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
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)
X Y Z : Mon.{v₁, u₁} C
M₁ M₂ : Bimod X Y
f : M₁ ⟶ M₂
N : Bimod Y Z
refine :
  Eq (α :=
    Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight Z.X) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ (M₂.tensorBimod N).X)
    (CategoryTheory.Functor.map (tensorRight Z.X)
        (coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft)) ≫
      (M₁.tensorBimod N).actRight ≫
        colimMap
          (parallelPairHom (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft) (M₂.actRight ▷ N.X)
            (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft) (f.hom ▷ Y.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯))
    (CategoryStruct.comp (Y :=
      Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight Z.X)
        (coequalizer (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft)))
      (CategoryTheory.Functor.map (tensorRight Z.X)
        (coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft)))
      (colimMap
            (parallelPairHom (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft) (M₂.actRight ▷ N.X)
              (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft) (f.hom ▷ Y.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷
          Z.X ≫
        (M₂.tensorBimod N).actRight))
⊢ (M₁.tensorBimod N).actRight ≫
      colimMap
        (parallelPairHom (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft) (M₂.actRight ▷ N.X)
          (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft) (f.hom ▷ Y.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) =
    colimMap
          (parallelPairHom (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft) (M₂.actRight ▷ N.X)
            (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft) (f.hom ▷ Y.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷
        Z.X ≫
      (M₂.tensorBimod N).actRight

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

Premises used: CategoryTheory.Functor.map, CategoryTheory.MonoidalCategory.tensorRight, Iff.mp, CategoryTheory.Limits.coequalizer.π, CategoryTheory.cancel_epi
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
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)
X Y Z : Mon.{v₁, u₁} C
M₁ M₂ : Bimod X Y
f : M₁ ⟶ M₂
N : Bimod Y Z
a :
  Iso.hom (α_ M₁.X N.X Z.X) ≫
    M₁.X ◁ N.actRight ≫ coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft)
| coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft) ▷ Z.X ≫ TensorBimod.actRight M₁ N

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
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)
X Y Z : Mon.{v₁, u₁} C
M₁ M₂ : Bimod X Y
f : M₁ ⟶ M₂
N : Bimod Y Z
slice_lhs :
  Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)
    (Iso.hom (α_ M₁.X N.X Z.X) ≫
        M₁.X ◁ N.actRight ≫
          f.hom ▷ N.X ≫
            colimit.ι (parallelPair (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
              WalkingParallelPair.one :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)
    (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft))
          Z.X)
        (coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft) ▷ Z.X)
        (colimMap
              (parallelPairHom (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft) (M₂.actRight ▷ N.X)
                (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft) (f.hom ▷ Y.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷
            Z.X ≫
          TensorBimod.actRight M₂ N) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)
    (CategoryStruct.comp (Y := TensorBimod.X M₁ N)
        (Iso.hom (α_ M₁.X N.X Z.X) ≫
          M₁.X ◁ N.actRight ≫ coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft))
        (colimMap
          (parallelPairHom (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft) (M₂.actRight ▷ N.X)
            (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft) (f.hom ▷ Y.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯)) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)
    (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft))
          Z.X)
        (coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft) ▷ Z.X)
        (colimMap
              (parallelPairHom (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft) (M₂.actRight ▷ N.X)
                (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft) (f.hom ▷ Y.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷
            Z.X ≫
          TensorBimod.actRight M₂ N) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)

Tactic:
slice_lhs 3 4 => rw [ι_colimMap, parallelPairHom_app_one]

Premises used: CategoryTheory.Limits.ι_colimMap, CategoryTheory.Limits.parallelPairHom_app_one
-- 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)
X Y Z : Mon.{v₁, u₁} C
M₁ M₂ : Bimod X Y
f : M₁ ⟶ M₂
N : Bimod Y Z
a : f.hom ▷ tensorObj.{v₁, u₁} (𝒞 := inst) N.X Z.X ≫ M₂.X ◁ N.actRight
| M₁.X ◁ N.actRight ≫ f.hom ▷ N.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)
X Y Z : Mon.{v₁, u₁} C
M₁ M₂ : Bimod X Y
f : M₁ ⟶ M₂
N : Bimod Y Z
slice_lhs :
  Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)
    (Iso.hom (α_ M₁.X N.X Z.X) ≫
        (f.hom ▷ tensorObj.{v₁, u₁} (𝒞 := inst) N.X Z.X ≫ M₂.X ◁ N.actRight) ≫
          colimit.ι (parallelPair (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
            WalkingParallelPair.one :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)
    (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft))
          Z.X)
        (coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft) ▷ Z.X)
        (colimMap
              (parallelPairHom (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft) (M₂.actRight ▷ N.X)
                (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft) (f.hom ▷ Y.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷
            Z.X ≫
          TensorBimod.actRight M₂ N) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)
    (Iso.hom (α_ M₁.X N.X Z.X) ≫
        M₁.X ◁ N.actRight ≫
          f.hom ▷ N.X ≫
            colimit.ι (parallelPair (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
              WalkingParallelPair.one :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)
    (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft))
          Z.X)
        (coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft) ▷ Z.X)
        (colimMap
              (parallelPairHom (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft) (M₂.actRight ▷ N.X)
                (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft) (f.hom ▷ Y.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷
            Z.X ≫
          TensorBimod.actRight M₂ N) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)

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
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)
X Y Z : Mon.{v₁, u₁} C
M₁ M₂ : Bimod X Y
f : M₁ ⟶ M₂
N : Bimod Y Z
a :
  f.hom ▷ N.X ▷ Z.X ≫
    colimit.ι (parallelPair (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
        WalkingParallelPair.one ▷
      Z.X
| CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft)) Z.X)
    (coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft) ▷ Z.X)
    (colimMap
        (parallelPairHom (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft) (M₂.actRight ▷ N.X)
          (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft) (f.hom ▷ Y.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷
      Z.X)

Tactic:
rw [← comp_whiskerRight, ι_colimMap, parallelPairHom_app_one,
      comp_whiskerRight]

Premises used: CategoryTheory.Limits.ι_colimMap, CategoryTheory.MonoidalCategory.comp_whiskerRight, CategoryTheory.Limits.parallelPairHom_app_one
-- 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)
X Y Z : Mon.{v₁, u₁} C
M₁ M₂ : Bimod X Y
f : M₁ ⟶ M₂
N : Bimod Y Z
slice_rhs :
  Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)
    (Iso.hom (α_ M₁.X N.X Z.X) ≫
        (f.hom ▷ tensorObj.{v₁, u₁} (𝒞 := inst) N.X Z.X ≫ M₂.X ◁ N.actRight) ≫
          colimit.ι (parallelPair (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
            WalkingParallelPair.one :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)
    ((f.hom ▷ N.X ▷ Z.X ≫
          colimit.ι (parallelPair (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
              WalkingParallelPair.one ▷
            Z.X) ≫
        TensorBimod.actRight M₂ N :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)
    (Iso.hom (α_ M₁.X N.X Z.X) ≫
        (f.hom ▷ tensorObj.{v₁, u₁} (𝒞 := inst) N.X Z.X ≫ M₂.X ◁ N.actRight) ≫
          colimit.ι (parallelPair (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
            WalkingParallelPair.one :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)
    (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft))
          Z.X)
        (coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft) ▷ Z.X)
        (colimMap
              (parallelPairHom (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft) (M₂.actRight ▷ N.X)
                (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft) (f.hom ▷ Y.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷
            Z.X ≫
          TensorBimod.actRight M₂ N) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)

Tactic:
slice_rhs 1 2 => rw [← comp_whiskerRight, ι_colimMap, parallelPairHom_app_one,
      comp_whiskerRight]

Premises used: CategoryTheory.Limits.ι_colimMap, CategoryTheory.MonoidalCategory.comp_whiskerRight, CategoryTheory.Limits.parallelPairHom_app_one
-- 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)
X Y Z : Mon.{v₁, u₁} C
M₁ M₂ : Bimod X Y
f : M₁ ⟶ M₂
N : Bimod Y Z
a :
  Iso.hom (α_ M₂.X N.X Z.X) ≫
    M₂.X ◁ N.actRight ≫ coequalizer.π (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft)
| colimit.ι (parallelPair (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft)) WalkingParallelPair.one ▷
      Z.X ≫
    TensorBimod.actRight M₂ N

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
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)
X Y Z : Mon.{v₁, u₁} C
M₁ M₂ : Bimod X Y
f : M₁ ⟶ M₂
N : Bimod Y Z
slice_rhs :
  Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)
    (Iso.hom (α_ M₁.X N.X Z.X) ≫
      (f.hom ▷ tensorObj.{v₁, u₁} (𝒞 := inst) N.X Z.X ≫ M₂.X ◁ N.actRight) ≫
        colimit.ι (parallelPair (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
          WalkingParallelPair.one)
    (f.hom ▷ N.X ▷ Z.X ≫
      Iso.hom (α_ M₂.X N.X Z.X) ≫
        M₂.X ◁ N.actRight ≫ coequalizer.π (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)
    (Iso.hom (α_ M₁.X N.X Z.X) ≫
        (f.hom ▷ tensorObj.{v₁, u₁} (𝒞 := inst) N.X Z.X ≫ M₂.X ◁ N.actRight) ≫
          colimit.ι (parallelPair (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
            WalkingParallelPair.one :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)
    ((f.hom ▷ N.X ▷ Z.X ≫
          colimit.ι (parallelPair (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
              WalkingParallelPair.one ▷
            Z.X) ≫
        TensorBimod.actRight M₂ N :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)

Tactic:
slice_rhs 2 3 => 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
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)
X Y Z : Mon.{v₁, u₁} C
M₁ M₂ : Bimod X Y
f : M₁ ⟶ M₂
N : Bimod Y Z
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) Z.X ⟶ TensorBimod.X M₂ N)
    (Iso.hom (α_ M₁.X N.X Z.X) ≫
      (f.hom ▷ tensorObj.{v₁, u₁} (𝒞 := inst) N.X Z.X ≫ M₂.X ◁ N.actRight) ≫
        colimit.ι (parallelPair (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
          WalkingParallelPair.one)
    (f.hom ▷ N.X ▷ Z.X ≫
      Iso.hom (α_ M₂.X N.X Z.X) ≫
        M₂.X ◁ N.actRight ≫ coequalizer.π (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.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
a :
  Iso.hom (α_ P.X Q.X T.X) ▷ L.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
| (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)

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

Premises used: Bimod.TensorBimod.π_tensor_id_actRight, 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_ 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.hom (α_ P.X Q.X L.X)

Tactic:
rw [associator_naturality_middle]

Premises used: CategoryTheory.MonoidalCategory.associator_naturality_middle
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
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) ≫
    P.X ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)
| P.X ◁ Q.actRight ▷ 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 [← whiskerLeft_comp, coequalizer.condition,
          whiskerLeft_comp, whiskerLeft_comp]

Premises used: CategoryTheory.Limits.coequalizer.condition, 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.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.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ T.X ▷ 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)

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
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 :
  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
| 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

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
dsimp :
  Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X (P.tensorBimod Q) L) U.X ⟶ TensorBimod.X P (Q.tensorBimod L))
    (TensorBimod.actRight (P.tensorBimod Q) L ≫ hom P Q L) (hom P Q L ▷ U.X ≫ TensorBimod.actRight P (Q.tensorBimod L))
⊢ ((P.tensorBimod Q).tensorBimod L).actRight ≫ hom P Q L = hom P Q L ▷ U.X ≫ (P.tensorBimod (Q.tensorBimod L)).actRight

Tactic:
dsimp