mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᵒᵖᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a b : B
f g : a ⟶ b
β : f ⟶ g
⊢ η.vCompNaturality θ f ≫ F.map₂ β ▷ η.vCompApp θ b =
    𝟙 ((η.app a ≫ θ.app a) ≫ H.map f) ⊗≫
      η.app a ◁ θ.naturality f ⊗≫ (η.naturality f ≫ F.map₂ β ▷ η.app b) ▷ θ.app b ⊗≫ 𝟙 (F.map g ≫ η.vCompApp θ b)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.Modification.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G : B ⥤ᵒᵖᴸ C
η θ : F ⟶ G
app : (a : B) → η.app a ≅ θ.app a
naturality : ∀ {a b : B} (f : a ⟶ b), (app a).hom ▷ G.map f ≫ θ.naturality f = η.naturality f ≫ F.map f ◁ (app b).hom
a b : B
f : a ⟶ b
⊢ (app a).inv ▷ G.map f ≫ η.naturality f = θ.naturality f ≫ F.map f ◁ (app b).inv

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.Modification.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G : B ⥤ᵒᵖᴸ C
η θ : F ⟶ G
Γ : Modification η θ
b c : B
a' : C
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

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Lax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a b : B
f g : a ⟶ b
β : f ⟶ g
⊢ 𝟙 ((η.app a ≫ θ.app a) ≫ H.map f) ⊗≫
      η.app a ◁ (θ.naturality f ≫ G.map₂ β ▷ θ.app b) ⊗≫ η.naturality g ▷ θ.app b ⊗≫ 𝟙 (F.map g ≫ η.vCompApp θ b) =
    η.vCompApp θ a ◁ H.map₂ β ≫ η.vCompNaturality θ g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.Modification.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G : B ⥤ᵒᵖᴸ C
η θ : F ⟶ G
Γ : Modification η θ
a b : B
a' : C
f : a ⟶ b
g : G.obj b ⟶ a'
⊢ F.map f ◁ Γ.app b ▷ g ≫ (α_ (F.map f) (θ.app b) g).inv ≫ θ.naturality f ▷ g =
    (α_ (F.map f) (η.app b) g).inv ≫ η.naturality f ▷ g ≫ Γ.app a ▷ G.map f ▷ g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.Modification.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G : B ⥤ᵒᵖᴸ C
η θ : F ⟶ G
app : (a : B) → η.app a ≅ θ.app a
naturality : ∀ {a b : B} (f : a ⟶ b), F.map f ◁ (app b).hom ≫ θ.naturality f = η.naturality f ≫ (app a).hom ▷ G.map f
a b : B
f : a ⟶ b
⊢ F.map f ◁ (app b).inv ≫ η.naturality f = θ.naturality f ≫ (app a).inv ▷ G.map f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.Modification.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G : B ⥤ᵒᵖᴸ C
η θ : F ⟶ G
Γ✝ : Modification η θ
Γ : (StrongTrans.toOplax η).Modification (StrongTrans.toOplax θ)
a✝ b✝ : B
f : a✝ ⟶ b✝
⊢ F.map f ◁ Γ.app b✝ ≫ (θ.naturality f).hom = (η.naturality f).hom ≫ Γ.app a✝ ▷ G.map f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.Modification.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G : B ⥤ᵒᵖᴸ C
