awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
R S T : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
a : Iso.hom (α_ P.X Q.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ P.X ◁ Q.X ◁ η
| tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X ◁ η ≫ Iso.hom (α_ P.X Q.X T.X)

Tactic:
rw [associator_naturality_right]

Premises used: CategoryTheory.MonoidalCategory.associator_naturality_right
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
R S T : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
slice_lhs :
  ((Iso.hom (α_ P.X Q.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ P.X ◁ Q.X ◁ η) ≫ P.X ◁ Q.actRight) ≫
      coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) =
    coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst) ≫
      Iso.hom (ρ_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))
⊢ tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X ◁ η ≫
      Iso.hom (α_ P.X Q.X T.X) ≫
        P.X ◁ Q.actRight ≫ coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) =
    coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst) ≫
      Iso.hom (ρ_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))

Tactic:
slice_lhs 1 2 => rw [associator_naturality_right]

Premises used: CategoryTheory.MonoidalCategory.associator_naturality_right
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
R S T : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
a : P.X ◁ Iso.hom (ρ_ Q.X)
| P.X ◁ Q.X ◁ η ≫ P.X ◁ Q.actRight

Tactic:
rw [← whiskerLeft_comp, actRight_one]

Premises used: Bimod.actRight_one, CategoryTheory.MonoidalCategory.whiskerLeft_comp
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
R S T : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
slice_lhs :
  Iso.hom (α_ P.X Q.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫
      P.X ◁ Iso.hom (ρ_ Q.X) ≫ coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) =
    coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst) ≫
      Iso.hom (ρ_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))
⊢ ((Iso.hom (α_ P.X Q.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ P.X ◁ Q.X ◁ η) ≫ P.X ◁ Q.actRight) ≫
      coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) =
    coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst) ≫
      Iso.hom (ρ_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))

Tactic:
slice_lhs 2 3 => rw [← whiskerLeft_comp, actRight_one]

Premises used: Bimod.actRight_one, CategoryTheory.MonoidalCategory.whiskerLeft_comp
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
R S T : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ Iso.hom (α_ P.X Q.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫
      P.X ◁ Iso.hom (ρ_ Q.X) ≫ coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) =
    coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst) ≫
      Iso.hom (ρ_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
R S T : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X (X P Q)) T.X ⟶ X P Q)
    (actLeft P Q ▷ T.X ≫ actRight P Q) (Iso.hom (α_ R.X (X P Q) T.X) ≫ R.X ◁ actRight P Q ≫ actLeft P Q)

Tactic:
(cancel_epi ((tensorLeft _ ⋙ tensorRight _).map (coequalizer.π _ _))).1 ?_

Premises used: CategoryTheory.Functor.comp, CategoryTheory.Limits.colimit, CategoryTheory.Functor.map, CategoryTheory.MonoidalCategory.tensorLeft, CategoryTheory.MonoidalCategory.tensorRight, Bimod.TensorBimod.X, Iff.mp, CategoryTheory.Limits.coequalizer.π, CategoryTheory.cancel_epi
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
R S T : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
refine :
  Eq (α :=
    Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft R.X ⋙ tensorRight T.X) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) ⟶ X P Q)
    (CategoryTheory.Functor.map (tensorLeft R.X ⋙ tensorRight T.X)
        (coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) ≫
      actLeft P Q ▷ T.X ≫ actRight P Q)
    (CategoryTheory.Functor.map (tensorLeft R.X ⋙ tensorRight T.X)
        (coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) ≫
      Iso.hom (α_ R.X (X P Q) T.X) ≫ R.X ◁ actRight P Q ≫ actLeft P Q)
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X (X P Q)) T.X ⟶ X P Q)
    (actLeft P Q ▷ T.X ≫ actRight P Q) (Iso.hom (α_ R.X (X P Q) T.X) ≫ R.X ◁ actRight P Q ≫ actLeft P Q)

Tactic:
refine (cancel_epi ((tensorLeft _ ⋙ tensorRight _).map (coequalizer.π _ _))).1 ?_

Premises used: CategoryTheory.Functor.comp, CategoryTheory.Functor.map, CategoryTheory.MonoidalCategory.tensorLeft, CategoryTheory.MonoidalCategory.tensorRight, Iff.mp, CategoryTheory.Limits.coequalizer.π, CategoryTheory.cancel_epi
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
R S T : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
a :
  Iso.inv (α_ R.X P.X Q.X) ▷ T.X ≫
    P.actLeft ▷ Q.X ▷ T.X ≫ coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ T.X
| (CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst)
        (tensorObj.{v₁, u₁} (𝒞 := inst) R.X
          (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))
        T.X)
      ((R.X ◁ coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) ▷ T.X :
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X))
            T.X ⟶
          tensorObj.{v₁, u₁} (𝒞 := inst)
            (tensorObj.{v₁, u₁} (𝒞 := inst) R.X
              (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))
            T.X)
      (actLeft P Q ▷ T.X :
        tensorObj.{v₁, u₁} (𝒞 := inst)
            (tensorObj.{v₁, u₁} (𝒞 := inst) R.X
              (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))
            T.X ⟶
          tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
            T.X) :
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X)) T.X ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X)

