awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Monoidal (goal)
Goal:
I : Type u
inst : AddMonoid I
C : Type u_1
inst_1 : Category.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
X₁ X₂ X₃ X₄ : GradedObject I C
inst_3 : HasTensor.{u, v_1, u_1} X₃ X₄
inst_4 : HasTensor.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)
inst_5 : HasTensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄))
i₁ i₂ i₃ i₄ j : I
h : i₁ + i₂ + i₃ + i₄ = j
i₂₃₄ : I
hi : i₂ + i₃ + i₄ = i₂₃₄
⊢ Eq.{v_1 + 1} (α :=
    MonoidalCategoryStruct.tensorObj.{v_1, u_1} (𝒞 := inst_1) (X₁ i₁)
        (MonoidalCategoryStruct.tensorObj.{v_1, u_1} (𝒞 := inst_1) (X₂ i₂)
          (MonoidalCategoryStruct.tensorObj.{v_1, u_1} (𝒞 := inst_1) (X₃ i₃) (X₄ i₄))) ⟶
      tensorObj.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)) j)
    (ιTensorObj₄ X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h)
    (X₁ i₁ ◁ ιTensorObj₃ X₂ X₃ X₄ i₂ i₃ i₄ i₂₃₄ hi ≫
      ιTensorObj X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)) i₁ i₂₃₄ j ⋯)
have subst : Eq.{v_1 + 1} (α :=
  MonoidalCategoryStruct.tensorObj.{v_1, u_1} (𝒞 := inst_1) (X₁ i₁)
      (MonoidalCategoryStruct.tensorObj.{v_1, u_1} (𝒞 := inst_1) (X₂ i₂)
        (MonoidalCategoryStruct.tensorObj.{v_1, u_1} (𝒞 := inst_1) (X₃ i₃) (X₄ i₄))) ⟶
    tensorObj.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)) j)
  (ιTensorObj₄ X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h)
  (X₁ i₁ ◁ ιTensorObj₃ X₂ X₃ X₄ i₂ i₃ i₄ (i₂ + i₃ + i₄) sorry ≫
    ιTensorObj X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)) i₁ (i₂ + i₃ + i₄) j sorry)
Tactic: subst hi
Cleared hypotheses: i₂₃₄, hi
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Monoidal (goal)
Goal:
I : Type u
inst : AddMonoid I
C : Type u_1
inst_1 : Category.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
X₁ X₂ X₃ X₄ : GradedObject I C
inst_3 : HasTensor.{u, v_1, u_1} X₃ X₄
inst_4 : HasTensor.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)
inst_5 : HasTensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄))
j : I
A : C
f g : tensorObj.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)) j ⟶ A
inst_6 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₂ X₃ X₄
H : HasTensor₄ObjExt.{u, v_1, u_1} X₁ X₂ X₃ X₄
h :
  ∀ (i₁ i₂ i₃ i₄ : I) (h : i₁ + i₂ + i₃ + i₄ = j),
    ιTensorObj₄ X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h ≫ f = ιTensorObj₄ X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h ≫ g
⊢ f = g
have h₁ : ∀ (i₁ i₂ : I) (hi : i₁ + i₂ = j),
  ιTensorObj X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)) i₁ i₂ j hi ≫ f =
    ιTensorObj X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)) i₁ i₂ j hi ≫ g
Tactic: apply tensorObj_ext
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Monoidal (goal)
Goal:
I : Type u
inst : AddMonoid I
C : Type u_1
inst_1 : Category.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
X₁ X₂ X₃ X₄ : GradedObject I C
inst_3 : HasTensor.{u, v_1, u_1} X₃ X₄
inst_4 : HasTensor.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)
inst_5 : HasTensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄))
j : I
A : C
f g : tensorObj.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)) j ⟶ A
inst_6 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₂ X₃ X₄
H : HasTensor₄ObjExt.{u, v_1, u_1} X₁ X₂ X₃ X₄
