ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.Functor.ReflectsIsomorphisms.reflects
class Functor.ReflectsIsomorphisms (F : C ⥤ D) : Prop where
    reflects : ∀ {A B : C} (f : A ⟶ B) [IsIso (F.map f)], IsIso f
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : C ⥤ D) [self : F.ReflectsIsomorphisms] {A B : C} (f : A ⟶ B) [CategoryTheory.IsIso (F.map f)]
Goal: CategoryTheory.IsIso f
-- CategoryTheory.isIso_of_reflects_iso
theorem isIso_of_reflects_iso {A B : C} (f : A ⟶ B) (F : C ⥤ D) [IsIso (F.map f)]
    [F.ReflectsIsomorphisms] : IsIso f 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {A B : C} (f : A ⟶ B) (F : C ⥤ D) [CategoryTheory.IsIso (F.map f)] [F.ReflectsIsomorphisms]
Goal: CategoryTheory.IsIso f
-- CategoryTheory.isIso_iff_of_reflects_iso
lemma isIso_iff_of_reflects_iso {A B : C} (f : A ⟶ B) (F : C ⥤ D) [F.ReflectsIsomorphisms] :
    IsIso (F.map f) ↔ IsIso f 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {A B : C} (f : A ⟶ B) (F : C ⥤ D) [F.ReflectsIsomorphisms]
Goal: CategoryTheory.IsIso (F.map f) ↔ CategoryTheory.IsIso f
-- CategoryTheory.Functor.FullyFaithful.reflectsIsomorphisms
lemma Functor.FullyFaithful.reflectsIsomorphisms {F : C ⥤ D} (hF : F.FullyFaithful) :
    F.ReflectsIsomorphisms 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : C ⥤ D} (hF : F.FullyFaithful)
Goal: F.ReflectsIsomorphisms
-- CategoryTheory.reflectsIsomorphisms_of_comp
lemma reflectsIsomorphisms_of_comp (F : C ⥤ D) (G : D ⥤ E)
    [(F ⋙ G).ReflectsIsomorphisms] : F.ReflectsIsomorphisms 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] (F : C ⥤ D) (G : D ⥤ E) [(F ⋙ G).ReflectsIsomorphisms]
Goal: F.ReflectsIsomorphisms
-- CategoryTheory.Functor.balanced_of_preserves
lemma Functor.balanced_of_preserves (F : C ⥤ D)
    [F.ReflectsIsomorphisms] [F.PreservesEpimorphisms] [F.PreservesMonomorphisms] [Balanced D] :
    Balanced C 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : C ⥤ D) [F.ReflectsIsomorphisms] [F.PreservesEpimorphisms] [F.PreservesMonomorphisms] [CategoryTheory.Balanced D]
Goal: CategoryTheory.Balanced C
-- CategoryTheory.OplaxNatTrans.naturality_naturality
structure OplaxNatTrans (F G : OplaxFunctor B C) where
  app (a : B) : F.obj a ⟶ G.obj a
  naturality {a b : B} (f : a ⟶ b) : F.map f ≫ app b ⟶ app a ≫ G.map f
  naturality_naturality :
    ∀ {a b : B} {f g : a ⟶ b} (η : f ⟶ g),
      F.map₂ η ▷ app b ≫ naturality g = naturality f ≫ app a ◁ G.map₂ η := by
    aesop_cat
  naturality_id :
    ∀ a : B,
      naturality (𝟙 a) ≫ app a ◁ G.mapId a =
        F.mapId a ▷ app a ≫ (λ_ (app a)).hom ≫ (ρ_ (app a)).inv := by
    aesop_cat
  naturality_comp :
    ∀ {a b c : B} (f : a ⟶ b) (g : b ⟶ c),
      naturality (f ≫ g) ≫ app a ◁ G.mapComp f g =
        F.mapComp f g ▷ app c ≫
          (α_ _ _ _).hom ≫
            F.map f ◁ naturality g ≫ (α_ _ _ _).inv ≫ naturality f ▷ G.map g ≫ (α_ _ _ _).hom := by
    aesop_cat
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G : CategoryTheory.OplaxFunctor B C} (self : CategoryTheory.OplaxNatTrans F G) {a b : B} {f g : a ⟶ b} (η : f ⟶ g)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (F.map₂ η) (self.app b))
    (self.naturality g) =
  CategoryTheory.CategoryStruct.comp (self.naturality f) (CategoryTheory.Bicategory.whiskerLeft (self.app a) (G.map₂ η))
-- CategoryTheory.OplaxNatTrans.naturality_id
structure OplaxNatTrans (F G : OplaxFunctor B C) where
  app (a : B) : F.obj a ⟶ G.obj a
  naturality {a b : B} (f : a ⟶ b) : F.map f ≫ app b ⟶ app a ≫ G.map f
  naturality_naturality :
    ∀ {a b : B} {f g : a ⟶ b} (η : f ⟶ g),
      F.map₂ η ▷ app b ≫ naturality g = naturality f ≫ app a ◁ G.map₂ η := by
    aesop_cat
  naturality_id :
    ∀ a : B,
      naturality (𝟙 a) ≫ app a ◁ G.mapId a =
        F.mapId a ▷ app a ≫ (λ_ (app a)).hom ≫ (ρ_ (app a)).inv := by
    aesop_cat
  naturality_comp :
    ∀ {a b c : B} (f : a ⟶ b) (g : b ⟶ c),
      naturality (f ≫ g) ≫ app a ◁ G.mapComp f g =
        F.mapComp f g ▷ app c ≫
          (α_ _ _ _).hom ≫
            F.map f ◁ naturality g ≫ (α_ _ _ _).inv ≫ naturality f ▷ G.map g ≫ (α_ _ _ _).hom := by
    aesop_cat
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G : CategoryTheory.OplaxFunctor B C} (self : CategoryTheory.OplaxNatTrans F G) (a : B)
Goal: CategoryTheory.CategoryStruct.comp (self.naturality (CategoryTheory.CategoryStruct.id a))
    (CategoryTheory.Bicategory.whiskerLeft (self.app a) (G.mapId a)) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (F.mapId a) (self.app a))
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.leftUnitor (self.app a)).hom
      (CategoryTheory.Bicategory.rightUnitor (self.app a)).inv)
-- CategoryTheory.OplaxNatTrans.naturality_comp
structure OplaxNatTrans (F G : OplaxFunctor B C) where
  app (a : B) : F.obj a ⟶ G.obj a
  naturality {a b : B} (f : a ⟶ b) : F.map f ≫ app b ⟶ app a ≫ G.map f
  naturality_naturality :
    ∀ {a b : B} {f g : a ⟶ b} (η : f ⟶ g),
      F.map₂ η ▷ app b ≫ naturality g = naturality f ≫ app a ◁ G.map₂ η := by
    aesop_cat
  naturality_id :
    ∀ a : B,
      naturality (𝟙 a) ≫ app a ◁ G.mapId a =
        F.mapId a ▷ app a ≫ (λ_ (app a)).hom ≫ (ρ_ (app a)).inv := by
    aesop_cat
  naturality_comp :
    ∀ {a b c : B} (f : a ⟶ b) (g : b ⟶ c),
      naturality (f ≫ g) ≫ app a ◁ G.mapComp f g =
        F.mapComp f g ▷ app c ≫
          (α_ _ _ _).hom ≫
            F.map f ◁ naturality g ≫ (α_ _ _ _).inv ≫ naturality f ▷ G.map g ≫ (α_ _ _ _).hom := by
    aesop_cat
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G : CategoryTheory.OplaxFunctor B C} (self : CategoryTheory.OplaxNatTrans F G) {a b c : B} (f : a ⟶ b) (g : b ⟶ c)
Goal: CategoryTheory.CategoryStruct.comp (self.naturality (CategoryTheory.CategoryStruct.comp f g))
    (CategoryTheory.Bicategory.whiskerLeft (self.app a) (G.mapComp f g)) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (F.mapComp f g) (self.app c))
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (F.map f) (F.map g) (self.app c)).hom
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft (F.map f) (self.naturality g))
        (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (F.map f) (self.app b) (G.map g)).inv
          (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (self.naturality f) (G.map g))
            (CategoryTheory.Bicategory.associator (self.app a) (G.map f) (G.map g)).hom))))
-- CategoryTheory.OplaxNatTrans.id_app
@[simps]
def id : OplaxNatTrans F F 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] (F : CategoryTheory.OplaxFunctor B C) (a : B)
Goal: (CategoryTheory.OplaxNatTrans.id F).app a = CategoryTheory.CategoryStruct.id (F.obj a)
-- CategoryTheory.OplaxNatTrans.id_naturality
@[simps]
def id : OplaxNatTrans F F 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] (F : CategoryTheory.OplaxFunctor B C) {a b : B} (f : a ⟶ b)
Goal: (CategoryTheory.OplaxNatTrans.id F).naturality f =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.rightUnitor (F.map f)).hom
    (CategoryTheory.Bicategory.leftUnitor (F.map f)).inv
-- CategoryTheory.OplaxNatTrans.whiskerLeft_naturality_naturality
