module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Galois.Prorepresentability | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 47
} | {
"line": 178,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{u₂, u₁} C\ninst✝¹ : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝ : FiberFunctor F\nx✝³ x✝² : PointedGaloisObject F\nA : C\na : (F.obj A).obj\nisGalois✝¹ : IsGalois A\nB : C\nb : (F.obj B).obj\nisGalois✝ : IsGalois B\nx✝¹ x✝ : { obj := A, pt := a, isGalois := isGalois✝¹ } ... | [
"C : Type u₁\ninst✝² : Category.{u₂, u₁} C\ninst✝¹ : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝ : FiberFunctor F\nx✝³ x✝² : PointedGaloisObject F\nA : C\na : (F.obj A).obj\nisGalois✝¹ : IsGalois A\nB : C\nb : (F.obj B).obj\nisGalois✝ : IsGalois B\nx✝¹ x✝ : { obj := A, pt := a, isGalois := isGalois✝¹ } ⟶ { obj := B... | refine ⟨⟨Z, z, hgal⟩, ⟨h, hhz⟩, hom_ext ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Galois.Prorepresentability | {
"line": 191,
"column": 66
} | {
"line": 191,
"column": 93
} | {
"line": 192,
"column": 4
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝² : Category.{u₂, u₁} C\ninst✝¹ : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝ : FiberFunctor F\nX A : C\na : (F.obj A).obj\nisGalois✝¹ : IsGalois A\nB : C\nb : (F.obj B).obj\nisGalois✝ : IsGalois B\nu : A ⟶ X\nv : B ⟶ X\n⊢ (ConcreteCategory.hom\n ((((evaluation... | [
"case refine_2\nC : Type u₁\ninst✝² : Category.{u₂, u₁} C\ninst✝¹ : GaloisCategory C\nF : C ⥤ FintypeCat\ninst✝ : FiberFunctor F\nX A : C\na : (F.obj A).obj\nisGalois✝¹ : IsGalois A\nB : C\nb : (F.obj B).obj\nisGalois✝ : IsGalois B\nu : A ⟶ X\nv : B ⟶ X\nh : (ConcreteCategory.hom (F.map u)) a = (ConcreteCategory.ho... | (h : F.map u a = F.map v b) | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.typeAscription |
Mathlib.CategoryTheory.Galois.IsFundamentalgroup | {
"line": 91,
"column": 51
} | {
"line": 94,
"column": 47
} | {
"line": 95,
"column": 2
} | [
{
"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\ng : G\n⊢ ∀ {X Y : C} (f : X ⟶ Y), F.map f ≫ (isoOnObj F g Y).hom = (isoOnObj F g X).hom ≫ F.map f",
"ppTerm": "?m.37",
"assigne... | [] | by
intro X Y f
ext
exact (IsNaturalSMul.naturality _ _ _).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.GradedObject.Braiding | {
"line": 76,
"column": 14
} | {
"line": 82,
"column": 54
} | {
"line": 83,
"column": 2
} | [
{
"pp": "I : Type u_1\ninst✝²⁰ : AddCommMonoid I\nC : Type u_2\ninst✝¹⁹ : Category.{v_1, u_2} C\ninst✝¹⁸ : MonoidalCategory C\nX Y Z : GradedObject I C\ninst✝¹⁷ : BraidedCategory C\ninst✝¹⁶ : X.HasTensor Y\ninst✝¹⁵ : Y.HasTensor X\ninst✝¹⁴ : Y.HasTensor Z\ninst✝¹³ : Z.HasTensor X\ninst✝¹² : X.HasTensor Z\ninst✝... | [
"I : Type u_1\ninst✝²⁰ : AddCommMonoid I\nC : Type u_2\ninst✝¹⁹ : Category.{v_1, u_2} C\ninst✝¹⁸ : MonoidalCategory C\nX Y Z : GradedObject I C\ninst✝¹⁷ : BraidedCategory C\ninst✝¹⁶ : X.HasTensor Y\ninst✝¹⁵ : Y.HasTensor X\ninst✝¹⁴ : Y.HasTensor Z\ninst✝¹³ : Z.HasTensor X\ninst✝¹² : X.HasTensor Z\ninst✝¹¹ : (tensor... | rw [ιTensorObj₃'_associator_hom_assoc, ιTensorObj₃_eq X Y Z i₁ i₂ i₃ k h _ rfl,
assoc, ι_tensorObjDesc_assoc, assoc, ← MonoidalCategory.id_tensorHom,
BraidedCategory.braiding_naturality_assoc,
BraidedCategory.braiding_tensor_right_hom, assoc, assoc, assoc, assoc, Iso.hom_inv_id_assoc,
MonoidalCategory.t... | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.CategoryTheory.GradedObject.Braiding | {
"line": 76,
"column": 14
} | {
"line": 82,
"column": 54
} | {
"line": 83,
"column": 2
} | [
{
"pp": "I : Type u_1\ninst✝²⁰ : AddCommMonoid I\nC : Type u_2\ninst✝¹⁹ : Category.{v_1, u_2} C\ninst✝¹⁸ : MonoidalCategory C\nX Y Z : GradedObject I C\ninst✝¹⁷ : BraidedCategory C\ninst✝¹⁶ : X.HasTensor Y\ninst✝¹⁵ : Y.HasTensor X\ninst✝¹⁴ : Y.HasTensor Z\ninst✝¹³ : Z.HasTensor X\ninst✝¹² : X.HasTensor Z\ninst✝... | [
"I : Type u_1\ninst✝²⁰ : AddCommMonoid I\nC : Type u_2\ninst✝¹⁹ : Category.{v_1, u_2} C\ninst✝¹⁸ : MonoidalCategory C\nX Y Z : GradedObject I C\ninst✝¹⁷ : BraidedCategory C\ninst✝¹⁶ : X.HasTensor Y\ninst✝¹⁵ : Y.HasTensor X\ninst✝¹⁴ : Y.HasTensor Z\ninst✝¹³ : Z.HasTensor X\ninst✝¹² : X.HasTensor Z\ninst✝¹¹ : (tensor... | rw [ιTensorObj₃'_associator_hom_assoc, ιTensorObj₃_eq X Y Z i₁ i₂ i₃ k h _ rfl,
assoc, ι_tensorObjDesc_assoc, assoc, ← MonoidalCategory.id_tensorHom,
BraidedCategory.braiding_naturality_assoc,
BraidedCategory.braiding_tensor_right_hom, assoc, assoc, assoc, assoc, Iso.hom_inv_id_assoc,
MonoidalCategory.t... | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.CategoryTheory.GradedObject.Braiding | {
"line": 76,
"column": 14
} | {
"line": 82,
"column": 54
} | {
"line": 83,
"column": 2
} | [
{
"pp": "I : Type u_1\ninst✝²⁰ : AddCommMonoid I\nC : Type u_2\ninst✝¹⁹ : Category.{v_1, u_2} C\ninst✝¹⁸ : MonoidalCategory C\nX Y Z : GradedObject I C\ninst✝¹⁷ : BraidedCategory C\ninst✝¹⁶ : X.HasTensor Y\ninst✝¹⁵ : Y.HasTensor X\ninst✝¹⁴ : Y.HasTensor Z\ninst✝¹³ : Z.HasTensor X\ninst✝¹² : X.HasTensor Z\ninst✝... | [
"I : Type u_1\ninst✝²⁰ : AddCommMonoid I\nC : Type u_2\ninst✝¹⁹ : Category.{v_1, u_2} C\ninst✝¹⁸ : MonoidalCategory C\nX Y Z : GradedObject I C\ninst✝¹⁷ : BraidedCategory C\ninst✝¹⁶ : X.HasTensor Y\ninst✝¹⁵ : Y.HasTensor X\ninst✝¹⁴ : Y.HasTensor Z\ninst✝¹³ : Z.HasTensor X\ninst✝¹² : X.HasTensor Z\ninst✝¹¹ : (tensor... | rw [ιTensorObj₃'_associator_hom_assoc, ιTensorObj₃_eq X Y Z i₁ i₂ i₃ k h _ rfl,
assoc, ι_tensorObjDesc_assoc, assoc, ← MonoidalCategory.id_tensorHom,
BraidedCategory.braiding_naturality_assoc,
BraidedCategory.braiding_tensor_right_hom, assoc, assoc, assoc, assoc, Iso.hom_inv_id_assoc,
MonoidalCategory.t... | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.CategoryTheory.GradedObject.Braiding | {
"line": 83,
"column": 18
} | {
"line": 83,
"column": 59
} | {
"line": 83,
"column": 60
} | [
{
"pp": "I : Type u_1\ninst✝²⁰ : AddCommMonoid I\nC : Type u_2\ninst✝¹⁹ : Category.{v_1, u_2} C\ninst✝¹⁸ : MonoidalCategory C\nX Y Z : GradedObject I C\ninst✝¹⁷ : BraidedCategory C\ninst✝¹⁶ : X.HasTensor Y\ninst✝¹⁵ : Y.HasTensor X\ninst✝¹⁴ : Y.HasTensor Z\ninst✝¹³ : Z.HasTensor X\ninst✝¹² : X.HasTensor Z\ninst✝... | [
"I : Type u_1\ninst✝²⁰ : AddCommMonoid I\nC : Type u_2\ninst✝¹⁹ : Category.{v_1, u_2} C\ninst✝¹⁸ : MonoidalCategory C\nX Y Z : GradedObject I C\ninst✝¹⁷ : BraidedCategory C\ninst✝¹⁶ : X.HasTensor Y\ninst✝¹⁵ : Y.HasTensor X\ninst✝¹⁴ : Y.HasTensor Z\ninst✝¹³ : Z.HasTensor X\ninst✝¹² : X.HasTensor Z\ninst✝¹¹ : (tensor... | ιTensorObj₃'_eq X Y Z i₁ i₂ i₃ k h _ rfl, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Groupoid.FreeGroupoid | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 37
} | {
"line": 97,
"column": 4
} | [
{
"pp": "case cons.y\nV : Type u\ninst✝ : Quiver V\nX Y b✝ c✝ : Paths (Symmetrify V)\nq : Path X b✝\nf : b✝ ⟶ c✝\nih : EqvGen (HomRel.CompClosure redStep) (q ≫ q.reverse) (𝟙 X)\n⊢ X ⟶ X",
"ppTerm": "?cons.y",
"assigned": true,
"usedConstants": [
"CategoryTheory.Paths.categoryPaths",
"Ca... | [
"case cons.a\nV : Type u\ninst✝ : Quiver V\nX Y b✝ c✝ : Paths (Symmetrify V)\nq : Path X b✝\nf : b✝ ⟶ c✝\nih : EqvGen (HomRel.CompClosure redStep) (q ≫ q.reverse) (𝟙 X)\n⊢ EqvGen (HomRel.CompClosure redStep) (q.cons f ≫ (reverse f).toPath.comp q.reverse) (q ≫ q.reverse)",
"case cons.a\nV : Type u\ninst✝ : Quiver... | · exact q ≫ Quiver.Path.reverse q | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Groupoid.FreeGroupoid | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 8
} | {
"line": 153,
"column": 4
} | [
{
"pp": "V : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ : V ⥤q V'\nX✝ Y✝ : Paths (Symmetrify V)\nX Y : Symmetrify V\nf : X ⟶ Y\n⊢ 𝟙 ((Symmetrify.lift φ).obj ((Paths.of (Symmetrify V)).obj X)) =\n (Symmetrify.lift φ).map f ≫ reverse ((Symmetrify.lift φ).map f)",
"ppTerm": "?m.149",
... | [
