awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
X Y : Functor.{v, v', u, u'} Cᵒᵖ A
x : X ⟶ Y
x_1 : Functor.{v, v', u, u'} Cᵒᵖ A
⊢ Eq (α :=
    obj Φ.presheafFiber (tensorObj (𝒞 := category) X x_1) ⟶
      tensorObj.{v', u'} (𝒞 := inst_1) (obj Φ.presheafFiber Y) (obj Φ.presheafFiber x_1))
    (Φ.presheafFiberDesc
        (fun (X_1 : C) (x : Φ.fiber.obj X_1) ↦ Φ.toPresheafFiber X_1 x X ⊗ₘ Φ.toPresheafFiber X_1 x x_1) ⋯ ≫
      map Φ.presheafFiber x ▷ obj Φ.presheafFiber x_1)
    (map Φ.presheafFiber (x ▷ x_1) ≫
      Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x Y ⊗ₘ Φ.toPresheafFiber X x x_1) ⋯)
have h : ∀ (X_1 : C) (x_2 : Φ.fiber.obj X_1),
  Eq (α :=
    (tensorObj (𝒞 := category) X x_1).obj (Opposite.op X_1) ⟶
      tensorObj.{v', u'} (𝒞 := inst_1) (obj Φ.presheafFiber Y) (obj Φ.presheafFiber x_1))
    (CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) X x_1))
      (Φ.toPresheafFiber X_1 x_2 (tensorObj (𝒞 := category) X x_1))
      (Φ.presheafFiberDesc
          (fun (X_2 : C) (x : Φ.fiber.obj X_2) ↦ Φ.toPresheafFiber X_2 x X ⊗ₘ Φ.toPresheafFiber X_2 x x_1) sorry ≫
        map Φ.presheafFiber x ▷ obj Φ.presheafFiber x_1))
    (CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) X x_1))
      (Φ.toPresheafFiber X_1 x_2 (tensorObj (𝒞 := category) X x_1))
      (map Φ.presheafFiber (x ▷ x_1) ≫
        Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x Y ⊗ₘ Φ.toPresheafFiber X x x_1) sorry))
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
X Y P : Functor.{v, v', u, u'} Cᵒᵖ A
f : X ⟶ Y
⊢ Eq (α :=
    obj Φ.presheafFiber (tensorObj (𝒞 := category) P X) ⟶
      tensorObj.{v', u'} (𝒞 := inst_1) (obj Φ.presheafFiber P) (obj Φ.presheafFiber Y))
    (Φ.presheafFiberDesc (fun (X_1 : C) (x : Φ.fiber.obj X_1) ↦ Φ.toPresheafFiber X_1 x P ⊗ₘ Φ.toPresheafFiber X_1 x X)
        ⋯ ≫
      obj Φ.presheafFiber P ◁ map Φ.presheafFiber f)
    (map Φ.presheafFiber (P ◁ f) ≫
      Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P ⊗ₘ Φ.toPresheafFiber X x Y) ⋯)
have h : ∀ (X_1 : C) (x : Φ.fiber.obj X_1),
  CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) P X))
      (Φ.toPresheafFiber X_1 x (tensorObj (𝒞 := category) P X))
      (Φ.presheafFiberDesc
          (fun (X_2 : C) (x : Φ.fiber.obj X_2) ↦ Φ.toPresheafFiber X_2 x P ⊗ₘ Φ.toPresheafFiber X_2 x X) sorry ≫
        obj Φ.presheafFiber P ◁ map Φ.presheafFiber f) =
    CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) P X))
      (Φ.toPresheafFiber X_1 x (tensorObj (𝒞 := category) P X))
      (map Φ.presheafFiber (P ◁ f) ≫
        Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P ⊗ₘ Φ.toPresheafFiber X x Y) sorry)
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ P₃ : Functor.{v, v', u, u'} Cᵒᵖ A
⊢ Eq (α :=
    obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃) ⟶
      tensorObj.{v', u'} (𝒞 := inst_1) (obj Φ.presheafFiber P₁)
        (tensorObj.{v', u'} (𝒞 := inst_1) (obj Φ.presheafFiber P₂) (obj Φ.presheafFiber P₃)))
    (Φ.presheafFiberDesc
        (fun (X : C) (x : Φ.fiber.obj X) ↦
          Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₁ P₂) ⊗ₘ Φ.toPresheafFiber X x P₃)
        ⋯ ≫
      Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂) ⋯ ▷
          obj Φ.presheafFiber P₃ ≫
        Iso.hom (α_ (obj Φ.presheafFiber P₁) (obj Φ.presheafFiber P₂) (obj Φ.presheafFiber P₃)))
    (map Φ.presheafFiber (Iso.hom (α_ P₁ P₂ P₃)) ≫
      Φ.presheafFiberDesc
          (fun (X : C) (x : Φ.fiber.obj X) ↦
            Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₂ P₃))
          ⋯ ≫
        obj Φ.presheafFiber P₁ ◁
          Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
            ⋯)
have h : ∀ (X : C) (x : Φ.fiber.obj X),
  CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
      (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
      (Φ.presheafFiberDesc
          (fun (X : C) (x : Φ.fiber.obj X) ↦
            Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₁ P₂) ⊗ₘ Φ.toPresheafFiber X x P₃)
          sorry ≫
        Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂) sorry ▷
            obj Φ.presheafFiber P₃ ≫
          Iso.hom (α_ (obj Φ.presheafFiber P₁) (obj Φ.presheafFiber P₂) (obj Φ.presheafFiber P₃))) =
    CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
      (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
      (map Φ.presheafFiber (Iso.hom (α_ P₁ P₂ P₃)) ≫
        Φ.presheafFiberDesc
            (fun (X : C) (x : Φ.fiber.obj X) ↦
              Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₂ P₃))
            sorry ≫
          obj Φ.presheafFiber P₁ ◁
            Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
              sorry)
Tactic: ext X x
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ P₃ : Functor.{v, v', u, u'} Cᵒᵖ A
X : C
x : Φ.fiber.obj X
| CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
    (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
    (Φ.presheafFiberDesc
        (fun (X : C) (x : Φ.fiber.obj X) ↦
          Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₁ P₂) ⊗ₘ Φ.toPresheafFiber X x P₃)
        ⋯ ≫
      Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂) ⋯ ▷
          obj Φ.presheafFiber P₃ ≫
        Iso.hom (α_ (obj Φ.presheafFiber P₁) (obj Φ.presheafFiber P₂) (obj Φ.presheafFiber P₃)))
have simp : tensorObj.{v', u'} (𝒞 := inst_1) (P₁.obj (Opposite.op X)) (P₂.obj (Opposite.op X)) ◁ Φ.toPresheafFiber X x P₃ ≫
  (Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂) ▷ obj Φ.presheafFiber P₃ ≫
    Iso.hom (α_ (obj Φ.presheafFiber P₁) (obj Φ.presheafFiber P₂) (obj Φ.presheafFiber P₃))
Tactic: simp only [toPresheafFiber_presheafFiberDesc_assoc, tensorHom_def'_assoc,
        ← comp_whiskerRight_assoc, toPresheafFiber_presheafFiberDesc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ P₃ : Functor.{v, v', u, u'} Cᵒᵖ A
X : C
x : Φ.fiber.obj X
| tensorObj.{v', u'} (𝒞 := inst_1) (P₁.obj (Opposite.op X)) (P₂.obj (Opposite.op X)) ◁ Φ.toPresheafFiber X x P₃ ≫
    (Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂) ▷ obj Φ.presheafFiber P₃ ≫
      Iso.hom (α_ (obj Φ.presheafFiber P₁) (obj Φ.presheafFiber P₂) (obj Φ.presheafFiber P₃))
have rw : Iso.hom (α_ (P₁.obj (Opposite.op X)) (P₂.obj (Opposite.op X)) (P₃.obj (Opposite.op X))) ≫
  (Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
Tactic: rw [← tensorHom_def'_assoc, associator_naturality]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ P₃ : Functor.{v, v', u, u'} Cᵒᵖ A
X : C
x : Φ.fiber.obj X
| CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
    (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
    (Φ.presheafFiberDesc
        (fun (X : C) (x : Φ.fiber.obj X) ↦
          Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₁ P₂) ⊗ₘ Φ.toPresheafFiber X x P₃)
        ⋯ ≫
      Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂) ⋯ ▷
          obj Φ.presheafFiber P₃ ≫
        Iso.hom (α_ (obj Φ.presheafFiber P₁) (obj Φ.presheafFiber P₂) (obj Φ.presheafFiber P₃)))
have simp : Iso.hom (α_ (P₁.obj (Opposite.op X)) (P₂.obj (Opposite.op X)) (P₃.obj (Opposite.op X))) ≫
  (Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
