ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.Pretriangulated.TriangleOpEquivalence.counitIso_hom_app_hom₁
@[simps!]
noncomputable def counitIso : inverse C ⋙ functor C ≅ 𝟭 _ 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ] (X : CategoryTheory.Pretriangulated.Triangle Cᵒᵖ)
Goal: ((CategoryTheory.Pretriangulated.TriangleOpEquivalence.counitIso C).hom.app X).hom₁ =
  CategoryTheory.CategoryStruct.id X.obj₁
-- CategoryTheory.Pretriangulated.triangleOpEquivalence_counitIso
@[simps]
noncomputable def triangleOpEquivalence :
    (Triangle C)ᵒᵖ ≌ Triangle Cᵒᵖ 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ]
Goal: (CategoryTheory.Pretriangulated.triangleOpEquivalence C).counitIso =
  CategoryTheory.Pretriangulated.TriangleOpEquivalence.counitIso C
-- CategoryTheory.Pretriangulated.triangleOpEquivalence_inverse
@[simps]
noncomputable def triangleOpEquivalence :
    (Triangle C)ᵒᵖ ≌ Triangle Cᵒᵖ 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ]
Goal: (CategoryTheory.Pretriangulated.triangleOpEquivalence C).inverse =
  CategoryTheory.Pretriangulated.TriangleOpEquivalence.inverse C
-- CategoryTheory.Pretriangulated.triangleOpEquivalence_unitIso
@[simps]
noncomputable def triangleOpEquivalence :
    (Triangle C)ᵒᵖ ≌ Triangle Cᵒᵖ 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ]
Goal: (CategoryTheory.Pretriangulated.triangleOpEquivalence C).unitIso =
  CategoryTheory.Pretriangulated.TriangleOpEquivalence.unitIso C
-- CategoryTheory.Pretriangulated.triangleOpEquivalence_functor
@[simps]
noncomputable def triangleOpEquivalence :
    (Triangle C)ᵒᵖ ≌ Triangle Cᵒᵖ 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ]
Goal: (CategoryTheory.Pretriangulated.triangleOpEquivalence C).functor =
  CategoryTheory.Pretriangulated.TriangleOpEquivalence.functor C
-- CategoryTheory.Pretriangulated.Opposite.mem_distinguishedTriangles_iff
lemma mem_distinguishedTriangles_iff (T : Triangle Cᵒᵖ) :
    T ∈ distinguishedTriangles C ↔
      ((triangleOpEquivalence C).inverse.obj T).unop ∈ distTriang C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle Cᵒᵖ)
Goal: T ∈ CategoryTheory.Pretriangulated.Opposite.distinguishedTriangles C ↔
  Opposite.unop ((CategoryTheory.Pretriangulated.triangleOpEquivalence C).inverse.obj T) ∈
    CategoryTheory.Pretriangulated.distinguishedTriangles
-- CategoryTheory.Pretriangulated.Opposite.mem_distinguishedTriangles_iff'
lemma mem_distinguishedTriangles_iff' (T : Triangle Cᵒᵖ) :
    T ∈ distinguishedTriangles C ↔
      ∃ (T' : Triangle C) (_ : T' ∈ distTriang C),
        Nonempty (T ≅ (triangleOpEquivalence C).functor.obj (Opposite.op T')) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle Cᵒᵖ)
Goal: T ∈ CategoryTheory.Pretriangulated.Opposite.distinguishedTriangles C ↔
  ∃ T',
    ∃ (_ : T' ∈ CategoryTheory.Pretriangulated.distinguishedTriangles),
      Nonempty (T ≅ (CategoryTheory.Pretriangulated.triangleOpEquivalence C).functor.obj (Opposite.op T'))
-- CategoryTheory.Pretriangulated.Opposite.isomorphic_distinguished
lemma isomorphic_distinguished (T₁ : Triangle Cᵒᵖ)
    (hT₁ : T₁ ∈ distinguishedTriangles C) (T₂ : Triangle Cᵒᵖ) (e : T₂ ≅ T₁) :
    T₂ ∈ distinguishedTriangles C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (T₁ : CategoryTheory.Pretriangulated.Triangle Cᵒᵖ) (hT₁ : T₁ ∈ CategoryTheory.Pretriangulated.Opposite.distinguishedTriangles C) (T₂ : CategoryTheory.Pretriangulated.Triangle Cᵒᵖ) (e : T₂ ≅ T₁)
Goal: T₂ ∈ CategoryTheory.Pretriangulated.Opposite.distinguishedTriangles C
-- CategoryTheory.Pretriangulated.Opposite.contractibleTriangleIso_inv_hom₃
@[simps!]
