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)
W X Y Z : Mon.{v₁, u₁} C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
| CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft)) P.X)
    (coequalizer.π (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ▷ P.X)
    (colimMap
        (parallelPairHom (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) (M'.actRight ▷ N.X)
          (Iso.hom (α_ M'.X X.X N.X) ≫ M'.X ◁ N.actLeft) (f.hom ▷ X.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷
      P.X)
have a : (f.hom ▷ N.X ≫
    colimit.ι (parallelPair (M'.actRight ▷ N.X) (Iso.hom (α_ M'.X X.X N.X) ≫ M'.X ◁ N.actLeft))
      WalkingParallelPair.one) ▷
  P.X
Tactic: rw [← comp_whiskerRight, ι_colimMap, parallelPairHom_app_one]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
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)
W X Y Z : Mon.{v₁, u₁} C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
| (CategoryStruct.comp (Y :=
      coequalizer (TensorBimod.actRight M' N ▷ P.X)
        (Iso.hom (α_ (coequalizer (M'.actRight ▷ N.X) (Iso.hom (α_ M'.X X.X N.X) ≫ M'.X ◁ N.actLeft)) Y.X P.X) ≫
          coequalizer (M'.actRight ▷ N.X) (Iso.hom (α_ M'.X X.X N.X) ≫ M'.X ◁ N.actLeft) ◁ P.actLeft))
      (colimit.ι
        (parallelPair (TensorBimod.actRight M' N ▷ P.X)
          (Iso.hom (α_ (coequalizer (M'.actRight ▷ N.X) (Iso.hom (α_ M'.X X.X N.X) ≫ M'.X ◁ N.actLeft)) Y.X P.X) ≫
            coequalizer (M'.actRight ▷ N.X) (Iso.hom (α_ M'.X X.X N.X) ≫ M'.X ◁ N.actLeft) ◁ P.actLeft))
        WalkingParallelPair.one)
      (coequalizer.desc (AssociatorBimod.homAux M' N P) ⋯) :
    (parallelPair (TensorBimod.actRight M' N ▷ P.X)
            (Iso.hom (α_ (coequalizer (M'.actRight ▷ N.X) (Iso.hom (α_ M'.X X.X N.X) ≫ M'.X ◁ N.actLeft)) Y.X P.X) ≫
              coequalizer (M'.actRight ▷ N.X) (Iso.hom (α_ M'.X X.X N.X) ≫ M'.X ◁ N.actLeft) ◁ P.actLeft)).obj
        WalkingParallelPair.one ⟶
      colimit
        (parallelPair (X :=
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M'.X X.X)
            (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft)))
          (Y :=
          tensorObj.{v₁, u₁} (𝒞 := inst) M'.X
            (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft)))
          (M'.actRight ▷ coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft) :
            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M'.X X.X)
                (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft)) ⟶
              tensorObj.{v₁, u₁} (𝒞 := inst) M'.X
                (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft)))
          (Iso.hom (α_ M'.X X.X (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft))) ≫
              M'.X ◁ TensorBimod.actLeft N P :
            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M'.X X.X)
                (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft)) ⟶
              tensorObj.{v₁, u₁} (𝒞 := inst) M'.X
                (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft)))))
have a : AssociatorBimod.homAux M' N P
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
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)
W X Y Z : Mon.{v₁, u₁} C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
| (f.hom ▷ N.X ≫
      colimit.ι (parallelPair (M'.actRight ▷ N.X) (Iso.hom (α_ M'.X X.X N.X) ≫ M'.X ◁ N.actLeft))
        WalkingParallelPair.one) ▷
    P.X
have a : f.hom ▷ N.X ▷ P.X ≫
  colimit.ι (parallelPair (M'.actRight ▷ N.X) (Iso.hom (α_ M'.X X.X N.X) ≫ M'.X ◁ N.actLeft)) WalkingParallelPair.one ▷
    P.X
Tactic: rw [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
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)
W X Y Z : Mon.{v₁, u₁} C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
