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 :
  Δ ▷ A ≫
      tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁ 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 ≫
              A ◁ Iso.hom (α_ A A A) ≫
                (((((((𝟙
                                  (tensorObj.{v₁, u₁} (𝒞 := inst) A
                                    (tensorObj.{v₁, u₁} (𝒞 := inst) A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))) ≫
                                A ◁ A ◁ Iso.hom (β_ A A)) ≫
                              Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))) ≫
                            tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁
                              (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _)) ≫
                          Iso.inv (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) A A) A A)) ≫
                        Iso.hom (α_ A A A) ▷ A) ≫
                      (A ◁ μ) ▷ A) ≫
                    μ ▷ 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 ▷ A ≫ tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁ 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 ≫
            A ◁ Iso.hom (α_ A A A) ≫
              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 ◁ A ◁ Iso.hom (β_ A A) ≫ Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))) ≫
                  tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁
                      (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _) ≫
                    Iso.inv (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) A A) A A) ≫
                      (Iso.hom (α_ A A A) ▷ A ≫ (A ◁ μ) ▷ A ≫ μ ▷ A) ≫ μ)
    ((ε ⊗ₘ ε) ≫ Iso.hom (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η)

Tactic:
slice_lhs 7 8 =>
    rw [Iso.inv_hom_id]

Premises used: CategoryTheory.Iso.inv_hom_id
-- 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 :
  Δ ▷ A ≫
      tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁ 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 ≫
              A ◁ Iso.hom (α_ A A A) ≫
                ((((((A ◁ A ◁ Iso.hom (β_ A A) ≫ Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))) ≫
                            tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁
                              (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _)) ≫
                          Iso.inv (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) A A) A A)) ≫
                        Iso.hom (α_ A A A) ▷ A) ≫
                      (A ◁ μ) ▷ A) ≫
                    μ ▷ A) ≫
                  μ =
    (ε ⊗ₘ ε) ≫ Iso.hom (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η
⊢ Δ ▷ A ≫
      tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁ 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 ≫
              A ◁ Iso.hom (α_ A A A) ≫
                (((((((𝟙
                                  (tensorObj.{v₁, u₁} (𝒞 := inst) A
                                    (tensorObj.{v₁, u₁} (𝒞 := inst) A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))) ≫
                                A ◁ A ◁ Iso.hom (β_ A A)) ≫
                              Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))) ≫
                            tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁
                              (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _)) ≫
                          Iso.inv (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) A A) A A)) ≫
                        Iso.hom (α_ A A A) ▷ A) ≫
                      (A ◁ μ) ▷ A) ≫
                    μ ▷ A) ≫
                  μ =
    (ε ⊗ₘ ε) ≫ Iso.hom (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η

Tactic:
rw [Category.id_comp]

Premises used: CategoryTheory.Category.id_comp
-- 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 ◁ (Iso.hom (β_ A A) ▷ A ≫ Iso.hom (α_ A A A) ≫ A ◁ Iso.hom (β_ A A))
| A ◁ Iso.hom (β_ A A) ▷ A ≫ A ◁ Iso.hom (α_ A A A) ≫ A ◁ A ◁ Iso.hom (β_ A A)

Tactic:
simp only [← whiskerLeft_comp]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp
-- 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 ◁ (Iso.hom (α_ A A A) ≫ Iso.hom (β_ A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫ Iso.hom (α_ A A A))
| A ◁ (Iso.hom (β_ A A) ▷ A ≫ Iso.hom (α_ A A A) ≫ A ◁ Iso.hom (β_ A A))

Tactic:
rw [← BraidedCategory.hexagon_forward]

Premises used: CategoryTheory.BraidedCategory.hexagon_forward
-- 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 ◁ Iso.hom (α_ A A A) ≫ A ◁ Iso.hom (β_ A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫ A ◁ Iso.hom (α_ A A A)
| A ◁ (Iso.hom (α_ A A A) ≫ Iso.hom (β_ A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫ Iso.hom (α_ A A A))

Tactic:
simp only [whiskerLeft_comp]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp
-- 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 ◁ Iso.hom (α_ A A A) ≫ A ◁ Iso.hom (β_ A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫ A ◁ Iso.hom (α_ A A A)
| A ◁ Iso.hom (β_ A A) ▷ A ≫ A ◁ Iso.hom (α_ A A A) ≫ A ◁ A ◁ Iso.hom (β_ A A)

