-- Module: Mathlib.Algebra.Homology.HomotopyCategory.MappingCone
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v, u_1} C
inst_1 : Category.{v', u_2} D
inst_2 : Preadditive.{v, u_1} C
inst_3 : Preadditive.{v', u_2} D
F G : CochainComplex.{v, u_1, 0} C ℤ
φ : F ⟶ G
inst_4 : HasHomotopyCofiber φ
H : CategoryTheory.Functor.{v, v', u_1, u_2} C D
inst_5 : H.Additive
inst_6 :
  HasHomotopyCofiber (F := Functor.obj (H.mapHomologicalComplex (ComplexShape.up ℤ)) F)
    (CategoryTheory.Functor.map (H.mapHomologicalComplex (ComplexShape.up ℤ)) φ)
n : ℤ
⊢ HAdd.hAdd (α :=
        H.obj ((mappingCone φ).X n) ⟶
          X
            (mappingCone (F := Functor.obj (H.mapHomologicalComplex (ComplexShape.up ℤ)) F)
              (CategoryTheory.Functor.map (H.mapHomologicalComplex (ComplexShape.up ℤ)) φ))
            n)
        (H.map (Cochain.v (↑(fst φ) : Cochain (mappingCone φ) F 1) n (n + 1) ⋯) ≫
          Cochain.v (inl (CategoryTheory.Functor.map (H.mapHomologicalComplex (ComplexShape.up ℤ)) φ)) (n + 1) n ⋯)
        (H.map (Cochain.v (snd φ) n n ⋯) ≫
          Hom.f (inr (CategoryTheory.Functor.map (H.mapHomologicalComplex (ComplexShape.up ℤ)) φ)) n) ≫
      (-Cochain.v
            (↑(fst (CategoryTheory.Functor.map (H.mapHomologicalComplex (ComplexShape.up ℤ)) φ)) :
              Cochain (mappingCone ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ))
                ((H.mapHomologicalComplex (ComplexShape.up ℤ)).obj F) 1)
            n (n + 1) ⋯ ≫
          H.map (F.d (n + 1) (n + 2))) =
    H.map ((mappingCone φ).d n (n + 1)) ≫
      HAdd.hAdd (α :=
          H.obj ((mappingCone φ).X (n + 1)) ⟶
            X
              (mappingCone (F := Functor.obj (H.mapHomologicalComplex (ComplexShape.up ℤ)) F)
                (CategoryTheory.Functor.map (H.mapHomologicalComplex (ComplexShape.up ℤ)) φ))
              (n + 1))
          (H.map (Cochain.v (↑(fst φ) : Cochain (mappingCone φ) F 1) (n + 1) (n + 2) ⋯) ≫
            Cochain.v (inl (CategoryTheory.Functor.map (H.mapHomologicalComplex (ComplexShape.up ℤ)) φ)) (n + 2) (n + 1)
              ⋯)
          (H.map (Cochain.v (snd φ) (n + 1) (n + 1) ⋯) ≫
            Hom.f (inr (CategoryTheory.Functor.map (H.mapHomologicalComplex (ComplexShape.up ℤ)) φ)) (n + 1)) ≫
        Cochain.v
          (↑(fst (CategoryTheory.Functor.map (H.mapHomologicalComplex (ComplexShape.up ℤ)) φ)) :
            Cochain (mappingCone ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ))
              ((H.mapHomologicalComplex (ComplexShape.up ℤ)).obj F) 1)
          (n + 1) (n + 2) ⋯

Tactic:
simp only [Functor.mapHomologicalComplex_obj_X, comp_neg, add_comp, assoc, inl_v_fst_v_assoc,
        inr_f_fst_v_assoc, zero_comp, comp_zero, add_zero, inl_v_fst_v, comp_id, inr_f_fst_v,
        ← H.map_comp, d_fst_v φ n (n + 1) (n + 2) rfl (by lia), Functor.map_neg]

Premises used: CategoryTheory.Category.assoc, CochainComplex.mappingCone.inr_f_fst_v, CategoryTheory.Preadditive.comp_neg, CategoryTheory.Limits.comp_zero, CategoryTheory.Functor.mapHomologicalComplex_obj_X, CategoryTheory.Category.comp_id, CochainComplex.mappingCone.inl_v_fst_v, CategoryTheory.Preadditive.add_comp, CochainComplex.mappingCone.inr_f_fst_v_assoc, HAdd.hAdd, CategoryTheory.Functor.map_comp, CategoryTheory.Functor.map_neg, CategoryTheory.Limits.zero_comp, add_zero, OfNat.ofNat, CochainComplex.mappingCone.inl_v_fst_v_assoc, rfl, CochainComplex.mappingCone.d_fst_v