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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
⊢ Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)) ≫
      tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X ◁ η ≫
        (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ≫ coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) =
    coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ≫
      𝟙 (coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ))
have h : Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)) ≫
    ((Iso.hom (α_ P.X S.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ P.X ◁ S.X ◁ η) ≫ P.X ◁ μ) ≫
      coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) =
  coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ≫
    𝟙 (coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ))
Tactic: slice_lhs 2 3 => rw [associator_naturality_right]
-- 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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
| P.X ◁ S.X ◁ η ≫ P.X ◁ μ
have a : P.X ◁ Iso.hom (ρ_ S.X)
Tactic: rw [← whiskerLeft_comp, MonObj.mul_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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
⊢ Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)) ≫
      ((Iso.hom (α_ P.X S.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ P.X ◁ S.X ◁ η) ≫ P.X ◁ μ) ≫
        coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) =
    coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ≫
      𝟙 (coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ))
have h : Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)) ≫
    Iso.hom (α_ P.X S.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫
      P.X ◁ Iso.hom (ρ_ S.X) ≫ coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) =
  coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ≫
    𝟙 (coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ))
Tactic: slice_lhs 3 4 => rw [← whiskerLeft_comp, MonObj.mul_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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
| coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ≫
    𝟙 (coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ))
have rw : coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)
Tactic: rw [Category.comp_id]
-- 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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
⊢ Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)) ≫
      Iso.hom (α_ P.X S.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫
        P.X ◁ Iso.hom (ρ_ S.X) ≫ coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) =
    coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ≫
      𝟙 (coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ))
have h : Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X ⟶ coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ))
  (Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)) ≫
    Iso.hom (α_ P.X S.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫
      P.X ◁ Iso.hom (ρ_ S.X) ≫ coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ))
  (coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ))
Tactic: slice_rhs 1 2 => rw [Category.comp_id]
-- 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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
⊢ CategoryStruct.comp (Y := TensorBimod.X P (regular S)) (inv P) (hom P) = 𝟙 P.X
have dsimp : CategoryStruct.comp (Y := TensorBimod.X P (regular S))
    (Iso.inv (ρ_ P.X) ≫ P.X ◁ η ≫ coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ))
    (coequalizer.desc P.actRight sorry) =
  𝟙 P.X
Tactic: dsimp [hom, inv]
-- 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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
| CategoryStruct.comp (Y := TensorBimod.X P (regular S))
    (coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)) (coequalizer.desc P.actRight ⋯)
have a : P.actRight
Tactic: rw [coequalizer.π_desc]
-- 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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
⊢ CategoryStruct.comp (Y := TensorBimod.X P (regular S))
      (Iso.inv (ρ_ P.X) ≫ P.X ◁ η ≫ coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ))
      (coequalizer.desc P.actRight ⋯) =
    𝟙 P.X
have slice_lhs : Iso.inv (ρ_ P.X) ≫ P.X ◁ η ≫ P.actRight = 𝟙 P.X
Tactic: slice_lhs 3 4 => rw [coequalizer.π_desc]
-- 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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
⊢ Iso.inv (ρ_ P.X) ≫ P.X ◁ η ≫ P.actRight = 𝟙 P.X
have rw : Eq.{v₁ + 1} (α := P.X ⟶ P.X)
  (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) P.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (Iso.inv (ρ_ P.X))
