awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
slice_lhs :
  Eq (α := A ⟶ tensorObj.{v₁, u₁} (𝒞 := inst) A A)
    (Δ ≫
      A ◁ Iso.inv (λ_ A) ≫
        A ◁ Iso.hom (β_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) A) ≫
          A ◁ @antipode _ _ inst_1 inst_2 _ _ ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst) ≫
            Iso.inv (α_ A A (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ μ ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst) ≫ A ◁ η)
    (ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ (η ⊗ₘ η))
⊢ Δ ≫
      A ◁ Iso.inv (λ_ A) ≫
        A ◁ Iso.hom (β_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) A) ≫
          ((A ◁ @antipode _ _ inst_1 inst_2 _ _ ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst) ≫ A ◁ A ◁ η) ≫ Iso.inv (α_ A A A)) ≫
            μ ▷ A =
    ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ (η ⊗ₘ η)

Tactic:
slice_lhs 5 7 =>
    rw [associator_inv_naturality_right_assoc, whisker_exchange]

Premises used: CategoryTheory.MonoidalCategory.whisker_exchange, CategoryTheory.MonoidalCategory.associator_inv_naturality_right_assoc
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
simp :
  Eq (α := A ⟶ tensorObj.{v₁, u₁} (𝒞 := inst) A A)
    (CategoryStruct.comp (Y := tensorUnit.{v₁, u₁} C (𝒞 := inst)) ε (η ≫ Iso.inv (ρ_ A) ≫ A ◁ η))
    (ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ (η ⊗ₘ η))
⊢ Eq (α := A ⟶ tensorObj.{v₁, u₁} (𝒞 := inst) A A)
    (Δ ≫
      A ◁ Iso.inv (λ_ A) ≫
        A ◁ Iso.hom (β_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) A) ≫
          A ◁ @antipode _ _ inst_1 inst_2 _ _ ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst) ≫
            Iso.inv (α_ A A (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ μ ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst) ≫ A ◁ η)
    (ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ (η ⊗ₘ η))

Tactic:
simp only [braiding_tensorUnit_left,
    whiskerLeft_comp, whiskerLeft_rightUnitor_inv,
    whiskerRight_id, whiskerLeft_rightUnitor, Category.assoc, Iso.hom_inv_id_assoc,
    Iso.inv_hom_id_assoc, whiskerLeft_inv_hom_assoc, antipode_right_assoc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.MonoidalCategory.whiskerLeft_rightUnitor_inv, CategoryTheory.MonoidalCategory.whiskerLeft_comp, CategoryTheory.Iso.inv_hom_id_assoc, CategoryTheory.MonoidalCategory.whiskerLeft_rightUnitor, CategoryTheory.braiding_tensorUnit_left, CategoryTheory.MonoidalCategory.whiskerLeft_inv_hom_assoc, CategoryTheory.HopfObj.antipode_right_assoc, CategoryTheory.MonoidalCategory.whiskerRight_id, CategoryTheory.Iso.hom_inv_id_assoc
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
rw :
  Eq (α := A ⟶ tensorObj.{v₁, u₁} (𝒞 := inst) A A)
    (ε ≫ Iso.inv (ρ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst) ≫ A ◁ η)
    (ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst) ≫ A ◁ η)
⊢ Eq (α := A ⟶ tensorObj.{v₁, u₁} (𝒞 := inst) A A)
    (CategoryStruct.comp (Y := tensorUnit.{v₁, u₁} C (𝒞 := inst)) ε (η ≫ Iso.inv (ρ_ A) ≫ A ◁ η))
    (ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ (η ⊗ₘ η))

Tactic:
rw [rightUnitor_inv_naturality_assoc, tensorHom_def]