"V : Type u\ninst✝¹ : Quiver V\nV' : Type u'\ninst✝ : Groupoid V'\nφ : V ⥤q V'\nX✝ Y✝ : Paths (Symmetrify V)\nX Y : Symmetrify V\nf : X ⟶ Y\n⊢ (Symmetrify.lift φ).map f ≫ reverse ((Symmetrify.lift φ).map f) =\n 𝟙 ((Symmetrify.lift φ).obj ((Paths.of (Symmetrify V)).obj X))"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Groupoid.FreeGroupoid | {
"line": 191,
"column": 36
} | {
"line": 191,
"column": 40
} | {
"line": 192,
"column": 2
} | [
{
"pp": "V : Type u\ninst✝ : Quiver V\n⊢ lift (𝟭q V ⋙q of V) = 𝟭 (Quiver.FreeGroupoid V)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"CategoryTheory.CategoryStruct.toQuiver",
"Prefunctor.id",
"CategoryTheory.Functor.id",
"Prefun... | [
"V : Type u\ninst✝ : Quiver V\n⊢ 𝟭 (Quiver.FreeGroupoid V) = lift (𝟭q V ⋙q of V)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Groupoid.FreeGroupoid | {
"line": 196,
"column": 36
} | {
"line": 196,
"column": 40
} | {
"line": 197,
"column": 2
} | [
{
"pp": "V : Type u\ninst✝² : Quiver V\nV' : Type u'\ninst✝¹ : Quiver V'\nV'' : Type u''\ninst✝ : Quiver V''\nφ : V ⥤q V'\nφ' : V' ⥤q V''\n⊢ lift (φ ⋙q φ' ⋙q of V'') = lift (φ ⋙q of V') ⋙ lift (φ' ⋙q of V'')",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
... | [
"V : Type u\ninst✝² : Quiver V\nV' : Type u'\ninst✝¹ : Quiver V'\nV'' : Type u''\ninst✝ : Quiver V''\nφ : V ⥤q V'\nφ' : V' ⥤q V''\n⊢ lift (φ ⋙q of V') ⋙ lift (φ' ⋙q of V'') = lift (φ ⋙q φ' ⋙q of V'')"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Groupoid.FreeGroupoidOfCategory | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 6
} | {
"line": 136,
"column": 6
} | [
{
"pp": "G : Type u₁\ninst✝ : Groupoid G\n⊢ lift (of G) = 𝟭 (FreeGroupoid G)",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"CategoryTheory.FreeGroupoid",
"CategoryTheory.Functor",
"CategoryTheory.Functor.id",
"CategoryTheory.instGroupoidFreeGroupoid",
"Cate... | [
"G : Type u₁\ninst✝ : Groupoid G\n⊢ 𝟭 (FreeGroupoid G) = lift (of G)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Groupoid.FreeGroupoidOfCategory | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 6
} | {
"line": 142,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nG : Type u₁\ninst✝¹ : Groupoid G\nH : Type u₂\ninst✝ : Groupoid H\nφ : C ⥤ G\nψ : G ⥤ H\n⊢ lift (φ ⋙ ψ) = lift φ ⋙ ψ",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"CategoryTheory.FreeGroupoid",
"CategoryTheory.Functor",
"... | [
"C : Type u\ninst✝² : Category.{v, u} C\nG : Type u₁\ninst✝¹ : Groupoid G\nH : Type u₂\ninst✝ : Groupoid H\nφ : C ⥤ G\nψ : G ⥤ H\n⊢ lift φ ⋙ ψ = lift (φ ⋙ ψ)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Groupoid.FreeGroupoidOfCategory | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 6
} | {
"line": 203,
"column": 6
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\n⊢ map (𝟭 C) = 𝟭 (FreeGroupoid C)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"CategoryTheory.FreeGroupoid",
"CategoryTheory.Functor",
"CategoryTheory.FreeGroupoid.map",
"CategoryTheory.Functor.id",
"Category... | [
"C : Type u\ninst✝ : Category.{v, u} C\n⊢ 𝟭 (FreeGroupoid C) = map (𝟭 C)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Groupoid.FreeGroupoidOfCategory | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 6
} | {
"line": 218,
"column": 6
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nE : Type u₂\ninst✝ : Category.{v₂, u₂} E\nφ : C ⥤ D\nφ' : D ⥤ E\n⊢ map (φ ⋙ φ') = map φ ⋙ map φ'",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"CategoryTheory.FreeGroupoid",
"CategoryT... | [
"C : Type u\ninst✝² : Category.{v, u} C\nD : Type u₁\ninst✝¹ : Category.{v₁, u₁} D\nE : Type u₂\ninst✝ : Category.{v₂, u₂} E\nφ : C ⥤ D\nφ' : D ⥤ E\n⊢ map φ ⋙ map φ' = map (φ ⋙ φ')"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.GradedObject.Braiding | {
"line": 114,
"column": 52
} | {
"line": 114,
"column": 93
} | {
"line": 114,
"column": 94
} | [
{
"pp": "I : Type u_1\ninst✝²⁰ : AddCommMonoid I\nC : Type u_2\ninst✝¹⁹ : Category.{v_1, u_2} C\ninst✝¹⁸ : MonoidalCategory C\nX Y Z : GradedObject I C\ninst✝¹⁷ : BraidedCategory C\ninst✝¹⁶ : X.HasTensor Y\ninst✝¹⁵ : Y.HasTensor Z\ninst✝¹⁴ : Z.HasTensor X\ninst✝¹³ : Z.HasTensor Y\ninst✝¹² : X.HasTensor Z\ninst✝... | [
"I : Type u_1\ninst✝²⁰ : AddCommMonoid I\nC : Type u_2\ninst✝¹⁹ : Category.{v_1, u_2} C\ninst✝¹⁸ : MonoidalCategory C\nX Y Z : GradedObject I C\ninst✝¹⁷ : BraidedCategory C\ninst✝¹⁶ : X.HasTensor Y\ninst✝¹⁵ : Y.HasTensor Z\ninst✝¹⁴ : Z.HasTensor X\ninst✝¹³ : Z.HasTensor Y\ninst✝¹² : X.HasTensor Z\ninst✝¹¹ : (tensor... | ιTensorObj₃'_eq X Y Z i₁ i₂ i₃ k h _ rfl, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Groupoid.Subgroupoid | {
"line": 504,
"column": 30
} | {
"line": 504,
"column": 33
} | {
"line": 504,
"column": 33
} | [
{
"pp": "C : Type u\ninst✝¹ : Groupoid C\nD : Type u_1\ninst✝ : Groupoid D\nφ : C ⥤ D\nhφ : Function.Injective φ.obj\nhφ' : im φ hφ = ⊤\nd : D\n⊢ d ∈ (im φ hφ).objs",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CategoryTheory.Subgroupoid.im",
... | [
"C : Type u\ninst✝¹ : Groupoid C\nD : Type u_1\ninst✝ : Groupoid D\nφ : C ⥤ D\nhφ : Function.Injective φ.obj\nhφ' : im φ hφ = ⊤\nd : D\n⊢ d ∈ ⊤.objs"
] | hφ' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Groupoid.Subgroupoid | {
"line": 516,
"column": 43
} | {
"line": 516,
"column": 46
} | {
"line": 516,
"column": 46
} | [
{
"pp": "C : 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\n⊢ d' ∈ (im φ hφ).objs",
"ppTerm": "?m.207",... | [
"C : 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\n⊢ d' ∈ ⊤.objs"
] | hφ' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Groupoid.Subgroupoid | {
"line": 519,
"column": 65
} | {
"line": 519,
"column": 68
} | {
"line": 519,
"column": 68
} | [
{
"pp": "C : 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\nc : C\nγ : c ⟶ c\nγS : γ ∈ S.arrows c c\ncd' : φ.obj c = φ.obj c\nc' : C\ng : φ.obj c ⟶ φ.obj c'\n⊢ g ∈ (im φ hφ).arrows (φ.obj c) (φ.obj c... | [
"C : 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\nc : C\nγ : c ⟶ c\nγS : γ ∈ S.arrows c c\ncd' : φ.obj c = φ.obj c\nc' : C\ng : φ.obj c ⟶ φ.obj c'\n⊢ g ∈ ⊤.arrows (φ.obj c) (φ.obj c')"
] | hφ' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.FormalCoproducts.Cech | {
"line": 203,
"column": 21
} | {
"line": 203,
"column": 34
} | {
"line": 205,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasFiniteProducts C\nX✝ Y✝ Z✝ : FormalCoproduct C\nx✝¹ : X✝ ⟶ Y✝\nx✝ : Y✝ ⟶ Z✝\n⊢ { app := fun x ↦ powerMap (x✝¹ ≫ x✝) (ToType (Opposite.unop x)), naturality := ⋯ } =\n { app := fun x ↦ powerMap x✝¹ (ToType (Opposite.unop x)), naturality := ⋯ } ≫\n ... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.FormalCoproducts.Cech | {
"line": 203,
"column": 21
} | {
"line": 203,
"column": 34
} | {
"line": 205,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasFiniteProducts C\nX✝ Y✝ Z✝ : FormalCoproduct C\nx✝¹ : X✝ ⟶ Y✝\nx✝ : Y✝ ⟶ Z✝\n⊢ { app := fun x ↦ powerMap (x✝¹ ≫ x✝) (ToType (Opposite.unop x)), naturality := ⋯ } =\n { app := fun x ↦ powerMap x✝¹ (ToType (Opposite.unop x)), naturality := ⋯ } ≫\n ... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Pi | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 40
} | {
"line": 84,
"column": 2
} | [
{
"pp": "I : Type v₁\nC : I → Type u₁\ninst✝¹ : (i : I) → Category.{v₁, u₁} (C i)\nJ : Type v₁\ninst✝ : SmallCategory J\nF : J ⥤ ((i : I) → C i)\nc : (i : I) → Cone (F ⋙ Pi.eval C i)\nP : (i : I) → IsLimit (c i)\ns : Cone F\nj : J\ni : I\n⊢ ((fun i ↦ (P i).lift (coneCompEval s i)) ≫ (coneOfConeCompEval c).π.app... | [] | exact (P i).fac (coneCompEval s i) j | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.Preserves.BifunctorCokernel | {
"line": 80,
"column": 44
} | {
"line": 80,
"column": 70
} | {
"line": 80,
"column": 71
