awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Bicategory.Free
Goal:
B : Type u₁
inst : Quiver.{v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : Prefunctor.{v₁, v₂, u₁, u₂} B C
a b : FreeBicategory B
f g h : a ⟶ b
η : f ⟶ g
θ : g ⟶ h
f₁ :
  ∀ (a_1 : Hom₂ f g),
    Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel a_1 ≫ θ) =
      Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel a_1) ≫ Quot.lift (r := Rel) (liftHom₂ F) ⋯ θ
h₁ :
  ∀ (a_1 b_1 : Hom₂ f g) (p : Rel a_1 b_1),
    Eq (α :=
      Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel b_1 ≫ θ) =
        Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel b_1) ≫ Quot.lift (r := Rel) (liftHom₂ F) ⋯ θ)
      ⋯ ⋯
⊢ Quot.lift (r := Rel) (liftHom₂ F) ⋯ (η ≫ θ) =
    Quot.lift (r := Rel) (liftHom₂ F) ⋯ η ≫ Quot.lift (r := Rel) (liftHom₂ F) ⋯ θ

Tactic:
induction η using Quot.rec
-- Module: Mathlib.CategoryTheory.Bicategory.Free
Goal:
B : Type u₁
inst : Quiver.{v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : Prefunctor.{v₁, v₂, u₁, u₂} B C
a b : FreeBicategory B
f g h : a ⟶ b
θ : g ⟶ h
a_1 : Hom₂ f g
f₁ :
  ∀ (a_2 : Hom₂ g h),
    Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel a_1 ≫ Quot.mk Rel a_2) =
      Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel a_1) ≫ Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel a_2)
f₂ :
  ∀ (a_2 b_1 : Hom₂ g h) (p : Rel a_2 b_1),
    Eq (α :=
      Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel a_1 ≫ Quot.mk Rel b_1) =
        Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel a_1) ≫ Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel b_1))
      ⋯ ⋯
⊢ Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel a_1 ≫ θ) =
    Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel a_1) ≫ Quot.lift (r := Rel) (liftHom₂ F) ⋯ θ

Tactic:
induction θ using Quot.rec
-- Module: Mathlib.CategoryTheory.Bicategory.Free
Goal:
B : Type u₁
inst : Quiver.{v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : Prefunctor.{v₁, v₂, u₁, u₂} B C
a b : FreeBicategory B
f g h : a ⟶ b
a_1 : Hom₂ f g
a_2 : Hom₂ g h
⊢ Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel a_1 ≫ Quot.mk Rel a_2) =
    Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel a_1) ≫ Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel a_2)

Tactic:
rfl
-- Module: Mathlib.CategoryTheory.Bicategory.Free
Goal:
B : Type u₁
inst : Quiver.{v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : Prefunctor.{v₁, v₂, u₁, u₂} B C
a b : FreeBicategory B
f g h : a ⟶ b
θ : g ⟶ h
a_1 : Hom₂ f g
a_2 b_1 : Hom₂ g h
p : Rel a_2 b_1
⊢ Eq (α :=
    Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel a_1 ≫ Quot.mk Rel b_1) =
      Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel a_1) ≫ Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel b_1))
    ⋯ ⋯

Tactic:
rfl
-- Module: Mathlib.CategoryTheory.Bicategory.Free
Goal:
B : Type u₁
inst : Quiver.{v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : Prefunctor.{v₁, v₂, u₁, u₂} B C
a b : FreeBicategory B
f g h : a ⟶ b
η : f ⟶ g
θ : g ⟶ h
a_1 b_1 : Hom₂ f g
p : Rel a_1 b_1
⊢ Eq (α :=
    Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel b_1 ≫ θ) =
      Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel b_1) ≫ Quot.lift (r := Rel) (liftHom₂ F) ⋯ θ)
    ⋯ ⋯

Tactic:
rfl
-- Module: Mathlib.CategoryTheory.Bicategory.Free
Goal:
B : Type u₁
inst : Quiver.{v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : Prefunctor.{v₁, v₂, u₁, u₂} B C
a b c : FreeBicategory B
f : a ⟶ b
g h : b ⟶ c
η : g ⟶ h
f₁ :
  ∀ (a_1 : Hom₂ g h),
    Quot.lift (r := Rel) (liftHom₂ F) ⋯ (f ◁ Quot.mk Rel a_1) =
      Iso.hom (Iso.refl (liftHom F (f ≫ g))) ≫
        liftHom F f ◁ Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel a_1) ≫ Iso.inv (Iso.refl (liftHom F (f ≫ h)))
h₁ : ∀ (a_1 b_1 : Hom₂ g h) (p : Rel a_1 b_1), ⋯ = ⋯
⊢ Quot.lift (r := Rel) (liftHom₂ F) ⋯ (f ◁ η) =
    Iso.hom (Iso.refl (liftHom F (f ≫ g))) ≫
      liftHom F f ◁ Quot.lift (r := Rel) (liftHom₂ F) ⋯ η ≫ Iso.inv (Iso.refl (liftHom F (f ≫ h)))

