awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y : F C
f' : X ⟶ᵐ Y
mk :
  ∀ {X Y Z : F C} (f : X ⟶ᵐ Y) (g : Y ⟶ᵐ Z),
    (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Y.normalizeObj →
      (fun (n : NormalMonoidalObject C) ↦ Y.normalizeObj n) = Z.normalizeObj →
        (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Z.normalizeObj
mk₁ :
  ∀ (X : F C) {Y₁ Y₂ : F C} (f : Y₁ ⟶ᵐ Y₂)
    (ih : (fun (n : NormalMonoidalObject C) ↦ Y₁.normalizeObj n) = Y₂.normalizeObj),
    (fun (n : NormalMonoidalObject C) ↦ (X.tensor Y₁).normalizeObj n) = (X.tensor Y₂).normalizeObj
mk₂ :
  ∀ {X₁ X₂ : F C} (f : X₁ ⟶ᵐ X₂) (Y : F C)
    (ih : (fun (n : NormalMonoidalObject C) ↦ X₁.normalizeObj n) = X₂.normalizeObj),
    (fun (n : NormalMonoidalObject C) ↦ (X₁.tensor Y).normalizeObj n) = (X₂.tensor Y).normalizeObj
mk₃ :
  ∀ {W X Y Z : F C} (f : W ⟶ᵐ Y) (g : X ⟶ᵐ Z)
    (ih₁ : (fun (n : NormalMonoidalObject C) ↦ W.normalizeObj n) = Y.normalizeObj)
    (ih₂ : (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Z.normalizeObj),
    (fun (n : NormalMonoidalObject C) ↦ (W.tensor X).normalizeObj n) = (Y.tensor Z).normalizeObj
mk₄ : ∀ (X : F C), (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = X.normalizeObj
mk₅ :
  ∀ (X Y Z : F C),
    (fun (n : NormalMonoidalObject C) ↦ ((X.tensor Y).tensor Z).normalizeObj n) = (X.tensor (Y.tensor Z)).normalizeObj
mk₆ :
  ∀ (X Y Z : F C),
    (fun (n : NormalMonoidalObject C) ↦ (X.tensor (Y.tensor Z)).normalizeObj n) = ((X.tensor Y).tensor Z).normalizeObj
mk₇ : ∀ (X : F C), (fun (n : NormalMonoidalObject C) ↦ (tensor unit X).normalizeObj n) = X.normalizeObj
⊢ (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Y.normalizeObj
have mk₈ : ∀ (X : F C), (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = (tensor unit X).normalizeObj
Tactic: induction f' with
  | comp _ _ _ _ => apply Eq.trans <;> assumption
  | whiskerLeft _ _ ih => funext; apply congr_fun ih
  | whiskerRight _ _ ih => funext; apply congr_arg₂ _ rfl (congr_fun ih _)
  | @tensor W X Y Z _ _ ih₁ ih₂ =>
      funext n
      simp [congr_fun ih₁ n, congr_fun ih₂ (normalizeObj Y n)]
  | _ => funext; rfl
Cleared hypotheses: f'
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y : F C
f' : X ⟶ᵐ Y
mk :
  ∀ {X Y Z : F C} (f : X ⟶ᵐ Y) (g : Y ⟶ᵐ Z),
    (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Y.normalizeObj →
      (fun (n : NormalMonoidalObject C) ↦ Y.normalizeObj n) = Z.normalizeObj →
        (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Z.normalizeObj
mk₁ :
  ∀ (X : F C) {Y₁ Y₂ : F C} (f : Y₁ ⟶ᵐ Y₂)
    (ih : (fun (n : NormalMonoidalObject C) ↦ Y₁.normalizeObj n) = Y₂.normalizeObj),
    (fun (n : NormalMonoidalObject C) ↦ (X.tensor Y₁).normalizeObj n) = (X.tensor Y₂).normalizeObj
mk₂ :
  ∀ {X₁ X₂ : F C} (f : X₁ ⟶ᵐ X₂) (Y : F C)
    (ih : (fun (n : NormalMonoidalObject C) ↦ X₁.normalizeObj n) = X₂.normalizeObj),
