theorem σ_comp_P_eq_zero (X : SimplicialObject C) {n q : ℕ} (i : Fin (n + 1)) (hi : n + 1 ≤ i + q) :
    X.σ i ≫ (P q).f (n + 1) = 0 := by
  revert i hi
  induction' q with q hq
  · intro i (hi : n + 1 ≤ i)
    exfalso
    linarith [Fin.is_lt i]
  · intro i (hi : n + 1 ≤ i + q + 1)
    by_cases h : n + 1 ≤ (i : ℕ) + q
    · rw [P_succ, HomologicalComplex.comp_f, ← assoc, hq i h, zero_comp]
    · replace hi : n = i + q := by
        obtain ⟨j, hj⟩ := le_iff_exists_add.mp hi
        rw [← Nat.lt_succ_iff, Nat.succ_eq_add_one, hj, not_lt, add_le_iff_nonpos_right,
          nonpos_iff_eq_zero] at h
        rw [← add_left_inj 1, hj, self_eq_add_right, h]
      rcases n with _|n
      · fin_cases i
        dsimp at h hi
        rw [show q = 0 by linarith]
        change X.σ 0 ≫ (P 1).f 1 = 0
        simp only [P_succ, HomologicalComplex.add_f_apply, comp_add,
          HomologicalComplex.id_f, AlternatingFaceMapComplex.obj_d_eq, Hσ,
          HomologicalComplex.comp_f, Homotopy.nullHomotopicMap'_f (c_mk 2 1 rfl) (c_mk 1 0 rfl),
          comp_id]
        erw [hσ'_eq' (zero_add 0).symm, hσ'_eq' (add_zero 1).symm, comp_id, Fin.sum_univ_two,
          Fin.sum_univ_succ, Fin.sum_univ_two]
        simp only [Fin.val_zero, pow_zero, pow_one, pow_add, one_smul, neg_smul, Fin.mk_one,
          Fin.val_succ, Fin.val_one, Fin.succ_one_eq_two, P_zero, HomologicalComplex.id_f,
          Fin.val_two, pow_two, mul_neg, one_mul, neg_mul, neg_neg, id_comp, add_comp,
          comp_add, Fin.mk_zero, neg_comp, comp_neg, Fin.succ_zero_eq_one]
        erw [SimplicialObject.δ_comp_σ_self, SimplicialObject.δ_comp_σ_self_assoc,
          SimplicialObject.δ_comp_σ_succ, comp_id,
          SimplicialObject.δ_comp_σ_of_le X
            (show (0 : Fin 2) ≤ Fin.castSucc 0 by rw [Fin.castSucc_zero]),
          SimplicialObject.δ_comp_σ_self_assoc, SimplicialObject.δ_comp_σ_succ_assoc]
        simp only [add_right_neg, add_zero, zero_add]
      · rw [← id_comp (X.σ i), ← (P_add_Q_f q n.succ : _ = 𝟙 (X.obj _)), add_comp, add_comp,
          P_succ]
        have v : HigherFacesVanish q ((P q).f n.succ ≫ X.σ i) :=
          (HigherFacesVanish.of_P q n).comp_σ hi
        erw [← assoc, v.comp_P_eq_self, HomologicalComplex.add_f_apply, Preadditive.comp_add,
          comp_id, v.comp_Hσ_eq hi, assoc, SimplicialObject.δ_comp_σ_succ_assoc, Fin.eta,
          decomposition_Q n q, sum_comp, sum_comp, Finset.sum_eq_zero, add_zero, add_neg_eq_zero]
        intro j hj
        simp only [true_and_iff, Finset.mem_univ, Finset.mem_filter] at hj
        simp only [Nat.succ_eq_add_one] at hi
        obtain ⟨k, hk⟩ := Nat.le.dest (Nat.lt_succ_iff.mp (Fin.is_lt j))
        rw [add_comm] at hk
        have hi' : i = Fin.castSucc ⟨i, by linarith⟩ := by
          ext
          simp only [Fin.castSucc_mk, Fin.eta]
        

Proof state:
case neg.succ.intro
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
q n : ℕ
hq :
  ∀ (i : Fin (Nat.succ n + 1)),
    Nat.succ n + 1 ≤ ↑i + q → SimplicialObject.σ X i ≫ HomologicalComplex.Hom.f (P q) (Nat.succ n + 1) = 0
i : Fin (Nat.succ n + 1)
h : ¬Nat.succ n + 1 ≤ ↑i + q
hi : n + 1 = ↑i + q
v : HigherFacesVanish q (HomologicalComplex.Hom.f (P q) (Nat.succ n) ≫ SimplicialObject.σ X i)
j : Fin (n + 1)
hj : ↑j < q
k : ℕ
hk : k + ↑j = n
hi' : i = Fin.castSucc { val := ↑i, isLt := (_ : ↑i < n + 1) }
⊢ ((HomologicalComplex.Hom.f (P ↑j) (n + 1) ≫
          SimplicialObject.δ X (Fin.succ (Fin.rev j)) ≫ SimplicialObject.σ X (Fin.rev j)) ≫
        SimplicialObject.σ X i) ≫
      HomologicalComplex.Hom.f (P q ≫ (𝟙 (AlternatingFaceMapComplex.obj X) + Hσ q)) (Nat.succ n + 1) =
    0

Next tactic:
have eq := hq j.rev.succ (by
          simp only [← hk, Fin.rev_eq j hk.symm, Nat.succ_eq_add_one, Fin.succ_mk, Fin.val_mk]
          linarith)