@[reassoc (attr := simp)]
theorem whiskerLeft_naturality_naturality (f : a' ⟶ G.obj a) {g h : a ⟶ b} (β : g ⟶ h) :
    f ◁ G.map₂ β ▷ θ.app b ≫ f ◁ θ.naturality h =
      f ◁ θ.naturality g ≫ f ◁ θ.app a ◁ H.map₂ β 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {G H : CategoryTheory.OplaxFunctor B C} (θ : CategoryTheory.OplaxNatTrans G H) {a b : B} {a' : C} (f : a' ⟶ G.obj a) {g h : a ⟶ b} (β : g ⟶ h)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerRight (G.map₂ β) (θ.app b)))
    (CategoryTheory.Bicategory.whiskerLeft f (θ.naturality h)) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft f (θ.naturality g))
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerLeft (θ.app a) (H.map₂ β)))
-- CategoryTheory.OplaxNatTrans.whiskerRight_naturality_naturality
@[reassoc (attr := simp)]
theorem whiskerRight_naturality_naturality {f g : a ⟶ b} (β : f ⟶ g) (h : G.obj b ⟶ a') :
    F.map₂ β ▷ η.app b ▷ h ≫ η.naturality g ▷ h =
      η.naturality f ▷ h ≫ (α_ _ _ _).hom ≫ η.app a ◁ G.map₂ β ▷ h ≫ (α_ _ _ _).inv 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G : CategoryTheory.OplaxFunctor B C} (η : CategoryTheory.OplaxNatTrans F G) {a b : B} {a' : C} {f g : a ⟶ b} (β : f ⟶ g) (h : G.obj b ⟶ a')
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerRight (F.map₂ β) (η.app b)) h)
    (CategoryTheory.Bicategory.whiskerRight (η.naturality g) h) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (η.naturality f) h)
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (η.app a) (G.map f) h).hom
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Bicategory.whiskerLeft (η.app a) (CategoryTheory.Bicategory.whiskerRight (G.map₂ β) h))
        (CategoryTheory.Bicategory.associator (η.app a) (G.map g) h).inv))
-- CategoryTheory.OplaxNatTrans.whiskerLeft_naturality_comp
@[reassoc (attr := simp)]
theorem whiskerLeft_naturality_comp (f : a' ⟶ G.obj a) (g : a ⟶ b) (h : b ⟶ c) :
    f ◁ θ.naturality (g ≫ h) ≫ f ◁ θ.app a ◁ H.mapComp g h =
      f ◁ G.mapComp g h ▷ θ.app c ≫
        f ◁ (α_ _ _ _).hom ≫
          f ◁ G.map g ◁ θ.naturality h ≫
            f ◁ (α_ _ _ _).inv ≫ f ◁ θ.naturality g ▷ H.map h ≫ f ◁ (α_ _ _ _).hom 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {G H : CategoryTheory.OplaxFunctor B C} (θ : CategoryTheory.OplaxNatTrans G H) {a b c : B} {a' : C} (f : a' ⟶ G.obj a) (g : a ⟶ b) (h : b ⟶ c)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (θ.naturality (CategoryTheory.CategoryStruct.comp g h)))
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerLeft (θ.app a) (H.mapComp g h))) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerRight (G.mapComp g h) (θ.app c)))
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator (G.map g) (G.map h) (θ.app c)).hom)
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerLeft (G.map g) (θ.naturality h)))
        (CategoryTheory.CategoryStruct.comp
          (CategoryTheory.Bicategory.whiskerLeft f
            (CategoryTheory.Bicategory.associator (G.map g) (θ.app b) (H.map h)).inv)
          (CategoryTheory.CategoryStruct.comp
            (CategoryTheory.Bicategory.whiskerLeft f
              (CategoryTheory.Bicategory.whiskerRight (θ.naturality g) (H.map h)))
            (CategoryTheory.Bicategory.whiskerLeft f
              (CategoryTheory.Bicategory.associator (θ.app a) (H.map g) (H.map h)).hom)))))
-- CategoryTheory.OplaxNatTrans.whiskerRight_naturality_comp
@[reassoc (attr := simp)]
theorem whiskerRight_naturality_comp (f : a ⟶ b) (g : b ⟶ c) (h : G.obj c ⟶ a') :