η θ : F ⟶ G
app : (a : B) → η.app a ≅ θ.app a
naturality :
  ∀ {a b : B} (f : a ⟶ b), F.map f ◁ (app b).hom ≫ (θ.naturality f).hom = (η.naturality f).hom ≫ (app a).hom ▷ G.map f
a b : B
f : a ⟶ b
⊢ F.map f ◁ (app b).inv ≫ (η.naturality f).hom = (θ.naturality f).hom ≫ (app a).inv ▷ G.map f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᵒᵖᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a b : B
f g : a ⟶ b
β : f ⟶ g
⊢ 𝟙 ((η.app a ≫ θ.app a) ≫ H.map f) ⊗≫
      η.app a ◁ θ.naturality f ⊗≫ (η.naturality f ≫ F.map₂ β ▷ η.app b) ▷ θ.app b ⊗≫ 𝟙 (F.map g ≫ η.vCompApp θ b) =
    𝟙 ((η.app a ≫ θ.app a) ≫ H.map f) ⊗≫
      η.app a ◁ (θ.naturality f ≫ G.map₂ β ▷ θ.app b) ⊗≫ η.naturality g ▷ θ.app b ⊗≫ 𝟙 (F.map g ≫ η.vCompApp θ b)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Lax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a : B
⊢ η.vCompApp θ a ◁ H.mapId a ≫ η.vCompNaturality θ (𝟙 a) =
    𝟙 (η.vCompApp θ a ≫ 𝟙 (H.obj a)) ⊗≫
      η.app a ◁ (θ.app a ◁ H.mapId a ≫ θ.naturality (𝟙 a)) ⊗≫
        η.naturality (𝟙 a) ▷ θ.app a ⊗≫ 𝟙 (F.map (𝟙 a) ≫ η.app a ≫ θ.app a)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᵒᵖᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a b : B
f g : a ⟶ b
β : f ⟶ g
⊢ 𝟙 ((η.app a ≫ θ.app a) ≫ H.map f) ⊗≫
      η.app a ◁ (θ.naturality f ≫ G.map₂ β ▷ θ.app b) ⊗≫ η.naturality g ▷ θ.app b ⊗≫ 𝟙 (F.map g ≫ η.vCompApp θ b) =
    η.vCompApp θ a ◁ H.map₂ β ≫ η.vCompNaturality θ g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Lax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a : B
⊢ 𝟙 (η.vCompApp θ a ≫ 𝟙 (H.obj a)) ⊗≫
      η.app a ◁ (θ.app a ◁ H.mapId a ≫ θ.naturality (𝟙 a)) ⊗≫
        η.naturality (𝟙 a) ▷ θ.app a ⊗≫ 𝟙 (F.map (𝟙 a) ≫ η.app a ≫ θ.app a) =
    𝟙 (η.vCompApp θ a ≫ 𝟙 (H.obj a)) ⊗≫
      (η.app a ◁ G.mapId a ≫ η.naturality (𝟙 a)) ▷ θ.app a ⊗≫ 𝟙 (F.map (𝟙 a) ≫ η.app a ≫ θ.app a)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᵒᵖᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a : B
⊢ η.vCompNaturality θ (𝟙 a) ≫ F.mapId a ▷ η.vCompApp θ a =
    𝟙 ((η.app a ≫ θ.app a) ≫ H.map (𝟙 a)) ⊗≫
      η.app a ◁ θ.naturality (𝟙 a) ⊗≫
        (η.naturality (𝟙 a) ≫ F.mapId a ▷ η.app a) ▷ θ.app a ⊗≫ 𝟙 (𝟙 (F.obj a) ≫ η.vCompApp θ a)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Lax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a : B
⊢ 𝟙 (η.vCompApp θ a ≫ 𝟙 (H.obj a)) ⊗≫
      (η.app a ◁ G.mapId a ≫ η.naturality (𝟙 a)) ▷ θ.app a ⊗≫ 𝟙 (F.map (𝟙 a) ≫ η.app a ≫ θ.app a) =
    (ρ_ (η.vCompApp θ a)).hom ≫ (λ_ (η.vCompApp θ a)).inv ≫ F.mapId a ▷ η.vCompApp θ a