Tactic: simp only [toPresheafFiber_presheafFiberDesc_assoc, tensorHom_def'_assoc,
        ← comp_whiskerRight_assoc, toPresheafFiber_presheafFiberDesc]
      rw [← tensorHom_def'_assoc, associator_naturality]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ P₃ : Functor.{v, v', u, u'} Cᵒᵖ A
X : C
x : Φ.fiber.obj X
⊢ CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
      (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
      (Φ.presheafFiberDesc
          (fun (X : C) (x : Φ.fiber.obj X) ↦
            Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₁ P₂) ⊗ₘ Φ.toPresheafFiber X x P₃)
          ⋯ ≫
        Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂) ⋯ ▷
            obj Φ.presheafFiber P₃ ≫
          Iso.hom (α_ (obj Φ.presheafFiber P₁) (obj Φ.presheafFiber P₂) (obj Φ.presheafFiber P₃))) =
    CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
      (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
      (map Φ.presheafFiber (Iso.hom (α_ P₁ P₂ P₃)) ≫
        Φ.presheafFiberDesc
            (fun (X : C) (x : Φ.fiber.obj X) ↦
              Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₂ P₃))
            ⋯ ≫
          obj Φ.presheafFiber P₁ ◁
            Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
              ⋯)
have h : Eq (α :=
  tensorObj.{v', u'} (𝒞 := inst_1) (tensorObj.{v', u'} (𝒞 := inst_1) (P₁.obj (Opposite.op X)) (P₂.obj (Opposite.op X)))
      (P₃.obj (Opposite.op X)) ⟶
    tensorObj.{v', u'} (𝒞 := inst_1) (obj Φ.presheafFiber P₁)
      (tensorObj.{v', u'} (𝒞 := inst_1) (obj Φ.presheafFiber P₂) (obj Φ.presheafFiber P₃)))
  (Iso.hom (α_ (P₁.obj (Opposite.op X)) (P₂.obj (Opposite.op X)) (P₃.obj (Opposite.op X))) ≫
    (Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃))
  (CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
    (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
    (map Φ.presheafFiber (Iso.hom (α_ P₁ P₂ P₃)) ≫
      Φ.presheafFiberDesc
          (fun (X : C) (x : Φ.fiber.obj X) ↦
            Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₂ P₃))
          sorry ≫
        obj Φ.presheafFiber P₁ ◁
          Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
            sorry))
Tactic: conv_lhs =>
      simp only [toPresheafFiber_presheafFiberDesc_assoc, tensorHom_def'_assoc,
        ← comp_whiskerRight_assoc, toPresheafFiber_presheafFiberDesc]
      rw [← tensorHom_def'_assoc, associator_naturality]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ P₃ : Functor.{v, v', u, u'} Cᵒᵖ A
X : C
x : Φ.fiber.obj X
| CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
    (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
    (map Φ.presheafFiber (Iso.hom (α_ P₁ P₂ P₃)) ≫
      Φ.presheafFiberDesc
          (fun (X : C) (x : Φ.fiber.obj X) ↦
            Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₂ P₃))
          ⋯ ≫
        obj Φ.presheafFiber P₁ ◁
          Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
            ⋯)
have simp : NatTrans.app (Iso.hom (α_ P₁ P₂ P₃)) (Opposite.op X) ≫
  Φ.toPresheafFiber X x P₁ ▷ tensorObj.{v', u'} (𝒞 := inst_1) (P₂.obj (Opposite.op X)) (P₃.obj (Opposite.op X)) ≫
    obj Φ.presheafFiber P₁ ◁ (Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
Tactic: simp only [toPresheafFiber_naturality_assoc, toPresheafFiber_presheafFiberDesc_assoc,
        tensorHom_def_assoc, ← MonoidalCategory.whiskerLeft_comp,
        toPresheafFiber_presheafFiberDesc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ P₃ : Functor.{v, v', u, u'} Cᵒᵖ A
X : C
x : Φ.fiber.obj X
| NatTrans.app (Iso.hom (α_ P₁ P₂ P₃)) (Opposite.op X) ≫