Tactic:
induction η using Quot.rec
-- Module: Mathlib.CategoryTheory.Bicategory.Free
Goal:
B : Type u₁
inst : Quiver.{v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : Prefunctor.{v₁, v₂, u₁, u₂} B C
a b c : FreeBicategory B
f : a ⟶ b
g h : b ⟶ c
η : g ⟶ h
a_1 b_1 : Hom₂ g h
p : Rel a_1 b_1
⊢ ⋯ = ⋯

Tactic:
rfl
-- Module: Mathlib.CategoryTheory.Bicategory.Free
Goal:
B : Type u₁
inst : Quiver.{v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : Prefunctor.{v₁, v₂, u₁, u₂} B C
a : FreeBicategory B
_ :
  ∀ {b c : FreeBicategory B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c),
    Quot.lift (r := Rel) (liftHom₂ F) ⋯ (η ▷ h) =
      Iso.hom (Iso.refl (liftHom F (f ≫ h))) ≫
        Quot.lift (r := Rel) (liftHom₂ F) ⋯ η ▷ liftHom F h ≫ Iso.inv (Iso.refl (liftHom F (g ≫ h)))
⊢ ∀ {b c : FreeBicategory B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c),
    Quot.lift (r := Rel) (liftHom₂ F) ⋯ (η ▷ h) =
      Iso.hom (Iso.refl (liftHom F (f ≫ h))) ≫
        Quot.lift (r := Rel) (liftHom₂ F) ⋯ η ▷ liftHom F h ≫ Iso.inv (Iso.refl (liftHom F (g ≫ h)))

Tactic:
_
-- Module: Mathlib.CategoryTheory.Bicategory.Free
Goal:
B : Type u₁
inst : Quiver.{v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : Prefunctor.{v₁, v₂, u₁, u₂} B C
a b : FreeBicategory B
_ :
  ∀ {c : FreeBicategory B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c),
    Quot.lift (r := Rel) (liftHom₂ F) ⋯ (η ▷ h) =
      Iso.hom (Iso.refl (liftHom F (f ≫ h))) ≫
        Quot.lift (r := Rel) (liftHom₂ F) ⋯ η ▷ liftHom F h ≫ Iso.inv (Iso.refl (liftHom F (g ≫ h)))
⊢ ∀ {c : FreeBicategory B} {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c),
    Quot.lift (r := Rel) (liftHom₂ F) ⋯ (η ▷ h) =
      Iso.hom (Iso.refl (liftHom F (f ≫ h))) ≫
        Quot.lift (r := Rel) (liftHom₂ F) ⋯ η ▷ liftHom F h ≫ Iso.inv (Iso.refl (liftHom F (g ≫ h)))

Tactic:
_
-- Module: Mathlib.CategoryTheory.Bicategory.Free
Goal:
B : Type u₁
inst : Quiver.{v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : Prefunctor.{v₁, v₂, u₁, u₂} B C
a b c : FreeBicategory B
_ :
  ∀ {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c),
    Quot.lift (r := Rel) (liftHom₂ F) ⋯ (η ▷ h) =
      Iso.hom (Iso.refl (liftHom F (f ≫ h))) ≫
        Quot.lift (r := Rel) (liftHom₂ F) ⋯ η ▷ liftHom F h ≫ Iso.inv (Iso.refl (liftHom F (g ≫ h)))
⊢ ∀ {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c),
    Quot.lift (r := Rel) (liftHom₂ F) ⋯ (η ▷ h) =
      Iso.hom (Iso.refl (liftHom F (f ≫ h))) ≫
        Quot.lift (r := Rel) (liftHom₂ F) ⋯ η ▷ liftHom F h ≫ Iso.inv (Iso.refl (liftHom F (g ≫ h)))

