-- Mathlib/NumberTheory/PellMatiyasevic.lean
theorem x_sub_y_dvd_pow (y : ℕ) :
    ∀ n, (2 * a * y - y * y - 1 : ℤ) ∣ yz a1 n * (a - y) + ↑(y ^ n) - xz a1 n
  | 0 => by simp [xz, yz, Int.ofNat_zero, Int.ofNat_one]
  | 1 => by simp [xz, yz, Int.ofNat_zero, Int.ofNat_one]
  | n + 2 => by
    have : (2 * a * y - y * y - 1 : ℤ) ∣ ↑(y ^ (n + 2)) - ↑(2 * a) * ↑(y ^ (n + 1)) + ↑(y ^ n) :=
      ⟨-↑(y ^ n), by
        simp [_root_.pow_succ, mul_add, Int.ofNat_mul, show ((2 : ℕ) : ℤ) = 2 from rfl, mul_comm,
          mul_left_comm]
        ring⟩
    rw [xz_succ_succ, yz_succ_succ, x_sub_y_dvd_pow_lem ↑(y ^ (n + 2)) ↑(y ^ (n + 1)) ↑(y ^ n)]
    exact _root_.dvd_sub (dvd_add this <| (x_sub_y_dvd_pow _ (n + 1)).mul_left _)
      (x_sub_y_dvd_pow _ n)

Proof state:
a : ℕ
a1 : 1 < a
y n : ℕ
this : 2 * ↑a * ↑y - ↑y * ↑y - 1 ∣ ↑(y ^ (n + 2)) - ↑(2 * a) * ↑(y ^ (n + 1)) + ↑(y ^ n)
⊢ 2 * ↑a * ↑y - ↑y * ↑y - 1 ∣
    ↑(y ^ (n + 2)) - ↑(2 * a) * ↑(y ^ (n + 1)) + ↑(y ^ n) +
        ↑(2 * a) * (yz a1 (n + 1) * (↑a - ↑y) + ↑(y ^ (n + 1)) - xz a1 (n + 1)) -
      (yz a1 n * (↑a - ↑y) + ↑(y ^ n) - xz a1 n)

Tactic:
exact _root_.dvd_sub (dvd_add this <| (x_sub_y_dvd_pow _ (n + 1)).mul_left _)
  (x_sub_y_dvd_pow _ n)

Result:
no goals

Proof state:
a : ℕ
a1 : 1 < a
y n : ℕ
⊢ ↑y * (↑y * ↑y ^ n) - ↑a * (↑y * ↑y ^ n * 2) + ↑y ^ n = -(↑y ^ n * (↑a * (↑y * 2) - ↑y * ↑y - 1))

Tactic:
ring

Result:
no goals

Proof state:
a : ℕ
a1 : 1 < a
y n : ℕ
⊢ ↑(y ^ (n + 2)) - ↑(2 * a) * ↑(y ^ (n + 1)) + ↑(y ^ n) = (2 * ↑a * ↑y - ↑y * ↑y - 1) * -↑(y ^ n)

Tactic:
simp [_root_.pow_succ, mul_add, Int.ofNat_mul, show ((2 : ℕ) : ℤ) = 2 from rfl, mul_comm,
  mul_left_comm]

Result:
a : ℕ
a1 : 1 < a
y n : ℕ
⊢ ↑y * (↑y * ↑y ^ n) - ↑a * (↑y * ↑y ^ n * 2) + ↑y ^ n = -(↑y ^ n * (↑a * (↑y * 2) - ↑y * ↑y - 1))

Proof state:
a : ℕ
a1 : 1 < a
y n : ℕ
⊢ 2 * ↑a * ↑y - ↑y * ↑y - 1 ∣ yz a1 (n + 2) * (↑a - ↑y) + ↑(y ^ (n + 2)) - xz a1 (n + 2)

Tactic:
have : (2 * a * y - y * y - 1 : ℤ) ∣ ↑(y ^ (n + 2)) - ↑(2 * a) * ↑(y ^ (n + 1)) + ↑(y ^ n) :=
  ⟨-↑(y ^ n), by
    simp [_root_.pow_succ, mul_add, Int.ofNat_mul, show ((2 : ℕ) : ℤ) = 2 from rfl, mul_comm,
      mul_left_comm]
    ring⟩

Result:
a : ℕ
a1 : 1 < a
y n : ℕ
this : 2 * ↑a * ↑y - ↑y * ↑y - 1 ∣ ↑(y ^ (n + 2)) - ↑(2 * a) * ↑(y ^ (n + 1)) + ↑(y ^ n)
⊢ 2 * ↑a * ↑y - ↑y * ↑y - 1 ∣ yz a1 (n + 2) * (↑a - ↑y) + ↑(y ^ (n + 2)) - xz a1 (n + 2)

Proof state:
a : ℕ
a1 : 1 < a
y n : ℕ
this : 2 * ↑a * ↑y - ↑y * ↑y - 1 ∣ ↑(y ^ (n + 2)) - ↑(2 * a) * ↑(y ^ (n + 1)) + ↑(y ^ n)
⊢ 2 * ↑a * ↑y - ↑y * ↑y - 1 ∣ yz a1 (n + 2) * (↑a - ↑y) + ↑(y ^ (n + 2)) - xz a1 (n + 2)

Tactic:
rw [xz_succ_succ, yz_succ_succ, x_sub_y_dvd_pow_lem ↑(y ^ (n + 2)) ↑(y ^ (n + 1)) ↑(y ^ n)]

Result:
a : ℕ
a1 : 1 < a
y n : ℕ
this : 2 * ↑a * ↑y - ↑y * ↑y - 1 ∣ ↑(y ^ (n + 2)) - ↑(2 * a) * ↑(y ^ (n + 1)) + ↑(y ^ n)
⊢ 2 * ↑a * ↑y - ↑y * ↑y - 1 ∣
    ↑(y ^ (n + 2)) - ↑(2 * a) * ↑(y ^ (n + 1)) + ↑(y ^ n) +
        ↑(2 * a) * (yz a1 (n + 1) * (↑a - ↑y) + ↑(y ^ (n + 1)) - xz a1 (n + 1)) -
      (yz a1 n * (↑a - ↑y) + ↑(y ^ n) - xz a1 n)

Proof state:
a : ℕ
a1 : 1 < a
y : ℕ
⊢ 2 * ↑a * ↑y - ↑y * ↑y - 1 ∣ yz a1 0 * (↑a - ↑y) + ↑(y ^ 0) - xz a1 0

Tactic:
simp [xz, yz, Int.ofNat_zero, Int.ofNat_one]

Result:
no goals

Proof state:
a : ℕ
a1 : 1 < a
y : ℕ
⊢ 2 * ↑a * ↑y - ↑y * ↑y - 1 ∣ yz a1 1 * (↑a - ↑y) + ↑(y ^ 1) - xz a1 1

Tactic:
simp [xz, yz, Int.ofNat_zero, Int.ofNat_one]

Result:
no goals