awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Grothendieck (goal)
Goal:
𝒮 : Type u₁
inst : Category.{v₁, u₁} 𝒮
F G H : LocallyDiscrete 𝒮ᵒᵖ ⥤ᵖ Cat
α : F ⟶ G
a b c : ∫ᶜ F
f : a ⟶ b
g : b ⟶ c
| Functor.map (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ }))
      (Functor.map (Cat.Hom.toFunctor (F.map f.base.op.toLoc)) g.fiber) ≫
    Functor.map (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ }))
        (NatTrans.app (Cat.Hom₂.toNatTrans (Iso.inv (F.mapComp g.base.op.toLoc f.base.op.toLoc))) c.fiber) ≫
      Functor.map (Cat.Hom.toFunctor (α.app { as := op a.base }))
        (NatTrans.app (Cat.Hom₂.toNatTrans (Iso.hom (F.mapComp g.base.op.toLoc f.base.op.toLoc))) c.fiber)
have a₁ : Functor.map (Cat.Hom.toFunctor (α.app { as := op a.base }))
  (Functor.map (Cat.Hom.toFunctor (F.map f.base.op.toLoc)) g.fiber)
Tactic: simp [← Cat.Hom.toNatIso_inv, Cat.Hom.comp_toFunctor,
        ← Cat.Hom.toNatIso_hom, ← map_comp, Iso.inv_hom_id_app, comp_obj, map_id, comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Grothendieck (goal)
Goal:
𝒮 : Type u₁
inst : Category.{v₁, u₁} 𝒮
F : LocallyDiscrete 𝒮ᵒᵖ ⥤ᵖ Cat
x y : ∫ᶜ F
f : x ⟶ y
⊢ (map (𝟙 F)).map f = f
have hfg₁ : ((map (𝟙 F)).map f).base = f.base
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Grothendieck (goal)
Goal:
𝒮 : Type u₁
inst : Category.{v₁, u₁} 𝒮
F : LocallyDiscrete 𝒮ᵒᵖ ⥤ᵖ Cat
x y : ∫ᶜ F
f : x ⟶ y
hfg₁ : ((map (𝟙 F)).map f).base = f.base
⊢ (map (𝟙 F)).map f = f
have hfg₂ : ((map (𝟙 F)).map f).fiber = f.fiber ≫ eqToHom sorry
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Grothendieck (goal)
Goal:
𝒮 : Type u₁
inst : Category.{v₁, u₁} 𝒮
F G H : LocallyDiscrete 𝒮ᵒᵖ ⥤ᵖ Cat
α : F ⟶ G
β : G ⟶ H
X Y : ∫ᶜ F
f : X ⟶ Y
⊢ (map (α ≫ β)).map f ≫ Iso.hom ((fun (x : ∫ᶜ F) ↦ (eqToIso ⋯ : (map (α ≫ β)).obj x ≅ (map α ⋙ map β).obj x)) Y) =
    Iso.hom ((fun (x : ∫ᶜ F) ↦ (eqToIso ⋯ : (map (α ≫ β)).obj x ≅ (map α ⋙ map β).obj x)) X) ≫ (map α ⋙ map β).map f
have dsimp : (map (α ≫ β)).map f ≫ 𝟙 ((map (α ≫ β)).obj Y) = 𝟙 ((map (α ≫ β)).obj X) ≫ (map β).map ((map α).map f)
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Grothendieck (goal)
Goal:
𝒮 : Type u₁
inst : Category.{v₁, u₁} 𝒮
F G H : LocallyDiscrete 𝒮ᵒᵖ ⥤ᵖ Cat
α : F ⟶ G
β : G ⟶ H
X Y : ∫ᶜ F
f : X ⟶ Y
⊢ (map (α ≫ β)).map f ≫ 𝟙 ((map (α ≫ β)).obj Y) = 𝟙 ((map (α ≫ β)).obj X) ≫ (map β).map ((map α).map f)
have simp : (map (α ≫ β)).map f = (map β).map ((map α).map f)
Tactic: simp only [comp_id, id_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Grothendieck (goal)
Goal:
𝒮 : Type u₁
inst : Category.{v₁, u₁} 𝒮
F G H : LocallyDiscrete 𝒮ᵒᵖ ⥤ᵖ Cat
α : F ⟶ G
β : G ⟶ H
X Y : ∫ᶜ F
f : X ⟶ Y
⊢ (map (α ≫ β)).map f = (map β).map ((map α).map f)
have hfg₁ : ((map (α ≫ β)).map f).base = ((map β).map ((map α).map f)).base
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Grothendieck (goal)
Goal:
𝒮 : Type u₁
inst : Category.{v₁, u₁} 𝒮
F G H : LocallyDiscrete 𝒮ᵒᵖ ⥤ᵖ Cat
α : F ⟶ G
β : G ⟶ H
X Y : ∫ᶜ F
f : X ⟶ Y
hfg₁ : ((map (α ≫ β)).map f).base = ((map β).map ((map α).map f)).base
⊢ (map (α ≫ β)).map f = (map β).map ((map α).map f)
have hfg₂ : ((map (α ≫ β)).map f).fiber = ((map β).map ((map α).map f)).fiber ≫ eqToHom sorry
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.InducedBicategory (goal)
Goal:
B : Type u_1
C : Type u_2
inst : Bicategory.{u_3, u_4, u_2} C
F : B → C
x x_1 x_2 : InducedBicategory C F
x_3 x_4 : x ⟶ x_1
x_5 x_6 : x_1 ⟶ x_2
η : x_3 ⟶ x_4
θ : x_5 ⟶ x_6
⊢ mkHom₂ (x_3.hom ◁ θ.hom) ≫ mkHom₂ (η.hom ▷ x_6.hom) = mkHom₂ (η.hom ▷ x_5.hom) ≫ mkHom₂ (x_4.hom ◁ θ.hom)
have h : (mkHom₂ (x_3.hom ◁ θ.hom) ≫ mkHom₂ (η.hom ▷ x_6.hom)).hom = (mkHom₂ (η.hom ▷ x_5.hom) ≫ mkHom₂ (x_4.hom ◁ θ.hom)).hom
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.InducedBicategory (goal)
Goal:
B : Type u_1
C : Type u_2
inst : Bicategory.{u_4, u_3, u_2} C
F : B → C
X Y : InducedBicategory C F
f g : X ⟶ Y
h : f = g
⊢ Hom₂.hom (eqToHom h) = eqToHom ⋯
have subst : Hom₂.hom (eqToHom sorry) = eqToHom sorry
Tactic: subst h
Cleared hypotheses: g, h
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.InducedBicategory (goal)
Goal:
B : Type u_1
C : Type u_2
inst : Bicategory.{u_4, u_3, u_2} C
F : B → C
X Y : InducedBicategory C F
f g : F X ⟶ F Y
h : f = g
⊢ mkHom₂ (eqToHom h) = eqToHom ⋯
have h₁ : (mkHom₂ (eqToHom h)).hom = Hom₂.hom (eqToHom sorry)
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.InducedBicategory (goal)
Goal:
B : Type u_1
C : Type u_2
inst : Bicategory.{u_4, u_3, u_2} C
F : B → C
X Y : InducedBicategory C F
f g : F X ⟶ F Y
h : f = g
⊢ (mkHom₂ (eqToHom h)).hom = Hom₂.hom (eqToHom ⋯)
have h₁ : (mkHom₂ (eqToHom sorry)).hom = Hom₂.hom (eqToHom sorry)
Tactic: subst h
Cleared hypotheses: g, h
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Kan.Adjunction (goal)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c : B
f : a ⟶ b
u : b ⟶ a
adj : f ⊣ u
x : B
h : a ⟶ x
s : LeftExtension f (𝟙 a ≫ h)
⊢ ((LeftExtension.mk u adj.unit).whisker h).unit ≫
      f ◁ (𝟙 (u ≫ h) ⊗≫ u ◁ s.unit ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 s.extension) =
    𝟙 (𝟙 a ≫ h) ⊗≫ (adj.unit ▷ (𝟙 a ≫ h) ≫ (f ≫ u) ◁ s.unit) ⊗≫ f ◁ adj.counit ▷ s.extension ⊗≫ 𝟙 (f ≫ s.extension)
have dsimp : (adj.unit ▷ h ≫ (α_ f u h).hom) ≫ f ◁ (𝟙 (u ≫ h) ⊗≫ u ◁ s.unit ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 s.extension) =
  𝟙 (𝟙 a ≫ h) ⊗≫ (adj.unit ▷ (𝟙 a ≫ h) ≫ (f ≫ u) ◁ s.unit) ⊗≫ f ◁ adj.counit ▷ s.extension ⊗≫ 𝟙 (f ≫ s.extension)
Tactic: dsimp only [whisker_extension, StructuredArrow.mk_right, whisker_unit,
            StructuredArrow.mk_hom_eq_self]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Kan.Adjunction (goal)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c : B
f : a ⟶ b
u : b ⟶ a
adj : f ⊣ u
x : B
h : a ⟶ x
s : LeftExtension f (𝟙 a ≫ h)
⊢ 𝟙 (𝟙 a ≫ h) ⊗≫ (adj.unit ▷ (𝟙 a ≫ h) ≫ (f ≫ u) ◁ s.unit) ⊗≫ f ◁ adj.counit ▷ s.extension ⊗≫ 𝟙 (f ≫ s.extension) =
    𝟙 (𝟙 a ≫ h) ⊗≫ s.unit ⊗≫ leftZigzag adj.unit adj.counit ▷ s.extension ⊗≫ 𝟙 (f ≫ s.extension)
have rw : 𝟙 (𝟙 a ≫ h) ⊗≫ (𝟙 a ◁ s.unit ≫ adj.unit ▷ (f ≫ s.extension)) ⊗≫ f ◁ adj.counit ▷ s.extension ⊗≫ 𝟙 (f ≫ s.extension) =
  𝟙 (𝟙 a ≫ h) ⊗≫ s.unit ⊗≫ leftZigzag adj.unit adj.counit ▷ s.extension ⊗≫ 𝟙 (f ≫ s.extension)
Tactic: rw [← whisker_exchange]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Kan.Adjunction (goal)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c : B
f : a ⟶ b
u : b ⟶ a
adj : f ⊣ u
x : B
h : a ⟶ x
s : LeftExtension f (𝟙 a ≫ h)
⊢ 𝟙 (𝟙 a ≫ h) ⊗≫ s.unit ⊗≫ leftZigzag adj.unit adj.counit ▷ s.extension ⊗≫ 𝟙 (f ≫ s.extension) = s.unit
have rw : 𝟙 (𝟙 a ≫ h) ⊗≫ s.unit ⊗≫ ((λ_ f).hom ≫ (ρ_ f).inv) ▷ s.extension ⊗≫ 𝟙 (f ≫ s.extension) = s.unit
Tactic: rw [adj.left_triangle]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Kan.Adjunction (goal)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c : B
f : a ⟶ b
u : b ⟶ a
adj : f ⊣ u
x : B
h : a ⟶ x
s : LeftExtension f (𝟙 a ≫ h)
τ₀ : (LeftExtension.mk u adj.unit).whisker h ⟶ s
⊢ τ₀ =
    (fun (s : LeftExtension f (𝟙 a ≫ h)) ↦
        homMk (𝟙 (u ≫ h) ⊗≫ u ◁ s.unit ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 s.extension) ⋯)
      s
have h₁ : StructuredArrow.Hom.right τ₀ =
  StructuredArrow.Hom.right
    ((fun (s : LeftExtension f (𝟙 a ≫ h)) ↦
        homMk (𝟙 (u ≫ h) ⊗≫ u ◁ s.unit ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 s.extension) sorry)
      s)
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Kan.Adjunction (hypothesis)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c : B
f : a ⟶ b
u : b ⟶ a
adj : f ⊣ u
x : B
h : a ⟶ x
s : LeftExtension f (𝟙 a ≫ h)
τ₀ : (LeftExtension.mk u adj.unit).whisker h ⟶ s
τ : u ≫ h ⟶ s.extension := StructuredArrow.Hom.right τ₀
⊢ StructuredArrow.Hom.right τ₀ =
    StructuredArrow.Hom.right
      ((fun (s : LeftExtension f (𝟙 a ≫ h)) ↦
          homMk (𝟙 (u ≫ h) ⊗≫ u ◁ s.unit ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 s.extension) ⋯)
        s)
have hτ : adj.unit ▷ h ⊗≫ f ◁ τ = s.unit
Tactic: have hτ : adj.unit ▷ h ⊗≫ f ◁ τ = s.unit := by
      simpa [bicategoricalComp] using LeftExtension.w τ₀
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Kan.Adjunction (goal)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c : B
f : a ⟶ b
u : b ⟶ a
adj : f ⊣ u
x : B
h : a ⟶ x
s : LeftExtension f (𝟙 a ≫ h)
τ₀ : (LeftExtension.mk u adj.unit).whisker h ⟶ s
τ : u ≫ h ⟶ s.extension := StructuredArrow.Hom.right τ₀
hτ : adj.unit ▷ h ⊗≫ f ◁ τ = s.unit
⊢ τ = 𝟙 (u ≫ h) ⊗≫ rightZigzag adj.unit adj.counit ▷ h ⊗≫ τ ⊗≫ 𝟙 s.extension
have rw : τ = 𝟙 (u ≫ h) ⊗≫ ((ρ_ u).hom ≫ (λ_ u).inv) ▷ h ⊗≫ τ ⊗≫ 𝟙 s.extension
Tactic: rw [adj.right_triangle]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Kan.Adjunction (goal)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c : B
f : a ⟶ b
u : b ⟶ a
adj : f ⊣ u
x : B
h : a ⟶ x
s : LeftExtension f (𝟙 a ≫ h)
τ₀ : (LeftExtension.mk u adj.unit).whisker h ⟶ s
