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

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

Tactic:
slice_lhs 2 3 => rw [middle_assoc]

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

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

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

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

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

Tactic:
rw [right_assoc]

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

Tactic:
slice_lhs 2 3 => rw [right_assoc]

Premises used: Bimod.right_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 : P.actRight ▷ S.X
| CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)) S.X)
    (coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ▷ S.X) (coequalizer.desc P.actRight ⋯ ▷ 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_ P.X S.X S.X) ≫ Iso.inv (α_ P.X S.X S.X) ≫ P.actRight ▷ S.X ≫ P.actRight = P.actRight ▷ S.X ≫ P.actRight
⊢ Iso.hom (α_ P.X S.X S.X) ≫ Iso.inv (α_ P.X S.X S.X) ≫ P.actRight ▷ S.X ≫ P.actRight =
    CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)) S.X)
      (coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ▷ S.X)
      (coequalizer.desc P.actRight ⋯ ▷ 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 rw₁ : P.actRight ▷ S.X ≫ P.actRight = P.actRight ▷ S.X ≫ P.actRight
⊢ Iso.hom (α_ P.X S.X S.X) ≫ Iso.inv (α_ P.X S.X S.X) ≫ P.actRight ▷ S.X ≫ P.actRight = P.actRight ▷ S.X ≫ P.actRight

Tactic:
rw [Iso.hom_inv_id_assoc]

Premises used: CategoryTheory.Iso.hom_inv_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
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)
X Y Z : Mon.{v₁, u₁} C
M : Bimod X Y
N : Bimod Y Z
h : Hom.hom (M.whiskerLeft (𝟙 N)) = Hom.hom (𝟙 (M.tensorBimod N))
⊢ M.whiskerLeft (𝟙 N) = 𝟙 (M.tensorBimod N)

Tactic:
ext
-- 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
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)
X Y Z : Mon.{v₁, u₁} C
M : Bimod X Y
N : Bimod Y Z
h :
  coequalizer.π (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft) ≫ Hom.hom (M.whiskerLeft (𝟙 N)) =
    coequalizer.π (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft) ≫ Hom.hom (𝟙 (M.tensorBimod N))
⊢ Hom.hom (M.whiskerLeft (𝟙 N)) = Hom.hom (𝟙 (M.tensorBimod N))

Tactic:
apply Limits.coequalizer.hom_ext

Premises used: CategoryTheory.Limits.coequalizer.hom_ext
-- 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
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)
X Y Z : Mon.{v₁, u₁} C
M : Bimod X Y
N : Bimod Y Z
⊢ coequalizer.π (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft) ≫ Hom.hom (M.whiskerLeft (𝟙 N)) =
    coequalizer.π (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft) ≫ Hom.hom (𝟙 (M.tensorBimod N))

Tactic:
simp [TensorBimod.X]

Premises used: Bimod.TensorBimod.X
-- 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
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)
X Y Z : Mon.{v₁, u₁} C
M : Bimod X Y
N : Bimod Y Z
h : Hom.hom (whiskerRight (𝟙 M) N) = Hom.hom (𝟙 (M.tensorBimod N))
⊢ whiskerRight (𝟙 M) N = 𝟙 (M.tensorBimod N)

Tactic:
ext