h :
  ∀ (i₁ i₂ i₃ i₄ : I) (h : i₁ + i₂ + i₃ + i₄ = j),
    ιTensorObj₄ X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h ≫ f = ιTensorObj₄ X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h ≫ g
i₁ i₂₃₄ : I
h' : i₁ + i₂₃₄ = j
⊢ ιTensorObj X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)) i₁ i₂₃₄ j h' ≫ f =
    ιTensorObj X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)) i₁ i₂₃₄ j h' ≫ g
have h₁ : ∀ (i₁_1 i₂ i₃ : I) (h : i₁_1 + i₂ + i₃ = i₂₃₄),
  X₁ i₁ ◁ ιTensorObj₃ X₂ X₃ X₄ i₁_1 i₂ i₃ i₂₃₄ h ≫
      ιTensorObj X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)) i₁ i₂₃₄ j h' ≫ f =
    X₁ i₁ ◁ ιTensorObj₃ X₂ X₃ X₄ i₁_1 i₂ i₃ i₂₃₄ h ≫
      ιTensorObj X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)) i₁ i₂₃₄ j h' ≫ g
Tactic: apply left_tensor_tensorObj₃_ext
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Monoidal (hypothesis)
Goal:
I : Type u
inst : AddMonoid I
C : Type u_1
inst_1 : Category.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
X₁ X₂ X₃ X₄ : GradedObject I C
inst_3 : HasTensor.{u, v_1, u_1} X₃ X₄
inst_4 : HasTensor.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)
inst_5 : HasTensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄))
j : I
A : C
f g : tensorObj.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)) j ⟶ A
inst_6 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₂ X₃ X₄
H : HasTensor₄ObjExt.{u, v_1, u_1} X₁ X₂ X₃ X₄
h :
  ∀ (i₁ i₂ i₃ i₄ : I) (h : i₁ + i₂ + i₃ + i₄ = j),
    ιTensorObj₄ X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h ≫ f = ιTensorObj₄ X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h ≫ g
i₁ i₂₃₄ : I
h' : i₁ + i₂₃₄ = j
i₂ i₃ i₄ : I
h'' : i₂ + i₃ + i₄ = i₂₃₄
⊢ X₁ i₁ ◁ ιTensorObj₃ X₂ X₃ X₄ i₂ i₃ i₄ i₂₃₄ h'' ≫
      ιTensorObj X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)) i₁ i₂₃₄ j h' ≫ f =
    X₁ i₁ ◁ ιTensorObj₃ X₂ X₃ X₄ i₂ i₃ i₄ i₂₃₄ h'' ≫
      ιTensorObj X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)) i₁ i₂₃₄ j h' ≫ g
have hj : i₁ + i₂ + i₃ + i₄ = j
Tactic: have hj : i₁ + i₂ + i₃ + i₄ = j := by simp only [← h', ← h'', add_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Monoidal (goal)
Goal:
I : Type u
inst : AddMonoid I
C : Type u_1
inst_1 : Category.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
X₁ X₂ X₃ X₄ : GradedObject I C
inst_3 : HasTensor.{u, v_1, u_1} X₁ X₂
inst_4 : HasTensor.{u, v_1, u_1} X₂ X₃
inst_5 : HasTensor.{u, v_1, u_1} X₃ X₄
inst_6 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) X₃
inst_7 : HasTensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ X₃)
inst_8 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₂ X₃) X₄
inst_9 : HasTensor.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)
inst_10 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) X₃) X₄
inst_11 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ X₃)) X₄
inst_12 : HasTensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₂ X₃) X₄)
inst_13 : HasTensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄))
inst_14 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) (tensorObj.{u, v_1, u_1} X₃ X₄)
inst_15 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} X₁ X₂ X₃
inst_16 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₁ X₂ X₃
inst_17 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ X₃) X₄