noncomputable def contractibleTriangleIso (X : Cᵒᵖ) :
    contractibleTriangle X ≅ (triangleOpEquivalence C).functor.obj
      (Opposite.op (contractibleTriangle X.unop).invRotate) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] (X : Cᵒᵖ)
Goal: (CategoryTheory.Pretriangulated.Opposite.contractibleTriangleIso X).inv.hom₃ = (⋯.iso ⋯).inv
-- CategoryTheory.Pretriangulated.Opposite.contractibleTriangleIso_hom_hom₁
@[simps!]
noncomputable def contractibleTriangleIso (X : Cᵒᵖ) :
    contractibleTriangle X ≅ (triangleOpEquivalence C).functor.obj
      (Opposite.op (contractibleTriangle X.unop).invRotate) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] (X : Cᵒᵖ)
Goal: (CategoryTheory.Pretriangulated.Opposite.contractibleTriangleIso X).hom.hom₁ = CategoryTheory.CategoryStruct.id X
-- CategoryTheory.Pretriangulated.Opposite.contractibleTriangleIso_inv_hom₁
@[simps!]
noncomputable def contractibleTriangleIso (X : Cᵒᵖ) :
    contractibleTriangle X ≅ (triangleOpEquivalence C).functor.obj
      (Opposite.op (contractibleTriangle X.unop).invRotate) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] (X : Cᵒᵖ)
Goal: (CategoryTheory.Pretriangulated.Opposite.contractibleTriangleIso X).inv.hom₁ = CategoryTheory.CategoryStruct.id X
-- CategoryTheory.Pretriangulated.Opposite.contractibleTriangleIso_hom_hom₃
@[simps!]
noncomputable def contractibleTriangleIso (X : Cᵒᵖ) :
    contractibleTriangle X ≅ (triangleOpEquivalence C).functor.obj
      (Opposite.op (contractibleTriangle X.unop).invRotate) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] (X : Cᵒᵖ)
Goal: (CategoryTheory.Pretriangulated.Opposite.contractibleTriangleIso X).hom.hom₃ = (⋯.iso ⋯).hom
-- CategoryTheory.Pretriangulated.Opposite.contractibleTriangleIso_inv_hom₂
@[simps!]
noncomputable def contractibleTriangleIso (X : Cᵒᵖ) :
    contractibleTriangle X ≅ (triangleOpEquivalence C).functor.obj
      (Opposite.op (contractibleTriangle X.unop).invRotate) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] (X : Cᵒᵖ)
Goal: (CategoryTheory.Pretriangulated.Opposite.contractibleTriangleIso X).inv.hom₂ = CategoryTheory.CategoryStruct.id X
-- CategoryTheory.Pretriangulated.Opposite.contractibleTriangleIso_hom_hom₂
@[simps!]
