/-- The augmented Čech nerve associated to a split epimorphism has an extra degeneracy. -/
noncomputable def extraDegeneracy : SimplicialObject.Augmented.ExtraDegeneracy f.augmentedCechNerve
    where
  s' := S.section_ ≫ WidePullback.lift f.hom (fun _ => 𝟙 _) fun i => by rw [id_comp]
  s n := ExtraDegeneracy.s f S n
  s'_comp_ε := by
    dsimp
    simp only [augmentedCechNerve_hom_app, assoc, WidePullback.lift_base, SplitEpi.id]
  s₀_comp_δ₁ := by
    dsimp [cechNerve, SimplicialObject.δ, SimplexCategory.δ]
    ext j
    · fin_cases j
      simpa only [assoc, WidePullback.lift_π, comp_id] using ExtraDegeneracy.s_comp_π_0 f S 0
    · simpa only [assoc, WidePullback.lift_base, SplitEpi.id, comp_id] using
        ExtraDegeneracy.s_comp_base f S 0
  s_comp_δ₀ n := by
    dsimp [cechNerve, SimplicialObject.δ, SimplexCategory.δ]
    ext j
    · simpa only [assoc, WidePullback.lift_π, id_comp] using ExtraDegeneracy.s_comp_π_succ f S n j
    · simpa only [assoc, WidePullback.lift_base, id_comp] using ExtraDegeneracy.s_comp_base f S n
  s_comp_δ n i := by
    dsimp [cechNerve, SimplicialObject.δ, SimplexCategory.δ]
    ext j
    · simp only [assoc, WidePullback.lift_π]
      by_cases h : j = 0
      · subst h
        erw [Fin.succ_succAbove_zero, ExtraDegeneracy.s_comp_π_0, ExtraDegeneracy.s_comp_π_0]
        dsimp
        simp only [WidePullback.lift_base_assoc]
      · cases' Fin.eq_succ_of_ne_zero h with k hk
        subst hk
        erw [Fin.succ_succAbove_succ, ExtraDegeneracy.s_comp_π_succ,
          ExtraDegeneracy.s_comp_π_succ]
        simp only [WidePullback.lift_π]
    · simp only [assoc, WidePullback.lift_base]
      erw [ExtraDegeneracy.s_comp_base, ExtraDegeneracy.s_comp_base]
      dsimp
      simp only [WidePullback.lift_base]
  s_comp_σ n i := by
    dsimp [cechNerve, SimplicialObject.σ, SimplexCategory.σ]
    ext j
    · simp only [assoc, WidePullback.lift_π]
      by_cases h : j = 0
      · subst h
        erw [ExtraDegeneracy.s_comp_π_0, ExtraDegeneracy.s_comp_π_0]
        dsimp
        simp only [WidePullback.lift_base_assoc]
      · cases' Fin.eq_succ_of_ne_zero h with k hk
        subst hk
        erw [Fin.succ_predAbove_succ, ExtraDegeneracy.s_comp_π_succ,
          ExtraDegeneracy.s_comp_π_succ]
        simp only [WidePullback.lift_π]
    · simp only [assoc, WidePullback.lift_base]
      erw [ExtraDegeneracy.s_comp_base, ExtraDegeneracy.s_comp_base]
      dsimp
      

Proof state:
case a
C : Type u_1
inst✝¹ : Category.{?u.323909, u_1} C
f : Arrow C
inst✝ : ∀ (n : ℕ), HasWidePullback f.right (fun x => f.left) fun x => f.hom
S : SplitEpi f.hom
n : ℕ
i : Fin (n + 1)
⊢ (WidePullback.base fun x => f.hom) =
    WidePullback.lift (WidePullback.base fun x => f.hom)
        (fun i_1 => WidePullback.π (fun x => f.hom) (Fin.predAbove i i_1))
        (_ :
          ∀ (j : Fin (SimplexCategory.len (op [n + 1]).unop + 1)),
            WidePullback.π (fun x => f.hom)
                  ((SimplexCategory.Hom.toOrderHom
                      (SimplexCategory.Hom.mk
                            { toFun := Fin.predAbove i, monotone' := (_ : Monotone (Fin.predAbove i)) }).op.unop)
                    j) ≫
                f.hom =
              WidePullback.base fun x => f.hom) ≫
      WidePullback.base fun x => f.hom

Next tactic:
simp only [WidePullback.lift_base]