awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
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)
⊢ 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)
have 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)
Tactic: rw [← comp_whiskerRight, Hom.right_act_hom, comp_whiskerRight]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
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)
⊢ 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)
have rw₂ : 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)
Tactic: rw [← comp_whiskerRight, Hom.right_act_hom, comp_whiskerRight]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
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
| M₁.X ◁ N.actLeft ≫ f.hom ▷ N.X
have a : f.hom ▷ tensorObj.{v₁, u₁} (𝒞 := inst) Y.X N.X ≫ M₂.X ◁ N.actLeft
Tactic: rw [whisker_exchange]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
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
⊢ (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
have 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
Tactic: slice_lhs 2 3 => rw [whisker_exchange]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
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
⊢ (M₁.tensorBimod 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) ⋯ ⋯) =
    X.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) ⋯ ⋯) ≫
      (M₂.tensorBimod N).actLeft
have refine : Eq (α := Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft X.X) (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ (M₂.tensorBimod N).X)
  (CategoryTheory.Functor.map (tensorLeft X.X)
      (coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft)) ≫
    (M₁.tensorBimod 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) sorry sorry))
  (CategoryStruct.comp (Y :=
    Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft X.X)
      (coequalizer (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft)))
    (CategoryTheory.Functor.map (tensorLeft X.X)
      (coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft)))
    (X.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) sorry sorry) ≫
      (M₂.tensorBimod N).actLeft))
Tactic: refine (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
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
| X.X ◁ coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft) ≫ TensorBimod.actLeft M₁ N
have a : Iso.inv (α_ X.X M₁.X N.X) ≫
  M₁.actLeft ▷ N.X ≫ coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft)
Tactic: rw [TensorBimod.whiskerLeft_π_actLeft]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
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
| CategoryStruct.comp (Y := TensorBimod.X M₁ N)
    (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) ⋯ ⋯))
have a : 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
Tactic: rw [ι_colimMap, parallelPairHom_app_one]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
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) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
    (CategoryStruct.comp (Y := TensorBimod.X M₁ N)
        (Iso.inv (α_ X.X M₁.X N.X) ≫
          M₁.actLeft ▷ N.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) ⋯ ⋯)) :
      tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
    (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) X.X
          (coequalizer (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft)))
        (X.X ◁ coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft))
        (X.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) ⋯ ⋯) ≫
          TensorBimod.actLeft M₂ N) :
      tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
have slice_lhs : Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
  (Iso.inv (α_ X.X M₁.X N.X) ≫
      M₁.actLeft ▷ N.X ≫
        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) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
  (CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) X.X
        (coequalizer (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft)))
      (X.X ◁ coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft))
      (X.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) sorry sorry) ≫
        TensorBimod.actLeft M₂ N) :
    tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
Tactic: slice_lhs 3 4 => rw [ι_colimMap, parallelPairHom_app_one]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
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
| M₁.actLeft ▷ N.X ≫ f.hom ▷ N.X
have a : (X.X ◁ f.hom ≫ M₂.actLeft) ▷ N.X
Tactic: rw [← comp_whiskerRight, Hom.left_act_hom]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
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) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
    (Iso.inv (α_ X.X M₁.X N.X) ≫
        M₁.actLeft ▷ N.X ≫
          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) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
    (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) X.X
          (coequalizer (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft)))
        (X.X ◁ coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft))
        (X.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) ⋯ ⋯) ≫
          TensorBimod.actLeft M₂ N) :
      tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
have slice_lhs : Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