    η.naturality (f ≫ g) ▷ h ≫ (α_ _ _ _).hom ≫ η.app a ◁ G.mapComp f g ▷ h =
      F.mapComp f g ▷ η.app c ▷ h ≫
        (α_ _ _ _).hom ▷ h ≫
          (α_ _ _ _).hom ≫
            F.map f ◁ η.naturality g ▷ h ≫
              (α_ _ _ _).inv ≫
                (α_ _ _ _).inv ▷ h ≫
                  η.naturality f ▷ G.map g ▷ h ≫ (α_ _ _ _).hom ▷ h ≫ (α_ _ _ _).hom 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G : CategoryTheory.OplaxFunctor B C} (η : CategoryTheory.OplaxNatTrans F G) {a b c : B} {a' : C} (f : a ⟶ b) (g : b ⟶ c) (h : G.obj c ⟶ a')
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (η.naturality (CategoryTheory.CategoryStruct.comp f g)) h)
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.associator (η.app a) (G.map (CategoryTheory.CategoryStruct.comp f g)) h).hom
      (CategoryTheory.Bicategory.whiskerLeft (η.app a) (CategoryTheory.Bicategory.whiskerRight (G.mapComp f g) h))) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerRight (F.mapComp f g) (η.app c)) h)
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator (F.map f) (F.map g) (η.app c)).hom
        h)
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Bicategory.associator (F.map f) (CategoryTheory.CategoryStruct.comp (F.map g) (η.app c)) h).hom
        (CategoryTheory.CategoryStruct.comp
          (CategoryTheory.Bicategory.whiskerLeft (F.map f) (CategoryTheory.Bicategory.whiskerRight (η.naturality g) h))
          (CategoryTheory.CategoryStruct.comp
            (CategoryTheory.Bicategory.associator (F.map f) (CategoryTheory.CategoryStruct.comp (η.app b) (G.map g))
                h).inv
            (CategoryTheory.CategoryStruct.comp
              (CategoryTheory.Bicategory.whiskerRight
                (CategoryTheory.Bicategory.associator (F.map f) (η.app b) (G.map g)).inv h)
              (CategoryTheory.CategoryStruct.comp
                (CategoryTheory.Bicategory.whiskerRight
                  (CategoryTheory.Bicategory.whiskerRight (η.naturality f) (G.map g)) h)
                (CategoryTheory.CategoryStruct.comp
                  (CategoryTheory.Bicategory.whiskerRight
                    (CategoryTheory.Bicategory.associator (η.app a) (G.map f) (G.map g)).hom h)
                  (CategoryTheory.Bicategory.associator (η.app a)
                      (CategoryTheory.CategoryStruct.comp (G.map f) (G.map g)) h).hom)))))))
-- CategoryTheory.OplaxNatTrans.whiskerLeft_naturality_id
@[reassoc (attr := simp)]
theorem whiskerLeft_naturality_id (f : a' ⟶ G.obj a) :
    f ◁ θ.naturality (𝟙 a) ≫ f ◁ θ.app a ◁ H.mapId a =
      f ◁ G.mapId a ▷ θ.app a ≫ f ◁ (λ_ (θ.app a)).hom ≫ f ◁ (ρ_ (θ.app a)).inv 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {G H : CategoryTheory.OplaxFunctor B C} (θ : CategoryTheory.OplaxNatTrans G H) {a : B} {a' : C} (f : a' ⟶ G.obj a)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (θ.naturality (CategoryTheory.CategoryStruct.id a)))
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerLeft (θ.app a) (H.mapId a))) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerRight (G.mapId a) (θ.app a)))
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.leftUnitor (θ.app a)).hom)
      (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.rightUnitor (θ.app a)).inv))
-- CategoryTheory.OplaxNatTrans.whiskerRight_naturality_id
@[reassoc (attr := simp)]
theorem whiskerRight_naturality_id (f : G.obj a ⟶ a') :
    η.naturality (𝟙 a) ▷ f ≫ (α_ _ _ _).hom ≫ η.app a ◁ G.mapId a ▷ f =