| f.hom ▷ N.X ▷ P.X ≫ Iso.hom (α_ M'.X N.X P.X)
have a : Iso.hom (α_ M.X N.X P.X) ≫ f.hom ▷ tensorObj.{v₁, u₁} (𝒞 := inst) N.X P.X
Tactic: rw [associator_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
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)
W X Y Z : Mon.{v₁, u₁} C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
| Iso.inv (α_ M.X N.X P.X) ≫ Iso.hom (α_ M.X N.X P.X) ≫ f.hom ▷ tensorObj.{v₁, u₁} (𝒞 := inst) N.X P.X
have a : f.hom ▷ tensorObj.{v₁, u₁} (𝒞 := inst) N.X P.X
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
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)
W X Y Z : Mon.{v₁, u₁} C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
| M.X ◁ coequalizer.π (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft) ≫
    f.hom ▷ coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft)
have a : f.hom ▷ tensorObj.{v₁, u₁} (𝒞 := inst) N.X P.X ≫
  M'.X ◁ coequalizer.π (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.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
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)
W X Y Z : Mon.{v₁, u₁} C
M : Bimod W X
N N' : Bimod X Y
f : N ⟶ N'
P : Bimod Y Z
| CategoryStruct.comp (Y :=
    coequalizer (TensorBimod.actRight M N ▷ P.X)
      (Iso.hom (α_ (TensorBimod.X M N) Y.X P.X) ≫ TensorBimod.X M N ◁ P.actLeft))
    (coequalizer.π (TensorBimod.actRight M N ▷ P.X)
      (Iso.hom (α_ (TensorBimod.X M N) Y.X P.X) ≫ TensorBimod.X M N ◁ P.actLeft))
    (colimMap
      (parallelPairHom (TensorBimod.actRight M N ▷ P.X)
        (Iso.hom (α_ (TensorBimod.X M N) Y.X P.X) ≫ TensorBimod.X M N ◁ P.actLeft) (TensorBimod.actRight M N' ▷ P.X)
        (Iso.hom (α_ (TensorBimod.X M N') Y.X P.X) ≫ TensorBimod.X M N' ◁ P.actLeft)
        (colimMap
              (parallelPairHom (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)
                (Iso.hom (α_ M.X X.X N'.X) ≫ M.X ◁ N'.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X X.X ◁ f.hom)
                (M.X ◁ f.hom) ⋯ ⋯) ▷
            Y.X ▷
          P.X)
        (colimMap
            (parallelPairHom (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)
              (Iso.hom (α_ M.X X.X N'.X) ≫ M.X ◁ N'.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X X.X ◁ f.hom)
              (M.X ◁ f.hom) ⋯ ⋯) ▷
          P.X)
        ⋯ ⋯))
have rw : colimMap
      (parallelPairHom (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)
        (Iso.hom (α_ M.X X.X N'.X) ≫ M.X ◁ N'.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X X.X ◁ f.hom) (M.X ◁ f.hom) sorry
        sorry) ▷
    P.X ≫
  colimit.ι
    (parallelPair (TensorBimod.actRight M N' ▷ P.X)
      (Iso.hom (α_ (TensorBimod.X M N') Y.X P.X) ≫ TensorBimod.X M N' ◁ P.actLeft))
    WalkingParallelPair.one
Tactic: rw [ι_colimMap, parallelPairHom_app_one]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
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)
W X Y Z : Mon.{v₁, u₁} C
M : Bimod W X
N N' : Bimod X Y
f : N ⟶ N'
P : Bimod Y Z
| CategoryStruct.comp (Y :=
    coequalizer (TensorBimod.actRight M N ▷ P.X)
      (Iso.hom (α_ (TensorBimod.X M N) Y.X P.X) ≫ TensorBimod.X M N ◁ P.actLeft))
    (coequalizer.π (TensorBimod.actRight M N ▷ P.X)
      (Iso.hom (α_ (TensorBimod.X M N) Y.X P.X) ≫ TensorBimod.X M N ◁ P.actLeft))
    (coequalizer.desc (AssociatorBimod.homAux M N P) ⋯)
have a : AssociatorBimod.homAux M N P
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
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)
W X Y Z : Mon.{v₁, u₁} C
M : Bimod W X
N N' : Bimod X Y
f : N ⟶ N'
P : Bimod Y Z
| CategoryStruct.comp (Y :=
    Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight P.X)
      (coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft)))
