ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.Bicategory.whiskerRightIso_inv
@[simps!]
def whiskerRightIso {f g : a ⟶ b} (η : f ≅ g) (h : b ⟶ c) : f ≫ h ≅ g ≫ h 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f g : a ⟶ b} (η : f ≅ g) (h : b ⟶ c)
Goal: (CategoryTheory.Bicategory.whiskerRightIso η h).inv = CategoryTheory.Bicategory.whiskerRight η.inv h
-- CategoryTheory.Bicategory.inv_whiskerRight
@[simp]
theorem inv_whiskerRight {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) [IsIso η] :
    inv (η ▷ h) = inv η ▷ h 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) [CategoryTheory.IsIso η]
Goal: CategoryTheory.inv (CategoryTheory.Bicategory.whiskerRight η h) =
  CategoryTheory.Bicategory.whiskerRight (CategoryTheory.inv η) h
-- CategoryTheory.Bicategory.pentagon_inv
@[reassoc (attr := simp)]
theorem pentagon_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
    f ◁ (α_ g h i).inv ≫ (α_ f (g ≫ h) i).inv ≫ (α_ f g h).inv ▷ i =
      (α_ f g (h ≫ i)).inv ≫ (α_ (f ≫ g) h i).inv 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c d e : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator g h i).inv)
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.comp g h) i).inv
      (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator f g h).inv i)) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.comp h i)).inv
    (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.comp f g) h i).inv
-- CategoryTheory.Bicategory.pentagon_inv_inv_hom_hom_inv
@[reassoc (attr := simp)]
theorem pentagon_inv_inv_hom_hom_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
    (α_ f (g ≫ h) i).inv ≫ (α_ f g h).inv ▷ i ≫ (α_ (f ≫ g) h i).hom =
    f ◁ (α_ g h i).hom ≫ (α_ f g (h ≫ i)).inv 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c d e : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.comp g h) i).inv
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator f g h).inv i)
      (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.comp f g) h i).hom) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator g h i).hom)
    (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.comp h i)).inv
-- CategoryTheory.Bicategory.pentagon_inv_hom_hom_hom_inv
@[reassoc (attr := simp)]
theorem pentagon_inv_hom_hom_hom_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
    (α_ (f ≫ g) h i).inv ≫ (α_ f g h).hom ▷ i ≫ (α_ f (g ≫ h) i).hom =
      (α_ f g (h ≫ i)).hom ≫ f ◁ (α_ g h i).inv 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c d e : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.comp f g) h i).inv
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator f g h).hom i)
      (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.comp g h) i).hom) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.comp h i)).hom
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator g h i).inv)
-- CategoryTheory.Bicategory.pentagon_hom_inv_inv_inv_inv
@[reassoc (attr := simp)]
theorem pentagon_hom_inv_inv_inv_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
    f ◁ (α_ g h i).hom ≫ (α_ f g (h ≫ i)).inv ≫ (α_ (f ≫ g) h i).inv =
      (α_ f (g ≫ h) i).inv ≫ (α_ f g h).inv ▷ i 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c d e : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator g h i).hom)
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.comp h i)).inv
      (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.comp f g) h i).inv) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.comp g h) i).inv
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator f g h).inv i)
-- CategoryTheory.Bicategory.pentagon_hom_hom_inv_hom_hom
@[reassoc (attr := simp)]
theorem pentagon_hom_hom_inv_hom_hom (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
    (α_ (f ≫ g) h i).hom ≫ (α_ f g (h ≫ i)).hom ≫ f ◁ (α_ g h i).inv =
      (α_ f g h).hom ▷ i ≫ (α_ f (g ≫ h) i).hom 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c d e : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.comp f g) h i).hom
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.comp h i)).hom
      (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator g h i).inv)) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator f g h).hom i)
    (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.comp g h) i).hom
-- CategoryTheory.Bicategory.pentagon_hom_inv_inv_inv_hom
@[reassoc (attr := simp)]
theorem pentagon_hom_inv_inv_inv_hom (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
    (α_ f g (h ≫ i)).hom ≫ f ◁ (α_ g h i).inv ≫ (α_ f (g ≫ h) i).inv =
    (α_ (f ≫ g) h i).inv ≫ (α_ f g h).hom ▷ i 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c d e : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.comp h i)).hom
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator g h i).inv)
      (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.comp g h) i).inv) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.comp f g) h i).inv
