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
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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
a :
  Iso.inv (α_ P.X Q.X L.X) ≫
    coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X ≫
      coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
        (Iso.hom (α_ (TensorBimod.X P Q) T.X L.X) ≫ TensorBimod.X P Q ◁ L.actLeft)
| CategoryStruct.comp (Y :=
    Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft P.X)
      (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
    (P.X ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))
    (CategoryStruct.comp (Y :=
      coequalizer (X :=
        tensorObj.{v₁, u₁} (𝒞 := inst) P.X
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X))
        (P.X ◁ Q.actRight ▷ L.X) (P.X ◁ (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
      (Iso.inv
        (PreservesCoequalizer.iso (tensorLeft P.X) (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
      (coequalizer.desc
        (Iso.inv (α_ P.X Q.X L.X) ≫
          coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X ≫
            coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
              (Iso.hom (α_ (TensorBimod.X P Q) T.X L.X) ≫ TensorBimod.X P Q ◁ L.actLeft))
        ⋯))

Tactic:
rw [id_tensor_π_preserves_coequalizer_inv_desc]

Premises used: id_tensor_π_preserves_coequalizer_inv_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
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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
a : P.X ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)
| Iso.inv (α_ P.X Q.X L.X) ≫
    Iso.hom (α_ P.X Q.X L.X) ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.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
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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
a :
  coequalizer.π (P.actRight ▷ (Q.tensorBimod L).X)
    (Iso.hom (α_ P.X S.X (Q.tensorBimod L).X) ≫ P.X ◁ (Q.tensorBimod L).actLeft)
| coequalizer.π (P.actRight ▷ (Q.tensorBimod L).X)
      (Iso.hom (α_ P.X S.X (Q.tensorBimod L).X) ≫ P.X ◁ (Q.tensorBimod L).actLeft) ≫
    𝟙
      (coequalizer (P.actRight ▷ (Q.tensorBimod L).X)
        (Iso.hom (α_ P.X S.X (Q.tensorBimod L).X) ≫ P.X ◁ (Q.tensorBimod L).actLeft))

Tactic:
rw [Category.comp_id]

Premises used: CategoryTheory.Category.comp_id
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
A B : Mon.{v₁, u₁} C
M : Bimod A B
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
dsimp : μ ▷ P.X ≫ P.actLeft = (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫ P.actLeft
⊢ (regular R).actRight ▷ P.X ≫ P.actLeft = (Iso.hom (α_ (regular R).X R.X P.X) ≫ (regular R).X ◁ P.actLeft) ≫ P.actLeft

Tactic:
dsimp
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
A B : Mon.{v₁, u₁} C
M : Bimod A B
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
rw : μ ▷ P.X ≫ P.actLeft = Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft ≫ P.actLeft
rw₁ rw₂ :
  Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft ≫ P.actLeft = Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft ≫ P.actLeft
⊢ μ ▷ P.X ≫ P.actLeft = (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫ P.actLeft

Tactic:
rw [Category.assoc, left_assoc]

Premises used: CategoryTheory.Category.assoc, 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
dsimp :
  coequalizer.desc P.actLeft ⋯ ≫
      Iso.inv (λ_ P.X) ≫
        η ▷ P.X ≫
          coequalizer.π ((regular R).actRight ▷ P.X) (Iso.hom (α_ (regular R).X R.X P.X) ≫ (regular R).X ◁ P.actLeft) =
    𝟙 (coequalizer ((regular R).actRight ▷ P.X) (Iso.hom (α_ (regular R).X R.X P.X) ≫ (regular R).X ◁ P.actLeft))
⊢ hom P ≫ inv P = 𝟙 (TensorBimod.X (regular R) P)

Tactic:
dsimp only [hom, inv, TensorBimod.X]