    (fun (n : NormalMonoidalObject C) ↦ (X₁.tensor Y).normalizeObj n) = (X₂.tensor Y).normalizeObj
mk₃ :
  ∀ {W X Y Z : F C} (f : W ⟶ᵐ Y) (g : X ⟶ᵐ Z)
    (ih₁ : (fun (n : NormalMonoidalObject C) ↦ W.normalizeObj n) = Y.normalizeObj)
    (ih₂ : (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Z.normalizeObj),
    (fun (n : NormalMonoidalObject C) ↦ (W.tensor X).normalizeObj n) = (Y.tensor Z).normalizeObj
mk₄ : ∀ (X : F C), (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = X.normalizeObj
mk₅ :
  ∀ (X Y Z : F C),
    (fun (n : NormalMonoidalObject C) ↦ ((X.tensor Y).tensor Z).normalizeObj n) = (X.tensor (Y.tensor Z)).normalizeObj
mk₆ :
  ∀ (X Y Z : F C),
    (fun (n : NormalMonoidalObject C) ↦ (X.tensor (Y.tensor Z)).normalizeObj n) = ((X.tensor Y).tensor Z).normalizeObj
mk₇ : ∀ (X : F C), (fun (n : NormalMonoidalObject C) ↦ (tensor unit X).normalizeObj n) = X.normalizeObj
mk₈ : ∀ (X : F C), (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = (tensor unit X).normalizeObj
⊢ (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Y.normalizeObj
have mk₉ : ∀ (X : F C), (fun (n : NormalMonoidalObject C) ↦ (X.tensor unit).normalizeObj n) = X.normalizeObj
Tactic: induction f' with
  | comp _ _ _ _ => apply Eq.trans <;> assumption
  | whiskerLeft _ _ ih => funext; apply congr_fun ih
  | whiskerRight _ _ ih => funext; apply congr_arg₂ _ rfl (congr_fun ih _)
  | @tensor W X Y Z _ _ ih₁ ih₂ =>
      funext n
      simp [congr_fun ih₁ n, congr_fun ih₂ (normalizeObj Y n)]
  | _ => funext; rfl
Cleared hypotheses: f'
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y : F C
f' : X ⟶ᵐ Y
mk :
  ∀ {X Y Z : F C} (f : X ⟶ᵐ Y) (g : Y ⟶ᵐ Z),
    (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Y.normalizeObj →
      (fun (n : NormalMonoidalObject C) ↦ Y.normalizeObj n) = Z.normalizeObj →
        (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Z.normalizeObj
mk₁ :
  ∀ (X : F C) {Y₁ Y₂ : F C} (f : Y₁ ⟶ᵐ Y₂)
    (ih : (fun (n : NormalMonoidalObject C) ↦ Y₁.normalizeObj n) = Y₂.normalizeObj),
    (fun (n : NormalMonoidalObject C) ↦ (X.tensor Y₁).normalizeObj n) = (X.tensor Y₂).normalizeObj
mk₂ :
  ∀ {X₁ X₂ : F C} (f : X₁ ⟶ᵐ X₂) (Y : F C)
    (ih : (fun (n : NormalMonoidalObject C) ↦ X₁.normalizeObj n) = X₂.normalizeObj),
    (fun (n : NormalMonoidalObject C) ↦ (X₁.tensor Y).normalizeObj n) = (X₂.tensor Y).normalizeObj
mk₃ :
  ∀ {W X Y Z : F C} (f : W ⟶ᵐ Y) (g : X ⟶ᵐ Z)
    (ih₁ : (fun (n : NormalMonoidalObject C) ↦ W.normalizeObj n) = Y.normalizeObj)
    (ih₂ : (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Z.normalizeObj),
    (fun (n : NormalMonoidalObject C) ↦ (W.tensor X).normalizeObj n) = (Y.tensor Z).normalizeObj
mk₄ : ∀ (X : F C), (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = X.normalizeObj
mk₅ :
  ∀ (X Y Z : F C),
    (fun (n : NormalMonoidalObject C) ↦ ((X.tensor Y).tensor Z).normalizeObj n) = (X.tensor (Y.tensor Z)).normalizeObj
mk₆ :