Tactic:
_
-- Module: Mathlib.CategoryTheory.Bicategory.Free
Goal:
B : Type u₁
inst : Quiver.{v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : Prefunctor.{v₁, v₂, u₁, u₂} B C
a b c : FreeBicategory B
f : a ⟶ b
_ :
  ∀ {g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c),
    Quot.lift (r := Rel) (liftHom₂ F) ⋯ (η ▷ h) =
      Iso.hom (Iso.refl (liftHom F (f ≫ h))) ≫
        Quot.lift (r := Rel) (liftHom₂ F) ⋯ η ▷ liftHom F h ≫ Iso.inv (Iso.refl (liftHom F (g ≫ h)))
⊢ ∀ {g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c),
    Quot.lift (r := Rel) (liftHom₂ F) ⋯ (η ▷ h) =
      Iso.hom (Iso.refl (liftHom F (f ≫ h))) ≫
        Quot.lift (r := Rel) (liftHom₂ F) ⋯ η ▷ liftHom F h ≫ Iso.inv (Iso.refl (liftHom F (g ≫ h)))

Tactic:
_
-- Module: Mathlib.CategoryTheory.Bicategory.Free
Goal:
B : Type u₁
inst : Quiver.{v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : Prefunctor.{v₁, v₂, u₁, u₂} B C
a b c : FreeBicategory B
f g : a ⟶ b
η : f ⟶ g
h : b ⟶ c
dsimp :
  Quot.lift (r := Rel) (liftHom₂ F) ⋯ (η ▷ h) =
    𝟙 (liftHom F f ≫ liftHom F h) ≫ Quot.lift (r := Rel) (liftHom₂ F) ⋯ η ▷ liftHom F h ≫ 𝟙 (liftHom F g ≫ liftHom F h)
⊢ Quot.lift (r := Rel) (liftHom₂ F) ⋯ (η ▷ h) =
    Iso.hom (Iso.refl (liftHom F (f ≫ h))) ≫
      Quot.lift (r := Rel) (liftHom₂ F) ⋯ η ▷ liftHom F h ≫ Iso.inv (Iso.refl (liftHom F (g ≫ h)))

Tactic:
dsimp
-- Module: Mathlib.CategoryTheory.Bicategory.Free
Goal:
B : Type u₁
inst : Quiver.{v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : Prefunctor.{v₁, v₂, u₁, u₂} B C
a b c : FreeBicategory B
f_1 g : a ⟶ b
η : f_1 ⟶ g
h : b ⟶ c
f :
  ∀ (a_1 : Hom₂ f_1 g),
    Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel a_1 ▷ h) =
      𝟙 (liftHom F f_1 ≫ liftHom F h) ≫
        Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel a_1) ▷ liftHom F h ≫ 𝟙 (liftHom F g ≫ liftHom F h)
h₁ :
  ∀ (a_1 b_1 : Hom₂ f_1 g) (p : Rel a_1 b_1),
    Eq (α :=
      Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel b_1 ▷ h) =
        𝟙 (liftHom F f_1 ≫ liftHom F h) ≫
          Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel b_1) ▷ liftHom F h ≫ 𝟙 (liftHom F g ≫ liftHom F h))
      ⋯ ⋯
⊢ Quot.lift (r := Rel) (liftHom₂ F) ⋯ (η ▷ h) =
    𝟙 (liftHom F f_1 ≫ liftHom F h) ≫
      Quot.lift (r := Rel) (liftHom₂ F) ⋯ η ▷ liftHom F h ≫ 𝟙 (liftHom F g ≫ liftHom F h)

Tactic:
induction η using Quot.rec
-- Module: Mathlib.CategoryTheory.Bicategory.Free
Goal:
B : Type u₁
inst : Quiver.{v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : Prefunctor.{v₁, v₂, u₁, u₂} B C
a b c : FreeBicategory B
f g : a ⟶ b
η : f ⟶ g
h : b ⟶ c
a_1 b_1 : Hom₂ f g
p : Rel a_1 b_1
h₁ h₂ h₃ h₄ :
  Eq (α :=
    Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel b_1 ▷ h) =
      𝟙 (liftHom F f ≫ liftHom F h) ≫
        Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel b_1) ▷ liftHom F h ≫ 𝟙 (liftHom F g ≫ liftHom F h))
    ⋯ ⋯
⊢ Eq (α :=
    Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel b_1 ▷ h) =
      𝟙 (liftHom F f ≫ liftHom F h) ≫
        Quot.lift (r := Rel) (liftHom₂ F) ⋯ (Quot.mk Rel b_1) ▷ liftHom F h ≫ 𝟙 (liftHom F g ≫ liftHom F h))
    ⋯ ⋯

