u₂} C\nJ : Type u₃\ninst✝⁵ : Category.{v₃, u₃} J\nK : Type u₄\ninst✝⁴ : Category.{v₄, u₄} K\ninst✝³ : EnrichedOrdinaryCategory V C\nF₁ F₂ F₃ F₄ : J ⥤ C\ninst✝² : HasEnrichedHom V F₁ F₂\ninst✝¹ : HasEnrichedH... | rw [assoc, tensorHom_def_assoc] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.CategoryTheory.Enriched.FunctorCategory | {
"line": 148,
"column": 18
} | {
"line": 148,
"column": 49
} | {
"line": 149,
"column": 6
} | [
{
"pp": "V : Type u₁\ninst✝⁸ : Category.{v₁, u₁} V\ninst✝⁷ : MonoidalCategory V\nC : Type u₂\ninst✝⁶ : Category.{v₂, u₂} C\nJ : Type u₃\ninst✝⁵ : Category.{v₃, u₃} J\nK : Type u₄\ninst✝⁴ : Category.{v₄, u₄} K\ninst✝³ : EnrichedOrdinaryCategory V C\nF₁ F₂ F₃ F₄ : J ⥤ C\ninst✝² : HasEnrichedHom V F₁ F₂\ninst✝¹ : ... | [
"V : Type u₁\ninst✝⁸ : Category.{v₁, u₁} V\ninst✝⁷ : MonoidalCategory V\nC : Type u₂\ninst✝⁶ : Category.{v₂, u₂} C\nJ : Type u₃\ninst✝⁵ : Category.{v₃, u₃} J\nK : Type u₄\ninst✝⁴ : Category.{v₄, u₄} K\ninst✝³ : EnrichedOrdinaryCategory V C\nF₁ F₂ F₃ F₄ : J ⥤ C\ninst✝² : HasEnrichedHom V F₁ F₂\ninst✝¹ : HasEnrichedH... | rw [assoc, tensorHom_def_assoc] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.CategoryTheory.FiberedCategory.HasFibers | {
"line": 118,
"column": 65
} | {
"line": 119,
"column": 51
} | {
"line": 121,
"column": 0
} | [
{
"pp": "𝒮 : Type u₁\n𝒳 : Type u₂\ninst✝² : Category.{v₁, u₁} 𝒮\ninst✝¹ : Category.{v₂, u₂} 𝒳\np : 𝒳 ⥤ 𝒮\ninst✝ : HasFibers p\nS : 𝒮\na : Fib p S\n⊢ p.obj ((ι S).obj a) = S",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"congrArg",
"Categ... | [] | by
simp only [← comp_obj, comp_const, const_obj_obj] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Enriched.FunctorCategory | {
"line": 233,
"column": 6
} | {
"line": 233,
"column": 13
} | {
"line": 233,
"column": 13
} | [
{
"pp": "V : Type u₁\ninst✝¹⁰ : Category.{v₁, u₁} V\ninst✝⁹ : MonoidalCategory V\nC : Type u₂\ninst✝⁸ : Category.{v₂, u₂} C\nJ : Type u₃\ninst✝⁷ : Category.{v₃, u₃} J\ninst✝⁶ : EnrichedOrdinaryCategory V C\nF₁ F₂ F₃ F₄ : J ⥤ C\ninst✝⁵ : HasEnrichedHom V F₁ F₂\ninst✝⁴ : HasEnrichedHom V F₁ F₃\ninst✝³ : HasEnrich... | [
"V : Type u₁\ninst✝¹⁰ : Category.{v₁, u₁} V\ninst✝⁹ : MonoidalCategory V\nC : Type u₂\ninst✝⁸ : Category.{v₂, u₂} C\nJ : Type u₃\ninst✝⁷ : Category.{v₃, u₃} J\ninst✝⁶ : EnrichedOrdinaryCategory V C\nF₁ F₂ F₃ F₄ : J ⥤ C\ninst✝⁵ : HasEnrichedHom V F₁ F₂\ninst✝⁴ : HasEnrichedHom V F₁ F₃\ninst✝³ : HasEnrichedHom V F₁ F... | e_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Enriched.FunctorCategory | {
"line": 348,
"column": 4
} | {
"line": 360,
"column": 7
} | {
"line": 360,
"column": 7
} | [
{
"pp": "V : Type u₁\ninst✝⁷ : Category.{v₁, u₁} V\ninst✝⁶ : MonoidalCategory V\nC : Type u₂\ninst✝⁵ : Category.{v₂, u₂} C\nJ : Type u₃\ninst✝⁴ : Category.{v₃, u₃} J\nK : Type u₄\ninst✝³ : Category.{v₄, u₄} K\ninst✝² : EnrichedOrdinaryCategory V C\nF₁ F₂ F₃ F₄ : J ⥤ C\ninst✝¹ : HasFunctorEnrichedHom V F₁ F₂\nin... | [] | dsimp
rw [← s.w f, assoc, assoc, assoc]
-- this was produced by `simp?`
simp only [functorEnrichedHom_obj, functorEnrichedHom_map, end_.lift_π_assoc, diagram_obj_obj,
Functor.comp_obj, Under.forget_obj, Under.mk_right, Under.map_obj_right, Iso.refl_inv,
NatTrans.id_app, eHomWhiskerRight_id, Iso.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Enriched.FunctorCategory | {
"line": 348,
"column": 4
} | {
"line": 360,
"column": 7
} | {
"line": 360,
"column": 7
} | [
{
"pp": "V : Type u₁\ninst✝⁷ : Category.{v₁, u₁} V\ninst✝⁶ : MonoidalCategory V\nC : Type u₂\ninst✝⁵ : Category.{v₂, u₂} C\nJ : Type u₃\ninst✝⁴ : Category.{v₃, u₃} J\nK : Type u₄\ninst✝³ : Category.{v₄, u₄} K\ninst✝² : EnrichedOrdinaryCategory V C\nF₁ F₂ F₃ F₄ : J ⥤ C\ninst✝¹ : HasFunctorEnrichedHom V F₁ F₂\nin... | [] | dsimp
rw [← s.w f, assoc, assoc, assoc]
-- this was produced by `simp?`
simp only [functorEnrichedHom_obj, functorEnrichedHom_map, end_.lift_π_assoc, diagram_obj_obj,
Functor.comp_obj, Under.forget_obj, Under.mk_right, Under.map_obj_right, Iso.refl_inv,
NatTrans.id_app, eHomWhiskerRight_id, Iso.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Galois.Examples | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 34
} | {
"line": 84,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝ : Group G\n⊢ PreservesFiniteLimits (Action.forget FintypeCat G ⋙ FintypeCat.incl)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Finite",
"FintypeCat",
"CategoryTheory.Limits.FintypeCat.hasFiniteLimits",
"Action.instPreservesFiniteLimits... | [] | apply comp_preservesFiniteLimits | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Galois.GaloisObjects | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 47
} | {
"line": 62,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\nX : C\ng h : Aut X\na : (F.obj X).obj\n⊢ (ConcreteCategory.hom (F.map (h.hom ≫ g.hom))) a = (ConcreteCategory.hom (F.map h.hom ≫ F.map g.hom)) a",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"CategoryTheory.Cate... | [] | simp only [map_comp, FintypeCat.comp_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Galois.Basic | {
"line": 233,
"column": 2
} | {
"line": 235,
"column": 62
} | {
"line": 237,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\ninst✝¹ : PreGaloisCategory C\ninst✝ : FiberFunctor F\nX : C\n⊢ (∀ (a : IsInitial X), False) ↔ Nonempty (F.obj X).obj",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"False",
... | [] | rw [← not_isEmpty_iff]
refine ⟨fun h he ↦ ?_, fun h hin ↦ h <| (initial_iff_fiber_empty F X).mp ⟨hin⟩⟩
exact Nonempty.elim ((initial_iff_fiber_empty F X).mpr he) h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Galois.Basic | {
"line": 233,
"column": 2
} | {
"line": 235,
"column": 62
} | {
"line": 237,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\ninst✝¹ : PreGaloisCategory C\ninst✝ : FiberFunctor F\nX : C\n⊢ (∀ (a : IsInitial X), False) ↔ Nonempty (F.obj X).obj",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"False",
... | [] | rw [← not_isEmpty_iff]
refine ⟨fun h he ↦ ?_, fun h hin ↦ h <| (initial_iff_fiber_empty F X).mp ⟨hin⟩⟩
exact Nonempty.elim ((initial_iff_fiber_empty F X).mpr he) h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Galois.Full | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 28
} | {
