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_1 : 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
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂
slice_rhs :
  Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) ⟶ TensorBimod.X M N₂)
    (Iso.inv (α_ X.X M.X N₁.X) ≫
      M.actLeft ▷ N₁.X ≫
        M.X ◁ f.hom ≫
          colimit.ι (parallelPair (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft))
            WalkingParallelPair.one)
    (Iso.inv (α_ X.X M.X N₁.X) ≫
      (M.actLeft ▷ N₁.X ≫ M.X ◁ f.hom) ≫
        coequalizer.π (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft))
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) ⟶ TensorBimod.X M N₂)
    (Iso.inv (α_ X.X M.X N₁.X) ≫
      M.actLeft ▷ N₁.X ≫
        M.X ◁ f.hom ≫
          colimit.ι (parallelPair (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft))
            WalkingParallelPair.one)
    (((Iso.inv (α_ X.X M.X N₁.X) ≫ tensorObj.{v₁, u₁} (𝒞 := inst) X.X M.X ◁ f.hom) ≫ M.actLeft ▷ N₂.X) ≫
      coequalizer.π (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft))

Tactic:
slice_rhs 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_1 : 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
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) ⟶ TensorBimod.X M N₂)
    (Iso.inv (α_ X.X M.X N₁.X) ≫
      M.actLeft ▷ N₁.X ≫
        M.X ◁ f.hom ≫
          colimit.ι (parallelPair (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft))
            WalkingParallelPair.one)
    (Iso.inv (α_ X.X M.X N₁.X) ≫
      (M.actLeft ▷ N₁.X ≫ M.X ◁ f.hom) ≫
        coequalizer.π (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.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_1 : 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
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂
⊢ (M.tensorBimod N₁).actRight ≫
      colimMap
        (parallelPairHom (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X)
          (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ f.hom) (M.X ◁ f.hom)
          ⋯ ⋯) =
    colimMap
          (parallelPairHom (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X)
            (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ f.hom)
            (M.X ◁ f.hom) ⋯ ⋯) ▷
        Z.X ≫
      (M.tensorBimod N₂).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
A B : Mon.{v₁, u₁} C
M_1 : 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
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂
refine :
  Eq (α :=
    Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight Z.X) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) ⟶ (M.tensorBimod N₂).X)
    (CategoryTheory.Functor.map (tensorRight Z.X)
        (coequalizer.π (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft)) ≫
      (M.tensorBimod N₁).actRight ≫
        colimMap
          (parallelPairHom (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X)
            (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ f.hom)
            (M.X ◁ f.hom) ⋯ ⋯))
    (CategoryStruct.comp (Y :=
      Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight Z.X)
        (coequalizer (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft)))
      (CategoryTheory.Functor.map (tensorRight Z.X)
        (coequalizer.π (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft)))
      (colimMap
            (parallelPairHom (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X)
              (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ f.hom)
              (M.X ◁ f.hom) ⋯ ⋯) ▷
          Z.X ≫
        (M.tensorBimod N₂).actRight))
⊢ (M.tensorBimod N₁).actRight ≫
      colimMap
        (parallelPairHom (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X)
          (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ f.hom) (M.X ◁ f.hom)
          ⋯ ⋯) =
    colimMap
          (parallelPairHom (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X)
            (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ f.hom)
            (M.X ◁ f.hom) ⋯ ⋯) ▷
        Z.X ≫
      (M.tensorBimod N₂).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_1 : 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
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂
a :
  Iso.hom (α_ M.X N₁.X Z.X) ≫
    M.X ◁ N₁.actRight ≫ coequalizer.π (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft)
| coequalizer.π (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) ▷ Z.X ≫ TensorBimod.actRight M N₁

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_1 : 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
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂
slice_lhs :
  Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)
    (Iso.hom (α_ M.X N₁.X Z.X) ≫
        M.X ◁ N₁.actRight ≫
          M.X ◁ f.hom ≫
            colimit.ι (parallelPair (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft))
              WalkingParallelPair.one :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)
    (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft))
          Z.X)
        (coequalizer.π (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) ▷ Z.X)
        (colimMap
              (parallelPairHom (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X)
                (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ f.hom)
                (M.X ◁ f.hom) ⋯ ⋯) ▷
            Z.X ≫
          TensorBimod.actRight M N₂) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)
    (CategoryStruct.comp (Y := TensorBimod.X M N₁)
        (Iso.hom (α_ M.X N₁.X Z.X) ≫
          M.X ◁ N₁.actRight ≫ coequalizer.π (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft))
        (colimMap
          (parallelPairHom (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X)
            (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ f.hom)
            (M.X ◁ f.hom) ⋯ ⋯)) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)
    (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft))
          Z.X)
        (coequalizer.π (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) ▷ Z.X)
        (colimMap
              (parallelPairHom (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X)
                (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ f.hom)
                (M.X ◁ f.hom) ⋯ ⋯) ▷
            Z.X ≫
          TensorBimod.actRight M N₂) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)

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_1 : 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
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂
a : M.X ◁ (f.hom ▷ Z.X ≫ N₂.actRight)
| M.X ◁ N₁.actRight ≫ M.X ◁ f.hom

Tactic:
rw [← whiskerLeft_comp, Hom.right_act_hom]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp, Bimod.Hom.right_act_hom
-- 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_1 : 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
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂
slice_lhs :
  Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)
    (Iso.hom (α_ M.X N₁.X Z.X) ≫
        M.X ◁ (f.hom ▷ Z.X ≫ N₂.actRight) ≫
          colimit.ι (parallelPair (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft))
            WalkingParallelPair.one :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)
    (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft))
          Z.X)
        (coequalizer.π (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) ▷ Z.X)
        (colimMap
              (parallelPairHom (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X)
                (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ f.hom)
                (M.X ◁ f.hom) ⋯ ⋯) ▷
            Z.X ≫
          TensorBimod.actRight M N₂) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)
    (Iso.hom (α_ M.X N₁.X Z.X) ≫
        M.X ◁ N₁.actRight ≫
          M.X ◁ f.hom ≫
            colimit.ι (parallelPair (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft))
              WalkingParallelPair.one :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)
    (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft))
          Z.X)
        (coequalizer.π (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) ▷ Z.X)
        (colimMap
              (parallelPairHom (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X)
                (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ f.hom)
                (M.X ◁ f.hom) ⋯ ⋯) ▷
            Z.X ≫
          TensorBimod.actRight M N₂) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)

Tactic:
slice_lhs 2 3 => rw [← whiskerLeft_comp, Hom.right_act_hom]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp, Bimod.Hom.right_act_hom
-- 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_1 : 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
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂
a :
  (M.X ◁ f.hom) ▷ Z.X ≫
    colimit.ι (parallelPair (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft))
        WalkingParallelPair.one ▷
      Z.X
| CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft)) Z.X)
    (coequalizer.π (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) ▷ Z.X)
    (colimMap
        (parallelPairHom (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X)
          (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ f.hom) (M.X ◁ f.hom)
          ⋯ ⋯) ▷
      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_1 : 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
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂
slice_rhs :
  Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)
    (Iso.hom (α_ M.X N₁.X Z.X) ≫
        M.X ◁ (f.hom ▷ Z.X ≫ N₂.actRight) ≫
          colimit.ι (parallelPair (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft))
            WalkingParallelPair.one :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)
    (((M.X ◁ f.hom) ▷ Z.X ≫
          colimit.ι (parallelPair (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft))
              WalkingParallelPair.one ▷
            Z.X) ≫
        TensorBimod.actRight M N₂ :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)
    (Iso.hom (α_ M.X N₁.X Z.X) ≫
        M.X ◁ (f.hom ▷ Z.X ≫ N₂.actRight) ≫
          colimit.ι (parallelPair (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft))
            WalkingParallelPair.one :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)
    (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft))
          Z.X)
        (coequalizer.π (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) ▷ Z.X)
        (colimMap
              (parallelPairHom (M.actRight ▷ N₁.X) (Iso.hom (α_ M.X Y.X N₁.X) ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X)
                (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ f.hom)
                (M.X ◁ f.hom) ⋯ ⋯) ▷
            Z.X ≫
          TensorBimod.actRight M N₂) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)

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_1 : 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
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂
a :
  Iso.hom (α_ M.X N₂.X Z.X) ≫
    M.X ◁ N₂.actRight ≫ coequalizer.π (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft)
| colimit.ι (parallelPair (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft)) WalkingParallelPair.one ▷
      Z.X ≫
    TensorBimod.actRight M N₂

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_1 : 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
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂
slice_rhs :
  Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)
    (Iso.hom (α_ M.X N₁.X Z.X) ≫
      M.X ◁ (f.hom ▷ Z.X ≫ N₂.actRight) ≫
        colimit.ι (parallelPair (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft))
          WalkingParallelPair.one)
    ((M.X ◁ f.hom) ▷ Z.X ≫
      Iso.hom (α_ M.X N₂.X Z.X) ≫
        M.X ◁ N₂.actRight ≫ coequalizer.π (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft))
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)
    (Iso.hom (α_ M.X N₁.X Z.X) ≫
        M.X ◁ (f.hom ▷ Z.X ≫ N₂.actRight) ≫
          colimit.ι (parallelPair (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft))
            WalkingParallelPair.one :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)
    (((M.X ◁ f.hom) ▷ Z.X ≫
          colimit.ι (parallelPair (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft))
              WalkingParallelPair.one ▷
            Z.X) ≫
        TensorBimod.actRight M N₂ :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)

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_1 : 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
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N₁.X) Z.X ⟶ TensorBimod.X M N₂)
    (Iso.hom (α_ M.X N₁.X Z.X) ≫
      M.X ◁ (f.hom ▷ Z.X ≫ N₂.actRight) ≫
        colimit.ι (parallelPair (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.actLeft))
          WalkingParallelPair.one)
    ((M.X ◁ f.hom) ▷ Z.X ≫
      Iso.hom (α_ M.X N₂.X Z.X) ≫
        M.X ◁ N₂.actRight ≫ coequalizer.π (M.actRight ▷ N₂.X) (Iso.hom (α_ M.X Y.X N₂.X) ≫ M.X ◁ N₂.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)
X Y Z : Mon.{v₁, u₁} C
M₁ M₂ : Bimod X Y
f : M₁ ⟶ M₂
N : Bimod Y Z
rw :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X Y.X) N.X ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) M₂.X N.X)
    ((M₁.actRight ≫ f.hom) ▷ N.X) (f.hom ▷ Y.X ▷ N.X ≫ M₂.actRight ▷ N.X)
rw₁ :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X Y.X) N.X ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) M₂.X N.X)
    ((f.hom ▷ Y.X ≫ M₂.actRight) ▷ N.X) (f.hom ▷ Y.X ▷ N.X ≫ M₂.actRight ▷ N.X)
rw₂ rw₃ :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X Y.X) N.X ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) M₂.X N.X)
    (f.hom ▷ Y.X ▷ N.X ≫ M₂.actRight ▷ N.X) (f.hom ▷ Y.X ▷ N.X ≫ M₂.actRight ▷ N.X)
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X Y.X) N.X ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) M₂.X N.X)
    (M₁.actRight ▷ N.X ≫ f.hom ▷ N.X) (f.hom ▷ Y.X ▷ N.X ≫ M₂.actRight ▷ N.X)

Tactic:
rw [← comp_whiskerRight, Hom.right_act_hom, comp_whiskerRight]

Premises used: CategoryTheory.MonoidalCategory.comp_whiskerRight, Bimod.Hom.right_act_hom
-- 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) Y.X N.X ≫ M₂.X ◁ N.actLeft
| M₁.X ◁ N.actLeft ≫ 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 :
  Iso.hom (α_ M₁.X Y.X N.X) ≫ f.hom ▷ tensorObj.{v₁, u₁} (𝒞 := inst) Y.X N.X ≫ M₂.X ◁ N.actLeft =
    f.hom ▷ Y.X ▷ N.X ≫ Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft
⊢ (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft) ≫ f.hom ▷ N.X =
    f.hom ▷ Y.X ▷ N.X ≫ Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft

Tactic:
slice_lhs 2 3 => rw [whisker_exchange]

Premises used: CategoryTheory.MonoidalCategory.whisker_exchange