    (Iso.hom (ρ_ P.X)))
  (𝟙 P.X)
Tactic: rw [actRight_one, Iso.inv_hom_id]
-- 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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
rw :
  Eq.{v₁ + 1} (α := P.X ⟶ P.X)
    (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) P.X (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
      (Iso.inv (ρ_ P.X)) (Iso.hom (ρ_ P.X)))
    (𝟙 P.X)
⊢ Iso.inv (ρ_ P.X) ≫ P.X ◁ η ≫ P.actRight = 𝟙 P.X
have rw₁ : Eq.{v₁ + 1} (α := P.X ⟶ P.X) (𝟙 P.X) (𝟙 P.X)
Tactic: rw [actRight_one, Iso.inv_hom_id]
-- 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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
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)
⊢ (P.tensorBimod (regular S)).actLeft ≫ hom P = R.X ◁ hom P ≫ P.actLeft
have dsimp : Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) R.X (TensorBimod.X P (regular S)) ⟶ P.X)
  (TensorBimod.actLeft P (regular S) ≫ hom P) (R.X ◁ hom P ≫ P.actLeft)
Tactic: dsimp
-- 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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
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)
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) R.X (TensorBimod.X P (regular S)) ⟶ P.X)
    (TensorBimod.actLeft P (regular S) ≫ hom P) (R.X ◁ hom P ≫ P.actLeft)
have dsimp : Eq (α :=
  tensorObj.{v₁, u₁} (𝒞 := inst) R.X
      (TensorBimod.X P
        { X := S.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry, right_assoc := sorry,
          middle_assoc := sorry : Bimod S S }) ⟶
    P.X)
  (CategoryStruct.comp (Y :=
    TensorBimod.X P
      { X := S.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry, right_assoc := sorry,
        middle_assoc := sorry : Bimod S S })
    (CategoryStruct.comp (Y :=
      coequalizer (X :=
        tensorObj.{v₁, u₁} (𝒞 := inst) R.X
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) S.X))
        (R.X ◁ P.actRight ▷ S.X) (R.X ◁ (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)))
      (Iso.inv (PreservesCoequalizer.iso (tensorLeft R.X) (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)))
      (colimMap
        (parallelPairHom (R.X ◁ P.actRight ▷ S.X) (R.X ◁ (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)) (P.actRight ▷ S.X)
          (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)
          (Iso.inv (α_ R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) S.X) ≫
            Iso.inv (α_ R.X P.X S.X) ▷ S.X ≫ P.actLeft ▷ S.X ▷ S.X)
          (Iso.inv (α_ R.X P.X S.X) ≫ P.actLeft ▷ S.X) sorry sorry)))
    (coequalizer.desc P.actRight sorry))
  (R.X ◁ coequalizer.desc P.actRight sorry ≫ P.actLeft)
Tactic: dsimp [hom, TensorBimod.actLeft, regular]
-- 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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
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)
| CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)))
    (R.X ◁ coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ))
    (CategoryStruct.comp (Y :=
      coequalizer (X :=
        tensorObj.{v₁, u₁} (𝒞 := inst) R.X
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) S.X))
        (R.X ◁ P.actRight ▷ S.X) (R.X ◁ (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)))
      (Iso.inv (PreservesCoequalizer.iso (tensorLeft R.X) (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)))
      (CategoryStruct.comp (Y :=
        TensorBimod.X P
          { X := S.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod S S })
        (colimMap
          (parallelPairHom (R.X ◁ P.actRight ▷ S.X) (R.X ◁ (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)) (P.actRight ▷ S.X)
            (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)
            (Iso.inv (α_ R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) S.X) ≫
              Iso.inv (α_ R.X P.X S.X) ▷ S.X ≫ P.actLeft ▷ S.X ▷ S.X)
            (Iso.inv (α_ R.X P.X S.X) ≫ P.actLeft ▷ S.X) ⋯ ⋯))
        (coequalizer.desc P.actRight ⋯)))
have rw : (Iso.inv (α_ R.X P.X S.X) ≫ P.actLeft ▷ S.X) ≫ P.actRight
Tactic: rw [id_tensor_π_preserves_coequalizer_inv_colimMap_desc]
-- 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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
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)
⊢ CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)))
      (R.X ◁ coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ))
      (CategoryStruct.comp (Y :=
        TensorBimod.X P
          { X := S.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod S S })
        (CategoryStruct.comp (Y :=
          coequalizer (X :=
            tensorObj.{v₁, u₁} (𝒞 := inst) R.X