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Lax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a b c : B
f : a ⟶ b
g : b ⟶ c
⊢ η.vCompApp θ a ◁ H.mapComp f g ≫ η.vCompNaturality θ (f ≫ g) =
    𝟙 (η.vCompApp θ a ≫ H.map f ≫ H.map g) ⊗≫
      η.app a ◁ (θ.app a ◁ H.mapComp f g ≫ θ.naturality (f ≫ g)) ⊗≫
        η.naturality (f ≫ g) ▷ θ.app c ⊗≫ 𝟙 (F.map (f ≫ g) ≫ η.app c ≫ θ.app c)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᵒᵖᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a : B
⊢ 𝟙 ((η.app a ≫ θ.app a) ≫ H.map (𝟙 a)) ⊗≫
      η.app a ◁ θ.naturality (𝟙 a) ⊗≫
        (η.naturality (𝟙 a) ≫ F.mapId a ▷ η.app a) ▷ θ.app a ⊗≫ 𝟙 (𝟙 (F.obj a) ≫ η.vCompApp θ a) =
    𝟙 ((η.app a ≫ θ.app a) ≫ H.map (𝟙 a)) ⊗≫
      η.app a ◁ (θ.naturality (𝟙 a) ≫ G.mapId a ▷ θ.app a) ⊗≫ 𝟙 (𝟙 (F.obj a) ≫ η.vCompApp θ a)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᵒᵖᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a : B
⊢ 𝟙 ((η.app a ≫ θ.app a) ≫ H.map (𝟙 a)) ⊗≫
      η.app a ◁ (θ.naturality (𝟙 a) ≫ G.mapId a ▷ θ.app a) ⊗≫ 𝟙 (𝟙 (F.obj a) ≫ η.vCompApp θ a) =
    η.vCompApp θ a ◁ H.mapId a ≫ (ρ_ (η.vCompApp θ a)).hom ≫ (λ_ (η.vCompApp θ a)).inv

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᵒᵖᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a : B
⊢ η.vCompNaturality θ (𝟙 a) ≫ F.mapId a ▷ η.vCompApp θ a =
    η.vCompApp θ a ◁ H.mapId a ≫ (ρ_ (η.vCompApp θ a)).hom ≫ (λ_ (η.vCompApp θ a)).inv