noncomputable def contractibleTriangleIso (X : Cᵒᵖ) :
    contractibleTriangle X ≅ (triangleOpEquivalence C).functor.obj
      (Opposite.op (contractibleTriangle X.unop).invRotate) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] (X : Cᵒᵖ)
Goal: (CategoryTheory.Pretriangulated.Opposite.contractibleTriangleIso X).hom.hom₂ = CategoryTheory.CategoryStruct.id X
-- CategoryTheory.Pretriangulated.Opposite.contractible_distinguished
lemma contractible_distinguished (X : Cᵒᵖ) :
    contractibleTriangle X ∈ distinguishedTriangles C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (X : Cᵒᵖ)
Goal: CategoryTheory.Pretriangulated.contractibleTriangle X ∈ CategoryTheory.Pretriangulated.Opposite.distinguishedTriangles C
-- CategoryTheory.Pretriangulated.Opposite.rotate_distinguished_triangle
lemma rotate_distinguished_triangle (T : Triangle Cᵒᵖ) :
    T ∈ distinguishedTriangles C ↔ T.rotate ∈ distinguishedTriangles C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle Cᵒᵖ)
Goal: T ∈ CategoryTheory.Pretriangulated.Opposite.distinguishedTriangles C ↔
  T.rotate ∈ CategoryTheory.Pretriangulated.Opposite.distinguishedTriangles C
-- CategoryTheory.Pretriangulated.Opposite.distinguished_cocone_triangle
lemma distinguished_cocone_triangle {X Y : Cᵒᵖ} (f : X ⟶ Y) :
    ∃ (Z : Cᵒᵖ) (g : Y ⟶ Z) (h : Z ⟶ X⟦(1 : ℤ)⟧),
      Triangle.mk f g h ∈ distinguishedTriangles C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] {X Y : Cᵒᵖ} (f : X ⟶ Y)
Goal: ∃ Z g h,
  CategoryTheory.Pretriangulated.Triangle.mk f g h ∈ CategoryTheory.Pretriangulated.Opposite.distinguishedTriangles C
-- CategoryTheory.Pretriangulated.Opposite.complete_distinguished_triangle_morphism
lemma complete_distinguished_triangle_morphism (T₁ T₂ : Triangle Cᵒᵖ)
    (hT₁ : T₁ ∈ distinguishedTriangles C) (hT₂ : T₂ ∈ distinguishedTriangles C)
    (a : T₁.obj₁ ⟶ T₂.obj₁) (b : T₁.obj₂ ⟶ T₂.obj₂) (comm : T₁.mor₁ ≫ b = a ≫ T₂.mor₁) :
    ∃ (c : T₁.obj₃ ⟶ T₂.obj₃), T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧
      T₁.mor₃ ≫ a⟦1⟧' = c ≫ T₂.mor₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (T₁ T₂ : CategoryTheory.Pretriangulated.Triangle Cᵒᵖ) (hT₁ : T₁ ∈ CategoryTheory.Pretriangulated.Opposite.distinguishedTriangles C) (hT₂ : T₂ ∈ CategoryTheory.Pretriangulated.Opposite.distinguishedTriangles C) (a : T₁.obj₁ ⟶ T₂.obj₁) (b : T₁.obj₂ ⟶ T₂.obj₂) (comm : CategoryTheory.CategoryStruct.comp T₁.mor₁ b = CategoryTheory.CategoryStruct.comp a T₂.mor₁)
Goal: ∃ c,
  CategoryTheory.CategoryStruct.comp T₁.mor₂ c = CategoryTheory.CategoryStruct.comp b T₂.mor₂ ∧
    CategoryTheory.CategoryStruct.comp T₁.mor₃ ((CategoryTheory.shiftFunctor Cᵒᵖ 1).map a) =
      CategoryTheory.CategoryStruct.comp c T₂.mor₃
-- CategoryTheory.Pretriangulated.mem_distTriang_op_iff
lemma mem_distTriang_op_iff (T : Triangle Cᵒᵖ) :
    (T ∈ distTriang Cᵒᵖ) ↔ ((triangleOpEquivalence C).inverse.obj T).unop ∈ distTriang C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle Cᵒᵖ)
