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
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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X ◁ L.actRight ≫ Iso.hom (α_ P.X Q.X L.X)
have a : Iso.hom (α_ P.X Q.X (tensorObj.{v₁, u₁} (𝒞 := inst) L.X U.X)) ≫ P.X ◁ Q.X ◁ L.actRight
Tactic: 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
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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| (P.X ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ▷ U.X ≫
    Iso.hom (α_ P.X (Q.tensorBimod L).X U.X)
have a : Iso.hom (α_ P.X (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X L.X) U.X) ≫
  P.X ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) ▷ U.X
Tactic: erw [associator_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
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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| P.X ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) ▷ U.X ≫
    P.X ◁ TensorBimod.actRight Q L
have a : P.X ◁ Iso.hom (α_ Q.X L.X U.X) ≫
  P.X ◁ Q.X ◁ L.actRight ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)
Tactic: rw [← whiskerLeft_comp, TensorBimod.π_tensor_id_actRight, whiskerLeft_comp, 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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
⊢ Eq (α :=
    Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft P.X)
        (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ⟶
      ((P.tensorBimod Q).tensorBimod L).X)
    (CategoryTheory.Functor.map (tensorLeft P.X) (Q.actRight ▷ L.X) ≫
      Iso.inv (α_ P.X Q.X L.X) ≫
        coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X ≫
          coequalizer.π ((P.tensorBimod Q).actRight ▷ L.X)
            (Iso.hom (α_ (P.tensorBimod Q).X T.X L.X) ≫ (P.tensorBimod Q).X ◁ L.actLeft))
    (CategoryTheory.Functor.map (tensorLeft P.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) ≫
      Iso.inv (α_ P.X Q.X L.X) ≫
        coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X ≫
          coequalizer.π ((P.tensorBimod Q).actRight ▷ L.X)
            (Iso.hom (α_ (P.tensorBimod Q).X T.X L.X) ≫ (P.tensorBimod Q).X ◁ L.actLeft))
have dsimp : Eq (α :=
  tensorObj.{v₁, u₁} (𝒞 := inst) P.X (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ⟶
    TensorBimod.X (P.tensorBimod Q) L)
  (P.X ◁ Q.actRight ▷ L.X ≫
    Iso.inv (α_ P.X Q.X L.X) ≫
      coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X ≫
        coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
          (Iso.hom (α_ (TensorBimod.X P Q) T.X L.X) ≫ TensorBimod.X P Q ◁ L.actLeft))
  (P.X ◁ (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) ≫
    Iso.inv (α_ P.X Q.X L.X) ≫
      coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X ≫
        coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
          (Iso.hom (α_ (TensorBimod.X P Q) T.X L.X) ≫ TensorBimod.X P Q ◁ L.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
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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| P.X ◁ Q.actRight ▷ L.X ≫ Iso.inv (α_ P.X Q.X L.X)
have a : Iso.inv (α_ P.X (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ≫ (P.X ◁ Q.actRight) ▷ L.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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| ((Iso.hom (α_ P.X Q.X T.X) ≫ P.X ◁ Q.actRight) ▷ L.X ≫
      coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X :
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) L.X ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) L.X)
have a : (CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst)
      (tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X)
      L.X)
    (coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ T.X ▷ L.X :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) L.X ⟶
        tensorObj.{v₁, u₁} (𝒞 := inst)
          (tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
            T.X)
          L.X)
    (TensorBimod.actRight P Q ▷ L.X :
      tensorObj.{v₁, u₁} (𝒞 := inst)
          (tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
            T.X)
          L.X ⟶
        tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
          L.X) :
  tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) L.X ⟶
    tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) L.X)
Tactic: rw [← comp_whiskerRight, Category.assoc, ← TensorBimod.π_tensor_id_actRight,
            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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| (CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) L.X)
      (TensorBimod.actRight P Q ▷ L.X :
        tensorObj.{v₁, u₁} (𝒞 := inst)
            (tensorObj.{v₁, u₁} (𝒞 := inst)
              (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X)
            L.X ⟶
          tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
            L.X)
      (coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
          (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
            coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) :
        tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
            L.X ⟶
          coequalizer (TensorBimod.actRight P Q ▷ L.X)
            (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
              coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft)) :
    tensorObj.{v₁, u₁} (𝒞 := inst)
        (tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
          T.X)
        L.X ⟶
      coequalizer (TensorBimod.actRight P Q ▷ L.X)
        (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
          coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft))
have a : ((Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
      coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) ≫
    coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
      (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
        coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) :
  tensorObj.{v₁, u₁} (𝒞 := inst)
      (tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X)
      L.X ⟶
    coequalizer (TensorBimod.actRight P Q ▷ L.X)
      (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
        coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft))
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
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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| P.X ◁ (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) ≫ Iso.inv (α_ P.X Q.X L.X)
have a : (P.X ◁ Iso.hom (α_ Q.X T.X L.X) ≫ P.X ◁ Q.X ◁ L.actLeft) ≫ Iso.inv (α_ P.X Q.X L.X)
Tactic: rw [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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| P.X ◁ Q.X ◁ L.actLeft ≫ Iso.inv (α_ P.X Q.X L.X)
have a : Iso.inv (α_ P.X Q.X (tensorObj.{v₁, u₁} (𝒞 := inst) T.X L.X)) ≫ tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X ◁ L.actLeft
Tactic: rw [associator_inv_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
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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X ◁ L.actLeft ≫
    coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X
have a : coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ tensorObj.{v₁, u₁} (𝒞 := inst) T.X L.X ≫
  coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
⊢ P.actRight ▷ (Q.tensorBimod L).X ≫ invAux P Q L =
    (Iso.hom (α_ P.X S.X (Q.tensorBimod L).X) ≫ P.X ◁ (Q.tensorBimod L).actLeft) ≫ invAux P Q L
have dsimp : Eq (α :=
  tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) (TensorBimod.X Q L) ⟶
    TensorBimod.X (P.tensorBimod Q) L)
  (P.actRight ▷ TensorBimod.X Q L ≫
      CategoryStruct.comp (Y :=
        coequalizer (X :=
          tensorObj.{v₁, u₁} (𝒞 := inst) P.X
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X))
          (P.X ◁ Q.actRight ▷ L.X) (P.X ◁ (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
        (Iso.inv
          (PreservesCoequalizer.iso (tensorLeft P.X) (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
        (coequalizer.desc
          (Iso.inv (α_ P.X Q.X L.X) ≫
            coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X ≫
              coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
                (Iso.hom (α_ (TensorBimod.X P Q) T.X L.X) ≫ TensorBimod.X P Q ◁ L.actLeft))
          sorry) :
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) (TensorBimod.X Q L) ⟶
      TensorBimod.X (P.tensorBimod Q) L)
  ((Iso.hom (α_ P.X S.X (TensorBimod.X Q L)) ≫ P.X ◁ TensorBimod.actLeft Q L) ≫
      CategoryStruct.comp (Y :=
        coequalizer (X :=
          tensorObj.{v₁, u₁} (𝒞 := inst) P.X
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X))
          (P.X ◁ Q.actRight ▷ L.X) (P.X ◁ (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
        (Iso.inv
          (PreservesCoequalizer.iso (tensorLeft P.X) (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
        (coequalizer.desc
          (Iso.inv (α_ P.X Q.X L.X) ≫
            coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X ≫
              coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
                (Iso.hom (α_ (TensorBimod.X P Q) T.X L.X) ≫ TensorBimod.X P Q ◁ L.actLeft))
          sorry) :
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) (TensorBimod.X Q L) ⟶
      TensorBimod.X (P.tensorBimod Q) L)
Tactic: dsimp [invAux]
-- 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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X ◁
      coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) ≫
    P.actRight ▷ coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)
have a : P.actRight ▷ tensorObj.{v₁, u₁} (𝒞 := inst) Q.X L.X ≫
  P.X ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| P.actRight ▷ tensorObj.{v₁, u₁} (𝒞 := inst) Q.X L.X ≫ Iso.inv (α_ P.X Q.X L.X)
have a : Iso.inv (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) Q.X L.X) ≫ P.actRight ▷ Q.X ▷ L.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
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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| P.actRight ▷ Q.X ▷ L.X ≫ coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X
have a : (Iso.hom (α_ P.X S.X Q.X) ▷ L.X ≫ (P.X ◁ Q.actLeft) ▷ L.X) ≫
  coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X
Tactic: rw [← comp_whiskerRight, coequalizer.condition, comp_whiskerRight, 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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X ◁
      coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) ≫
    Iso.hom (α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
have a : Iso.hom (α_ P.X S.X (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X L.X)) ≫
  P.X ◁ S.X ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)
Tactic: 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
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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| (CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) P.X
        (tensorObj.{v₁, u₁} (𝒞 := inst) S.X
          (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))))
      (P.X ◁ S.X ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) :
        tensorObj.{v₁, u₁} (𝒞 := inst) P.X
            (tensorObj.{v₁, u₁} (𝒞 := inst) S.X (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X L.X)) ⟶
          tensorObj.{v₁, u₁} (𝒞 := inst) P.X
            (tensorObj.{v₁, u₁} (𝒞 := inst) S.X
              (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))))
      (P.X ◁ TensorBimod.actLeft Q L :
        tensorObj.{v₁, u₁} (𝒞 := inst) P.X
            (tensorObj.{v₁, u₁} (𝒞 := inst) S.X
              (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))) ⟶
          tensorObj.{v₁, u₁} (𝒞 := inst) P.X
            (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))) :
    tensorObj.{v₁, u₁} (𝒞 := inst) P.X (tensorObj.{v₁, u₁} (𝒞 := inst) S.X (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X L.X)) ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) P.X (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
have a : P.X ◁ Iso.inv (α_ S.X Q.X L.X) ≫
  P.X ◁ Q.actLeft ▷ L.X ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)
Tactic: rw [← whiskerLeft_comp, TensorBimod.whiskerLeft_π_actLeft,
          whiskerLeft_comp, 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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| P.X ◁ Q.actLeft ▷ L.X ≫ Iso.inv (α_ P.X Q.X L.X)
have a : Iso.inv (α_ P.X (tensorObj.{v₁, u₁} (𝒞 := inst) S.X Q.X) L.X) ≫ (P.X ◁ Q.actLeft) ▷ L.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
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)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
⊢ hom P Q L ≫ inv P Q L = 𝟙 ((P.tensorBimod Q).tensorBimod L).X
have dsimp : coequalizer.desc
      (CategoryStruct.comp (Y :=
        coequalizer (X :=
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) Q.X)
            L.X)
          (P.actRight ▷ Q.X ▷ L.X) ((Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X))
        (Iso.inv
          (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))
        (coequalizer.desc
          (Iso.hom (α_ P.X Q.X L.X) ≫
            CategoryStruct.comp (Y :=
              tensorObj.{v₁, u₁} (𝒞 := inst) P.X
                (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
              (P.X ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) :
                tensorObj.{v₁, u₁} (𝒞 := inst) P.X (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X L.X) ⟶
                  tensorObj.{v₁, u₁} (𝒞 := inst) P.X
                    (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
              (coequalizer.π (P.actRight ▷ TensorBimod.X Q L)
                  (Iso.hom (α_ P.X S.X (TensorBimod.X Q L)) ≫ P.X ◁ TensorBimod.actLeft Q L) :
                tensorObj.{v₁, u₁} (𝒞 := inst) P.X
                    (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
                  coequalizer (X :=
                    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) (TensorBimod.X Q L))
                    (P.actRight ▷ TensorBimod.X Q L)
                    (Iso.hom (α_ P.X S.X (TensorBimod.X Q L)) ≫ P.X ◁ TensorBimod.actLeft Q L)))
          sorry))
      sorry ≫
    coequalizer.desc
      (CategoryStruct.comp (Y :=
        coequalizer (X :=
          tensorObj.{v₁, u₁} (𝒞 := inst) P.X
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X))
          (P.X ◁ Q.actRight ▷ L.X) (P.X ◁ (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
        (Iso.inv
          (PreservesCoequalizer.iso (tensorLeft P.X) (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
        (coequalizer.desc
          (Iso.inv (α_ P.X Q.X L.X) ≫
            coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X ≫
              coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
                (Iso.hom (α_ (TensorBimod.X P Q) T.X L.X) ≫ TensorBimod.X P Q ◁ L.actLeft))
          sorry))
      sorry =
  𝟙 (TensorBimod.X (P.tensorBimod Q) L)
Tactic: dsimp [hom, homAux, inv, invAux]