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.hom (α_ (P.tensorBimod Q).X L.X U.X) ≫
    (P.tensorBimod Q).X ◁ L.actRight ≫
      coequalizer.π ((P.tensorBimod Q).actRight ▷ L.X)
        (Iso.hom (α_ (P.tensorBimod Q).X T.X L.X) ≫ (P.tensorBimod Q).X ◁ L.actLeft)
| coequalizer.π ((P.tensorBimod Q).actRight ▷ L.X)
        (Iso.hom (α_ (P.tensorBimod Q).X T.X L.X) ≫ (P.tensorBimod Q).X ◁ L.actLeft) ▷
      U.X ≫
    TensorBimod.actRight (P.tensorBimod Q) L

Tactic:
rw [TensorBimod.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
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.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) L.X U.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) L.X U.X
| coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X ▷ U.X ≫
    Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) L.X U.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
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 :
  tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X ◁ L.actRight ≫
    coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.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) L.X U.X ≫
    coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actRight

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
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.hom (α_ P.X Q.X (tensorObj.{v₁, u₁} (𝒞 := inst) L.X U.X)) ≫ P.X ◁ Q.X ◁ L.actRight
| tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X ◁ L.actRight ≫ Iso.hom (α_ P.X Q.X L.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
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.hom (α_ P.X (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X L.X) U.X) ≫
    P.X ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) ▷ U.X
| (P.X ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ▷ U.X ≫
    Iso.hom (α_ P.X (Q.tensorBimod L).X U.X)

Tactic:
erw [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
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 ◁ Iso.hom (α_ Q.X L.X U.X) ≫
    P.X ◁ Q.X ◁ L.actRight ≫ P.X ◁ coequalizer.π (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) ▷ U.X ≫
    P.X ◁ TensorBimod.actRight Q L

Tactic:
rw [← whiskerLeft_comp, TensorBimod.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
A B : Mon.{v₁, u₁} C
M : Bimod A B
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
dsimp :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) P.X (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ⟶
      TensorBimod.X (P.tensorBimod Q) L)
    (P.X ◁ Q.actRight ▷ L.X ≫
      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))
    (P.X ◁ (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) ≫
      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))
⊢ Eq (α :=
    Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft P.X)
        (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ⟶
      ((P.tensorBimod Q).tensorBimod L).X)
    (CategoryTheory.Functor.map (tensorLeft P.X) (Q.actRight ▷ L.X) ≫
      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 ((P.tensorBimod Q).actRight ▷ L.X)
            (Iso.hom (α_ (P.tensorBimod Q).X T.X L.X) ≫ (P.tensorBimod Q).X ◁ L.actLeft))
    (CategoryTheory.Functor.map (tensorLeft P.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) ≫
      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 ((P.tensorBimod Q).actRight ▷ L.X)
            (Iso.hom (α_ (P.tensorBimod Q).X T.X L.X) ≫ (P.tensorBimod Q).X ◁ L.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
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 (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ≫ (P.X ◁ Q.actRight) ▷ L.X
| P.X ◁ Q.actRight ▷ L.X ≫ Iso.inv (α_ P.X Q.X L.X)

Tactic:
rw [associator_inv_naturality_middle]

Premises used: CategoryTheory.MonoidalCategory.associator_inv_naturality_middle
-- 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
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 :
  (CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst)
        (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)
        L.X)
      (coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ T.X ▷ L.X :
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X)
            L.X ⟶
          tensorObj.{v₁, u₁} (𝒞 := inst)
            (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)
            L.X)
      (TensorBimod.actRight P Q ▷ L.X :
        tensorObj.{v₁, u₁} (𝒞 := inst)
            (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)
            L.X ⟶
          tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
            L.X) :
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) L.X ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) L.X)
| ((Iso.hom (α_ P.X Q.X T.X) ≫ P.X ◁ Q.actRight) ▷ L.X ≫
      coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X :
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) L.X ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) L.X)

Tactic:
rw [← comp_whiskerRight, Category.assoc, ← TensorBimod.π_tensor_id_actRight,
            comp_whiskerRight]

