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
⊢ CategoryStruct.comp (Y :=
      coequalizer ((regular R).actRight ▷ P.X) (Iso.hom (α_ (regular R).X R.X P.X) ≫ (regular R).X ◁ P.actLeft))
      (coequalizer.π ((regular R).actRight ▷ P.X) (Iso.hom (α_ (regular R).X R.X P.X) ≫ (regular R).X ◁ P.actLeft))
      (coequalizer.desc P.actLeft ⋯ ≫
        Iso.inv (λ_ P.X) ≫
          η ▷ P.X ≫
            coequalizer.π ((regular R).actRight ▷ P.X)
              (Iso.hom (α_ (regular R).X R.X P.X) ≫ (regular R).X ◁ P.actLeft)) =
    coequalizer.π ((regular R).actRight ▷ P.X) (Iso.hom (α_ (regular R).X R.X P.X) ≫ (regular R).X ◁ P.actLeft) ≫
      𝟙 (coequalizer ((regular R).actRight ▷ P.X) (Iso.hom (α_ (regular R).X R.X P.X) ≫ (regular R).X ◁ P.actLeft))
have h : CategoryStruct.comp (Y := coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
    (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
    (coequalizer.desc P.actLeft sorry ≫
      Iso.inv (λ_ P.X) ≫ η ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) =
  coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
    𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ 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
| CategoryStruct.comp (Y := coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
    (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) (coequalizer.desc P.actLeft ⋯)
have a : P.actLeft
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 := coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
      (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
      (coequalizer.desc P.actLeft ⋯ ≫
        Iso.inv (λ_ P.X) ≫ η ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
have h : ((P.actLeft ≫ Iso.inv (λ_ P.X)) ≫ η ▷ P.X) ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
  coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
    𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
Tactic: slice_lhs 1 2 => 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
| P.actLeft ≫ Iso.inv (λ_ P.X)
have a : Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫ tensorUnit.{v₁, u₁} C (𝒞 := inst) ◁ P.actLeft
Tactic: rw [leftUnitor_inv_naturality]
-- 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.actLeft ≫ Iso.inv (λ_ P.X)) ≫ η ▷ P.X) ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
have h : ((Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫ tensorUnit.{v₁, u₁} C (𝒞 := inst) ◁ P.actLeft) ≫ η ▷ P.X) ≫
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
  coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
    𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
Tactic: slice_lhs 1 2 => rw [leftUnitor_inv_naturality]
-- 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
| tensorUnit.{v₁, u₁} C (𝒞 := inst) ◁ P.actLeft ≫ η ▷ P.X
have a : η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ≫ R.X ◁ P.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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
⊢ ((Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫ tensorUnit.{v₁, u₁} C (𝒞 := inst) ◁ P.actLeft) ≫ η ▷ P.X) ≫
      coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
have h : Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
    (η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ≫ R.X ◁ P.actLeft) ≫
      coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
  coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
    𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
| R.X ◁ P.actLeft
have a : Iso.inv (α_ R.X R.X P.X) ≫ Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft
Tactic: rw [← Iso.inv_hom_id_assoc (α_ R.X R.X P.X) (R.X ◁ P.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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
⊢ Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
      (η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ≫ R.X ◁ P.actLeft) ≫
        coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
have h : Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
    η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ≫
      (Iso.inv (α_ R.X R.X P.X) ≫ Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
        coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
  coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
    𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
Tactic: slice_lhs 3 3 => rw [← Iso.inv_hom_id_assoc (α_ R.X R.X P.X) (R.X ◁ P.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
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
| Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
have a : μ ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
Tactic: rw [← Category.assoc, ← coequalizer.condition]
-- 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 (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
      η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ≫
        (Iso.inv (α_ R.X R.X P.X) ≫ Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
          coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
have h : Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
    η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ≫
      Iso.inv (α_ R.X R.X P.X) ≫ μ ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
  coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
    𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
Tactic: slice_lhs 4 6 => rw [← Category.assoc, ← coequalizer.condition]
-- 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
| η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ≫ Iso.inv (α_ R.X R.X P.X)
have a : Iso.inv (α_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) R.X P.X) ≫ η ▷ R.X ▷ P.X
Tactic: rw [associator_inv_naturality_left]
-- 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 (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
      η ▷ tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ≫
        Iso.inv (α_ R.X R.X P.X) ≫ μ ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
have h : Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
    ((Iso.inv (α_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) R.X P.X) ≫ η ▷ R.X ▷ P.X) ≫ μ ▷ P.X) ≫
      coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
  coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
    𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
Tactic: slice_lhs 2 3 => rw [associator_inv_naturality_left]
-- 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
| η ▷ R.X ▷ P.X ≫ μ ▷ P.X
have a : Iso.hom (λ_ R.X) ▷ P.X
Tactic: rw [← comp_whiskerRight, MonObj.one_mul]
-- 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 (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
      ((Iso.inv (α_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) R.X P.X) ≫ η ▷ R.X ▷ P.X) ≫ μ ▷ P.X) ≫
        coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
have h : Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
    Iso.inv (α_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) R.X P.X) ≫
      Iso.hom (λ_ R.X) ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
  coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
    𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
Tactic: slice_lhs 3 4 => rw [← comp_whiskerRight, MonObj.one_mul]
-- 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 (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
    𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
have rw : coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
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 (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
      Iso.inv (α_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) R.X P.X) ≫
        Iso.hom (λ_ R.X) ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) =
    coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ≫
      𝟙 (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
have h : Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X ⟶ coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
  (Iso.inv (λ_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X P.X)) ≫
    Iso.inv (α_ (tensorUnit.{v₁, u₁} C (𝒞 := inst)) R.X P.X) ≫
      Iso.hom (λ_ R.X) ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
  (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
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 (regular R) P) (inv P) (hom P) = 𝟙 P.X
have dsimp : CategoryStruct.comp (Y := TensorBimod.X (regular R) P)
    (Iso.inv (λ_ P.X) ≫ η ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
    (coequalizer.desc P.actLeft 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 (regular R) P)
    (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) (coequalizer.desc P.actLeft ⋯)
have a : P.actLeft
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 (regular R) P)
      (Iso.inv (λ_ P.X) ≫ η ▷ P.X ≫ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
      (coequalizer.desc P.actLeft ⋯) =
    𝟙 P.X
have slice_lhs : Iso.inv (λ_ P.X) ≫ η ▷ P.X ≫ P.actLeft = 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 (λ_ P.X) ≫ η ▷ P.X ≫ P.actLeft = 𝟙 P.X
have rw : Eq.{v₁ + 1} (α := P.X ⟶ P.X)
  (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) P.X) (Iso.inv (λ_ P.X))
    (Iso.hom (λ_ P.X)))
  (𝟙 P.X)
Tactic: rw [one_actLeft, 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) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) P.X)
      (Iso.inv (λ_ P.X)) (Iso.hom (λ_ P.X)))
    (𝟙 P.X)
⊢ Iso.inv (λ_ P.X) ≫ η ▷ P.X ≫ P.actLeft = 𝟙 P.X
have rw₁ : Eq.{v₁ + 1} (α := P.X ⟶ P.X) (𝟙 P.X) (𝟙 P.X)
Tactic: rw [one_actLeft, 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)
⊢ ((regular R).tensorBimod P).actLeft ≫ hom P = R.X ◁ hom P ≫ P.actLeft
have dsimp : Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) R.X (TensorBimod.X (regular R) P) ⟶ P.X)
  (TensorBimod.actLeft (regular R) P ≫ 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 (regular R) P) ⟶ P.X)