    (coequalizer.π (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ▷ P.X)
    (colimMap
        (parallelPairHom (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) (M.actRight ▷ N'.X)
          (Iso.hom (α_ M.X X.X N'.X) ≫ M.X ◁ N'.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X X.X ◁ f.hom) (M.X ◁ f.hom)
          ⋯ ⋯) ▷
      P.X)
have a : (M.X ◁ f.hom ≫
    colimit.ι (parallelPair (M.actRight ▷ N'.X) (Iso.hom (α_ M.X X.X N'.X) ≫ M.X ◁ N'.actLeft))
      WalkingParallelPair.one) ▷
  P.X
Tactic: rw [← comp_whiskerRight, ι_colimMap, parallelPairHom_app_one]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
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)
W X Y Z : Mon.{v₁, u₁} C
M : Bimod W X
N N' : Bimod X Y
f : N ⟶ N'
P : Bimod Y Z
| (CategoryStruct.comp (Y :=
      TensorBimod.X M
        { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯,
          actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ : Bimod X Z })
      (coequalizer.π (M.actRight ▷ TensorBimod.X N P)
          (Iso.hom (α_ M.X X.X (TensorBimod.X N P)) ≫ M.X ◁ TensorBimod.actLeft N P) :
        tensorObj.{v₁, u₁} (𝒞 := inst) M.X
            (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft)) ⟶
          coequalizer (X := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X X.X) (TensorBimod.X N P))
            (M.actRight ▷ TensorBimod.X N P) (Iso.hom (α_ M.X X.X (TensorBimod.X N P)) ≫ M.X ◁ TensorBimod.actLeft N P))
      (colimMap
        (parallelPairHom (M.actRight ▷ TensorBimod.X N P)
          (Iso.hom (α_ M.X X.X (TensorBimod.X N P)) ≫ M.X ◁ TensorBimod.actLeft N P) (M.actRight ▷ TensorBimod.X N' P)
          (Iso.hom (α_ M.X X.X (TensorBimod.X N' P)) ≫ M.X ◁ TensorBimod.actLeft N' P)
          (tensorObj.{v₁, u₁} (𝒞 := inst) M.X X.X ◁
            colimMap
              (parallelPairHom (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)
                (Iso.hom (α_ N'.X Y.X P.X) ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))
          (M.X ◁
            colimMap
              (parallelPairHom (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)
                (Iso.hom (α_ N'.X Y.X P.X) ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))
          ⋯ ⋯)) :
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft)) ⟶
      TensorBimod.X M
        { X := TensorBimod.X N' P, actLeft := TensorBimod.actLeft N' P, one_actLeft := ⋯, left_assoc := ⋯,
          actRight := TensorBimod.actRight N' P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ : Bimod X Z })
have a : M.X ◁
    colimMap
      (parallelPairHom (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)
        (Iso.hom (α_ N'.X Y.X P.X) ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) sorry sorry) ≫
  colimit.ι
    (parallelPair (X := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X X.X) (TensorBimod.X N' P))
      (M.actRight ▷ TensorBimod.X N' P) (Iso.hom (α_ M.X X.X (TensorBimod.X N' P)) ≫ M.X ◁ TensorBimod.actLeft N' P))
    WalkingParallelPair.one
Tactic: rw [ι_colimMap, parallelPairHom_app_one]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
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)
W X Y Z : Mon.{v₁, u₁} C
M : Bimod W X
N N' : Bimod X Y
f : N ⟶ N'
P : Bimod Y Z
| CategoryStruct.comp (Y :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft)))
    (M.X ◁ coequalizer.π (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft))
    (M.X ◁
      colimMap
        (parallelPairHom (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft) (N'.actRight ▷ P.X)
          (Iso.hom (α_ N'.X Y.X P.X) ≫ N'.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) ⋯ ⋯))
have a : M.X ◁
  (f.hom ▷ P.X ≫
    colimit.ι (parallelPair (N'.actRight ▷ P.X) (Iso.hom (α_ N'.X Y.X P.X) ≫ N'.X ◁ P.actLeft)) WalkingParallelPair.one)
