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
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)) (R.X ◁ coequalizer.desc P.actLeft ⋯)
have a : R.X ◁ P.actLeft
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 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 ⋯ ≫ P.actLeft)
have slice_rhs : Iso.inv (α_ R.X R.X P.X) ≫ Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft ≫ P.actLeft = R.X ◁ P.actLeft ≫ 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 R.X P.X) ≫ Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft ≫ P.actLeft = R.X ◁ P.actLeft ≫ P.actLeft
have rw : R.X ◁ P.actLeft ≫ P.actLeft = R.X ◁ P.actLeft ≫ 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)
⊢ ((regular R).tensorBimod P).actRight ≫ hom P = hom P ▷ S.X ≫ P.actRight
have dsimp : Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X (regular R) P) S.X ⟶ P.X)
  (TensorBimod.actRight (regular R) P ≫ 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 (regular R) P) S.X ⟶ P.X)
    (TensorBimod.actRight (regular R) P ≫ hom P) (hom P ▷ S.X ≫ P.actRight)
have dsimp : Eq (α :=
  tensorObj.{v₁, u₁} (𝒞 := inst)
      (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)
      S.X ⟶
    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) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X)
          S.X)
        (μ ▷ P.X ▷ S.X) ((Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X))
      (Iso.inv (PreservesCoequalizer.iso (tensorRight S.X) (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
      (colimMap
        (parallelPairHom (μ ▷ P.X ▷ S.X) ((Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X) (μ ▷ P.X)
          (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
          (Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X S.X) ≫
            Iso.hom (α_ R.X R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)) ≫
              R.X ◁ R.X ◁ P.actRight ≫ Iso.inv (α_ R.X R.X P.X))
          (Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight) sorry sorry)))
    (coequalizer.desc P.actLeft sorry))
  (coequalizer.desc P.actLeft 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 (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) S.X)
    (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X)
    (CategoryStruct.comp (Y :=
      coequalizer (X :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X)
          S.X)
        (μ ▷ P.X ▷ S.X) ((Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X))
      (Iso.inv (PreservesCoequalizer.iso (tensorRight S.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 (μ ▷ P.X ▷ S.X) ((Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X) (μ ▷ P.X)
            (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
            (Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X S.X) ≫
              Iso.hom (α_ R.X R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)) ≫
                R.X ◁ R.X ◁ P.actRight ≫ Iso.inv (α_ R.X R.X P.X))
            (Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight) ⋯ ⋯))
        (coequalizer.desc P.actLeft ⋯)))
have rw : (Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight) ≫ P.actLeft
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 (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) S.X)
      (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X)
      (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) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X)
              S.X)
            (μ ▷ P.X ▷ S.X) ((Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X))
          (Iso.inv (PreservesCoequalizer.iso (tensorRight S.X) (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)))
          (colimMap
            (parallelPairHom (μ ▷ P.X ▷ S.X) ((Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X) (μ ▷ P.X)
              (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)
              (Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) R.X R.X) P.X S.X) ≫
                Iso.hom (α_ R.X R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)) ≫
                  R.X ◁ R.X ◁ P.actRight ≫ Iso.inv (α_ R.X R.X P.X))
              (Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight) ⋯ ⋯)))
        (coequalizer.desc P.actLeft ⋯)) =
    CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) S.X)