Goal: T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles ↔
  Opposite.unop ((CategoryTheory.Pretriangulated.triangleOpEquivalence C).inverse.obj T) ∈
    CategoryTheory.Pretriangulated.distinguishedTriangles
-- CategoryTheory.Pretriangulated.mem_distTriang_op_iff'
lemma mem_distTriang_op_iff' (T : Triangle Cᵒᵖ) :
    (T ∈ distTriang Cᵒᵖ) ↔ ∃ (T' : Triangle C) (_ : T' ∈ distTriang C),
      Nonempty (T ≅ (triangleOpEquivalence C).functor.obj (Opposite.op T')) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle Cᵒᵖ)
Goal: T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles ↔
  ∃ T',
    ∃ (_ : T' ∈ CategoryTheory.Pretriangulated.distinguishedTriangles),
      Nonempty (T ≅ (CategoryTheory.Pretriangulated.triangleOpEquivalence C).functor.obj (Opposite.op T'))
-- CategoryTheory.Pretriangulated.op_distinguished
lemma op_distinguished (T : Triangle C) (hT : T ∈ distTriang C) :
    ((triangleOpEquivalence C).functor.obj (Opposite.op T)) ∈ distTriang Cᵒᵖ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: (CategoryTheory.Pretriangulated.triangleOpEquivalence C).functor.obj (Opposite.op T) ∈
  CategoryTheory.Pretriangulated.distinguishedTriangles
-- CategoryTheory.Pretriangulated.unop_distinguished
lemma unop_distinguished (T : Triangle Cᵒᵖ) (hT : T ∈ distTriang Cᵒᵖ) :
    ((triangleOpEquivalence C).inverse.obj T).unop ∈ distTriang C 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_2, u_1} C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] (T : CategoryTheory.Pretriangulated.Triangle Cᵒᵖ) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: Opposite.unop ((CategoryTheory.Pretriangulated.triangleOpEquivalence C).inverse.obj T) ∈
  CategoryTheory.Pretriangulated.distinguishedTriangles
-- CategoryTheory.Functor.map_distinguished_op_exact
lemma map_distinguished_op_exact [HasShift C ℤ] [HasZeroObject C] [Preadditive C]
    [∀ (n : ℤ), (shiftFunctor C n).Additive]
    [Pretriangulated C]{A : Type*} [Category A] [Abelian A] (F : Cᵒᵖ ⥤ A)
    [F.IsHomological] (T : Triangle C) (hT : T ∈ distTriang C) :
    ((shortComplexOfDistTriangle T hT).op.map F).Exact 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.HasShift C ℤ] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive] [CategoryTheory.Pretriangulated C] {A : Type u_2} [CategoryTheory.Category.{u_4, u_2} A] [CategoryTheory.Abelian A] (F : CategoryTheory.Functor Cᵒᵖ A) [F.IsHomological] (T : CategoryTheory.Pretriangulated.Triangle C) (hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles)
Goal: ((CategoryTheory.Pretriangulated.shortComplexOfDistTriangle T hT).op.map F).Exact
-- CategoryTheory.LocallyDiscrete.mk_as
@[simp]
theorem mk_as (a : LocallyDiscrete C) : mk a.as = a 
Arguments: {C : Type u} (a : CategoryTheory.LocallyDiscrete C)
Goal: { as := a.as } = a
-- CategoryTheory.LocallyDiscrete.locallyDiscreteEquiv_symm_apply_as
@[simps]
def locallyDiscreteEquiv : LocallyDiscrete C ≃ C 