"line": 117,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nF : C ⥤ FintypeCat\ninst✝¹ : GaloisCategory C\ninst✝ : FiberFunctor F\nX Y : C\nf : (functorToAction F).obj X ⟶ (functorToAction F).obj Y\nu : (functorToAction F).obj X ⟶ (functorToAction F).obj X ⨯ (functorToAction F).obj Y :=\n prod.lift (𝟙 ((functorToA... | [
"case h\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nF : C ⥤ FintypeCat\ninst✝¹ : GaloisCategory C\ninst✝ : FiberFunctor F\nX Y : C\nf : (functorToAction F).obj X ⟶ (functorToAction F).obj Y\nu : (functorToAction F).obj X ⟶ (functorToAction F).obj X ⨯ (functorToAction F).obj Y := ⋯\ni : (functorToAction F).obj X ... | use inv ψ ≫ g ≫ prod.snd | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.CategoryTheory.Galois.Topology | {
"line": 97,
"column": 46
} | {
"line": 98,
"column": 63
} | {
"line": 98,
"column": 63
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\na : (X : C) → Aut (F.obj X)\nh : ∀ (i : Arrow C), F.map i.hom ≫ (a i.right).hom = (a i.left).hom ≫ F.map i.hom\nX Y : C\nf : X ⟶ Y\n⊢ F.map f ≫ (a Y).hom = (a X).hom ≫ F.map f",
"ppTerm": "?m.102",
"assigned": true,
"usedConstant... | [] | by
ext; simpa using ConcreteCategory.congr_hom (h ⟨X, Y, f⟩) _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Galois.Decomposition | {
"line": 288,
"column": 4
} | {
"line": 288,
"column": 42
} | {
"line": 289,
"column": 4
} | [
{
"pp": "case h\nC : Type u₁\ninst✝² : Category.{u₂, u₁} C\ninst✝¹ : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝ : FiberFunctor F\nX A : C\nu : A ⟶ selfProd F X\na : (F.obj A).obj\nh1 : (ConcreteCategory.hom (F.map u)) a = mkSelfProdFib F X\nh2 : IsConnected A\nh3 : Mono u\nx y : (F.obj A).obj\nfi1 : A ≅ A\nhfi... | [
"case h\nC : Type u₁\ninst✝² : Category.{u₂, u₁} C\ninst✝¹ : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝ : FiberFunctor F\nX A : C\nu : A ⟶ selfProd F X\na : (F.obj A).obj\nh1 : (ConcreteCategory.hom (F.map u)) a = mkSelfProdFib F X\nh2 : IsConnected A\nh3 : Mono u\nx y : (F.obj A).obj\nfi1 : A ≅ A\nhfi1 : (Concret... | change F.map (fi1.hom ≫ fi2.inv) x = y | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.CategoryTheory.Galois.Decomposition | {
"line": 289,
"column": 4
} | {
"line": 289,
"column": 47
} | {
"line": 290,
"column": 4
} | [
{
"pp": "case h\nC : Type u₁\ninst✝² : Category.{u₂, u₁} C\ninst✝¹ : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝ : FiberFunctor F\nX A : C\nu : A ⟶ selfProd F X\na : (F.obj A).obj\nh1 : (ConcreteCategory.hom (F.map u)) a = mkSelfProdFib F X\nh2 : IsConnected A\nh3 : Mono u\nx y : (F.obj A).obj\nfi1 : A ≅ A\nhfi... | [
"case h\nC : Type u₁\ninst✝² : Category.{u₂, u₁} C\ninst✝¹ : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝ : FiberFunctor F\nX A : C\nu : A ⟶ selfProd F X\na : (F.obj A).obj\nh1 : (ConcreteCategory.hom (F.map u)) a = mkSelfProdFib F X\nh2 : IsConnected A\nh3 : Mono u\nx y : (F.obj A).obj\nfi1 : A ≅ A\nhfi1 : (Concret... | simp only [map_comp, FintypeCat.comp_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Galois.IsFundamentalgroup | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 20
} | {
"line": 98,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : (X : C) → MulAction G (F.obj X).obj\ninst✝ : IsNaturalSMul F G\nx✝¹ : C\nx✝ : (F.obj x✝¹).obj\n⊢ (ConcreteCategory.hom ((NatIso.ofComponents (isoOnObj F 1) ⋯).hom.app x✝¹)) x✝ =\n (ConcreteCategor... | [
"C : Type u₁\ninst✝³ : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : (X : C) → MulAction G (F.obj X).obj\ninst✝ : IsNaturalSMul F G\nx✝¹ : C\nx✝ : (F.obj x✝¹).obj\n⊢ 1 • x✝ = (ConcreteCategory.hom ((Iso.hom 1).app x✝¹)) x✝"
] | dsimp [isoOnObj] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Galois.Prorepresentability | {
"line": 397,
"column": 4
} | {
"line": 397,
"column": 91
} | {
"line": 398,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{u₂, u₁} C\ninst✝¹ : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝ : FiberFunctor F\nt : (AutGalois F)ᵐᵒᵖ\n⊢ ((↑(endMulEquivAutGalois F)).comp (Aut.toEnd F)) ((fun t ↦ asIso ((endMulEquivAutGalois F).symm t)) t) = t",
"ppTerm": "?m.63",
"assigned": true,
"usedCo... | [
"C : Type u₁\ninst✝² : Category.{u₂, u₁} C\ninst✝¹ : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝ : FiberFunctor F\nt : (AutGalois F)ᵐᵒᵖ\n⊢ (endMulEquivAutGalois F) ↑((Aut.unitsEndEquivAut F).symm (asIso ((endMulEquivAutGalois F).symm t))) = t"
] | simp only [MonoidHom.coe_comp, MonoidHom.coe_coe, Function.comp_apply, Aut.toEnd_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Galois.Prorepresentability | {
"line": 436,
"column": 2
} | {
"line": 436,
"column": 24
} | {
"line": 437,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{u₂, u₁} C\ninst✝² : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝¹ : FiberFunctor F\nX : C\ninst✝ : IsConnected X\nA : C\nf : A ⟶ X\nhgal : IsGalois A\nhs : Function.Surjective ⇑(ConcreteCategory.hom (F.map f))\nx y : (F.obj X).obj\na : (F.obj A).obj\nha : (ConcreteCategor... | [
"C : Type u₁\ninst✝³ : Category.{u₂, u₁} C\ninst✝² : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝¹ : FiberFunctor F\nX : C\ninst✝ : IsConnected X\nA : C\nf : A ⟶ X\nhgal : IsGalois A\nhs : Function.Surjective ⇑(ConcreteCategory.hom (F.map f))\nx y : (F.obj X).obj\na : (F.obj A).obj\nha : (ConcreteCategory.hom (F.map... | obtain ⟨b, hb⟩ := hs y | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Galois.EssSurj | {
"line": 93,
"column": 10
} | {
"line": 93,
"column": 22
} | {
"line": 93,
"column": 22
} | [
{
"pp": "G : Type u_1\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalSpace G\ninst✝⁴ : IsTopologicalGroup G\ninst✝³ : CompactSpace G\nX : Action FintypeCat G\ninst✝² : TopologicalSpace X.V.obj\ninst✝¹ : DiscreteTopology X.V.obj\ninst✝ : ContinuousSMul G X.V.obj\nι : Type\nhf : Finite ι\nf : ι → Action FintypeCat G\nu :... | [
"G : Type u_1\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalSpace G\ninst✝⁴ : IsTopologicalGroup G\ninst✝³ : CompactSpace G\nX : Action FintypeCat G\ninst✝² : TopologicalSpace X.V.obj\ninst✝¹ : DiscreteTopology X.V.obj\ninst✝ : ContinuousSMul G X.V.obj\nι : Type\nhf : Finite ι\nf : ι → Action FintypeCat G\nu : ∐ f ≅ X\nhc... | ← le_bot_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.GradedObject.Braiding | {