      (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X)
      (coequalizer.desc P.actLeft ⋯ ▷ S.X ≫ P.actRight)
have slice_lhs : (Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight) ≫ P.actLeft =
  CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) S.X)
    (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X)
    (coequalizer.desc P.actLeft sorry ▷ S.X ≫ P.actRight)
Tactic: slice_lhs 1 4 => 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 (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) S.X)
    (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X) (coequalizer.desc P.actLeft ⋯ ▷ S.X)
have a : P.actLeft ▷ S.X
Tactic: rw [← comp_whiskerRight, 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.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight) ≫ P.actLeft =
    CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft)) S.X)
      (coequalizer.π (μ ▷ P.X) (Iso.hom (α_ R.X R.X P.X) ≫ R.X ◁ P.actLeft) ▷ S.X)
      (coequalizer.desc P.actLeft ⋯ ▷ S.X ≫ P.actRight)
have slice_rhs : (Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight) ≫ P.actLeft = P.actLeft ▷ S.X ≫ P.actRight
Tactic: slice_rhs 1 2 => rw [← comp_whiskerRight, 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)
| P.actLeft ▷ S.X ≫ P.actRight
have rw : 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.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight) ≫ P.actLeft = P.actLeft ▷ S.X ≫ P.actRight
have slice_rhs : (Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight) ≫ P.actLeft = Iso.hom (α_ R.X P.X S.X) ≫ R.X ◁ P.actRight ≫ P.actLeft
Tactic: slice_rhs 1 2 => 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
A B : Mon.{v₁, u₁} C
M : Bimod A B
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
⊢ P.actRight ▷ (regular S).X ≫ P.actRight =
    (Iso.hom (α_ P.X S.X (regular S).X) ≫ P.X ◁ (regular S).actLeft) ≫ P.actRight
have dsimp : P.actRight ▷ S.X ≫ P.actRight = (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ≫ 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
A B : Mon.{v₁, u₁} C
M : Bimod A B
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
⊢ P.actRight ▷ S.X ≫ P.actRight = (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ≫ P.actRight
have rw : P.actRight ▷ S.X ≫ P.actRight = Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ ≫ P.actRight
Tactic: rw [Category.assoc, right_assoc, Iso.hom_inv_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
A B : Mon.{v₁, u₁} C
M : Bimod A B
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
rw : P.actRight ▷ S.X ≫ P.actRight = Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ ≫ P.actRight
⊢ P.actRight ▷ S.X ≫ P.actRight = (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ≫ P.actRight
have rw₁ : P.actRight ▷ S.X ≫ P.actRight = Iso.hom (α_ P.X S.X S.X) ≫ Iso.inv (α_ P.X S.X S.X) ≫ P.actRight ▷ S.X ≫ P.actRight
Tactic: rw [Category.assoc, right_assoc, Iso.hom_inv_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
A B : Mon.{v₁, u₁} C
M : Bimod A B
inst_2 : HasCoequalizers.{v₁, u₁} C
R S : Mon.{v₁, u₁} C
P : Bimod R S
rw : P.actRight ▷ S.X ≫ P.actRight = Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ ≫ P.actRight
rw₁ :
  P.actRight ▷ S.X ≫ P.actRight = Iso.hom (α_ P.X S.X S.X) ≫ Iso.inv (α_ P.X S.X S.X) ≫ P.actRight ▷ S.X ≫ P.actRight
⊢ P.actRight ▷ S.X ≫ P.actRight = (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ) ≫ P.actRight
have rw₂ : P.actRight ▷ S.X ≫ P.actRight = P.actRight ▷ S.X ≫ P.actRight
Tactic: rw [Category.assoc, right_assoc, Iso.hom_inv_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
⊢ hom P ≫ inv P = 𝟙 (TensorBimod.X P (regular S))
have dsimp : coequalizer.desc P.actRight sorry ≫
    Iso.inv (ρ_ P.X) ≫
      P.X ◁ η ≫
        coequalizer.π (P.actRight ▷ (regular S).X) (Iso.hom (α_ P.X S.X (regular S).X) ≫ P.X ◁ (regular S).actLeft) =
  𝟙 (coequalizer (P.actRight ▷ (regular S).X) (Iso.hom (α_ P.X S.X (regular S).X) ≫ P.X ◁ (regular S).actLeft))
Tactic: dsimp only [hom, inv, TensorBimod.X]
-- 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.desc P.actRight ⋯ ≫
      Iso.inv (ρ_ P.X) ≫
        P.X ◁ η ≫
          coequalizer.π (P.actRight ▷ (regular S).X) (Iso.hom (α_ P.X S.X (regular S).X) ≫ P.X ◁ (regular S).actLeft) =