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator f g h).hom i)
-- CategoryTheory.Bicategory.pentagon_hom_hom_inv_inv_hom
@[reassoc (attr := simp)]
theorem pentagon_hom_hom_inv_inv_hom (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
    (α_ f (g ≫ h) i).hom ≫ f ◁ (α_ g h i).hom ≫ (α_ f g (h ≫ i)).inv =
      (α_ f g h).inv ▷ i ≫ (α_ (f ≫ g) h i).hom 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c d e : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.comp g h) i).hom
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator g h i).hom)
      (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.comp h i)).inv) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator f g h).inv i)
    (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.comp f g) h i).hom
-- CategoryTheory.Bicategory.pentagon_inv_hom_hom_hom_hom
@[reassoc (attr := simp)]
theorem pentagon_inv_hom_hom_hom_hom (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
    (α_ f g h).inv ▷ i ≫ (α_ (f ≫ g) h i).hom ≫ (α_ f g (h ≫ i)).hom =
      (α_ f (g ≫ h) i).hom ≫ f ◁ (α_ g h i).hom 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c d e : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator f g h).inv i)
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.comp f g) h i).hom
      (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.comp h i)).hom) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.comp g h) i).hom
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator g h i).hom)
-- CategoryTheory.Bicategory.pentagon_inv_inv_hom_inv_inv
@[reassoc (attr := simp)]
theorem pentagon_inv_inv_hom_inv_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
    (α_ f g (h ≫ i)).inv ≫ (α_ (f ≫ g) h i).inv ≫ (α_ f g h).hom ▷ i =
      f ◁ (α_ g h i).inv ≫ (α_ f (g ≫ h) i).inv 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c d e : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.comp h i)).inv
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.comp f g) h i).inv
      (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.associator f g h).hom i)) =
  CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.associator g h i).inv)
    (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.comp g h) i).inv
-- CategoryTheory.Bicategory.triangle_assoc_comp_left
theorem triangle_assoc_comp_left (f : a ⟶ b) (g : b ⟶ c) :
    (α_ f (𝟙 b) g).hom ≫ f ◁ (λ_ g).hom = (ρ_ f).hom ▷ g 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} (f : a ⟶ b) (g : b ⟶ c)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.id b) g).hom
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.leftUnitor g).hom) =
  CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.rightUnitor f).hom g
-- CategoryTheory.Bicategory.triangle_assoc_comp_right
@[reassoc (attr := simp)]
theorem triangle_assoc_comp_right (f : a ⟶ b) (g : b ⟶ c) :
    (α_ f (𝟙 b) g).inv ≫ (ρ_ f).hom ▷ g = f ◁ (λ_ g).hom 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} (f : a ⟶ b) (g : b ⟶ c)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.id b) g).inv
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.rightUnitor f).hom g) =
  CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.leftUnitor g).hom
-- CategoryTheory.Bicategory.triangle_assoc_comp_right_inv
@[reassoc (attr := simp)]
theorem triangle_assoc_comp_right_inv (f : a ⟶ b) (g : b ⟶ c) :
    (ρ_ f).inv ▷ g ≫ (α_ f (𝟙 b) g).hom = f ◁ (λ_ g).inv 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} (f : a ⟶ b) (g : b ⟶ c)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.rightUnitor f).inv g)
    (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.id b) g).hom =
  CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.leftUnitor g).inv
-- CategoryTheory.Bicategory.triangle_assoc_comp_left_inv
@[reassoc (attr := simp)]
theorem triangle_assoc_comp_left_inv (f : a ⟶ b) (g : b ⟶ c) :
    f ◁ (λ_ g).inv ≫ (α_ f (𝟙 b) g).inv = (ρ_ f).inv ▷ g 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} (f : a ⟶ b) (g : b ⟶ c)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.leftUnitor g).inv)
    (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.id b) g).inv =
  CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.rightUnitor f).inv g
-- CategoryTheory.Bicategory.associator_naturality_left
@[reassoc]
theorem associator_naturality_left {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d) :
    η ▷ g ▷ h ≫ (α_ f' g h).hom = (α_ f g h).hom ≫ η ▷ (g ≫ h) 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c d : B} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerRight η g) h)
    (CategoryTheory.Bicategory.associator f' g h).hom =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).hom
    (CategoryTheory.Bicategory.whiskerRight η (CategoryTheory.CategoryStruct.comp g h))
-- CategoryTheory.Bicategory.associator_inv_naturality_left