"line": 45,
"column": 26
} | {
"line": 45,
"column": 64
} | {
"line": 45,
"column": 64
} | [
{
"pp": "I : Type u_1\ninst✝⁵ : AddCommMonoid I\nC : Type u_2\ninst✝⁴ : Category.{v_1, u_2} C\ninst✝³ : MonoidalCategory C\nX Y Z : GradedObject I C\ninst✝² : BraidedCategory C\ninst✝¹ : X.HasTensor Y\ninst✝ : Y.HasTensor X\nk i j : I\nhij : i + j = k\n⊢ j + i = k",
"ppTerm": "?m.114",
"assigned": true,... | [] | by simpa only [add_comm j i] using hij | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.GradedObject.Braiding | {
"line": 47,
"column": 26
} | {
"line": 47,
"column": 64
} | {
"line": 47,
"column": 64
} | [
{
"pp": "I : Type u_1\ninst✝⁵ : AddCommMonoid I\nC : Type u_2\ninst✝⁴ : Category.{v_1, u_2} C\ninst✝³ : MonoidalCategory C\nX Y Z : GradedObject I C\ninst✝² : BraidedCategory C\ninst✝¹ : X.HasTensor Y\ninst✝ : Y.HasTensor X\nk i j : I\nhij : i + j = k\n⊢ j + i = k",
"ppTerm": "?m.121",
"assigned": true,... | [] | by simpa only [add_comm j i] using hij | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Galois.EssSurj | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 34
} | {
"line": 193,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\ninst✝⁵ : GaloisCategory C\ninst✝⁴ : FiberFunctor F\nG : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : CompactSpace G\nV U : OpenSubgroup (Aut F)\nh : (↑U).Normal\nA : C\nu : (functorToAction... | [
"C : Type u₁\ninst✝⁶ : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\ninst✝⁵ : GaloisCategory C\ninst✝⁴ : FiberFunctor F\nG : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : CompactSpace G\nV U : OpenSubgroup (Aut F)\nh : (↑U).Normal\nA : C\nu : (functorToAction F).obj A ≅ ... | ext (x : Aut F ⧸ V.toSubgroup) | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.CategoryTheory.Galois.EssSurj | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 34
} | {
"line": 219,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\ninst✝⁵ : GaloisCategory C\ninst✝⁴ : FiberFunctor F\nG : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : CompactSpace G\nV U : OpenSubgroup (Aut F)\nh : (↑U).Normal\nA : C\nu : (functorToAction... | [
"C : Type u₁\ninst✝⁶ : Category.{u₂, u₁} C\nF : C ⥤ FintypeCat\ninst✝⁵ : GaloisCategory C\ninst✝⁴ : FiberFunctor F\nG : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : CompactSpace G\nV U : OpenSubgroup (Aut F)\nh : (↑U).Normal\nA : C\nu : (functorToAction F).obj A ≅ ... | ext (x : Aut F ⧸ V.toSubgroup) | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.CategoryTheory.Idempotents.Biproducts | {
"line": 70,
"column": 6
} | {
"line": 71,
"column": 64
} | {
"line": 73,
"column": 0
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝³ : Category.{v, u_1} C\ninst✝² : Preadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Finite J\nF : J → Karoubi C\nj j' : J\nh : ¬j = j'\n⊢ { f := biproduct.ι (fun j ↦ (F j).X) j ≫ biproduct.map fun j ↦ (F j).p, comm := ⋯ } ≫\n { f := (biproduct.map fun j ... | [] | simp only [biproduct.ι_map, biproduct.map_π, hom_ext_iff, comp_f,
assoc, biproduct.ι_π_ne_assoc _ h, zero_comp, comp_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Groupoid.Subgroupoid | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 30
} | {
"line": 348,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝ : Groupoid C\nX : (c d : C) → Set (c ⟶ d)\nc d : C\n⊢ X c d ⊆ (generated X).arrows c d",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"id",
"LE.le",
"CategoryTheory.Subgroupoid... | [
"C : Type u\ninst✝ : Groupoid C\nX : (c d : C) → Set (c ⟶ d)\nc d : C\n⊢ X c d ⊆ ⋂ S ∈ {S | ∀ (c d : C), X c d ⊆ S.arrows c d}, S.arrows c d"
] | dsimp only [generated, sInf] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Groupoid.Subgroupoid | {
"line": 517,
"column": 6
} | {
"line": 517,
"column": 34
} | {
"line": 518,
"column": 6
} | [
{
"pp": "case refl\nC : Type u\ninst✝¹ : Groupoid C\nS : Subgroupoid C\nD : Type u_1\ninst✝ : Groupoid D\nφ : C ⥤ D\nhφ : Function.Injective φ.obj\nhφ' : im φ hφ = ⊤\nSn : S.IsNormal\nd' : D\nc : C\ng : φ.obj c ⟶ d'\nγ : c ⟶ c\nγS : γ ∈ S.arrows c c\ncd' : φ.obj c = φ.obj c\nthis : d' ∈ (im φ hφ).objs\n⊢ Groupo... | [
"case refl\nC : Type u\ninst✝¹ : Groupoid C\nS : Subgroupoid C\nD : Type u_1\ninst✝ : Groupoid D\nφ : C ⥤ D\nhφ : Function.Injective φ.obj\nhφ' : im φ hφ = ⊤\nSn : S.IsNormal\nd' : D\nc : C\ng : φ.obj c ⟶ d'\nγ : c ⟶ c\nγS : γ ∈ S.arrows c c\ncd' : φ.obj c = φ.obj c\nthis : ∃ c, φ.obj c = d'\n⊢ Groupoid.inv g ≫ (eq... | rw [mem_im_objs_iff] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Final.ParallelPair | {
"line": 42,
"column": 10
} | {
"line": 43,
"column": 42
} | {
"line": 44,
"column": 4
} | [
{
"pp": "case one\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y : C\nf g : X ⟶ Y\nh₁ : ∀ (Z : C), Nonempty (X ⟶ Z)\nh₂ : ∀ ⦃Z : C⦄ (i j : X ⟶ Z), Zigzag (mk i) (mk j)\nZ : C\nthis : Nonempty (CostructuredArrow (parallelPair f g) Z)\nas✝ : PUnit.{1}\nφ : (parallelPair f g).obj one ⟶ (Functor.fromPUnit Z).obj... | [] | refine Zigzag.trans ?_ (h₂ (f ≫ φ) _)
exact Zigzag.of_inv (homMk left) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Final.ParallelPair | {
"line": 42,
"column": 10
} | {
"line": 43,
"column": 42
} | {
"line": 44,
"column": 4
} | [
{
"pp": "case one\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y : C\nf g : X ⟶ Y\nh₁ : ∀ (Z : C), Nonempty (X ⟶ Z)\nh₂ : ∀ ⦃Z : C⦄ (i j : X ⟶ Z), Zigzag (mk i) (mk j)\nZ : C\nthis : Nonempty (CostructuredArrow (parallelPair f g) Z)\nas✝ : PUnit.{1}\nφ : (parallelPair f g).obj one ⟶ (Functor.fromPUnit Z).obj... | [] | refine Zigzag.trans ?_ (h₂ (f ≫ φ) _)
exact Zigzag.of_inv (homMk left) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.FormalCoproducts.Basic | {
"line": 247,
"column": 11
} | {
"line": 247,
"column": 22
} | {
"line": 247,
"column": 22
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u₁\ninst✝ : Category.{v₁, u₁} A\n𝒜 : Type w\nf : 𝒜 → FormalCoproduct C\nt X✝ X : FormalCoproduct C\ni j : X.I\nhij : i = j\n⊢ X.obj i = X.obj j",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [] | by rw [hij] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.PiProd | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 40
} | {
"line": 72,
"column": 2
} | [
{
"pp": "C : Type u_1\nI : Type u_2\ninst✝⁷ : Category.{v_1, u_1} C\nX Y : I → C\nf : (i : I) → X i ⟶ Y i\nP : I → Prop\ninst✝⁶ : HasProduct X\ninst✝⁵ : HasProduct Y\ninst✝⁴ : HasProduct fun i ↦ X ↑i\ninst✝³ : HasProduct fun i ↦ X ↑i\ninst✝² : HasProduct fun i ↦ Y ↑i\ninst✝¹ : HasProduct fun i ↦ Y ↑i\ninst✝ : (... | [
"C : Type u_1\nI : Type u_2\ninst✝⁷ : Category.{v_1, u_1} C\nX Y : I → C\nf : (i : I) → X i ⟶ Y i\nP : I → Prop\ninst✝⁶ : HasProduct X\ninst✝⁵ : HasProduct Y\ninst✝⁴ : HasProduct fun i ↦ X ↑i\ninst✝³ : HasProduct fun i ↦ X ↑i\ninst✝² : HasProduct fun i ↦ Y ↑i\ninst✝¹ : HasProduct fun i ↦ Y ↑i\ninst✝ : (i : I) → Dec... | rw [← Category.assoc, Iso.eq_comp_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.ChosenPullback | {
"line": 188,
"column": 30
} | {
"line": 189,
"column": 56
} | {
"line": 191,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX₁ X₂ X₃ S : C\nf₁ : X₁ ⟶ S\nf₂ : X₂ ⟶ S\nf₃ : X₃ ⟶ S\nh₁₂ : ChosenPullback f₁ f₂\nh₂₃ : ChosenPullback f₂ f₃\nh₁₃ : ChosenPullback f₁ f₃\nh : ChosenPullback₃ h₁₂ h₂₃ h₁₃\n⊢ h.p₂ ≫ f₂ = h.p",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
... | [] | by
rw [← p₂₃_p₂_assoc, h₂₃.condition, ← w₃, p₂₃_p₃_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Localization.Pi | {
"line": 69,
"column": 6
} | {
"line": 69,
"column": 77
} | {
"line": 70,
"column": 6
} | [
{
"pp": "case h_option.refine_1\nJ : Type w\ninst✝⁴ : Fintype J\nhJ :\n ∀ {C : J → Type u₁} {D : J → Type u₂} [inst : (j : J) → Category.{v₁, u₁} (C j)]\n [inst_1 : (j : J) → Category.{v₂, u₂} (D j)] (L : (j : J) → C j ⥤ D j) (W : (j : J) → MorphismProperty (C j))\n [∀ (j : J), (W j).ContainsIdentities] ... | [
"case h_option.refine_1\nJ : Type w\ninst✝⁴ : Fintype J\nhJ :\n ∀ {C : J → Type u₁} {D : J → Type u₂} [inst : (j : J) → Category.{v₁, u₁} (C j)]\n [inst_1 : (j : J) → Category.{v₂, u₂} (D j)] (L : (j : J) → C j ⥤ D j) (W : (j : J) → MorphismProperty (C j))\n [∀ (j : J), (W j).ContainsIdentities] [∀ (j : J), ... | refine ⟨_, _, (Pi.optionEquivalence C).inverse.map f, ?_, ⟨Iso.refl _⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Monoidal.Free.Basic | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 54
} | {
"line": 173,
"column": 2
} | [
{
"pp": "case mk.mk.mk\nC : Type u\nX₁ X₂ X₃ Y₁ Y₂ Y₃ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\nf₁✝ : X₁ ⟶ Y₁\nf₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\nf₂✝ : X₂ ⟶ Y₂\nf₂ : failed to pretty print e... | [] | exact Quotient.sound (associator_naturality _ _ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Localization.Monoidal.Braided | {
"line": 119,
"column": 2
} | {
"line": 123,
"column": 42
} | {
"line": 125,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝³ : MonoidalCategory C\ninst✝² : W.IsMonoidal\ninst✝¹ : L.IsLocalization W\nunit : D\nε : L.obj (𝟙_ C) ≅ unit\ninst✝ : BraidedCategory C\n⊢ BraidedCategory (LocalizedMono... | [] | refine .ofBifunctor (braidingNatIso L W ε) ?_ ?_
· apply natTrans₃_ext (L') (L') (L') W W W
simpa using! map_hexagon_forward _ _ _
· apply natTrans₃_ext (L') (L') (L') W W W
simpa using! map_hexagon_reverse _ _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Localization.Monoidal.Braided | {
"line": 119,
"column": 2
} | {
"line": 123,
"column": 42
} | {
"line": 125,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝³ : MonoidalCategory C\ninst✝² : W.IsMonoidal\ninst✝¹ : L.IsLocalization W\nunit : D\nε : L.obj (𝟙_ C) ≅ unit\ninst✝ : BraidedCategory C\n⊢ BraidedCategory (LocalizedMono... | [] | refine .ofBifunctor (braidingNatIso L W ε) ?_ ?_
· apply natTrans₃_ext (L') (L') (L') W W W
simpa using! map_hexagon_forward _ _ _
· apply natTrans₃_ext (L') (L') (L') W W W
simpa using! map_hexagon_reverse _ _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Mod | {
"line": 370,
"column": 4
} | {
"line": 377,
"column": 54
} | {
"line": 379,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹² : Category.{v₁, u₁} C\ninst✝¹¹ : MonoidalCategory C\nD : Type u₂\ninst✝¹⁰ : Category.{v₂, u₂} D\ninst✝⁹ : MonoidalLeftAction C D\nM' N' O' : D\nA✝¹ : C\ninst✝⁸ : MonObj A✝¹\nM✝¹ N O : D\ninst✝⁷ : ModObj A✝¹ M✝¹\ninst✝⁶ : ModObj A✝¹ N\ninst✝⁵ : ModObj A✝¹ O\nA✝ : C\ninst✝⁴ : MonObj ... | [] | slice_rhs 2 3 => rw [action_exchange]
simp only [actionHomLeft_action_assoc, Category.assoc, Iso.hom_inv_id_assoc,
actionHomRight_comp]
slice_rhs 4 6 => rw [ModObj.assoc_flip]
slice_rhs 2 4 => rw [← whiskerLeft_actionHomLeft]
slice_rhs 1 2 => rw [← comp_actionHomLeft]
rw [← comp_actionHomLeft,... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Mod | {
"line": 370,
"column": 4
} | {
"line": 377,
"column": 54
} | {
"line": 379,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹² : Category.{v₁, u₁} C\ninst✝¹¹ : MonoidalCategory C\nD : Type u₂\ninst✝¹⁰ : Category.{v₂, u₂} D\ninst✝⁹ : MonoidalLeftAction C D\nM' N' O' : D\nA✝¹ : C\ninst✝⁸ : MonObj A✝¹\nM✝¹ N O : D\ninst✝⁷ : ModObj A✝¹ M✝¹\ninst✝⁶ : ModObj A✝¹ N\ninst✝⁵ : ModObj A✝¹ O\nA✝ : C\ninst✝⁴ : MonObj ... | [] | slice_rhs 2 3 => rw [action_exchange]
simp only [actionHomLeft_action_assoc, Category.assoc, Iso.hom_inv_id_assoc,
actionHomRight_comp]
slice_rhs 4 6 => rw [ModObj.assoc_flip]
slice_rhs 2 4 => rw [← whiskerLeft_actionHomLeft]
slice_rhs 1 2 => rw [← comp_actionHomLeft]
rw [← comp_actionHomLeft,... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 314,
"column": 19
} | {
"line": 315,
"column": 22
} | {
"line": 316,
"column": 2
} | [
{
"pp": "case a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : HasCoequalizers C\nR S T : Mon C\nP : Bimod R S\nQ : Bimod S T\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n| coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ ... | [
"case a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : HasCoequalizers C\nR S T : Mon C\nP : Bimod R S\nQ : Bimod S T\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n| (α_ P.X Q.X T.X).hom ▷ T.X ≫\n (P.X ◁ Q.actRight) ▷ T.X ≫ coequalizer.π ... | rw [← comp_whiskerRight, π_tensor_id_actRight, comp_whiskerRight,
comp_whiskerRight] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 314,
"column": 19
} | {
"line": 315,
"column": 22
} | {
"line": 316,
"column": 2
} | [
{
