Module: Mathlib.Algebra.Homology.BifunctorAssociator

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝²⁰ : Category.{v_1, u_1} C₁
inst✝¹⁹ : Category.{v_2, u_2} C₂
inst✝¹⁸ : Category.{v_3, u_5} C₃
inst✝¹⁷ : Category.{v_4, u_6} C₄
inst✝¹⁶ : Category.{v_5, u_3} C₁₂
inst✝¹⁵ : HasZeroMorphisms C₁
inst✝¹⁴ : HasZeroMorphisms C₂
inst✝¹³ : HasZeroMorphisms C₃
inst✝¹² : Preadditive C₁₂
inst✝¹¹ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝¹⁰ : F₁₂.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁸ : G.Additive
inst✝⁷ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁶ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₄ : ComplexShape ι₄
inst✝⁵ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁴ : TotalComplexShape c₁₂ c₃ c₄
inst✝³ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝² : DecidableEq ι₁₂
inst✝¹ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
j j' : ι₄
inst✝ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
⊢ ((K₁.mapBifunctor K₂ F₁₂ c₁₂).mapBifunctor K₃ G c₄).d j j' =
    D₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' + D₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' + D₃ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j'

Tactic:
rw [mapBifunctor.d_eq]
  congr 1
  ext i₁ i₂ i₃ h
  simp only [Preadditive.comp_add, ι_D₁, ι_D₂]
  rw [ι_eq _ _ _ _ _ _ _ _ _ _ _ _ rfl h, assoc, mapBifunctor.ι_D₁]
  set i₁₂ := ComplexShape.π c₁ c₂ c₁₂ ⟨i₁, i₂⟩
  by_cases h₁ : c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
  · by_cases h₂ : ComplexShape.π c₁₂ c₃ c₄ (c₁₂.next i₁₂, i₃) = j'
    · rw [mapBifunctor.d₁_eq _ _ _ _ h₁ _ _ h₂]
      simp only [i₁₂, mapBifunctor.d_eq, Functor.map_add, NatTrans.app_add,
        Preadditive.add_comp, smul_add, Preadditive.comp_add, Linear.comp_units_smul]
      congr 1
      · rw [← NatTrans.comp_app_assoc, ← Functor.map_comp,
          mapBifunctor.ι_D₁]
        by_cases h₃ : c₁.Rel i₁ (c₁.next i₁)
        · have h₄ := (ComplexShape.next_π₁ c₂ c₁₂ h₃ i₂).symm
          rw [mapBifunctor.d₁_eq _ _ _ _ h₃ _ _ h₄,
            d₁_eq _ _ _ _ _ _ _ h₃,
            ιOrZero_eq _ _ _ _ _ _ _ _ _ _ _ (by rw [← h₂, ← h₄]; rfl),
            ι_eq _ _ _ _ _ _ _ _ _ _ (c₁₂.next i₁₂) _ h₄ h₂,
            Functor.map_units_smul, Functor.map_comp, NatTrans.app_units_zsmul,
            NatTrans.comp_app, Linear.units_smul_comp, assoc, smul_smul]
        · rw [d₁_eq_zero _ _ _ _ _ _ _ _ _ _ _ h₃,
            mapBifunctor.d₁_eq_zero _ _ _ _ _ _ _ h₃,
            Functor.map_zero, zero_app, zero_comp, smul_zero]
      · rw [← NatTrans.comp_app_assoc, ← Functor.map_comp,
          mapBifunctor.ι_D₂]
        by_cases h₃ : c₂.Rel i₂ (c₂.next i₂)
        · have h₄ := (ComplexShape.next_π₂ c₁ c₁₂ i₁ h₃).symm
          rw [mapBifunctor.d₂_eq _ _ _ _ _ h₃ _ h₄,
            d₂_eq _ _ _ _ _ _ _ _ h₃,
            ιOrZero_eq _ _ _ _ _ _ _ _ _ _ _ (by rw [← h₂, ← h₄]; rfl),
            ι_eq _ _ _ _ _ _ _ _ _ _ (c₁₂.next i₁₂) _ h₄ h₂,
            Functor.map_units_smul, Functor.map_comp, NatTrans.app_units_zsmul,
            NatTrans.comp_app, Linear.units_smul_comp, assoc, smul_smul]
        · rw [d₂_eq_zero _ _ _ _ _ _ _ _ _ _ _ h₃,
            mapBifunctor.d₂_eq_zero _ _ _ _ _ _ _ h₃,
            Functor.map_zero, zero_app, zero_comp, smul_zero]
    · rw [mapBifunctor.d₁_eq_zero' _ _ _ _ h₁ _ _ h₂, comp_zero]
      trans 0 + 0
      · simp
      · congr 1
        · by_cases h₃ : c₁.Rel i₁ (c₁.next i₁)
          · rw [d₁_eq _ _ _ _ _ _ _ h₃, ιOrZero_eq_zero, comp_zero, smul_zero]
            dsimp [ComplexShape.r]
            intro h₄
            apply h₂
            rw [← h₄, ComplexShape.next_π₁ c₂ c₁₂ h₃ i₂]
          · rw [d₁_eq_zero _ _ _ _ _ _ _ _ _ _ _ h₃]
        · by_cases h₃ : c₂.Rel i₂ (c₂.next i₂)
          · rw [d₂_eq _ _ _ _ _ _ _ _ h₃, ιOrZero_eq_zero, comp_zero, smul_zero]
            dsimp [ComplexShape.r]
            intro h₄
            apply h₂
            rw [← h₄, ComplexShape.next_π₂ c₁ c₁₂ i₁ h₃]
          · rw [d₂_eq_zero _ _ _ _ _ _ _ _ _ _ _ h₃]
  · rw [mapBifunctor.d₁_eq_zero _ _ _ _ _ _ _ h₁, comp_zero,
      d₁_eq_zero, d₂_eq_zero, zero_add]
    · intro h₂
      apply h₁
      have := ComplexShape.rel_π₂ c₁ c₁₂ i₁ h₂
      rw [c₁₂.next_eq' this]
      exact this
    · intro h₂
      apply h₁
      have := ComplexShape.rel_π₁ c₂ c₁₂ h₂ i₂
      rw [c₁₂.next_eq' this]
      exact this

Goal state after: