awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Coherence (goal)
Goal:
B : Type u
inst : Quiver.{v, u} B
a b c : B
f g : Hom.{v, u} b c
η' : Hom₂ f g
mk :
  ∀ {a_1 b : FreeBicategory B} {f g h : a_1 ⟶ b} (η : Hom₂ f g) (θ : Hom₂ g h),
    (Eq (α := (x : Path.{v, u} a a_1) → Path.{v, u} a b) (fun (p : Path.{v, u} a a_1) ↦ normalizeAux p f)
        fun (p : Path.{v, u} a a_1) ↦ normalizeAux p g) →
      (Eq (α := (x : Path.{v, u} a a_1) → Path.{v, u} a b) (fun (p : Path.{v, u} a a_1) ↦ normalizeAux p g)
          fun (p : Path.{v, u} a a_1) ↦ normalizeAux p h) →
        Eq (α := (x : Path.{v, u} a a_1) → Path.{v, u} a b) (fun (p : Path.{v, u} a a_1) ↦ normalizeAux p f)
          fun (p : Path.{v, u} a a_1) ↦ normalizeAux p h
mk₁ :
  ∀ {a_1 b c : FreeBicategory B} (f : a_1 ⟶ b) {g h : b ⟶ c} (η : Hom₂ g h)
    (ih :
      Eq (α := (x : Path.{v, u} a b) → Path.{v, u} a c) (fun (p : Path.{v, u} a b) ↦ normalizeAux p g)
        fun (p : Path.{v, u} a b) ↦ normalizeAux p h),
    Eq (α := (x : Path.{v, u} a a_1) → Path.{v, u} a c) (fun (p : Path.{v, u} a a_1) ↦ normalizeAux p (f ≫ g))
      fun (p : Path.{v, u} a a_1) ↦ normalizeAux p (f ≫ h)
mk₂ :
  ∀ {a_1 b c : FreeBicategory B} {f g : a_1 ⟶ b} (h : b ⟶ c) (η : Hom₂ f g)
    (ih :
      Eq (α := (x : Path.{v, u} a a_1) → Path.{v, u} a b) (fun (p : Path.{v, u} a a_1) ↦ normalizeAux p f)
        fun (p : Path.{v, u} a a_1) ↦ normalizeAux p g),
    Eq (α := (x : Path.{v, u} a a_1) → Path.{v, u} a c) (fun (p : Path.{v, u} a a_1) ↦ normalizeAux p (Hom.comp f h))
      fun (p : Path.{v, u} a a_1) ↦ normalizeAux p (Hom.comp g h)
mk₃ :
  ∀ {a_1 b : FreeBicategory B} (f : a_1 ⟶ b),
    Eq (α := (x : Path.{v, u} a a_1) → Path.{v, u} a b) (fun (p : Path.{v, u} a a_1) ↦ normalizeAux p f)
      fun (p : Path.{v, u} a a_1) ↦ normalizeAux p f
mk₄ :
  ∀ {a_1 b c d : FreeBicategory B} (f : a_1 ⟶ b) (g : b ⟶ c) (h : c ⟶ d),
    Eq (α := (x : Path.{v, u} a a_1) → Path.{v, u} a d) (fun (p : Path.{v, u} a a_1) ↦ normalizeAux p ((f ≫ g) ≫ h))
      fun (p : Path.{v, u} a a_1) ↦ normalizeAux p (f ≫ g ≫ h)
mk₅ :
  ∀ {a_1 b c d : FreeBicategory B} (f : a_1 ⟶ b) (g : b ⟶ c) (h : c ⟶ d),
    Eq (α := (x : Path.{v, u} a a_1) → Path.{v, u} a d) (fun (p : Path.{v, u} a a_1) ↦ normalizeAux p (f ≫ g ≫ h))
      fun (p : Path.{v, u} a a_1) ↦ normalizeAux p ((f ≫ g) ≫ h)
mk₆ :
  ∀ {a_1 b : FreeBicategory B} (f : a_1 ⟶ b),
    Eq (α := (x : Path.{v, u} a a_1) → Path.{v, u} a b) (fun (p : Path.{v, u} a a_1) ↦ normalizeAux p (f ≫ 𝟙 b))
      fun (p : Path.{v, u} a a_1) ↦ normalizeAux p f
⊢ Eq (α := (x : Path.{v, u} a b) → Path.{v, u} a c) (fun (p : Path.{v, u} a b) ↦ normalizeAux p f)
    fun (p : Path.{v, u} a b) ↦ normalizeAux p g
have mk₇ : ∀ {a_1 b : FreeBicategory B} (f : a_1 ⟶ b),
  Eq (α := (x : Path.{v, u} a a_1) → Path.{v, u} a b) (fun (p : Path.{v, u} a a_1) ↦ normalizeAux p f)
    fun (p : Path.{v, u} a a_1) ↦ normalizeAux p (f ≫ 𝟙 b)
Tactic: induction η' with
  | vcomp _ _ _ _ => apply Eq.trans <;> assumption
  | whisker_left _ _ ih => funext; apply congr_fun ih
  | whisker_right _ _ ih => funext; apply congr_arg₂ _ (congr_fun ih _) rfl
  | _ => funext; rfl
Cleared hypotheses: η'
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Coherence (goal)
Goal:
B : Type u
inst : Quiver.{v, u} B
a b c : B
p : Path.{v, u} a b
f g : Hom.{v, u} b c
η : f ⟶ g
⊢ Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B))) { as := p } ◁
        η ≫
      (normalizeIso p g).hom =
    (normalizeIso p f).hom ≫ PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯)
have mk : ∀ (η' : Hom₂ f g),
  Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B))) { as := p } ◁
        Quot.mk Rel η' ≫
      (normalizeIso p g).hom =
    (normalizeIso p f).hom ≫ PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom sorry)
Tactic: rcases η with ⟨η'⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Coherence (goal)
Goal:
B : Type u
inst : Quiver.{v, u} B
a b c : B
f g : Hom.{v, u} b c
a_1 b_1 : FreeBicategory B
f_1 g_1 h : a_1 ⟶ b_1
η : Hom₂ f_1 g_1
θ : Hom₂ g_1 h
ihf :
  ∀ (p : Path.{v, u} a a_1),
    Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := p } ◁
          Quot.mk Rel η ≫
        (normalizeIso p g_1).hom =
      (normalizeIso p f_1).hom ≫
        PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯)
ihg :
  ∀ (p : Path.{v, u} a a_1),
    Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := p } ◁
          Quot.mk Rel θ ≫
        (normalizeIso p h).hom =
      (normalizeIso p g_1).hom ≫
        PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯)
p : Path.{v, u} a a_1
⊢ Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B))) { as := p } ◁
        Quot.mk Rel (η.vcomp θ) ≫
      (normalizeIso p h).hom =
    (normalizeIso p f_1).hom ≫
      PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯)
have mk : CategoryStruct.comp (Y :=
      Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
          { as := p } ≫
        g_1)
      (Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
          { as := p } ◁
        η.mk)
      (Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
          { as := p } ◁
        θ.mk) ≫
    (normalizeIso p h).hom =
  (normalizeIso p f_1).hom ≫ PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom sorry)
Tactic: simp only [mk_vcomp, whiskerLeft_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Coherence (goal)
Goal:
B : Type u
inst : Quiver.{v, u} B
a b c : B
f g : Hom.{v, u} b c
a_1 b_1 : FreeBicategory B
f_1 g_1 h : a_1 ⟶ b_1
η : Hom₂ f_1 g_1
θ : Hom₂ g_1 h
ihf :
  ∀ (p : Path.{v, u} a a_1),
    Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := p } ◁
          Quot.mk Rel η ≫
        (normalizeIso p g_1).hom =
      (normalizeIso p f_1).hom ≫
        PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯)
ihg :
  ∀ (p : Path.{v, u} a a_1),
    Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := p } ◁
          Quot.mk Rel θ ≫
        (normalizeIso p h).hom =
      (normalizeIso p g_1).hom ≫
        PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯)
p : Path.{v, u} a a_1
| Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B))) { as := p } ◁
      θ.mk ≫
    (normalizeIso p h).hom
have a₁ : (normalizeIso p g_1).hom ≫ PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom sorry)
Tactic: rw [ihg]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Coherence (goal)
Goal:
B : Type u
inst : Quiver.{v, u} B
a b c : B
f g : Hom.{v, u} b c
a_1 b_1 : FreeBicategory B
f_1 g_1 h : a_1 ⟶ b_1
η : Hom₂ f_1 g_1
θ : Hom₂ g_1 h
ihf :
  ∀ (p : Path.{v, u} a a_1),
    Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := p } ◁
          Quot.mk Rel η ≫
        (normalizeIso p g_1).hom =
      (normalizeIso p f_1).hom ≫
        PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯)
ihg :
  ∀ (p : Path.{v, u} a a_1),
    Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := p } ◁
          Quot.mk Rel θ ≫
        (normalizeIso p h).hom =
      (normalizeIso p g_1).hom ≫
        PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯)
p : Path.{v, u} a a_1
⊢ CategoryStruct.comp (Y :=
        Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := p } ≫
          g_1)
        (Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := p } ◁
          η.mk)
        (Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := p } ◁
          θ.mk) ≫
      (normalizeIso p h).hom =
    (normalizeIso p f_1).hom ≫
      PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯)
have mk : Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B))) { as := p } ◁
      η.mk ≫
    (normalizeIso p g_1).hom ≫
      PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom sorry) =
  (normalizeIso p f_1).hom ≫ PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom sorry)
Tactic: slice_lhs 2 3 => rw [ihg]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Coherence (goal)
Goal:
B : Type u
inst : Quiver.{v, u} B
a b c : B
f g : Hom.{v, u} b c
a_1 b_1 : FreeBicategory B
f_1 g_1 h : a_1 ⟶ b_1
η : Hom₂ f_1 g_1
θ : Hom₂ g_1 h
ihf :
  ∀ (p : Path.{v, u} a a_1),
    Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := p } ◁
          Quot.mk Rel η ≫
        (normalizeIso p g_1).hom =
      (normalizeIso p f_1).hom ≫
        PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯)
ihg :
  ∀ (p : Path.{v, u} a a_1),
    Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := p } ◁
          Quot.mk Rel θ ≫
        (normalizeIso p h).hom =
      (normalizeIso p g_1).hom ≫
        PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯)
p : Path.{v, u} a a_1
| Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B))) { as := p } ◁
      η.mk ≫
    (normalizeIso p g_1).hom
have a₁ : (normalizeIso p f_1).hom ≫ PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom sorry)
Tactic: rw [ihf]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Coherence (goal)
Goal:
B : Type u
inst : Quiver.{v, u} B
a b c : B
f g : Hom.{v, u} b c
a_1 b_1 : FreeBicategory B
f_1 g_1 h : a_1 ⟶ b_1
η : Hom₂ f_1 g_1
θ : Hom₂ g_1 h
ihf :
  ∀ (p : Path.{v, u} a a_1),
    Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := p } ◁
          Quot.mk Rel η ≫
        (normalizeIso p g_1).hom =
      (normalizeIso p f_1).hom ≫
        PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯)
ihg :
  ∀ (p : Path.{v, u} a a_1),
    Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := p } ◁
          Quot.mk Rel θ ≫
        (normalizeIso p h).hom =
      (normalizeIso p g_1).hom ≫
        PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯)
p : Path.{v, u} a a_1
⊢ Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B))) { as := p } ◁
        η.mk ≫
      (normalizeIso p g_1).hom ≫
        PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯) =
    (normalizeIso p f_1).hom ≫
      PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯)
have mk : ((normalizeIso p f_1).hom ≫
      PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom sorry)) ≫
    PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom sorry) =
  (normalizeIso p f_1).hom ≫ PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom sorry)
Tactic: slice_lhs 1 2 => rw [ihf]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Coherence (goal)
Goal:
B : Type u
inst : Quiver.{v, u} B
a b c : B
f g : Hom.{v, u} b c
a_1 b_1 c_1 : FreeBicategory B
f_1 : a_1 ⟶ b_1
g_1 h : b_1 ⟶ c_1
η : Hom₂ g_1 h
ih :
  ∀ (p : Path.{v, u} a b_1),
    Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := p } ◁
          Quot.mk Rel η ≫
        (normalizeIso p h).hom =
      (normalizeIso p g_1).hom ≫
        PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯)
p : Path.{v, u} a a_1
⊢ Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B))) { as := p } ◁
        Quot.mk Rel (Hom₂.whisker_left f_1 η) ≫
      (normalizeIso p (f_1 ≫ h)).hom =
    (normalizeIso p (f_1 ≫ g_1)).hom ≫
      PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯)
have mk : Eq (α :=
  Quiver.Hom (V := a ⟶ c_1)
    (Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
        { as := p } ≫
      f_1 ≫ g_1)
    (Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
      { as := normalizeAux (normalizeAux p f_1) h }))
  (Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
        { as := p } ◁
      whiskerLeft (B := FreeBicategory B) (a := a_1) (b := b_1) (c := c_1) f_1 (g := g_1) (h := h) η.mk ≫
    Iso.inv
        (α_
          (Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := p : Discrete (a ⟶ a_1) })
          f_1 h) ≫
      whiskerRight (B := FreeBicategory B) (a := a) (b := b_1) (c := c_1) (f :=
          Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
              { as := p : Discrete (a ⟶ a_1) } ≫
            f_1)
          (g :=
          Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := normalizeAux p f_1 : Discrete (a ⟶ b_1) })
          (normalizeIso p f_1).hom h ≫
        (normalizeIso (normalizeAux p f_1) h).hom)
  ((Iso.inv
        (α_
          (Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := p : Discrete (a ⟶ a_1) })
          f_1 g_1) ≫
      whiskerRight (B := FreeBicategory B) (a := a) (b := b_1) (c := c_1) (f :=
          Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
              { as := p : Discrete (a ⟶ a_1) } ≫
            f_1)
          (g :=
          Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := normalizeAux p f_1 : Discrete (a ⟶ b_1) })
          (normalizeIso p f_1).hom g_1 ≫
        (normalizeIso (normalizeAux p f_1) g_1).hom) ≫
    eqToHom sorry)
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Coherence (hypothesis)
Goal:
B : Type u
inst : Quiver.{v, u} B
a b c : B
f g : Hom.{v, u} b c
a_1 b_1 c_1 : FreeBicategory B
f_1 g_1 : a_1 ⟶ b_1
h : b_1 ⟶ c_1
η' : Hom₂ f_1 g_1
ih :
  ∀ (p : Path.{v, u} a a_1),
    Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := p } ◁
          Quot.mk Rel η' ≫
        (normalizeIso p g_1).hom =
      (normalizeIso p f_1).hom ≫
        PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯)
p : Path.{v, u} a a_1
⊢ Eq (α :=
    Quiver.Hom (V := a ⟶ c_1)
      (Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
          { as := p } ≫
        Hom.comp f_1 h)
      (Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
        { as := normalizeAux (normalizeAux p g_1) h }))
    (Iso.inv
        (α_
          (Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := p : Discrete (a ⟶ a_1) })
          f_1 h) ≫
      ((normalizeIso p f_1).hom ▷ h ≫
          PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯) ▷ h) ≫
        (normalizeIso (normalizeAux p g_1) h).hom)
    ((Iso.inv
          (α_
            (Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
              { as := p : Discrete (a ⟶ a_1) })
            f_1 h) ≫
        whiskerRight (B := FreeBicategory B) (a := a) (b := b_1) (c := c_1) (f :=
            Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
                { as := p : Discrete (a ⟶ a_1) } ≫
              f_1)
            (g :=
            Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
              { as := normalizeAux p f_1 : Discrete (a ⟶ b_1) })
            (normalizeIso p f_1).hom h ≫
          (normalizeIso (normalizeAux p f_1) h).hom) ≫
      eqToHom ⋯)
have this : (normalizeIso (normalizeAux p f_1) h).hom = eqToHom sorry ≫ (normalizeIso (normalizeAux p g_1) h).hom ≫ eqToHom sorry
Tactic: have := dcongr_arg (fun x => (normalizeIso x h).hom) (normalizeAux_congr p (Quot.mk _ η'))
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Coherence (goal)
Goal:
B : Type u
inst : Quiver.{v, u} B
a b c : B
p : Path.{v, u} a b
f g : Hom.{v, u} b c
η' : Hom₂ f g
⊢ Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B))) { as := p } ◁
        Quot.mk Rel η' ≫
      (normalizeIso p g).hom =
    (normalizeIso p f).hom ≫ PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯)
have mk : ∀ {a_1 b : FreeBicategory B} (f : a_1 ⟶ b) (p : Path.{v, u} a a_1),
  Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B))) { as := p } ◁
        Quot.mk Rel (Hom₂.id f) ≫
      (normalizeIso p f).hom =
    (normalizeIso p f).hom ≫ PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom sorry)
Tactic: induction η' with
  | id => simp
  | vcomp η θ ihf ihg =>
    simp only [mk_vcomp, whiskerLeft_comp]
    slice_lhs 2 3 => rw [ihg]
    slice_lhs 1 2 => rw [ihf]
    simp
  -- p ≠ nil required! See the docstring of `normalizeAux`.
  | whisker_left _ _ ih =>
    dsimp
    rw [associator_inv_naturality_right_assoc, whisker_exchange_assoc, ih]
    simp
  | whisker_right h η' ih =>
    dsimp
    rw [associator_inv_naturality_middle_assoc, ← comp_whiskerRight_assoc, ih, comp_whiskerRight]
    have := dcongr_arg (fun x => (normalizeIso x h).hom) (normalizeAux_congr p (Quot.mk _ η'))
    dsimp at this; simp [this]
  | _ => simp
Cleared hypotheses: p, η'
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Coherence (goal)
Goal:
B : Type u
inst : Quiver.{v, u} B
a b c : B
p : Path.{v, u} a b
f g : Hom.{v, u} b c
η' : Hom₂ f g
mk :
  ∀ {a_1 b : FreeBicategory B} (f : a_1 ⟶ b) (p : Path.{v, u} a a_1),
    Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
            { as := p } ◁
          Quot.mk Rel (Hom₂.id f) ≫
        (normalizeIso p f).hom =
      (normalizeIso p f).hom ≫
        PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯)
⊢ Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B))) { as := p } ◁
        Quot.mk Rel η' ≫
      (normalizeIso p g).hom =
    (normalizeIso p f).hom ≫ PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom ⋯)
have mk₁ : ∀ {a_1 b : FreeBicategory B} {f g h : a_1 ⟶ b} (η : Hom₂ f g) (θ : Hom₂ g h)
  (ihf :
    ∀ (p : Path.{v, u} a a_1),
      Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
              { as := p } ◁
            Quot.mk Rel η ≫
          (normalizeIso p g).hom =
        (normalizeIso p f).hom ≫
          PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom sorry))
  (ihg :
    ∀ (p : Path.{v, u} a a_1),
      Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)))
              { as := p } ◁
            Quot.mk Rel θ ≫
          (normalizeIso p h).hom =
        (normalizeIso p g).hom ≫
          PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom sorry))
  (p : Path.{v, u} a a_1),
  Prefunctor.map (PrelaxFunctorStruct.toPrefunctor (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B))) { as := p } ◁
        Quot.mk Rel (η.vcomp θ) ≫
      (normalizeIso p h).hom =
    (normalizeIso p f).hom ≫ PrelaxFunctorStruct.map₂ (PrelaxFunctor.toPrelaxFunctorStruct (preinclusion B)) (eqToHom sorry)
Tactic: induction η' with
  | id => simp
  | vcomp η θ ihf ihg =>
    simp only [mk_vcomp, whiskerLeft_comp]
    slice_lhs 2 3 => rw [ihg]
    slice_lhs 1 2 => rw [ihf]
    simp
  -- p ≠ nil required! See the docstring of `normalizeAux`.
  | whisker_left _ _ ih =>
    dsimp
    rw [associator_inv_naturality_right_assoc, whisker_exchange_assoc, ih]
    simp
  | whisker_right h η' ih =>
    dsimp
    rw [associator_inv_naturality_middle_assoc, ← comp_whiskerRight_assoc, ih, comp_whiskerRight]
    have := dcongr_arg (fun x => (normalizeIso x h).hom) (normalizeAux_congr p (Quot.mk _ η'))
    dsimp at this; simp [this]
  | _ => simp
Cleared hypotheses: p, η'
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Coherence (goal)
Goal:
B : Type u
inst : Quiver.{v, u} B
b c a_1 b_1 c_1 : B
g : Hom.{v, u} a_1 b_1
g_1 : Hom.{v, u} b_1 c_1
ihf : ∀ {a : B} (f : Hom.{v, u} a a_1), normalizeAux nil (f.comp g) = (normalizeAux nil f).comp (normalizeAux nil g)
ihg : ∀ {a : B} (f : Hom.{v, u} a b_1), normalizeAux nil (f.comp g_1) = (normalizeAux nil f).comp (normalizeAux nil g_1)
a : B
f : Hom.{v, u} a a_1
⊢ normalizeAux nil (f.comp (g.comp g_1)) = (normalizeAux nil f).comp (normalizeAux nil (g.comp g_1))
have comp : (normalizeAux nil (f.comp g)).comp (normalizeAux nil g_1) = (normalizeAux nil f).comp (normalizeAux nil (g.comp g_1))
Tactic: erw [ihg (f.comp g), ihf f, ihg g, comp_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Coherence (goal)
Goal:
B : Type u
inst : Quiver.{v, u} B
b c a_1 b_1 c_1 : B
g : Hom.{v, u} a_1 b_1
g_1 : Hom.{v, u} b_1 c_1
ihf : ∀ {a : B} (f : Hom.{v, u} a a_1), normalizeAux nil (f.comp g) = (normalizeAux nil f).comp (normalizeAux nil g)
ihg : ∀ {a : B} (f : Hom.{v, u} a b_1), normalizeAux nil (f.comp g_1) = (normalizeAux nil f).comp (normalizeAux nil g_1)
a : B
f : Hom.{v, u} a a_1
comp :
  (normalizeAux nil (f.comp g)).comp (normalizeAux nil g_1) = (normalizeAux nil f).comp (normalizeAux nil (g.comp g_1))
⊢ normalizeAux nil (f.comp (g.comp g_1)) = (normalizeAux nil f).comp (normalizeAux nil (g.comp g_1))
have comp₁ : ((normalizeAux nil f).comp (normalizeAux nil g)).comp (normalizeAux nil g_1) =
  (normalizeAux nil f).comp (normalizeAux nil (g.comp g_1))
Tactic: erw [ihg (f.comp g), ihf f, ihg g, comp_assoc]