@[reassoc]
theorem associator_inv_naturality_left {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d) :
    η ▷ (g ≫ h) ≫ (α_ f' g h).inv = (α_ f g h).inv ≫ η ▷ g ▷ h 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c d : B} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight η (CategoryTheory.CategoryStruct.comp g h))
    (CategoryTheory.Bicategory.associator f' g h).inv =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).inv
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerRight η g) h)
-- CategoryTheory.Bicategory.whiskerRight_comp_symm
@[reassoc]
theorem whiskerRight_comp_symm {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d) :
    η ▷ g ▷ h = (α_ f g h).hom ≫ η ▷ (g ≫ h) ≫ (α_ f' g h).inv 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c d : B} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d)
Goal: CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerRight η g) h =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).hom
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerRight η (CategoryTheory.CategoryStruct.comp g h))
      (CategoryTheory.Bicategory.associator f' g h).inv)
-- CategoryTheory.Bicategory.associator_naturality_middle
@[reassoc]
theorem associator_naturality_middle (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d) :
    (f ◁ η) ▷ h ≫ (α_ f g' h).hom = (α_ f g h).hom ≫ f ◁ η ▷ h 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c d : B} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerLeft f η) h)
    (CategoryTheory.Bicategory.associator f g' h).hom =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).hom
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerRight η h))
-- CategoryTheory.Bicategory.associator_inv_naturality_middle
@[reassoc]
theorem associator_inv_naturality_middle (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d) :
    f ◁ η ▷ h ≫ (α_ f g' h).inv = (α_ f g h).inv ≫ (f ◁ η) ▷ h 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c d : B} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerRight η h))
    (CategoryTheory.Bicategory.associator f g' h).inv =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).inv
    (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerLeft f η) h)
-- CategoryTheory.Bicategory.whisker_assoc_symm
@[reassoc]
theorem whisker_assoc_symm (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d) :
    f ◁ η ▷ h = (α_ f g h).inv ≫ (f ◁ η) ▷ h ≫ (α_ f g' h).hom 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c d : B} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d)
Goal: CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerRight η h) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).inv
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.whiskerLeft f η) h)
      (CategoryTheory.Bicategory.associator f g' h).hom)
-- CategoryTheory.Bicategory.associator_naturality_right
@[reassoc]
theorem associator_naturality_right (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h') :
    (f ≫ g) ◁ η ≫ (α_ f g h').hom = (α_ f g h).hom ≫ f ◁ g ◁ η 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h')
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft (CategoryTheory.CategoryStruct.comp f g) η)
    (CategoryTheory.Bicategory.associator f g h').hom =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).hom
    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerLeft g η))
-- CategoryTheory.Bicategory.associator_inv_naturality_right
@[reassoc]
theorem associator_inv_naturality_right (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h') :
    f ◁ g ◁ η ≫ (α_ f g h').inv = (α_ f g h).inv ≫ (f ≫ g) ◁ η 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h')
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerLeft g η))
    (CategoryTheory.Bicategory.associator f g h').inv =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).inv
    (CategoryTheory.Bicategory.whiskerLeft (CategoryTheory.CategoryStruct.comp f g) η)
-- CategoryTheory.Bicategory.comp_whiskerLeft_symm
@[reassoc]
theorem comp_whiskerLeft_symm (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h') :
    f ◁ g ◁ η = (α_ f g h).inv ≫ (f ≫ g) ◁ η ≫ (α_ f g h').hom 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) {h h' : c ⟶ d} (η : h ⟶ h')
Goal: CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.whiskerLeft g η) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g h).inv
    (CategoryTheory.CategoryStruct.comp
      (CategoryTheory.Bicategory.whiskerLeft (CategoryTheory.CategoryStruct.comp f g) η)
      (CategoryTheory.Bicategory.associator f g h').hom)
-- CategoryTheory.Bicategory.leftUnitor_naturality
@[reassoc]
theorem leftUnitor_naturality {f g : a ⟶ b} (η : f ⟶ g) :
    𝟙 a ◁ η ≫ (λ_ g).hom = (λ_ f).hom ≫ η 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f g : a ⟶ b} (η : f ⟶ g)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft (CategoryTheory.CategoryStruct.id a) η)