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) S.X))
            (R.X ◁ P.actRight ▷ S.X) (R.X ◁ (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)))
          (Iso.inv (PreservesCoequalizer.iso (tensorLeft R.X) (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)))
          (colimMap
            (parallelPairHom (R.X ◁ P.actRight ▷ S.X) (R.X ◁ (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)) (P.actRight ▷ S.X)
              (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)
              (Iso.inv (α_ R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) S.X) ≫
                Iso.inv (α_ R.X P.X S.X) ▷ S.X ≫ P.actLeft ▷ S.X ▷ S.X)
              (Iso.inv (α_ R.X P.X S.X) ≫ P.actLeft ▷ S.X) ⋯ ⋯)))
        (coequalizer.desc P.actRight ⋯)) =
    CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)))
      (R.X ◁ coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ))
      (R.X ◁ coequalizer.desc P.actRight ⋯ ≫ P.actLeft)
have slice_lhs : (Iso.inv (α_ R.X P.X S.X) ≫ P.actLeft ▷ S.X) ≫ P.actRight =
  CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)))
    (R.X ◁ coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ))
    (R.X ◁ coequalizer.desc P.actRight sorry ≫ P.actLeft)
Tactic: slice_lhs 1 4 => rw [id_tensor_π_preserves_coequalizer_inv_colimMap_desc]
-- 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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
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)
| P.actLeft ▷ S.X ≫ P.actRight
have a : Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight ≫ P.actLeft
Tactic: rw [middle_assoc]
-- 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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
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)
⊢ (Iso.inv (α_ R.X P.X S.X) ≫ P.actLeft ▷ S.X) ≫ P.actRight =
    CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)))
      (R.X ◁ coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ))
      (R.X ◁ coequalizer.desc P.actRight ⋯ ≫ P.actLeft)
have slice_lhs : Iso.inv (α_ R.X P.X S.X) ≫ Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight ≫ P.actLeft =
  CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)))
    (R.X ◁ coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ))
    (R.X ◁ coequalizer.desc P.actRight sorry ≫ P.actLeft)
Tactic: slice_lhs 2 3 => rw [middle_assoc]
-- 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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
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)
| CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)))
    (R.X ◁ coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)) (R.X ◁ coequalizer.desc P.actRight ⋯)
have a : R.X ◁ P.actRight
Tactic: rw [← whiskerLeft_comp, coequalizer.π_desc]
-- 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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
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)
⊢ Iso.inv (α_ R.X P.X S.X) ≫ Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight ≫ P.actLeft =
    CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)))
      (R.X ◁ coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ))
      (R.X ◁ coequalizer.desc P.actRight ⋯ ≫ P.actLeft)
have slice_rhs : Iso.inv (α_ R.X P.X S.X) ≫ Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight ≫ P.actLeft = R.X ◁ P.actRight ≫ P.actLeft
Tactic: slice_rhs 1 2 => rw [← whiskerLeft_comp, coequalizer.π_desc]
-- 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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
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)
⊢ Iso.inv (α_ R.X P.X S.X) ≫ Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight ≫ P.actLeft = R.X ◁ P.actRight ≫ P.actLeft
have rw : R.X ◁ P.actRight ≫ P.actLeft = R.X ◁ P.actRight ≫ P.actLeft
Tactic: rw [Iso.inv_hom_id_assoc]
-- 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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
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)
⊢ (P.tensorBimod (regular S)).actRight ≫ hom P = hom P ▷ S.X ≫ P.actRight
have dsimp : Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X P (regular S)) S.X ⟶ P.X)
  (TensorBimod.actRight P (regular S) ≫ hom P) (hom P ▷ S.X ≫ P.actRight)
Tactic: dsimp
-- 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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
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)
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X P (regular S)) S.X ⟶ P.X)
    (TensorBimod.actRight P (regular S) ≫ hom P) (hom P ▷ S.X ≫ P.actRight)
have dsimp : Eq (α :=
  tensorObj.{v₁, u₁} (𝒞 := inst)
      (TensorBimod.X P