inst_18 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ X₃) X₄
inst_19 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} X₂ X₃ X₄
inst_20 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₂ X₃ X₄
inst_21 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) X₃ X₄
inst_22 : HasGoodTensorTensor₂₃.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) X₃ X₄
inst_23 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} X₁ X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)
inst_24 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₁ X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)
inst_25 : HasTensor₄ObjExt.{u, v_1, u_1} X₁ X₂ X₃ X₄
j i₁ i₂ i₃ i₄ : I
h : i₁ + i₂ + i₃ + i₄ = j
⊢ i₁ + (i₂ + i₃) = i₁ + i₂ + i₃
have rw : i₁ + (i₂ + i₃) = i₁ + (i₂ + i₃)
Tactic: rw [add_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Monoidal (goal)
Goal:
I : Type u
inst : AddMonoid I
C : Type u_1
inst_1 : Category.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
X₁ X₂ X₃ X₄ : GradedObject I C
inst_3 : HasTensor.{u, v_1, u_1} X₁ X₂
inst_4 : HasTensor.{u, v_1, u_1} X₂ X₃
inst_5 : HasTensor.{u, v_1, u_1} X₃ X₄
inst_6 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) X₃
inst_7 : HasTensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ X₃)
inst_8 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₂ X₃) X₄
inst_9 : HasTensor.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)
inst_10 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) X₃) X₄
inst_11 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ X₃)) X₄
inst_12 : HasTensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₂ X₃) X₄)
inst_13 : HasTensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄))
inst_14 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) (tensorObj.{u, v_1, u_1} X₃ X₄)
inst_15 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} X₁ X₂ X₃
inst_16 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₁ X₂ X₃
inst_17 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ X₃) X₄
inst_18 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ X₃) X₄
inst_19 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} X₂ X₃ X₄
inst_20 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₂ X₃ X₄
inst_21 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) X₃ X₄
inst_22 : HasGoodTensorTensor₂₃.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) X₃ X₄
inst_23 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} X₁ X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)
inst_24 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₁ X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)
inst_25 : HasTensor₄ObjExt.{u, v_1, u_1} X₁ X₂ X₃ X₄
j i₁ i₂ i₃ i₄ : I
h rw : i₁ + i₂ + i₃ + i₄ = j
⊢ i₁ + i₂ + (i₃ + i₄) = j
have rw₁ : j = j
Tactic: rw [← add_assoc, h]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Monoidal (goal)
Goal:
I : Type u
inst : AddMonoid I
C : Type u_1
inst_1 : Category.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
X₁ X₂ X₃ X₄ : GradedObject I C
inst_3 : HasTensor.{u, v_1, u_1} X₁ X₂
inst_4 : HasTensor.{u, v_1, u_1} X₂ X₃
inst_5 : HasTensor.{u, v_1, u_1} X₃ X₄
inst_6 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) X₃
inst_7 : HasTensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ X₃)