    Φ.toPresheafFiber X x P₁ ▷ tensorObj.{v', u'} (𝒞 := inst_1) (P₂.obj (Opposite.op X)) (P₃.obj (Opposite.op X)) ≫
      obj Φ.presheafFiber P₁ ◁ (Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
have rw : NatTrans.app (Iso.hom (α_ P₁ P₂ P₃)) (Opposite.op X) ≫
  (Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
Tactic: rw [← tensorHom_def]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ P₃ : Functor.{v, v', u, u'} Cᵒᵖ A
X : C
x : Φ.fiber.obj X
| NatTrans.app (Iso.hom (α_ P₁ P₂ P₃)) (Opposite.op X) ≫
    (Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
have dsimp : Iso.hom (α_ (P₁.obj (Opposite.op X)) (P₂.obj (Opposite.op X)) (P₃.obj (Opposite.op X))) ≫
  (Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ P₃ : Functor.{v, v', u, u'} Cᵒᵖ A
X : C
x : Φ.fiber.obj X
| CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
    (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
    (map Φ.presheafFiber (Iso.hom (α_ P₁ P₂ P₃)) ≫
      Φ.presheafFiberDesc
          (fun (X : C) (x : Φ.fiber.obj X) ↦
            Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₂ P₃))
          ⋯ ≫
        obj Φ.presheafFiber P₁ ◁
          Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
            ⋯)
have simp : Iso.hom (α_ (P₁.obj (Opposite.op X)) (P₂.obj (Opposite.op X)) (P₃.obj (Opposite.op X))) ≫
  (Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
Tactic: simp only [toPresheafFiber_naturality_assoc, toPresheafFiber_presheafFiberDesc_assoc,
        tensorHom_def_assoc, ← MonoidalCategory.whiskerLeft_comp,
        toPresheafFiber_presheafFiberDesc]
      rw [← tensorHom_def]
      dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ P₃ : Functor.{v, v', u, u'} Cᵒᵖ A
X : C
x : Φ.fiber.obj X
⊢ Eq (α :=
    tensorObj.{v', u'} (𝒞 := inst_1)
        (tensorObj.{v', u'} (𝒞 := inst_1) (P₁.obj (Opposite.op X)) (P₂.obj (Opposite.op X))) (P₃.obj (Opposite.op X)) ⟶
      tensorObj.{v', u'} (𝒞 := inst_1) (obj Φ.presheafFiber P₁)
        (tensorObj.{v', u'} (𝒞 := inst_1) (obj Φ.presheafFiber P₂) (obj Φ.presheafFiber P₃)))
    (Iso.hom (α_ (P₁.obj (Opposite.op X)) (P₂.obj (Opposite.op X)) (P₃.obj (Opposite.op X))) ≫
      (Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃))
    (CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
      (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
      (map Φ.presheafFiber (Iso.hom (α_ P₁ P₂ P₃)) ≫
        Φ.presheafFiberDesc
            (fun (X : C) (x : Φ.fiber.obj X) ↦
              Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₂ P₃))
            ⋯ ≫
          obj Φ.presheafFiber P₁ ◁
            Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
              ⋯))
have conv_rhs : Eq (α :=
  tensorObj.{v', u'} (𝒞 := inst_1) (tensorObj.{v', u'} (𝒞 := inst_1) (P₁.obj (Opposite.op X)) (P₂.obj (Opposite.op X)))
      (P₃.obj (Opposite.op X)) ⟶
    tensorObj.{v', u'} (𝒞 := inst_1) (obj Φ.presheafFiber P₁)
      (tensorObj.{v', u'} (𝒞 := inst_1) (obj Φ.presheafFiber P₂) (obj Φ.presheafFiber P₃)))
  (Iso.hom (α_ (P₁.obj (Opposite.op X)) (P₂.obj (Opposite.op X)) (P₃.obj (Opposite.op X))) ≫
    (Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃))
  (CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
    (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
    (map Φ.presheafFiber (Iso.hom (α_ P₁ P₂ P₃)) ≫
      Φ.presheafFiberDesc
          (fun (X : C) (x : Φ.fiber.obj X) ↦
            Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₂ P₃))
          sorry ≫
        obj Φ.presheafFiber P₁ ◁
          Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
