l3lab/ntp-mathlib huggingface dataset:
/-- Left action for the tensor product of two bimodules. -/
noncomputable def actLeft : R.X ⊗ X P Q ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorLeft R.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((𝟙 _ ⊗ (α_ _ _ _).hom) ≫ (α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 S.X ⊗ 𝟙 Q.X) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 Q.X))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_inv_naturality]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 4 5 => rw [← tensor_comp, middle_assoc, tensor_comp, comp_tensor_id]
          coherence)
        (by
          

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (tensorLeft R.X).map ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ≫ (α_ R.X P.X Q.X).inv ≫ (P.actLeft ⊗ 𝟙 Q.X) =
    ((𝟙 R.X ⊗ (α_ P.X S.X Q.X).hom) ≫
        (α_ R.X P.X (S.X ⊗ Q.X)).inv ≫ (P.actLeft ⊗ 𝟙 S.X ⊗ 𝟙 Q.X) ≫ (α_ P.X S.X Q.X).inv) ≫
      (α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)

Next tactic:
dsimp
/-- Left action for the tensor product of two bimodules. -/
noncomputable def actLeft : R.X ⊗ X P Q ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorLeft R.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((𝟙 _ ⊗ (α_ _ _ _).hom) ≫ (α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 S.X ⊗ 𝟙 Q.X) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 Q.X))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_inv_naturality]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 4 5 => rw [← tensor_comp, middle_assoc, tensor_comp, comp_tensor_id]
          coherence)
        (by
          dsimp
          

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (𝟙 R.X ⊗ (α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ≫ (α_ R.X P.X Q.X).inv ≫ (P.actLeft ⊗ 𝟙 Q.X) =
    ((𝟙 R.X ⊗ (α_ P.X S.X Q.X).hom) ≫
        (α_ R.X P.X (S.X ⊗ Q.X)).inv ≫ (P.actLeft ⊗ 𝟙 S.X ⊗ 𝟙 Q.X) ≫ (α_ P.X S.X Q.X).inv) ≫
      (α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)

Next tactic:
slice_lhs 1 1 => rw [id_tensor_comp]
/-- Left action for the tensor product of two bimodules. -/
noncomputable def actLeft : R.X ⊗ X P Q ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorLeft R.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((𝟙 _ ⊗ (α_ _ _ _).hom) ≫ (α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 S.X ⊗ 𝟙 Q.X) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 Q.X))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_inv_naturality]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 4 5 => rw [← tensor_comp, middle_assoc, tensor_comp, comp_tensor_id]
          coherence)
        (by
          dsimp
          slice_lhs 1 1 => 

Proof state:
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| 𝟙 R.X ⊗ (α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| (α_ R.X P.X Q.X).inv
case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| P.actLeft ⊗ 𝟙 Q.X

Next tactic:
rw [id_tensor_comp]
/-- Left action for the tensor product of two bimodules. -/
noncomputable def actLeft : R.X ⊗ X P Q ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorLeft R.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((𝟙 _ ⊗ (α_ _ _ _).hom) ≫ (α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 S.X ⊗ 𝟙 Q.X) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 Q.X))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_inv_naturality]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 4 5 => rw [← tensor_comp, middle_assoc, tensor_comp, comp_tensor_id]
          coherence)
        (by
          dsimp
          slice_lhs 1 1 => rw [id_tensor_comp]
          

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (((𝟙 R.X ⊗ (α_ P.X S.X Q.X).hom) ≫ (𝟙 R.X ⊗ 𝟙 P.X ⊗ Q.actLeft)) ≫ (α_ R.X P.X Q.X).inv) ≫ (P.actLeft ⊗ 𝟙 Q.X) =
    ((𝟙 R.X ⊗ (α_ P.X S.X Q.X).hom) ≫
        (α_ R.X P.X (S.X ⊗ Q.X)).inv ≫ (P.actLeft ⊗ 𝟙 S.X ⊗ 𝟙 Q.X) ≫ (α_ P.X S.X Q.X).inv) ≫
      (α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)

Next tactic:
slice_lhs 2 3 => rw [associator_inv_naturality]
/-- Left action for the tensor product of two bimodules. -/
noncomputable def actLeft : R.X ⊗ X P Q ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorLeft R.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((𝟙 _ ⊗ (α_ _ _ _).hom) ≫ (α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 S.X ⊗ 𝟙 Q.X) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 Q.X))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_inv_naturality]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 4 5 => rw [← tensor_comp, middle_assoc, tensor_comp, comp_tensor_id]
          coherence)
        (by
          dsimp
          slice_lhs 1 1 => rw [id_tensor_comp]
          slice_lhs 2 3 => 