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC : Type u_3\ninst✝¹¹ : Category.{v_1, u_1} C₁\ninst✝¹⁰ : Category.{v_2, u_2} C₂\ninst✝⁹ : Category.{v_3, u_3} C\ninst✝⁸ : HasZeroMorphisms C₁\ninst✝⁷ : HasZeroMorphisms C₂\ninst✝⁶ : HasZeroMorphisms C\nX₁ Y₁ : C₁\nf₁ : X₁ ⟶ Y₁\nc₁ : CokernelCofork f₁\nhc₁ : IsColimit c₁\n... | [
"C₁ : Type u_1\nC₂ : Type u_2\nC : Type u_3\ninst✝¹¹ : Category.{v_1, u_1} C₁\ninst✝¹⁰ : Category.{v_2, u_2} C₂\ninst✝⁹ : Category.{v_3, u_3} C\ninst✝⁸ : HasZeroMorphisms C₁\ninst✝⁷ : HasZeroMorphisms C₂\ninst✝⁶ : HasZeroMorphisms C\nX₁ Y₁ : C₁\nf₁ : X₁ ⟶ Y₁\nc₁ : CokernelCofork f₁\nhc₁ : IsColimit c₁\nX₂ Y₂ : C₂\n... | NatTrans.naturality_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Preserves.BifunctorCokernel | {
"line": 78,
"column": 2
} | {
"line": 82,
"column": 31
} | {
"line": 83,
"column": 2
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC : Type u_3\ninst✝¹¹ : Category.{v_1, u_1} C₁\ninst✝¹⁰ : Category.{v_2, u_2} C₂\ninst✝⁹ : Category.{v_3, u_3} C\ninst✝⁸ : HasZeroMorphisms C₁\ninst✝⁷ : HasZeroMorphisms C₂\ninst✝⁶ : HasZeroMorphisms C\nX₁ Y₁ : C₁\nf₁ : X₁ ⟶ Y₁\nc₁ : CokernelCofork f₁\nhc₁ : IsColimit c₁\n... | [
"C₁ : Type u_1\nC₂ : Type u_2\nC : Type u_3\ninst✝¹¹ : Category.{v_1, u_1} C₁\ninst✝¹⁰ : Category.{v_2, u_2} C₂\ninst✝⁹ : Category.{v_3, u_3} C\ninst✝⁸ : HasZeroMorphisms C₁\ninst✝⁷ : HasZeroMorphisms C₂\ninst✝⁶ : HasZeroMorphisms C\nX₁ Y₁ : C₁\nf₁ : X₁ ⟶ Y₁\nc₁ : CokernelCofork f₁\nhc₁ : IsColimit c₁\nX₂ Y₂ : C₂\n... | obtain ⟨l', hl'⟩ := Cofork.IsColimit.desc' (mapIsColimit _ hc₂ (F.obj c₁.pt)) l (by
have := coprod.inr ≫= s.condition
rw [coprod.inr_desc_assoc, ← dsimp% hl, NatTrans.naturality_assoc, comp_zero] at this
apply Cofork.IsColimit.hom_ext (mapIsColimit _ hc₁ (F.flip.obj X₂))
rwa [zero_comp, comp_zero]) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Limits.Types.PreservesLimit | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 19
} | {
"line": 134,
"column": 4
} | [
{
"pp": "case pos\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : Type u'\ninst✝² : Category.{v', u'} J\ninst✝¹ : LocallySmall.{w, v, u} C\nF : J ⥤ Cᵒᵖ\ninst✝ : Small.{w, u'} J\nx✝ : Cᵒᵖ ⥤ Type w\nhF : HasLimit F\n⊢ (MorphismProperty.single\n (coconePtToShrinkYoneda (limit.cone F) (colimit.isColimit (F.le... | [
"case e'_2\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : Type u'\ninst✝² : Category.{v', u'} J\ninst✝¹ : LocallySmall.{w, v, u} C\nF : J ⥤ Cᵒᵖ\ninst✝ : Small.{w, u'} J\nx✝ : Cᵒᵖ ⥤ Type w\nhF : HasLimit F\n⊢ MorphismProperty.ofHoms (preservesLimitHomFamily F) =\n MorphismProperty.single\n (coconePtToShrinkYon... | convert Iff.rfl | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.CatCospanTransform | {
"line": 264,
"column": 2
} | {
"line": 264,
"column": 6
} | {
"line": 265,
"column": 2
} | [
{
"pp": "A : Type u₁\nB : Type u₂\nC : Type u₃\nA' : Type u₄\nB' : Type u₅\nC' : Type u₆\ninst✝⁶ : Category.{v₁, u₁} A\ninst✝⁵ : Category.{v₂, u₂} B\ninst✝⁴ : Category.{v₃, u₃} C\nF : A ⥤ B\nG : C ⥤ B\ninst✝³ : Category.{v₄, u₄} A'\ninst✝² : Category.{v₅, u₅} B'\ninst✝¹ : Category.{v₆, u₆} C'\nF' : A' ⥤ B'\nG' ... | [
"A : Type u₁\nB : Type u₂\nC : Type u₃\nA' : Type u₄\nB' : Type u₅\nC' : Type u₆\ninst✝⁶ : Category.{v₁, u₁} A\ninst✝⁵ : Category.{v₂, u₂} B\ninst✝⁴ : Category.{v₃, u₃} C\nF : A ⥤ B\nG : C ⥤ B\ninst✝³ : Category.{v₄, u₄} A'\ninst✝² : Category.{v₅, u₅} B'\ninst✝¹ : Category.{v₆, u₆} C'\nF' : A' ⥤ B'\nG' : C' ⥤ B'\nψ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.CatCospanTransform | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 6
} | {
"line": 271,
"column": 2
} | [
{
"pp": "A : Type u₁\nB : Type u₂\nC : Type u₃\nA' : Type u₄\nB' : Type u₅\nC' : Type u₆\ninst✝⁶ : Category.{v₁, u₁} A\ninst✝⁵ : Category.{v₂, u₂} B\ninst✝⁴ : Category.{v₃, u₃} C\nF : A ⥤ B\nG : C ⥤ B\ninst✝³ : Category.{v₄, u₄} A'\ninst✝² : Category.{v₅, u₅} B'\ninst✝¹ : Category.{v₆, u₆} C'\nF' : A' ⥤ B'\nG' ... | [
"A : Type u₁\nB : Type u₂\nC : Type u₃\nA' : Type u₄\nB' : Type u₅\nC' : Type u₆\ninst✝⁶ : Category.{v₁, u₁} A\ninst✝⁵ : Category.{v₂, u₂} B\ninst✝⁴ : Category.{v₃, u₃} C\nF : A ⥤ B\nG : C ⥤ B\ninst✝³ : Category.{v₄, u₄} A'\ninst✝² : Category.{v₅, u₅} B'\ninst✝¹ : Category.{v₆, u₆} C'\nF' : A' ⥤ B'\nG' : C' ⥤ B'\nψ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.CatCospanTransform | {
"line": 276,
"column": 2
} | {
"line": 276,
"column": 6
} | {
"line": 277,
"column": 2
} | [
{
"pp": "A : Type u₁\nB : Type u₂\nC : Type u₃\nA' : Type u₄\nB' : Type u₅\nC' : Type u₆\ninst✝⁶ : Category.{v₁, u₁} A\ninst✝⁵ : Category.{v₂, u₂} B\ninst✝⁴ : Category.{v₃, u₃} C\nF : A ⥤ B\nG : C ⥤ B\ninst✝³ : Category.{v₄, u₄} A'\ninst✝² : Category.{v₅, u₅} B'\ninst✝¹ : Category.{v₆, u₆} C'\nF' : A' ⥤ B'\nG' ... | [
"A : Type u₁\nB : Type u₂\nC : Type u₃\nA' : Type u₄\nB' : Type u₅\nC' : Type u₆\ninst✝⁶ : Category.{v₁, u₁} A\ninst✝⁵ : Category.{v₂, u₂} B\ninst✝⁴ : Category.{v₃, u₃} C\nF : A ⥤ B\nG : C ⥤ B\ninst✝³ : Category.{v₄, u₄} A'\ninst✝² : Category.{v₅, u₅} B'\ninst✝¹ : Category.{v₆, u₆} C'\nF' : A' ⥤ B'\nG' : C' ⥤ B'\nψ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Limits.Shapes.SequentialProduct | {
"line": 222,
"column": 55
} | {
"line": 235,
"column": 28
} | {
"line": 236,
"column": 0
} | [
{
"pp": "C : Type u_1\nM N : ℕ → C\ninst✝⁴ : Category.{v_1, u_1} C\nf : (n : ℕ) → M n ⟶ N n\ninst✝³ : HasCountableProducts C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasFiniteBiproducts C\ninst✝ : ∀ (n : ℕ), Epi (f n)\nn : ℕ\n⊢ Epi (functorMap f n)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants... | [] | by
rw [functorMap, Pi.map_eq_prod_map (P := fun m : ℕ ↦ m < n + 1)]
apply +allowSynthFailures epi_comp
apply +allowSynthFailures epi_comp
apply +allowSynthFailures prod.map_epi
· apply +allowSynthFailures Pi.map_epi
intro ⟨_, _⟩
split
all_goals infer_instance
· apply +allowSynthFailures IsIso.ep... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Localization.BousfieldTransfiniteComposition | {
"line": 58,
"column": 12
} | {
"line": 58,
"column": 86
} | {
"line": 58,
"column": 86
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nP : ObjectProperty C\nJ : Type w\ninst✝³ : LinearOrder J\ninst✝² : SuccOrder J\ninst✝¹ : OrderBot J\ninst✝ : WellFoundedLT J\nX Y : C\nf : X ⟶ Y\nx✝ : P.isLocal.transfiniteCompositionsOfShape J f\nZ : C\nhZ : P Z\nhf : P.isLocal.TransfiniteCompositionOfShape J f\... | [] | by simpa using! (hf.F.isColimitOfIsWellOrderContinuous j hj).fac c ⟨k, hk⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 6
} | {
"line": 180,
"column": 2
} | [
{
"pp": "A : Type u₁\nB : Type u₂\nC : Type u₃\ninst✝³ : Category.{v₁, u₁} A\ninst✝² : Category.{v₂, u₂} B\ninst✝¹ : Category.{v₃, u₃} C\nF : A ⥤ B\nG : C ⥤ B\nx y : F ⊡ G\nf : x ⟶ y\ninst✝ : IsIso f\n⊢ (inv f).fst = inv f.fst",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Category... | [
"A : Type u₁\nB : Type u₂\nC : Type u₃\ninst✝³ : Category.{v₁, u₁} A\ninst✝² : Category.{v₂, u₂} B\ninst✝¹ : Category.{v₃, u₃} C\nF : A ⥤ B\nG : C ⥤ B\nx y : F ⊡ G\nf : x ⟶ y\ninst✝ : IsIso f\n⊢ inv f.fst = (inv f).fst"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Categorical.Basic | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 6
} | {
"line": 186,
"column": 2
} | [
{
"pp": "A : Type u₁\nB : Type u₂\nC : Type u₃\ninst✝³ : Category.{v₁, u₁} A\ninst✝² : Category.{v₂, u₂} B\ninst✝¹ : Category.{v₃, u₃} C\nF : A ⥤ B\nG : C ⥤ B\nx y : F ⊡ G\nf : x ⟶ y\ninst✝ : IsIso f\n⊢ (inv f).snd = inv f.snd",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Category... | [
"A : Type u₁\nB : Type u₂\nC : Type u₃\ninst✝³ : Category.{v₁, u₁} A\ninst✝² : Category.{v₂, u₂} B\ninst✝¹ : Category.{v₃, u₃} C\nF : A ⥤ B\nG : C ⥤ B\nx y : F ⊡ G\nf : x ⟶ y\ninst✝ : IsIso f\n⊢ inv f.snd = (inv f).snd"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Monoidal.Free.Basic | {
"line": 329,
