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)
X Y : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
a : Iso.inv (α_ X.X X.X N.X) ≫ Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft
| X.X ◁ N.actLeft

Tactic:
rw [← Iso.inv_hom_id_assoc (α_ X.X X.X N.X) (X.X ◁ N.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
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 : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
h :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) X.X M.X ⟶
      TensorBimod.X
        { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod X X }
        N)
    (X.X ◁ f.hom ≫
      colimit.ι (parallelPair (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft)) WalkingParallelPair.one)
    (X.X ◁ f.hom ≫
      Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) X.X N.X)) ≫
        η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) X.X N.X ≫
          (Iso.inv (α_ X.X X.X N.X) ≫ Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft) ≫
            coequalizer.π (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft))
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) X.X M.X ⟶
      TensorBimod.X
        { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod X X }
        N)
    (X.X ◁ f.hom ≫
      colimit.ι (parallelPair (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft)) WalkingParallelPair.one)
    (X.X ◁ f.hom ≫
      Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) X.X N.X)) ≫
        (η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) X.X N.X ≫ X.X ◁ N.actLeft) ≫
          coequalizer.π (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft))

Tactic:
slice_rhs 4 4 => rw [← Iso.inv_hom_id_assoc (α_ X.X X.X N.X) (X.X ◁ N.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
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 : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
a : μ ▷ N.X ≫ coequalizer.π (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft)
| Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft ≫ coequalizer.π (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.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
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 : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
h :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) X.X M.X ⟶
      TensorBimod.X
        { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod X X }
        N)
    (X.X ◁ f.hom ≫
      colimit.ι (parallelPair (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft)) WalkingParallelPair.one)
    (X.X ◁ f.hom ≫
      Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) X.X N.X)) ≫
        η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) X.X N.X ≫
          Iso.inv (α_ X.X X.X N.X) ≫ μ ▷ N.X ≫ coequalizer.π (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft))
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) X.X M.X ⟶
      TensorBimod.X
        { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod X X }
        N)
    (X.X ◁ f.hom ≫
      colimit.ι (parallelPair (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft)) WalkingParallelPair.one)
    (X.X ◁ f.hom ≫
      Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) X.X N.X)) ≫
        η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) X.X N.X ≫
          (Iso.inv (α_ X.X X.X N.X) ≫ Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft) ≫
            coequalizer.π (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft))

Tactic:
slice_rhs 5 7 => 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
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 : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
a : Iso.inv (α_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) X.X N.X) ≫ η ▷ X.X ▷ N.X
| η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) X.X N.X ≫ Iso.inv (α_ X.X X.X N.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
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 : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
h :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) X.X M.X ⟶
      TensorBimod.X
        { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod X X }
        N)
    (X.X ◁ f.hom ≫
      colimit.ι (parallelPair (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft)) WalkingParallelPair.one)
    (X.X ◁ f.hom ≫
      Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) X.X N.X)) ≫
        ((Iso.inv (α_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) X.X N.X) ≫ η ▷ X.X ▷ N.X) ≫ μ ▷ N.X) ≫
          coequalizer.π (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft))
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) X.X M.X ⟶
      TensorBimod.X
        { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod X X }
        N)
    (X.X ◁ f.hom ≫
      colimit.ι (parallelPair (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft)) WalkingParallelPair.one)
    (X.X ◁ f.hom ≫
      Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) X.X N.X)) ≫
        η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) X.X N.X ≫
          Iso.inv (α_ X.X X.X N.X) ≫ μ ▷ N.X ≫ coequalizer.π (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft))

Tactic:
slice_rhs 3 4 => 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
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 : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
a : Iso.hom (λ_ X.X) ▷ N.X
| η ▷ X.X ▷ N.X ≫ μ ▷ N.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
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 : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
h :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) X.X M.X ⟶
      TensorBimod.X
        { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod X X }
        N)
    (X.X ◁ f.hom ≫
      colimit.ι (parallelPair (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft)) WalkingParallelPair.one)
    (X.X ◁ f.hom ≫
      Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) X.X N.X)) ≫
        Iso.inv (α_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) X.X N.X) ≫
          Iso.hom (λ_ X.X) ▷ N.X ≫ coequalizer.π (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft))
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) X.X M.X ⟶
      TensorBimod.X
        { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod X X }
        N)
    (X.X ◁ f.hom ≫
      colimit.ι (parallelPair (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft)) WalkingParallelPair.one)
    (X.X ◁ f.hom ≫
      Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) X.X N.X)) ≫
        ((Iso.inv (α_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) X.X N.X) ≫ η ▷ X.X ▷ N.X) ≫ μ ▷ N.X) ≫
          coequalizer.π (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft))

