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 : Mon.{v₁, u₁} C
P : Bimod R S
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)
dsimp :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) R.X
        (TensorBimod.X
          { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }
          P) ⟶
      P.X)
    (CategoryStruct.comp (Y :=
      TensorBimod.X
        { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod R R }
        P)
      (CategoryStruct.comp (Y :=
        coequalizer (X :=
          tensorObj.{v₁, u₁} (𝒞 := inst) R.X
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X))
          (R.X ◁ μ ▷ P.X) (R.X ◁ (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
        (Iso.inv (PreservesCoequalizer.iso (tensorLeft R.X) (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
        (colimMap
          (parallelPairHom (R.X ◁ μ ▷ P.X) (R.X ◁ (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) (μ ▷ P.X)
            (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
            (Iso.inv (α_ R.X (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X) ≫
              Iso.inv (α_ R.X R.X R.X) ▷ P.X ≫ μ ▷ R.X ▷ P.X)
            (Iso.inv (α_ R.X R.X P.X) ≫ μ ▷ P.X) ⋯ ⋯)))
      (coequalizer.desc P.actLeft ⋯))
    (R.X ◁ coequalizer.desc P.actLeft ⋯ ≫ P.actLeft)
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) R.X (TensorBimod.X (regular R) P) ⟶ P.X)
    (TensorBimod.actLeft (regular R) P ≫ hom P) (R.X ◁ hom P ≫ P.actLeft)

Tactic:
dsimp [hom, TensorBimod.actLeft, regular]

Premises used: Bimod.TensorBimod.actLeft, Bimod.LeftUnitorBimod.hom, Bimod.regular
-- 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 : Mon.{v₁, u₁} C
P : Bimod R S
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) R.X
        (TensorBimod.X
          { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }
          P) ⟶
      P.X)
    (CategoryStruct.comp (Y :=
      TensorBimod.X
        { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod R R }
        P)
      (CategoryStruct.comp (Y :=
        coequalizer (X :=
          tensorObj.{v₁, u₁} (𝒞 := inst) R.X
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X))
          (R.X ◁ μ ▷ P.X) (R.X ◁ (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
        (Iso.inv (PreservesCoequalizer.iso (tensorLeft R.X) (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
        (colimMap
          (parallelPairHom (R.X ◁ μ ▷ P.X) (R.X ◁ (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) (μ ▷ P.X)
            (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
            (Iso.inv (α_ R.X (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X) ≫
              Iso.inv (α_ R.X R.X R.X) ▷ P.X ≫ μ ▷ R.X ▷ P.X)
            (Iso.inv (α_ R.X R.X P.X) ≫ μ ▷ P.X) ⋯ ⋯)))
      (coequalizer.desc P.actLeft ⋯))
    (R.X ◁ coequalizer.desc P.actLeft ⋯ ≫ P.actLeft)

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

Premises used: CategoryTheory.Limits.colimit, CategoryTheory.Functor.map, CategoryTheory.MonoidalCategory.tensorLeft, 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 : Mon.{v₁, u₁} C
P : Bimod R S
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)
rw : (Iso.inv (α_ R.X R.X P.X) ≫ μ ▷ P.X) ≫ P.actLeft
| CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
    (R.X ◁ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
    (CategoryStruct.comp (Y :=
      coequalizer (X :=
        tensorObj.{v₁, u₁} (𝒞 := inst) R.X
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X))
        (R.X ◁ μ ▷ P.X) (R.X ◁ (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
      (Iso.inv (PreservesCoequalizer.iso (tensorLeft R.X) (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
      (CategoryStruct.comp (Y :=
        TensorBimod.X
          { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }
          P)
        (colimMap
          (parallelPairHom (R.X ◁ μ ▷ P.X) (R.X ◁ (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) (μ ▷ P.X)
            (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
            (Iso.inv (α_ R.X (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X) ≫
              Iso.inv (α_ R.X R.X R.X) ▷ P.X ≫ μ ▷ R.X ▷ P.X)
            (Iso.inv (α_ R.X R.X P.X) ≫ μ ▷ P.X) ⋯ ⋯))
        (coequalizer.desc P.actLeft ⋯)))

