File: Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean

full_name: CochainComplex.homologySequenceδ_quotient_mapTriangle_obj
code: @[reassoc]
lemma homologySequenceδ_quotient_mapTriangle_obj
    (T : Triangle (CochainComplex C ℤ)) (n₀ n₁ : ℤ) (h : n₀ + 1 = n₁) :
    (homologyFunctor C (up ℤ) 0).homologySequenceδ
        ((quotient C (up ℤ)).mapTriangle.obj T) n₀ n₁ h =
      (homologyFunctorFactors C (up ℤ) n₀).hom.app _ ≫
        (HomologicalComplex.homologyFunctor C (up ℤ) 0).shiftMap T.mor₃ n₀ n₁ (by omega) ≫
        (homologyFunctorFactors C (up ℤ) n₁).inv.app _ 
goal: (HomotopyCategory.homologyFunctor C (ComplexShape.up ℤ) 0).homologySequenceδ
    ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).mapTriangle.obj T) n₀ n₁ h =
  (HomotopyCategory.homologyFunctorFactors C (ComplexShape.up ℤ) n₀).hom.app T.obj₃ ≫
    (HomologicalComplex.homologyFunctor C (ComplexShape.up ℤ) 0).shiftMap T.mor₃ n₀ n₁ ⋯ ≫
      (HomotopyCategory.homologyFunctorFactors C (ComplexShape.up ℤ) n₁).inv.app T.obj₁

full_name: CochainComplex.mappingCone.descShortComplex
code: /-- The canonical morphism `mappingCone S.f ⟶ S.X₃` when `S` is a short complex
of cochain complexes. -/
noncomputable def descShortComplex : mappingCone S.f ⟶ S.X₃ 
goal: CochainComplex.mappingCone S.f ⟶ S.X₃

full_name: CochainComplex.mappingCone.inr_descShortComplex
code: @[reassoc (attr := simp)]
lemma inr_descShortComplex : inr S.f ≫ descShortComplex S = S.g 
goal: CochainComplex.mappingCone.inr S.f ≫ CochainComplex.mappingCone.descShortComplex S = S.g

full_name: CochainComplex.mappingCone.inr_f_descShortComplex_f
code: @[reassoc (attr := simp)]
lemma inr_f_descShortComplex_f (n : ℤ) : (inr S.f).f n ≫ (descShortComplex S).f n = S.g.f n 
goal: (CochainComplex.mappingCone.inr S.f).f n ≫ (CochainComplex.mappingCone.descShortComplex S).f n = S.g.f n

full_name: CochainComplex.mappingCone.inl_v_descShortComplex_f
code: @[reassoc (attr := simp)]
lemma inl_v_descShortComplex_f (i j : ℤ) (h : i + (-1) = j) :
    (inl S.f).v i j h ≫ (descShortComplex S).f j = 0 
goal: (CochainComplex.mappingCone.inl S.f).v i j h ≫ (CochainComplex.mappingCone.descShortComplex S).f j = 0

full_name: CochainComplex.mappingCone.homologySequenceδ_triangleh
code: lemma homologySequenceδ_triangleh (n₀ : ℤ) (n₁ : ℤ) (h : n₀ + 1 = n₁) :
    (homologyFunctor C (up ℤ) 0).homologySequenceδ (triangleh S.f) n₀ n₁ h =
      (homologyFunctorFactors C (up ℤ) n₀).hom.app _ ≫
        HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫ hS.δ n₀ n₁ h ≫
          (homologyFunctorFactors C (up ℤ) n₁).inv.app _ 
goal: (HomotopyCategory.homologyFunctor C (ComplexShape.up ℤ) 0).homologySequenceδ (CochainComplex.mappingCone.triangleh S.f)
    n₀ n₁ h =
  (HomotopyCategory.homologyFunctorFactors C (ComplexShape.up ℤ) n₀).hom.app (CochainComplex.mappingCone S.f) ≫
    HomologicalComplex.homologyMap (CochainComplex.mappingCone.descShortComplex S) n₀ ≫
      hS.δ n₀ n₁ h ≫ (HomotopyCategory.homologyFunctorFactors C (ComplexShape.up ℤ) n₁).inv.app S.X₁

full_name: CochainComplex.mappingCone.quasiIso_descShortComplex
code: lemma quasiIso_descShortComplex : QuasiIso (descShortComplex S) 
goal: QuasiIso (CochainComplex.mappingCone.descShortComplex S)