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Lax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a b c : B
f : a ⟶ b
g : b ⟶ c
⊢ 𝟙 (η.vCompApp θ a ≫ H.map f ≫ H.map g) ⊗≫
      η.app a ◁ (θ.app a ◁ H.mapComp f g ≫ θ.naturality (f ≫ g)) ⊗≫
        η.naturality (f ≫ g) ▷ θ.app c ⊗≫ 𝟙 (F.map (f ≫ g) ≫ η.app c ≫ θ.app c) =
    𝟙 (η.vCompApp θ a ≫ H.map f ≫ H.map g) ⊗≫
      η.app a ◁ (θ.naturality f ▷ H.map g ⊗≫ G.map f ◁ θ.naturality g) ⊗≫
        (η.app a ◁ G.mapComp f g ≫ η.naturality (f ≫ g)) ▷ θ.app c ⊗≫ 𝟙 (F.map (f ≫ g) ≫ η.app c ≫ θ.app c)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᵒᵖᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a b c : B
f : a ⟶ b
g : b ⟶ c
⊢ η.vCompNaturality θ (f ≫ g) ≫ F.mapComp f g ▷ η.vCompApp θ c =
    𝟙 ((η.app a ≫ θ.app a) ≫ H.map (f ≫ g)) ⊗≫
      η.app a ◁ θ.naturality (f ≫ g) ⊗≫
        (η.naturality (f ≫ g) ≫ F.mapComp f g ▷ η.app c) ▷ θ.app c ⊗≫ 𝟙 ((F.map f ≫ F.map g) ≫ η.vCompApp θ c)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.FunctorBicategory.Lax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H I : B ⥤ᴸ C
η : F ⟶ G
θ ι : G ⟶ H
Γ : θ ⟶ ι
a b : B
f : a ⟶ b
⊢ (η.app a ◁ Γ.as.app a) ▷ H.map f ≫
      (α_ (η.app a) (ι.app a) (H.map f)).hom ≫
        η.app a ◁ ι.naturality f ≫
          (α_ (η.app a) (G.map f) (ι.app b)).inv ≫ η.naturality f ▷ ι.app b ≫ (α_ (F.map f) (η.app b) (ι.app b)).hom =
    𝟙 ((η.app a ≫ θ.app a) ≫ H.map f) ⊗≫
      η.app a ◁ (Γ.as.app a ▷ H.map f ≫ ι.naturality f) ⊗≫ η.naturality f ▷ ι.app b ⊗≫ 𝟙 (F.map f ≫ η.app b ≫ ι.app b)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Lax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a b c : B
f : a ⟶ b
g : b ⟶ c
⊢ 𝟙 (η.vCompApp θ a ≫ H.map f ≫ H.map g) ⊗≫
      η.app a ◁ (θ.naturality f ▷ H.map g ⊗≫ G.map f ◁ θ.naturality g) ⊗≫
        (η.app a ◁ G.mapComp f g ≫ η.naturality (f ≫ g)) ▷ θ.app c ⊗≫ 𝟙 (F.map (f ≫ g) ≫ η.app c ≫ θ.app c) =
    𝟙 (η.vCompApp θ a ≫ H.map f ≫ H.map g) ⊗≫
      η.app a ◁ θ.naturality f ▷ H.map g ⊗≫
        ((η.app a ≫ G.map f) ◁ θ.naturality g ≫ η.naturality f ▷ (G.map g ≫ θ.app c)) ⊗≫
          F.map f ◁ η.naturality g ▷ θ.app c ⊗≫
            F.mapComp f g ▷ η.app c ▷ θ.app c ⊗≫ 𝟙 (F.map (f ≫ g) ≫ η.app c ≫ θ.app c)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.FunctorBicategory.Lax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H I : B ⥤ᴸ C
η : F ⟶ G
θ ι : G ⟶ H
Γ : θ ⟶ ι
a b : B
f : a ⟶ b
⊢ 𝟙 ((η.app a ≫ θ.app a) ≫ H.map f) ⊗≫
      η.app a ◁ (Γ.as.app a ▷ H.map f ≫ ι.naturality f) ⊗≫ η.naturality f ▷ ι.app b ⊗≫ 𝟙 (F.map f ≫ η.app b ≫ ι.app b) =
    𝟙 ((η.app a ≫ θ.app a) ≫ H.map f) ⊗≫
      η.app a ◁ θ.naturality f ⊗≫
        ((η.app a ≫ G.map f) ◁ Γ.as.app b ≫ η.naturality f ▷ ι.app b) ⊗≫ 𝟙 (F.map f ≫ η.app b ≫ ι.app b)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᵒᵖᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a b c : B
f : a ⟶ b
g : b ⟶ c
⊢ 𝟙 ((η.app a ≫ θ.app a) ≫ H.map (f ≫ g)) ⊗≫
      η.app a ◁ θ.naturality (f ≫ g) ⊗≫
        (η.naturality (f ≫ g) ≫ F.mapComp f g ▷ η.app c) ▷ θ.app c ⊗≫ 𝟙 ((F.map f ≫ F.map g) ≫ η.vCompApp θ c) =
    𝟙 ((η.app a ≫ θ.app a) ≫ H.map (f ≫ g)) ⊗≫
      η.app a ◁ (θ.naturality (f ≫ g) ≫ G.mapComp f g ▷ θ.app c) ⊗≫
        (η.naturality f ▷ G.map g ⊗≫ F.map f ◁ η.naturality g) ▷ θ.app c ⊗≫ 𝟙 ((F.map f ≫ F.map g) ≫ η.vCompApp θ c)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.FunctorBicategory.Lax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H I : B ⥤ᴸ C
η : F ⟶ G
θ ι : G ⟶ H
Γ : θ ⟶ ι
a b : B
f : a ⟶ b
⊢ 𝟙 ((η.app a ≫ θ.app a) ≫ H.map f) ⊗≫
      η.app a ◁ θ.naturality f ⊗≫
        ((η.app a ≫ G.map f) ◁ Γ.as.app b ≫ η.naturality f ▷ ι.app b) ⊗≫ 𝟙 (F.map f ≫ η.app b ≫ ι.app b) =
    ((α_ (η.app a) (θ.app a) (H.map f)).hom ≫
        η.app a ◁ θ.naturality f ≫
          (α_ (η.app a) (G.map f) (θ.app b)).inv ≫ η.naturality f ▷ θ.app b ≫ (α_ (F.map f) (η.app b) (θ.app b)).hom) ≫
      F.map f ◁ η.app b ◁ Γ.as.app b