Tactic:
rw [id_tensor_π_preserves_coequalizer_inv_colimMap_desc]

Premises used: id_tensor_π_preserves_coequalizer_inv_colimMap_desc
-- 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 : Mon.{v₁, u₁} C
P : Bimod R S
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_lhs :
  (Iso.inv (α_ R.X R.X P.X) ≫ μ ▷ P.X) ≫ P.actLeft =
    CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
      (R.X ◁ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
      (R.X ◁ coequalizer.desc P.actLeft ⋯ ≫ P.actLeft)
⊢ CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
      (R.X ◁ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
      (CategoryStruct.comp (Y :=
        TensorBimod.X
          { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }
          P)
        (CategoryStruct.comp (Y :=
          coequalizer (X :=
            tensorObj.{v₁, u₁} (𝒞 := inst) R.X
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X))
            (R.X ◁ μ ▷ P.X) (R.X ◁ (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
          (Iso.inv (PreservesCoequalizer.iso (tensorLeft R.X) (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
          (colimMap
            (parallelPairHom (R.X ◁ μ ▷ P.X) (R.X ◁ (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) (μ ▷ P.X)
              (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
              (Iso.inv (α_ R.X (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X) ≫
                Iso.inv (α_ R.X R.X R.X) ▷ P.X ≫ μ ▷ R.X ▷ P.X)
              (Iso.inv (α_ R.X R.X P.X) ≫ μ ▷ P.X) ⋯ ⋯)))
        (coequalizer.desc P.actLeft ⋯)) =
    CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
      (R.X ◁ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
      (R.X ◁ coequalizer.desc P.actLeft ⋯ ≫ P.actLeft)

Tactic:
slice_lhs 1 4 => rw [id_tensor_π_preserves_coequalizer_inv_colimMap_desc]

Premises used: id_tensor_π_preserves_coequalizer_inv_colimMap_desc
-- 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 : Mon.{v₁, u₁} C
P : Bimod R S
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 R.X P.X) ≫ R.X ◁ P.actLeft ≫ P.actLeft
| μ ▷ P.X ≫ P.actLeft

Tactic:
rw [left_assoc]

Premises used: Bimod.left_assoc
-- 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 : Mon.{v₁, u₁} C
P : Bimod R S
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_lhs :
  Iso.inv (α_ R.X R.X P.X) ≫ Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft ≫ P.actLeft =
    CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
      (R.X ◁ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
      (R.X ◁ coequalizer.desc P.actLeft ⋯ ≫ P.actLeft)
⊢ (Iso.inv (α_ R.X R.X P.X) ≫ μ ▷ P.X) ≫ P.actLeft =
    CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
      (R.X ◁ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
      (R.X ◁ coequalizer.desc P.actLeft ⋯ ≫ P.actLeft)

Tactic:
slice_lhs 2 3 => rw [left_assoc]

Premises used: Bimod.left_assoc
-- 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 : Mon.{v₁, u₁} C
P : Bimod R S
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 ◁ P.actLeft
| CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
    (R.X ◁ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) (R.X ◁ coequalizer.desc P.actLeft ⋯)

Tactic:
rw [← whiskerLeft_comp, coequalizer.π_desc]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp, CategoryTheory.Limits.coequalizer.π_desc
-- 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 : Mon.{v₁, u₁} C
P : Bimod R S
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 :
  Iso.inv (α_ R.X R.X P.X) ≫ Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft ≫ P.actLeft = R.X ◁ P.actLeft ≫ P.actLeft
⊢ Iso.inv (α_ R.X R.X P.X) ≫ Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft ≫ P.actLeft =
    CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
      (R.X ◁ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
      (R.X ◁ coequalizer.desc P.actLeft ⋯ ≫ P.actLeft)