Tactic: rw [← whiskerLeft_comp, ι_colimMap, parallelPairHom_app_one]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
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)
W X Y Z : Mon.{v₁, u₁} C
M : Bimod W X
N N' : Bimod X Y
f : N ⟶ N'
P : Bimod Y Z
| (CategoryStruct.comp (Y :=
      TensorBimod.X M
        { X := TensorBimod.X N' P, actLeft := TensorBimod.actLeft N' P, one_actLeft := ⋯, left_assoc := ⋯,
          actRight := TensorBimod.actRight N' P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ : Bimod X Z })
      (colimit.ι
        (parallelPair (X :=
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X X.X) (TensorBimod.X N' P))
          (M.actRight ▷ TensorBimod.X N' P)
          (Iso.hom (α_ M.X X.X (TensorBimod.X N' P)) ≫ M.X ◁ TensorBimod.actLeft N' P))
        WalkingParallelPair.one)
      (coequalizer.desc (AssociatorBimod.invAux M N' P) ⋯) :
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X (TensorBimod.X N' P) ⟶
      colimit
        (parallelPair (TensorBimod.actRight M N' ▷ P.X)
          (Iso.hom (α_ (TensorBimod.X M N') Y.X P.X) ≫ TensorBimod.X M N' ◁ P.actLeft)))
have a : AssociatorBimod.invAux M N' P
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
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)
W X Y Z : Mon.{v₁, u₁} C
M : Bimod W X
N N' : Bimod X Y
f : N ⟶ N'
P : Bimod Y Z
| M.X ◁
    (f.hom ▷ P.X ≫
      colimit.ι (parallelPair (N'.actRight ▷ P.X) (Iso.hom (α_ N'.X Y.X P.X) ≫ N'.X ◁ P.actLeft))
        WalkingParallelPair.one)
have a : M.X ◁ f.hom ▷ P.X ≫
  M.X ◁
    colimit.ι (parallelPair (N'.actRight ▷ P.X) (Iso.hom (α_ N'.X Y.X P.X) ≫ N'.X ◁ P.actLeft)) WalkingParallelPair.one
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
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)
W X Y Z : Mon.{v₁, u₁} C
M : Bimod W X
N N' : Bimod X Y
f : N ⟶ N'
P : Bimod Y Z
| (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) M.X (TensorBimod.X N' P))
      (M.X ◁
        colimit.ι (parallelPair (N'.actRight ▷ P.X) (Iso.hom (α_ N'.X Y.X P.X) ≫ N'.X ◁ P.actLeft))
          WalkingParallelPair.one)
      (CategoryStruct.comp (Y :=
        coequalizer (X :=
          tensorObj.{v₁, u₁} (𝒞 := inst) M.X
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N'.X Y.X) P.X))
          (M.X ◁ N'.actRight ▷ P.X) (M.X ◁ (Iso.hom (α_ N'.X Y.X P.X) ≫ N'.X ◁ P.actLeft)))
        (Iso.inv
          (PreservesCoequalizer.iso (tensorLeft M.X) (N'.actRight ▷ P.X)
            (Iso.hom (α_ N'.X Y.X P.X) ≫ N'.X ◁ P.actLeft)))
        (coequalizer.desc
          (Iso.inv (α_ M.X N'.X P.X) ≫
            coequalizer.π (M.actRight ▷ N'.X) (Iso.hom (α_ M.X X.X N'.X) ≫ M.X ◁ N'.actLeft) ▷ P.X ≫
              coequalizer.π (TensorBimod.actRight M N' ▷ P.X)
                (Iso.hom (α_ (TensorBimod.X M N') Y.X P.X) ≫ TensorBimod.X M N' ◁ P.actLeft))
          ⋯)) :
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X (tensorObj.{v₁, u₁} (𝒞 := inst) N'.X P.X) ⟶
      colimit
        (parallelPair (TensorBimod.actRight M N' ▷ P.X)
          (Iso.hom (α_ (TensorBimod.X M N') Y.X P.X) ≫ TensorBimod.X M N' ◁ P.actLeft)))
have a : Iso.inv (α_ M.X N'.X P.X) ≫
  coequalizer.π (M.actRight ▷ N'.X) (Iso.hom (α_ M.X X.X N'.X) ≫ M.X ◁ N'.actLeft) ▷ P.X ≫
    coequalizer.π (TensorBimod.actRight M N' ▷ P.X)
      (Iso.hom (α_ (TensorBimod.X M N') Y.X P.X) ≫ TensorBimod.X M N' ◁ P.actLeft)
Tactic: rw [id_tensor_π_preserves_coequalizer_inv_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
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon.{v₁, u₁} C
M N : Bimod X Y
P Q : Bimod Y Z
f : M ⟶ N
g : P ⟶ Q
⊢ Eq (α := M.tensorBimod P ⟶ N.tensorBimod Q) (M.whiskerLeft g ≫ whiskerRight f Q) (whiskerRight f P ≫ N.whiskerLeft g)
