awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
Z : F C
as as_1 : NormalMonoidalObject C
h : as = as_1
⊢ ((tensorFunc C).obj Z).map
      {
        down :=
          { down := h :
            PLift
              ({ as := as : Discrete (NormalMonoidalObject C) }.as =
                { as := as_1 : Discrete (NormalMonoidalObject C) }.as) } :
        ULift.{u, 0}
          (PLift
            ({ as := as : Discrete (NormalMonoidalObject C) }.as =
              { as := as_1 : Discrete (NormalMonoidalObject C) }.as)) } =
    map inclusion { down := { down := h } } ▷ Z
have mk : ((tensorFunc C).obj Z).map
    {
      down :=
        { down := sorry :
          PLift
            ({ as := as : Discrete (NormalMonoidalObject C) }.as =
              { as := as : Discrete (NormalMonoidalObject C) }.as) } :
      ULift.{u, 0}
        (PLift
          ({ as := as : Discrete (NormalMonoidalObject C) }.as =
            { as := as : Discrete (NormalMonoidalObject C) }.as)) } =
  map inclusion { down := { down := sorry } } ▷ Z
Tactic: subst h
Cleared hypotheses: as_1, h
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y : F C
n : (Discrete ∘ NormalMonoidalObject) C
⊢ Iso.symm (α_ (obj inclusion { as := n.as : Discrete (NormalMonoidalObject C) }) X Y) ≪≫
      normalizeIsoApp C X n ▷ᵢ Y ≪≫
        normalizeIsoApp C Y { as := X.normalizeObj n.as : Discrete (NormalMonoidalObject C) } =
    Iso.symm (α_ (inclusionObj n.as) X Y) ≪≫
      normalizeIsoApp' C X n.as ▷ᵢ Y ≪≫ normalizeIsoApp' C Y (X.normalizeObj n.as)
have rw : Iso.symm (α_ (obj inclusion { as := n.as : Discrete (NormalMonoidalObject C) }) X Y) ≪≫
    normalizeIsoApp' C X n.as ▷ᵢ Y ≪≫
      normalizeIsoApp C Y { as := X.normalizeObj n.as : Discrete (NormalMonoidalObject C) } =
  Iso.symm (α_ (inclusionObj n.as) X Y) ≪≫ normalizeIsoApp' C X n.as ▷ᵢ Y ≪≫ normalizeIsoApp' C Y (X.normalizeObj n.as)
Tactic: rw [normalizeIsoApp_eq X n]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y : F C
n : (Discrete ∘ NormalMonoidalObject) C
⊢ Iso.symm (α_ (obj inclusion { as := n.as : Discrete (NormalMonoidalObject C) }) X Y) ≪≫
      normalizeIsoApp' C X n.as ▷ᵢ Y ≪≫
        normalizeIsoApp C Y { as := X.normalizeObj n.as : Discrete (NormalMonoidalObject C) } =
    Iso.symm (α_ (inclusionObj n.as) X Y) ≪≫
      normalizeIsoApp' C X n.as ▷ᵢ Y ≪≫ normalizeIsoApp' C Y (X.normalizeObj n.as)
have rw : Iso.symm (α_ (obj inclusion { as := n.as : Discrete (NormalMonoidalObject C) }) X Y) ≪≫
    normalizeIsoApp' C X n.as ▷ᵢ Y ≪≫
      normalizeIsoApp' C Y { as := X.normalizeObj n.as : Discrete (NormalMonoidalObject C) }.as =
  Iso.symm (α_ (inclusionObj n.as) X Y) ≪≫ normalizeIsoApp' C X n.as ▷ᵢ Y ≪≫ normalizeIsoApp' C Y (X.normalizeObj n.as)
Tactic: rw [normalizeIsoApp_eq Y ⟨normalizeObj X n.as⟩]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X_1 : F C
X Y : NormalMonoidalObject C
f : X = Y
⊢ ((tensorFunc C).obj X_1).map
        {
          down :=
            { down := f :
              PLift
                ({ as := X : Discrete (NormalMonoidalObject C) }.as =
                  { as := Y : Discrete (NormalMonoidalObject C) }.as) } :
          ULift.{u, 0}
            (PLift
              ({ as := X : Discrete (NormalMonoidalObject C) }.as =
                { as := Y : Discrete (NormalMonoidalObject C) }.as)) } ≫
      Iso.hom (normalizeIsoApp C X_1 { as := Y : Discrete (NormalMonoidalObject C) }) =
    Iso.hom (normalizeIsoApp C X_1 { as := X }) ≫
      ((normalize' C).obj X_1).map
        {
          down :=
            { down := f :
              PLift
                ({ as := X : Discrete (NormalMonoidalObject C) }.as =
                  { as := Y : Discrete (NormalMonoidalObject C) }.as) } :
          ULift.{u, 0}
            (PLift
              ({ as := X : Discrete (NormalMonoidalObject C) }.as =
                { as := Y : Discrete (NormalMonoidalObject C) }.as)) }
have subst : ((tensorFunc C).obj X_1).map
      {
        down :=
          { down := sorry :
            PLift
              ({ as := X : Discrete (NormalMonoidalObject C) }.as =