Tactic:
whisker_exchange

Goal state after:
B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H I : B ⥤ᴸ C
η : F ⟶ G
θ ι : G ⟶ H
Γ : θ ⟶ ι
a b : B
f : a ⟶ b
⊢ 𝟙 ((η.app a ≫ θ.app a) ≫ H.map f) ⊗≫
      η.app a ◁ θ.naturality f ⊗≫
        (η.naturality f ▷ θ.app b ≫ (F.map f ≫ η.app b) ◁ Γ.as.app b) ⊗≫ 𝟙 (F.map f ≫ η.app b ≫ ι.app b) =
    ((α_ (η.app a) (θ.app a) (H.map f)).hom ≫
        η.app a ◁ θ.naturality f ≫
          (α_ (η.app a) (G.map f) (θ.app b)).inv ≫ η.naturality f ▷ θ.app b ≫ (α_ (F.map f) (η.app b) (θ.app b)).hom) ≫
      F.map f ◁ η.app b ◁ Γ.as.app b
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Lax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a b c : B
f : a ⟶ b
g : b ⟶ c
⊢ 𝟙 (η.vCompApp θ a ≫ H.map f ≫ H.map g) ⊗≫
      η.app a ◁ θ.naturality f ▷ H.map g ⊗≫
        ((η.app a ≫ G.map f) ◁ θ.naturality g ≫ η.naturality f ▷ (G.map g ≫ θ.app c)) ⊗≫
          F.map f ◁ η.naturality g ▷ θ.app c ⊗≫
            F.mapComp f g ▷ η.app c ▷ θ.app c ⊗≫ 𝟙 (F.map (f ≫ g) ≫ η.app c ≫ θ.app c) =
    (α_ (η.vCompApp θ a) (H.map f) (H.map g)).inv ≫
      η.vCompNaturality θ f ▷ H.map g ≫
        (α_ (F.map f) (η.vCompApp θ b) (H.map g)).hom ≫
          F.map f ◁ η.vCompNaturality θ g ≫
            (α_ (F.map f) (F.map g) (η.vCompApp θ c)).inv ≫ F.mapComp f g ▷ η.vCompApp θ c

Tactic:
whisker_exchange

Goal state after:
B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a b c : B
f : a ⟶ b
g : b ⟶ c
⊢ 𝟙 (η.vCompApp θ a ≫ H.map f ≫ H.map g) ⊗≫
      η.app a ◁ θ.naturality f ▷ H.map g ⊗≫
        (η.naturality f ▷ (θ.app b ≫ H.map g) ≫ (F.map f ≫ η.app b) ◁ θ.naturality g) ⊗≫
          F.map f ◁ η.naturality g ▷ θ.app c ⊗≫
            F.mapComp f g ▷ η.app c ▷ θ.app c ⊗≫ 𝟙 (F.map (f ≫ g) ≫ η.app c ≫ θ.app c) =
    (α_ (η.vCompApp θ a) (H.map f) (H.map g)).inv ≫
      η.vCompNaturality θ f ▷ H.map g ≫
        (α_ (F.map f) (η.vCompApp θ b) (H.map g)).hom ≫
          F.map f ◁ η.vCompNaturality θ g ≫
            (α_ (F.map f) (F.map g) (η.vCompApp θ c)).inv ≫ F.mapComp f g ▷ η.vCompApp θ c
Module: Mathlib.CategoryTheory.Bicategory.FunctorBicategory.Lax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H I : B ⥤ᴸ C
η : F ⟶ G
θ ι : G ⟶ H
Γ : θ ⟶ ι
a b : B
f : a ⟶ b
⊢ 𝟙 ((η.app a ≫ θ.app a) ≫ H.map f) ⊗≫
      η.app a ◁ θ.naturality f ⊗≫
        ((η.app a ≫ G.map f) ◁ Γ.as.app b ≫ η.naturality f ▷ ι.app b) ⊗≫ 𝟙 (F.map f ≫ η.app b ≫ ι.app b) =
    ((α_ (η.app a) (θ.app a) (H.map f)).hom ≫
        η.app a ◁ θ.naturality f ≫
          (α_ (η.app a) (G.map f) (θ.app b)).inv ≫ η.naturality f ▷ θ.app b ≫ (α_ (F.map f) (η.app b) (θ.app b)).hom) ≫
      F.map f ◁ η.app b ◁ Γ.as.app b