inst_8 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₂ X₃) X₄
inst_9 : HasTensor.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)
inst_10 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) X₃) X₄
inst_11 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ X₃)) X₄
inst_12 : HasTensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₂ X₃) X₄)
inst_13 : HasTensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ (tensorObj.{u, v_1, u_1} X₃ X₄))
inst_14 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) (tensorObj.{u, v_1, u_1} X₃ X₄)
inst_15 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} X₁ X₂ X₃
inst_16 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₁ X₂ X₃
inst_17 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ X₃) X₄
inst_18 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} X₂ X₃) X₄
inst_19 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} X₂ X₃ X₄
inst_20 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₂ X₃ X₄
inst_21 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) X₃ X₄
inst_22 : HasGoodTensorTensor₂₃.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ X₂) X₃ X₄
inst_23 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} X₁ X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)
inst_24 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₁ X₂ (tensorObj.{u, v_1, u_1} X₃ X₄)
inst_25 : HasTensor₄ObjExt.{u, v_1, u_1} X₁ X₂ X₃ X₄
j i₁ i₂ i₃ i₄ : I
h : i₁ + i₂ + i₃ + i₄ = j
⊢ i₂ + (i₃ + i₄) = i₂ + i₃ + i₄
have rw : i₂ + (i₃ + i₄) = i₂ + (i₃ + i₄)
Tactic: rw [add_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Monoidal (goal)
Goal:
I : Type u
inst : AddMonoid I
C : Type u_1
inst_1 : Category.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
inst_3 : DecidableEq I
inst_4 : HasInitial.{v_1, u_1} C
inst_5 : ∀ (X₁ : C), PreservesColimit.{0, 0, v_1, v_1, u_1, u_1} (Functor.empty C) (Functor.obj (curriedTensor C) X₁)
inst_6 :
  ∀ (X₂ : C),
    PreservesColimit.{0, 0, v_1, v_1, u_1, u_1} (Functor.empty C) (Functor.obj (Functor.flip (curriedTensor C)) X₂)
X₁ X₃ : GradedObject I C
inst_7 : HasTensor.{u, v_1, u_1} X₁ X₃
inst_8 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ tensorUnit.{u, v_1, u_1}) X₃
inst_9 : HasTensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} tensorUnit.{u, v_1, u_1} X₃)
inst_10 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} X₁ tensorUnit.{u, v_1, u_1} X₃
inst_11 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₁ tensorUnit.{u, v_1, u_1} X₃
⊢ Eq (α :=
    tensorObj.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ tensorUnit.{u, v_1, u_1}) X₃ ⟶ tensorObj.{u, v_1, u_1} X₁ X₃)
    (CategoryStruct.comp (Y := tensorObj.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} tensorUnit.{u, v_1, u_1} X₃))
        (Iso.hom (associator X₁ tensorUnit.{u, v_1, u_1} X₃) :
          tensorObj.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ tensorUnit.{u, v_1, u_1}) X₃ ⟶
            tensorObj.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} tensorUnit.{u, v_1, u_1} X₃))
        (tensorHom (𝟙 X₁) (Iso.hom (leftUnitor X₃)) :
          tensorObj.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} tensorUnit.{u, v_1, u_1} X₃) ⟶
            tensorObj.{u, v_1, u_1} X₁ X₃) :
      tensorObj.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ tensorUnit.{u, v_1, u_1}) X₃ ⟶ tensorObj.{u, v_1, u_1} X₁ X₃)