Premises used: CategoryTheory.MonoidalCategory.rightUnitor_inv_naturality_assoc, CategoryTheory.MonoidalCategory.tensorHom_def
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
⊢ Eq (α := A ⟶ tensorObj.{v₁, u₁} (𝒞 := inst) A A)
    (ε ≫ Iso.inv (ρ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst) ≫ A ◁ η)
    (ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst) ≫ A ◁ η)

Tactic:
monoidal
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
hba : HMul.hMul (α := Conv A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) (@antipode _ _ inst_1 inst_2 _ _ ≫ Δ) Δ = 1
hac :
  HMul.hMul (α := Conv A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) Δ
      (Δ ≫ Iso.hom (β_ A A) ≫ (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _)) =
    1
⊢ Eq.{v₁ + 1} (α := A ⟶ tensorObj.{v₁, u₁} (𝒞 := inst) A A) (@antipode _ _ inst_1 inst_2 _ _ ≫ Δ)
    (Δ ≫ Iso.hom (β_ A A) ≫ (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _))

Tactic:
apply left_inv_eq_right_inv
    (M := Conv A (A ⊗ A))
    (a := Δ[A])

Premises used: left_inv_eq_right_inv
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
hba :
  CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) A A) Δ
      ((@antipode _ _ inst_1 inst_2 _ _ ≫ Δ) ▷ A ≫ tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁ Δ ≫ μ) =
    CategoryStruct.comp (Y := tensorUnit.{v₁, u₁} C (𝒞 := inst)) ε η
⊢ HMul.hMul (α := Conv A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) (@antipode _ _ inst_1 inst_2 _ _ ≫ Δ) Δ = 1

Tactic:
rw [Conv.mul_eq, Conv.one_eq]

Premises used: CategoryTheory.Conv.one_eq, CategoryTheory.Conv.mul_eq
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
hba :
  Eq (α := Conv A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))
    (Δ ≫
      @antipode _ _ inst_1 inst_2 _ _ ▷ A ≫
        Δ ▷ A ≫
          Iso.hom (α_ A A A) ≫
            A ◁ A ◁ Δ ≫ Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫ tensorμ A A A A ≫ (μ ⊗ₘ μ))
    (ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ (η ⊗ₘ η))
⊢ CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) A A) Δ
      ((@antipode _ _ inst_1 inst_2 _ _ ≫ Δ) ▷ A ≫ tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁ Δ ≫ μ) =
    CategoryStruct.comp (Y := tensorUnit.{v₁, u₁} C (𝒞 := inst)) ε η

Tactic:
simp only [comp_whiskerRight, tensor_whiskerLeft, MonObj.tensorObj.mul_def, Category.assoc,
      MonObj.tensorObj.one_def]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.MonObj.tensorObj.mul_def, CategoryTheory.MonoidalCategory.comp_whiskerRight, CategoryTheory.MonObj.tensorObj.one_def, CategoryTheory.MonoidalCategory.tensor_whiskerLeft
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
hba :
  Eq (α := Conv A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))
    (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) A A) Δ
      (@antipode _ _ inst_1 inst_2 _ _ ▷ A ≫
        Δ ▷ A ≫
          Iso.hom (α_ A A A) ≫
            A ◁ A ◁ Δ ≫
              Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫
                (Iso.hom (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫
                    A ◁ Iso.inv (α_ A A A) ≫
                      A ◁ Iso.hom (β_ A A) ▷ A ≫
                        A ◁ Iso.hom (α_ A A A) ≫ Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))) ≫
                  (μ ⊗ₘ μ)))
    (ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ (η ⊗ₘ η))
⊢ Eq (α := Conv A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))
    (Δ ≫
      @antipode _ _ inst_1 inst_2 _ _ ▷ A ≫
        Δ ▷ A ≫
          Iso.hom (α_ A A A) ≫
            A ◁ A ◁ Δ ≫ Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫ tensorμ A A A A ≫ (μ ⊗ₘ μ))
    (ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ (η ⊗ₘ η))

Tactic:
simp only [tensorμ]