    𝟙 (coequalizer (P.actRight ▷ (regular S).X) (Iso.hom (α_ P.X S.X (regular S).X) ≫ P.X ◁ (regular S).actLeft))
have h : CategoryStruct.comp (Y :=
    coequalizer (P.actRight ▷ (regular S).X) (Iso.hom (α_ P.X S.X (regular S).X) ≫ P.X ◁ (regular S).actLeft))
    (coequalizer.π (P.actRight ▷ (regular S).X) (Iso.hom (α_ P.X S.X (regular S).X) ≫ P.X ◁ (regular S).actLeft))
    (coequalizer.desc P.actRight sorry ≫
      Iso.inv (ρ_ P.X) ≫
        P.X ◁ η ≫
          coequalizer.π (P.actRight ▷ (regular S).X) (Iso.hom (α_ P.X S.X (regular S).X) ≫ P.X ◁ (regular S).actLeft)) =
  coequalizer.π (P.actRight ▷ (regular S).X) (Iso.hom (α_ P.X S.X (regular S).X) ≫ P.X ◁ (regular S).actLeft) ≫
    𝟙 (coequalizer (P.actRight ▷ (regular S).X) (Iso.hom (α_ P.X S.X (regular S).X) ≫ P.X ◁ (regular S).actLeft))
Tactic: ext
-- 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 (P.actRight ▷ (regular S).X) (Iso.hom (α_ P.X S.X (regular S).X) ≫ P.X ◁ (regular S).actLeft))
      (coequalizer.π (P.actRight ▷ (regular S).X) (Iso.hom (α_ P.X S.X (regular S).X) ≫ P.X ◁ (regular S).actLeft))
      (coequalizer.desc P.actRight ⋯ ≫
        Iso.inv (ρ_ P.X) ≫
          P.X ◁ η ≫
            coequalizer.π (P.actRight ▷ (regular S).X)
              (Iso.hom (α_ P.X S.X (regular S).X) ≫ P.X ◁ (regular S).actLeft)) =
    coequalizer.π (P.actRight ▷ (regular S).X) (Iso.hom (α_ P.X S.X (regular S).X) ≫ P.X ◁ (regular S).actLeft) ≫
      𝟙 (coequalizer (P.actRight ▷ (regular S).X) (Iso.hom (α_ P.X S.X (regular S).X) ≫ P.X ◁ (regular S).actLeft))
have h : CategoryStruct.comp (Y := coequalizer (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))
    (coequalizer.desc P.actRight sorry ≫
      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.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: 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 (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)) (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 := coequalizer (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))
      (coequalizer.desc P.actRight ⋯ ≫
        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.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 : ((P.actRight ≫ 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.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 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.actRight ≫ Iso.inv (ρ_ P.X)
have a : Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)) ≫ P.actRight ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst)
Tactic: rw [rightUnitor_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.actRight ≫ 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.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)) ≫ P.actRight ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst)) ≫ P.X ◁ η) ≫
    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 1 2 => rw [rightUnitor_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.actRight ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst) ≫ P.X ◁ η
have a : tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X ◁ η ≫ P.actRight ▷ S.X
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 (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)) ≫ P.actRight ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst)) ≫ P.X ◁ η) ≫
      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)) ≫
    (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X ◁ η ≫ P.actRight ▷ 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 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
| P.actRight ▷ S.X ≫ coequalizer.π (P.actRight ▷ S.X) (Iso.hom (α_ P.X S.X S.X) ≫ P.X ◁ μ)
have a : (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: rw [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 (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)) ≫
      (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X ◁ η ≫ P.actRight ▷ 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 : 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))
Tactic: slice_lhs 3 4 => rw [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
| tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X ◁ η ≫ Iso.hom (α_ P.X S.X S.X)
have a : Iso.hom (α_ P.X S.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ P.X ◁ S.X ◁ η
Tactic: rw [associator_naturality_right]