    (tensorHom (Iso.hom (rightUnitor X₁)) (𝟙 X₃) :
      tensorObj.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ tensorUnit.{u, v_1, u_1}) X₃ ⟶ tensorObj.{u, v_1, u_1} X₁ X₃)
have convert_3 : HasGoodTrifunctor₁₂Obj.{v_1, u_1, v_1, u_1, v_1, u_1, v_1, u_1, v_1, u_1, u, u, u, u, u} (curriedTensor C)
  (curriedTensor C) (r := r₁₂₃) (TriangleIndexData.ρ₁₂ (triangleIndexData I)) X₁
  (Functor.obj (single₀ I) (MonoidalCategoryStruct.tensorUnit.{v_1, u_1} C (𝒞 := inst_1))) X₃
Tactic: convert!
    mapBifunctor_triangle (curriedAssociatorNatIso C) (𝟙_ C) (rightUnitorNatIso C)
      (leftUnitorNatIso C) (triangleIndexData I) X₁ X₃ (by simp)
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Monoidal (goal)
Goal:
I : Type u
inst : AddMonoid I
C : Type u_1
inst_1 : Category.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
inst_3 : DecidableEq I
inst_4 : HasInitial.{v_1, u_1} C
inst_5 : ∀ (X₁ : C), PreservesColimit.{0, 0, v_1, v_1, u_1, u_1} (Functor.empty C) (Functor.obj (curriedTensor C) X₁)
inst_6 :
  ∀ (X₂ : C),
    PreservesColimit.{0, 0, v_1, v_1, u_1, u_1} (Functor.empty C) (Functor.obj (Functor.flip (curriedTensor C)) X₂)
X₁ X₃ : GradedObject I C
inst_7 : HasTensor.{u, v_1, u_1} X₁ X₃
inst_8 : HasTensor.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ tensorUnit.{u, v_1, u_1}) X₃
inst_9 : HasTensor.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} tensorUnit.{u, v_1, u_1} X₃)
inst_10 : HasGoodTensor₁₂Tensor.{u, v_1, u_1} X₁ tensorUnit.{u, v_1, u_1} X₃
inst_11 : HasGoodTensorTensor₂₃.{u, v_1, u_1} X₁ tensorUnit.{u, v_1, u_1} X₃
convert_3 :
  HasGoodTrifunctor₁₂Obj.{v_1, u_1, v_1, u_1, v_1, u_1, v_1, u_1, v_1, u_1, u, u, u, u, u} (curriedTensor C)
    (curriedTensor C) (r := r₁₂₃) (TriangleIndexData.ρ₁₂ (triangleIndexData I)) X₁
    (Functor.obj (single₀ I) (MonoidalCategoryStruct.tensorUnit.{v_1, u_1} C (𝒞 := inst_1))) X₃
⊢ Eq (α :=
    tensorObj.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ tensorUnit.{u, v_1, u_1}) X₃ ⟶ tensorObj.{u, v_1, u_1} X₁ X₃)
    (CategoryStruct.comp (Y := tensorObj.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} tensorUnit.{u, v_1, u_1} X₃))
        (Iso.hom (associator X₁ tensorUnit.{u, v_1, u_1} X₃) :
          tensorObj.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ tensorUnit.{u, v_1, u_1}) X₃ ⟶
            tensorObj.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} tensorUnit.{u, v_1, u_1} X₃))
        (tensorHom (𝟙 X₁) (Iso.hom (leftUnitor X₃)) :
          tensorObj.{u, v_1, u_1} X₁ (tensorObj.{u, v_1, u_1} tensorUnit.{u, v_1, u_1} X₃) ⟶
            tensorObj.{u, v_1, u_1} X₁ X₃) :
      tensorObj.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ tensorUnit.{u, v_1, u_1}) X₃ ⟶ tensorObj.{u, v_1, u_1} X₁ X₃)
    (tensorHom (Iso.hom (rightUnitor X₁)) (𝟙 X₃) :
      tensorObj.{u, v_1, u_1} (tensorObj.{u, v_1, u_1} X₁ tensorUnit.{u, v_1, u_1}) X₃ ⟶ tensorObj.{u, v_1, u_1} X₁ X₃)
have convert_4 : HasGoodTrifunctor₂₃Obj.{v_1, u_1, v_1, u_1, v_1, u_1, v_1, u_1, v_1, u_1, u, u, u, u, u} (curriedTensor C)
  (curriedTensor C) (r := r₁₂₃) (TriangleIndexData.ρ₂₃ (triangleIndexData I)) X₁
  (Functor.obj (single₀ I) (MonoidalCategoryStruct.tensorUnit.{v_1, u_1} C (𝒞 := inst_1))) X₃
Tactic: convert!
