∀ {a : ℕ} (a1 : 1 < a) {y : ℕ} (motive : (x : ℕ) → x * x - Pell.d✝ a1 * y * y = 1 → Prop) (x : ℕ)
  (hp : x * x - Pell.d✝¹ a1 * y * y = 1),
  (∀ (hp : 0 - Pell.d✝² a1 * y * y = 1), motive 0 hp) →
    (∀ (x : ℕ) (_hp : (x + 1) * (x + 1) - Pell.d✝³ a1 * y * y = 1), motive x.succ _hp) → motive x hp