"pp": "case a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : HasCoequalizers C\nR S T : Mon C\nP : Bimod R S\nQ : Bimod S T\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n| coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ ... | [
"case a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : HasCoequalizers C\nR S T : Mon C\nP : Bimod R S\nQ : Bimod S T\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n| (α_ P.X Q.X T.X).hom ▷ T.X ≫\n (P.X ◁ Q.actRight) ▷ T.X ≫ coequalizer.π ... | rw [← comp_whiskerRight, π_tensor_id_actRight, comp_whiskerRight,
comp_whiskerRight] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 314,
"column": 19
} | {
"line": 315,
"column": 22
} | {
"line": 316,
"column": 2
} | [
{
"pp": "case a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : HasCoequalizers C\nR S T : Mon C\nP : Bimod R S\nQ : Bimod S T\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n| coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ ... | [
"case a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : HasCoequalizers C\nR S T : Mon C\nP : Bimod R S\nQ : Bimod S T\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n| (α_ P.X Q.X T.X).hom ▷ T.X ≫\n (P.X ◁ Q.actRight) ▷ T.X ≫ coequalizer.π ... | rw [← comp_whiskerRight, π_tensor_id_actRight, comp_whiskerRight,
comp_whiskerRight] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.CategoryTheory.Monoidal.Closed.Functor | {
"line": 112,
"column": 64
} | {
"line": 129,
"column": 6
} | {
"line": 131,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nD : Type u'\ninst✝⁵ : Category.{v, u'} D\ninst✝⁴ : CartesianMonoidalCategory C\ninst✝³ : CartesianMonoidalCategory D\nF : C ⥤ D\ninst✝² : MonoidalClosed C\ninst✝¹ : MonoidalClosed D\ninst✝ : Limits.PreservesLimitsOfShape (Discrete Limits.WalkingPair) F\nA A' : C\... | [] | by
unfold expComparison MonoidalClosed.pre
have vcomp1 := mateEquiv_conjugateEquiv_vcomp
(ihom.adjunction A) (ihom.adjunction (F.obj A)) (ihom.adjunction (F.obj A'))
((prodComparisonNatIso F A).inv) (((curriedTensor D).map (F.map f)))
have vcomp2 := conjugateEquiv_mateEquiv_vcomp
(ihom.adjunction A) (... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Internal.Module | {
"line": 51,
"column": 24
} | {
"line": 53,
"column": 58
} | {
"line": 54,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nA : ModuleCat R\ninst✝ : MonObj A\nx : ↑A\n⊢ 1 * x = x",
"ppTerm": "?m.236",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Mul.mk",
"instHSMul",
"One.mk",
"HMul.hMul",
"CategoryTheory.CategoryStruct... | [] | by
convert! LinearMap.congr_fun (ModuleCat.hom_ext_iff.mp (one_mul A)) ((1 : R) ⊗ₜ x)
rw [MonoidalCategory.leftUnitor_hom_apply, one_smul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.ObjectProperty.Ind | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 31
} | {
"line": 147,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nP : ObjectProperty C\nι : Type u_1\ninst✝³ : Small.{w, u_1} ι\ninst✝² : P.IsClosedUnderLimitsOfShape (Discrete ι)\ninst✝¹ : HasProductsOfShape ι C\ninst✝ : IsIPCOfShape.{w, u_1, v, u} ι C\n⊢ P.ind.IsClosedUnderLimitsOfShape (Discrete ι)",
"ppTerm": "?m.13",
... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\nP : ObjectProperty C\nι : Type u_1\ninst✝³ : Small.{w, u_1} ι\ninst✝² : P.IsClosedUnderLimitsOfShape (Discrete ι)\ninst✝¹ : HasProductsOfShape ι C\ninst✝ : IsIPCOfShape.{w, u_1, v, u} ι C\nX : C\nx✝ : P.ind.strictLimitsOfShape (Discrete ι) X\nY : Discrete ι ⥤ C\nh : ∀ (j : D... | refine .mk' fun X ⟨Y, h⟩ ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.MorphismProperty.LocalClosure | {
"line": 111,
"column": 6
} | {
"line": 111,
"column": 58
} | {
"line": 112,
"column": 6
} | [
{
"pp": "case refine_1.of_iso\nC : Type u\ninst✝⁶ : Category.{v, u} C\nK : Precoverage C\nP : MorphismProperty C\ninst✝⁵ : P.RespectsIso\ninst✝⁴ : P.RespectsLeft K.morphismProperty\ninst✝³ : K.HasIsos\ninst✝² : K.IsStableUnderBaseChange\ninst✝¹ : K.IsStableUnderComposition\ninst✝ : K.HasPullbacks\nX Y : C\nf✝ :... | [
"case refine_1.of_iso\nC : Type u\ninst✝⁶ : Category.{v, u} C\nK : Precoverage C\nP : MorphismProperty C\ninst✝⁵ : P.RespectsIso\ninst✝⁴ : P.RespectsLeft K.morphismProperty\ninst✝³ : K.HasIsos\ninst✝² : K.IsStableUnderBaseChange\ninst✝¹ : K.IsStableUnderComposition\ninst✝ : K.HasPullbacks\nX Y : C\nf✝ : X ⟶ Y\nX✝ Y... | rw [← Category.assoc, P.cancel_right_of_respectsIso] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Subobject.ArtinianObject | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 45
} | {
"line": 119,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\nhX : IsZero X\nthis : Subsingleton (Subobject X)\nf : ℕ →o (Subobject X)ᵒᵈ\n⊢ ∃ n, ∀ (m : ℕ), n ≤ m → f n = f m",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"instOfNatNat",
"LE.le",
... | [] | exact ⟨0, fun m hm ↦ Subsingleton.elim _ _⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Subobject.NoetherianObject | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 45
} | {
"line": 115,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\nhX : IsZero X\nthis : Subsingleton (Subobject X)\nf : ℕ →o Subobject X\n⊢ ∃ n, ∀ (m : ℕ), n ≤ m → f n = f m",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"instOfNatNat",
"LE.le",
"Ca... | [] | exact ⟨0, fun m hm ↦ Subsingleton.elim _ _⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd | {
"line": 199,
"column": 6
} | {
"line": 199,
"column": 54
} | {
"line": 200,
"column": 6
} | [
{
"pp": "V : Type u_1\ninst✝² : Category.{v_1, u_1} V\ninst✝¹ : Preadditive V\nu✝ v✝ : Arrow V\nf✝ g✝¹ : u✝ ⟶ v✝\ninst✝ : HasBinaryBiproducts V\nu v : Arrow V\nf : u ⟶ v\nw : Arrow V\ng✝ : v ⟶ w\nh : RightHomotopy (f ≫ g✝) 0\nA✝ : RightFreyd V\ng : (quotient V).obj v ⟶ A✝\nhg : (quotient V).map f ≫ g = 0\n⊢ Non... | [
"V : Type u_1\ninst✝² : Category.{v_1, u_1} V\ninst✝¹ : Preadditive V\nu✝ v✝ : Arrow V\nf✝ g✝¹ : u✝ ⟶ v✝\ninst✝ : HasBinaryBiproducts V\nu v : Arrow V\nf : u ⟶ v\nw : Arrow V\ng✝ : v ⟶ w\nh : RightHomotopy (f ≫ g✝) 0\nA✝ : RightFreyd V\ng : v ⟶ A✝.as\nhg : (quotient V).map f ≫ (quotient V).map g = 0\n⊢ Nonempty { l... | obtain ⟨g, rfl⟩ := (quotient V).map_surjective g | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Preadditive.Mat | {
"line": 598,
"column": 49
} | {
"line": 598,
"column": 56
} | {
"line": 598,
