l3lab/ntp-mathlib huggingface dataset:
theorem left_assoc' :
    (R.mul ⊗ 𝟙 _) ≫ actLeft P Q = (α_ R.X R.X _).hom ≫ (𝟙 R.X ⊗ actLeft P Q) ≫ actLeft P Q := by
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  dsimp [X]
  -- porting note: had to replace some `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]
  slice_lhs 1 2 => rw [← MonoidalCategory.tensor_id, associator_inv_naturality]
  slice_lhs 2 3 => rw [← comp_tensor_id, left_assoc, comp_tensor_id, comp_tensor_id]
  slice_rhs 1 2 => erw [← MonoidalCategory.tensor_id, associator_naturality]
  slice_rhs 2 3 => rw [← id_tensor_comp, id_tensor_π_actLeft, id_tensor_comp, 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)
⊢ (α_ (R.X ⊗ R.X) P.X Q.X).inv ≫
      (((α_ R.X R.X P.X).hom ⊗ 𝟙 Q.X) ≫ ((𝟙 R.X ⊗ P.actLeft) ⊗ 𝟙 Q.X) ≫ (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 R.X (P.X ⊗ Q.X)).hom ≫
      ((𝟙 R.X ⊗ (α_ R.X P.X Q.X).inv) ≫
          (𝟙 R.X ⊗ P.actLeft ⊗ 𝟙 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

Next tactic:
slice_rhs 4 5 => rw [id_tensor_π_actLeft]
theorem left_assoc' :
    (R.mul ⊗ 𝟙 _) ≫ actLeft P Q = (α_ R.X R.X _).hom ≫ (𝟙 R.X ⊗ actLeft P Q) ≫ actLeft P Q := by
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  dsimp [X]
  -- porting note: had to replace some `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]
  slice_lhs 1 2 => rw [← MonoidalCategory.tensor_id, associator_inv_naturality]
  slice_lhs 2 3 => rw [← comp_tensor_id, left_assoc, comp_tensor_id, comp_tensor_id]
  slice_rhs 1 2 => erw [← MonoidalCategory.tensor_id, associator_naturality]
  slice_rhs 2 3 => rw [← id_tensor_comp, id_tensor_π_actLeft, id_tensor_comp, id_tensor_comp]
  slice_rhs 4 5 => 

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)
| (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)
| (α_ R.X R.X (P.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)
| 𝟙 R.X ⊗ (α_ R.X P.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)
| 𝟙 R.X ⊗ P.actLeft ⊗ 𝟙 Q.X

Next tactic:
rw [id_tensor_π_actLeft]
theorem left_assoc' :
    (R.mul ⊗ 𝟙 _) ≫ actLeft P Q = (α_ R.X R.X _).hom ≫ (𝟙 R.X ⊗ actLeft P Q) ≫ actLeft P Q := by
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  dsimp [X]
  -- porting note: had to replace some `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]
  slice_lhs 1 2 => rw [← MonoidalCategory.tensor_id, associator_inv_naturality]
  slice_lhs 2 3 => rw [← comp_tensor_id, left_assoc, comp_tensor_id, comp_tensor_id]
  slice_rhs 1 2 => erw [← MonoidalCategory.tensor_id, associator_naturality]
  slice_rhs 2 3 => rw [← id_tensor_comp, id_tensor_π_actLeft, id_tensor_comp, id_tensor_comp]
  slice_rhs 4 5 => 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)
⊢ (α_ (R.X ⊗ R.X) P.X Q.X).inv ≫
      (((α_ R.X R.X P.X).hom ⊗ 𝟙 Q.X) ≫ ((𝟙 R.X ⊗ P.actLeft) ⊗ 𝟙 Q.X) ≫ (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 R.X (P.X ⊗ Q.X)).hom ≫
      (𝟙 R.X ⊗ (α_ R.X P.X Q.X).inv) ≫
        (𝟙 R.X ⊗ P.actLeft ⊗ 𝟙 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))

Next tactic:
slice_rhs 3 4 => rw [associator_inv_naturality]
theorem left_assoc' :
    (R.mul ⊗ 𝟙 _) ≫ actLeft P Q = (α_ R.X R.X _).hom ≫ (𝟙 R.X ⊗ actLeft P Q) ≫ actLeft P Q := by
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  dsimp [X]
  -- porting note: had to replace some `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]
  slice_lhs 1 2 => rw [← MonoidalCategory.tensor_id, associator_inv_naturality]
  slice_lhs 2 3 => rw [← comp_tensor_id, left_assoc, comp_tensor_id, comp_tensor_id]
  slice_rhs 1 2 => erw [← MonoidalCategory.tensor_id, associator_naturality]
  slice_rhs 2 3 => rw [← id_tensor_comp, id_tensor_π_actLeft, id_tensor_comp, id_tensor_comp]
  slice_rhs 4 5 => rw [id_tensor_π_actLeft]
  slice_rhs 3 4 => 