Premises used: CategoryTheory.MonoidalCategory.tensorμ
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
hba :
  Eq (α := Conv A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))
    (Δ ≫
      @antipode _ _ inst_1 inst_2 _ _ ▷ A ≫
        Δ ▷ A ≫
          Iso.hom (α_ A A A) ≫
            A ◁ A ◁ Δ ≫
              A ◁ Iso.inv (α_ A A A) ≫
                A ◁ Iso.hom (β_ A A) ▷ A ≫
                  A ◁ Iso.hom (α_ A A A) ≫ Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫ (μ ⊗ₘ μ))
    (ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ (η ⊗ₘ η))
⊢ Eq (α := Conv A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))
    (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) A A) Δ
      (@antipode _ _ inst_1 inst_2 _ _ ▷ A ≫
        Δ ▷ A ≫
          Iso.hom (α_ A A A) ≫
            A ◁ A ◁ Δ ≫
              Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫
                (Iso.hom (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫
                    A ◁ Iso.inv (α_ A A A) ≫
                      A ◁ Iso.hom (β_ A A) ▷ A ≫
                        A ◁ Iso.hom (α_ A A A) ≫ Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))) ≫
                  (μ ⊗ₘ μ)))
    (ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ (η ⊗ₘ η))

Tactic:
simp only [Category.assoc, Iso.inv_hom_id_assoc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Iso.inv_hom_id_assoc
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
⊢ Eq (α := Conv A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))
    (Δ ≫
      @antipode _ _ inst_1 inst_2 _ _ ▷ A ≫
        Δ ▷ A ≫
          Iso.hom (α_ A A A) ≫
            A ◁ A ◁ Δ ≫
              A ◁ Iso.inv (α_ A A A) ≫
                A ◁ Iso.hom (β_ A A) ▷ A ≫
                  A ◁ Iso.hom (α_ A A A) ≫ Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫ (μ ⊗ₘ μ))
    (ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ (η ⊗ₘ η))

Tactic:
antipode_comul₁ A

Premises used: CategoryTheory.HopfObj.antipode_comul₁, CategoryTheory.Conv
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
⊢ Eq (α := Conv A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))
    (Δ ≫
      @antipode _ _ inst_1 inst_2 _ _ ▷ A ≫
        Δ ▷ A ≫
          Iso.hom (α_ A A A) ≫
            A ◁ A ◁ Δ ≫
              A ◁ Iso.inv (α_ A A A) ≫
                A ◁ Iso.hom (β_ A A) ▷ A ≫
                  A ◁ Iso.hom (α_ A A A) ≫ Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫ (μ ⊗ₘ μ))
    (ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ (η ⊗ₘ η))

Tactic:
exact antipode_comul₁ A

Premises used: CategoryTheory.HopfObj.antipode_comul₁
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
hac :
  CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) A A) Δ
      (Δ ▷ A ≫
        tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁
            (Δ ≫ Iso.hom (β_ A A) ≫ (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _)) ≫
          μ) =
    CategoryStruct.comp (Y := tensorUnit.{v₁, u₁} C (𝒞 := inst)) ε η
⊢ HMul.hMul (α := Conv A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) Δ
      (Δ ≫ Iso.hom (β_ A A) ≫ (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _)) =
    1

Tactic:
rw [Conv.mul_eq, Conv.one_eq]

Premises used: CategoryTheory.Conv.one_eq, CategoryTheory.Conv.mul_eq
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
hac :
  Eq (α := Conv A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))
    (Δ ≫
      Δ ▷ A ≫
        Iso.hom (α_ A A A) ≫
          A ◁ A ◁ Δ ≫
            A ◁ A ◁ Iso.hom (β_ A A) ≫
              A ◁ A ◁ (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _) ≫
                Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫ tensorμ A A A A ≫ (μ ⊗ₘ μ))
    (ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ (η ⊗ₘ η))
⊢ CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) A A) Δ
      (Δ ▷ A ≫
        tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁
            (Δ ≫ Iso.hom (β_ A A) ≫ (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _)) ≫
          μ) =
    CategoryStruct.comp (Y := tensorUnit.{v₁, u₁} C (𝒞 := inst)) ε η