Tactic:
slice_rhs 4 5 => 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
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 : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) X.X M.X ⟶
      TensorBimod.X
        { X := X.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod X X }
        N)
    (X.X ◁ f.hom ≫
      colimit.ι (parallelPair (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.X ◁ N.actLeft)) WalkingParallelPair.one)
    (X.X ◁ f.hom ≫
      Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) X.X N.X)) ≫
        Iso.inv (α_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) X.X N.X) ≫
          Iso.hom (λ_ X.X) ▷ N.X ≫ coequalizer.π (μ ▷ N.X) (Iso.hom (α_ X.X X.X N.X) ≫ X.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
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)
W X Y Z : Mon.{v₁, u₁} C
M : Bimod W X
N : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
dsimp :
  { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯,
          actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.whiskerLeft
      f =
    Iso.hom
        (isoOfIso
          { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ :
            TensorBimod.X
                { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯,
                  actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ :
                  Bimod W Y }
                P ≅
              TensorBimod.X M
                { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯,
                  actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ :
                  Bimod X Z } }
          ⋯ ⋯) ≫
      M.whiskerLeft (N.whiskerLeft f) ≫
        Iso.inv
          (isoOfIso
            { hom := AssociatorBimod.hom M N P', inv := AssociatorBimod.inv M N P', hom_inv_id := ⋯, inv_hom_id := ⋯ :
              TensorBimod.X
                  { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯,
                    actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ :
                    Bimod W Y }
                  P' ≅
                TensorBimod.X M
                  { X := TensorBimod.X N P', actLeft := TensorBimod.actLeft N P', one_actLeft := ⋯, left_assoc := ⋯,
                    actRight := TensorBimod.actRight N P', actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ :
                    Bimod X Z } }
            ⋯ ⋯)
⊢ (M.tensorBimod N).whiskerLeft f =
    Iso.hom (M.associatorBimod N P) ≫ M.whiskerLeft (N.whiskerLeft f) ≫ Iso.inv (M.associatorBimod N P')

Tactic:
dsimp [tensorHom, tensorBimod, associatorBimod]

Premises used: Bimod.tensorBimod, Bimod.associatorBimod, CategoryTheory.MonoidalCategoryStruct.tensorHom
-- 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)
W X Y Z : Mon.{v₁, u₁} C
M : Bimod W X
N : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
rw :
  TensorBimod.X M N ◁ f.hom ≫
    colimit.ι
      (parallelPair (TensorBimod.actRight M N ▷ P'.X)
        (Iso.hom (α_ (TensorBimod.X M N) Y.X P'.X) ≫ TensorBimod.X M N ◁ P'.actLeft))
      WalkingParallelPair.one
| CategoryStruct.comp (Y :=
    coequalizer (TensorBimod.actRight M N ▷ P.X)
      (Iso.hom (α_ (TensorBimod.X M N) Y.X P.X) ≫ TensorBimod.X M N ◁ P.actLeft))
    (coequalizer.π (TensorBimod.actRight M N ▷ P.X)
      (Iso.hom (α_ (TensorBimod.X M N) Y.X P.X) ≫ TensorBimod.X M N ◁ P.actLeft))
    (colimMap
      (parallelPairHom (TensorBimod.actRight M N ▷ P.X)
        (Iso.hom (α_ (TensorBimod.X M N) Y.X P.X) ≫ TensorBimod.X M N ◁ P.actLeft) (TensorBimod.actRight M N ▷ P'.X)
        (Iso.hom (α_ (TensorBimod.X M N) Y.X P'.X) ≫ TensorBimod.X M N ◁ P'.actLeft)
        (tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M N) Y.X ◁ f.hom) (TensorBimod.X M N ◁ f.hom) ⋯ ⋯))

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

Tactic:
rw [← whiskerLeft_comp, ι_colimMap, parallelPairHom_app_one]

Premises used: CategoryTheory.Limits.ι_colimMap, CategoryTheory.MonoidalCategory.whiskerLeft_comp, 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
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)
W X Y Z : Mon.{v₁, u₁} C
M : Bimod W X
N : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
a :
  M.X ◁ N.X ◁ f.hom ≫
    M.X ◁
      colimit.ι (parallelPair (N.actRight ▷ P'.X) (Iso.hom (α_ N.X Y.X P'.X) ≫ N.X ◁ P'.actLeft))
        WalkingParallelPair.one
| M.X ◁
    (N.X ◁ f.hom ≫
      colimit.ι (parallelPair (N.actRight ▷ P'.X) (Iso.hom (α_ N.X Y.X P'.X) ≫ N.X ◁ P'.actLeft))
        WalkingParallelPair.one)