Tactic:
slice_rhs 1 2 => rw [← whiskerLeft_comp, coequalizer.π_desc]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp, CategoryTheory.Limits.coequalizer.π_desc
-- 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 : Mon.{v₁, u₁} C
P : Bimod R S
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)
rw rw₁ : R.X ◁ P.actLeft ≫ P.actLeft = R.X ◁ P.actLeft ≫ P.actLeft
⊢ Iso.inv (α_ R.X R.X P.X) ≫ Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft ≫ P.actLeft = R.X ◁ P.actLeft ≫ P.actLeft

Tactic:
rw [Iso.inv_hom_id_assoc]

Premises used: CategoryTheory.Iso.inv_hom_id_assoc
-- 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 : Mon.{v₁, u₁} C
P : Bimod R S
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)
dsimp :
  Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X (regular R) P) S.X ⟶ P.X)
    (TensorBimod.actRight (regular R) P ≫ hom P) (hom P ▷ S.X ≫ P.actRight)
⊢ ((regular R).tensorBimod P).actRight ≫ hom P = hom P ▷ S.X ≫ P.actRight

Tactic:
dsimp
-- 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 : Mon.{v₁, u₁} C
P : Bimod R S
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)
dsimp :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst)
        (TensorBimod.X
          { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }
          P)
        S.X ⟶
      P.X)
    (CategoryStruct.comp (Y :=
      TensorBimod.X
        { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod R R }
        P)
      (CategoryStruct.comp (Y :=
        coequalizer (X :=
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X)
            S.X)
          (μ ▷ P.X ▷ S.X) ((Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X))
        (Iso.inv (PreservesCoequalizer.iso (tensorRight S.X) (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
        (colimMap
          (parallelPairHom (μ ▷ P.X ▷ S.X) ((Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X) (μ ▷ P.X)
            (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
            (Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X S.X) ≫
              Iso.hom (α_ R.X R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)) ≫
                R.X ◁ R.X ◁ P.actRight ≫ Iso.inv (α_ R.X R.X P.X))
            (Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight) ⋯ ⋯)))
      (coequalizer.desc P.actLeft ⋯))
    (coequalizer.desc P.actLeft ⋯ ▷ S.X ≫ P.actRight)
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X (regular R) P) S.X ⟶ P.X)
    (TensorBimod.actRight (regular R) P ≫ hom P) (hom P ▷ S.X ≫ P.actRight)

Tactic:
dsimp [hom, TensorBimod.actRight, regular]

Premises used: Bimod.TensorBimod.actRight, Bimod.LeftUnitorBimod.hom, Bimod.regular
-- 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 : Mon.{v₁, u₁} C
P : Bimod R S
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)
        (TensorBimod.X
          { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }
          P)
        S.X ⟶
      P.X)
    (CategoryStruct.comp (Y :=
      TensorBimod.X
        { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod R R }
        P)
      (CategoryStruct.comp (Y :=
        coequalizer (X :=
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X)
            S.X)
          (μ ▷ P.X ▷ S.X) ((Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X))
        (Iso.inv (PreservesCoequalizer.iso (tensorRight S.X) (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
        (colimMap
          (parallelPairHom (μ ▷ P.X ▷ S.X) ((Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X) (μ ▷ P.X)
            (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
            (Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X S.X) ≫
              Iso.hom (α_ R.X R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)) ≫
                R.X ◁ R.X ◁ P.actRight ≫ Iso.inv (α_ R.X R.X P.X))
            (Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight) ⋯ ⋯)))
      (coequalizer.desc P.actLeft ⋯))
    (coequalizer.desc P.actLeft ⋯ ▷ S.X ≫ P.actRight)