Premises used: Bimod.LeftUnitorBimod.inv, Bimod.LeftUnitorBimod.hom, 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
R S : Mon.{v₁, u₁} C
P : Bimod R S
h :
  CategoryStruct.comp (Y :=
      coequalizer ((regular R).actRight ▷ P.X) (Iso.hom (α_ (regular R).X R.X P.X) ≫ (regular R).X ◁ P.actLeft))
      (coequalizer.π ((regular R).actRight ▷ P.X) (Iso.hom (α_ (regular R).X R.X P.X) ≫ (regular R).X ◁ P.actLeft))
      (coequalizer.desc P.actLeft ⋯ ≫
        Iso.inv (λ_ P.X) ≫
          η ▷ P.X ≫
            coequalizer.π ((regular R).actRight ▷ P.X)
              (Iso.hom (α_ (regular R).X R.X P.X) ≫ (regular R).X ◁ P.actLeft)) =
    coequalizer.π ((regular R).actRight ▷ P.X) (Iso.hom (α_ (regular R).X R.X P.X) ≫ (regular R).X ◁ P.actLeft) ≫
      𝟙 (coequalizer ((regular R).actRight ▷ P.X) (Iso.hom (α_ (regular R).X R.X P.X) ≫ (regular R).X ◁ P.actLeft))
⊢ coequalizer.desc P.actLeft ⋯ ≫
      Iso.inv (λ_ P.X) ≫
        η ▷ P.X ≫
          coequalizer.π ((regular R).actRight ▷ P.X) (Iso.hom (α_ (regular R).X R.X P.X) ≫ (regular R).X ◁ P.actLeft) =
    𝟙 (coequalizer ((regular R).actRight ▷ P.X) (Iso.hom (α_ (regular R).X R.X P.X) ≫ (regular R).X ◁ P.actLeft))

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
R S : Mon.{v₁, u₁} C
P : Bimod R S
h :
  CategoryStruct.comp (Y := coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
      (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
      (coequalizer.desc P.actLeft ⋯ ≫
        Iso.inv (λ_ P.X) ≫ η ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
⊢ CategoryStruct.comp (Y :=
      coequalizer ((regular R).actRight ▷ P.X) (Iso.hom (α_ (regular R).X R.X P.X) ≫ (regular R).X ◁ P.actLeft))
      (coequalizer.π ((regular R).actRight ▷ P.X) (Iso.hom (α_ (regular R).X R.X P.X) ≫ (regular R).X ◁ P.actLeft))
      (coequalizer.desc P.actLeft ⋯ ≫
        Iso.inv (λ_ P.X) ≫
          η ▷ P.X ≫
            coequalizer.π ((regular R).actRight ▷ P.X)
              (Iso.hom (α_ (regular R).X R.X P.X) ≫ (regular R).X ◁ P.actLeft)) =
    coequalizer.π ((regular R).actRight ▷ P.X) (Iso.hom (α_ (regular R).X R.X P.X) ≫ (regular R).X ◁ P.actLeft) ≫
      𝟙 (coequalizer ((regular R).actRight ▷ P.X) (Iso.hom (α_ (regular R).X R.X P.X) ≫ (regular R).X ◁ P.actLeft))

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
a : P.actLeft
| CategoryStruct.comp (Y := coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
    (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) (coequalizer.desc P.actLeft ⋯)

Tactic:
rw [coequalizer.π_desc]

Premises used: 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
h :
  ((P.actLeft ≫ Iso.inv (λ_ P.X)) ≫ η ▷ P.X) ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
⊢ CategoryStruct.comp (Y := coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
      (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
      (coequalizer.desc P.actLeft ⋯ ≫
        Iso.inv (λ_ P.X) ≫ η ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))

Tactic:
slice_lhs 1 2 => rw [coequalizer.π_desc]

Premises used: 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
a : Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫ tensorUnit.{v₁, u₁} C (𝒞 := inst) ◁ P.actLeft
| P.actLeft ≫ Iso.inv (λ_ P.X)

Tactic:
rw [leftUnitor_inv_naturality]

Premises used: CategoryTheory.MonoidalCategory.leftUnitor_inv_naturality
-- 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
h :
  ((Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫ tensorUnit.{v₁, u₁} C (𝒞 := inst) ◁ P.actLeft) ≫ η ▷ P.X) ≫
      coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
⊢ ((P.actLeft ≫ Iso.inv (λ_ P.X)) ≫ η ▷ P.X) ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))

Tactic:
slice_lhs 1 2 => rw [leftUnitor_inv_naturality]

Premises used: CategoryTheory.MonoidalCategory.leftUnitor_inv_naturality
-- 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
a : η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ≫ R.X ◁ P.actLeft
| tensorUnit.{v₁, u₁} C (𝒞 := inst) ◁ P.actLeft ≫ η ▷ P.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 : Mon.{v₁, u₁} C
P : Bimod R S
h :
  Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
      (η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ≫ R.X ◁ P.actLeft) ≫
        coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
⊢ ((Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫ tensorUnit.{v₁, u₁} C (𝒞 := inst) ◁ P.actLeft) ≫ η ▷ P.X) ≫
      coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))

Tactic:
slice_lhs 2 3 => 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 : Mon.{v₁, u₁} C
P : Bimod R S
a : Iso.inv (α_ R.X R.X P.X) ≫ Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft
| R.X ◁ P.actLeft

Tactic:
rw [← Iso.inv_hom_id_assoc (α_ R.X R.X P.X) (R.X ◁ P.actLeft)]

Premises used: CategoryTheory.MonoidalCategoryStruct.whiskerLeft, CategoryTheory.Iso.inv_hom_id_assoc, Bimod.X, Bimod.actLeft, CategoryTheory.Mon.X, CategoryTheory.MonoidalCategoryStruct.associator
-- 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
h :
  Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
      η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ≫
        (Iso.inv (α_ R.X R.X P.X) ≫ Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
          coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
⊢ Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
      (η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ≫ R.X ◁ P.actLeft) ≫
        coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))

Tactic:
slice_lhs 3 3 => rw [← Iso.inv_hom_id_assoc (α_ R.X R.X P.X) (R.X ◁ P.actLeft)]

Premises used: CategoryTheory.MonoidalCategoryStruct.whiskerLeft, CategoryTheory.Iso.inv_hom_id_assoc, Bimod.X, Bimod.actLeft, CategoryTheory.Mon.X, CategoryTheory.MonoidalCategoryStruct.associator
-- 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
a : μ ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
| Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)

Tactic:
rw [← Category.assoc, ← coequalizer.condition]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Limits.coequalizer.condition
-- 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
h :
  Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
      η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ≫
        Iso.inv (α_ R.X R.X P.X) ≫ μ ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
⊢ Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
      η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ≫
        (Iso.inv (α_ R.X R.X P.X) ≫ Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
          coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))

Tactic:
slice_lhs 4 6 => rw [← Category.assoc, ← coequalizer.condition]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Limits.coequalizer.condition
-- 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
a : Iso.inv (α_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) R.X P.X) ≫ η ▷ R.X ▷ P.X
| η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ≫ Iso.inv (α_ R.X R.X P.X)

Tactic:
rw [associator_inv_naturality_left]

Premises used: CategoryTheory.MonoidalCategory.associator_inv_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 : Mon.{v₁, u₁} C
P : Bimod R S
h :
  Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
      ((Iso.inv (α_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) R.X P.X) ≫ η ▷ R.X ▷ P.X) ≫ μ ▷ P.X) ≫
        coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
⊢ Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
      η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ≫
        Iso.inv (α_ R.X R.X P.X) ≫ μ ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))

Tactic:
slice_lhs 2 3 => rw [associator_inv_naturality_left]

Premises used: CategoryTheory.MonoidalCategory.associator_inv_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 : Mon.{v₁, u₁} C
P : Bimod R S
a : Iso.hom (λ_ R.X) ▷ P.X
| η ▷ R.X ▷ P.X ≫ μ ▷ P.X

Tactic:
rw [← comp_whiskerRight, MonObj.one_mul]

Premises used: CategoryTheory.MonoidalCategory.comp_whiskerRight, CategoryTheory.MonObj.one_mul
-- 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
h :
  Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
      Iso.inv (α_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) R.X P.X) ≫
        Iso.hom (λ_ R.X) ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
⊢ Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
      ((Iso.inv (α_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) R.X P.X) ≫ η ▷ R.X ▷ P.X) ≫ μ ▷ P.X) ≫
        coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))

Tactic:
slice_lhs 3 4 => rw [← comp_whiskerRight, MonObj.one_mul]

Premises used: CategoryTheory.MonoidalCategory.comp_whiskerRight, CategoryTheory.MonObj.one_mul
-- 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
rw : coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
| coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
    𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))

Tactic:
rw [Category.comp_id]

Premises used: CategoryTheory.Category.comp_id
-- 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
h :
  Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ⟶ coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
    (Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
      Iso.inv (α_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) R.X P.X) ≫
        Iso.hom (λ_ R.X) ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
    (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
⊢ Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
      Iso.inv (α_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) R.X P.X) ≫
        Iso.hom (λ_ R.X) ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))

Tactic:
slice_rhs 1 2 => rw [Category.comp_id]

Premises used: CategoryTheory.Category.comp_id
-- 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
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ⟶ coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
    (Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
      Iso.inv (α_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) R.X P.X) ≫
        Iso.hom (λ_ R.X) ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
    (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))

Tactic:
monoidal
-- 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
dsimp :
  CategoryStruct.comp (Y := TensorBimod.X (regular R) P)
      (Iso.inv (λ_ P.X) ≫ η ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
      (coequalizer.desc P.actLeft ⋯) =
    𝟙 P.X
⊢ CategoryStruct.comp (Y := TensorBimod.X (regular R) P) (inv P) (hom P) = 𝟙 P.X

Tactic:
dsimp [hom, inv]

Premises used: Bimod.LeftUnitorBimod.inv, Bimod.LeftUnitorBimod.hom
-- 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
a : P.actLeft
| CategoryStruct.comp (Y := TensorBimod.X (regular R) P)
    (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) (coequalizer.desc P.actLeft ⋯)

Tactic:
rw [coequalizer.π_desc]

Premises used: 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
slice_lhs : Iso.inv (λ_ P.X) ≫ η ▷ P.X ≫ P.actLeft = 𝟙 P.X
⊢ CategoryStruct.comp (Y := TensorBimod.X (regular R) P)
      (Iso.inv (λ_ P.X) ≫ η ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
      (coequalizer.desc P.actLeft ⋯) =
    𝟙 P.X

Tactic:
slice_lhs 3 4 => rw [coequalizer.π_desc]

Premises used: 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
rw :
  Eq.{v₁ + 1} (α := P.X ⟶ P.X)
    (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) P.X)
      (Iso.inv (λ_ P.X)) (Iso.hom (λ_ P.X)))
    (𝟙 P.X)
rw₁ rw₂ : Eq.{v₁ + 1} (α := P.X ⟶ P.X) (𝟙 P.X) (𝟙 P.X)
⊢ Iso.inv (λ_ P.X) ≫ η ▷ P.X ≫ P.actLeft = 𝟙 P.X

Tactic:
rw [one_actLeft, Iso.inv_hom_id]

Premises used: CategoryTheory.Iso.inv_hom_id, Bimod.one_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 : 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 (regular R) P) ⟶ P.X)
    (TensorBimod.actLeft (regular R) P ≫ hom P) (R.X ◁ hom P ≫ P.actLeft)
⊢ ((regular R).tensorBimod P).actLeft ≫ hom P = R.X ◁ hom P ≫ P.actLeft

Tactic:
dsimp