Arguments: {C : Type u} (as : C)
Goal: (CategoryTheory.LocallyDiscrete.locallyDiscreteEquiv.symm as).as = as
-- CategoryTheory.LocallyDiscrete.locallyDiscreteEquiv_apply
@[simps]
def locallyDiscreteEquiv : LocallyDiscrete C ≃ C 
Arguments: {C : Type u} (self : CategoryTheory.LocallyDiscrete C)
Goal: CategoryTheory.LocallyDiscrete.locallyDiscreteEquiv self = self.as
-- CategoryTheory.LocallyDiscrete.id_as
@[simp]
lemma id_as (a : LocallyDiscrete C) : (𝟙 a : Discrete (a.as ⟶ a.as)).as = 𝟙 a.as 
Arguments: {C : Type u} [CategoryTheory.CategoryStruct.{v, u} C] (a : CategoryTheory.LocallyDiscrete C)
Goal: (CategoryTheory.CategoryStruct.id a).as = CategoryTheory.CategoryStruct.id a.as
-- CategoryTheory.LocallyDiscrete.comp_as
@[simp]
lemma comp_as {a b c : LocallyDiscrete C} (f : a ⟶ b) (g : b ⟶ c) : (f ≫ g).as = f.as ≫ g.as 
Arguments: {C : Type u} [CategoryTheory.CategoryStruct.{v, u} C] {a b c : CategoryTheory.LocallyDiscrete C} (f : a ⟶ b) (g : b ⟶ c)
Goal: (CategoryTheory.CategoryStruct.comp f g).as = CategoryTheory.CategoryStruct.comp f.as g.as
-- CategoryTheory.LocallyDiscrete.eq_of_hom
theorem eq_of_hom {X Y : LocallyDiscrete C} {f g : X ⟶ Y} (η : f ⟶ g) : f = g 
Arguments: {C : Type u} [CategoryTheory.CategoryStruct.{v, u} C] {X Y : CategoryTheory.LocallyDiscrete C} {f g : X ⟶ Y} (η : f ⟶ g)
Goal: f = g
-- CategoryTheory.Functor.toPseudoFunctor_toPrelaxFunctor_toPrelaxFunctorStruct_map₂
@[simps]
def Functor.toPseudoFunctor (F : I ⥤ B) : Pseudofunctor (LocallyDiscrete I) B 
Arguments: {I : Type u₁} [CategoryTheory.Category.{v₁, u₁} I] {B : Type u₂} [CategoryTheory.Bicategory B] [CategoryTheory.Bicategory.Strict B] (F : CategoryTheory.Functor I B)
Goal: ∀ {a b : CategoryTheory.LocallyDiscrete I} {f g : a ⟶ b} (η : f ⟶ g),
  F.toPseudoFunctor.map₂ η = CategoryTheory.eqToHom ⋯
-- CategoryTheory.Functor.toPseudoFunctor_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_map
@[simps]
def Functor.toPseudoFunctor (F : I ⥤ B) : Pseudofunctor (LocallyDiscrete I) B 
Arguments: {I : Type u₁} [CategoryTheory.Category.{v₁, u₁} I] {B : Type u₂} [CategoryTheory.Bicategory B] [CategoryTheory.Bicategory.Strict B] (F : CategoryTheory.Functor I B)
Goal: ∀ {X Y : CategoryTheory.LocallyDiscrete I} (f : X ⟶ Y), F.toPseudoFunctor.map f = F.map f.as
-- CategoryTheory.Functor.toPseudoFunctor_mapId
@[simps]
def Functor.toPseudoFunctor (F : I ⥤ B) : Pseudofunctor (LocallyDiscrete I) B 
Arguments: {I : Type u₁} [CategoryTheory.Category.{v₁, u₁} I] {B : Type u₂} [CategoryTheory.Bicategory B] [CategoryTheory.Bicategory.Strict B] (F : CategoryTheory.Functor I B) (i : CategoryTheory.LocallyDiscrete I)
Goal: F.toPseudoFunctor.mapId i = CategoryTheory.eqToIso ⋯
-- CategoryTheory.Functor.toPseudoFunctor_mapComp
@[simps]
def Functor.toPseudoFunctor (F : I ⥤ B) : Pseudofunctor (LocallyDiscrete I) B 
Arguments: {I : Type u₁} [CategoryTheory.Category.{v₁, u₁} I] {B : Type u₂} [CategoryTheory.Bicategory B] [CategoryTheory.Bicategory.Strict B] (F : CategoryTheory.Functor I B)