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.FunctorBicategory.Lax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H I : B ⥤ᴸ C
η : F ⟶ G
θ ι : G ⟶ H
Γ : θ ⟶ ι
a b : B
f : a ⟶ b
⊢ (η.app a ◁ Γ.as.app a) ▷ H.map f ≫ (η ≫ ι).naturality f = (η ≫ θ).naturality f ≫ F.map f ◁ η.app b ◁ Γ.as.app b

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᵒᵖᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a b c : B
f : a ⟶ b
g : b ⟶ c
⊢ 𝟙 ((η.app a ≫ θ.app a) ≫ H.map (f ≫ g)) ⊗≫
      η.app a ◁ (θ.naturality (f ≫ g) ≫ G.mapComp f g ▷ θ.app c) ⊗≫
        (η.naturality f ▷ G.map g ⊗≫ F.map f ◁ η.naturality g) ▷ θ.app c ⊗≫ 𝟙 ((F.map f ≫ F.map g) ≫ η.vCompApp θ c) =
    𝟙 ((η.app a ≫ θ.app a) ≫ H.map (f ≫ g)) ⊗≫
      η.app a ◁ (θ.app a ◁ H.mapComp f g ⊗≫ θ.naturality f ▷ H.map g) ⊗≫
        ((η.app a ≫ G.map f) ◁ θ.naturality g ≫ η.naturality f ▷ (G.map g ≫ θ.app c)) ⊗≫
          F.map f ◁ η.naturality g ▷ θ.app c ⊗≫ 𝟙 ((F.map f ≫ F.map g) ≫ η.vCompApp θ c)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᵒᵖᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a b c : B
f : a ⟶ b
g : b ⟶ c
⊢ 𝟙 ((η.app a ≫ θ.app a) ≫ H.map (f ≫ g)) ⊗≫
      η.app a ◁ (θ.app a ◁ H.mapComp f g ⊗≫ θ.naturality f ▷ H.map g) ⊗≫
        ((η.app a ≫ G.map f) ◁ θ.naturality g ≫ η.naturality f ▷ (G.map g ≫ θ.app c)) ⊗≫
          F.map f ◁ η.naturality g ▷ θ.app c ⊗≫ 𝟙 ((F.map f ≫ F.map g) ≫ η.vCompApp θ c) =
    η.vCompApp θ a ◁ H.mapComp f g ≫
      (α_ (η.vCompApp θ a) (H.map f) (H.map g)).inv ≫
        η.vCompNaturality θ f ▷ H.map g ≫
          (α_ (F.map f) (η.vCompApp θ b) (H.map g)).hom ≫
            F.map f ◁ η.vCompNaturality θ g ≫ (α_ (F.map f) (F.map g) (η.vCompApp θ c)).inv