"column": 56
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nR : Type\ninst✝ : Ring R\nX : Mat R\n⊢ (equivalenceSingleObjInverse R).obj { ι := X.obj, fintype := FintypeCat.fintype, X := fun x ↦ PUnit.unit } = X",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"CategoryT... | [
"case mk\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nR : Type\ninst✝ : Ring R\nobj✝ : Type\nproperty✝ : Finite obj✝\n⊢ (equivalenceSingleObjInverse R).obj\n { ι := { obj := obj✝, property := property✝ }.obj, fintype := FintypeCat.fintype, X := fun x ↦ PUnit.unit } =\n { obj := obj✝, ... | cases X | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.CategoryTheory.Presentable.SharplyLT.Basic | {
"line": 144,
"column": 37
} | {
"line": 144,
"column": 49
} | {
"line": 144,
"column": 49
} | [
{
"pp": "κ₁ κ₂ : Cardinal.{w}\nX : Type w\ninst✝ : PartialOrder X\nY : (B : Set X) → HasCardinalLT (↑B) κ₂ → Set (SetCardinalLT κ₁ ↑B)\nhY' : ∀ (B : Set X) (hB : HasCardinalLT (↑B) κ₂), IsCofinal (Y B hB)\nm : (B : Set X) → (hB : HasCardinalLT (↑B) κ₂) → (C : SetCardinalLT κ₁ ↑B) → C ∈ Y B hB → X\nhm : ∀ (B : S... | [] | by simp [C₀] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 363,
"column": 4
} | {
"line": 363,
"column": 42
} | {
"line": 364,
"column": 4
} | [
{
"pp": "case inr\nJ : Type w\ninst✝¹ : SmallCategory J\nκ : Cardinal.{w}\ninst✝ : Fact κ.IsRegular\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm : J\nu : (i : ι) → (D i).top ⟶ m\nhD : ∀ {i : ι}, ¬(D i).P m\nf : m ⟶ m\nhf : MorphismProperty.ofHoms (fun x ↦ (D x.fst).isTerminal.li... | [
"case inr\nJ : Type w\ninst✝¹ : SmallCategory J\nκ : Cardinal.{w}\ninst✝ : Fact κ.IsRegular\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm : J\nu : (i : ι) → (D i).top ⟶ m\nhD : ∀ {i : ι}, ¬(D i).P m\nf : m ⟶ m\nhf : ∃ i, Arrow.mk f = Arrow.mk ((D i.fst).isTerminal.lift ⋯ ≫ u i.fst)\n... | rw [MorphismProperty.ofHoms_iff] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 356,
"column": 15
} | {
"line": 366,
"column": 22
} | {
"line": 368,
"column": 0
} | [
{
"pp": "J : Type w\ninst✝¹ : SmallCategory J\nκ : Cardinal.{w}\ninst✝ : Fact κ.IsRegular\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm : J\nu : (i : ι) → (D i).top ⟶ m\nhD : ∀ {i : ι}, ¬(D i).P m\nf : m ⟶ m\nhf : (D₂ D hι u).W f\n⊢ f = 𝟙 m",
"ppTerm": "?m.35",
"assigned... | [] | by
simp only [D₂_W] at hf
obtain ((hf | ⟨⟨⟩⟩) | hf) := hf
· simp only [MorphismProperty.iSup_iff] at hf
obtain ⟨i, hi⟩ := hf
exact (hD ((D i).src hi)).elim
· rfl
· rw [MorphismProperty.ofHoms_iff] at hf
obtain ⟨⟨i, j, hj⟩, hi⟩ := hf
obtain rfl : m = j := congr_arg Arrow.leftFunc.obj hi
exa... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Profunctor.Basic | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 21
} | {
"line": 92,
"column": 0
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Category.{v₂, u₂} D\nP : ProfunctorCore.{w, v₁, v₂, u₁, u₂} C D\nX X' : C\nY Y' : D\nf : X ⟶ X'\ng : Y ⟶ Y'\n⊢ P.map (𝟙 X) g ≫ P.map f (𝟙 Y) = P.map f g",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Categor... | [] | simp [← P.map_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Profunctor.Basic | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 21
} | {
"line": 92,
"column": 0
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Category.{v₂, u₂} D\nP : ProfunctorCore.{w, v₁, v₂, u₁, u₂} C D\nX X' : C\nY Y' : D\nf : X ⟶ X'\ng : Y ⟶ Y'\n⊢ P.map (𝟙 X) g ≫ P.map f (𝟙 Y) = P.map f g",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Categor... | [] | simp [← P.map_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Profunctor.Basic | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 21
} | {
"line": 92,
"column": 0
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Category.{v₂, u₂} D\nP : ProfunctorCore.{w, v₁, v₂, u₁, u₂} C D\nX X' : C\nY Y' : D\nf : X ⟶ X'\ng : Y ⟶ Y'\n⊢ P.map (𝟙 X) g ≫ P.map f (𝟙 Y) = P.map f g",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Categor... | [] | simp [← P.map_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Profunctor.Basic | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 21
} | {
"line": 97,
"column": 0
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Category.{v₂, u₂} D\nP : ProfunctorCore.{w, v₁, v₂, u₁, u₂} C D\nX X' : C\nY Y' : D\nf : X ⟶ X'\ng : Y ⟶ Y'\n⊢ P.map f (𝟙 Y') ≫ P.map (𝟙 X') g = P.map f g",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Categ... | [] | simp [← P.map_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Profunctor.Basic | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 21
} | {
"line": 97,
"column": 0
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Category.{v₂, u₂} D\nP : ProfunctorCore.{w, v₁, v₂, u₁, u₂} C D\nX X' : C\nY Y' : D\nf : X ⟶ X'\ng : Y ⟶ Y'\n⊢ P.map f (𝟙 Y') ≫ P.map (𝟙 X') g = P.map f g",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Categ... | [] | simp [← P.map_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Profunctor.Basic | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 21
} | {
"line": 97,
"column": 0
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Category.{v₂, u₂} D\nP : ProfunctorCore.{w, v₁, v₂, u₁, u₂} C D\nX X' : C\nY Y' : D\nf : X ⟶ X'\ng : Y ⟶ Y'\n⊢ P.map f (𝟙 Y') ≫ P.map (𝟙 X') g = P.map f g",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Categ... | [] | simp [← P.map_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 476,
"column": 6
} | {
"line": 476,
"column": 44
} | {
"line": 477,
"column": 6
} | [
{
"pp": "case inr\nJ : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nthis✝¹ : IsCardinalFiltered (DiagramWithUniqueTerminal J κ) κ\nthis✝ : IsFiltered J\nthis : IsFiltered (DiagramWithUniqueTerminal J κ)\nj ... | [
"case inr\nJ : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nthis✝¹ : IsCardinalFiltered (DiagramWithUniqueTerminal J κ) κ\nthis✝ : IsFiltered J\nthis : IsFiltered (DiagramWithUniqueTerminal J κ)\nj : J\nD : Dia... | rw [MorphismProperty.ofHoms_iff] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves | {
"line": 90,
"column": 4
} | {
"line": 91,
"column": 67
} | {
"line": 92,
"column": 4
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : FinitaryPreExtensive C\ninst✝ : FinitaryExtensive C\nF : Cᵒᵖ ⥤ Type w\nhF : ∀ {X : C}, ∀ R ∈ (extensiveCoverage C).coverings X, IsSheafFor F R\nn : ℕ\nK : Discrete (Fin n) ⥤ Cᵒᵖ\nZ : Fin n → C := fun i ↦ unop (K.obj { as := i })\n⊢ P... | [
"case refine_1\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : FinitaryPreExtensive C\ninst✝ : FinitaryExtensive C\nF : Cᵒᵖ ⥤ Type w\nhF : ∀ {X : C}, ∀ R ∈ (extensiveCoverage C).coverings X, IsSheafFor F R\nn : ℕ\nK : Discrete (Fin n) ⥤ Cᵒᵖ\nZ : Fin n → C := fun i ↦ unop (K.obj { as := i })\nthis : (ofArrow... | have : (ofArrows Z (Cofan.mk (∐ Z) (Sigma.ι Z)).inj).HasPairwisePullbacks :=
inferInstanceAs (ofArrows Z (Sigma.ι Z)).HasPairwisePullbacks | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Sites.Coherent.SheafComparison | {
"line": 89,
"column": 48
} | {
"line": 93,
"column": 64
} | {
"line": 95,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Category.{v_2, u_2} D\nF : C ⥤ D\ninst✝⁵ : F.PreservesFiniteEffectiveEpiFamilies\ninst✝⁴ : F.ReflectsFiniteEffectiveEpiFamilies\ninst✝³ : F.Full\ninst✝² : F.Faithful\ninst✝¹ : F.EffectivelyEnough\ninst✝ : Precoherent D\n⊢ coherentTopo... | [] | by
ext X S
have := F.reflects_precoherent
rw [← exists_effectiveEpiFamily_iff_mem_induced F X]
rw [← coherentTopology.mem_sieves_iff_hasEffectiveEpiFamily S] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 26
} | {
"line": 219,
"column": 0
} | [
{
"pp": "case h₂.refine_1\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX B : C\nπ : X ⟶ B\nc : PullbackCone π π\nhc : IsLimit c\nZ : (Sieve.ofArrows (fun x ↦ X) fun x ↦ π).arrows.category\ni j : unop (op Z) ⟶ unop (op ((Sieve.ofArrows (fun x ↦ X) fun x ↦ π).arrows.categoryMk π ⋯))\nhi : Over.Hom.left i.hom ≫ π... | [] | all_goals congr; aesop | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.CategoryTheory.Sites.Descent.DescentDataPrime | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 73
} | {
"line": 98,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nsq : (i j : ι) → ChosenPullback (f i) (f j)\nobj obj' : (i : ι) → ↑(F.obj { as := op (X i) })\nhom :\n (i j : ι) → (F.map (sq i j).p₁.op.toLoc).toFunctor.obj (obj i)... | [
"C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nsq : (i j : ι) → ChosenPullback (f i) (f j)\nobj obj' : (i : ι) → ↑(F.obj { as := op (X i) })\nhom :\n (i j : ι) → (F.map (sq i j).p₁.op.toLoc).toFunctor.obj (obj i) ⟶ (F.map (s... | rw [pullHom'_eq_pullHom _ _ _ _ p, pullHom'_eq_pullHom _ _ _ _ (g ≫ p)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 286,
"column": 2
} | {
"line": 286,
"column": 10
} | {
"line": 287,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j\nhf... | subst hp | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.CategoryTheory.Sites.Descent.DescentDataPrime | {
"line": 298,
"column": 6
} | {
"line": 298,
"column": 35
} | {
"line": 298,
"column": 35
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nsq : (i j : ι) → ChosenPullback (f i) (f j)\nsq₃ : (i₁ i₂ i₃ : ι) → ChosenPullback₃ (sq i₁ i₂) (sq i₂ i₃) (sq i₁ i₃)\nD : F.DescentData f\nY : C\nq : Y ⟶ S\ni₁ i₂ : ι... | [
"C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nsq : (i j : ι) → ChosenPullback (f i) (f j)\nsq₃ : (i₁ i₂ i₃ : ι) → ChosenPullback₃ (sq i₁ i₂) (sq i₂ i₃) (sq i₁ i₃)\nD : F.DescentData f\nY : C\nq : Y ⟶ S\ni₁ i₂ : ι\nf₁ : Y ⟶ X... | pullHom'_eq_pullHom _ _ _ _ p | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Point.Comap | {
"line": 44,
"column": 37
} | {
"line": 48,
"column": 8
} | {
"line": 50,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\nK : GrothendieckTopology D\nΦ : K.Point\nF : C ⥤ D\ninst✝¹ : RepresentablyFlat F\nJ : GrothendieckTopology C\nhF : CoverPreserving J K F\ninst✝ : InitiallySmall (F ⋙ Φ.fiber).Elements\nX : C\nR : Sieve X\nhR : R... | [] | by
obtain ⟨Y, f, ⟨W, g, h, hg, rfl⟩, y, rfl⟩ :=
Φ.jointly_surjective (Sieve.functorPushforward F R) (hF.cover_preserve hR) x
use W, g, hg, Φ.fiber.map h y
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered | {
"line": 100,
"column": 6
} | {
"line": 100,
"column": 75
} | {
"line": 100,
"column": 75
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : LocallySmall.{w, v, u} C\nN : Type u'\ninst✝¹ : Category.{v', u'} N\np : N ⥤ C\ninst✝ : InitiallySmall N\nU : N\nX : C\nf : p.obj U ⟶ X\nY : C\ng : X ⟶ Y\n⊢ (hom\n ((p.op.whiskerLeft (shrinkYoneda.{w, v, u}.map g)).app (op U) ≫\n colimit.... | [] | simp [fiberMk, shrinkYoneda_map_app_shrinkYonedaObjObjEquiv_symm.{w}] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered | {
"line": 100,
"column": 6
} | {
"line": 100,
"column": 75
} | {
"line": 100,
"column": 75
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : LocallySmall.{w, v, u} C\nN : Type u'\ninst✝¹ : Category.{v', u'} N\np : N ⥤ C\ninst✝ : InitiallySmall N\nU : N\nX : C\nf : p.obj U ⟶ X\nY : C\ng : X ⟶ Y\n⊢ (hom\n ((p.op.whiskerLeft (shrinkYoneda.{w, v, u}.map g)).app (op U) ≫\n colimit.... | [] | simp [fiberMk, shrinkYoneda_map_app_shrinkYonedaObjObjEquiv_symm.{w}] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered | {
"line": 100,
"column": 6
} | {
"line": 100,
"column": 75
} | {
"line": 100,
"column": 75
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : LocallySmall.{w, v, u} C\nN : Type u'\ninst✝¹ : Category.{v', u'} N\np : N ⥤ C\ninst✝ : InitiallySmall N\nU : N\nX : C\nf : p.obj U ⟶ X\nY : C\ng : X ⟶ Y\n⊢ (hom\n ((p.op.whiskerLeft (shrinkYoneda.{w, v, u}.map g)).app (op U) ≫\n colimit.... | [] | simp [fiberMk, shrinkYoneda_map_app_shrinkYonedaObjObjEquiv_symm.{w}] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Precoverage.Generates | {
"line": 126,
"column": 2
} | {
"line": 133,
"column": 82
} | {
"line": 135,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nK : Precoverage C\nJ : GrothendieckTopology C\nH : K.Generates J\n⊢ K.toGrothendieck = J",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"CategoryTheory.Presieve.IsSheaf",
"Eq.mpr",
"CategoryTheory.Precoverage.toGrothendieck... | [] | refine le_antisymm ?_ ?_
· rw [toGrothendieck_le_iff_le_toPrecoverage]
exact H.le_toPrecoverage
· apply CategoryTheory.le_topology_of_closedSieves_isSheaf
rw [H.isSheaf_type_iff]
intro X R hR
rw [Presieve.isSheafFor_iff_generate]
exact classifier_isSheaf K.toGrothendieck _ (K.generate_mem_toGrot... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Precoverage.Generates | {