Tactic:
cat_disch
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Cat.ObjectProperty
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
F : B ⥤ᵖ Cat
P : F.ObjectProperty
inst_1 : P.IsClosedUnderMapObj
X : B
M : P.Obj X
⊢ Eq (α := (P.map (𝟙 X)).obj M ⟶ Functor.obj.{v', v', u', u'} (𝟭 (P.Obj X)) M) (NatTrans.app (Iso.hom (P.mapId X)) M)
    (ObjectProperty.homMk (P := P.prop X) ((F.mapId X).hom.toNatTrans.app (M.obj (P := P.prop X))))

Tactic:
rfl

Premises used: CategoryTheory.ObjectProperty.homMk, rfl
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Cat.ObjectProperty
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
F : B ⥤ᵖ Cat
P : F.ObjectProperty
inst_1 : P.IsClosedUnderMapObj
X : B
M : P.Obj X
⊢ Eq (α := Functor.obj.{v', v', u', u'} (𝟭 (P.Obj X)) M ⟶ (P.map (𝟙 X)).obj M) (NatTrans.app (Iso.inv (P.mapId X)) M)
    (ObjectProperty.homMk (P := P.prop X) ((F.mapId X).inv.toNatTrans.app (M.obj (P := P.prop X))))

Tactic:
rfl

Premises used: CategoryTheory.ObjectProperty.homMk, rfl
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Cat.ObjectProperty
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
F : B ⥤ᵖ Cat
P : F.ObjectProperty
inst_1 : P.IsClosedUnderMapObj
X Y Z : B
f : X ⟶ Y
g : Y ⟶ Z
M : P.Obj X
⊢ Eq (α := (P.map (f ≫ g)).obj M ⟶ (P.map f ⋙ P.map g).obj M) (NatTrans.app (Iso.hom (P.mapComp f g)) M)
    (ObjectProperty.homMk (P := P.prop Z) ((F.mapComp f g).hom.toNatTrans.app (M.obj (P := P.prop X))))

Tactic:
rfl

Premises used: CategoryTheory.ObjectProperty.homMk, rfl
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Cat.ObjectProperty
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
F : B ⥤ᵖ Cat
P : F.ObjectProperty
inst_1 : P.IsClosedUnderMapObj
X Y Z : B
f : X ⟶ Y
g : Y ⟶ Z
M : P.Obj X
⊢ Eq (α := (P.map f ⋙ P.map g).obj M ⟶ (P.map (f ≫ g)).obj M) (NatTrans.app (Iso.inv (P.mapComp f g)) M)
    (ObjectProperty.homMk (P := P.prop Z) ((F.mapComp f g).inv.toNatTrans.app (M.obj (P := P.prop X))))

Tactic:
rfl

Premises used: CategoryTheory.ObjectProperty.homMk, rfl
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Cat
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
F : B ⥤ᵖ Cat
b₀ : B
X Y : (↑(F.obj b₀) : Type u')
f : b₀ ⟶ b₀
hf : f = 𝟙 b₀
a : X ⟶ Y
⊢ (F.map f).toFunctor.map a ≫ (F.mapId' f hf).hom.toNatTrans.app Y = (F.mapId' f hf).hom.toNatTrans.app X ≫ a

Tactic:
(F.mapId' f hf).hom.toNatTrans.naturality a

Premises used: CategoryTheory.Pseudofunctor.mapId', CategoryTheory.NatTrans.naturality, CategoryTheory.Cat.Hom₂.toNatTrans, CategoryTheory.Iso.hom
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Cat
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
F : B ⥤ᵖ Cat
b₀ : B
X Y : (↑(F.obj b₀) : Type u')
f : b₀ ⟶ b₀
hf : f = 𝟙 b₀
a : X ⟶ Y
⊢ (F.mapId' f hf).inv.toNatTrans.app X ≫ (F.map f).toFunctor.map a = a ≫ (F.mapId' f hf).inv.toNatTrans.app Y

