-- Mathlib/Algebra/Homology/Monoidal.lean
@[simp]
lemma tensor_unit_d₂ (i₁ i₂ j : I) :
    mapBifunctor.d₂ K (tensorUnit C c) (curriedTensor C) c i₁ i₂ j = 0

Proof state:
case pos
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
i₁ i₂ j : I
h₁ : c.Rel i₂ (c.next i₂)
⊢ mapBifunctor.d₂ K (tensorUnit C c) (curriedTensor C) c i₁ i₂ j = 0

Tactic:
by_cases h₂ : ComplexShape.π c c c (i₁, c.next i₂) = j

Result:
case pos
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
i₁ i₂ j : I
h₁ : c.Rel i₂ (c.next i₂)
h₂ : c.π c c (i₁, c.next i₂) = j
⊢ mapBifunctor.d₂ K (tensorUnit C c) (curriedTensor C) c i₁ i₂ j = 0

case neg
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
i₁ i₂ j : I
h₁ : c.Rel i₂ (c.next i₂)
h₂ : ¬c.π c c (i₁, c.next i₂) = j
⊢ mapBifunctor.d₂ K (tensorUnit C c) (curriedTensor C) c i₁ i₂ j = 0

Proof state:
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
i₁ i₂ j : I
⊢ mapBifunctor.d₂ K (tensorUnit C c) (curriedTensor C) c i₁ i₂ j = 0

Tactic:
by_cases h₁ : c.Rel i₂ (c.next i₂)

Result:
case pos
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
i₁ i₂ j : I
h₁ : c.Rel i₂ (c.next i₂)
⊢ mapBifunctor.d₂ K (tensorUnit C c) (curriedTensor C) c i₁ i₂ j = 0

case neg
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
i₁ i₂ j : I
h₁ : ¬c.Rel i₂ (c.next i₂)
⊢ mapBifunctor.d₂ K (tensorUnit C c) (curriedTensor C) c i₁ i₂ j = 0

Proof state:
case neg
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
i₁ i₂ j : I
h₁ : ¬c.Rel i₂ (c.next i₂)
⊢ mapBifunctor.d₂ K (tensorUnit C c) (curriedTensor C) c i₁ i₂ j = 0

Tactic:
rw [mapBifunctor.d₂_eq_zero _ _ _ _ _ _ _ h₁]

Result:
no goals

Proof state:
case pos
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
i₁ i₂ j : I
h₁ : c.Rel i₂ (c.next i₂)
h₂ : c.π c c (i₁, c.next i₂) = j
⊢ mapBifunctor.d₂ K (tensorUnit C c) (curriedTensor C) c i₁ i₂ j = 0

Tactic:
rw [mapBifunctor.d₂_eq _ _ _ _ _ h₁ _ h₂, single_obj_d, Functor.map_zero,
  zero_comp, smul_zero]

Result:
no goals

Proof state:
case neg
C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
i₁ i₂ j : I
h₁ : c.Rel i₂ (c.next i₂)
h₂ : ¬c.π c c (i₁, c.next i₂) = j
⊢ mapBifunctor.d₂ K (tensorUnit C c) (curriedTensor C) c i₁ i₂ j = 0

Tactic:
rw [mapBifunctor.d₂_eq_zero' _ _ _ _ _ h₁ _ h₂]

Result:
no goals