Proof state:
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| (𝟙 R.X ⊗ 𝟙 P.X ⊗ Q.actLeft) ≫ (α_ R.X P.X Q.X).inv
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| P.actLeft ⊗ 𝟙 Q.X
case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| 𝟙 R.X ⊗ (α_ P.X S.X Q.X).hom

Next tactic:
rw [associator_inv_naturality]
/-- Left action for the tensor product of two bimodules. -/
noncomputable def actLeft : R.X ⊗ X P Q ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorLeft R.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((𝟙 _ ⊗ (α_ _ _ _).hom) ≫ (α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 S.X ⊗ 𝟙 Q.X) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 Q.X))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_inv_naturality]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 4 5 => rw [← tensor_comp, middle_assoc, tensor_comp, comp_tensor_id]
          coherence)
        (by
          dsimp
          slice_lhs 1 1 => rw [id_tensor_comp]
          slice_lhs 2 3 => rw [associator_inv_naturality]
          

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (𝟙 R.X ⊗ (α_ P.X S.X Q.X).hom) ≫
      ((α_ R.X P.X (S.X ⊗ Q.X)).inv ≫ ((𝟙 R.X ⊗ 𝟙 P.X) ⊗ Q.actLeft)) ≫ (P.actLeft ⊗ 𝟙 Q.X) =
    ((𝟙 R.X ⊗ (α_ P.X S.X Q.X).hom) ≫
        (α_ R.X P.X (S.X ⊗ Q.X)).inv ≫ (P.actLeft ⊗ 𝟙 S.X ⊗ 𝟙 Q.X) ≫ (α_ P.X S.X Q.X).inv) ≫
      (α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)

Next tactic:
slice_lhs 3 4 => rw [tensor_id, id_tensor_comp_tensor_id]
/-- Left action for the tensor product of two bimodules. -/
noncomputable def actLeft : R.X ⊗ X P Q ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorLeft R.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((𝟙 _ ⊗ (α_ _ _ _).hom) ≫ (α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 S.X ⊗ 𝟙 Q.X) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 Q.X))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_inv_naturality]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 4 5 => rw [← tensor_comp, middle_assoc, tensor_comp, comp_tensor_id]
          coherence)
        (by
          dsimp
          slice_lhs 1 1 => rw [id_tensor_comp]
          slice_lhs 2 3 => rw [associator_inv_naturality]
          slice_lhs 3 4 => 

Proof state:
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| ((𝟙 R.X ⊗ 𝟙 P.X) ⊗ Q.actLeft) ≫ (P.actLeft ⊗ 𝟙 Q.X)
case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| 𝟙 R.X ⊗ (α_ P.X S.X Q.X).hom
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| (α_ R.X P.X (S.X ⊗ Q.X)).inv

Next tactic:
rw [tensor_id, id_tensor_comp_tensor_id]
/-- Left action for the tensor product of two bimodules. -/
noncomputable def actLeft : R.X ⊗ X P Q ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorLeft R.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((𝟙 _ ⊗ (α_ _ _ _).hom) ≫ (α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 S.X ⊗ 𝟙 Q.X) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 Q.X))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_inv_naturality]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 4 5 => rw [← tensor_comp, middle_assoc, tensor_comp, comp_tensor_id]
          coherence)
        (by
          dsimp
          slice_lhs 1 1 => rw [id_tensor_comp]
          slice_lhs 2 3 => rw [associator_inv_naturality]
          slice_lhs 3 4 => rw [tensor_id, id_tensor_comp_tensor_id]
          

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (𝟙 R.X ⊗ (α_ P.X S.X Q.X).hom) ≫ (α_ R.X P.X (S.X ⊗ Q.X)).inv ≫ (P.actLeft ⊗ Q.actLeft) =
    ((𝟙 R.X ⊗ (α_ P.X S.X Q.X).hom) ≫
        (α_ R.X P.X (S.X ⊗ Q.X)).inv ≫ (P.actLeft ⊗ 𝟙 S.X ⊗ 𝟙 Q.X) ≫ (α_ P.X S.X Q.X).inv) ≫
      (α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)