"line": 126,
"column": 2
} | {
"line": 133,
"column": 82
} | {
"line": 135,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nK : Precoverage C\nJ : GrothendieckTopology C\nH : K.Generates J\n⊢ K.toGrothendieck = J",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"CategoryTheory.Presieve.IsSheaf",
"Eq.mpr",
"CategoryTheory.Precoverage.toGrothendieck... | [] | refine le_antisymm ?_ ?_
· rw [toGrothendieck_le_iff_le_toPrecoverage]
exact H.le_toPrecoverage
· apply CategoryTheory.le_topology_of_closedSieves_isSheaf
rw [H.isSheaf_type_iff]
intro X R hR
rw [Presieve.isSheafFor_iff_generate]
exact classifier_isSheaf K.toGrothendieck _ (K.generate_mem_toGrot... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 523,
"column": 68
} | {
"line": 525,
"column": 16
} | {
"line": 527,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nS : C\nR : Presieve S\nh : F.IsStackFor R\n⊢ (F.toDescentData fun f ↦ f.obj.hom).EssSurj",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"CategoryTheory.Over",
"Opposite",
"Catego... | [] | by
have := h.isEquivalence
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Topos.Sheaf | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 40
} | {
"line": 74,
"column": 40
} | [
{
"pp": "case h\nC : Type u\ninst✝ : Category.{v, u} C\nF G : Cᵒᵖ ⥤ Type (max u v)\nm : F ⟶ G\nX : Cᵒᵖ\nx : G.obj X\nY Z : C\nf : Y ⟶ Opposite.unop X\na : (fun X ↦ X) (F.obj (Opposite.op Y))\nha : (ConcreteCategory.hom (G.map f.op)) x = (ConcreteCategory.hom (m.app (Opposite.op Y))) a\ng : Z ⟶ Y\n⊢ (ConcreteCat... | [] | simp [ha, NatTrans.naturality_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Triangulated.Opposite.Functor | {
"line": 57,
"column": 68
} | {
"line": 70,
"column": 42
} | {
"line": 72,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Category.{v_2, u_2} D\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasShift D ℤ\nF : C ⥤ D\ninst✝² : F.CommShift ℤ\nG : D ⥤ C\ninst✝¹ : G.CommShift ℤ\nadj : F ⊣ G\ninst✝ : adj.CommShift ℤ\n⊢ adj.op.CommShift ℤ",
"ppTerm": "?m.55",
"assign... | [] | by
have eq : adj.op = PullbackShift.adjunction
(AddMonoidHom.mk' (fun (n : ℤ) => -n) (by intros; lia))
(OppositeShift.adjunction ℤ adj) := by
ext
dsimp [PullbackShift.adjunction, NatTrans.PullbackShift.natIsoId,
NatTrans.PullbackShift.natIsoComp, PullbackShift.functor, PullbackShift.natTrans,
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Triangulated.Generators | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 24
} | {
"line": 157,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nP : ObjectProperty C\nX Y : C\nr : Retract X Y\nhY : ∃ n, P.triangEnvelopeIter n Y\n⊢ ∃ n, P.triangEnvelopeI... | [
"C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nP : ObjectProperty C\nX Y : C\nr : Retract X Y\nn : ℕ\nhn : P.triangEnvelopeIter n Y\n⊢ ∃ n, P.triangEnvelopeIter n X"
] | obtain ⟨n, hn⟩ := hY | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Triangulated.Generators | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 55
} | {
"line": 200,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nP : ObjectProperty C\nn : ℕ\nhn : P.triangEnvelopeIter n = ⊤\n⊢ P.IsClassicalTriangulatedGenerator",
"pp... | [
"C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : HasZeroObject C\ninst✝³ : HasShift C ℤ\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nP : ObjectProperty C\nn : ℕ\nhn : P.triangEnvelopeIter n = ⊤\n⊢ ⊤ ≤ P.triangEnvelope"
] | rw [isClassicalTriangulatedGenerator_iff, eq_top_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Triangulated.Opposite.OpOp | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 55
} | {
"line": 109,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasShift C ℤ\nX : C\n⊢ (iso C 0).hom.app X = (Functor.CommShift.isoZero (opOp C) ℤ).hom.app X",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"Opposite",
"CategoryTheory.CategoryStruc... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasShift C ℤ\nX : C\n⊢ ((iso C 0).hom.app X).unop.unop = ((Functor.CommShift.isoZero (opOp C) ℤ).hom.app X).unop.unop"
] | refine Quiver.Hom.unop_inj (Quiver.Hom.unop_inj ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Triangulated.Opposite.OpOp | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 55
} | {
"line": 114,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasShift C ℤ\np q : ℤ\nX : C\n⊢ (iso C (p + q)).hom.app X = (Functor.CommShift.isoAdd (iso C p) (iso C q)).hom.app X",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"Opposite",
"AddMo... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasShift C ℤ\np q : ℤ\nX : C\n⊢ ((iso C (p + q)).hom.app X).unop.unop = ((Functor.CommShift.isoAdd (iso C p) (iso C q)).hom.app X).unop.unop"
] | refine Quiver.Hom.unop_inj (Quiver.Hom.unop_inj ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Enumerative.DoubleCounting | {
"line": 83,
"column": 95
} | {
"line": 85,
"column": 17
} | {
"line": 87,
"column": 0
} | [
{
"pp": "R : Type u_1\nα : Type u_2\nβ : Type u_3\nr : α → β → Prop\ns : Finset α\nt : Finset β\ninst✝¹ : CommMonoid R\nf : α → β → R\ninst✝ : (a : α) → (b : β) → Decidable (r a b)\n⊢ ∏ a ∈ s, ∏ b ∈ bipartiteAbove r t a, f a b = ∏ b ∈ t, ∏ a ∈ bipartiteBelow r s b, f a b",
"ppTerm": "?m.34",
"assigned":... | [] | by
simp_rw [bipartiteAbove, bipartiteBelow, prod_filter]
exact prod_comm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Enumerative.DoubleCounting | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 81
} | {
"line": 213,
"column": 2
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nB : α → Finset β\ns : Finset α\n⊢ ∑ j ∈ s, #(B j) = ∑ x ∈ s.biUnion B, #{j | j ∈ s ∧ x ∈ B j}",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"... | [
"case e'_2.a\nα : Type u_2\nβ : Type u_3\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nB : α → Finset β\ns : Finset α\nx✝ : α\na✝ : x✝ ∈ s\n⊢ B x✝ = bipartiteAbove (fun j x ↦ x ∈ B j) (s.biUnion B) x✝",
"case e'_3.a\nα : Type u_2\nβ : Type u_3\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nins... | convert sum_card_bipartiteAbove_eq_sum_card_bipartiteBelow (fun j x => x ∈ B j) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
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