Tactic:
rw [← comp_whiskerRight, whiskerLeft_π_actLeft, comp_whiskerRight,
    comp_whiskerRight]

Premises used: Bimod.TensorBimod.whiskerLeft_π_actLeft, CategoryTheory.MonoidalCategory.comp_whiskerRight
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
R S T : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
a :
  Iso.hom (α_ P.X Q.X T.X) ≫
    P.X ◁ Q.actRight ≫ coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)
| (CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X)
      (coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ T.X :
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X ⟶
          tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
            T.X)
      (actRight P Q : tensorObj.{v₁, u₁} (𝒞 := inst) (X P Q) T.X ⟶ X P Q) :
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X ⟶
      coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))

Tactic:
rw [π_tensor_id_actRight]

Premises used: Bimod.TensorBimod.π_tensor_id_actRight
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
R S T : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
a : Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X) Q.X T.X) ≫ P.actLeft ▷ tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X
| P.actLeft ▷ Q.X ▷ T.X ≫ Iso.hom (α_ P.X Q.X T.X)

Tactic:
rw [associator_naturality_left]

Premises used: CategoryTheory.MonoidalCategory.associator_naturality_left
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
R S T : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
a :
  Iso.hom (α_ R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) ≫
    R.X ◁ coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ T.X
| (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫
    Iso.hom (α_ R.X (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X)

Tactic:
rw [associator_naturality_middle]

Premises used: CategoryTheory.MonoidalCategory.associator_naturality_middle
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
R S T : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
a :
  R.X ◁ Iso.hom (α_ P.X Q.X T.X) ≫
    R.X ◁ P.X ◁ Q.actRight ≫ R.X ◁ coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)
| (CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) R.X
        (tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
          T.X))
      (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ T.X :
        tensorObj.{v₁, u₁} (𝒞 := inst) R.X
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) ⟶
          tensorObj.{v₁, u₁} (𝒞 := inst) R.X
            (tensorObj.{v₁, u₁} (𝒞 := inst)
              (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X))
      (R.X ◁ actRight P Q :
        tensorObj.{v₁, u₁} (𝒞 := inst) R.X
            (tensorObj.{v₁, u₁} (𝒞 := inst)
              (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X) ⟶
          tensorObj.{v₁, u₁} (𝒞 := inst) R.X
            (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))) :
    tensorObj.{v₁, u₁} (𝒞 := inst) R.X (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))

Tactic:
rw [← whiskerLeft_comp, π_tensor_id_actRight,
    whiskerLeft_comp, whiskerLeft_comp]

Premises used: Bimod.TensorBimod.π_tensor_id_actRight, CategoryTheory.MonoidalCategory.whiskerLeft_comp
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
R S T : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
a :
  Iso.inv (α_ R.X P.X Q.X) ≫
    P.actLeft ▷ Q.X ≫ coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)
| (CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))
      (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) :
        tensorObj.{v₁, u₁} (𝒞 := inst) R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) ⟶
          tensorObj.{v₁, u₁} (𝒞 := inst) R.X
            (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))
      (actLeft P Q : tensorObj.{v₁, u₁} (𝒞 := inst) R.X (X P Q) ⟶ X P Q) :
    tensorObj.{v₁, u₁} (𝒞 := inst) R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) ⟶
      coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))

Tactic:
rw [whiskerLeft_π_actLeft]

Premises used: Bimod.TensorBimod.whiskerLeft_π_actLeft
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
R S T : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
slice_rhs :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X)) T.X ⟶
      coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
    (Iso.inv (α_ R.X P.X Q.X) ▷ T.X ≫
      ((Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X) Q.X T.X) ≫
            P.actLeft ▷ tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) ≫
          P.X ◁ Q.actRight) ≫
        coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
    (Iso.hom (α_ R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) ≫
      R.X ◁ Iso.hom (α_ P.X Q.X T.X) ≫
        R.X ◁ P.X ◁ Q.actRight ≫
          Iso.inv (α_ R.X P.X Q.X) ≫
            P.actLeft ▷ Q.X ≫ coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X)) T.X ⟶
      coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
    (Iso.inv (α_ R.X P.X Q.X) ▷ T.X ≫
        ((Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X) Q.X T.X) ≫
              P.actLeft ▷ tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) ≫
            P.X ◁ Q.actRight) ≫
          coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X)) T.X ⟶
        coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
    (Iso.hom (α_ R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) ≫
        CategoryStruct.comp (Y :=
          tensorObj.{v₁, u₁} (𝒞 := inst) R.X
            (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))
          (R.X ◁ Iso.hom (α_ P.X Q.X T.X) ≫
              R.X ◁ P.X ◁ Q.actRight ≫
                R.X ◁ coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) :
            tensorObj.{v₁, u₁} (𝒞 := inst) R.X
                (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) ⟶
              tensorObj.{v₁, u₁} (𝒞 := inst) R.X
                (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))
          (actLeft P Q : tensorObj.{v₁, u₁} (𝒞 := inst) R.X (X P Q) ⟶ X P Q) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X)) T.X ⟶
        coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))

Tactic:
slice_rhs 4 5 => rw [whiskerLeft_π_actLeft]

Premises used: Bimod.TensorBimod.whiskerLeft_π_actLeft
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
R S T : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
a : Iso.inv (α_ R.X P.X (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X)) ≫ tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ◁ Q.actRight
| R.X ◁ P.X ◁ Q.actRight ≫ Iso.inv (α_ R.X P.X Q.X)

Tactic:
rw [associator_inv_naturality_right]

Premises used: CategoryTheory.MonoidalCategory.associator_inv_naturality_right
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
R S T : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
slice_rhs :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X)) T.X ⟶
      coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
    (Iso.inv (α_ R.X P.X Q.X) ▷ T.X ≫
      ((Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X) Q.X T.X) ≫
            P.actLeft ▷ tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) ≫
          P.X ◁ Q.actRight) ≫
        coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
    (Iso.hom (α_ R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) ≫
      R.X ◁ Iso.hom (α_ P.X Q.X T.X) ≫
        ((Iso.inv (α_ R.X P.X (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X)) ≫
              tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ◁ Q.actRight) ≫
            P.actLeft ▷ Q.X) ≫
          coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X)) T.X ⟶
      coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
    (Iso.inv (α_ R.X P.X Q.X) ▷ T.X ≫
      ((Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X) Q.X T.X) ≫
            P.actLeft ▷ tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) ≫
          P.X ◁ Q.actRight) ≫
        coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
    (Iso.hom (α_ R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) ≫
      R.X ◁ Iso.hom (α_ P.X Q.X T.X) ≫
        R.X ◁ P.X ◁ Q.actRight ≫
          Iso.inv (α_ R.X P.X Q.X) ≫
            P.actLeft ▷ Q.X ≫ coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))

Tactic:
slice_rhs 3 4 => rw [associator_inv_naturality_right]

Premises used: CategoryTheory.MonoidalCategory.associator_inv_naturality_right
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
R S T : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
a : P.actLeft ▷ tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X ≫ P.X ◁ Q.actRight
| tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ◁ Q.actRight ≫ P.actLeft ▷ Q.X

Tactic:
rw [whisker_exchange]

Premises used: CategoryTheory.MonoidalCategory.whisker_exchange
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
R S T : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
slice_rhs :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X)) T.X ⟶
      coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
    (Iso.inv (α_ R.X P.X Q.X) ▷ T.X ≫
      ((Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X) Q.X T.X) ≫
            P.actLeft ▷ tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) ≫
          P.X ◁ Q.actRight) ≫
        coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
    (Iso.hom (α_ R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) ≫
      R.X ◁ Iso.hom (α_ P.X Q.X T.X) ≫
        Iso.inv (α_ R.X P.X (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X)) ≫
          (P.actLeft ▷ tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X ≫ P.X ◁ Q.actRight) ≫
            coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X)) T.X ⟶
      coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
    (Iso.inv (α_ R.X P.X Q.X) ▷ T.X ≫
      ((Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X) Q.X T.X) ≫
            P.actLeft ▷ tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) ≫
          P.X ◁ Q.actRight) ≫
        coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
    (Iso.hom (α_ R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) ≫
      R.X ◁ Iso.hom (α_ P.X Q.X T.X) ≫
        ((Iso.inv (α_ R.X P.X (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X)) ≫
              tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ◁ Q.actRight) ≫
            P.actLeft ▷ Q.X) ≫
          coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))

Tactic:
slice_rhs 4 5 => rw [whisker_exchange]

Premises used: CategoryTheory.MonoidalCategory.whisker_exchange