Next tactic:
slice_rhs 4 6 => rw [Iso.inv_hom_id_assoc]
/-- Left action for the tensor product of two bimodules. -/
noncomputable def actLeft : R.X ⊗ X P Q ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorLeft R.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((𝟙 _ ⊗ (α_ _ _ _).hom) ≫ (α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 S.X ⊗ 𝟙 Q.X) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 Q.X))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_inv_naturality]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 4 5 => rw [← tensor_comp, middle_assoc, tensor_comp, comp_tensor_id]
          coherence)
        (by
          dsimp
          slice_lhs 1 1 => rw [id_tensor_comp]
          slice_lhs 2 3 => rw [associator_inv_naturality]
          slice_lhs 3 4 => rw [tensor_id, id_tensor_comp_tensor_id]
          slice_rhs 4 6 => 

Proof state:
case a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| (α_ P.X S.X Q.X).inv ≫ (α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)
case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| 𝟙 R.X ⊗ (α_ P.X S.X Q.X).hom
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| (α_ R.X P.X (S.X ⊗ Q.X)).inv
case a.a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| P.actLeft ⊗ 𝟙 S.X ⊗ 𝟙 Q.X

Next tactic:
rw [Iso.inv_hom_id_assoc]
/-- Left action for the tensor product of two bimodules. -/
noncomputable def actLeft : R.X ⊗ X P Q ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorLeft R.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((𝟙 _ ⊗ (α_ _ _ _).hom) ≫ (α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 S.X ⊗ 𝟙 Q.X) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 Q.X))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_inv_naturality]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 4 5 => rw [← tensor_comp, middle_assoc, tensor_comp, comp_tensor_id]
          coherence)
        (by
          dsimp
          slice_lhs 1 1 => rw [id_tensor_comp]
          slice_lhs 2 3 => rw [associator_inv_naturality]
          slice_lhs 3 4 => rw [tensor_id, id_tensor_comp_tensor_id]
          slice_rhs 4 6 => rw [Iso.inv_hom_id_assoc]
          

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (𝟙 R.X ⊗ (α_ P.X S.X Q.X).hom) ≫ (α_ R.X P.X (S.X ⊗ Q.X)).inv ≫ (P.actLeft ⊗ Q.actLeft) =
    (𝟙 R.X ⊗ (α_ P.X S.X Q.X).hom) ≫ (α_ R.X P.X (S.X ⊗ Q.X)).inv ≫ (P.actLeft ⊗ 𝟙 S.X ⊗ 𝟙 Q.X) ≫ (𝟙 P.X ⊗ Q.actLeft)

Next tactic:
slice_rhs 3 4 => rw [tensor_id, tensor_id_comp_id_tensor]
/-- Left action for the tensor product of two bimodules. -/
noncomputable def actLeft : R.X ⊗ X P Q ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorLeft R.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((𝟙 _ ⊗ (α_ _ _ _).hom) ≫ (α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 S.X ⊗ 𝟙 Q.X) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 Q.X))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_inv_naturality]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 4 5 => rw [← tensor_comp, middle_assoc, tensor_comp, comp_tensor_id]
          coherence)
        (by
          dsimp
          slice_lhs 1 1 => rw [id_tensor_comp]
          slice_lhs 2 3 => rw [associator_inv_naturality]
          slice_lhs 3 4 => rw [tensor_id, id_tensor_comp_tensor_id]
          slice_rhs 4 6 => rw [Iso.inv_hom_id_assoc]
          slice_rhs 3 4 => 

Proof state:
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| (P.actLeft ⊗ 𝟙 S.X ⊗ 𝟙 Q.X) ≫ (𝟙 P.X ⊗ Q.actLeft)
case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| 𝟙 R.X ⊗ (α_ P.X S.X Q.X).hom
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| (α_ R.X P.X (S.X ⊗ Q.X)).inv

Next tactic:
rw [tensor_id, tensor_id_comp_id_tensor]
theorem id_tensor_π_actLeft :
    (𝟙 R.X ⊗ coequalizer.π _ _) ≫ actLeft P Q =
      (α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 Q.X) ≫ coequalizer.π _ _ := by
  

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (𝟙 R.X ⊗ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫ actLeft P Q =
    (α_ R.X P.X Q.X).inv ≫
      (P.actLeft ⊗ 𝟙 Q.X) ≫ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))