Premises used: Bimod.TensorBimod.π_tensor_id_actRight, CategoryTheory.Category.assoc, CategoryTheory.MonoidalCategory.comp_whiskerRight
-- 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
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.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
        coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) ≫
      coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
        (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
          coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) :
    tensorObj.{v₁, u₁} (𝒞 := inst)
        (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)
        L.X ⟶
      coequalizer (TensorBimod.actRight P Q ▷ L.X)
        (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
          coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.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)) L.X)
      (TensorBimod.actRight P Q ▷ L.X :
        tensorObj.{v₁, u₁} (𝒞 := inst)
            (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)
            L.X ⟶
          tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (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_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
            coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) :
        tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
            L.X ⟶
          coequalizer (TensorBimod.actRight P Q ▷ L.X)
            (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
              coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft)) :
    tensorObj.{v₁, u₁} (𝒞 := inst)
        (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)
        L.X ⟶
      coequalizer (TensorBimod.actRight P Q ▷ L.X)
        (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
          coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft))

Tactic:
rw [coequalizer.condition]

Premises used: CategoryTheory.Limits.coequalizer.condition
-- 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
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 ◁ Iso.hom (α_ Q.X T.X L.X) ≫ P.X ◁ Q.X ◁ L.actLeft) ≫ Iso.inv (α_ P.X Q.X L.X)
| P.X ◁ (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) ≫ Iso.inv (α_ P.X Q.X L.X)

Tactic:
rw [whiskerLeft_comp]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp
-- 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
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 (tensorObj.{v₁, u₁} (𝒞 := inst) T.X L.X)) ≫ tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X ◁ L.actLeft
| P.X ◁ Q.X ◁ L.actLeft ≫ Iso.inv (α_ P.X Q.X L.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
A B : Mon.{v₁, u₁} C
M : Bimod A B
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.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷
      tensorObj.{v₁, u₁} (𝒞 := inst) T.X L.X ≫
    coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft
| tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X ◁ L.actLeft ≫
    coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.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
A B : Mon.{v₁, u₁} C
M : Bimod A B
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
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) P.X (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ⟶
      coequalizer (TensorBimod.actRight P Q ▷ L.X)
        (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
          coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft))
    (Iso.inv (α_ P.X (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ≫
        Iso.inv (α_ P.X Q.X T.X) ▷ L.X ≫
          ((Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X L.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) T.X L.X) ≫
              coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) ≫
            coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
              (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
                coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) :
      tensorObj.{v₁, u₁} (𝒞 := inst) P.X (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ⟶
        coequalizer (TensorBimod.actRight P Q ▷ L.X)
          (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
            coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft))
    (P.X ◁ Iso.hom (α_ Q.X T.X L.X) ≫
        Iso.inv (α_ P.X Q.X (tensorObj.{v₁, u₁} (𝒞 := inst) T.X L.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) T.X L.X ≫
              coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) ≫
            coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
              (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
                coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) :
      tensorObj.{v₁, u₁} (𝒞 := inst) P.X (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ⟶
        coequalizer (TensorBimod.actRight P Q ▷ L.X)
          (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
            coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft))

Tactic:
simp
-- 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
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
dsimp :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) (TensorBimod.X Q L) ⟶
      TensorBimod.X (P.tensorBimod Q) L)
    (P.actRight ▷ TensorBimod.X Q L ≫
        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))
            ⋯) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) (TensorBimod.X Q L) ⟶
        TensorBimod.X (P.tensorBimod Q) L)
    ((Iso.hom (α_ P.X S.X (TensorBimod.X Q L)) ≫ P.X ◁ TensorBimod.actLeft Q L) ≫
        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))
            ⋯) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) (TensorBimod.X Q L) ⟶
        TensorBimod.X (P.tensorBimod Q) L)
⊢ P.actRight ▷ (Q.tensorBimod L).X ≫ invAux P Q L =
    (Iso.hom (α_ P.X S.X (Q.tensorBimod L).X) ≫ P.X ◁ (Q.tensorBimod L).actLeft) ≫ invAux P Q L

Tactic:
dsimp [invAux]

Premises used: Bimod.AssociatorBimod.invAux
-- 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
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
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) (TensorBimod.X Q L) ⟶
      TensorBimod.X (P.tensorBimod Q) L)
    (P.actRight ▷ TensorBimod.X Q L ≫
        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))
            ⋯) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) (TensorBimod.X Q L) ⟶
        TensorBimod.X (P.tensorBimod Q) L)
    ((Iso.hom (α_ P.X S.X (TensorBimod.X Q L)) ≫ P.X ◁ TensorBimod.actLeft Q L) ≫
        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))
            ⋯) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) (TensorBimod.X Q L) ⟶
        TensorBimod.X (P.tensorBimod Q) L)

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