Tactic:
whisker_exchange

Goal state after:
B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᵒᵖᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a b c : B
f : a ⟶ b
g : b ⟶ c
⊢ 𝟙 ((η.app a ≫ θ.app a) ≫ H.map (f ≫ g)) ⊗≫
      η.app a ◁ (θ.app a ◁ H.mapComp f g ⊗≫ θ.naturality f ▷ H.map g) ⊗≫
        (η.naturality f ▷ (θ.app b ≫ H.map g) ≫ (F.map f ≫ η.app b) ◁ θ.naturality g) ⊗≫
          F.map f ◁ η.naturality g ▷ θ.app c ⊗≫ 𝟙 ((F.map f ≫ F.map g) ≫ η.vCompApp θ c) =
    η.vCompApp θ a ◁ H.mapComp f g ≫
      (α_ (η.vCompApp θ a) (H.map f) (H.map g)).inv ≫
        η.vCompNaturality θ f ▷ H.map g ≫
          (α_ (F.map f) (η.vCompApp θ b) (H.map g)).hom ≫
            F.map f ◁ η.vCompNaturality θ g ≫ (α_ (F.map f) (F.map g) (η.vCompApp θ c)).inv
Module: Mathlib.CategoryTheory.Bicategory.FunctorBicategory.Lax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H I : B ⥤ᴸ C
η θ : F ⟶ G
Γ : η ⟶ θ
ι : G ⟶ H
a b : B
f : a ⟶ b
⊢ Γ.as.app a ▷ ι.app a ▷ H.map f ≫
      (α_ (θ.app a) (ι.app a) (H.map f)).hom ≫
        θ.app a ◁ ι.naturality f ≫
          (α_ (θ.app a) (G.map f) (ι.app b)).inv ≫ θ.naturality f ▷ ι.app b ≫ (α_ (F.map f) (θ.app b) (ι.app b)).hom =
    𝟙 ((η.app a ≫ ι.app a) ≫ H.map f) ⊗≫
      (Γ.as.app a ▷ (ι.app a ≫ H.map f) ≫ θ.app a ◁ ι.naturality f) ⊗≫
        θ.naturality f ▷ ι.app b ⊗≫ 𝟙 (F.map f ≫ θ.app b ≫ ι.app b)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Lax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a b c : B
f : a ⟶ b
g : b ⟶ c
⊢ 𝟙 (η.vCompApp θ a ≫ H.map f ≫ H.map g) ⊗≫
      η.app a ◁ θ.naturality f ▷ H.map g ⊗≫
        ((η.app a ≫ G.map f) ◁ θ.naturality g ≫ η.naturality f ▷ (G.map g ≫ θ.app c)) ⊗≫
          F.map f ◁ η.naturality g ▷ θ.app c ⊗≫
            F.mapComp f g ▷ η.app c ▷ θ.app c ⊗≫ 𝟙 (F.map (f ≫ g) ≫ η.app c ≫ θ.app c) =
    (α_ (η.vCompApp θ a) (H.map f) (H.map g)).inv ≫
      η.vCompNaturality θ f ▷ H.map g ≫
        (α_ (F.map f) (η.vCompApp θ b) (H.map g)).hom ≫
          F.map f ◁ η.vCompNaturality θ g ≫
            (α_ (F.map f) (F.map g) (η.vCompApp θ c)).inv ≫ F.mapComp f g ▷ η.vCompApp θ c