Goal: ∀ {a b c : CategoryTheory.LocallyDiscrete I} (f : a ⟶ b) (g : b ⟶ c),
  F.toPseudoFunctor.mapComp f g = CategoryTheory.eqToIso ⋯
-- CategoryTheory.Functor.toPseudoFunctor_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_obj
@[simps]
def Functor.toPseudoFunctor (F : I ⥤ B) : Pseudofunctor (LocallyDiscrete I) B 
Arguments: {I : Type u₁} [CategoryTheory.Category.{v₁, u₁} I] {B : Type u₂} [CategoryTheory.Bicategory B] [CategoryTheory.Bicategory.Strict B] (F : CategoryTheory.Functor I B) (i : CategoryTheory.LocallyDiscrete I)
Goal: F.toPseudoFunctor.obj i = F.obj i.as
-- CategoryTheory.Functor.toOplaxFunctor_toPrelaxFunctor_toPrelaxFunctorStruct_map₂
@[simps]
def Functor.toOplaxFunctor (F : I ⥤ B) : OplaxFunctor (LocallyDiscrete I) B 
Arguments: {I : Type u₁} [CategoryTheory.Category.{v₁, u₁} I] {B : Type u₂} [CategoryTheory.Bicategory B] [CategoryTheory.Bicategory.Strict B] (F : CategoryTheory.Functor I B)
Goal: ∀ {a b : CategoryTheory.LocallyDiscrete I} {f g : a ⟶ b} (η : f ⟶ g), F.toOplaxFunctor.map₂ η = CategoryTheory.eqToHom ⋯
-- CategoryTheory.Functor.toOplaxFunctor_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_obj
@[simps]
def Functor.toOplaxFunctor (F : I ⥤ B) : OplaxFunctor (LocallyDiscrete I) B 
Arguments: {I : Type u₁} [CategoryTheory.Category.{v₁, u₁} I] {B : Type u₂} [CategoryTheory.Bicategory B] [CategoryTheory.Bicategory.Strict B] (F : CategoryTheory.Functor I B) (i : CategoryTheory.LocallyDiscrete I)
Goal: F.toOplaxFunctor.obj i = F.obj i.as
-- CategoryTheory.Functor.toOplaxFunctor_mapId
@[simps]
def Functor.toOplaxFunctor (F : I ⥤ B) : OplaxFunctor (LocallyDiscrete I) B 
Arguments: {I : Type u₁} [CategoryTheory.Category.{v₁, u₁} I] {B : Type u₂} [CategoryTheory.Bicategory B] [CategoryTheory.Bicategory.Strict B] (F : CategoryTheory.Functor I B) (i : CategoryTheory.LocallyDiscrete I)
Goal: F.toOplaxFunctor.mapId i = CategoryTheory.eqToHom ⋯
-- CategoryTheory.Functor.toOplaxFunctor_mapComp
@[simps]
def Functor.toOplaxFunctor (F : I ⥤ B) : OplaxFunctor (LocallyDiscrete I) B 
Arguments: {I : Type u₁} [CategoryTheory.Category.{v₁, u₁} I] {B : Type u₂} [CategoryTheory.Bicategory B] [CategoryTheory.Bicategory.Strict B] (F : CategoryTheory.Functor I B)
Goal: ∀ {a b c : CategoryTheory.LocallyDiscrete I} (f : a ⟶ b) (g : b ⟶ c),
  F.toOplaxFunctor.mapComp f g = CategoryTheory.eqToHom ⋯
-- CategoryTheory.Functor.toOplaxFunctor_toPrelaxFunctor_toPrelaxFunctorStruct_toPrefunctor_map
@[simps]
def Functor.toOplaxFunctor (F : I ⥤ B) : OplaxFunctor (LocallyDiscrete I) B 
Arguments: {I : Type u₁} [CategoryTheory.Category.{v₁, u₁} I] {B : Type u₂} [CategoryTheory.Bicategory B] [CategoryTheory.Bicategory.Strict B] (F : CategoryTheory.Functor I B)
Goal: ∀ {X Y : CategoryTheory.LocallyDiscrete I} (f : X ⟶ Y), F.toOplaxFunctor.map f = F.map f.as
-- CategoryTheory.PrelaxFunctor.map₂_eqToHom
@[simp]