  ∀ (X Y Z : F C),
    (fun (n : NormalMonoidalObject C) ↦ (X.tensor (Y.tensor Z)).normalizeObj n) = ((X.tensor Y).tensor Z).normalizeObj
mk₇ : ∀ (X : F C), (fun (n : NormalMonoidalObject C) ↦ (tensor unit X).normalizeObj n) = X.normalizeObj
mk₈ : ∀ (X : F C), (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = (tensor unit X).normalizeObj
mk₉ : ∀ (X : F C), (fun (n : NormalMonoidalObject C) ↦ (X.tensor unit).normalizeObj n) = X.normalizeObj
⊢ (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Y.normalizeObj
have mk₁₀ : ∀ (X : F C), (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = (X.tensor unit).normalizeObj
Tactic: induction f' with
  | comp _ _ _ _ => apply Eq.trans <;> assumption
  | whiskerLeft _ _ ih => funext; apply congr_fun ih
  | whiskerRight _ _ ih => funext; apply congr_arg₂ _ rfl (congr_fun ih _)
  | @tensor W X Y Z _ _ ih₁ ih₂ =>
      funext n
      simp [congr_fun ih₁ n, congr_fun ih₂ (normalizeObj Y n)]
  | _ => funext; rfl
Cleared hypotheses: f'
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
n : NormalMonoidalObject C
X Y : F C
f : X ⟶ Y
⊢ inclusionObj n ◁ f ≫ (normalizeIsoApp' C Y n).hom = (normalizeIsoApp' C X n).hom ≫ map inclusion (eqToHom ⋯)
have revert : ∀ (n : NormalMonoidalObject C),
  inclusionObj n ◁ f ≫ (normalizeIsoApp' C Y n).hom = (normalizeIsoApp' C X n).hom ≫ map inclusion (eqToHom sorry)
Tactic: revert n
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y : F C
f : X ⟶ Y
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ f ≫ (normalizeIsoApp' C Y n).hom = (normalizeIsoApp' C X n).hom ≫ map inclusion (eqToHom ⋯)
have id : ∀ (X : F C) (n : NormalMonoidalObject C),
  inclusionObj n ◁ 𝟙 X ≫ (normalizeIsoApp' C X n).hom = (normalizeIsoApp' C X n).hom ≫ map inclusion (eqToHom sorry)
Tactic: induction f using Hom.inductionOn
Cleared hypotheses: f
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y : F C
f : X ⟶ Y
id :
  ∀ (X : F C) (n : NormalMonoidalObject C),
    inclusionObj n ◁ 𝟙 X ≫ (normalizeIsoApp' C X n).hom = (normalizeIsoApp' C X n).hom ≫ map inclusion (eqToHom ⋯)
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ f ≫ (normalizeIsoApp' C Y n).hom = (normalizeIsoApp' C X n).hom ≫ map inclusion (eqToHom ⋯)
have α_hom : ∀ (X Y Z : F C) (n : NormalMonoidalObject C),
  Eq (α :=
    tensorObj (𝒞 := categoryFreeMonoidalCategory) (inclusionObj n)
        (tensorObj (𝒞 := categoryFreeMonoidalCategory) (tensorObj (𝒞 := categoryFreeMonoidalCategory) X Y) Z) ⟶
      inclusionObj
        ((tensorObj (𝒞 := categoryFreeMonoidalCategory) X
              (tensorObj (𝒞 := categoryFreeMonoidalCategory) Y Z)).normalizeObj
          n))
    (inclusionObj n ◁ Iso.hom (α_ X Y Z) ≫
      Iso.hom
        (normalizeIsoApp' C
          (tensorObj (𝒞 := categoryFreeMonoidalCategory) X (tensorObj (𝒞 := categoryFreeMonoidalCategory) Y Z)) n))
    (Iso.hom
        (normalizeIsoApp' C
          (tensorObj (𝒞 := categoryFreeMonoidalCategory) (tensorObj (𝒞 := categoryFreeMonoidalCategory) X Y) Z) n) ≫
      map inclusion (eqToHom sorry))
Tactic: induction f using Hom.inductionOn
Cleared hypotheses: f
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y : F C
f : X ⟶ Y
id :
  ∀ (X : F C) (n : NormalMonoidalObject C),
    inclusionObj n ◁ 𝟙 X ≫ (normalizeIsoApp' C X n).hom = (normalizeIsoApp' C X n).hom ≫ map inclusion (eqToHom ⋯)
α_hom :
  ∀ (X Y Z : F C) (n : NormalMonoidalObject C),
    Eq (α :=
      tensorObj (𝒞 := categoryFreeMonoidalCategory) (inclusionObj n)
          (tensorObj (𝒞 := categoryFreeMonoidalCategory) (tensorObj (𝒞 := categoryFreeMonoidalCategory) X Y) Z) ⟶
        inclusionObj
          ((tensorObj (𝒞 := categoryFreeMonoidalCategory) X
                (tensorObj (𝒞 := categoryFreeMonoidalCategory) Y Z)).normalizeObj
            n))
      (inclusionObj n ◁ Iso.hom (α_ X Y Z) ≫
        Iso.hom
          (normalizeIsoApp' C
            (tensorObj (𝒞 := categoryFreeMonoidalCategory) X (tensorObj (𝒞 := categoryFreeMonoidalCategory) Y Z)) n))
      (Iso.hom
          (normalizeIsoApp' C
            (tensorObj (𝒞 := categoryFreeMonoidalCategory) (tensorObj (𝒞 := categoryFreeMonoidalCategory) X Y) Z) n) ≫
        map inclusion (eqToHom ⋯))
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ f ≫ (normalizeIsoApp' C Y n).hom = (normalizeIsoApp' C X n).hom ≫ map inclusion (eqToHom ⋯)
have α_inv : ∀ (X Y Z : F C) (n : NormalMonoidalObject C),
  Eq (α :=
    tensorObj (𝒞 := categoryFreeMonoidalCategory) (inclusionObj n)
        (tensorObj (𝒞 := categoryFreeMonoidalCategory) X (tensorObj (𝒞 := categoryFreeMonoidalCategory) Y Z)) ⟶
      inclusionObj
        ((tensorObj (𝒞 := categoryFreeMonoidalCategory) (tensorObj (𝒞 := categoryFreeMonoidalCategory) X Y)
              Z).normalizeObj
          n))
    (inclusionObj n ◁ Iso.inv (α_ X Y Z) ≫
      Iso.hom
        (normalizeIsoApp' C
          (tensorObj (𝒞 := categoryFreeMonoidalCategory) (tensorObj (𝒞 := categoryFreeMonoidalCategory) X Y) Z) n))
    (Iso.hom
        (normalizeIsoApp' C
          (tensorObj (𝒞 := categoryFreeMonoidalCategory) X (tensorObj (𝒞 := categoryFreeMonoidalCategory) Y Z)) n) ≫
      map inclusion (eqToHom sorry))
Tactic: induction f using Hom.inductionOn
Cleared hypotheses: f
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y : F C
f : X ⟶ Y
id :
  ∀ (X : F C) (n : NormalMonoidalObject C),
    inclusionObj n ◁ 𝟙 X ≫ (normalizeIsoApp' C X n).hom = (normalizeIsoApp' C X n).hom ≫ map inclusion (eqToHom ⋯)
α_hom :
  ∀ (X Y Z : F C) (n : NormalMonoidalObject C),
    Eq (α :=
      tensorObj (𝒞 := categoryFreeMonoidalCategory) (inclusionObj n)
          (tensorObj (𝒞 := categoryFreeMonoidalCategory) (tensorObj (𝒞 := categoryFreeMonoidalCategory) X Y) Z) ⟶
        inclusionObj
          ((tensorObj (𝒞 := categoryFreeMonoidalCategory) X
                (tensorObj (𝒞 := categoryFreeMonoidalCategory) Y Z)).normalizeObj
            n))
      (inclusionObj n ◁ Iso.hom (α_ X Y Z) ≫
        Iso.hom
          (normalizeIsoApp' C
            (tensorObj (𝒞 := categoryFreeMonoidalCategory) X (tensorObj (𝒞 := categoryFreeMonoidalCategory) Y Z)) n))
      (Iso.hom
          (normalizeIsoApp' C
            (tensorObj (𝒞 := categoryFreeMonoidalCategory) (tensorObj (𝒞 := categoryFreeMonoidalCategory) X Y) Z) n) ≫
        map inclusion (eqToHom ⋯))
α_inv :
  ∀ (X Y Z : F C) (n : NormalMonoidalObject C),
    Eq (α :=
      tensorObj (𝒞 := categoryFreeMonoidalCategory) (inclusionObj n)
          (tensorObj (𝒞 := categoryFreeMonoidalCategory) X (tensorObj (𝒞 := categoryFreeMonoidalCategory) Y Z)) ⟶
        inclusionObj
          ((tensorObj (𝒞 := categoryFreeMonoidalCategory) (tensorObj (𝒞 := categoryFreeMonoidalCategory) X Y)
                Z).normalizeObj
            n))
      (inclusionObj n ◁ Iso.inv (α_ X Y Z) ≫
        Iso.hom
          (normalizeIsoApp' C
            (tensorObj (𝒞 := categoryFreeMonoidalCategory) (tensorObj (𝒞 := categoryFreeMonoidalCategory) X Y) Z) n))
      (Iso.hom
          (normalizeIsoApp' C
            (tensorObj (𝒞 := categoryFreeMonoidalCategory) X (tensorObj (𝒞 := categoryFreeMonoidalCategory) Y Z)) n) ≫
        map inclusion (eqToHom ⋯))
⊢ ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ f ≫ (normalizeIsoApp' C Y n).hom = (normalizeIsoApp' C X n).hom ≫ map inclusion (eqToHom ⋯)
have l_hom : ∀ (X : F C) (n : NormalMonoidalObject C),
  Eq (α :=
    tensorObj (𝒞 := categoryFreeMonoidalCategory) (inclusionObj n)
        (tensorObj (𝒞 := categoryFreeMonoidalCategory) ((𝟙_ (F C)) (𝒞 := categoryFreeMonoidalCategory)) X) ⟶
      inclusionObj (X.normalizeObj n))
    (inclusionObj n ◁ Iso.hom (λ_ X) ≫ (normalizeIsoApp' C X n).hom)
    (Iso.hom
        (normalizeIsoApp' C
          (tensorObj (𝒞 := categoryFreeMonoidalCategory) ((𝟙_ (F C)) (𝒞 := categoryFreeMonoidalCategory)) X) n) ≫
      map inclusion (eqToHom sorry))
Tactic: induction f using Hom.inductionOn
Cleared hypotheses: f
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X_1 Y_1 X' X Y : F C
f : X ⟶ Y
ih :
  ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ f ≫ (normalizeIsoApp' C Y n).hom = (normalizeIsoApp' C X n).hom ≫ map inclusion (eqToHom ⋯)
n : NormalMonoidalObject C
⊢ Eq (α :=
    tensorObj (𝒞 := categoryFreeMonoidalCategory) (inclusionObj n)
        (tensorObj (𝒞 := categoryFreeMonoidalCategory) X' X) ⟶
      inclusionObj ((tensorObj (𝒞 := categoryFreeMonoidalCategory) X' Y).normalizeObj n))
    (inclusionObj n ◁ X' ◁ f ≫ Iso.hom (normalizeIsoApp' C (tensorObj (𝒞 := categoryFreeMonoidalCategory) X' Y) n))
    (Iso.hom (normalizeIsoApp' C (tensorObj (𝒞 := categoryFreeMonoidalCategory) X' X) n) ≫ map inclusion (eqToHom ⋯))
have dsimp : inclusionObj n ◁ X' ◁ f ≫
    Iso.inv (α_ (inclusionObj n) X' Y) ≫
      (normalizeIsoApp' C X' n).hom ▷ Y ≫ Iso.hom (normalizeIsoApp' C Y (X'.normalizeObj n)) =
  (Iso.inv (α_ (inclusionObj n) X' X) ≫
      (normalizeIsoApp' C X' n).hom ▷ X ≫ Iso.hom (normalizeIsoApp' C X (X'.normalizeObj n))) ≫
    map inclusion (eqToHom sorry)
Tactic: dsimp only [normalizeObj_tensor, normalizeIsoApp', tensor_eq_tensor, Iso.trans_hom,
      Iso.symm_hom, whiskerRightIso_hom, Function.comp_apply, inclusion_obj]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X_1 Y_1 X' X Y : F C
f : X ⟶ Y
ih :
  ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ f ≫ (normalizeIsoApp' C Y n).hom = (normalizeIsoApp' C X n).hom ≫ map inclusion (eqToHom ⋯)
n : NormalMonoidalObject C
⊢ inclusionObj n ◁ X' ◁ f ≫
      Iso.inv (α_ (inclusionObj n) X' Y) ≫
        (normalizeIsoApp' C X' n).hom ▷ Y ≫ Iso.hom (normalizeIsoApp' C Y (X'.normalizeObj n)) =
    (Iso.inv (α_ (inclusionObj n) X' X) ≫
        (normalizeIsoApp' C X' n).hom ▷ X ≫ Iso.hom (normalizeIsoApp' C X (X'.normalizeObj n))) ≫
      map inclusion (eqToHom ⋯)
have rw : Eq (α :=
  tensorObj (𝒞 := categoryFreeMonoidalCategory) (inclusionObj n) (tensorObj (𝒞 := categoryFreeMonoidalCategory) X' X) ⟶
    inclusionObj (Y.normalizeObj (X'.normalizeObj n)))
  (Iso.inv (α_ (inclusionObj n) X' X) ≫
    (normalizeIsoApp' C X' n).hom ▷ X ≫ Iso.hom (normalizeIsoApp' C X (X'.normalizeObj n)) ≫ map inclusion (eqToHom sorry))
  ((Iso.inv (α_ (inclusionObj n) X' X) ≫
      (normalizeIsoApp' C X' n).hom ▷ X ≫ Iso.hom (normalizeIsoApp' C X (X'.normalizeObj n))) ≫
    map inclusion (eqToHom sorry))
Tactic: rw [associator_inv_naturality_right_assoc, whisker_exchange_assoc, ih]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X_1 Y_1 X Y : F C
h : X ⟶ Y
η' : F C
ih :
  ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ h ≫ (normalizeIsoApp' C Y n).hom = (normalizeIsoApp' C X n).hom ≫ map inclusion (eqToHom ⋯)
n : NormalMonoidalObject C
⊢ Eq (α :=
    tensorObj (𝒞 := categoryFreeMonoidalCategory) (inclusionObj n)
        (tensorObj (𝒞 := categoryFreeMonoidalCategory) X η') ⟶
      inclusionObj ((tensorObj (𝒞 := categoryFreeMonoidalCategory) Y η').normalizeObj n))
    (inclusionObj n ◁ h ▷ η' ≫ Iso.hom (normalizeIsoApp' C (tensorObj (𝒞 := categoryFreeMonoidalCategory) Y η') n))
    (Iso.hom (normalizeIsoApp' C (tensorObj (𝒞 := categoryFreeMonoidalCategory) X η') n) ≫ map inclusion (eqToHom ⋯))
have dsimp : inclusionObj n ◁ h ▷ η' ≫
    Iso.inv (α_ (inclusionObj n) Y η') ≫
      (normalizeIsoApp' C Y n).hom ▷ η' ≫ Iso.hom (normalizeIsoApp' C η' (Y.normalizeObj n)) =
  (Iso.inv (α_ (inclusionObj n) X η') ≫
      (normalizeIsoApp' C X n).hom ▷ η' ≫ Iso.hom (normalizeIsoApp' C η' (X.normalizeObj n))) ≫
    map inclusion (eqToHom sorry)
Tactic: dsimp only [normalizeObj_tensor, normalizeIsoApp', tensor_eq_tensor, Iso.trans_hom,
      Iso.symm_hom, whiskerRightIso_hom, Function.comp_apply, inclusion_obj]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X_1 Y_1 X Y : F C
h : X ⟶ Y
η' : F C
ih :
  ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ h ≫ (normalizeIsoApp' C Y n).hom = (normalizeIsoApp' C X n).hom ≫ map inclusion (eqToHom ⋯)
n : NormalMonoidalObject C
⊢ inclusionObj n ◁ h ▷ η' ≫
      Iso.inv (α_ (inclusionObj n) Y η') ≫
        (normalizeIsoApp' C Y n).hom ▷ η' ≫ Iso.hom (normalizeIsoApp' C η' (Y.normalizeObj n)) =
    (Iso.inv (α_ (inclusionObj n) X η') ≫
        (normalizeIsoApp' C X n).hom ▷ η' ≫ Iso.hom (normalizeIsoApp' C η' (X.normalizeObj n))) ≫
      map inclusion (eqToHom ⋯)
have rw : Eq (α :=
  tensorObj (𝒞 := categoryFreeMonoidalCategory) (inclusionObj n) (tensorObj (𝒞 := categoryFreeMonoidalCategory) X η') ⟶
    inclusionObj (η'.normalizeObj (Y.normalizeObj n)))
  (Iso.inv (α_ (inclusionObj n) X η') ≫
    ((normalizeIsoApp' C X n).hom ≫ map inclusion (eqToHom sorry)) ▷ η' ≫
      Iso.hom (normalizeIsoApp' C η' (Y.normalizeObj n)))
  ((Iso.inv (α_ (inclusionObj n) X η') ≫
      (normalizeIsoApp' C X n).hom ▷ η' ≫ Iso.hom (normalizeIsoApp' C η' (X.normalizeObj n))) ≫
    map inclusion (eqToHom sorry))
Tactic: rw [associator_inv_naturality_middle_assoc, ← comp_whiskerRight_assoc, ih]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (hypothesis)
Goal:
C : Type u
X_1 Y_1 X Y : F C
h : X ⟶ Y
η' : F C
ih :
  ∀ (n : NormalMonoidalObject C),
    inclusionObj n ◁ h ≫ (normalizeIsoApp' C Y n).hom = (normalizeIsoApp' C X n).hom ≫ map inclusion (eqToHom ⋯)
n : NormalMonoidalObject C
⊢ Eq (α :=
    tensorObj (𝒞 := categoryFreeMonoidalCategory) (inclusionObj n)
        (tensorObj (𝒞 := categoryFreeMonoidalCategory) X η') ⟶
      inclusionObj (η'.normalizeObj (Y.normalizeObj n)))
    (Iso.inv (α_ (inclusionObj n) X η') ≫
      ((normalizeIsoApp' C X n).hom ≫ map inclusion (eqToHom ⋯)) ▷ η' ≫
        Iso.hom (normalizeIsoApp' C η' (Y.normalizeObj n)))
    ((Iso.inv (α_ (inclusionObj n) X η') ≫
        (normalizeIsoApp' C X n).hom ▷ η' ≫ Iso.hom (normalizeIsoApp' C η' (X.normalizeObj n))) ≫
      map inclusion (eqToHom ⋯))
have this : Eq (α :=
  tensorObj (𝒞 := categoryFreeMonoidalCategory) (inclusionObj (X.normalizeObj n)) η' ⟶
    inclusionObj (η'.normalizeObj (X.normalizeObj n)))
  (Iso.hom (normalizeIsoApp' C η' (X.normalizeObj n)))
  (CategoryStruct.comp (Y := tensorObj (𝒞 := categoryFreeMonoidalCategory) (inclusionObj (Y.normalizeObj n)) η')
    (eqToHom sorry) (Iso.hom (normalizeIsoApp' C η' (Y.normalizeObj n)) ≫ eqToHom sorry))
Tactic: have := dcongr_arg (fun x => (normalizeIsoApp' C η' x).hom) (normalizeObj_congr n h)
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y : F C
f : X ⟶ Y
⊢ (tensorFunc C).map f ≫ (normalizeIsoAux C Y).hom = (normalizeIsoAux C X).hom ≫ (normalize' C).map f
have w : ∀ (n : NormalMonoidalObject C),
  Eq (α :=
    ((tensorFunc C).obj X).obj { as := n : Discrete (NormalMonoidalObject C) } ⟶ ((normalize' C).obj Y).obj { as := n })
    (((tensorFunc C).map f ≫ (normalizeIsoAux C Y).hom).app { as := n })
    (((normalizeIsoAux C X).hom ≫ (normalize' C).map f).app { as := n })
Tactic: ext ⟨n⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y : F C
f : X ⟶ Y
n : NormalMonoidalObject C
⊢ Eq (α :=
    ((tensorFunc C).obj X).obj { as := n : Discrete (NormalMonoidalObject C) } ⟶ ((normalize' C).obj Y).obj { as := n })
    (((tensorFunc C).map f ≫ (normalizeIsoAux C Y).hom).app { as := n })
    (((normalizeIsoAux C X).hom ≫ (normalize' C).map f).app { as := n })
have h : (((tensorFunc C).obj X).obj { as := n : Discrete (NormalMonoidalObject C) } ⟶
      ((normalize' C).obj Y).obj { as := n : Discrete (NormalMonoidalObject C) }) =
    (tensorObj (𝒞 := categoryFreeMonoidalCategory) (inclusionObj n) X ⟶ inclusionObj (Y.normalizeObj n)) →
  ((tensorFunc C).map f ≫ (normalizeIsoAux C Y).hom).app { as := n } = inclusionObj n ◁ f ≫ (normalizeIsoApp' C Y n).hom
Tactic: convert! normalize_naturality n f using 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y : F C
f : X ⟶ Y
n : NormalMonoidalObject C
h :
  (((tensorFunc C).obj X).obj { as := n : Discrete (NormalMonoidalObject C) } ⟶
        ((normalize' C).obj Y).obj { as := n : Discrete (NormalMonoidalObject C) }) =
      (tensorObj (𝒞 := categoryFreeMonoidalCategory) (inclusionObj n) X ⟶ inclusionObj (Y.normalizeObj n)) →
    ((tensorFunc C).map f ≫ (normalizeIsoAux C Y).hom).app { as := n } =
      inclusionObj n ◁ f ≫ (normalizeIsoApp' C Y n).hom
⊢ Eq (α :=
    ((tensorFunc C).obj X).obj { as := n : Discrete (NormalMonoidalObject C) } ⟶ ((normalize' C).obj Y).obj { as := n })
    (((tensorFunc C).map f ≫ (normalizeIsoAux C Y).hom).app { as := n })
    (((normalizeIsoAux C X).hom ≫ (normalize' C).map f).app { as := n })
have h₁ : (((tensorFunc C).obj X).obj { as := n : Discrete (NormalMonoidalObject C) } ⟶
      ((normalize' C).obj Y).obj { as := n : Discrete (NormalMonoidalObject C) }) =
    (tensorObj (𝒞 := categoryFreeMonoidalCategory) (inclusionObj n) X ⟶ inclusionObj (Y.normalizeObj n)) →
  ((normalizeIsoAux C X).hom ≫ (normalize' C).map f).app { as := n } =
    (normalizeIsoApp' C X n).hom ≫ map inclusion (eqToHom sorry)
Tactic: convert! normalize_naturality n f using 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y : F C
f : X ⟶ Y
n : NormalMonoidalObject C
e_1 :
  (((tensorFunc C).obj X).obj { as := n : Discrete (NormalMonoidalObject C) } ⟶
      ((normalize' C).obj Y).obj { as := n : Discrete (NormalMonoidalObject C) }) =
    (tensorObj (𝒞 := categoryFreeMonoidalCategory) (inclusionObj n) X ⟶ inclusionObj (Y.normalizeObj n))
⊢ ((tensorFunc C).map f ≫ (normalizeIsoAux C Y).hom).app { as := n } = inclusionObj n ◁ f ≫ (normalizeIsoApp' C Y n).hom
have h : inclusionObj n ◁ f ≫ Iso.hom (normalizeIsoApp C Y { as := n : Discrete (NormalMonoidalObject C) }) =
  inclusionObj n ◁ f ≫ (normalizeIsoApp' C Y n).hom
Tactic: dsimp