case succ k IH =>
  intro n hn hk H
  rcases H.succ_elim (le_trans (by simp) hk) with ⟨n', rfl, H'⟩
  have IH : NuonAux (2 * A) (k + 1) n' := IH n' H'.le (le_trans (by simp) hk) H'
  simpa using IH.succ (le_trans hk (length_monotone $ by simp))
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚫₀ + 𝛀₁
A k : M
IH : ∀ n ≤ k, k ≤ ‖A‖ → NuonAux A k n → NuonAux (2 * A) (k + 1) n
⊢ ∀ n ≤ k + 1, k + 1 ≤ ‖A‖ → NuonAux A (k + 1) n → NuonAux (2 * A) (k + 1 + 1) n
no goals