Tactic:
((F.mapId' f hf).inv.toNatTrans.naturality a).symm

Premises used: CategoryTheory.Pseudofunctor.mapId', CategoryTheory.NatTrans.naturality, CategoryTheory.Cat.Hom₂.toNatTrans, Eq.symm, CategoryTheory.Iso.inv
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Cat
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
F : B ⥤ᵖ Cat
b₀ b₁ b₂ : B
X Y : (↑(F.obj b₀) : Type u')
f : b₀ ⟶ b₁
g : b₁ ⟶ b₂
fg : b₀ ⟶ b₂
hfg : f ≫ g = fg
a : X ⟶ Y
⊢ (F.map fg).toFunctor.map a ≫ (F.mapComp' f g fg hfg).hom.toNatTrans.app Y =
    (F.mapComp' f g fg hfg).hom.toNatTrans.app X ≫ (F.map g).toFunctor.map ((F.map f).toFunctor.map a)

Tactic:
(F.mapComp' f g fg hfg).hom.toNatTrans.naturality a

Premises used: CategoryTheory.NatTrans.naturality, CategoryTheory.Pseudofunctor.mapComp', CategoryTheory.Cat.Hom₂.toNatTrans, CategoryTheory.Iso.hom
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Cat
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
F : B ⥤ᵖ Cat
b₀ b₁ b₂ : B
X Y : (↑(F.obj b₀) : Type u')
f : b₀ ⟶ b₁
g : b₁ ⟶ b₂
fg : b₀ ⟶ b₂
hfg : f ≫ g = fg
a : X ⟶ Y
⊢ (F.map g).toFunctor.map ((F.map f).toFunctor.map a) ≫ (F.mapComp' f g fg hfg).inv.toNatTrans.app Y =
    (F.mapComp' f g fg hfg).inv.toNatTrans.app X ≫ (F.map fg).toFunctor.map a

Tactic:
(F.mapComp' f g fg hfg).inv.toNatTrans.naturality a

Premises used: CategoryTheory.NatTrans.naturality, CategoryTheory.Pseudofunctor.mapComp', CategoryTheory.Cat.Hom₂.toNatTrans, CategoryTheory.Iso.inv
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Cat
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
F : B ⥤ᵖ Cat
b₀ b₁ b₂ : B
X Y : (↑(F.obj b₀) : Type u')
f : b₀ ⟶ b₁
g : b₁ ⟶ b₂
fg : b₀ ⟶ b₂
hfg : f ≫ g = fg
a : X ⟶ Y
⊢ (F.mapComp' f g fg hfg).inv.toNatTrans.app X ≫
      (F.map fg).toFunctor.map a ≫ (F.mapComp' f g fg hfg).hom.toNatTrans.app Y =
    (F.map g).toFunctor.map ((F.map f).toFunctor.map a)

Tactic:
NatIso.naturality_1 (Cat.Hom.toNatIso (F.mapComp' f g fg hfg)) a

Premises used: CategoryTheory.NatIso.naturality_1, CategoryTheory.Cat.Hom.toNatIso, CategoryTheory.Pseudofunctor.mapComp'
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Cat
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
F : B ⥤ᵖ Cat
b₀ b₁ b₂ : B
X Y : (↑(F.obj b₀) : Type u')
f : b₀ ⟶ b₁
g : b₁ ⟶ b₂
fg : b₀ ⟶ b₂
hfg : f ≫ g = fg
a : X ⟶ Y
⊢ (F.mapComp' f g fg hfg).hom.toNatTrans.app X ≫
      (F.map g).toFunctor.map ((F.map f).toFunctor.map a) ≫ (F.mapComp' f g fg hfg).inv.toNatTrans.app Y =
    (F.map fg).toFunctor.map a

Tactic:
NatIso.naturality_2 (Cat.Hom.toNatIso (F.mapComp' f g fg hfg)) a

Premises used: CategoryTheory.Cat.Hom.toNatIso, CategoryTheory.NatIso.naturality_2, CategoryTheory.Pseudofunctor.mapComp'
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Lax
Goal:
B : Type u₁
inst : Bicategory.{w₁, v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : LaxFunctor.{w₁, w₂, v₁, v₂, u₁, u₂} B C
a b c d : B
f : a ⟶ b
g : b ⟶ c
h : c ⟶ d
rw :
  F.mapComp f g ▷ F.map h ≫ F.mapComp (f ≫ g) h =
    F.mapComp f g ▷ F.map h ≫ F.mapComp (f ≫ g) h ≫ F.map₂ ((α_ f g h).hom ≫ (α_ f g h).inv)
⊢ F.mapComp f g ▷ F.map h ≫ F.mapComp (f ≫ g) h =
    (α_ (F.map f) (F.map g) (F.map h)).hom ≫ F.map f ◁ F.mapComp g h ≫ F.mapComp f (g ≫ h) ≫ F.map₂ (α_ f g h).inv