  (Iso.inv (α_ X.X M₁.X N.X) ≫
      (X.X ◁ f.hom ≫ M₂.actLeft) ▷ 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) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
  (CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) X.X
        (coequalizer (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft)))
      (X.X ◁ coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft))
      (X.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) sorry sorry) ≫
        TensorBimod.actLeft M₂ N) :
    tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
Tactic: slice_lhs 2 3 => rw [← comp_whiskerRight, Hom.left_act_hom]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
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
| CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst) X.X (coequalizer (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft)))
    (X.X ◁ coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft))
    (X.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) ⋯ ⋯))
have a : X.X ◁ f.hom ▷ N.X ≫
  X.X ◁
    colimit.ι (parallelPair (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft)) WalkingParallelPair.one
Tactic: rw [← whiskerLeft_comp, ι_colimMap, parallelPairHom_app_one, whiskerLeft_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
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) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
    (Iso.inv (α_ X.X M₁.X N.X) ≫
        (X.X ◁ f.hom ≫ M₂.actLeft) ▷ 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) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
    (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) X.X
          (coequalizer (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft)))
        (X.X ◁ coequalizer.π (M₁.actRight ▷ N.X) (Iso.hom (α_ M₁.X Y.X N.X) ≫ M₁.X ◁ N.actLeft))
        (X.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) ⋯ ⋯) ≫
          TensorBimod.actLeft M₂ N) :
      tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
have slice_rhs : Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
  (Iso.inv (α_ X.X M₁.X N.X) ≫
      (X.X ◁ f.hom ≫ M₂.actLeft) ▷ 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) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
  ((X.X ◁ f.hom ▷ N.X ≫
        X.X ◁
          colimit.ι (parallelPair (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
            WalkingParallelPair.one) ≫
      TensorBimod.actLeft M₂ N :
    tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
Tactic: slice_rhs 1 2 => rw [← whiskerLeft_comp, ι_colimMap, parallelPairHom_app_one, whiskerLeft_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
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
| X.X ◁
      colimit.ι (parallelPair (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
        WalkingParallelPair.one ≫
    TensorBimod.actLeft M₂ N
have a : Iso.inv (α_ X.X M₂.X N.X) ≫
  M₂.actLeft ▷ N.X ≫ coequalizer.π (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft)
Tactic: rw [TensorBimod.whiskerLeft_π_actLeft]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
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) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
    (Iso.inv (α_ X.X M₁.X N.X) ≫
        (X.X ◁ f.hom ≫ M₂.actLeft) ▷ 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) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
    ((X.X ◁ f.hom ▷ N.X ≫
          X.X ◁
            colimit.ι (parallelPair (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
              WalkingParallelPair.one) ≫
        TensorBimod.actLeft M₂ N :
      tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
have slice_rhs : Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
  (Iso.inv (α_ X.X M₁.X N.X) ≫
    (X.X ◁ f.hom ≫ M₂.actLeft) ▷ N.X ≫
      colimit.ι (parallelPair (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
        WalkingParallelPair.one)
  (X.X ◁ f.hom ▷ N.X ≫
    Iso.inv (α_ X.X M₂.X N.X) ≫
      M₂.actLeft ▷ N.X ≫ coequalizer.π (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
Tactic: slice_rhs 2 3 => rw [TensorBimod.whiskerLeft_π_actLeft]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
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
| X.X ◁ f.hom ▷ N.X ≫ Iso.inv (α_ X.X M₂.X N.X)
have a : Iso.inv (α_ X.X M₁.X N.X) ≫ (X.X ◁ f.hom) ▷ N.X
Tactic: rw [associator_inv_naturality_middle]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
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) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
    (Iso.inv (α_ X.X M₁.X N.X) ≫
      (X.X ◁ f.hom ≫ M₂.actLeft) ▷ N.X ≫
        colimit.ι (parallelPair (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
          WalkingParallelPair.one)
    (X.X ◁ f.hom ▷ N.X ≫
      Iso.inv (α_ X.X M₂.X N.X) ≫
        M₂.actLeft ▷ N.X ≫ coequalizer.π (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
have slice_rhs : Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) X.X (tensorObj.{v₁, u₁} (𝒞 := inst) M₁.X N.X) ⟶ TensorBimod.X M₂ N)
  (Iso.inv (α_ X.X M₁.X N.X) ≫
    (X.X ◁ f.hom ≫ M₂.actLeft) ▷ N.X ≫
      colimit.ι (parallelPair (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
        WalkingParallelPair.one)
  (((Iso.inv (α_ X.X M₁.X N.X) ≫ (X.X ◁ f.hom) ▷ N.X) ≫ M₂.actLeft ▷ N.X) ≫
    coequalizer.π (M₂.actRight ▷ N.X) (Iso.hom (α_ M₂.X Y.X N.X) ≫ M₂.X ◁ N.actLeft))
Tactic: slice_rhs 1 2 => rw [associator_inv_naturality_middle]