lemma PrelaxFunctor.map₂_eqToHom (F : PrelaxFunctor B C) {a b : B} {f g : a ⟶ b} (h : f = g) :
    F.map₂ (eqToHom h) = eqToHom (F.congr_map h) 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] (F : CategoryTheory.PrelaxFunctor B C) {a b : B} {f g : a ⟶ b} (h : f = g)
Goal: F.map₂ (CategoryTheory.eqToHom h) = CategoryTheory.eqToHom ⋯
-- Quiver.Hom.toLoc_as
@[simps]
def toLoc {a b : C} (f : a ⟶ b) : LocallyDiscrete.mk a ⟶ LocallyDiscrete.mk b 
Arguments: {C : Type u} [CategoryTheory.CategoryStruct.{v, u} C] {a b : C} (f : a ⟶ b)
Goal: f.toLoc.as = f
-- Quiver.Hom.id_toLoc
@[simp]
lemma id_toLoc (a : C) : (𝟙 a).toLoc = 𝟙 (LocallyDiscrete.mk a) 
Arguments: {C : Type u} [CategoryTheory.CategoryStruct.{v, u} C] (a : C)
Goal: (CategoryTheory.CategoryStruct.id a).toLoc = CategoryTheory.CategoryStruct.id { as := a }
-- Quiver.Hom.comp_toLoc
@[simp]
lemma comp_toLoc {a b c : C} (f : a ⟶ b) (g : b ⟶ c) : (f ≫ g).toLoc = f.toLoc ≫ g.toLoc 
Arguments: {C : Type u} [CategoryTheory.CategoryStruct.{v, u} C] {a b c : C} (f : a ⟶ b) (g : b ⟶ c)
Goal: (CategoryTheory.CategoryStruct.comp f g).toLoc = CategoryTheory.CategoryStruct.comp f.toLoc g.toLoc
-- CategoryTheory.LocallyDiscrete.eqToHom_toLoc
@[simp]
lemma CategoryTheory.LocallyDiscrete.eqToHom_toLoc {C : Type u} [Category.{v} C] {a b : C}
    (h : a = b) : (eqToHom h).toLoc = eqToHom (congrArg LocallyDiscrete.mk h) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {a b : C} (h : a = b)
Goal: (CategoryTheory.eqToHom h).toLoc = CategoryTheory.eqToHom ⋯
-- CategoryTheory.UnbundledHom.hom_comp
class UnbundledHom {c : Type u → Type u} (hom : ∀ ⦃α β⦄, c α → c β → (α → β) → Prop) : Prop where
  hom_id : ∀ {α} (ia : c α), hom ia ia id
  hom_comp : ∀ {α β γ} {Iα : c α} {Iβ : c β} {Iγ : c γ} {g : β → γ} {f : α → β} (_ : hom Iβ Iγ g)
      (_ : hom Iα Iβ f), hom Iα Iγ (g ∘ f)
Arguments: {c : Type u → Type u} {hom : ⦃α β : Type u⦄ → c α → c β → (α → β) → Prop} [self : CategoryTheory.UnbundledHom hom] {α β γ : Type u} {Iα : c α} {Iβ : c β} {Iγ : c γ} {g : β → γ} {f : α → β}
Goal: hom Iβ Iγ g → hom Iα Iβ f → hom Iα Iγ (g ∘ f)
-- CategoryTheory.UnbundledHom.hom_id
class UnbundledHom {c : Type u → Type u} (hom : ∀ ⦃α β⦄, c α → c β → (α → β) → Prop) : Prop where
  hom_id : ∀ {α} (ia : c α), hom ia ia id
  hom_comp : ∀ {α β γ} {Iα : c α} {Iβ : c β} {Iγ : c γ} {g : β → γ} {f : α → β} (_ : hom Iβ Iγ g)
      (_ : hom Iα Iβ f), hom Iα Iγ (g ∘ f)
Arguments: {c : Type u → Type u} {hom : ⦃α β : Type u⦄ → c α → c β → (α → β) → Prop} [self : CategoryTheory.UnbundledHom hom] {α : Type u} (ia : c α)
Goal: hom ia ia id
-- CategoryTheory.Bicategory.LeftExtension.w
@[reassoc (attr := simp)]
theorem w (η : s ⟶ t) : s.unit ≫ f ◁ η.right = t.unit 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f : a ⟶ b} {g : a ⟶ c} {s t : CategoryTheory.Bicategory.LeftExtension f g} (η : s ⟶ t)
Goal: CategoryTheory.CategoryStruct.comp s.unit (CategoryTheory.Bicategory.whiskerLeft f η.right) = t.unit