"column": 34
} | {
"line": 329,
"column": 80
} | {
"line": 330,
"column": 4
} | [
{
"pp": "case tensorHom_comp_tensorHom\nC : Type u\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : MonoidalCategory D\nf✝ : C → D\nX Y : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\nf g : failed to pretty print expression (use 'set_option pp.rawOnError true... | [] | rw [MonoidalCategory.tensorHom_comp_tensorHom] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.MarkovCategory.Positive | {
"line": 80,
"column": 6
} | {
"line": 80,
"column": 52
} | {
"line": 81,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : MonoidalCategory C\ninst✝¹ : PositiveCategory C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsIso f\n⊢ Δ ≫ (𝟙 X ⊗ₘ f) ≫ (f ⊗ₘ 𝟙 Y) = Δ ≫ (𝟙 X ≫ f ⊗ₘ f ≫ 𝟙 Y)",
"ppTerm": "?m.231",
"assigned": true,
"usedConstants": [
"CategoryTheory.ComonObj.comul... | [] | rw [MonoidalCategory.tensorHom_comp_tensorHom] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.MarkovCategory.Positive | {
"line": 80,
"column": 6
} | {
"line": 80,
"column": 52
} | {
"line": 81,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : MonoidalCategory C\ninst✝¹ : PositiveCategory C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsIso f\n⊢ Δ ≫ (𝟙 X ⊗ₘ f) ≫ (f ⊗ₘ 𝟙 Y) = Δ ≫ (𝟙 X ≫ f ⊗ₘ f ≫ 𝟙 Y)",
"ppTerm": "?m.231",
"assigned": true,
"usedConstants": [
"CategoryTheory.ComonObj.comul... | [] | rw [MonoidalCategory.tensorHom_comp_tensorHom] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MarkovCategory.Positive | {
"line": 80,
"column": 6
} | {
"line": 80,
"column": 52
} | {
"line": 81,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : MonoidalCategory C\ninst✝¹ : PositiveCategory C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsIso f\n⊢ Δ ≫ (𝟙 X ⊗ₘ f) ≫ (f ⊗ₘ 𝟙 Y) = Δ ≫ (𝟙 X ≫ f ⊗ₘ f ≫ 𝟙 Y)",
"ppTerm": "?m.231",
"assigned": true,
"usedConstants": [
"CategoryTheory.ComonObj.comul... | [] | rw [MonoidalCategory.tensorHom_comp_tensorHom] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monad.Types | {
"line": 43,
"column": 21
} | {
"line": 43,
"column": 48
} | {
"line": 45,
"column": 0
} | [
{
"pp": "m : Type u → Type u\ninst✝¹ : _root_.Monad m\ninst✝ : LawfulMonad m\nx✝ : Type u\n⊢ (ofTypeFunctor m).map ({ app := fun X ↦ ↾pure, naturality := ⋯ }.app x✝) ≫\n { app := fun X ↦ ↾joinM, naturality := ⋯ }.app x✝ =\n 𝟙 ((ofTypeFunctor m).obj ((𝟭 (Type u)).obj x✝))",
"ppTerm": "?m.178",
... | [] | ext; exact joinM_map_pure _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monad.Types | {
"line": 43,
"column": 21
} | {
"line": 43,
"column": 48
} | {
"line": 45,
"column": 0
} | [
{
"pp": "m : Type u → Type u\ninst✝¹ : _root_.Monad m\ninst✝ : LawfulMonad m\nx✝ : Type u\n⊢ (ofTypeFunctor m).map ({ app := fun X ↦ ↾pure, naturality := ⋯ }.app x✝) ≫\n { app := fun X ↦ ↾joinM, naturality := ⋯ }.app x✝ =\n 𝟙 ((ofTypeFunctor m).obj ((𝟭 (Type u)).obj x✝))",
"ppTerm": "?m.178",
... | [] | ext; exact joinM_map_pure _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monad.Monadicity | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 39
} | {
"line": 125,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u₁\nD : Type u₂\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₁, u₂} D\nG : D ⥤ C\nF : C ⥤ D\nadj : F ⊣ G\ninst✝ : ∀ (A : adj.toMonad.Algebra), HasCoequalizer (F.map A.a) (adj.counit.app (F.obj A.A))\nA : adj.toMonad.Algebra\nB : D\n⊢ (comparisonLeftAdjointObj adj A ⟶ B) ≃ ... | [] | apply comparisonLeftAdjointHomEquiv | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Monad.Monadicity | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 39
} | {
"line": 125,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u₁\nD : Type u₂\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₁, u₂} D\nG : D ⥤ C\nF : C ⥤ D\nadj : F ⊣ G\ninst✝ : ∀ (A : adj.toMonad.Algebra), HasCoequalizer (F.map A.a) (adj.counit.app (F.obj A.A))\nA : adj.toMonad.Algebra\nB : D\n⊢ (comparisonLeftAdjointObj adj A ⟶ B) ≃ ... | [] | apply comparisonLeftAdjointHomEquiv | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monad.Monadicity | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 39
} | {
"line": 125,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u₁\nD : Type u₂\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₁, u₂} D\nG : D ⥤ C\nF : C ⥤ D\nadj : F ⊣ G\ninst✝ : ∀ (A : adj.toMonad.Algebra), HasCoequalizer (F.map A.a) (adj.counit.app (F.obj A.A))\nA : adj.toMonad.Algebra\nB : D\n⊢ (comparisonLeftAdjointObj adj A ⟶ B) ≃ ... | [] | apply comparisonLeftAdjointHomEquiv | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 234,
"column": 23
} | {
"line": 234,
"column": 53
} | {
"line": 234,
"column": 53
} | [
{
"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} (tensorLeft X)\n| η ▷ (P.X ⊗ Q.X) ≫ (α_ R.X P.X Q.X).inv",
"ppTerm": "?a.a✝",
... | [
"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} (tensorLeft X)\n| (α_ (𝟙_ C) P.X Q.X).inv ≫ η ▷ P.X ▷ Q.X",
"case a.a\nC : Type u₁\ninst✝³ ... | associator_inv_naturality_left | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 247,
"column": 23
} | {
"line": 247,
"column": 53
} | {
"line": 247,
"column": 53
} | [
{
"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} (tensorLeft X)\n| μ ▷ (P.X ⊗ Q.X) ≫ (α_ R.X P.X Q.X).inv",
"ppTerm": "?a.a✝",
... | [
"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} (tensorLeft X)\n| (α_ (R.X ⊗ R.X) P.X Q.X).inv ≫ μ ▷ P.X ▷ Q.X",
"case a.a\nC : Type u₁\nins... | associator_inv_naturality_left | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 313,
"column": 23
} | {
"line": 313,
"column": 53
} | {
"line": 313,
"column": 53
} | [
{
"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 T.X).inv ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.... | associator_inv_naturality_left | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Closed.Functor | {
"line": 89,
"column": 2
} | {
"line": 92,
"column": 21
} | {
"line": 94,
"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 B : C\n... | [] | convert! mateEquiv_counit _ _ (prodComparisonNatIso F A).inv B using 2
apply IsIso.inv_eq_of_hom_inv_id -- Porting note (https://github.com/leanprover-community/mathlib4/issues/11041): was `ext`
simp only [prodComparisonNatTrans_app, prodComparisonNatIso_inv, NatIso.isIso_inv_app,
IsIso.hom_inv_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Closed.Functor | {
"line": 89,
"column": 2
} | {
"line": 92,
"column": 21
} | {
"line": 94,
"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 B : C\n... | [] | convert! mateEquiv_counit _ _ (prodComparisonNatIso F A).inv B using 2
apply IsIso.inv_eq_of_hom_inv_id -- Porting note (https://github.com/leanprover-community/mathlib4/issues/11041): was `ext`
simp only [prodComparisonNatTrans_app, prodComparisonNatIso_inv, NatIso.isIso_inv_app,
IsIso.hom_inv_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Basic | {
"line": 66,
"column": 10
} | {
"line": 66,
"column": 36
} | {
"line": 66,
"column": 37
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalClosed C\nJ : Type u₂\ninst✝¹ : Category.{v₂, u₂} J\ninst✝ : ∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂\nF₁ F₂ F₂' F₃ F₃' : J ⥤ C\ng : F₂ ⟶ functorEnrichedHom C F₁ F₃\nj j' : J\nφ : j ⟶ j'\n⊢ uncurry\n (... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalClosed C\nJ : Type u₂\ninst✝¹ : Category.{v₂, u₂} J\ninst✝ : ∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂\nF₁ F₂ F₂' F₃ F₃' : J ⥤ C\ng : F₂ ⟶ functorEnrichedHom C F₁ F₃\nj j' : J\nφ : j ⟶ j'\n⊢ uncurry\n (g.app j ≫\n ... | NatTrans.naturality_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Basic | {
"line": 83,
"column": 4
} | {
"line": 88,
"column": 85
} | {
