-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four (goal)
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.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_epi_mono'._proof_1 = 0
hR₁' : Epi (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_epi_mono'._proof_1)
hR₂ : R₂.Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_2)
h₁ : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
ψ : mk₃ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ := R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
    (X₃ := R₁.obj' 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 ⟶
  mk₃ (X₀ := R₂.obj ⟨0, ⋯⟩) (X₁ := R₂.obj ⟨1, ⋯⟩) (X₃ := R₁.obj' 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 :=
  homMk₃ (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯)
    (𝟙
      ((mk₃ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ :=
            R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₃ := R₁.obj' 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).obj'
        3 _proof_300✝))
    ⋯ ⋯ ⋯
⊢ {
      X₁ :=
        (mk₂
              ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                mono_of_epi_of_mono_of_mono'._proof_4)
              ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4
                mono_of_epi_of_mono_of_mono'._proof_7)).obj
          ⟨0, ⋯⟩,
      X₂ :=
        (mk₂
              ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                mono_of_epi_of_mono_of_mono'._proof_4)
              ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4
                mono_of_epi_of_mono_of_mono'._proof_7)).obj
          ⟨1, ⋯⟩,
      X₃ :=
        (mk₂
              ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                mono_of_epi_of_mono_of_mono'._proof_4)
              ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4
                mono_of_epi_of_mono_of_mono'._proof_7)).obj
          ⟨2, ⋯⟩,
      f :=
        (mk₂
              ((mk₃ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ :=
                    R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₃ := R₁.obj' 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).map'
                1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
              ((mk₃ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ :=
                    R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₃ := R₁.obj' 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).map'
                2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).map'
          0 1 isComplex₂_iff._proof_2 isComplex₂_iff._proof_4,
      g :=
        (mk₂
              ((mk₃ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ :=
                    R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₃ := R₁.obj' 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).map'
                1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
              ((mk₃ (X₀ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ :=
                    R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₃ := R₁.obj' 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).map'
                2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).map'
          1 2 isComplex₂_iff._proof_4 isComplex₂_iff._proof_5,
      zero := ⋯ : ShortComplex.{v_1, u_1} C }.Exact
have refine_2 : Epi
  {
      X₁ :=
        (mk₂
              ((mk₃ (R₁.map' 0 1 sorry sorry) (R₁.map' 1 2 sorry sorry) 0).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                mono_of_epi_of_mono_of_mono'._proof_4)
              ((mk₃ (R₁.map' 0 1 sorry sorry) (R₁.map' 1 2 sorry sorry) 0).map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4
                mono_of_epi_of_mono_of_mono'._proof_7)).obj
          ⟨0, sorry⟩,
      X₂ :=
        (mk₂
              ((mk₃ (R₁.map' 0 1 sorry sorry) (R₁.map' 1 2 sorry sorry) 0).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                mono_of_epi_of_mono_of_mono'._proof_4)
              ((mk₃ (R₁.map' 0 1 sorry sorry) (R₁.map' 1 2 sorry sorry) 0).map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4
                mono_of_epi_of_mono_of_mono'._proof_7)).obj
          ⟨1, sorry⟩,
      X₃ :=
        (mk₂
              ((mk₃ (R₁.map' 0 1 sorry sorry) (R₁.map' 1 2 sorry sorry) 0).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                mono_of_epi_of_mono_of_mono'._proof_4)
              ((mk₃ (R₁.map' 0 1 sorry sorry) (R₁.map' 1 2 sorry sorry) 0).map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4
                mono_of_epi_of_mono_of_mono'._proof_7)).obj
          ⟨2, sorry⟩,
      f :=
        (mk₂
              ((mk₃ (X₀ := R₁.obj (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ :=
                    R₁.obj (⟨1, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₃ := R₁.obj' 0 sorry)
                    (R₁.map' 0 1 sorry sorry :
                      R₁.obj (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    (R₁.map' 1 2 sorry sorry :
                      R₁.obj (⟨1, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    0).map'
                1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
              ((mk₃ (X₀ := R₁.obj (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ :=
                    R₁.obj (⟨1, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₃ := R₁.obj' 0 sorry)
                    (R₁.map' 0 1 sorry sorry :
                      R₁.obj (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    (R₁.map' 1 2 sorry sorry :
                      R₁.obj (⟨1, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    0).map'
                2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).map'
          0 1 isComplex₂_iff._proof_2 isComplex₂_iff._proof_4,
      g :=
        (mk₂
              ((mk₃ (X₀ := R₁.obj (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ :=
                    R₁.obj (⟨1, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₃ := R₁.obj' 0 sorry)
                    (R₁.map' 0 1 sorry sorry :
                      R₁.obj (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    (R₁.map' 1 2 sorry sorry :
                      R₁.obj (⟨1, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    0).map'
                1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
              ((mk₃ (X₀ := R₁.obj (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₁ :=
                    R₁.obj (⟨1, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₃ := R₁.obj' 0 sorry)
                    (R₁.map' 0 1 sorry sorry :
                      R₁.obj (⟨0, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    (R₁.map' 1 2 sorry sorry :
                      R₁.obj (⟨1, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, sorry⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    0).map'
                2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).map'
          1 2 isComplex₂_iff._proof_4 isComplex₂_iff._proof_5,
      zero := sorry : ShortComplex.{v_1, u_1} C }.f
Tactic: rw [ShortComplex.exact_iff_epi _ (by simp)]