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)
X Y : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
      TensorBimod.X N
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod Y Y })
    (f.hom ▷ Y.X ≫
        colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one :
      tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
        colimit (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)))
    ((M.actRight ≫ f.hom) ≫ RightUnitorBimod.inv N :
      tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
        TensorBimod.X N
          { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
            right_assoc := ⋯, middle_assoc := ⋯ : Bimod Y Y })
have h : Eq (α :=
  tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
    TensorBimod.X N
      { X := Y.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry, right_assoc := sorry,
        middle_assoc := sorry : Bimod Y Y })
  (f.hom ▷ Y.X ≫
    colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
  ((M.actRight ≫ f.hom) ≫
    Iso.inv (ρ_ N.X) ≫ N.X ◁ η ≫ coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))
Tactic: dsimp [RightUnitorBimod.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
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 : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
| M.actRight ≫ f.hom
have a : f.hom ▷ Y.X ≫ N.actRight
Tactic: rw [Hom.right_act_hom]
-- 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 : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
      TensorBimod.X N
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod Y Y })
    (f.hom ▷ Y.X ≫
      colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
    ((M.actRight ≫ f.hom) ≫
      Iso.inv (ρ_ N.X) ≫ N.X ◁ η ≫ coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))
have h : Eq (α :=
  tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
    TensorBimod.X N
      { X := Y.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry, right_assoc := sorry,
        middle_assoc := sorry : Bimod Y Y })
  (f.hom ▷ Y.X ≫
    colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
  ((((f.hom ▷ Y.X ≫ N.actRight) ≫ Iso.inv (ρ_ N.X)) ≫ N.X ◁ η) ≫
    coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))
Tactic: slice_rhs 1 2 => rw [Hom.right_act_hom]
-- 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 : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
| N.actRight ≫ Iso.inv (ρ_ N.X)
have a : Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫ N.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
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 : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
      TensorBimod.X N
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod Y Y })
    (f.hom ▷ Y.X ≫
      colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
    ((((f.hom ▷ Y.X ≫ N.actRight) ≫ Iso.inv (ρ_ N.X)) ≫ N.X ◁ η) ≫
      coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))
have h : Eq (α :=
  tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
    TensorBimod.X N
      { X := Y.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry, right_assoc := sorry,
        middle_assoc := sorry : Bimod Y Y })
  (f.hom ▷ Y.X ≫
    colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
  (f.hom ▷ Y.X ≫
    ((Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫ N.actRight ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst)) ≫
        N.X ◁ η) ≫
      coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))
Tactic: slice_rhs 2 3 => 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
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 : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
| N.actRight ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst) ≫ N.X ◁ η
have a : tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X ◁ η ≫ N.actRight ▷ Y.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
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 : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
      TensorBimod.X N
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod Y Y })
    (f.hom ▷ Y.X ≫
      colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
    (f.hom ▷ Y.X ≫
      ((Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫ N.actRight ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst)) ≫
          N.X ◁ η) ≫
        coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))
have h : Eq (α :=
  tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
    TensorBimod.X N
      { X := Y.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry, right_assoc := sorry,
        middle_assoc := sorry : Bimod Y Y })
  (f.hom ▷ Y.X ≫
    colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
  (f.hom ▷ Y.X ≫
    Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫
      (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X ◁ η ≫ N.actRight ▷ Y.X) ≫
        coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))
Tactic: slice_rhs 3 4 => 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)
X Y : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
| N.actRight ▷ Y.X ≫ coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)
have a : (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ) ≫ coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.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
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 : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
      TensorBimod.X N
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod Y Y })
    (f.hom ▷ Y.X ≫
      colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
    (f.hom ▷ Y.X ≫
      Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫
        (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X ◁ η ≫ N.actRight ▷ Y.X) ≫
          coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))
have h : Eq (α :=
  tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
    TensorBimod.X N
      { X := Y.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry, right_assoc := sorry,
        middle_assoc := sorry : Bimod Y Y })
  (f.hom ▷ Y.X ≫
    colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
  (f.hom ▷ Y.X ≫
    Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫
      tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X ◁ η ≫
        (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ) ≫ coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))
Tactic: slice_rhs 4 5 => 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
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 : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
| tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X ◁ η ≫ Iso.hom (α_ N.X Y.X Y.X)
have a : Iso.hom (α_ N.X Y.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ N.X ◁ Y.X ◁ η
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)
X Y : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
      TensorBimod.X N
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod Y Y })
    (f.hom ▷ Y.X ≫
      colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
    (f.hom ▷ Y.X ≫
      Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫
        tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X ◁ η ≫
          (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ) ≫ coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))
have h : Eq (α :=
  tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
    TensorBimod.X N
      { X := Y.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry, right_assoc := sorry,
        middle_assoc := sorry : Bimod Y Y })
  (f.hom ▷ Y.X ≫
    colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
  (f.hom ▷ Y.X ≫
    Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫
      ((Iso.hom (α_ N.X Y.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ N.X ◁ Y.X ◁ η) ≫ N.X ◁ μ) ≫
        coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))
Tactic: slice_rhs 3 4 => 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)
X Y : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
| N.X ◁ Y.X ◁ η ≫ N.X ◁ μ
have a : N.X ◁ Iso.hom (ρ_ Y.X)
Tactic: rw [← whiskerLeft_comp, MonObj.mul_one]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
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 : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
      TensorBimod.X N
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod Y Y })
    (f.hom ▷ Y.X ≫
      colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
    (f.hom ▷ Y.X ≫
      Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫
        ((Iso.hom (α_ N.X Y.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ N.X ◁ Y.X ◁ η) ≫ N.X ◁ μ) ≫
          coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))
have h : Eq (α :=
  tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
    TensorBimod.X N
      { X := Y.X, actLeft := μ, one_actLeft := sorry, left_assoc := sorry, actRight := μ, actRight_one := sorry, right_assoc := sorry,
        middle_assoc := sorry : Bimod Y Y })
  (f.hom ▷ Y.X ≫
    colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
  (f.hom ▷ Y.X ≫
    Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫
      Iso.hom (α_ N.X Y.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫
        N.X ◁ Iso.hom (ρ_ Y.X) ≫ coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))
Tactic: slice_rhs 4 5 => rw [← whiskerLeft_comp, MonObj.mul_one]
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
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
⊢ whiskerRight f (N.tensorBimod P) =
    Iso.inv (M.associatorBimod N P) ≫ whiskerRight (whiskerRight f N) P ≫ Iso.hom (M'.associatorBimod N P)
have dsimp : whiskerRight f
    { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := sorry, left_assoc := sorry,
      actRight := TensorBimod.actRight N P, actRight_one := sorry, right_assoc := sorry, middle_assoc := sorry } =
  Iso.inv
      (isoOfIso
        { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := sorry, inv_hom_id := sorry :
          TensorBimod.X
              { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := sorry, left_assoc := sorry,
                actRight := TensorBimod.actRight M N, actRight_one := sorry, right_assoc := sorry, middle_assoc := sorry :
                Bimod W Y }
              P ≅
            TensorBimod.X M
              { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := sorry, left_assoc := sorry,
                actRight := TensorBimod.actRight N P, actRight_one := sorry, right_assoc := sorry, middle_assoc := sorry :
                Bimod X Z } }
        sorry sorry) ≫
    whiskerRight (whiskerRight f N) P ≫
      Iso.hom
        (isoOfIso
          { hom := AssociatorBimod.hom M' N P, inv := AssociatorBimod.inv M' N P, hom_inv_id := sorry, inv_hom_id := sorry :
            TensorBimod.X
                { X := TensorBimod.X M' N, actLeft := TensorBimod.actLeft M' N, one_actLeft := sorry, left_assoc := sorry,
                  actRight := TensorBimod.actRight M' N, actRight_one := sorry, right_assoc := sorry, middle_assoc := sorry :
                  Bimod W Y }
                P ≅
              TensorBimod.X M'
                { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := sorry, left_assoc := sorry,
                  actRight := TensorBimod.actRight N P, actRight_one := sorry, right_assoc := sorry, middle_assoc := sorry :
                  Bimod X Z } }
          sorry sorry)
Tactic: dsimp [tensorHom, tensorBimod, associatorBimod]
-- 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 (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))
    (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))
    (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) (f.hom ▷ X.X ▷ TensorBimod.X N P)
        (f.hom ▷ TensorBimod.X N P) ⋯ ⋯))
have rw : f.hom ▷ TensorBimod.X N P ≫
  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 M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
| (CategoryStruct.comp (Y :=
      Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft 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))
      (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))
          ⋯)) :
    Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft M.X) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X P.X) ⟶
      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))
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)
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))
      (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_ (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)
          (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)
          (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) ⋯ ⋯) ▷
              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) (f.hom ▷ X.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷
            P.X)
          ⋯ ⋯)) :
    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 (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))
have a : 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) sorry sorry) ▷
    P.X ≫
  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
Tactic: rw [ι_colimMap, parallelPairHom_app_one]