"line": 89,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalClosed C\nJ : Type u₂\ninst✝¹ : Category.{v₂, u₂} J\ninst✝ : ∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂\nF₁ F₂ F₂' F₃ F₃' : J ⥤ C\ng : F₂ ⟶ functorEnrichedHom C F₁ F₃\nj : J\nk : Under j\n⊢ end_.lift\n ... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalClosed C\nJ : Type u₂\ninst✝¹ : Category.{v₂, u₂} J\ninst✝ : ∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂\nF₁ F₂ F₂' F₃ F₃' : J ⥤ C\ng : F₂ ⟶ functorEnrichedHom C F₁ F₃\nj : J\nk : Under j\n⊢ g.app j ≫\n enrichedHomπ ... | simp only [diagram_obj_obj, Functor.comp_obj, Under.forget_obj, enrichedCategorySelf_hom,
curry_uncurry, NatTrans.naturality_assoc, functorEnrichedHom_obj, functorEnrichedHom_map,
Under.map, Comma.mapLeft, Functor.const_obj_obj, Functor.id_obj, Discrete.natTrans_app,
StructuredArrow.left_eq_id, end_.l... | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Subterminal | {
"line": 104,
"column": 33
} | {
"line": 104,
"column": 46
} | {
"line": 104,
"column": 46
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nA : C\ninst✝¹ : HasBinaryProduct A A\ninst✝ : IsIso (diag A)\nZ : C\nf g : Z ⟶ A\nthis : prod.fst = prod.snd\n⊢ prod.lift f g ≫ prod.snd = g",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStr... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nA : C\ninst✝¹ : HasBinaryProduct A A\ninst✝ : IsIso (diag A)\nZ : C\nf g : Z ⟶ A\nthis : prod.fst = prod.snd\n⊢ g = g"
] | prod.lift_snd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Closed.Ideal | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 6
} | {
"line": 88,
"column": 2
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝⁵ : Category.{v₁, u₁} C\ninst✝⁴ : Category.{v₁, u₂} D\ni : D ⥤ C\ninst✝³ : CartesianMonoidalCategory C\ninst✝² : MonoidalClosed C\nA : C\ninst✝¹ : Reflective i\ninst✝ : ExponentialIdeal i\n⊢ i ⋙ ihom A ⋙ reflector i ⋙ i ≅ i ⋙ ihom A",
"ppTerm": "?m.55",
"assigned"... | [
"C : Type u₁\nD : Type u₂\ninst✝⁵ : Category.{v₁, u₁} C\ninst✝⁴ : Category.{v₁, u₂} D\ni : D ⥤ C\ninst✝³ : CartesianMonoidalCategory C\ninst✝² : MonoidalClosed C\nA : C\ninst✝¹ : Reflective i\ninst✝ : ExponentialIdeal i\n⊢ i ⋙ ihom A ≅ i ⋙ ihom A ⋙ reflector i ⋙ i"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Basic | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 6
} | {
"line": 119,
"column": 2
} | [
{
"pp": "case e_a\nC : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\ninst✝⁴ : MonoidalCategory C\ninst✝³ : MonoidalClosed C\nJ : Type u₂\ninst✝² : Category.{v₂, u₂} J\ninst✝¹ : ∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂\nF₁ F₂ F₃ F₃' : J ⥤ C\ninst✝ : ∀ (F₁ F₂ : J ⥤ C), HasEnrichedHom C F₁ F₂\nf : F₁ ⊗ F₂ ⟶ F₃\nf... | [
"case e_a\nC : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\ninst✝⁴ : MonoidalCategory C\ninst✝³ : MonoidalClosed C\nJ : Type u₂\ninst✝² : Category.{v₂, u₂} J\ninst✝¹ : ∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂\nF₁ F₂ F₃ F₃' : J ⥤ C\ninst✝ : ∀ (F₁ F₂ : J ⥤ C), HasEnrichedHom C F₁ F₂\nf : F₁ ⊗ F₂ ⟶ F₃\nf₃ : F₃ ⟶ F₃'... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Monoidal.Closed.Ideal | {
"line": 213,
"column": 10
} | {
"line": 213,
"column": 14
} | {
"line": 214,
"column": 10
} | [
{
"pp": "case comm1\nC : Type u₁\nD : Type u₂\ninst✝⁶ : Category.{v₁, u₁} C\ninst✝⁵ : Category.{v₁, u₂} D\ni : D ⥤ C\ninst✝⁴ : CartesianMonoidalCategory C\ninst✝³ : Reflective i\ninst✝² : MonoidalClosed C\ninst✝¹ : CartesianMonoidalCategory D\ninst✝ : ExponentialIdeal i\nl : i.EssImageSubcategory ⥤ D\nφ : l ⋙ i... | [
"case comm1\nC : Type u₁\nD : Type u₂\ninst✝⁶ : Category.{v₁, u₁} C\ninst✝⁵ : Category.{v₁, u₂} D\ni : D ⥤ C\ninst✝⁴ : CartesianMonoidalCategory C\ninst✝³ : Reflective i\ninst✝² : MonoidalClosed C\ninst✝¹ : CartesianMonoidalCategory D\ninst✝ : ExponentialIdeal i\nl : i.EssImageSubcategory ⥤ D\nφ : l ⋙ i ≅ i.essImag... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed | {
"line": 188,
"column": 2
} | {
"line": 191,
"column": 60
} | {
"line": 193,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁵ : Category.{v₂, u₂} V\ninst✝⁴ : MonoidalCategory C\ninst✝³ : MonoidalCategory V\ninst✝² : MonoidalClosed V\nF G H : C ⥤ V\ninst✝¹ : DayConvolution F H\nℌ : DayConvolutionInternalHom F G H\nG' H' : C ⥤ V\nℌ' : DayConvolutionInternalHom F G' ... | [] | apply DayConvolution.corepresentableBy F H |>.homEquiv.injective
dsimp
ext ⟨x, y⟩
simp [MonoidalClosed.uncurry_eq, ← whiskerLeft_comp_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed | {
"line": 188,
"column": 2
} | {
"line": 191,
"column": 60
} | {
"line": 193,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁵ : Category.{v₂, u₂} V\ninst✝⁴ : MonoidalCategory C\ninst✝³ : MonoidalCategory V\ninst✝² : MonoidalClosed V\nF G H : C ⥤ V\ninst✝¹ : DayConvolution F H\nℌ : DayConvolutionInternalHom F G H\nG' H' : C ⥤ V\nℌ' : DayConvolutionInternalHom F G' ... | [] | apply DayConvolution.corepresentableBy F H |>.homEquiv.injective
dsimp
ext ⟨x, y⟩
simp [MonoidalClosed.uncurry_eq, ← whiskerLeft_comp_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.DayConvolution.DayFunctor | {
"line": 223,
"column": 13
} | {
"line": 228,
"column": 9
} | {
"line": 230,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁹ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁸ : Category.{v₂, u₂} V\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : MonoidalCategory V\nhasDayConvolution : ∀ (F G : C ⥤ V), (tensor C).HasPointwiseLeftKanExtension (F ⊠ G)\nhasDayConvolutionUnit : (Functor.fromPUnit (𝟙_ C)).HasPointwiseLeftKanEx... | [] | by
ext1
apply Functor.hom_ext_of_isLeftKanExtension
(𝟙_ (C ⊛⥤ V)).functor (νNatTrans C V)
ext
exact h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Free.Coherence | {
"line": 255,
"column": 6
} | {
"line": 255,
"column": 62
} | {
"line": 256,
"column": 2
} | [
{
"pp": "case mk.tensor\nC : Type u\nX✝ Y✝ W X Y Z : F C\nf✝ : W ⟶ᵐ Y\ng✝ : X ⟶ᵐ Z\nih₁ : (fun n ↦ W.normalizeObj n) = Y.normalizeObj\nih₂ : (fun n ↦ X.normalizeObj n) = Z.normalizeObj\nn : NormalMonoidalObject C\n⊢ (W.tensor X).normalizeObj n = (Y.tensor Z).normalizeObj n",
"ppTerm": "?mk.tensor",
"ass... | [] | simp [congr_fun ih₁ n, congr_fun ih₂ (normalizeObj Y n)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 346,
"column": 8
} | {
"line": 346,
"column": 38
} | {
"line": 346,
"column": 38
} | [
{
"pp": "case a.a.a.a.a.a.a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| μ ▷ (A ⊗ A) ≫ (α_ A A A).inv",
"ppTerm": "?a.a.a.a.a.a.a.a.a.a.a✝",
"assigned": true,
"usedConstants": [
"CategoryTheory.Category... | [
"case a.a.a.a.a.a.a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| (α_ (A ⊗ A) A A).inv ≫ μ ▷ A ▷ A",
"case a.a.a.a.a.a.a.a.a.a.a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCateg... | associator_inv_naturality_left | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 418,
"column": 4
} | {
"line": 418,
"column": 31
} | {
"line": 419,
"column": 2
} | [
{
"pp": "case a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| η ▷ 𝟙_ C ≫ (ρ_ A).hom",
"ppTerm": "?a",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"co... | [
"case a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| (ρ_ (𝟙_ C)).hom ≫ η",
"case a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| ε ⊗ₘ ε"
] | rw [rightUnitor_naturality] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 418,
"column": 4
} | {
"line": 418,
"column": 31
} | {
"line": 419,
"column": 2
} | [
{
"pp": "case a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| η ▷ 𝟙_ C ≫ (ρ_ A).hom",
"ppTerm": "?a",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"co... | [
"case a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| (ρ_ (𝟙_ C)).hom ≫ η",
"case a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| ε ⊗ₘ ε"
] | rw [rightUnitor_naturality] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Hopf_ | {
"line": 418,
"column": 4
} | {
"line": 418,
"column": 31
} | {
"line": 419,
"column": 2
} | [
{
"pp": "case a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| η ▷ 𝟙_ C ≫ (ρ_ A).hom",
"ppTerm": "?a",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"co... | [
"case a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| (ρ_ (𝟙_ C)).hom ≫ η",
"case a\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\ninst✝¹ : BraidedCategory C\nA : C\ninst✝ : HopfObj A\n| ε ⊗ₘ ε"
] | rw [rightUnitor_naturality] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 571,
"column": 27
} | {
"line": 571,
"column": 57
} | {
"line": 571,
"column": 57
} | [
{
"pp": "case a.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon C\nM : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\... | [
"case a.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon C\nM : Bimod A B\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nR S T U : M... | associator_inv_naturality_left | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Limits.Basic | {
"line": 65,
"column": 6
} | {
"line": 68,
"column": 61
} | {
"line": 69,
"column": 6
} | [
{
"pp": "J : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF G H : J ⥤ C\nj : J\n⊢ limit.lift (F ⊗ G) { pt := limit F ⊗ limit G, π := { app := fun j ↦ limit.π F j ⊗ₘ limit.π G j, naturality := ⋯ } } ▷\n limit H ≫\n ... | [
"J : Type w\ninst✝³ : SmallCategory J\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : MonoidalCategory C\nF G H : J ⥤ C\nj : J\n⊢ ({ pt := limit F ⊗ limit G, π := { app := fun j ↦ limit.π F j ⊗ₘ limit.π G j, naturality := ⋯ } }.π.app j ⊗ₘ\n limit.π H j) ≫\n (α_ (F.obj j) ... | conv_lhs => rw [tensorHom_def, Category.assoc, ← comp_whiskerRight_assoc,
limit.lift_π, tensor_whiskerLeft, Category.assoc, Category.assoc,
Iso.inv_hom_id, Category.comp_id,
← associator_naturality_right, ← tensorHom_def_assoc] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convLHS_1 | Mathlib.Tactic.Conv.convLHS |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 638,
"column": 23
} | {
"line": 638,
"column": 53
} | {
"line": 638,
"column": 53
} | [
{
"pp": "case a.a.a\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : HasCoequalizers C\nR S : Mon C\nP : Bimod R S\n| η ▷ (R.X ⊗ P.X) ≫ (α_ R.X R.X P.X).inv",
"ppTerm": "?a.a.a✝",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
... | [
"case a.a.a\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : HasCoequalizers C\nR S : Mon C\nP : Bimod R S\n| (α_ (𝟙_ C) R.X P.X).inv ≫ η ▷ R.X ▷ P.X",
"case a.a.a\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : HasCoequalizers C\nR S : Mon C\nP : ... | associator_inv_naturality_left | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.Monoidal.Rigid.Functor | {
"line": 61,
"column": 21
} | {
"line": 61,
"column": 49
} | {
"line": 63,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\ninst✝ : RightRigidCategory C\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n⊢ (f ≫ g)ᘁ.op.mop = fᘁ.op.mop ≫ gᘁ.op.mop",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"CategoryTheory.MonoidalOpposite.mop",
"... | [] | simp [comp_rightAdjointMate] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Monoidal.Rigid.Functor | {
"line": 61,
"column": 21
} | {
"line": 61,
"column": 49
} | {
"line": 63,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\ninst✝ : RightRigidCategory C\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n⊢ (f ≫ g)ᘁ.op.mop = fᘁ.op.mop ≫ gᘁ.op.mop",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"CategoryTheory.MonoidalOpposite.mop",
"... | [] | simp [comp_rightAdjointMate] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Rigid.Functor | {
"line": 61,
"column": 21
} | {
"line": 61,
"column": 49
} | {
"line": 63,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\ninst✝ : RightRigidCategory C\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n⊢ (f ≫ g)ᘁ.op.mop = fᘁ.op.mop ≫ gᘁ.op.mop",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"CategoryTheory.MonoidalOpposite.mop",
"... | [] | simp [comp_rightAdjointMate] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.ObjectProperty.Ind | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 63
} | {
"line": 106,
"column": 0
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝¹ : Category.{v, u} C\nP : ObjectProperty C\nH✝ : P ≤ isFinitelyPresentable C\ninst✝ : IsFinitelyAccessibleCategory C\nX : C\nhfac : ∀ {Z : C} (g : Z ⟶ X) [IsFinitelyPresentable Z], ∃ W u v, u ≫ v = g ∧ P W\nincl : P.FullSubcategory ⥤ (isFinitelyPresentable C).FullSubcat... | [] | exact of_essentiallySmall_index ⟨_, _, hc⟩ fun Y ↦ Y.left.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Monoidal.DayConvolution | {
"line": 885,
"column": 2
} | {
"line": 892,
"column": 37
} | {
"line": 894,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁶ : Category.{v₂, u₂} V\ninst✝⁵ : MonoidalCategory C\ninst✝⁴ : MonoidalCategory V\nD : Type u₃\ninst✝³ : Category.{v₃, u₃} D\ninst✝² : MonoidalCategoryStruct D\ninst✝¹ : LawfulDayConvolutionMonoidalCategoryStruct C V D\nd : D\ninst✝ :\n ∀ (v... | [] | apply corepresentableByLeft
(ι C V D |>.obj <| 𝟙_ D) (ι C V D |>.obj d) |>.homEquiv.injective
dsimp
ext ⟨_, x⟩
dsimp [corepresentableByLeft]
simp only [whiskerLeft_id, Category.comp_id,
DayConvolutionUnit.leftUnitor_hom_unit_app]
exact leftUnitor_hom_unit_app V d x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.DayConvolution | {
"line": 885,
"column": 2
} | {
"line": 892,
"column": 37
} | {
"line": 894,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\nV : Type u₂\ninst✝⁶ : Category.{v₂, u₂} V\ninst✝⁵ : MonoidalCategory C\ninst✝⁴ : MonoidalCategory V\nD : Type u₃\ninst✝³ : Category.{v₃, u₃} D\ninst✝² : MonoidalCategoryStruct D\ninst✝¹ : LawfulDayConvolutionMonoidalCategoryStruct C V D\nd : D\ninst✝ :\n ∀ (v... | [] | apply corepresentableByLeft
(ι C V D |>.obj <| 𝟙_ D) (ι C V D |>.obj d) |>.homEquiv.injective
dsimp
ext ⟨_, x⟩
dsimp [corepresentableByLeft]
simp only [whiskerLeft_id, Category.comp_id,
DayConvolutionUnit.leftUnitor_hom_unit_app]
exact leftUnitor_hom_unit_app V d x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 810,
"column": 23
} | {
"line": 810,
"column": 53
} | {
"line": 810,
"column": 53
} | [
{
"pp": "case a.a.a.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nX Y : Mon C\nM N : Bimo... | [
"case a.a.a.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nX Y : Mon C\nM N : Bimod X Y\nf : M... | associator_inv_naturality_left | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.CategoryTheory.MorphismProperty.Ind | {
"line": 72,
"column": 32
} | {
"line": 77,
"column": 9
} | {
"line": 79,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nP : MorphismProperty C\n⊢ P ≤ P.ind",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"ChainCompletePartialOrder.instOfCompleteLattice",
"and_true",
"CategoryTheory.CategoryStruct... | [] | by
intro X Y f hf
refine ⟨PUnit, inferInstance, inferInstance, (Functor.const PUnit).obj Y, ?_, 𝟙 _, ?_, ?_⟩
· exact { app _ := f }
· exact isColimitConstCocone _ _
· simpa | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.DayConvolution | {
"line": 1000,
"column": 6
} | {
"line": 1001,
"column": 51
} | {
"line": 1001,
"column": 51
} | [
{
"pp": "C✝ : Type u₁\ninst✝¹⁶ : Category.{v₁, u₁} C✝\nV✝ : Type u₂\ninst✝¹⁵ : Category.{v₂, u₂} V✝\ninst✝¹⁴ : MonoidalCategory C✝\ninst✝¹³ : MonoidalCategory V✝\nC : Type u₁\ninst✝¹² : Category.{v₁, u₁} C\nV : Type u₂\ninst✝¹¹ : Category.{v₂, u₂} V\ninst✝¹⁰ : MonoidalCategory C\ninst✝⁹ : MonoidalCategory V\nD ... | [] | exact DayConvolutionUnit.rightUnitor_naturality
(ι C V D |>.obj <| 𝟙_ D) (ι C V D |>.map f) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.MorphismProperty.LocalEpi | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 31
} | {