    mapBifunctor_triangle (curriedAssociatorNatIso C) (𝟙_ C) (rightUnitorNatIso C)
      (leftUnitorNatIso C) (triangleIndexData I) X₁ X₃ (by simp)
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Monoidal (goal)
Goal:
I : Type u
inst : AddMonoid I
C : Type u_1
inst_1 : Category.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
n : ℕ
⊢ Finite (↑(Set.preimage (β := ℕ) (fun (i : ℕ × ℕ) ↦ i.1 + i.2) {n}) : Type)
have refine : Function.Injective (β := Fin (n + 1) × Fin (n + 1))
  fun (x : (↑(Set.preimage (β := ℕ) (fun (i : ℕ × ℕ) ↦ i.1 + i.2) {n}) : Type)) ↦
  match (motive := (↑(Set.preimage (β := ℕ) (fun (i : ℕ × ℕ) ↦ i.1 + i.2) {n}) : Type) → Fin (n + 1) × Fin (n + 1))
    x with
  | ⟨(i₁, i₂), hi⟩ => (⟨i₁, sorry⟩, ⟨i₂, sorry⟩)
Tactic: refine Finite.of_injective (fun ⟨⟨i₁, i₂⟩, (hi : i₁ + i₂ = n)⟩ =>
    ((⟨i₁, by lia⟩, ⟨i₂, by lia⟩) : Fin (n + 1) × Fin (n + 1))) ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Monoidal (goal)
Goal:
I : Type u
inst : AddMonoid I
C : Type u_1
inst_1 : Category.{v_1, u_1} C
inst_2 : MonoidalCategory.{v_1, u_1} C
n : ℕ
⊢ Finite (↑{i : ℕ × ℕ × ℕ | i.1 + i.2.1 + i.2.2 = n} : Type)
have refine : Function.Injective (β := Fin (n + 1) × Fin (n + 1) × Fin (n + 1))
  fun (x : (↑{i : ℕ × ℕ × ℕ | i.1 + i.2.1 + i.2.2 = n} : Type)) ↦
  match (motive := (↑{i : ℕ × ℕ × ℕ | i.1 + i.2.1 + i.2.2 = n} : Type) → Fin (n + 1) × Fin (n + 1) × Fin (n + 1)) x with
  | ⟨(i₁, i₂, i₃), hi⟩ => (⟨i₁, sorry⟩, ⟨i₂, sorry⟩, ⟨i₃, sorry⟩)
Tactic: refine Finite.of_injective (fun ⟨⟨i₁, i₂, i₃⟩, (hi : i₁ + i₂ + i₃ = n)⟩ =>
    (⟨⟨i₁, by lia⟩, ⟨i₂, by lia⟩, ⟨i₃, by lia⟩⟩ :
      Fin (n + 1) × Fin (n + 1) × Fin (n + 1))) ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Single (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_2} C
inst_1 : HasInitial.{v_1, u_2} C
inst_2 : DecidableEq J
j : J
X₁ X₂ : C
f : X₁ ⟶ X₂
i : J
h : ¬i = j
⊢ (if i = j then X₁ else initial.{v_1, u_2} C) = if i = j then X₂ else initial.{v_1, u_2} C
have rw : initial.{v_1, u_2} C = if i = j then X₂ else initial.{v_1, u_2} C
Tactic: rw [if_neg h, if_neg h]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Single (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_2} C
inst_1 : HasInitial.{v_1, u_2} C
inst_2 : DecidableEq J
j : J
X₁ X₂ : C
f : X₁ ⟶ X₂
i : J
h : ¬i = j
rw : initial.{v_1, u_2} C = if i = j then X₂ else initial.{v_1, u_2} C
⊢ (if i = j then X₁ else initial.{v_1, u_2} C) = if i = j then X₂ else initial.{v_1, u_2} C
have rw₁ : initial.{v_1, u_2} C = initial.{v_1, u_2} C
Tactic: rw [if_neg h, if_neg h]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Single (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_2} C
inst_1 : HasInitial.{v_1, u_2} C
inst_2 : DecidableEq J
j : J
X : C
i : J
h : i ≠ j
⊢ IsInitial.{v_1, u_2} (Functor.obj (single j) X i)
have dsimp : IsInitial.{v_1, u_2} (if i = j then X else initial.{v_1, u_2} C)
Tactic: dsimp [single]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Single (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_2} C
inst_1 : HasInitial.{v_1, u_2} C
inst_2 : DecidableEq J
j : J
X : C
i : J
h : i ≠ j
⊢ IsInitial.{v_1, u_2} (if i = j then X else initial.{v_1, u_2} C)
have rw : IsInitial.{v_1, u_2} (initial.{v_1, u_2} C)
Tactic: rw [if_neg h]
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Single (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_2} C
inst_1 : HasInitial.{v_1, u_2} C
inst_2 : DecidableEq J
j : J
X Y : C
f : X ⟶ Y
i : J
h : i = j
⊢ Eq (α := X ⟶ Functor.obj (single j) Y i)
    (CategoryStruct.comp (Y := Functor.obj (single j) X i) (Iso.inv (singleObjApplyIsoOfEq j X i h))
      (Functor.map (single j) f i))
    (f ≫ Iso.inv (singleObjApplyIsoOfEq j Y i h))
have subst : Eq (α := X ⟶ Functor.obj (single i) Y i)
  (CategoryStruct.comp (Y := Functor.obj (single i) X i) (Iso.inv (singleObjApplyIsoOfEq i X i sorry))
    (Functor.map (single i) f i))
  (f ≫ Iso.inv (singleObjApplyIsoOfEq i Y i sorry))
Tactic: subst h
Cleared hypotheses: j, h
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Single (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_2} C
inst_1 : HasInitial.{v_1, u_2} C
inst_2 : DecidableEq J
j : J
X Y : C
f : X ⟶ Y
i : J
h : i = j
⊢ Eq.{v_1 + 1} (α := Functor.obj (single j) X i ⟶ Y)
    (CategoryStruct.comp (Y := Functor.obj (single j) Y i) (Functor.map (single j) f i)
        (Iso.hom (singleObjApplyIsoOfEq j Y i h)) :
      Functor.obj (single j) X i ⟶ Y)
    (Iso.hom (singleObjApplyIsoOfEq j X i h) ≫ f : Functor.obj (single j) X i ⟶ Y)
have subst : Eq.{v_1 + 1} (α := Functor.obj (single i) X i ⟶ Y)
  (CategoryStruct.comp (Y := Functor.obj (single i) Y i) (Functor.map (single i) f i)
      (Iso.hom (singleObjApplyIsoOfEq i Y i sorry)) :
    Functor.obj (single i) X i ⟶ Y)
  (Iso.hom (singleObjApplyIsoOfEq i X i sorry) ≫ f : Functor.obj (single i) X i ⟶ Y)
Tactic: subst h
Cleared hypotheses: j, h
-- Lean 4 proof step from Mathlib.CategoryTheory.GradedObject.Trifunctor (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₁₂ : Type u_5
C₂₃ : Type u_6
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
inst_2 : Category.{v_3, u_3} C₃
inst_3 : Category.{v_4, u_4} C₄
inst_4 : Category.{v_5, u_5} C₁₂
inst_5 : Category.{v_6, u_6} C₂₃
F F' : C₁ ⥤ C₂ ⥤ Functor.{v_3, v_4, u_3, u_4} C₃ C₄
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
X₁ Y₁ : GradedObject I₁ C₁
φ : X₁ ⟶ Y₁
X₂ Y₂ : GradedObject I₂ C₂
ψ : X₂ ⟶ Y₂
⊢ (mapTrifunctorObj F X₁ I₂ I₃).map ψ ≫
      {
        app := fun (X₃ : GradedObject I₃ C₃) (x : I₁ × I₂ × I₃) ↦
          NatTrans.app (NatTrans.app (F.map (φ x.1)) (Y₂ x.2.1)) (X₃ x.2.2),
        naturality := ⋯ } =
    {
        app := fun (X₃ : GradedObject I₃ C₃) (x : I₁ × I₂ × I₃) ↦
          NatTrans.app (NatTrans.app (F.map (φ x.1)) (X₂ x.2.1)) (X₃ x.2.2),
        naturality := ⋯ : NatTrans ((mapTrifunctorObj F X₁ I₂ I₃).obj X₂) ((mapTrifunctorObj F Y₁ I₂ I₃).obj X₂) } ≫
      (mapTrifunctorObj F Y₁ I₂ I₃).map ψ
have w : ∀ (X₃ : GradedObject I₃ C₃) (x : I₁ × I₂ × I₃),
  Eq.{v_4 + 1}
    (NatTrans.app
      ((mapTrifunctorObj F X₁ I₂ I₃).map ψ ≫
        {
          app := fun (X₃ : GradedObject I₃ C₃) (x : I₁ × I₂ × I₃) ↦
            NatTrans.app (NatTrans.app (F.map (φ x.1)) (Y₂ x.2.1)) (X₃ x.2.2),
          naturality := sorry })
      X₃ x)
    (NatTrans.app
      ({
          app := fun (X₃ : GradedObject I₃ C₃) (x : I₁ × I₂ × I₃) ↦
            NatTrans.app (NatTrans.app (F.map (φ x.1)) (X₂ x.2.1)) (X₃ x.2.2),
          naturality := sorry : NatTrans ((mapTrifunctorObj F X₁ I₂ I₃).obj X₂) ((mapTrifunctorObj F Y₁ I₂ I₃).obj X₂) } ≫
        (mapTrifunctorObj F Y₁ I₂ I₃).map ψ)
      X₃ x)
Tactic: ext X₃ x