Tactic:
rw [← F.map₂_associator_assoc, ← F.map₂_comp]

Premises used: CategoryTheory.LaxFunctor.map₂_associator_assoc, CategoryTheory.LaxFunctor.toPrelaxFunctor, CategoryTheory.PrelaxFunctor.map₂_comp
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Lax
Goal:
B : Type u₁
inst : Bicategory.{w₁, v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : LaxFunctor.{w₁, w₂, v₁, v₂, u₁, u₂} B C
a b c d : B
f : a ⟶ b
g : b ⟶ c
h : c ⟶ d
⊢ F.mapComp f g ▷ F.map h ≫ F.mapComp (f ≫ g) h =
    F.mapComp f g ▷ F.map h ≫ F.mapComp (f ≫ g) h ≫ F.map₂ ((α_ f g h).hom ≫ (α_ f g h).inv)

Tactic:
simp only [Iso.hom_inv_id, PrelaxFunctor.map₂_id, comp_id]

Premises used: CategoryTheory.Iso.hom_inv_id, CategoryTheory.Category.comp_id, CategoryTheory.PrelaxFunctor.map₂_id
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Lax
Goal:
B : Type u₁
inst : Bicategory.{w₁, v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : LaxFunctor.{w₁, w₂, v₁, v₂, u₁, u₂} B C
a b c d : B
f : a ⟶ b
g : b ⟶ c
h : c ⟶ d
⊢ F.map f ◁ F.mapComp g h ≫ F.mapComp f (g ≫ h) =
    (α_ (F.map f) (F.map g) (F.map h)).inv ≫ F.mapComp f g ▷ F.map h ≫ F.mapComp (f ≫ g) h ≫ F.map₂ (α_ f g h).hom

Tactic:
simp only [map₂_associator, Iso.inv_hom_id_assoc]

Premises used: CategoryTheory.Iso.inv_hom_id_assoc, CategoryTheory.LaxFunctor.map₂_associator
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Lax
Goal:
B : Type u₁
inst : Bicategory.{w₁, v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : LaxFunctor.{w₁, w₂, v₁, v₂, u₁, u₂} B C
a b : B
f : a ⟶ b
rw : (F.map₂Iso (λ_ f)).inv = (λ_ (F.map f)).inv ≫ F.mapId a ▷ F.map f ≫ F.mapComp (𝟙 a) f
⊢ (λ_ (F.map f)).hom = F.mapId a ▷ F.map f ≫ F.mapComp (𝟙 a) f ≫ F.map₂ (λ_ f).hom

Tactic:
rw [← PrelaxFunctor.map₂Iso_hom, ← assoc, ← Iso.comp_inv_eq, ← Iso.eq_inv_comp]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Iso.comp_inv_eq, CategoryTheory.PrelaxFunctor.map₂Iso_hom, CategoryTheory.Iso.eq_inv_comp
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Lax
Goal:
B : Type u₁
inst : Bicategory.{w₁, v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : LaxFunctor.{w₁, w₂, v₁, v₂, u₁, u₂} B C
a b : B
f : a ⟶ b
⊢ (F.map₂Iso (λ_ f)).inv = (λ_ (F.map f)).inv ≫ F.mapId a ▷ F.map f ≫ F.mapComp (𝟙 a) f

Tactic:
simp only [Functor.mapIso_inv, PrelaxFunctor.mapFunctor_map, map₂_leftUnitor]

Premises used: CategoryTheory.PrelaxFunctor.mapFunctor_map, CategoryTheory.LaxFunctor.map₂_leftUnitor, CategoryTheory.Functor.mapIso_inv
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Lax
Goal:
B : Type u₁
inst : Bicategory.{w₁, v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : LaxFunctor.{w₁, w₂, v₁, v₂, u₁, u₂} B C
a b : B
f : a ⟶ b
rw : (F.map₂Iso (ρ_ f)).inv = (ρ_ (F.map f)).inv ≫ F.map f ◁ F.mapId b ≫ F.mapComp f (𝟙 b)
⊢ (ρ_ (F.map f)).hom = F.map f ◁ F.mapId b ≫ F.mapComp f (𝟙 b) ≫ F.map₂ (ρ_ f).hom