"line": 115,
"column": 4
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝² : Category.{v_2, u_2} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝¹ : G.Faithful\ninst✝ : G.Full\nX Y : C\nf : X ⟶ Y\n⊢ Epi (F.map f) → localEpi (fun x ↦ x ∈ Set.range G.obj) (G.map (F.map f))",
"ppTerm": "?refine_2",
... | [
"case refine_2\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝² : Category.{v_2, u_2} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝¹ : G.Faithful\ninst✝ : G.Full\nX Y : C\nf : X ⟶ Y\nh : Epi (F.map f)\nZ : D\nu v : (F ⋙ G).obj Y ⟶ G.obj Z\nhuv : (fun g ↦ G.map (F.map f) ≫ g) u = (fun g ↦ G.map (F.m... | rintro h _ ⟨Z, rfl⟩ u v huv | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.Preadditive.EndoFunctor | {
"line": 87,
"column": 18
} | {
"line": 87,
"column": 94
} | {
"line": 87,
"column": 95
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Coalgebra F\nα β : A₁ ⟶ A₂\n⊢ A₁.str ≫ F.map (α.f - β.f) = (α.f - β.f) ≫ A₂.str",
"ppTerm": "?m.355",
"assigned": true,
"usedConstants": [
"CategoryTheory.Endofunctor.Coalgebra.st... | [] | simp only [Functor.map_sub, comp_sub, Endofunctor.Coalgebra.Hom.h, sub_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Preadditive.EndoFunctor | {
"line": 87,
"column": 18
} | {
"line": 87,
"column": 94
} | {
"line": 87,
"column": 95
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Coalgebra F\nα β : A₁ ⟶ A₂\n⊢ A₁.str ≫ F.map (α.f - β.f) = (α.f - β.f) ≫ A₂.str",
"ppTerm": "?m.355",
"assigned": true,
"usedConstants": [
"CategoryTheory.Endofunctor.Coalgebra.st... | [] | simp only [Functor.map_sub, comp_sub, Endofunctor.Coalgebra.Hom.h, sub_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Preadditive.EndoFunctor | {
"line": 87,
"column": 18
} | {
"line": 87,
"column": 94
} | {
"line": 87,
"column": 95
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Preadditive C\nF : C ⥤ C\ninst✝ : F.Additive\nA₁ A₂ : Coalgebra F\nα β : A₁ ⟶ A₂\n⊢ A₁.str ≫ F.map (α.f - β.f) = (α.f - β.f) ≫ A₂.str",
"ppTerm": "?m.355",
"assigned": true,
"usedConstants": [
"CategoryTheory.Endofunctor.Coalgebra.st... | [] | simp only [Functor.map_sub, comp_sub, Endofunctor.Coalgebra.Hom.h, sub_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Preadditive.HomOrthogonal | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 10
} | {
"line": 100,
"column": 6
} | [
{
"pp": "case neg\nC : Type u\ninst✝⁴ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteBiproducts C\no : HomOrthogonal s\nα β : Type\ninst✝¹ : Finite α\ninst✝ : Finite β\nf : α → ι\ng : β → ι\nz : (⨁ fun a ↦ s (f a)) ⟶ ⨁ fun b ↦ s (g b)\nj : β\nk : α\nh : ¬f k = g j\n... | [
"case neg\nC : Type u\ninst✝⁴ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasFiniteBiproducts C\no : HomOrthogonal s\nα β : Type\ninst✝¹ : Finite α\ninst✝ : Finite β\nf : α → ι\ng : β → ι\nz : (⨁ fun a ↦ s (f a)) ⟶ ⨁ fun b ↦ s (g b)\nj : β\nk : α\nh : ¬f k = g j\n⊢ biproduct.... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Pi.Monoidal | {
"line": 259,
"column": 20
} | {
"line": 259,
"column": 67
} | {
"line": 260,
"column": 2
} | [
{
"pp": "I : Type w₁\nC : I → Type u₁\ninst✝⁶ : (i : I) → Category.{v₁, u₁} (C i)\ninst✝⁵ : (i : I) → MonoidalCategory (C i)\nD : Type u_1\ninst✝⁴ : Category.{v_1, u_1} D\ninst✝³ : MonoidalCategory D\nF G : D ⥤ ((i : I) → C i)\ninst✝² : F.LaxMonoidal\ninst✝¹ : G.LaxMonoidal\nτ : (i : I) → F ⋙ eval C i ⟶ G ⋙ eva... | [] | simpa using NatTrans.IsMonoidal.unit (τ := τ i) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.EckmannHilton | {
"line": 69,
"column": 18
} | {
"line": 69,
"column": 65
} | {
"line": 71,
"column": 0
} | [
{
"pp": "X : Type u\nm₁ m₂ : X → X → X\ne₁ e₂ : X\nh₁ : IsUnital m₁ e₁\nh₂ : IsUnital m₂ e₂\ndistrib : ∀ (a b c d : X), m₁ (m₂ a b) (m₂ c d) = m₂ (m₁ a c) (m₁ b d)\na b : X\n⊢ m₁ (m₂ a e₁) (m₂ e₁ b) = m₂ a b",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Std.LawfulRightIdentity.rig... | [] | by simp only [distrib, h₁.left_id, h₁.right_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 886,
"column": 76
} | {
"line": 907,
"column": 5
} | {
"line": 909,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nW X Y Z : Mon C\nM M' : Bimod W X\nf ... | [] | by
dsimp [tensorHom, tensorBimod, associatorBimod]
ext
apply coequalizer.hom_ext
dsimp
slice_lhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]
dsimp [TensorBimod.X, AssociatorBimod.inv]
slice_rhs 1 2 => rw [coequalizer.π_desc]
dsimp [AssociatorBimod.invAux, AssociatorBimod.hom]
refine (cancel_epi ((... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Preadditive.OfBiproducts | {
"line": 94,
"column": 39
} | {
"line": 94,
"column": 47
} | {
"line": 94,
"column": 47
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nf g h k : X ⟶ Y\ndiag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)\nhd₁ : biprod.inl ≫ diag = biprod.lift f h\nhd₂ : biprod.inr ≫ diag = biprod.lift g k\nh₁ : biprod.lift (... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nf g h k : X ⟶ Y\ndiag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)\nhd₁ : biprod.inl ≫ diag = biprod.lift f h\nhd₂ : biprod.inr ≫ diag = biprod.lift g k\nh₁ : biprod.lift (rightAdd X Y... | rightAdd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Presentable.SharplyLT.Basic | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 29
} | {
"line": 97,
"column": 2
} | [
{
"pp": "κ₁ κ₂ : Cardinal.{w}\ninst✝² : Fact κ₁.IsRegular\ninst✝¹ : Fact κ₂.IsRegular\nh : κ₁ ≤ κ₂\ninst✝ : IsCardinalAccessibleCategory (CardinalDirectedPoset κ₁) κ₂\nX : Type w\nhX : HasCardinalLT X κ₂\nJ : Type w\nw✝¹ : SmallCategory J\nw✝ : IsCardinalFiltered J κ₂\np : (isCardinalPresentable (CardinalDirect... | [
"κ₁ κ₂ : Cardinal.{w}\ninst✝² : Fact κ₁.IsRegular\ninst✝¹ : Fact κ₂.IsRegular\nh : κ₁ ≤ κ₂\ninst✝ : IsCardinalAccessibleCategory (CardinalDirectedPoset κ₁) κ₂\nX : Type w\nhX : HasCardinalLT X κ₂\nJ : Type w\nw✝¹ : SmallCategory J\nw✝ : IsCardinalFiltered J κ₂\np : (isCardinalPresentable (CardinalDirectedPoset κ₁) ... | convert Set.mem_singleton b | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
Mathlib.CategoryTheory.Presentable.Type | {
"line": 111,
"column": 4
} | {
"line": 112,
"column": 23
} | {
"line": 114,
"column": 0
} | [
{
"pp": "case refine_2\nX : Type u\nκ : Cardinal.{u}\nhκ : Cardinal.aleph0 ≤ κ\nthis : IsFiltered (HasCardinalLT.Set X κ)\n⊢ ∀ (i : HasCardinalLT.Set X κ) (x y : (fun X ↦ X) ((functor X κ).obj i)),\n (hom ((cocone X κ).ι.app i)) x = (hom ((cocone X κ).ι.app i)) y →\n ∃ k f, (hom ((functor X κ).map f)) x... | [] | rintro A ⟨x, hx⟩ ⟨y, hy⟩ rfl
exact ⟨A, 𝟙 _, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Presentable.Type | {
"line": 111,
"column": 4
} | {
"line": 112,
"column": 23
} | {
"line": 114,
"column": 0
} | [
{
"pp": "case refine_2\nX : Type u\nκ : Cardinal.{u}\nhκ : Cardinal.aleph0 ≤ κ\nthis : IsFiltered (HasCardinalLT.Set X κ)\n⊢ ∀ (i : HasCardinalLT.Set X κ) (x y : (fun X ↦ X) ((functor X κ).obj i)),\n (hom ((cocone X κ).ι.app i)) x = (hom ((cocone X κ).ι.app i)) y →\n ∃ k f, (hom ((functor X κ).map f)) x... | [] | rintro A ⟨x, hx⟩ ⟨y, hy⟩ rfl
exact ⟨A, 𝟙 _, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.RepresentedBy | {
"line": 104,
"column": 2
} | {
"line": 106,
"column": 30
} | {
"line": 108,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nX : C\nx : F.obj (op X)\nh : F.IsRepresentedBy x\nF' : Cᵒᵖ ⥤ Type w\ne : F ≅ F'\n⊢ F'.IsRepresentedBy ((ConcreteCategory.hom (e.hom.app (op X))) x)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Categ... | [] | rw [iff_exists_representableBy]
use h.representableBy.ofIso e
simp [RepresentableBy.ofIso] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.RepresentedBy | {
"line": 104,
"column": 2
} | {
"line": 106,
"column": 30
} | {