        { X := S.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry, right_assoc := sorry,
          middle_assoc := sorry : Bimod S S })
      S.X ⟶
    P.X)
  (CategoryStruct.comp (Y :=
    TensorBimod.X P
      { X := S.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry, right_assoc := sorry,
        middle_assoc := sorry : Bimod S S })
    (CategoryStruct.comp (Y :=
      coequalizer (X :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) S.X)
          S.X)
        (P.actRight ▷ S.X ▷ S.X) ((Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ▷ S.X))
      (Iso.inv (PreservesCoequalizer.iso (tensorRight S.X) (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)))
      (colimMap
        (parallelPairHom (P.actRight ▷ S.X ▷ S.X) ((Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ▷ S.X) (P.actRight ▷ S.X)
          (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)
          (Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) S.X S.X) ≫
            Iso.hom (α_ P.X S.X (tensorObj.{v₁, u₁} (𝒞 := inst) S.X S.X)) ≫ P.X ◁ S.X ◁ μ ≫ Iso.inv (α_ P.X S.X S.X))
          (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) sorry sorry)))
    (coequalizer.desc P.actRight sorry))
  (coequalizer.desc P.actRight sorry ▷ S.X ≫ P.actRight)
Tactic: dsimp [hom, TensorBimod.actRight, regular]
-- 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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
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)
| CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)) S.X)
    (coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ▷ S.X)
    (CategoryStruct.comp (Y :=
      coequalizer (X :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) S.X)
          S.X)
        (P.actRight ▷ S.X ▷ S.X) ((Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ▷ S.X))
      (Iso.inv (PreservesCoequalizer.iso (tensorRight S.X) (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)))
      (CategoryStruct.comp (Y :=
        TensorBimod.X P
          { X := S.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod S S })
        (colimMap
          (parallelPairHom (P.actRight ▷ S.X ▷ S.X) ((Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ▷ S.X) (P.actRight ▷ S.X)
            (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)
            (Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) S.X S.X) ≫
              Iso.hom (α_ P.X S.X (tensorObj.{v₁, u₁} (𝒞 := inst) S.X S.X)) ≫ P.X ◁ S.X ◁ μ ≫ Iso.inv (α_ P.X S.X S.X))
            (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ⋯ ⋯))
        (coequalizer.desc P.actRight ⋯)))
have rw : (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ≫ P.actRight
Tactic: rw [π_tensor_id_preserves_coequalizer_inv_colimMap_desc]
-- 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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
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)
⊢ CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)) S.X)
      (coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ▷ S.X)
      (CategoryStruct.comp (Y :=
        TensorBimod.X P
          { X := S.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod S S })
        (CategoryStruct.comp (Y :=
          coequalizer (X :=
            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) S.X)
              S.X)
            (P.actRight ▷ S.X ▷ S.X) ((Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ▷ S.X))
          (Iso.inv (PreservesCoequalizer.iso (tensorRight S.X) (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)))
          (colimMap
            (parallelPairHom (P.actRight ▷ S.X ▷ S.X) ((Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ▷ S.X) (P.actRight ▷ S.X)
              (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)
              (Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) S.X S.X) ≫
                Iso.hom (α_ P.X S.X (tensorObj.{v₁, u₁} (𝒞 := inst) S.X S.X)) ≫
                  P.X ◁ S.X ◁ μ ≫ Iso.inv (α_ P.X S.X S.X))
              (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ⋯ ⋯)))
        (coequalizer.desc P.actRight ⋯)) =
    CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)) S.X)
      (coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ▷ S.X)
      (coequalizer.desc P.actRight ⋯ ▷ S.X ≫ P.actRight)
have slice_lhs : (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ≫ P.actRight =
  CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)) S.X)
    (coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ▷ S.X)
    (coequalizer.desc P.actRight sorry ▷ S.X ≫ P.actRight)
Tactic: slice_lhs 1 4 => rw [π_tensor_id_preserves_coequalizer_inv_colimMap_desc]