Next tactic:
erw [map_π_preserves_coequalizer_inv_colimMap (tensorLeft _)]
theorem id_tensor_π_actLeft :
    (𝟙 R.X ⊗ coequalizer.π _ _) ≫ actLeft P Q =
      (α_ _ _ _).inv ≫ (P.actLeft ⊗ 𝟙 Q.X) ≫ coequalizer.π _ _ := by
  erw [map_π_preserves_coequalizer_inv_colimMap (tensorLeft _)]
  

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ ((α_ R.X P.X Q.X).inv ≫ (P.actLeft ⊗ 𝟙 Q.X)) ≫
      coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) =
    (α_ R.X P.X Q.X).inv ≫
      (P.actLeft ⊗ 𝟙 Q.X) ≫ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))

Next tactic:
simp only [Category.assoc]
theorem one_act_left' : (R.one ⊗ 𝟙 _) ≫ actLeft P Q = (λ_ _).hom := by
  

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (R.one ⊗ 𝟙 (X P Q)) ≫ actLeft P Q = (λ_ (X P Q)).hom

Next tactic:
refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
theorem one_act_left' : (R.one ⊗ 𝟙 _) ≫ actLeft P Q = (λ_ _).hom := by
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (tensorLeft (𝟙_ C)).map (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
      (R.one ⊗ 𝟙 (X P Q)) ≫ actLeft P Q =
    (tensorLeft (𝟙_ C)).map (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
      (λ_ (X P Q)).hom

Next tactic:
dsimp [X]
theorem one_act_left' : (R.one ⊗ 𝟙 _) ≫ actLeft P Q = (λ_ _).hom := by
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  dsimp [X]
  -- porting note: had to replace `rw` by `erw`
  

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (𝟙 (𝟙_ C) ⊗ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
      (R.one ⊗ 𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)))) ≫ actLeft P Q =
    (𝟙 (𝟙_ C) ⊗ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
      (λ_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)))).hom

Next tactic:
slice_lhs 1 2 => erw [id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]
theorem one_act_left' : (R.one ⊗ 𝟙 _) ≫ actLeft P Q = (λ_ _).hom := by
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  dsimp [X]
  -- porting note: had to replace `rw` by `erw`
  slice_lhs 1 2 => 

Proof state:
case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| (𝟙 (𝟙_ C) ⊗ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
    (R.one ⊗ 𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))))
case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| actLeft P Q

Next tactic:
erw [id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]
theorem one_act_left' : (R.one ⊗ 𝟙 _) ≫ actLeft P Q = (λ_ _).hom := by
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  dsimp [X]
  -- porting note: had to replace `rw` by `erw`
  slice_lhs 1 2 => erw [id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]
  

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ ((R.one ⊗ 𝟙 (P.X ⊗ Q.X)) ≫
        (𝟙 R.X ⊗ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)))) ≫
      actLeft P Q =
    (𝟙 (𝟙_ C) ⊗ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
      (λ_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)))).hom

Next tactic:
slice_lhs 2 3 => rw [id_tensor_π_actLeft]
theorem one_act_left' : (R.one ⊗ 𝟙 _) ≫ actLeft P Q = (λ_ _).hom := by
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  dsimp [X]
  -- porting note: had to replace `rw` by `erw`
  slice_lhs 1 2 => erw [id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]
  slice_lhs 2 3 => 

Proof state:
case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| (𝟙 R.X ⊗ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫ actLeft P Q
case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| R.one ⊗ 𝟙 (P.X ⊗ Q.X)

Next tactic:
rw [id_tensor_π_actLeft]
theorem one_act_left' : (R.one ⊗ 𝟙 _) ≫ actLeft P Q = (λ_ _).hom := by
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  dsimp [X]
  -- porting note: had to replace `rw` by `erw`
  slice_lhs 1 2 => erw [id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]
  slice_lhs 2 3 => rw [id_tensor_π_actLeft]
  

Proof state:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (R.one ⊗ 𝟙 (P.X ⊗ Q.X)) ≫
      (α_ R.X P.X Q.X).inv ≫
        (P.actLeft ⊗ 𝟙 Q.X) ≫ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) =
    (𝟙 (𝟙_ C) ⊗ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
      (λ_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)))).hom

Next tactic:
slice_lhs 1 2 => rw [← MonoidalCategory.tensor_id, associator_inv_naturality]