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
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)
W X Y Z : Mon.{v₁, u₁} C
M : Bimod W X
N : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
a :
  Iso.inv (α_ M.X N.X P'.X) ≫
    coequalizer.π (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ▷ P'.X ≫
      coequalizer.π (TensorBimod.actRight M N ▷ P'.X)
        (Iso.hom (α_ (TensorBimod.X M N) Y.X P'.X) ≫ TensorBimod.X M N ◁ P'.actLeft)
| (CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) M.X
        (coequalizer (N.actRight ▷ P'.X) (Iso.hom (α_ N.X Y.X P'.X) ≫ N.X ◁ P'.actLeft)))
      (M.X ◁
        colimit.ι (parallelPair (N.actRight ▷ P'.X) (Iso.hom (α_ N.X Y.X P'.X) ≫ N.X ◁ P'.actLeft))
          WalkingParallelPair.one)
      (CategoryStruct.comp (Y :=
        coequalizer (X :=
          tensorObj.{v₁, u₁} (𝒞 := inst) M.X
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X) P'.X))
          (M.X ◁ N.actRight ▷ P'.X) (M.X ◁ (Iso.hom (α_ N.X Y.X P'.X) ≫ N.X ◁ P'.actLeft)))
        (Iso.inv
          (PreservesCoequalizer.iso (tensorLeft M.X) (N.actRight ▷ P'.X)
            (Iso.hom (α_ N.X Y.X P'.X) ≫ N.X ◁ P'.actLeft)))
        (coequalizer.desc
          (Iso.inv (α_ M.X N.X P'.X) ≫
            coequalizer.π (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ▷ P'.X ≫
              coequalizer.π (TensorBimod.actRight M N ▷ P'.X)
                (Iso.hom (α_ (TensorBimod.X M N) Y.X P'.X) ≫ TensorBimod.X M N ◁ P'.actLeft))
          ⋯)) :
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X (tensorObj.{v₁, u₁} (𝒞 := inst) N.X P'.X) ⟶
      colimit
        (parallelPair (TensorBimod.actRight M N ▷ P'.X)
          (Iso.hom (α_ (coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft)) Y.X P'.X) ≫
            coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ◁ P'.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)
W X Y Z : Mon.{v₁, u₁} C
M : Bimod W X
N : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
a : Iso.inv (α_ M.X N.X P.X) ≫ tensorObj.{v₁, u₁} (𝒞 := inst) M.X N.X ◁ f.hom
| M.X ◁ N.X ◁ f.hom ≫ Iso.inv (α_ M.X N.X P'.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
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)
W X Y Z : Mon.{v₁, u₁} C
M : Bimod W X
N : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
h :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N.X) P.X ⟶
      colimit
        (parallelPair (TensorBimod.actRight M N ▷ P'.X)
          (Iso.hom (α_ (coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft)) Y.X P'.X) ≫
            coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ◁ P'.actLeft)))
    (coequalizer.π (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ▷ P.X ≫
      coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ◁ f.hom ≫
        colimit.ι
          (parallelPair (TensorBimod.actRight M N ▷ P'.X)
            (Iso.hom (α_ (coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft)) Y.X P'.X) ≫
              coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ◁ P'.actLeft))
          WalkingParallelPair.one)
    (Iso.hom (α_ M.X N.X P.X) ≫
      ((Iso.inv (α_ M.X N.X P.X) ≫ tensorObj.{v₁, u₁} (𝒞 := inst) M.X N.X ◁ f.hom) ≫
          coequalizer.π (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ▷ P'.X) ≫
        coequalizer.π (TensorBimod.actRight M N ▷ P'.X)
          (Iso.hom (α_ (TensorBimod.X M N) Y.X P'.X) ≫ TensorBimod.X M N ◁ P'.actLeft))
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N.X) P.X ⟶
      colimit
        (parallelPair (TensorBimod.actRight M N ▷ P'.X)
          (Iso.hom (α_ (coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft)) Y.X P'.X) ≫
            coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ◁ P'.actLeft)))
    (coequalizer.π (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ▷ P.X ≫
      coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ◁ f.hom ≫
        colimit.ι
          (parallelPair (TensorBimod.actRight M N ▷ P'.X)
            (Iso.hom (α_ (coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft)) Y.X P'.X) ≫
              coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ◁ P'.actLeft))
          WalkingParallelPair.one)
    (Iso.hom (α_ M.X N.X P.X) ≫
      M.X ◁ N.X ◁ f.hom ≫
        Iso.inv (α_ M.X N.X P'.X) ≫
          coequalizer.π (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ▷ P'.X ≫
            coequalizer.π (TensorBimod.actRight M N ▷ P'.X)
              (Iso.hom (α_ (TensorBimod.X M N) Y.X P'.X) ≫ TensorBimod.X M N ◁ P'.actLeft))

Tactic:
slice_rhs 2 3 => 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
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)
W X Y Z : Mon.{v₁, u₁} C
M : Bimod W X
N : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
a : tensorObj.{v₁, u₁} (𝒞 := inst) M.X N.X ◁ f.hom
| Iso.hom (α_ M.X N.X P.X) ≫ Iso.inv (α_ M.X N.X P.X) ≫ tensorObj.{v₁, u₁} (𝒞 := inst) M.X N.X ◁ f.hom

Tactic:
rw [Iso.hom_inv_id_assoc]

Premises used: CategoryTheory.Iso.hom_inv_id_assoc