Tactic:
rw [← PrelaxFunctor.map₂Iso_hom, ← assoc, ← Iso.comp_inv_eq, ← Iso.eq_inv_comp]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Iso.comp_inv_eq, CategoryTheory.PrelaxFunctor.map₂Iso_hom, CategoryTheory.Iso.eq_inv_comp
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Lax
Goal:
B : Type u₁
inst : Bicategory.{w₁, v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : LaxFunctor.{w₁, w₂, v₁, v₂, u₁, u₂} B C
a b : B
f : a ⟶ b
⊢ (F.map₂Iso (ρ_ f)).inv = (ρ_ (F.map f)).inv ≫ F.map f ◁ F.mapId b ≫ F.mapComp f (𝟙 b)

Tactic:
simp only [Functor.mapIso_inv, PrelaxFunctor.mapFunctor_map, map₂_rightUnitor]

Premises used: CategoryTheory.PrelaxFunctor.mapFunctor_map, CategoryTheory.Functor.mapIso_inv, CategoryTheory.LaxFunctor.map₂_rightUnitor
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Lax
Goal:
B : Type u₁
inst : Bicategory.{w₁, v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : LaxFunctor.{w₁, w₂, v₁, v₂, u₁, u₂} B C
b : B
f : b ⟶ b
hf : f = 𝟙 b
rw rw₁ : Eq.{v₁ + 1} (α := b ⟶ b) (𝟙 b) (𝟙 b)
⊢ 𝟙 b = f

Tactic:
rw [hf]
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Lax
Goal:
B : Type u₁
inst : Bicategory.{w₁, v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : LaxFunctor.{w₁, w₂, v₁, v₂, u₁, u₂} B C
b : B
⊢ F.mapId' (𝟙 b) ⋯ = F.mapId b

Tactic:
simp [mapId']

Premises used: CategoryTheory.LaxFunctor.mapId'
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Lax
Goal:
B : Type u₁
inst : Bicategory.{w₁, v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : LaxFunctor.{w₁, w₂, v₁, v₂, u₁, u₂} B C
b₀ b₁ b₂ : B
f : b₀ ⟶ b₁
g : b₁ ⟶ b₂
fg : b₀ ⟶ b₂
h : f ≫ g = fg
rw rw₁ : fg = fg
⊢ f ≫ g = fg

Tactic:
rw [h]
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Lax
Goal:
B : Type u₁
inst : Bicategory.{w₁, v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F : LaxFunctor.{w₁, w₂, v₁, v₂, u₁, u₂} B C
b₀ b₁ b₂ : B
f : b₀ ⟶ b₁
g : b₁ ⟶ b₂
⊢ F.mapComp' f g (f ≫ g) ⋯ = F.mapComp f g

Tactic:
simp [mapComp']

Premises used: CategoryTheory.LaxFunctor.mapComp'
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Lax
Goal:
B : Type u₁
inst : Bicategory.{w₁, v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F_1 : LaxFunctor.{w₁, w₂, v₁, v₂, u₁, u₂} B C
D : Type u₃
inst_2 : Bicategory.{w₃, v₃, u₃} D
F : LaxFunctor.{w₁, w₂, v₁, v₂, u₁, u₂} B C
G : LaxFunctor.{w₂, w₃, v₂, v₃, u₂, u₃} C D
a b c : B
f f' : a ⟶ b
η : f ⟶ f'
g : b ⟶ c
dsimp :
  (G.mapComp (F.map f) (F.map g) ≫ G.map₂ (F.mapComp f g)) ≫ G.map₂ (F.map₂ (η ▷ g)) =
    G.map₂ (F.map₂ η) ▷ G.map (F.map g) ≫ G.mapComp (F.map f') (F.map g) ≫ G.map₂ (F.mapComp f' g)
⊢ (G.mapComp (F.map f) (F.map g) ≫ G.map₂ (F.mapComp f g)) ≫ (F.comp G.toPrelaxFunctor).map₂ (η ▷ g) =
    (F.comp G.toPrelaxFunctor).map₂ η ▷ (F.comp G.toPrelaxFunctor).map g ≫
      G.mapComp (F.map f') (F.map g) ≫ G.map₂ (F.mapComp f' g)