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.FunctorBicategory.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H I : B ⥤ᵒᵖᴸ C
η : F ⟶ G
θ ι : G ⟶ H
Γ : θ ⟶ ι
a b : B
f : a ⟶ b
⊢ (η.app a ◁ Γ.as.app a) ▷ H.map f ≫
      (α_ (η.app a) (ι.app a) (H.map f)).hom ≫
        η.app a ◁ ι.naturality f ≫
          (α_ (η.app a) (G.map f) (ι.app b)).inv ≫ η.naturality f ▷ ι.app b ≫ (α_ (F.map f) (η.app b) (ι.app b)).hom =
    𝟙 ((η.app a ≫ θ.app a) ≫ H.map f) ⊗≫
      η.app a ◁ (Γ.as.app a ▷ H.map f ≫ ι.naturality f) ⊗≫ η.naturality f ▷ ι.app b ⊗≫ 𝟙 (F.map f ≫ η.app b ≫ ι.app b)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᵒᵖᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a b c : B
f : a ⟶ b
g : b ⟶ c
⊢ 𝟙 ((η.app a ≫ θ.app a) ≫ H.map (f ≫ g)) ⊗≫
      η.app a ◁ (θ.app a ◁ H.mapComp f g ⊗≫ θ.naturality f ▷ H.map g) ⊗≫
        ((η.app a ≫ G.map f) ◁ θ.naturality g ≫ η.naturality f ▷ (G.map g ≫ θ.app c)) ⊗≫
          F.map f ◁ η.naturality g ▷ θ.app c ⊗≫ 𝟙 ((F.map f ≫ F.map g) ≫ η.vCompApp θ c) =
    η.vCompApp θ a ◁ H.mapComp f g ≫
      (α_ (η.vCompApp θ a) (H.map f) (H.map g)).inv ≫
        η.vCompNaturality θ f ▷ H.map g ≫
          (α_ (F.map f) (η.vCompApp θ b) (H.map g)).hom ≫
            F.map f ◁ η.vCompNaturality θ g ≫ (α_ (F.map f) (F.map g) (η.vCompApp θ c)).inv

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᵒᵖᴸ C
η : LaxTrans F G
θ : LaxTrans G H
a b c : B
f : a ⟶ b
g : b ⟶ c
⊢ η.vCompNaturality θ (f ≫ g) ≫ F.mapComp f g ▷ η.vCompApp θ c =
    η.vCompApp θ a ◁ H.mapComp f g ≫
      (α_ (η.vCompApp θ a) (H.map f) (H.map g)).inv ≫
        η.vCompNaturality θ f ▷ H.map g ≫
          (α_ (F.map f) (η.vCompApp θ b) (H.map g)).hom ≫
            F.map f ◁ η.vCompNaturality θ g ≫ (α_ (F.map f) (F.map g) (η.vCompApp θ c)).inv

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
G H : B ⥤ᵒᵖᴸ C
θ : OplaxTrans G H
a b : B
a' : C
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

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.FunctorBicategory.Lax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H I : B ⥤ᴸ C
η θ : F ⟶ G
Γ : η ⟶ θ
ι : G ⟶ H
a b : B
f : a ⟶ b
⊢ 𝟙 ((η.app a ≫ ι.app a) ≫ H.map f) ⊗≫
      (Γ.as.app a ▷ (ι.app a ≫ H.map f) ≫ θ.app a ◁ ι.naturality f) ⊗≫
        θ.naturality f ▷ ι.app b ⊗≫ 𝟙 (F.map f ≫ θ.app b ≫ ι.app b) =
    𝟙 ((η.app a ≫ ι.app a) ≫ H.map f) ⊗≫
      (η.app a ◁ ι.naturality f ⊗≫ (Γ.as.app a ▷ G.map f ≫ θ.naturality f) ▷ ι.app b) ⊗≫ 𝟙 (F.map f ≫ θ.app b ≫ ι.app b)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G : B ⥤ᵒᵖᴸ C
η : OplaxTrans F G
a b : B
a' : C
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_ (η.app a) (G.map f) h).hom ≫ η.app a ◁ G.map₂ β ▷ h ≫ (α_ (η.app a) (G.map g) h).inv

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Lax

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F G H : B ⥤ᴸ C
η : OplaxTrans F G
θ : OplaxTrans G H
a b : B
f g : a ⟶ b
β : f ⟶ g
⊢ F.map₂ β ▷ η.vCompApp θ b ≫ η.vCompNaturality θ g =
    𝟙 (F.map f ≫ η.vCompApp θ b) ⊗≫
      (F.map₂ β ▷ η.app b ≫ η.naturality g) ▷ θ.app b ⊗≫ η.app a ◁ θ.naturality g ⊗≫ 𝟙 ((η.app a ≫ θ.app a) ≫ H.map g)

Tactic:
by

Goal state after: