Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four

Goal state before:
pp: case refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
hR₁' : Mono (R₁.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₁ : IsIso (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
h₂ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) ⟶ mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) :=
  homMk₄ 0 0 (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) ⋯ ⋯ ⋯ ⋯
⊢ (mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯)).Exact

Tactic:
refine exact_of_δ₀ (exact₂_mk _ (by simp) ?_) (exact_of_δ₀ ?_ (exact₂_mk _ _ (hR₁.exact 0)))

Goal state after:
case refine_1.refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
hR₁' : Mono (R₁.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₁ : IsIso (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
h₂ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) ⟶ mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) :=
  homMk₄ 0 0 (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) ⋯ ⋯ ⋯ ⋯
⊢ {
      X₁ :=
        (mk₂ ((mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯)).map' 0 1 isComplex₂_iff._proof_2 ⋯)
              ((mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯)).map' 1 2 isComplex₂_iff._proof_4 ⋯)).obj
          ⟨0, ⋯⟩,
      X₂ :=
        (mk₂ ((mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯)).map' 0 1 isComplex₂_iff._proof_2 ⋯)
              ((mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯)).map' 1 2 isComplex₂_iff._proof_4 ⋯)).obj
          ⟨1, ⋯⟩,
      X₃ :=
        (mk₂ ((mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯)).map' 0 1 isComplex₂_iff._proof_2 ⋯)
              ((mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯)).map' 1 2 isComplex₂_iff._proof_4 ⋯)).obj
          ⟨2, ⋯⟩,
      f :=
        (mk₂ ((mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯)).map' 0 1 isComplex₂_iff._proof_2 ⋯)
              ((mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯)).map' 1 2 isComplex₂_iff._proof_4 ⋯)).map'
          0 1 isComplex₂_iff._proof_2 isComplex₂_iff._proof_4,
      g :=
        (mk₂ ((mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯)).map' 0 1 isComplex₂_iff._proof_2 ⋯)
              ((mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯)).map' 1 2 isComplex₂_iff._proof_4 ⋯)).map'
          1 2 isComplex₂_iff._proof_4 isComplex₂_iff._proof_5,
      zero := ⋯ }.Exact

case refine_1.refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
hR₁' : Mono (R₁.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₁ : IsIso (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
h₂ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) ⟶ mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) :=
  homMk₄ 0 0 (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) ⋯ ⋯ ⋯ ⋯
⊢ (mk₂ ((mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯)).δ₀.map' 0 1 isComplex₂_iff._proof_2 ⋯)
      ((mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯)).δ₀.map' 1 2 isComplex₂_iff._proof_4 ⋯)).Exact