τ : u ≫ h ⟶ s.extension := StructuredArrow.Hom.right τ₀
hτ : adj.unit ▷ h ⊗≫ f ◁ τ = s.unit
⊢ 𝟙 (u ≫ h) ⊗≫ rightZigzag adj.unit adj.counit ▷ h ⊗≫ τ ⊗≫ 𝟙 s.extension =
    𝟙 (u ≫ h) ⊗≫ u ◁ adj.unit ▷ h ⊗≫ (adj.counit ▷ (u ≫ h) ≫ 𝟙 b ◁ τ) ⊗≫ 𝟙 s.extension
have rw : 𝟙 (u ≫ h) ⊗≫ (u ◁ adj.unit ⊗≫ adj.counit ▷ u) ▷ h ⊗≫ τ ⊗≫ 𝟙 s.extension =
  𝟙 (u ≫ h) ⊗≫ u ◁ adj.unit ▷ h ⊗≫ (adj.counit ▷ (u ≫ h) ≫ 𝟙 b ◁ τ) ⊗≫ 𝟙 s.extension
Tactic: rw [rightZigzag]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Kan.Adjunction (goal)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c : B
f : a ⟶ b
u : b ⟶ a
adj : f ⊣ u
x : B
h : a ⟶ x
s : LeftExtension f (𝟙 a ≫ h)
τ₀ : (LeftExtension.mk u adj.unit).whisker h ⟶ s
τ : u ≫ h ⟶ s.extension := StructuredArrow.Hom.right τ₀
hτ : adj.unit ▷ h ⊗≫ f ◁ τ = s.unit
⊢ 𝟙 (u ≫ h) ⊗≫ u ◁ adj.unit ▷ h ⊗≫ (adj.counit ▷ (u ≫ h) ≫ 𝟙 b ◁ τ) ⊗≫ 𝟙 s.extension =
    𝟙 (u ≫ h) ⊗≫ u ◁ (adj.unit ▷ h ⊗≫ f ◁ τ) ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 s.extension
have rw : 𝟙 (u ≫ h) ⊗≫ u ◁ adj.unit ▷ h ⊗≫ ((u ≫ f) ◁ τ ≫ adj.counit ▷ s.extension) ⊗≫ 𝟙 s.extension =
  𝟙 (u ≫ h) ⊗≫ u ◁ (adj.unit ▷ h ⊗≫ f ◁ τ) ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 s.extension
Tactic: rw [← whisker_exchange]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Kan.Adjunction (goal)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c : B
f : a ⟶ b
u : b ⟶ a
adj : f ⊣ u
x : B
h : a ⟶ x
s : LeftExtension f (𝟙 a ≫ h)
τ₀ : (LeftExtension.mk u adj.unit).whisker h ⟶ s
τ : u ≫ h ⟶ s.extension := StructuredArrow.Hom.right τ₀
hτ : adj.unit ▷ h ⊗≫ f ◁ τ = s.unit
⊢ 𝟙 (u ≫ h) ⊗≫ u ◁ (adj.unit ▷ h ⊗≫ f ◁ τ) ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 s.extension =
    StructuredArrow.Hom.right
      ((fun (s : LeftExtension f (𝟙 a ≫ h)) ↦
          homMk (𝟙 (u ≫ h) ⊗≫ u ◁ s.unit ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 s.extension) ⋯)
        s)
have rw : 𝟙 (u ≫ h) ⊗≫ u ◁ s.unit ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 s.extension =
  StructuredArrow.Hom.right
    ((fun (s : LeftExtension f (𝟙 a ≫ h)) ↦
        homMk (𝟙 (u ≫ h) ⊗≫ u ◁ s.unit ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 s.extension) sorry)
      s)
Tactic: rw [hτ]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Kan.Adjunction (goal)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c : B
f : a ⟶ b
u : b ⟶ a
adj : f ⊣ u
x : B
h : a ⟶ x
s : LeftExtension f (𝟙 a ≫ h)
τ₀ : (LeftExtension.mk u adj.unit).whisker h ⟶ s
τ : u ≫ h ⟶ s.extension := StructuredArrow.Hom.right τ₀
hτ : adj.unit ▷ h ⊗≫ f ◁ τ = s.unit
⊢ StructuredArrow.Hom.right τ₀ =
    StructuredArrow.Hom.right
      ((fun (s : LeftExtension f (𝟙 a ≫ h)) ↦
          homMk (𝟙 (u ≫ h) ⊗≫ u ◁ s.unit ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 s.extension) ⋯)
        s)
have calc : τ = 𝟙 (u ≫ h) ⊗≫ rightZigzag adj.unit adj.counit ▷ h ⊗≫ τ ⊗≫ 𝟙 s.extension
Tactic: calc τ
      _ = 𝟙 _ ⊗≫ rightZigzag adj.unit adj.counit ▷ h ⊗≫ τ ⊗≫ 𝟙 _ := by
        rw [adj.right_triangle]; bicategory
      _ = 𝟙 _ ⊗≫ u ◁ adj.unit ▷ h ⊗≫ (adj.counit ▷ _ ≫ _ ◁ τ) ⊗≫ 𝟙 _ := by
        rw [rightZigzag]; bicategory
      _ = 𝟙 _ ⊗≫ u ◁ (adj.unit ▷ h ⊗≫ f ◁ τ) ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 _ := by
        rw [← whisker_exchange]; bicategory
      _ = _ := by
        rw [hτ]; dsimp only [StructuredArrow.homMk_right]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Kan.Adjunction (goal)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c : B
f : a ⟶ b
u : b ⟶ a
adj : f ⊣ u
x : B
h : a ⟶ x
s : LeftExtension f (𝟙 a ≫ h)
τ₀ : (LeftExtension.mk u adj.unit).whisker h ⟶ s
τ : u ≫ h ⟶ s.extension := StructuredArrow.Hom.right τ₀
hτ : adj.unit ▷ h ⊗≫ f ◁ τ = s.unit
calc : τ = 𝟙 (u ≫ h) ⊗≫ rightZigzag adj.unit adj.counit ▷ h ⊗≫ τ ⊗≫ 𝟙 s.extension
⊢ StructuredArrow.Hom.right τ₀ =
    StructuredArrow.Hom.right
      ((fun (s : LeftExtension f (𝟙 a ≫ h)) ↦
          homMk (𝟙 (u ≫ h) ⊗≫ u ◁ s.unit ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 s.extension) ⋯)
        s)
have calc₁ : 𝟙 (u ≫ h) ⊗≫ rightZigzag adj.unit adj.counit ▷ h ⊗≫ τ ⊗≫ 𝟙 s.extension =
  𝟙 (u ≫ h) ⊗≫ u ◁ adj.unit ▷ h ⊗≫ (adj.counit ▷ (u ≫ h) ≫ 𝟙 b ◁ τ) ⊗≫ 𝟙 s.extension
Tactic: calc τ
      _ = 𝟙 _ ⊗≫ rightZigzag adj.unit adj.counit ▷ h ⊗≫ τ ⊗≫ 𝟙 _ := by
        rw [adj.right_triangle]; bicategory
      _ = 𝟙 _ ⊗≫ u ◁ adj.unit ▷ h ⊗≫ (adj.counit ▷ _ ≫ _ ◁ τ) ⊗≫ 𝟙 _ := by
        rw [rightZigzag]; bicategory
      _ = 𝟙 _ ⊗≫ u ◁ (adj.unit ▷ h ⊗≫ f ◁ τ) ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 _ := by
        rw [← whisker_exchange]; bicategory
      _ = _ := by
        rw [hτ]; dsimp only [StructuredArrow.homMk_right]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Kan.Adjunction (goal)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c : B
f : a ⟶ b
u : b ⟶ a
adj : f ⊣ u
x : B
h : a ⟶ x
s : LeftExtension f (𝟙 a ≫ h)
τ₀ : (LeftExtension.mk u adj.unit).whisker h ⟶ s
τ : u ≫ h ⟶ s.extension := StructuredArrow.Hom.right τ₀
hτ : adj.unit ▷ h ⊗≫ f ◁ τ = s.unit
calc : τ = 𝟙 (u ≫ h) ⊗≫ rightZigzag adj.unit adj.counit ▷ h ⊗≫ τ ⊗≫ 𝟙 s.extension
calc₁ :
  𝟙 (u ≫ h) ⊗≫ rightZigzag adj.unit adj.counit ▷ h ⊗≫ τ ⊗≫ 𝟙 s.extension =
    𝟙 (u ≫ h) ⊗≫ u ◁ adj.unit ▷ h ⊗≫ (adj.counit ▷ (u ≫ h) ≫ 𝟙 b ◁ τ) ⊗≫ 𝟙 s.extension
⊢ StructuredArrow.Hom.right τ₀ =
    StructuredArrow.Hom.right
      ((fun (s : LeftExtension f (𝟙 a ≫ h)) ↦
          homMk (𝟙 (u ≫ h) ⊗≫ u ◁ s.unit ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 s.extension) ⋯)
        s)
have calc₂ : 𝟙 (u ≫ h) ⊗≫ u ◁ adj.unit ▷ h ⊗≫ (adj.counit ▷ (u ≫ h) ≫ 𝟙 b ◁ τ) ⊗≫ 𝟙 s.extension =
  𝟙 (u ≫ h) ⊗≫ u ◁ (adj.unit ▷ h ⊗≫ f ◁ τ) ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 s.extension
Tactic: calc τ
      _ = 𝟙 _ ⊗≫ rightZigzag adj.unit adj.counit ▷ h ⊗≫ τ ⊗≫ 𝟙 _ := by
        rw [adj.right_triangle]; bicategory
      _ = 𝟙 _ ⊗≫ u ◁ adj.unit ▷ h ⊗≫ (adj.counit ▷ _ ≫ _ ◁ τ) ⊗≫ 𝟙 _ := by
        rw [rightZigzag]; bicategory
      _ = 𝟙 _ ⊗≫ u ◁ (adj.unit ▷ h ⊗≫ f ◁ τ) ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 _ := by
        rw [← whisker_exchange]; bicategory
      _ = _ := by
        rw [hτ]; dsimp only [StructuredArrow.homMk_right]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Kan.Adjunction (goal)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c : B
f : a ⟶ b
u : b ⟶ a
adj : f ⊣ u
x : B
h : a ⟶ x
s : LeftExtension f (𝟙 a ≫ h)
τ₀ : (LeftExtension.mk u adj.unit).whisker h ⟶ s
τ : u ≫ h ⟶ s.extension := StructuredArrow.Hom.right τ₀
hτ : adj.unit ▷ h ⊗≫ f ◁ τ = s.unit
calc : τ = 𝟙 (u ≫ h) ⊗≫ rightZigzag adj.unit adj.counit ▷ h ⊗≫ τ ⊗≫ 𝟙 s.extension
calc₁ :
  𝟙 (u ≫ h) ⊗≫ rightZigzag adj.unit adj.counit ▷ h ⊗≫ τ ⊗≫ 𝟙 s.extension =
    𝟙 (u ≫ h) ⊗≫ u ◁ adj.unit ▷ h ⊗≫ (adj.counit ▷ (u ≫ h) ≫ 𝟙 b ◁ τ) ⊗≫ 𝟙 s.extension
calc₂ :
  𝟙 (u ≫ h) ⊗≫ u ◁ adj.unit ▷ h ⊗≫ (adj.counit ▷ (u ≫ h) ≫ 𝟙 b ◁ τ) ⊗≫ 𝟙 s.extension =
    𝟙 (u ≫ h) ⊗≫ u ◁ (adj.unit ▷ h ⊗≫ f ◁ τ) ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 s.extension
⊢ StructuredArrow.Hom.right τ₀ =
    StructuredArrow.Hom.right
      ((fun (s : LeftExtension f (𝟙 a ≫ h)) ↦
          homMk (𝟙 (u ≫ h) ⊗≫ u ◁ s.unit ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 s.extension) ⋯)
        s)
have calc₃ : 𝟙 (u ≫ h) ⊗≫ u ◁ (adj.unit ▷ h ⊗≫ f ◁ τ) ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 s.extension =
  StructuredArrow.Hom.right
    ((fun (s : LeftExtension f (𝟙 a ≫ h)) ↦
        homMk (𝟙 (u ≫ h) ⊗≫ u ◁ s.unit ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 s.extension) sorry)
      s)
Tactic: calc τ
      _ = 𝟙 _ ⊗≫ rightZigzag adj.unit adj.counit ▷ h ⊗≫ τ ⊗≫ 𝟙 _ := by
        rw [adj.right_triangle]; bicategory
      _ = 𝟙 _ ⊗≫ u ◁ adj.unit ▷ h ⊗≫ (adj.counit ▷ _ ≫ _ ◁ τ) ⊗≫ 𝟙 _ := by
        rw [rightZigzag]; bicategory
      _ = 𝟙 _ ⊗≫ u ◁ (adj.unit ▷ h ⊗≫ f ◁ τ) ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 _ := by
        rw [← whisker_exchange]; bicategory
      _ = _ := by
        rw [hτ]; dsimp only [StructuredArrow.homMk_right]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Kan.Adjunction (goal)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c : B
f : a ⟶ b
t : LeftExtension f (𝟙 a)
H : t.IsKan
H' : (t.whisker f).IsKan
ε : t.extension ≫ f ⟶ 𝟙 b := H'.desc (LeftExtension.mk (𝟙 b) ((λ_ f).hom ≫ (ρ_ f).inv))
Hε : leftZigzag t.unit ε = (λ_ f).hom ≫ (ρ_ f).inv
⊢ rightZigzag t.unit ε = (ρ_ t.extension).hom ≫ (λ_ t.extension).inv
have apply : (ρ_ t.extension).inv ≫ rightZigzag t.unit ε = (ρ_ t.extension).inv ≫ (ρ_ t.extension).hom ≫ (λ_ t.extension).inv
Tactic: apply (cancel_epi (ρ_ _).inv).mp
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Kan.Adjunction (goal)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c : B
f : a ⟶ b
t : LeftExtension f (𝟙 a)
H : t.IsKan
H' : (t.whisker f).IsKan
ε : t.extension ≫ f ⟶ 𝟙 b := H'.desc (LeftExtension.mk (𝟙 b) ((λ_ f).hom ≫ (ρ_ f).inv))
Hε : leftZigzag t.unit ε = (λ_ f).hom ≫ (ρ_ f).inv
⊢ (ρ_ t.extension).inv ≫ rightZigzag t.unit ε = (ρ_ t.extension).inv ≫ (ρ_ t.extension).hom ≫ (λ_ t.extension).inv
have apply : t.unit ≫ f ◁ ((ρ_ t.extension).inv ≫ rightZigzag t.unit ε) =
  t.unit ≫ f ◁ ((ρ_ t.extension).inv ≫ (ρ_ t.extension).hom ≫ (λ_ t.extension).inv)
Tactic: apply H.hom_ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Kan.Adjunction (goal)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c : B
f : a ⟶ b
t : LeftExtension f (𝟙 a)
H : t.IsKan
H' : (t.whisker f).IsKan
ε : t.extension ≫ f ⟶ 𝟙 b := H'.desc (LeftExtension.mk (𝟙 b) ((λ_ f).hom ≫ (ρ_ f).inv))
Hε : leftZigzag t.unit ε = (λ_ f).hom ≫ (ρ_ f).inv
⊢ Eq (α := 𝟙 a ⟶ f ≫ 𝟙 b ≫ t.extension) (𝟙 (𝟙 a) ⊗≫ t.unit ⊗≫ f ◁ rightZigzag t.unit ε ⊗≫ 𝟙 (f ≫ 𝟙 b ≫ t.extension))
    (𝟙 (𝟙 a) ⊗≫ (t.unit ▷ 𝟙 a ≫ (f ≫ t.extension) ◁ t.unit) ⊗≫ f ◁ ε ▷ t.extension ⊗≫ 𝟙 (f ≫ 𝟙 b ≫ t.extension))
have rw : Eq (α := 𝟙 a ⟶ f ≫ 𝟙 b ≫ t.extension)
  (𝟙 (𝟙 a) ⊗≫ t.unit ⊗≫ f ◁ (t.extension ◁ t.unit ⊗≫ ε ▷ t.extension) ⊗≫ 𝟙 (f ≫ 𝟙 b ≫ t.extension))
  (𝟙 (𝟙 a) ⊗≫ (t.unit ▷ 𝟙 a ≫ (f ≫ t.extension) ◁ t.unit) ⊗≫ f ◁ ε ▷ t.extension ⊗≫ 𝟙 (f ≫ 𝟙 b ≫ t.extension))
Tactic: rw [rightZigzag]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Kan.Adjunction (goal)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c : B
f : a ⟶ b
t : LeftExtension f (𝟙 a)
H : t.IsKan
H' : (t.whisker f).IsKan
ε : t.extension ≫ f ⟶ 𝟙 b := H'.desc (LeftExtension.mk (𝟙 b) ((λ_ f).hom ≫ (ρ_ f).inv))
Hε : leftZigzag t.unit ε = (λ_ f).hom ≫ (ρ_ f).inv
⊢ 𝟙 (𝟙 a) ⊗≫ t.unit ⊗≫ (t.unit ▷ f ⊗≫ f ◁ ε) ▷ t.extension ⊗≫ 𝟙 (f ≫ 𝟙 b ≫ t.extension) =
    t.unit ≫ f ◁ ((ρ_ t.extension).inv ≫ (ρ_ t.extension).hom ≫ (λ_ t.extension).inv)
have rw : 𝟙 (𝟙 a) ⊗≫ t.unit ⊗≫ ((λ_ f).hom ≫ (ρ_ f).inv) ▷ t.extension ⊗≫ 𝟙 (f ≫ 𝟙 b ≫ t.extension) =
  t.unit ≫ f ◁ ((ρ_ t.extension).inv ≫ (ρ_ t.extension).hom ≫ (λ_ t.extension).inv)
Tactic: rw [← leftZigzag, Hε]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Kan.Adjunction (goal)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c : B
f : a ⟶ b
t : LeftExtension f (𝟙 a)
H : t.IsKan
H' : (t.whisker f).IsKan
ε : t.extension ≫ f ⟶ 𝟙 b := H'.desc (LeftExtension.mk (𝟙 b) ((λ_ f).hom ≫ (ρ_ f).inv))
Hε : leftZigzag t.unit ε = (λ_ f).hom ≫ (ρ_ f).inv
⊢ t.unit ≫ f ◁ ((ρ_ t.extension).inv ≫ rightZigzag t.unit ε) =
    t.unit ≫ f ◁ ((ρ_ t.extension).inv ≫ (ρ_ t.extension).hom ≫ (λ_ t.extension).inv)
have calc : t.unit ≫ f ◁ ((ρ_ t.extension).inv ≫ rightZigzag t.unit ε) =
  𝟙 (𝟙 a) ⊗≫ t.unit ⊗≫ f ◁ rightZigzag t.unit ε ⊗≫ 𝟙 (f ≫ 𝟙 b ≫ t.extension)
Tactic: calc _
        _ = 𝟙 _ ⊗≫ t.unit ⊗≫ f ◁ rightZigzag t.unit ε ⊗≫ 𝟙 _ := by
          bicategory
        _ = 𝟙 _ ⊗≫ (t.unit ▷ _ ≫ _ ◁ t.unit) ⊗≫ f ◁ ε ▷ t.extension ⊗≫ 𝟙 _ := by
          rw [rightZigzag]; bicategory
        _ = 𝟙 _ ⊗≫ t.unit ⊗≫ (t.unit ▷ f ⊗≫ f ◁ ε) ▷ t.extension ⊗≫ 𝟙 _ := by
          rw [← whisker_exchange]; bicategory
        _ = _ := by
          rw [← leftZigzag, Hε]; bicategory
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Kan.Adjunction (goal)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c : B
f : a ⟶ b
t : LeftExtension f (𝟙 a)
H : t.IsKan
H' : (t.whisker f).IsKan
ε : t.extension ≫ f ⟶ 𝟙 b := H'.desc (LeftExtension.mk (𝟙 b) ((λ_ f).hom ≫ (ρ_ f).inv))
Hε : leftZigzag t.unit ε = (λ_ f).hom ≫ (ρ_ f).inv
calc :
  t.unit ≫ f ◁ ((ρ_ t.extension).inv ≫ rightZigzag t.unit ε) =
    𝟙 (𝟙 a) ⊗≫ t.unit ⊗≫ f ◁ rightZigzag t.unit ε ⊗≫ 𝟙 (f ≫ 𝟙 b ≫ t.extension)
⊢ t.unit ≫ f ◁ ((ρ_ t.extension).inv ≫ rightZigzag t.unit ε) =
    t.unit ≫ f ◁ ((ρ_ t.extension).inv ≫ (ρ_ t.extension).hom ≫ (λ_ t.extension).inv)
have calc₁ : Eq (α := 𝟙 a ⟶ f ≫ 𝟙 b ≫ t.extension) (𝟙 (𝟙 a) ⊗≫ t.unit ⊗≫ f ◁ rightZigzag t.unit ε ⊗≫ 𝟙 (f ≫ 𝟙 b ≫ t.extension))
  (𝟙 (𝟙 a) ⊗≫ (t.unit ▷ 𝟙 a ≫ (f ≫ t.extension) ◁ t.unit) ⊗≫ f ◁ ε ▷ t.extension ⊗≫ 𝟙 (f ≫ 𝟙 b ≫ t.extension))
Tactic: calc _
        _ = 𝟙 _ ⊗≫ t.unit ⊗≫ f ◁ rightZigzag t.unit ε ⊗≫ 𝟙 _ := by
          bicategory
        _ = 𝟙 _ ⊗≫ (t.unit ▷ _ ≫ _ ◁ t.unit) ⊗≫ f ◁ ε ▷ t.extension ⊗≫ 𝟙 _ := by
          rw [rightZigzag]; bicategory
        _ = 𝟙 _ ⊗≫ t.unit ⊗≫ (t.unit ▷ f ⊗≫ f ◁ ε) ▷ t.extension ⊗≫ 𝟙 _ := by
          rw [← whisker_exchange]; bicategory
        _ = _ := by
          rw [← leftZigzag, Hε]; bicategory
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Kan.Adjunction (goal)
Goal:
B : Type u
inst : Bicategory.{w, v, u} B
a b c : B
f : a ⟶ b
t : LeftExtension f (𝟙 a)
H : t.IsKan
H' : (t.whisker f).IsKan
ε : t.extension ≫ f ⟶ 𝟙 b := H'.desc (LeftExtension.mk (𝟙 b) ((λ_ f).hom ≫ (ρ_ f).inv))
Hε : leftZigzag t.unit ε = (λ_ f).hom ≫ (ρ_ f).inv
calc :
  t.unit ≫ f ◁ ((ρ_ t.extension).inv ≫ rightZigzag t.unit ε) =
    𝟙 (𝟙 a) ⊗≫ t.unit ⊗≫ f ◁ rightZigzag t.unit ε ⊗≫ 𝟙 (f ≫ 𝟙 b ≫ t.extension)
calc₁ :
  Eq (α := 𝟙 a ⟶ f ≫ 𝟙 b ≫ t.extension) (𝟙 (𝟙 a) ⊗≫ t.unit ⊗≫ f ◁ rightZigzag t.unit ε ⊗≫ 𝟙 (f ≫ 𝟙 b ≫ t.extension))
    (𝟙 (𝟙 a) ⊗≫ (t.unit ▷ 𝟙 a ≫ (f ≫ t.extension) ◁ t.unit) ⊗≫ f ◁ ε ▷ t.extension ⊗≫ 𝟙 (f ≫ 𝟙 b ≫ t.extension))
⊢ t.unit ≫ f ◁ ((ρ_ t.extension).inv ≫ rightZigzag t.unit ε) =
    t.unit ≫ f ◁ ((ρ_ t.extension).inv ≫ (ρ_ t.extension).hom ≫ (λ_ t.extension).inv)
have calc₂ : Eq (α := 𝟙 a ⟶ f ≫ 𝟙 b ≫ t.extension)
  (𝟙 (𝟙 a) ⊗≫ (t.unit ▷ 𝟙 a ≫ (f ≫ t.extension) ◁ t.unit) ⊗≫ f ◁ ε ▷ t.extension ⊗≫ 𝟙 (f ≫ 𝟙 b ≫ t.extension))
  (𝟙 (𝟙 a) ⊗≫ t.unit ⊗≫ (t.unit ▷ f ⊗≫ f ◁ ε) ▷ t.extension ⊗≫ 𝟙 (f ≫ 𝟙 b ≫ t.extension))
Tactic: calc _
        _ = 𝟙 _ ⊗≫ t.unit ⊗≫ f ◁ rightZigzag t.unit ε ⊗≫ 𝟙 _ := by
          bicategory
        _ = 𝟙 _ ⊗≫ (t.unit ▷ _ ≫ _ ◁ t.unit) ⊗≫ f ◁ ε ▷ t.extension ⊗≫ 𝟙 _ := by
          rw [rightZigzag]; bicategory
        _ = 𝟙 _ ⊗≫ t.unit ⊗≫ (t.unit ▷ f ⊗≫ f ◁ ε) ▷ t.extension ⊗≫ 𝟙 _ := by
          rw [← whisker_exchange]; bicategory
        _ = _ := by
          rw [← leftZigzag, Hε]; bicategory