Proof state:
case a.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)
| (𝟙 R.X ⊗ P.actLeft ⊗ 𝟙 Q.X) ≫ (α_ R.X P.X Q.X).inv
case a.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 Q.X
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)
| coequalizer.π (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)
| (α_ R.X R.X (P.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)
| 𝟙 R.X ⊗ (α_ R.X P.X Q.X).inv

Next tactic:
rw [associator_inv_naturality]
theorem left_assoc' :
    (R.mul ⊗ 𝟙 _) ≫ actLeft P Q = (α_ R.X R.X _).hom ≫ (𝟙 R.X ⊗ actLeft P Q) ≫ actLeft P Q := by
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  dsimp [X]
  -- porting note: had to replace some `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]
  slice_lhs 1 2 => rw [← MonoidalCategory.tensor_id, associator_inv_naturality]
  slice_lhs 2 3 => rw [← comp_tensor_id, left_assoc, comp_tensor_id, comp_tensor_id]
  slice_rhs 1 2 => erw [← MonoidalCategory.tensor_id, associator_naturality]
  slice_rhs 2 3 => rw [← id_tensor_comp, id_tensor_π_actLeft, id_tensor_comp, id_tensor_comp]
  slice_rhs 4 5 => rw [id_tensor_π_actLeft]
  slice_rhs 3 4 => 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)
⊢ (α_ (R.X ⊗ R.X) P.X Q.X).inv ≫
      (((α_ R.X R.X P.X).hom ⊗ 𝟙 Q.X) ≫ ((𝟙 R.X ⊗ P.actLeft) ⊗ 𝟙 Q.X) ≫ (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 R.X (P.X ⊗ Q.X)).hom ≫
      (𝟙 R.X ⊗ (α_ R.X P.X Q.X).inv) ≫
        (((α_ R.X (R.X ⊗ P.X) Q.X).inv ≫ ((𝟙 R.X ⊗ P.actLeft) ⊗ 𝟙 Q.X)) ≫ (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:
coherence
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (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} (tensorRight X)
⊢ (tensorRight T.X).map (P.actRight ⊗ 𝟙 Q.X) ≫ (α_ P.X Q.X T.X).hom ≫ (𝟙 P.X ⊗ Q.actRight) =
    ((α_ (P.X ⊗ S.X) Q.X T.X).hom ≫
        (α_ P.X S.X (Q.X ⊗ T.X)).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ P.X S.X Q.X).inv) ≫
      (P.actRight ⊗ 𝟙 Q.X)

Next tactic:
dsimp
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (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} (tensorRight X)
⊢ ((P.actRight ⊗ 𝟙 Q.X) ⊗ 𝟙 T.X) ≫ (α_ P.X Q.X T.X).hom ≫ (𝟙 P.X ⊗ Q.actRight) =
    ((α_ (P.X ⊗ S.X) Q.X T.X).hom ≫
        (α_ P.X S.X (Q.X ⊗ T.X)).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ P.X S.X Q.X).inv) ≫
      (P.actRight ⊗ 𝟙 Q.X)

Next tactic:
slice_lhs 1 2 => rw [associator_naturality]
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          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} (tensorRight X)
| ((P.actRight ⊗ 𝟙 Q.X) ⊗ 𝟙 T.X) ≫ (α_ P.X Q.X T.X).hom
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} (tensorRight X)
| 𝟙 P.X ⊗ Q.actRight

Next tactic:
rw [associator_naturality]
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_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} (tensorRight X)
⊢ ((α_ (P.X ⊗ S.X) Q.X T.X).hom ≫ (P.actRight ⊗ 𝟙 Q.X ⊗ 𝟙 T.X)) ≫ (𝟙 P.X ⊗ Q.actRight) =
    ((α_ (P.X ⊗ S.X) Q.X T.X).hom ≫
        (α_ P.X S.X (Q.X ⊗ T.X)).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ P.X S.X Q.X).inv) ≫
      (P.actRight ⊗ 𝟙 Q.X)

Next tactic:
slice_lhs 2 3 => rw [tensor_id, tensor_id_comp_id_tensor]
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_naturality]
          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} (tensorRight X)
| (P.actRight ⊗ 𝟙 Q.X ⊗ 𝟙 T.X) ≫ (𝟙 P.X ⊗ Q.actRight)
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} (tensorRight X)
| (α_ (P.X ⊗ S.X) Q.X T.X).hom

Next tactic:
rw [tensor_id, tensor_id_comp_id_tensor]
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_naturality]
          slice_lhs 2 3 => rw [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} (tensorRight X)
⊢ (α_ (P.X ⊗ S.X) Q.X T.X).hom ≫ (P.actRight ⊗ Q.actRight) =
    ((α_ (P.X ⊗ S.X) Q.X T.X).hom ≫
        (α_ P.X S.X (Q.X ⊗ T.X)).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ P.X S.X Q.X).inv) ≫
      (P.actRight ⊗ 𝟙 Q.X)