have h : Eq (α := (M.tensorBimod P).X ⟶ (N.tensorBimod Q).X) (Hom.hom (M.whiskerLeft g ≫ whiskerRight f Q))
  (Hom.hom (whiskerRight f P ≫ N.whiskerLeft g))
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
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon.{v₁, u₁} C
M N : Bimod X Y
P Q : Bimod Y Z
f : M ⟶ N
g : P ⟶ Q
⊢ Eq (α := (M.tensorBimod P).X ⟶ (N.tensorBimod Q).X) (Hom.hom (M.whiskerLeft g ≫ whiskerRight f Q))
    (Hom.hom (whiskerRight f P ≫ N.whiskerLeft g))
have h : Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) M.X P.X ⟶ (N.tensorBimod Q).X)
  (coequalizer.π (M.actRight ▷ P.X) (Iso.hom (α_ M.X Y.X P.X) ≫ M.X ◁ P.actLeft) ≫
    Hom.hom (M.whiskerLeft g ≫ whiskerRight f Q))
  (coequalizer.π (M.actRight ▷ P.X) (Iso.hom (α_ M.X Y.X P.X) ≫ M.X ◁ P.actLeft) ≫
    Hom.hom (whiskerRight f P ≫ N.whiskerLeft g))
Tactic: apply coequalizer.hom_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
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon.{v₁, u₁} C
M N : Bimod X Y
P Q : Bimod Y Z
f : M ⟶ N
g : P ⟶ Q
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) M.X P.X ⟶ (N.tensorBimod Q).X)
    (coequalizer.π (M.actRight ▷ P.X) (Iso.hom (α_ M.X Y.X P.X) ≫ M.X ◁ P.actLeft) ≫
      Hom.hom (M.whiskerLeft g ≫ whiskerRight f Q))
    (coequalizer.π (M.actRight ▷ P.X) (Iso.hom (α_ M.X Y.X P.X) ≫ M.X ◁ P.actLeft) ≫
      Hom.hom (whiskerRight f P ≫ N.whiskerLeft g))
have h : Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) M.X P.X ⟶ TensorBimod.X N Q)
  (CategoryStruct.comp (Y := coequalizer (M.actRight ▷ P.X) (Iso.hom (α_ M.X Y.X P.X) ≫ M.X ◁ P.actLeft))
    (coequalizer.π (M.actRight ▷ P.X) (Iso.hom (α_ M.X Y.X P.X) ≫ M.X ◁ P.actLeft))
    (CategoryStruct.comp (Y := TensorBimod.X M Q)
      (colimMap
        (parallelPairHom (M.actRight ▷ P.X) (Iso.hom (α_ M.X Y.X P.X) ≫ M.X ◁ P.actLeft) (M.actRight ▷ Q.X)
          (Iso.hom (α_ M.X Y.X Q.X) ≫ M.X ◁ Q.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ g.hom) (M.X ◁ g.hom) sorry
          sorry))
      (colimMap
        (parallelPairHom (M.actRight ▷ Q.X) (Iso.hom (α_ M.X Y.X Q.X) ≫ M.X ◁ Q.actLeft) (N.actRight ▷ Q.X)
          (Iso.hom (α_ N.X Y.X Q.X) ≫ N.X ◁ Q.actLeft) (f.hom ▷ Y.X ▷ Q.X) (f.hom ▷ Q.X) sorry sorry))))
  (CategoryStruct.comp (Y := coequalizer (M.actRight ▷ P.X) (Iso.hom (α_ M.X Y.X P.X) ≫ M.X ◁ P.actLeft))
    (coequalizer.π (M.actRight ▷ P.X) (Iso.hom (α_ M.X Y.X P.X) ≫ M.X ◁ P.actLeft))
    (CategoryStruct.comp (Y := TensorBimod.X N P)
      (colimMap
        (parallelPairHom (M.actRight ▷ P.X) (Iso.hom (α_ M.X Y.X P.X) ≫ M.X ◁ P.actLeft) (N.actRight ▷ P.X)
          (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft) (f.hom ▷ Y.X ▷ P.X) (f.hom ▷ P.X) sorry sorry))
      (colimMap
        (parallelPairHom (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft) (N.actRight ▷ Q.X)
          (Iso.hom (α_ N.X Y.X Q.X) ≫ N.X ◁ Q.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X ◁ g.hom) (N.X ◁ g.hom) sorry
          sorry))))
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
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon.{v₁, u₁} C
M N : Bimod X Y
P Q : Bimod Y Z
f : M ⟶ N
g : P ⟶ Q
| CategoryStruct.comp (Y := coequalizer (M.actRight ▷ P.X) (Iso.hom (α_ M.X Y.X P.X) ≫ M.X ◁ P.actLeft))
    (coequalizer.π (M.actRight ▷ P.X) (Iso.hom (α_ M.X Y.X P.X) ≫ M.X ◁ P.actLeft))
    (colimMap
      (parallelPairHom (M.actRight ▷ P.X) (Iso.hom (α_ M.X Y.X P.X) ≫ M.X ◁ P.actLeft) (M.actRight ▷ Q.X)
        (Iso.hom (α_ M.X Y.X Q.X) ≫ M.X ◁ Q.actLeft) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ◁ g.hom) (M.X ◁ g.hom) ⋯
        ⋯))
have a : M.X ◁ g.hom ≫
  colimit.ι (parallelPair (M.actRight ▷ Q.X) (Iso.hom (α_ M.X Y.X Q.X) ≫ M.X ◁ Q.actLeft)) WalkingParallelPair.one
Tactic: rw [ι_colimMap, parallelPairHom_app_one]