                { as := X : Discrete (NormalMonoidalObject C) }.as) } :
        ULift.{u, 0}
          (PLift
            ({ as := X : Discrete (NormalMonoidalObject C) }.as =
              { as := X : Discrete (NormalMonoidalObject C) }.as)) } ≫
    Iso.hom (normalizeIsoApp C X_1 { as := X : Discrete (NormalMonoidalObject C) }) =
  Iso.hom (normalizeIsoApp C X_1 { as := X }) ≫
    ((normalize' C).obj X_1).map
      {
        down :=
          { down := sorry :
            PLift
              ({ as := X : Discrete (NormalMonoidalObject C) }.as =
                { as := X : Discrete (NormalMonoidalObject C) }.as) } :
        ULift.{u, 0}
          (PLift
            ({ as := X : Discrete (NormalMonoidalObject C) }.as = { as := X : Discrete (NormalMonoidalObject C) }.as)) }
Tactic: subst f
Cleared hypotheses: Y, f
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X_1 : F C
X : NormalMonoidalObject C
⊢ ((tensorFunc C).obj X_1).map
        {
          down :=
            { down := ⋯ :
              PLift
                ({ as := X : Discrete (NormalMonoidalObject C) }.as =
                  { as := X : Discrete (NormalMonoidalObject C) }.as) } :
          ULift.{u, 0}
            (PLift
              ({ as := X : Discrete (NormalMonoidalObject C) }.as =
                { as := X : Discrete (NormalMonoidalObject C) }.as)) } ≫
      Iso.hom (normalizeIsoApp C X_1 { as := X : Discrete (NormalMonoidalObject C) }) =
    Iso.hom (normalizeIsoApp C X_1 { as := X }) ≫
      ((normalize' C).obj X_1).map
        {
          down :=
            { down := ⋯ :
              PLift
                ({ as := X : Discrete (NormalMonoidalObject C) }.as =
                  { as := X : Discrete (NormalMonoidalObject C) }.as) } :
          ULift.{u, 0}
            (PLift
              ({ as := X : Discrete (NormalMonoidalObject C) }.as =
                { as := X : Discrete (NormalMonoidalObject C) }.as)) }
have dsimp : map
      (Discrete.functor fun (n : NormalMonoidalObject C) ↦
        tensorObj (𝒞 := categoryFreeMonoidalCategory) (inclusionObj n) X_1)
      (𝟙 { as := X : Discrete (NormalMonoidalObject C) }) ≫
    Iso.hom (normalizeIsoApp C X_1 { as := X : Discrete (NormalMonoidalObject C) }) =
  Iso.hom (normalizeIsoApp C X_1 { as := X : Discrete (NormalMonoidalObject C) }) ≫
    𝟙 (inclusionObj (X_1.normalizeObj'.obj { as := X }).as)
Tactic: dsimp
-- 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
⊢ X.normalizeObj n = Y.normalizeObj n
have mk : ∀ (f' : X ⟶ᵐ Y), X.normalizeObj n = Y.normalizeObj n
Tactic: rcases f with ⟨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
f' : X ⟶ᵐ Y
⊢ X.normalizeObj n = Y.normalizeObj n
have mk : (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Y.normalizeObj
Tactic: apply @congr_fun _ _ fun n => normalizeObj X n
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y X_1 Y_1 Z : F C
f : X_1 ⟶ᵐ Y_1
g : Y_1 ⟶ᵐ Z
f_ih : (fun (n : NormalMonoidalObject C) ↦ X_1.normalizeObj n) = Y_1.normalizeObj
g_ih : (fun (n : NormalMonoidalObject C) ↦ Y_1.normalizeObj n) = Z.normalizeObj
mk : (fun (n : NormalMonoidalObject C) ↦ X_1.normalizeObj n) = Y_1.normalizeObj
mk₁ : Y_1.normalizeObj = Z.normalizeObj
⊢ (fun (n : NormalMonoidalObject C) ↦ X_1.normalizeObj n) = Z.normalizeObj
have mk₂ : (x : NormalMonoidalObject C) → NormalMonoidalObject C
Tactic: apply Eq.trans
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y X_1 Y₁ Y₂ : F C
f : Y₁ ⟶ᵐ Y₂
ih : (fun (n : NormalMonoidalObject C) ↦ Y₁.normalizeObj n) = Y₂.normalizeObj
⊢ (fun (n : NormalMonoidalObject C) ↦ (X_1.tensor Y₁).normalizeObj n) = (X_1.tensor Y₂).normalizeObj
have mk : ∀ (x : NormalMonoidalObject C), (X_1.tensor Y₁).normalizeObj x = (X_1.tensor Y₂).normalizeObj x
Tactic: funext
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y X₁ X₂ : F C
f : X₁ ⟶ᵐ X₂
Y_1 : F C
ih : (fun (n : NormalMonoidalObject C) ↦ X₁.normalizeObj n) = X₂.normalizeObj
⊢ (fun (n : NormalMonoidalObject C) ↦ (X₁.tensor Y_1).normalizeObj n) = (X₂.tensor Y_1).normalizeObj