Tactic:
simp only [← whiskerLeft_comp]
    rw [← BraidedCategory.hexagon_forward]
    simp only [whiskerLeft_comp]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp, CategoryTheory.BraidedCategory.hexagon_forward
-- 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 :
  Δ ▷ A ≫
      tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁ Δ ≫
        Iso.hom (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫
          A ◁ Iso.inv (α_ A A A) ≫
            (((((((A ◁ Iso.hom (α_ A A A) ≫
                            A ◁ Iso.hom (β_ A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫ A ◁ Iso.hom (α_ A A A)) ≫
                          Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))) ≫
                        tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁
                          (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _)) ≫
                      Iso.inv (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) A A) A A)) ≫
                    Iso.hom (α_ A A A) ▷ A) ≫
                  (A ◁ μ) ▷ A) ≫
                μ ▷ A) ≫
              μ =
    (ε ⊗ₘ ε) ≫ Iso.hom (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η
⊢ Δ ▷ A ≫
      tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁ 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 ≫
              A ◁ Iso.hom (α_ A A A) ≫
                ((((((A ◁ A ◁ Iso.hom (β_ A A) ≫ Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))) ≫
                            tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁
                              (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _)) ≫
                          Iso.inv (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) A A) A A)) ≫
                        Iso.hom (α_ A A A) ▷ A) ≫
                      (A ◁ μ) ▷ A) ≫
                    μ ▷ A) ≫
                  μ =
    (ε ⊗ₘ ε) ≫ Iso.hom (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η

Tactic:
slice_lhs 5 7 =>
    simp only [← whiskerLeft_comp]
    rw [← BraidedCategory.hexagon_forward]
    simp only [whiskerLeft_comp]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp, CategoryTheory.BraidedCategory.hexagon_forward
-- 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 := tensorObj.{v₁, u₁} (𝒞 := inst) A A ⟶ A)
    (Δ ▷ A ≫
      Iso.hom (α_ A A A) ≫
        A ◁ A ◁ Δ ≫
          A ◁ Iso.hom (β_ A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫
            A ◁ Iso.hom (α_ A A A) ≫
              A ◁ A ◁ (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _) ≫
                A ◁ Iso.inv (α_ A A A) ≫ A ◁ μ ▷ A ≫ A ◁ μ ≫ μ)
    ((ε ⊗ₘ ε) ≫ Iso.hom (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η)
⊢ Δ ▷ A ≫
      tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁ Δ ≫
        Iso.hom (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫
          A ◁ Iso.inv (α_ A A A) ≫
            (((((((A ◁ Iso.hom (α_ A A A) ≫
                            A ◁ Iso.hom (β_ A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫ A ◁ Iso.hom (α_ A A A)) ≫
                          Iso.inv (α_ A A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))) ≫
                        tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁
                          (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _)) ≫
                      Iso.inv (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) A A) A A)) ≫
                    Iso.hom (α_ A A A) ▷ A) ≫
                  (A ◁ μ) ▷ A) ≫
                μ ▷ A) ≫
              μ =
    (ε ⊗ₘ ε) ≫ Iso.hom (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η

Tactic:
simp only [tensor_whiskerLeft, Category.assoc, Iso.inv_hom_id_assoc, pentagon_inv_inv_hom_inv_inv,
    whisker_assoc, MonObj.mul_assoc, whiskerLeft_inv_hom_assoc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.MonObj.mul_assoc, CategoryTheory.Iso.inv_hom_id_assoc, CategoryTheory.MonoidalCategory.whiskerLeft_inv_hom_assoc, CategoryTheory.MonoidalCategory.pentagon_inv_inv_hom_inv_inv, CategoryTheory.MonoidalCategory.tensor_whiskerLeft, CategoryTheory.MonoidalCategory.whisker_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
a : A ◁ (A ◁ Δ ≫ Iso.hom (β_ A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)))
| A ◁ A ◁ Δ ≫ A ◁ Iso.hom (β_ A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))

Tactic:
simp only [← whiskerLeft_comp]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp
-- 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 ◁ (Iso.hom (β_ A A) ≫ Δ ▷ A)
| A ◁ (A ◁ Δ ≫ Iso.hom (β_ A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)))

Tactic:
rw [BraidedCategory.braiding_naturality_right]

Premises used: CategoryTheory.BraidedCategory.braiding_naturality_right
-- 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 ◁ Iso.hom (β_ A A) ≫ A ◁ Δ ▷ A
| A ◁ (Iso.hom (β_ A A) ≫ Δ ▷ A)

Tactic:
simp only [whiskerLeft_comp]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp
-- 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 ◁ Iso.hom (β_ A A) ≫ A ◁ Δ ▷ A
| A ◁ A ◁ Δ ≫ A ◁ Iso.hom (β_ A (tensorObj.{v₁, u₁} (𝒞 := inst) A A))

Tactic:
simp only [← whiskerLeft_comp]
    rw [BraidedCategory.braiding_naturality_right]
    simp only [whiskerLeft_comp]

Premises used: CategoryTheory.BraidedCategory.braiding_naturality_right, CategoryTheory.MonoidalCategory.whiskerLeft_comp
-- 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 ▷ A ≫
      Iso.hom (α_ A A A) ≫
        ((((((A ◁ Iso.hom (β_ A A) ≫ A ◁ Δ ▷ A) ≫ A ◁ Iso.hom (α_ A A A)) ≫
                  A ◁ A ◁ (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _)) ≫
                A ◁ Iso.inv (α_ A A A)) ≫
              A ◁ μ ▷ 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 ▷ A ≫
      Iso.hom (α_ A A A) ≫
        A ◁ A ◁ Δ ≫
          A ◁ Iso.hom (β_ A (tensorObj.{v₁, u₁} (𝒞 := inst) A A)) ≫
            A ◁ Iso.hom (α_ A A A) ≫
              A ◁ A ◁ (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _) ≫
                A ◁ Iso.inv (α_ A A A) ≫ A ◁ μ ▷ A ≫ A ◁ μ ≫ μ)
    ((ε ⊗ₘ ε) ≫ Iso.hom (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η)

Tactic:
slice_lhs 3 4 =>
    simp only [← whiskerLeft_comp]
    rw [BraidedCategory.braiding_naturality_right]
    simp only [whiskerLeft_comp]

Premises used: CategoryTheory.BraidedCategory.braiding_naturality_right, CategoryTheory.MonoidalCategory.whiskerLeft_comp
-- 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 := tensorObj.{v₁, u₁} (𝒞 := inst) A A ⟶ A)
    (Δ ▷ A ≫
      Iso.hom (α_ A A A) ≫
        ((((((A ◁ Iso.hom (β_ A A) ≫ A ◁ Δ ▷ A) ≫ A ◁ Iso.hom (α_ A A A)) ≫
                  A ◁ A ◁ (A ◁ @antipode _ _ inst_1 inst_2 _ _ ≫ @antipode _ _ inst_1 inst_2 _ _ ▷ A)) ≫
                A ◁ Iso.inv (α_ A A A)) ≫
              A ◁ μ ▷ 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 ▷ A ≫
      Iso.hom (α_ A A A) ≫
        ((((((A ◁ Iso.hom (β_ A A) ≫ A ◁ Δ ▷ A) ≫ A ◁ Iso.hom (α_ A A A)) ≫
                  A ◁ A ◁ (@antipode _ _ inst_1 inst_2 _ _ ⊗ₘ @antipode _ _ inst_1 inst_2 _ _)) ≫
                A ◁ Iso.inv (α_ A A A)) ≫
              A ◁ μ ▷ A) ≫
            A ◁ μ) ≫
          μ)
    ((ε ⊗ₘ ε) ≫ Iso.hom (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η)

Tactic:
rw [tensorHom_def']

Premises used: 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
simp :
  Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) A A ⟶ A)
    (Δ ▷ A ≫
      Iso.hom (α_ A A A) ≫
        ((((((A ◁ Iso.hom (β_ A A) ≫ A ◁ Δ ▷ A) ≫ A ◁ Iso.hom (α_ A A A)) ≫
                  A ◁ A ◁ A ◁ @antipode _ _ inst_1 inst_2 _ _ ≫ A ◁ A ◁ @antipode _ _ inst_1 inst_2 _ _ ▷ A) ≫
                A ◁ Iso.inv (α_ A A A)) ≫
              A ◁ μ ▷ 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 ▷ A ≫
      Iso.hom (α_ A A A) ≫
        ((((((A ◁ Iso.hom (β_ A A) ≫ A ◁ Δ ▷ A) ≫ A ◁ Iso.hom (α_ A A A)) ≫
                  A ◁ A ◁ (A ◁ @antipode _ _ inst_1 inst_2 _ _ ≫ @antipode _ _ inst_1 inst_2 _ _ ▷ A)) ≫
                A ◁ Iso.inv (α_ A A A)) ≫
              A ◁ μ ▷ A) ≫
            A ◁ μ) ≫
          μ)
    ((ε ⊗ₘ ε) ≫ Iso.hom (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η)

Tactic:
simp only [whiskerLeft_comp]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp
-- 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 ◁ (Iso.hom (α_ A A A) ≫ A ◁ A ◁ @antipode _ _ inst_1 inst_2 _ _)
| A ◁ Iso.hom (α_ A A A) ≫ A ◁ A ◁ A ◁ @antipode _ _ inst_1 inst_2 _ _

Tactic:
simp only [← whiskerLeft_comp]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp
-- 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 ◁ (tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁ @antipode _ _ inst_1 inst_2 _ _ ≫ Iso.hom (α_ A A A))
| A ◁ (Iso.hom (α_ A A A) ≫ A ◁ A ◁ @antipode _ _ inst_1 inst_2 _ _)

Tactic:
rw [← associator_naturality_right]

Premises used: CategoryTheory.MonoidalCategory.associator_naturality_right
-- 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 ◁ tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁ @antipode _ _ inst_1 inst_2 _ _ ≫ A ◁ Iso.hom (α_ A A A)
| A ◁ (tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁ @antipode _ _ inst_1 inst_2 _ _ ≫ Iso.hom (α_ A A A))

Tactic:
simp only [whiskerLeft_comp]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp
-- 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 ◁ tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁ @antipode _ _ inst_1 inst_2 _ _ ≫ A ◁ Iso.hom (α_ A A A)
| A ◁ Iso.hom (α_ A A A) ≫ A ◁ A ◁ A ◁ @antipode _ _ inst_1 inst_2 _ _

Tactic:
simp only [← whiskerLeft_comp]
    rw [← associator_naturality_right]
    simp only [whiskerLeft_comp]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp, CategoryTheory.MonoidalCategory.associator_naturality_right
-- 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 ▷ A ≫
      Iso.hom (α_ A A A) ≫
        A ◁ Iso.hom (β_ A A) ≫
          A ◁ Δ ▷ A ≫
            (((((A ◁ tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁ @antipode _ _ inst_1 inst_2 _ _ ≫ A ◁ Iso.hom (α_ A A A)) ≫
                      A ◁ A ◁ @antipode _ _ inst_1 inst_2 _ _ ▷ A) ≫
                    A ◁ Iso.inv (α_ A A A)) ≫
                  A ◁ μ ▷ 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 ▷ A ≫
      Iso.hom (α_ A A A) ≫
        ((((((A ◁ Iso.hom (β_ A A) ≫ A ◁ Δ ▷ A) ≫ A ◁ Iso.hom (α_ A A A)) ≫
                  A ◁ A ◁ A ◁ @antipode _ _ inst_1 inst_2 _ _ ≫ A ◁ A ◁ @antipode _ _ inst_1 inst_2 _ _ ▷ A) ≫
                A ◁ Iso.inv (α_ A A A)) ≫
              A ◁ μ ▷ A) ≫
            A ◁ μ) ≫
          μ)
    ((ε ⊗ₘ ε) ≫ Iso.hom (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η)

Tactic:
slice_lhs 5 6 =>
    simp only [← whiskerLeft_comp]
    rw [← associator_naturality_right]
    simp only [whiskerLeft_comp]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp, CategoryTheory.MonoidalCategory.associator_naturality_right
-- 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 ◁ (Δ ▷ A ≫ tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁ @antipode _ _ inst_1 inst_2 _ _)
| A ◁ Δ ▷ A ≫ A ◁ tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁ @antipode _ _ inst_1 inst_2 _ _

Tactic:
simp only [← whiskerLeft_comp]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp
-- 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 ◁ (A ◁ @antipode _ _ inst_1 inst_2 _ _ ≫ Δ ▷ A)
| A ◁ (Δ ▷ A ≫ tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁ @antipode _ _ inst_1 inst_2 _ _)

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
a : A ◁ A ◁ @antipode _ _ inst_1 inst_2 _ _ ≫ A ◁ Δ ▷ A
| A ◁ (A ◁ @antipode _ _ inst_1 inst_2 _ _ ≫ Δ ▷ A)

Tactic:
simp only [whiskerLeft_comp]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp
-- 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 ◁ A ◁ @antipode _ _ inst_1 inst_2 _ _ ≫ A ◁ Δ ▷ A
| A ◁ Δ ▷ A ≫ A ◁ tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁ @antipode _ _ inst_1 inst_2 _ _

Tactic:
simp only [← whiskerLeft_comp]
    rw [← whisker_exchange]
    simp only [whiskerLeft_comp]

Premises used: CategoryTheory.MonoidalCategory.whisker_exchange, CategoryTheory.MonoidalCategory.whiskerLeft_comp
-- 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 ▷ A ≫
      Iso.hom (α_ A A A) ≫
        A ◁ Iso.hom (β_ A A) ≫
          ((((((A ◁ A ◁ @antipode _ _ inst_1 inst_2 _ _ ≫ A ◁ Δ ▷ A) ≫ A ◁ Iso.hom (α_ A A A)) ≫
                    A ◁ A ◁ @antipode _ _ inst_1 inst_2 _ _ ▷ A) ≫
                  A ◁ Iso.inv (α_ A A A)) ≫
                A ◁ μ ▷ 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 ▷ A ≫
      Iso.hom (α_ A A A) ≫
        A ◁ Iso.hom (β_ A A) ≫
          A ◁ Δ ▷ A ≫
            (((((A ◁ tensorObj.{v₁, u₁} (𝒞 := inst) A A ◁ @antipode _ _ inst_1 inst_2 _ _ ≫ A ◁ Iso.hom (α_ A A A)) ≫
                      A ◁ A ◁ @antipode _ _ inst_1 inst_2 _ _ ▷ A) ≫
                    A ◁ Iso.inv (α_ A A A)) ≫
                  A ◁ μ ▷ A) ≫
                A ◁ μ) ≫
              μ)
    ((ε ⊗ₘ ε) ≫ Iso.hom (λ_ (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ η)

Tactic:
slice_lhs 4 5 =>
    simp only [← whiskerLeft_comp]
    rw [← whisker_exchange]
    simp only [whiskerLeft_comp]

Premises used: CategoryTheory.MonoidalCategory.whisker_exchange, CategoryTheory.MonoidalCategory.whiskerLeft_comp
-- 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 ◁
    CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) A A) A) (Δ ▷ A)
      ((A ◁ @antipode _ _ inst_1 inst_2 _ _) ▷ A ≫ μ ▷ A)
| A ◁ (Δ ▷ A ≫ Iso.hom (α_ A A A) ≫ A ◁ @antipode _ _ inst_1 inst_2 _ _ ▷ A ≫ Iso.inv (α_ A A A) ≫ μ ▷ A)

Tactic:
rw [associator_inv_naturality_middle_assoc, Iso.hom_inv_id_assoc]

Premises used: CategoryTheory.MonoidalCategory.associator_inv_naturality_middle_assoc, 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
a :
  CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst) A (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) A)) (A ◁ ε ▷ A)
    (A ◁ η ▷ A)
| A ◁ Δ ▷ A ≫ A ◁ Iso.hom (α_ A A A) ≫ A ◁ A ◁ @antipode _ _ inst_1 inst_2 _ _ ▷ A ≫ A ◁ Iso.inv (α_ A A A) ≫ A ◁ μ ▷ A

Tactic:
simp only [← whiskerLeft_comp]
    rw [associator_inv_naturality_middle_assoc, Iso.hom_inv_id_assoc]
    simp only [← comp_whiskerRight]
    rw [antipode_right]
    simp only [comp_whiskerRight]
    simp only [whiskerLeft_comp]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp, CategoryTheory.MonoidalCategory.comp_whiskerRight, CategoryTheory.MonoidalCategory.associator_inv_naturality_middle_assoc, CategoryTheory.Iso.hom_inv_id_assoc, CategoryTheory.HopfObj.antipode_right