Tactic:
dsimp
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Lax
Goal:
B : Type u₁
inst : Bicategory.{w₁, v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F_1 : LaxFunctor.{w₁, w₂, v₁, v₂, u₁, u₂} B C
D : Type u₃
inst_2 : Bicategory.{w₃, v₃, u₃} D
F : LaxFunctor.{w₁, w₂, v₁, v₂, u₁, u₂} B C
G : LaxFunctor.{w₂, w₃, v₂, v₃, u₂, u₃} C D
a b c : B
f f' : a ⟶ b
η : f ⟶ f'
g : b ⟶ c
rw :
  G.mapComp (F.map f) (F.map g) ≫ G.map₂ (F.mapComp f g) ≫ G.map₂ (F.map₂ (η ▷ g)) =
    G.map₂ (F.map₂ η) ▷ G.map (F.map g) ≫ G.mapComp (F.map f') (F.map g) ≫ G.map₂ (F.mapComp f' g)
rw₁ :
  G.mapComp (F.map f) (F.map g) ≫ G.map₂ (F.mapComp f g ≫ F.map₂ (η ▷ g)) =
    G.map₂ (F.map₂ η) ▷ G.map (F.map g) ≫ G.mapComp (F.map f') (F.map g) ≫ G.map₂ (F.mapComp f' g)
rw₂ :
  G.mapComp (F.map f) (F.map g) ≫ G.map₂ (F.map₂ η ▷ F.map g ≫ F.mapComp f' g) =
    G.map₂ (F.map₂ η) ▷ G.map (F.map g) ≫ G.mapComp (F.map f') (F.map g) ≫ G.map₂ (F.mapComp f' g)
rw₃ :
  G.mapComp (F.map f) (F.map g) ≫ G.map₂ (F.map₂ η ▷ F.map g) ≫ G.map₂ (F.mapComp f' g) =
    G.map₂ (F.map₂ η) ▷ G.map (F.map g) ≫ G.mapComp (F.map f') (F.map g) ≫ G.map₂ (F.mapComp f' g)
rw₄ rw₅ :
  G.map₂ (F.map₂ η) ▷ G.map (F.map g) ≫ G.mapComp (F.map f') (F.map g) ≫ G.map₂ (F.mapComp f' g) =
    G.map₂ (F.map₂ η) ▷ G.map (F.map g) ≫ G.mapComp (F.map f') (F.map g) ≫ G.map₂ (F.mapComp f' g)
⊢ (G.mapComp (F.map f) (F.map g) ≫ G.map₂ (F.mapComp f g)) ≫ G.map₂ (F.map₂ (η ▷ g)) =
    G.map₂ (F.map₂ η) ▷ G.map (F.map g) ≫ G.mapComp (F.map f') (F.map g) ≫ G.map₂ (F.mapComp f' g)

Tactic:
rw [assoc, ← G.map₂_comp, mapComp_naturality_left, G.map₂_comp, mapComp_naturality_left_assoc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.LaxFunctor.toPrelaxFunctor, CategoryTheory.PrelaxFunctor.map₂_comp, CategoryTheory.LaxFunctor.mapComp_naturality_left, CategoryTheory.LaxFunctor.mapComp_naturality_left_assoc
-- Module: Mathlib.CategoryTheory.Bicategory.Functor.Lax
Goal:
B : Type u₁
inst : Bicategory.{w₁, v₁, u₁} B
C : Type u₂
inst_1 : Bicategory.{w₂, v₂, u₂} C
F_1 : LaxFunctor.{w₁, w₂, v₁, v₂, u₁, u₂} B C
D : Type u₃
inst_2 : Bicategory.{w₃, v₃, u₃} D
F : LaxFunctor.{w₁, w₂, v₁, v₂, u₁, u₂} B C
G : LaxFunctor.{w₂, w₃, v₂, v₃, u₂, u₃} C D
a b c : B
f : a ⟶ b
x x_1 : b ⟶ c
η : x ⟶ x_1
dsimp :
  (G.mapComp (F.map f) (F.map x) ≫ G.map₂ (F.mapComp f x)) ≫ G.map₂ (F.map₂ (f ◁ η)) =
    G.map (F.map f) ◁ G.map₂ (F.map₂ η) ≫ G.mapComp (F.map f) (F.map x_1) ≫ G.map₂ (F.mapComp f x_1)
⊢ (G.mapComp (F.map f) (F.map x) ≫ G.map₂ (F.mapComp f x)) ≫ (F.comp G.toPrelaxFunctor).map₂ (f ◁ η) =
    (F.comp G.toPrelaxFunctor).map f ◁ (F.comp G.toPrelaxFunctor).map₂ η ≫
      G.mapComp (F.map f) (F.map x_1) ≫ G.map₂ (F.mapComp f x_1)

Tactic:
dsimp