"line": 108,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nX : C\nx : F.obj (op X)\nh : F.IsRepresentedBy x\nF' : Cᵒᵖ ⥤ Type w\ne : F ≅ F'\n⊢ F'.IsRepresentedBy ((ConcreteCategory.hom (e.hom.app (op X))) x)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Categ... | [] | rw [iff_exists_representableBy]
use h.representableBy.ofIso e
simp [RepresentableBy.ofIso] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Coherent.LocallySurjective | {
"line": 132,
"column": 70
} | {
"line": 132,
"column": 91
} | {
"line": 132,
"column": 91
} | [
{
"pp": "case e'_2\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Preregular C\ninst✝² : FinitaryPreExtensive C\nF G : Cᵒᵖ ⥤ Type w\nf : F ⟶ G\ninst✝¹ : PreservesFiniteProducts F\ninst✝ : PreservesFiniteProducts G\nU✝ : C\ny : ToType (G.obj (op U✝))\nα : Type\nw✝ : Finite α\nZ : α → C\nπ : (a : α) → Z ... | [
"case e'_2\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Preregular C\ninst✝² : FinitaryPreExtensive C\nF G : Cᵒᵖ ⥤ Type w\nf : F ⟶ G\ninst✝¹ : PreservesFiniteProducts F\ninst✝ : PreservesFiniteProducts G\nU✝ : C\ny : ToType (G.obj (op U✝))\nα : Type\nw✝ : Finite α\nZ : α → C\nπ : (a : α) → Z a ⟶ U✝\nh : ... | ← piComparison_comp_π | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Descent.DescentDataPrime | {
"line": 101,
"column": 20
} | {
"line": 101,
"column": 80
} | {
"line": 101,
"column": 80
} | [
{
"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)... | [] | rw [← Quiver.Hom.comp_toLoc, ← op_comp, IsPullback.lift_fst] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.Descent.DescentDataPrime | {
"line": 101,
"column": 20
} | {
"line": 101,
"column": 80
} | {
"line": 101,
"column": 80
} | [
{
"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)... | [] | rw [← Quiver.Hom.comp_toLoc, ← op_comp, IsPullback.lift_fst] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Descent.DescentDataPrime | {
"line": 101,
"column": 20
} | {
"line": 101,
"column": 80
} | {
"line": 101,
"column": 80
} | [
{
"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)... | [] | rw [← Quiver.Hom.comp_toLoc, ← op_comp, IsPullback.lift_fst] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Descent.DescentDataPrime | {
"line": 100,
"column": 2
} | {
"line": 104,
"column": 91
} | {
"line": 105,
"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 [F.mapComp'₀₂₃_hom_comp_mapComp'_hom_whiskerRight_app_assoc
_ _ _ _ _ _ (by rw [← Quiver.Hom.comp_toLoc, ← op_comp, IsPullback.lift_fst])
rfl (by cat_disch),
F.mapComp'_inv_whiskerRight_mapComp'₀₂₃_inv_app _ _ _ _ _ _
(by rw [← Quiver.Hom.comp_toLoc, ← op_comp, IsPullback.lift_snd]) rfl (by cat_dis... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.MayerVietorisSquare | {
"line": 144,
"column": 12
} | {
"line": 144,
"column": 63
} | {
"line": 144,
"column": 63
} | [
{
"pp": "case refine_3.left\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\nsq : Square C\ninst✝¹ : Mono sq.f₂₄\ninst✝ : Mono sq.f₃₄\nh₁ : sq.IsPullback\nh₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄\nthis : Mono sq.f₁₃\nF : Sheaf J (Type v)\ns : Pullba... | [
"case refine_3.left\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\nsq : Square C\ninst✝¹ : Mono sq.f₂₄\ninst✝ : Mono sq.f₃₄\nh₁ : sq.IsPullback\nh₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄\nthis : Mono sq.f₁₃\nF : Sheaf J (Type v)\ns : PullbackCone (sq.o... | F.2.amalgamateOfArrows_map _ _ _ _ WalkingPair.left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 315,
"column": 4
} | {
"line": 315,
"column": 71
} | {
"line": 316,
"column": 4
} | [
{
"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\nS' : C\np : S' ⟶ 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 ≫ p\nβ : 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\nS' : C\np : S' ⟶ 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 ≫ p\nβ : ι' → ι\np'' :... | simp only [Cat.Hom.hom_inv_id_toNatTrans_app_assoc, Category.assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Sites.Types | {
"line": 181,
"column": 4
} | {
"line": 184,
"column": 57
} | {
"line": 186,
"column": 0
} | [
{
"pp": "X : Type u\n⊢ yoneda'.map\n ((NatIso.ofComponents\n (fun _α ↦\n { hom := ↾fun x ↦ ↾fun x_1 ↦ x, inv := ↾fun f ↦ (TypeCat.Hom.hom f) PUnit.unit, hom_inv_id := ⋯,\n inv_hom_id := ⋯ })\n ⋯).hom.app\n X) ≫\n (NatIso.ofCo... | [] | ext1
apply yonedaEquiv.injective
dsimp [yoneda', yonedaEquiv, equivYoneda, evalEquiv]
simpa using! typesGlue_eval (S := yoneda.obj X) (𝟙 X) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Types | {
"line": 181,
"column": 4
} | {
"line": 184,
"column": 57
} | {
"line": 186,
"column": 0
} | [
{
"pp": "X : Type u\n⊢ yoneda'.map\n ((NatIso.ofComponents\n (fun _α ↦\n { hom := ↾fun x ↦ ↾fun x_1 ↦ x, inv := ↾fun f ↦ (TypeCat.Hom.hom f) PUnit.unit, hom_inv_id := ⋯,\n inv_hom_id := ⋯ })\n ⋯).hom.app\n X) ≫\n (NatIso.ofCo... | [] | ext1
apply yonedaEquiv.injective
dsimp [yoneda', yonedaEquiv, equivYoneda, evalEquiv]
simpa using! typesGlue_eval (S := yoneda.obj X) (𝟙 X) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 518,
"column": 4
} | {
"line": 519,
"column": 28
} | {
"line": 519,
"column": 28
} | [
{
"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).FullyFaithful",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"CategoryTheory.Over",
"Opposite",
"... | [] | rw [isStackFor_iff] at h
exact .ofFullyFaithful _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 518,
"column": 4
} | {
"line": 519,
"column": 28
} | {
"line": 519,
"column": 28
} | [
{
"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).FullyFaithful",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"CategoryTheory.Over",
"Opposite",
"... | [] | rw [isStackFor_iff] at h
exact .ofFullyFaithful _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sums.Products | {
"line": 189,
"column": 6
} | {
"line": 191,
"column": 75
} | {
"line": 191,
"column": 75
} | [
{
"pp": "case h₁\nA : Type u_1\ninst✝³ : Category.{v_1, u_1} A\nA' : Type u_2\ninst✝² : Category.{v_2, u_2} A'\nB : Type u\ninst✝¹ : Category.{v, u} B\nT : Type u_3\ninst✝ : Category.{v_3, u_3} T\nX✝ Y✝ : (A ⊕ A') ⊕ T ⥤ B\nf✝ : X✝ ⟶ Y✝\nx✝ : A\n⊢ (((sum.associativity A A' T).congrLeft.trans\n ... | [] | all_goals
dsimp
simp only [Category.comp_id, Category.id_comp, NatTrans.naturality] | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.CategoryTheory.Sites.SheafCohomology.MayerVietoris | {
"line": 136,
"column": 13
} | {
"line": 136,
"column": 17
} | {
"line": 136,
"column": 17
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\ninst✝¹ : HasSheafify J AddCommGrpCat\ninst✝ : HasExt (Sheaf J AddCommGrpCat)\nS : J.MayerVietorisSquare\nF : Sheaf J AddCommGrpCat\nn₀ n₁ : ℕ\nh : n₀ + 1 = n₁\nx✝ : ↑((S.sequence F n₀ n₁ h).obj ⟨1, ... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\ninst✝¹ : HasSheafify J AddCommGrpCat\ninst✝ : HasExt (Sheaf J AddCommGrpCat)\nS : J.MayerVietorisSquare\nF : Sheaf J AddCommGrpCat\nn₀ n₁ : ℕ\nh : n₀ + 1 = n₁\nx✝ : ↑((S.sequence F n₀ n₁ h).obj ⟨1, ⋯⟩)\n⊢ (AddC... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Sites.SheafCohomology.MayerVietoris | {
"line": 139,
"column": 13
} | {
"line": 139,
"column": 17
} | {
"line": 139,
"column": 17
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\ninst✝¹ : HasSheafify J AddCommGrpCat\ninst✝ : HasExt (Sheaf J AddCommGrpCat)\nS : J.MayerVietorisSquare\nF : Sheaf J AddCommGrpCat\nn₀ n₁ : ℕ\nh : n₀ + 1 = n₁\nx✝ : ↑((S.sequence F n₀ n₁ h).obj ⟨4, ... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\ninst✝¹ : HasSheafify J AddCommGrpCat\ninst✝ : HasExt (Sheaf J AddCommGrpCat)\nS : J.MayerVietorisSquare\nF : Sheaf J AddCommGrpCat\nn₀ n₁ : ℕ\nh : n₀ + 1 = n₁\nx✝ : ↑((S.sequence F n₀ n₁ h).obj ⟨4, ⋯⟩)\n⊢ (AddC... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
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