Tactic:
simp only [whiskerLeft_comp, tensor_whiskerLeft, Category.assoc, Iso.inv_hom_id_assoc,
      MonObj.tensorObj.mul_def, MonObj.tensorObj.one_def]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.MonoidalCategory.whiskerLeft_comp, CategoryTheory.MonObj.tensorObj.mul_def, CategoryTheory.MonObj.tensorObj.one_def, CategoryTheory.Iso.inv_hom_id_assoc, CategoryTheory.MonoidalCategory.tensor_whiskerLeft
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
hac :
  Eq (α := Conv A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))
    (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) A A) Δ
      (Δ ▷ A ≫
        Iso.hom (α_ A A A) ≫
          A ◁ A ◁ Δ ≫
            A ◁ A ◁ Iso.hom (β_ A A) ≫
              A ◁ A ◁ (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _) ≫
                Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫
                  (Iso.hom (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫
                      A ◁ Iso.inv (α_ A A A) ≫
                        A ◁ Iso.hom (β_ A A) ▷ A ≫
                          A ◁ Iso.hom (α_ A A A) ≫ Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))) ≫
                    (μ ⊗ₘ μ)))
    (ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ (η ⊗ₘ η))
⊢ Eq (α := Conv A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))
    (Δ ≫
      Δ ▷ A ≫
        Iso.hom (α_ A A A) ≫
          A ◁ A ◁ Δ ≫
            A ◁ A ◁ Iso.hom (β_ A A) ≫
              A ◁ A ◁ (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _) ≫
                Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫ tensorμ A A A A ≫ (μ ⊗ₘ μ))
    (ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ (η ⊗ₘ η))

Tactic:
simp only [tensorμ]

Premises used: CategoryTheory.MonoidalCategory.tensorμ
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
hac :
  Eq (α := Conv A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))
    (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) A A) Δ
      (Δ ▷ A ≫
        Iso.hom (α_ A A A) ≫
          A ◁ A ◁ Δ ≫
            A ◁ A ◁ Iso.hom (β_ A A) ≫
              A ◁ A ◁ (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _) ≫
                A ◁ Iso.inv (α_ A A A) ≫
                  A ◁ Iso.hom (β_ A A) ▷ A ≫
                    A ◁ Iso.hom (α_ A A A) ≫ Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫ (μ ⊗ₘ μ)))
    (ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ (η ⊗ₘ η))
⊢ Eq (α := Conv A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))
    (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) A A) Δ
      (Δ ▷ A ≫
        Iso.hom (α_ A A A) ≫
          A ◁ A ◁ Δ ≫
            A ◁ A ◁ Iso.hom (β_ A A) ≫
              A ◁ A ◁ (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _) ≫
                Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫
                  (Iso.hom (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫
                      A ◁ Iso.inv (α_ A A A) ≫
                        A ◁ Iso.hom (β_ A A) ▷ A ≫
                          A ◁ Iso.hom (α_ A A A) ≫ Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))) ≫
                    (μ ⊗ₘ μ)))
    (ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ (η ⊗ₘ η))

Tactic:
simp only [Category.assoc, Iso.inv_hom_id_assoc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Iso.inv_hom_id_assoc
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
⊢ Eq (α := Conv A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))
    (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) A A) Δ
      (Δ ▷ A ≫
        Iso.hom (α_ A A A) ≫
          A ◁ A ◁ Δ ≫
            A ◁ A ◁ Iso.hom (β_ A A) ≫
              A ◁ A ◁ (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _) ≫
                A ◁ Iso.inv (α_ A A A) ≫
                  A ◁ Iso.hom (β_ A A) ▷ A ≫
                    A ◁ Iso.hom (α_ A A A) ≫ Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫ (μ ⊗ₘ μ)))
    (ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ (η ⊗ₘ η))