            sorry))
Tactic: conv_rhs =>
      simp only [toPresheafFiber_naturality_assoc, toPresheafFiber_presheafFiberDesc_assoc,
        tensorHom_def_assoc, ← MonoidalCategory.whiskerLeft_comp,
        toPresheafFiber_presheafFiberDesc]
      rw [← tensorHom_def]
      dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
x : Functor.{v, v', u, u'} Cᵒᵖ A
⊢ Eq (α :=
    obj Φ.presheafFiber x ⟶
      tensorObj.{v', u'} (𝒞 := inst_1) (tensorUnit.{v', u'} A (𝒞 := inst_1)) (obj Φ.presheafFiber x))
    (Iso.inv (λ_ (obj Φ.presheafFiber x)))
    (map Φ.presheafFiber (Iso.inv (λ_ x)) ≫
      Φ.presheafFiberDesc
          (fun (X : C) (x_1 : Φ.fiber.obj X) ↦
            Φ.toPresheafFiber X x_1 ((𝟙_ (Functor.{v, v', u, u'} Cᵒᵖ A)) (𝒞 := category)) ⊗ₘ Φ.toPresheafFiber X x_1 x)
          ⋯ ≫
        Φ.presheafFiberDesc
            (fun (x : C) (x_1 : Φ.fiber.obj x) ↦
              𝟙 (obj.{v, v', u, u'} ((𝟙_ (Functor.{v, v', u, u'} Cᵒᵖ A)) (𝒞 := category)) (Opposite.op x)))
            ⋯ ▷
          obj Φ.presheafFiber x)
have h : ∀ (X : C) (x_1 : Φ.fiber.obj X),
  Eq (α :=
    x.obj (Opposite.op X) ⟶
      tensorObj.{v', u'} (𝒞 := inst_1) (tensorUnit.{v', u'} A (𝒞 := inst_1)) (obj Φ.presheafFiber x))
    (Φ.toPresheafFiber X x_1 x ≫ Iso.inv (λ_ (obj Φ.presheafFiber x)))
    (CategoryStruct.comp (Y := obj Φ.presheafFiber x) (Φ.toPresheafFiber X x_1 x)
      (map Φ.presheafFiber (Iso.inv (λ_ x)) ≫
        Φ.presheafFiberDesc
            (fun (X : C) (x_2 : Φ.fiber.obj X) ↦
              Φ.toPresheafFiber X x_2 ((𝟙_ (Functor.{v, v', u, u'} Cᵒᵖ A)) (𝒞 := category)) ⊗ₘ
                Φ.toPresheafFiber X x_2 x)
            sorry ≫
          Φ.presheafFiberDesc
              (fun (x : C) (x_2 : Φ.fiber.obj x) ↦
                𝟙 (obj.{v, v', u, u'} ((𝟙_ (Functor.{v, v', u, u'} Cᵒᵖ A)) (𝒞 := category)) (Opposite.op x)))
              sorry ▷
            obj Φ.presheafFiber x))
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Point.Monoidal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
x : Functor.{v, v', u, u'} Cᵒᵖ A
⊢ Eq (α :=
    obj Φ.presheafFiber x ⟶
      tensorObj.{v', u'} (𝒞 := inst_1) (obj Φ.presheafFiber x) (tensorUnit.{v', u'} A (𝒞 := inst_1)))
    (Iso.inv (ρ_ (obj Φ.presheafFiber x)))
    (map Φ.presheafFiber (Iso.inv (ρ_ x)) ≫
      Φ.presheafFiberDesc
          (fun (X : C) (x_1 : Φ.fiber.obj X) ↦
            Φ.toPresheafFiber X x_1 x ⊗ₘ Φ.toPresheafFiber X x_1 ((𝟙_ (Functor.{v, v', u, u'} Cᵒᵖ A)) (𝒞 := category)))
          ⋯ ≫
        obj Φ.presheafFiber x ◁
          Φ.presheafFiberDesc
            (fun (x : C) (x_1 : Φ.fiber.obj x) ↦
              𝟙 (obj.{v, v', u, u'} ((𝟙_ (Functor.{v, v', u, u'} Cᵒᵖ A)) (𝒞 := category)) (Opposite.op x)))
            ⋯)
have h : ∀ (X : C) (x_1 : Φ.fiber.obj X),
  Eq (α :=
    x.obj (Opposite.op X) ⟶
      tensorObj.{v', u'} (𝒞 := inst_1) (obj Φ.presheafFiber x) (tensorUnit.{v', u'} A (𝒞 := inst_1)))
    (Φ.toPresheafFiber X x_1 x ≫ Iso.inv (ρ_ (obj Φ.presheafFiber x)))
    (CategoryStruct.comp (Y := obj Φ.presheafFiber x) (Φ.toPresheafFiber X x_1 x)
      (map Φ.presheafFiber (Iso.inv (ρ_ x)) ≫
        Φ.presheafFiberDesc
            (fun (X : C) (x_2 : Φ.fiber.obj X) ↦
              Φ.toPresheafFiber X x_2 x ⊗ₘ
                Φ.toPresheafFiber X x_2 ((𝟙_ (Functor.{v, v', u, u'} Cᵒᵖ A)) (𝒞 := category)))
            sorry ≫
          obj Φ.presheafFiber x ◁
            Φ.presheafFiberDesc
              (fun (x : C) (x_2 : Φ.fiber.obj x) ↦
                𝟙 (obj.{v, v', u, u'} ((𝟙_ (Functor.{v, v', u, u'} Cᵒᵖ A)) (𝒞 := category)) (Opposite.op x)))
              sorry))
Tactic: ext