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

Premises used: CategoryTheory.Limits.colimit, CategoryTheory.Functor.map, 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 : Mon.{v₁, u₁} C
P : Bimod R S
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)
rw : (Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight) ≫ P.actLeft
| CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) S.X)
    (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X)
    (CategoryStruct.comp (Y :=
      coequalizer (X :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X)
          S.X)
        (μ ▷ P.X ▷ S.X) ((Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X))
      (Iso.inv (PreservesCoequalizer.iso (tensorRight S.X) (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
      (CategoryStruct.comp (Y :=
        TensorBimod.X
          { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }
          P)
        (colimMap
          (parallelPairHom (μ ▷ P.X ▷ S.X) ((Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X) (μ ▷ P.X)
            (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
            (Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X S.X) ≫
              Iso.hom (α_ R.X R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)) ≫
                R.X ◁ R.X ◁ P.actRight ≫ Iso.inv (α_ R.X R.X P.X))
            (Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight) ⋯ ⋯))
        (coequalizer.desc P.actLeft ⋯)))

Tactic:
rw [π_tensor_id_preserves_coequalizer_inv_colimMap_desc]

Premises used: π_tensor_id_preserves_coequalizer_inv_colimMap_desc
-- 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 : Mon.{v₁, u₁} C
P : Bimod R S
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_lhs :
  (Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight) ≫ P.actLeft =
    CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) S.X)
      (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X)
      (coequalizer.desc P.actLeft ⋯ ▷ S.X ≫ P.actRight)
⊢ CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) S.X)
      (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X)
      (CategoryStruct.comp (Y :=
        TensorBimod.X
          { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }
          P)
        (CategoryStruct.comp (Y :=
          coequalizer (X :=
            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X)
              S.X)
            (μ ▷ P.X ▷ S.X) ((Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X))
          (Iso.inv (PreservesCoequalizer.iso (tensorRight S.X) (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
          (colimMap
            (parallelPairHom (μ ▷ P.X ▷ S.X) ((Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X) (μ ▷ P.X)
              (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
              (Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X S.X) ≫
                Iso.hom (α_ R.X R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)) ≫
                  R.X ◁ R.X ◁ P.actRight ≫ Iso.inv (α_ R.X R.X P.X))
              (Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight) ⋯ ⋯)))
        (coequalizer.desc P.actLeft ⋯)) =
    CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) S.X)
      (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X)
      (coequalizer.desc P.actLeft ⋯ ▷ S.X ≫ P.actRight)

Tactic:
slice_lhs 1 4 => rw [π_tensor_id_preserves_coequalizer_inv_colimMap_desc]

Premises used: π_tensor_id_preserves_coequalizer_inv_colimMap_desc
-- 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 : Mon.{v₁, u₁} C
P : Bimod R S
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 ▷ S.X
| CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) S.X)
    (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X) (coequalizer.desc P.actLeft ⋯ ▷ S.X)

Tactic:
rw [← comp_whiskerRight, coequalizer.π_desc]

Premises used: CategoryTheory.MonoidalCategory.comp_whiskerRight, CategoryTheory.Limits.coequalizer.π_desc
-- 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 : Mon.{v₁, u₁} C
P : Bimod R S
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 : (Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight) ≫ P.actLeft = P.actLeft ▷ S.X ≫ P.actRight
⊢ (Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight) ≫ P.actLeft =
    CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) S.X)
      (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X)
      (coequalizer.desc P.actLeft ⋯ ▷ S.X ≫ P.actRight)

Tactic:
slice_rhs 1 2 => rw [← comp_whiskerRight, coequalizer.π_desc]

Premises used: CategoryTheory.MonoidalCategory.comp_whiskerRight, CategoryTheory.Limits.coequalizer.π_desc
-- 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 : Mon.{v₁, u₁} C
P : Bimod R S
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)
rw : Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight ≫ P.actLeft
| P.actLeft ▷ S.X ≫ P.actRight

Tactic:
rw [middle_assoc]

Premises used: Bimod.middle_assoc