-- Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
R₁ R₂ : ComposableArrows.{v_1, u_1} C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
hR₁' : Epi (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_epi_mono'._proof_1)
hR₂' : Epi (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_epi_mono'._proof_1)
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_2)
h₁ : IsIso (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
ψ : mk₄ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ := R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
    (X₃ := 0) (X₄ := 0)
    (R₁.map' 0 1 ⋯ ⋯ : R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
    (R₁.map' 1 2 ⋯ ⋯ : R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
    0 0 ⟶
  mk₄ (X₀ := R₂.obj ⟨0, ⋯⟩) (X₁ := R₂.obj ⟨1, ⋯⟩) (X₃ := 0) (X₄ := 0)
    (R₂.map' 0 1 ⋯ ⋯ : R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj ⟨1, ⋯⟩)
    (R₂.map' 1 2 ⋯ ⋯ : R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj ⟨2, ⋯⟩) 0 0 :=
  homMk₄ (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) 0 0 ⋯ ⋯ ⋯ ⋯
refine_1 :
  Epi
    { X₁ := (mk₄ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0 0).δ₀.δ₀.obj ⟨0, ⋯⟩,
        X₂ := (mk₄ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0 0).δ₀.δ₀.obj ⟨1, ⋯⟩,
        X₃ := (mk₄ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0 0).δ₀.δ₀.obj ⟨2, ⋯⟩,
        f :=
          (mk₄ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ :=
                    R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₃ := 0) (X₄ := 0)
                    (R₁.map' 0 1 ⋯ ⋯ :
                      R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    (R₁.map' 1 2 ⋯ ⋯ :
                      R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    0 0).δ₀.δ₀.map'
            0 1 isComplex₂_iff._proof_2 isComplex₂_iff._proof_4,
        g :=
          (mk₄ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ :=
                    R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₃ := 0) (X₄ := 0)
                    (R₁.map' 0 1 ⋯ ⋯ :
                      R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    (R₁.map' 1 2 ⋯ ⋯ :
                      R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    0 0).δ₀.δ₀.map'
            1 2 isComplex₂_iff._proof_4 isComplex₂_iff._proof_5,
        zero := ⋯ : ShortComplex.{v_1, u_1} C }.f
⊢ { X₁ := (mk₄ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0 0).δ₀.δ₀.obj ⟨0, ⋯⟩,
      X₂ := (mk₄ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0 0).δ₀.δ₀.obj ⟨1, ⋯⟩,
      X₃ := (mk₄ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0 0).δ₀.δ₀.obj ⟨2, ⋯⟩,
      f :=
        (mk₄ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ :=
                  R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₃ := 0) (X₄ := 0)
                  (R₁.map' 0 1 ⋯ ⋯ :
                    R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                  (R₁.map' 1 2 ⋯ ⋯ :
                    R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                  0 0).δ₀.δ₀.map'
          0 1 isComplex₂_iff._proof_2 isComplex₂_iff._proof_4,
      g :=
        (mk₄ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ :=
                  R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₃ := 0) (X₄ := 0)
                  (R₁.map' 0 1 ⋯ ⋯ :
                    R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                  (R₁.map' 1 2 ⋯ ⋯ :
                    R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                  0 0).δ₀.δ₀.map'
          1 2 isComplex₂_iff._proof_4 isComplex₂_iff._proof_5,
      zero := ⋯ : ShortComplex.{v_1, u_1} C }.Exact

Tactic:
rw [ShortComplex.exact_iff_epi _ (by simp)]

Premises used: CategoryTheory.ShortComplex.exact_iff_epi