Tactic:
antipode_comul₂ A

Premises used: CategoryTheory.HopfObj.antipode_comul₂, CategoryTheory.Conv
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
⊢ Eq (α := Conv A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))
    (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) A A) Δ
      (Δ ▷ A ≫
        Iso.hom (α_ A A A) ≫
          A ◁ A ◁ Δ ≫
            A ◁ A ◁ Iso.hom (β_ A A) ≫
              A ◁ A ◁ (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _) ≫
                A ◁ Iso.inv (α_ A A A) ≫
                  A ◁ Iso.hom (β_ A A) ▷ A ≫
                    A ◁ Iso.hom (α_ A A A) ≫ Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫ (μ ⊗ₘ μ)))
    (ε ≫ Iso.inv (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ (η ⊗ₘ η))

Tactic:
exact antipode_comul₂ A

Premises used: CategoryTheory.HopfObj.antipode_comul₂
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
a : Iso.hom (α_ A A A) ≫ @antipode _ _ inst_1 inst_2 _ _ ▷ tensorObj.{v₁, u₁} (𝒞 := inst) A A
| @antipode _ _ inst_1 inst_2 _ _ ▷ A ▷ A ≫ Iso.hom (α_ A A A)

Tactic:
rw [associator_naturality_left]

Premises used: CategoryTheory.MonoidalCategory.associator_naturality_left
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
slice_lhs :
  Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) A A ⟶ A)
    ((Δ ⊗ₘ Δ) ≫
      Iso.hom (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫
        A ◁ Iso.inv (α_ A A A) ≫
          A ◁ Iso.hom (β_ A A) ▷ A ≫
            Iso.inv (α_ A (tensorObj.{v₁, u₁} (𝒞 := inst) A A) A) ≫
              Iso.inv (α_ A A A) ▷ A ≫
                μ ▷ A ▷ A ≫
                  ((Iso.hom (α_ A A A) ≫ @antipode _ _ inst_1 inst_2 _ _ ▷ tensorObj.{v₁, u₁} (𝒞 := inst) A A) ≫
                      A ◁ μ) ≫
                    μ)
    ((ε ⊗ₘ ε) ≫ Iso.hom (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η)
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) A A ⟶ A)
    ((Δ ⊗ₘ Δ) ≫
      Iso.hom (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫
        A ◁ Iso.inv (α_ A A A) ≫
          A ◁ Iso.hom (β_ A A) ▷ A ≫
            Iso.inv (α_ A (tensorObj.{v₁, u₁} (𝒞 := inst) A A) A) ≫
              Iso.inv (α_ A A A) ▷ A ≫
                μ ▷ A ▷ A ≫ @antipode _ _ inst_1 inst_2 _ _ ▷ A ▷ A ≫ Iso.hom (α_ A A A) ≫ A ◁ μ ≫ μ)
    ((ε ⊗ₘ ε) ≫ Iso.hom (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η)

Tactic:
slice_lhs 8 9 =>
    rw [associator_naturality_left]

Premises used: CategoryTheory.MonoidalCategory.associator_naturality_left
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
a : A ◁ μ ≫ @antipode _ _ inst_1 inst_2 _ _ ▷ A
| @antipode _ _ inst_1 inst_2 _ _ ▷ tensorObj.{v₁, u₁} (𝒞 := inst) A A ≫ A ◁ μ

Tactic:
rw [← whisker_exchange]

Premises used: CategoryTheory.MonoidalCategory.whisker_exchange
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
slice_lhs :
  Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) A A ⟶ A)
    ((Δ ⊗ₘ Δ) ≫
      Iso.hom (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫
        A ◁ Iso.inv (α_ A A A) ≫
          A ◁ Iso.hom (β_ A A) ▷ A ≫
            Iso.inv (α_ A (tensorObj.{v₁, u₁} (𝒞 := inst) A A) A) ≫
              Iso.inv (α_ A A A) ▷ A ≫
                μ ▷ A ▷ A ≫ Iso.hom (α_ A A A) ≫ (A ◁ μ ≫ @antipode _ _ inst_1 inst_2 _ _ ▷ A) ≫ μ)
    ((ε ⊗ₘ ε) ≫ Iso.hom (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η)
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) A A ⟶ A)
    ((Δ ⊗ₘ Δ) ≫
      Iso.hom (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫
        A ◁ Iso.inv (α_ A A A) ≫
          A ◁ Iso.hom (β_ A A) ▷ A ≫
            Iso.inv (α_ A (tensorObj.{v₁, u₁} (𝒞 := inst) A A) A) ≫
              Iso.inv (α_ A A A) ▷ A ≫
                μ ▷ A ▷ A ≫
                  ((Iso.hom (α_ A A A) ≫ @antipode _ _ inst_1 inst_2 _ _ ▷ tensorObj.{v₁, u₁} (𝒞 := inst) A A) ≫
                      A ◁ μ) ≫
                    μ)
    ((ε ⊗ₘ ε) ≫ Iso.hom (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η)

Tactic:
slice_lhs 9 10 =>
    rw [← whisker_exchange]

Premises used: CategoryTheory.MonoidalCategory.whisker_exchange
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
a : Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) A A) A A) ≫ μ ▷ tensorObj.{v₁, u₁} (𝒞 := inst) A A
| μ ▷ A ▷ A ≫ Iso.hom (α_ A A A)

Tactic:
rw [associator_naturality_left]

Premises used: CategoryTheory.MonoidalCategory.associator_naturality_left
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
slice_lhs :
  Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) A A ⟶ A)
    ((Δ ⊗ₘ Δ) ≫
      Iso.hom (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫
        A ◁ Iso.inv (α_ A A A) ≫
          A ◁ Iso.hom (β_ A A) ▷ A ≫
            Iso.inv (α_ A (tensorObj.{v₁, u₁} (𝒞 := inst) A A) A) ≫
              Iso.inv (α_ A A A) ▷ A ≫
                (((Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) A A) A A) ≫ μ ▷ tensorObj.{v₁, u₁} (𝒞 := inst) A A) ≫
                      A ◁ μ) ≫
                    @antipode _ _ inst_1 inst_2 _ _ ▷ A) ≫
                  μ)
    ((ε ⊗ₘ ε) ≫ Iso.hom (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η)
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) A A ⟶ A)
    ((Δ ⊗ₘ Δ) ≫
      Iso.hom (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫
        A ◁ Iso.inv (α_ A A A) ≫
          A ◁ Iso.hom (β_ A A) ▷ A ≫
            Iso.inv (α_ A (tensorObj.{v₁, u₁} (𝒞 := inst) A A) A) ≫
              Iso.inv (α_ A A A) ▷ A ≫
                μ ▷ A ▷ A ≫ Iso.hom (α_ A A A) ≫ (A ◁ μ ≫ @antipode _ _ inst_1 inst_2 _ _ ▷ A) ≫ μ)
    ((ε ⊗ₘ ε) ≫ Iso.hom (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η)

Tactic:
slice_lhs 7 8 =>
    rw [associator_naturality_left]

Premises used: CategoryTheory.MonoidalCategory.associator_naturality_left
-- Module: Mathlib.CategoryTheory.Monoidal.Hopf_
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : BraidedCategory.{v₁, u₁} C
A : C
inst_3 : HopfObj A
a : μ ⊗ₘ μ
| μ ▷ tensorObj.{v₁, u₁} (𝒞 := inst) A A ≫ A ◁ μ

Tactic:
rw [← tensorHom_def]

Premises used: CategoryTheory.MonoidalCategory.tensorHom_def