    (TensorBimod.actLeft (regular R) P ≫ hom P) (R.X ◁ hom P ≫ P.actLeft)
have dsimp : Eq (α :=
  tensorObj.{v₁, u₁} (𝒞 := inst) R.X
      (TensorBimod.X
        { X := R.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry, right_assoc := sorry,
          middle_assoc := sorry : Bimod R R }
        P) ⟶
    P.X)
  (CategoryStruct.comp (Y :=
    TensorBimod.X
      { X := R.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry, right_assoc := sorry,
        middle_assoc := sorry : Bimod R R }
      P)
    (CategoryStruct.comp (Y :=
      coequalizer (X :=
        tensorObj.{v₁, u₁} (𝒞 := inst) R.X
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X))
        (R.X ◁ μ ▷ P.X) (R.X ◁ (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
      (Iso.inv (PreservesCoequalizer.iso (tensorLeft R.X) (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
      (colimMap
        (parallelPairHom (R.X ◁ μ ▷ P.X) (R.X ◁ (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) (μ ▷ P.X)
          (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
          (Iso.inv (α_ R.X (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X) ≫
            Iso.inv (α_ R.X R.X R.X) ▷ P.X ≫ μ ▷ R.X ▷ P.X)
          (Iso.inv (α_ R.X R.X P.X) ≫ μ ▷ P.X) sorry sorry)))
    (coequalizer.desc P.actLeft sorry))
  (R.X ◁ coequalizer.desc P.actLeft 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 (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
    (R.X ◁ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
    (CategoryStruct.comp (Y :=
      coequalizer (X :=
        tensorObj.{v₁, u₁} (𝒞 := inst) R.X
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X))
        (R.X ◁ μ ▷ P.X) (R.X ◁ (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
      (Iso.inv (PreservesCoequalizer.iso (tensorLeft R.X) (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
      (CategoryStruct.comp (Y :=
        TensorBimod.X
          { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }
          P)
        (colimMap
          (parallelPairHom (R.X ◁ μ ▷ P.X) (R.X ◁ (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) (μ ▷ P.X)
            (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
            (Iso.inv (α_ R.X (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X) ≫
              Iso.inv (α_ R.X R.X R.X) ▷ P.X ≫ μ ▷ R.X ▷ P.X)
            (Iso.inv (α_ R.X R.X P.X) ≫ μ ▷ P.X) ⋯ ⋯))
        (coequalizer.desc P.actLeft ⋯)))
have rw : (Iso.inv (α_ R.X R.X P.X) ≫ μ ▷ P.X) ≫ P.actLeft
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 (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
      (R.X ◁ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
      (CategoryStruct.comp (Y :=
        TensorBimod.X
          { X := R.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod R R }
          P)
        (CategoryStruct.comp (Y :=
          coequalizer (X :=
            tensorObj.{v₁, u₁} (𝒞 := inst) R.X
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X))
            (R.X ◁ μ ▷ P.X) (R.X ◁ (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
          (Iso.inv (PreservesCoequalizer.iso (tensorLeft R.X) (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
          (colimMap
            (parallelPairHom (R.X ◁ μ ▷ P.X) (R.X ◁ (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) (μ ▷ P.X)
              (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
              (Iso.inv (α_ R.X (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X) ≫
                Iso.inv (α_ R.X R.X R.X) ▷ P.X ≫ μ ▷ R.X ▷ P.X)
              (Iso.inv (α_ R.X R.X P.X) ≫ μ ▷ P.X) ⋯ ⋯)))
        (coequalizer.desc P.actLeft ⋯)) =
    CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
      (R.X ◁ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
      (R.X ◁ coequalizer.desc P.actLeft ⋯ ≫ P.actLeft)
have slice_lhs : (Iso.inv (α_ R.X R.X P.X) ≫ μ ▷ P.X) ≫ P.actLeft =
  CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
    (R.X ◁ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
    (R.X ◁ coequalizer.desc P.actLeft 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)
| μ ▷ P.X ≫ P.actLeft
have a : Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft ≫ P.actLeft
Tactic: rw [left_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 R.X P.X) ≫ μ ▷ P.X) ≫ P.actLeft =
    CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
      (R.X ◁ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
      (R.X ◁ coequalizer.desc P.actLeft ⋯ ≫ P.actLeft)
have slice_lhs : Iso.inv (α_ R.X R.X P.X) ≫ Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft ≫ P.actLeft =
  CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst) R.X (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
    (R.X ◁ coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft))
    (R.X ◁ coequalizer.desc P.actLeft sorry ≫ P.actLeft)
Tactic: slice_lhs 2 3 => rw [left_assoc]