    F.mapId a ▷ η.app a ▷ f ≫ (λ_ (η.app a)).hom ▷ f ≫ (ρ_ (η.app a)).inv ▷ f ≫ (α_ _ _ _).hom 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G : CategoryTheory.OplaxFunctor B C} (η : CategoryTheory.OplaxNatTrans F G) {a : B} {a' : C} (f : G.obj a ⟶ a')
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (η.naturality (CategoryTheory.CategoryStruct.id a)) f)
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.associator (η.app a) (G.map (CategoryTheory.CategoryStruct.id a)) f).hom
      (CategoryTheory.Bicategory.whiskerLeft (η.app a) (CategoryTheory.Bicategory.whiskerRight (G.mapId a) f))) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerRight (F.mapId a) (η.app a)) f)
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.leftUnitor (η.app a)).hom f)
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.rightUnitor (η.app a)).inv f)
        (CategoryTheory.Bicategory.associator (η.app a) (CategoryTheory.CategoryStruct.id (G.obj a)) f).hom))
-- CategoryTheory.OplaxNatTrans.vcomp_app
@[simps]
def vcomp (η : OplaxNatTrans F G) (θ : OplaxNatTrans G H) : OplaxNatTrans F H 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G H : CategoryTheory.OplaxFunctor B C} (η : CategoryTheory.OplaxNatTrans F G) (θ : CategoryTheory.OplaxNatTrans G H) (a : B)
Goal: (η.vcomp θ).app a = CategoryTheory.CategoryStruct.comp (η.app a) (θ.app a)
-- CategoryTheory.OplaxNatTrans.vcomp_naturality
@[simps]
def vcomp (η : OplaxNatTrans F G) (θ : OplaxNatTrans G H) : OplaxNatTrans F H 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G H : CategoryTheory.OplaxFunctor B C} (η : CategoryTheory.OplaxNatTrans F G) (θ : CategoryTheory.OplaxNatTrans G H) {a b : B} (f : a ⟶ b)
Goal: (η.vcomp θ).naturality f =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (F.map f) (η.app b) (θ.app b)).inv
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (η.naturality f) (θ.app b))
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (η.app a) (G.map f) (θ.app b)).hom
        (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft (η.app a) (θ.naturality f))
          (CategoryTheory.Bicategory.associator (η.app a) (θ.app a) (H.map f)).inv)))
-- CategoryTheory.OplaxNatTrans.instCategoryStructOplaxFunctor_id
@[simps id comp]
instance : CategoryStruct (OplaxFunctor B C) 
Arguments: (B : Type u₁) [CategoryTheory.Bicategory B] (C : Type u₂) [CategoryTheory.Bicategory C] (F : CategoryTheory.OplaxFunctor B C)
Goal: CategoryTheory.CategoryStruct.id F = CategoryTheory.OplaxNatTrans.id F
-- CategoryTheory.OplaxNatTrans.instCategoryStructOplaxFunctor_comp
@[simps id comp]
instance : CategoryStruct (OplaxFunctor B C) 
Arguments: (B : Type u₁) [CategoryTheory.Bicategory B] (C : Type u₂) [CategoryTheory.Bicategory C]
Goal: ∀ {X Y Z : CategoryTheory.OplaxFunctor B C} (η : CategoryTheory.OplaxNatTrans X Y)
  (θ : CategoryTheory.OplaxNatTrans Y Z), CategoryTheory.CategoryStruct.comp η θ = η.vcomp θ
-- CategoryTheory.OplaxNatTrans.Modification.naturality
@[ext]
structure Modification (η θ : F ⟶ G) where
  app (a : B) : η.app a ⟶ θ.app a
  naturality :
    ∀ {a b : B} (f : a ⟶ b),
      F.map f ◁ app b ≫ θ.naturality f = η.naturality f ≫ app a ▷ G.map f := by
    aesop_cat
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G : CategoryTheory.OplaxFunctor B C} {η θ : F ⟶ G} (self : CategoryTheory.OplaxNatTrans.Modification η θ) {a b : B} (f : a ⟶ b)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft (F.map f) (self.app b)) (θ.naturality f) =
  CategoryTheory.CategoryStruct.comp (η.naturality f) (CategoryTheory.Bicategory.whiskerRight (self.app a) (G.map f))
-- CategoryTheory.OplaxNatTrans.Modification.id_app