Next tactic:
slice_rhs 3 4 => rw [associator_inv_naturality]
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_naturality]
          slice_lhs 2 3 => rw [tensor_id, tensor_id_comp_id_tensor]
          slice_rhs 3 4 => 

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} (tensorRight X)
| (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ 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} (tensorRight X)
| P.actRight ⊗ 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} (tensorRight X)
| (α_ (P.X ⊗ S.X) Q.X T.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} (tensorRight X)
| (α_ P.X S.X (Q.X ⊗ T.X)).hom

Next tactic:
rw [associator_inv_naturality]
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_naturality]
          slice_lhs 2 3 => rw [tensor_id, tensor_id_comp_id_tensor]
          slice_rhs 3 4 => 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} (tensorRight X)
⊢ (α_ (P.X ⊗ S.X) Q.X T.X).hom ≫ (P.actRight ⊗ Q.actRight) =
    (α_ (P.X ⊗ S.X) Q.X T.X).hom ≫
      (α_ P.X S.X (Q.X ⊗ T.X)).hom ≫
        ((α_ P.X S.X (Q.X ⊗ T.X)).inv ≫ ((𝟙 P.X ⊗ 𝟙 S.X) ⊗ Q.actRight)) ≫ (P.actRight ⊗ 𝟙 Q.X)

Next tactic:
slice_rhs 2 4 => rw [Iso.hom_inv_id_assoc]
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_naturality]
          slice_lhs 2 3 => rw [tensor_id, tensor_id_comp_id_tensor]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 2 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} (tensorRight X)
| (α_ P.X S.X (Q.X ⊗ T.X)).hom ≫ (α_ P.X S.X (Q.X ⊗ T.X)).inv ≫ ((𝟙 P.X ⊗ 𝟙 S.X) ⊗ Q.actRight)
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} (tensorRight X)
| P.actRight ⊗ 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} (tensorRight X)
| (α_ (P.X ⊗ S.X) Q.X T.X).hom

Next tactic:
rw [Iso.hom_inv_id_assoc]
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_naturality]
          slice_lhs 2 3 => rw [tensor_id, tensor_id_comp_id_tensor]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 2 4 => rw [Iso.hom_inv_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} (tensorRight X)
⊢ (α_ (P.X ⊗ S.X) Q.X T.X).hom ≫ (P.actRight ⊗ Q.actRight) =
    (α_ (P.X ⊗ S.X) Q.X T.X).hom ≫ ((𝟙 P.X ⊗ 𝟙 S.X) ⊗ Q.actRight) ≫ (P.actRight ⊗ 𝟙 Q.X)

Next tactic:
slice_rhs 2 3 => rw [tensor_id, id_tensor_comp_tensor_id]
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_naturality]
          slice_lhs 2 3 => rw [tensor_id, tensor_id_comp_id_tensor]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 2 4 => rw [Iso.hom_inv_id_assoc]
          slice_rhs 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} (tensorRight X)
| ((𝟙 P.X ⊗ 𝟙 S.X) ⊗ Q.actRight) ≫ (P.actRight ⊗ 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} (tensorRight X)
| (α_ (P.X ⊗ S.X) Q.X T.X).hom

Next tactic:
rw [tensor_id, id_tensor_comp_tensor_id]
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_naturality]
          slice_lhs 2 3 => rw [tensor_id, tensor_id_comp_id_tensor]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 2 4 => rw [Iso.hom_inv_id_assoc]
          slice_rhs 2 3 => rw [tensor_id, id_tensor_comp_tensor_id])
        (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} (tensorRight X)
⊢ (tensorRight T.X).map ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ≫ (α_ P.X Q.X T.X).hom ≫ (𝟙 P.X ⊗ Q.actRight) =
    ((α_ (P.X ⊗ S.X) Q.X T.X).hom ≫
        (α_ P.X S.X (Q.X ⊗ T.X)).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ P.X S.X Q.X).inv) ≫
      (α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)

Next tactic:
dsimp
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_naturality]
          slice_lhs 2 3 => rw [tensor_id, tensor_id_comp_id_tensor]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 2 4 => rw [Iso.hom_inv_id_assoc]
          slice_rhs 2 3 => rw [tensor_id, id_tensor_comp_tensor_id])
        (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} (tensorRight X)
⊢ ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft) ⊗ 𝟙 T.X) ≫ (α_ P.X Q.X T.X).hom ≫ (𝟙 P.X ⊗ Q.actRight) =
    ((α_ (P.X ⊗ S.X) Q.X T.X).hom ≫
        (α_ P.X S.X (Q.X ⊗ T.X)).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (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 [comp_tensor_id]