    (CategoryTheory.Bicategory.leftUnitor g).hom =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.leftUnitor f).hom η
-- CategoryTheory.Bicategory.leftUnitor_inv_naturality
@[reassoc]
theorem leftUnitor_inv_naturality {f g : a ⟶ b} (η : f ⟶ g) :
    η ≫ (λ_ g).inv = (λ_ f).inv ≫ 𝟙 a ◁ η 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f g : a ⟶ b} (η : f ⟶ g)
Goal: CategoryTheory.CategoryStruct.comp η (CategoryTheory.Bicategory.leftUnitor g).inv =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.leftUnitor f).inv
    (CategoryTheory.Bicategory.whiskerLeft (CategoryTheory.CategoryStruct.id a) η)
-- CategoryTheory.Bicategory.id_whiskerLeft_symm
theorem id_whiskerLeft_symm {f g : a ⟶ b} (η : f ⟶ g) : η = (λ_ f).inv ≫ 𝟙 a ◁ η ≫ (λ_ g).hom 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f g : a ⟶ b} (η : f ⟶ g)
Goal: η =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.leftUnitor f).inv
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerLeft (CategoryTheory.CategoryStruct.id a) η)
      (CategoryTheory.Bicategory.leftUnitor g).hom)
-- CategoryTheory.Bicategory.rightUnitor_naturality
@[reassoc]
theorem rightUnitor_naturality {f g : a ⟶ b} (η : f ⟶ g) :
    η ▷ 𝟙 b ≫ (ρ_ g).hom = (ρ_ f).hom ≫ η 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f g : a ⟶ b} (η : f ⟶ g)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight η (CategoryTheory.CategoryStruct.id b))
    (CategoryTheory.Bicategory.rightUnitor g).hom =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.rightUnitor f).hom η
-- CategoryTheory.Bicategory.rightUnitor_inv_naturality
@[reassoc]
theorem rightUnitor_inv_naturality {f g : a ⟶ b} (η : f ⟶ g) :
    η ≫ (ρ_ g).inv = (ρ_ f).inv ≫ η ▷ 𝟙 b 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f g : a ⟶ b} (η : f ⟶ g)
Goal: CategoryTheory.CategoryStruct.comp η (CategoryTheory.Bicategory.rightUnitor g).inv =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.rightUnitor f).inv
    (CategoryTheory.Bicategory.whiskerRight η (CategoryTheory.CategoryStruct.id b))
-- CategoryTheory.Bicategory.whiskerRight_id_symm
theorem whiskerRight_id_symm {f g : a ⟶ b} (η : f ⟶ g) : η = (ρ_ f).inv ≫ η ▷ 𝟙 b ≫ (ρ_ g).hom 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f g : a ⟶ b} (η : f ⟶ g)
Goal: η =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.rightUnitor f).inv
    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight η (CategoryTheory.CategoryStruct.id b))
      (CategoryTheory.Bicategory.rightUnitor g).hom)
-- CategoryTheory.Bicategory.whiskerLeft_iff
theorem whiskerLeft_iff {f g : a ⟶ b} (η θ : f ⟶ g) : 𝟙 a ◁ η = 𝟙 a ◁ θ ↔ η = θ 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f g : a ⟶ b} (η θ : f ⟶ g)
Goal: CategoryTheory.Bicategory.whiskerLeft (CategoryTheory.CategoryStruct.id a) η =
    CategoryTheory.Bicategory.whiskerLeft (CategoryTheory.CategoryStruct.id a) θ ↔
  η = θ
-- CategoryTheory.Bicategory.whiskerRight_iff
theorem whiskerRight_iff {f g : a ⟶ b} (η θ : f ⟶ g) : η ▷ 𝟙 b = θ ▷ 𝟙 b ↔ η = θ 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b : B} {f g : a ⟶ b} (η θ : f ⟶ g)
Goal: CategoryTheory.Bicategory.whiskerRight η (CategoryTheory.CategoryStruct.id b) =
    CategoryTheory.Bicategory.whiskerRight θ (CategoryTheory.CategoryStruct.id b) ↔
  η = θ
-- CategoryTheory.Bicategory.leftUnitor_whiskerRight
@[reassoc, simp]
theorem leftUnitor_whiskerRight (f : a ⟶ b) (g : b ⟶ c) :
    (λ_ f).hom ▷ g = (α_ (𝟙 a) f g).hom ≫ (λ_ (f ≫ g)).hom 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} (f : a ⟶ b) (g : b ⟶ c)
Goal: CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.leftUnitor f).hom g =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.id a) f g).hom
    (CategoryTheory.Bicategory.leftUnitor (CategoryTheory.CategoryStruct.comp f g)).hom
-- CategoryTheory.Bicategory.leftUnitor_inv_whiskerRight
@[reassoc, simp]
theorem leftUnitor_inv_whiskerRight (f : a ⟶ b) (g : b ⟶ c) :
    (λ_ f).inv ▷ g = (λ_ (f ≫ g)).inv ≫ (α_ (𝟙 a) f g).inv 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} (f : a ⟶ b) (g : b ⟶ c)
Goal: CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.leftUnitor f).inv g =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.leftUnitor (CategoryTheory.CategoryStruct.comp f g)).inv
    (CategoryTheory.Bicategory.associator (CategoryTheory.CategoryStruct.id a) f g).inv
-- CategoryTheory.Bicategory.whiskerLeft_rightUnitor
@[reassoc, simp]
theorem whiskerLeft_rightUnitor (f : a ⟶ b) (g : b ⟶ c) :
    f ◁ (ρ_ g).hom = (α_ f g (𝟙 c)).inv ≫ (ρ_ (f ≫ g)).hom 
Arguments: {B : Type u} [CategoryTheory.Bicategory B] {a b c : B} (f : a ⟶ b) (g : b ⟶ c)
Goal: CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.rightUnitor g).hom =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.id c)).inv
    (CategoryTheory.Bicategory.rightUnitor (CategoryTheory.CategoryStruct.comp f g)).hom