@[simps]
def id : Modification η η 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G : CategoryTheory.OplaxFunctor B C} (η : F ⟶ G) (a : B)
Goal: (CategoryTheory.OplaxNatTrans.Modification.id η).app a = CategoryTheory.CategoryStruct.id (η.app a)
-- CategoryTheory.OplaxNatTrans.Modification.whiskerLeft_naturality
@[reassoc (attr := simp)]
theorem whiskerLeft_naturality (f : a' ⟶ F.obj b) (g : b ⟶ c) :
    f ◁ F.map g ◁ Γ.app c ≫ f ◁ θ.naturality g = f ◁ η.naturality g ≫ f ◁ Γ.app b ▷ G.map g 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G : CategoryTheory.OplaxFunctor B C} {η θ : F ⟶ G} (Γ : CategoryTheory.OplaxNatTrans.Modification η θ) {b c : B} {a' : C} (f : a' ⟶ F.obj b) (g : b ⟶ c)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerLeft (F.map g) (Γ.app c)))
    (CategoryTheory.Bicategory.whiskerLeft f (θ.naturality g)) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft f (η.naturality g))
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerRight (Γ.app b) (G.map g)))
-- CategoryTheory.OplaxNatTrans.Modification.whiskerRight_naturality
@[reassoc (attr := simp)]
theorem whiskerRight_naturality (f : a ⟶ b) (g : G.obj b ⟶ a') :
    F.map f ◁ Γ.app b ▷ g ≫ (α_ _ _ _).inv ≫ θ.naturality f ▷ g =
      (α_ _ _ _).inv ≫ η.naturality f ▷ g ≫ Γ.app a ▷ G.map f ▷ g 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G : CategoryTheory.OplaxFunctor B C} {η θ : F ⟶ G} (Γ : CategoryTheory.OplaxNatTrans.Modification η θ) {a b : B} {a' : C} (f : a ⟶ b) (g : G.obj b ⟶ a')
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft (F.map f) (CategoryTheory.Bicategory.whiskerRight (Γ.app b) g))
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (F.map f) (θ.app b) g).inv
      (CategoryTheory.Bicategory.whiskerRight (θ.naturality f) g)) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (F.map f) (η.app b) g).inv
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (η.naturality f) g)
      (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerRight (Γ.app a) (G.map f)) g))
-- CategoryTheory.OplaxNatTrans.Modification.vcomp_app
@[simps]
def vcomp (Γ : Modification η θ) (Δ : Modification θ ι) : Modification η ι 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G : CategoryTheory.OplaxFunctor B C} {η θ ι : F ⟶ G} (Γ : CategoryTheory.OplaxNatTrans.Modification η θ) (Δ : CategoryTheory.OplaxNatTrans.Modification θ ι) (a : B)
Goal: (Γ.vcomp Δ).app a = CategoryTheory.CategoryStruct.comp (Γ.app a) (Δ.app a)
-- CategoryTheory.OplaxNatTrans.category_id
@[simps]
instance category (F G : OplaxFunctor B C) : Category (F ⟶ G) 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] (F G : CategoryTheory.OplaxFunctor B C) (η : F ⟶ G)
Goal: CategoryTheory.CategoryStruct.id η = CategoryTheory.OplaxNatTrans.Modification.id η
-- CategoryTheory.OplaxNatTrans.category_comp
@[simps]
instance category (F G : OplaxFunctor B C) : Category (F ⟶ G) 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] (F G : CategoryTheory.OplaxFunctor B C)
Goal: ∀ {X Y Z : F ⟶ G} (Γ : CategoryTheory.OplaxNatTrans.Modification X Y)
  (Δ : CategoryTheory.OplaxNatTrans.Modification Y Z), CategoryTheory.CategoryStruct.comp Γ Δ = Γ.vcomp Δ
-- CategoryTheory.OplaxNatTrans.ext
@[ext]
lemma ext {F G : OplaxFunctor B C} {α β : F ⟶ G} {m n : α ⟶ β} (w : ∀ b, m.app b = n.app b) :
    m = n 
Arguments: {B : Type u₁} [CategoryTheory.Bicategory B] {C : Type u₂} [CategoryTheory.Bicategory C] {F G : CategoryTheory.OplaxFunctor B C} {α β : F ⟶ G} {m n : α ⟶ β} (w : ∀ (b : B), m.app b = n.app b)
Goal: m = n