have mk : ∀ (x : NormalMonoidalObject C), (X₁.tensor Y_1).normalizeObj x = (X₂.tensor Y_1).normalizeObj x
Tactic: funext
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X_1 Y_1 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
have mk : ∀ (n : NormalMonoidalObject C), (W.tensor X).normalizeObj n = (Y.tensor Z).normalizeObj n
Tactic: funext n
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y X_1 : F C
⊢ (fun (n : NormalMonoidalObject C) ↦ X_1.normalizeObj n) = X_1.normalizeObj
have mk : ∀ (x : NormalMonoidalObject C), X_1.normalizeObj x = X_1.normalizeObj x
Tactic: funext
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y X_1 Y_1 Z : F C
⊢ (fun (n : NormalMonoidalObject C) ↦ ((X_1.tensor Y_1).tensor Z).normalizeObj n) =
    (X_1.tensor (Y_1.tensor Z)).normalizeObj
have mk : ∀ (x : NormalMonoidalObject C), ((X_1.tensor Y_1).tensor Z).normalizeObj x = (X_1.tensor (Y_1.tensor Z)).normalizeObj x
Tactic: funext
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y X_1 Y_1 Z : F C
⊢ (fun (n : NormalMonoidalObject C) ↦ (X_1.tensor (Y_1.tensor Z)).normalizeObj n) =
    ((X_1.tensor Y_1).tensor Z).normalizeObj
have mk : ∀ (x : NormalMonoidalObject C), (X_1.tensor (Y_1.tensor Z)).normalizeObj x = ((X_1.tensor Y_1).tensor Z).normalizeObj x
Tactic: funext
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y X_1 : F C
⊢ (fun (n : NormalMonoidalObject C) ↦ (tensor unit X_1).normalizeObj n) = X_1.normalizeObj
have mk : ∀ (x : NormalMonoidalObject C), (tensor unit X_1).normalizeObj x = X_1.normalizeObj x
Tactic: funext
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y X_1 : F C
⊢ (fun (n : NormalMonoidalObject C) ↦ X_1.normalizeObj n) = (tensor unit X_1).normalizeObj
have mk : ∀ (x : NormalMonoidalObject C), X_1.normalizeObj x = (tensor unit X_1).normalizeObj x
Tactic: funext
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y X_1 : F C
⊢ (fun (n : NormalMonoidalObject C) ↦ (X_1.tensor unit).normalizeObj n) = X_1.normalizeObj
have mk : ∀ (x : NormalMonoidalObject C), (X_1.tensor unit).normalizeObj x = X_1.normalizeObj x
Tactic: funext
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y X_1 : F C
⊢ (fun (n : NormalMonoidalObject C) ↦ X_1.normalizeObj n) = (X_1.tensor unit).normalizeObj
have mk : ∀ (x : NormalMonoidalObject C), X_1.normalizeObj x = (X_1.tensor unit).normalizeObj x
Tactic: funext
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Free.Coherence (goal)
Goal:
C : Type u
X Y : F C
f' : X ⟶ᵐ Y
⊢ (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Y.normalizeObj
have 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
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
⊢ (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Y.normalizeObj
have 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
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
⊢ (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Y.normalizeObj
have 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
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
⊢ (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Y.normalizeObj
have 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
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
⊢ (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Y.normalizeObj
have mk₄ : ∀ (X : F C), (fun (n : NormalMonoidalObject C) ↦ X.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
⊢ (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Y.normalizeObj
have mk₅ : ∀ (X Y Z : F C),
  (fun (n : NormalMonoidalObject C) ↦ ((X.tensor Y).tensor Z).normalizeObj n) = (X.tensor (Y.tensor Z)).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
⊢ (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Y.normalizeObj
have mk₆ : ∀ (X Y Z : F C),
  (fun (n : NormalMonoidalObject C) ↦ (X.tensor (Y.tensor Z)).normalizeObj n) = ((X.tensor Y).tensor Z).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
⊢ (fun (n : NormalMonoidalObject C) ↦ X.normalizeObj n) = Y.normalizeObj
have mk₇ : ∀ (X : F C), (fun (n : NormalMonoidalObject C) ↦ (tensor unit X).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'