-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Adjunction.Mate (goal)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c d e f : B
g₁ : a ⟶ c
g₂ : c ⟶ e
h₁ : b ⟶ d
h₂ : d ⟶ f
l₁ : a ⟶ b
r₁ : b ⟶ a
l₂ : c ⟶ d
r₂ : d ⟶ c
l₃ : e ⟶ f
r₃ : f ⟶ e
adj₁ : l₁ ⊣ r₁
adj₂ : l₂ ⊣ r₂
adj₃ : l₃ ⊣ r₃
α : g₁ ≫ l₂ ⟶ l₁ ≫ h₁
β : g₂ ≫ l₃ ⟶ l₂ ≫ h₂
_ :
  (α_ r₁ g₁ g₂).inv ≫
      (𝟙 (r₁ ≫ g₁) ⊗≫ r₁ ◁ g₁ ◁ adj₂.unit ⊗≫ r₁ ◁ α ▷ r₂ ⊗≫ adj₁.counit ▷ h₁ ▷ r₂ ⊗≫ 𝟙 (h₁ ≫ r₂)) ▷ g₂ ≫
        (α_ h₁ r₂ g₂).hom ≫
          h₁ ◁ (𝟙 (r₂ ≫ g₂) ⊗≫ r₂ ◁ g₂ ◁ adj₃.unit ⊗≫ r₂ ◁ β ▷ r₃ ⊗≫ adj₂.counit ▷ h₂ ▷ r₃ ⊗≫ 𝟙 (h₂ ≫ r₃)) ≫
            (α_ h₁ h₂ r₃).inv =
    𝟙 (r₁ ≫ g₁ ≫ g₂) ⊗≫
      r₁ ◁ g₁ ◁ adj₂.unit ▷ g₂ ⊗≫
        r₁ ◁ α ▷ r₂ ▷ g₂ ⊗≫
          (adj₁.counit ▷ (h₁ ≫ r₂ ≫ g₂ ≫ 𝟙 e) ≫ 𝟙 b ◁ h₁ ◁ r₂ ◁ g₂ ◁ adj₃.unit) ⊗≫
            h₁ ◁ r₂ ◁ β ▷ r₃ ⊗≫ h₁ ◁ adj₂.counit ▷ h₂ ▷ r₃ ⊗≫ 𝟙 ((h₁ ≫ h₂) ≫ r₃)
⊢ (α_ r₁ g₁ g₂).inv ≫
      (𝟙 (r₁ ≫ g₁) ⊗≫ r₁ ◁ g₁ ◁ adj₂.unit ⊗≫ r₁ ◁ α ▷ r₂ ⊗≫ adj₁.counit ▷ h₁ ▷ r₂ ⊗≫ 𝟙 (h₁ ≫ r₂)) ▷ g₂ ≫
        (α_ h₁ r₂ g₂).hom ≫
          h₁ ◁ (𝟙 (r₂ ≫ g₂) ⊗≫ r₂ ◁ g₂ ◁ adj₃.unit ⊗≫ r₂ ◁ β ▷ r₃ ⊗≫ adj₂.counit ▷ h₂ ▷ r₃ ⊗≫ 𝟙 (h₂ ≫ r₃)) ≫
            (α_ h₁ h₂ r₃).inv =
    𝟙 (r₁ ≫ g₁ ≫ g₂) ⊗≫
      r₁ ◁ (g₁ ≫ g₂) ◁ adj₃.unit ⊗≫
        r₁ ◁ ((α_ g₁ g₂ l₃).hom ≫ g₁ ◁ β ≫ (α_ g₁ l₂ h₂).inv ≫ α ▷ h₂ ≫ (α_ l₁ h₁ h₂).hom) ▷ r₃ ⊗≫
          adj₁.counit ▷ (h₁ ≫ h₂) ▷ r₃ ⊗≫ 𝟙 ((h₁ ≫ h₂) ≫ r₃)
have _₁ : 𝟙 (r₁ ≫ g₁ ≫ g₂) ⊗≫
    r₁ ◁ g₁ ◁ adj₂.unit ▷ g₂ ⊗≫
      r₁ ◁ α ▷ r₂ ▷ g₂ ⊗≫
        (adj₁.counit ▷ (h₁ ≫ r₂ ≫ g₂ ≫ 𝟙 e) ≫ 𝟙 b ◁ h₁ ◁ r₂ ◁ g₂ ◁ adj₃.unit) ⊗≫
          h₁ ◁ r₂ ◁ β ▷ r₃ ⊗≫ h₁ ◁ adj₂.counit ▷ h₂ ▷ r₃ ⊗≫ 𝟙 ((h₁ ≫ h₂) ≫ r₃) =
  𝟙 (r₁ ≫ g₁ ≫ g₂) ⊗≫
    r₁ ◁ g₁ ◁ adj₂.unit ▷ g₂ ⊗≫
      r₁ ◁ (α ▷ (r₂ ≫ g₂ ≫ 𝟙 e) ≫ (l₁ ≫ h₁) ◁ r₂ ◁ g₂ ◁ adj₃.unit) ⊗≫
        (adj₁.counit ▷ (h₁ ≫ r₂) ▷ (g₂ ≫ l₃) ≫ (𝟙 b ≫ h₁ ≫ r₂) ◁ β) ▷ r₃ ⊗≫
          h₁ ◁ adj₂.counit ▷ h₂ ▷ r₃ ⊗≫ 𝟙 ((h₁ ≫ h₂) ≫ r₃)
Tactic: calc
    _ = 𝟙 _ ⊗≫ r₁ ◁ g₁ ◁ adj₂.unit ▷ g₂ ⊗≫ r₁ ◁ α ▷ r₂ ▷ g₂ ⊗≫
          ((adj₁.counit ▷ (h₁ ≫ r₂ ≫ g₂ ≫ 𝟙 e)) ≫ 𝟙 b ◁ (h₁ ◁ r₂ ◁ g₂ ◁ adj₃.unit)) ⊗≫
            h₁ ◁ r₂ ◁ β ▷ r₃ ⊗≫ h₁ ◁ adj₂.counit ▷ h₂ ▷ r₃ ⊗≫ 𝟙 _ := by
      bicategory
    _ = 𝟙 _ ⊗≫ r₁ ◁ g₁ ◁ adj₂.unit ▷ g₂ ⊗≫
          (r₁ ◁ (α ▷ (r₂ ≫ g₂ ≫ 𝟙 e) ≫ (l₁ ≫ h₁) ◁ r₂ ◁ g₂ ◁ adj₃.unit)) ⊗≫
            ((adj₁.counit ▷ (h₁ ≫ r₂) ▷ (g₂ ≫ l₃) ≫ (𝟙 b ≫ h₁ ≫ r₂) ◁ β) ▷ r₃) ⊗≫
              h₁ ◁ adj₂.counit ▷ h₂ ▷ r₃ ⊗≫ 𝟙 _ := by
      rw [← whisker_exchange]
      bicategory
    _ = 𝟙 _ ⊗≫ r₁ ◁ g₁ ◁ (adj₂.unit ▷ (g₂ ≫ 𝟙 e) ≫ (l₂ ≫ r₂) ◁ g₂ ◁ adj₃.unit) ⊗≫
          (r₁ ◁ (α ▷ (r₂ ≫ g₂ ≫ l₃) ≫ (l₁ ≫ h₁) ◁ r₂ ◁ β) ▷ r₃) ⊗≫
            (adj₁.counit ▷ h₁ ▷ (r₂ ≫ l₂) ≫ (𝟙 b ≫ h₁) ◁ adj₂.counit) ▷ h₂ ▷ r₃ ⊗≫ 𝟙 _ := by
      rw [← whisker_exchange, ← whisker_exchange]
      bicategory
    _ = 𝟙 _ ⊗≫ r₁ ◁ g₁ ◁ g₂ ◁ adj₃.unit ⊗≫
          r₁ ◁ g₁ ◁ (adj₂.unit ▷ (g₂ ≫ l₃) ≫ (l₂ ≫ r₂) ◁ β) ▷ r₃ ⊗≫
            r₁ ◁ (α ▷ (r₂ ≫ l₂) ≫ (l₁ ≫ h₁) ◁ adj₂.counit) ▷ h₂ ▷ r₃ ⊗≫
              adj₁.counit ▷ h₁ ▷ h₂ ▷ r₃ ⊗≫ 𝟙 _ := by
      rw [← whisker_exchange, ← whisker_exchange, ← whisker_exchange]
      bicategory
    _ = 𝟙 _ ⊗≫ r₁ ◁ g₁ ◁ g₂ ◁ adj₃.unit ⊗≫ r₁ ◁ g₁ ◁ β ▷ r₃ ⊗≫
          ((r₁ ≫ g₁) ◁ leftZigzag adj₂.unit adj₂.counit ▷ (h₂ ≫ r₃)) ⊗≫
            r₁ ◁ α ▷ h₂ ▷ r₃ ⊗≫ adj₁.counit ▷ h₁ ▷ h₂ ▷ r₃ ⊗≫ 𝟙 _ := by
      rw [← whisker_exchange, ← whisker_exchange]
      bicategory
    _ = _ := by
      rw [adj₂.left_triangle]
      bicategory