l3lab/ntp-mathlib huggingface dataset:
theorem eq_pow_of_pell_lem {a y k : ℕ} (hy0 : y ≠ 0) (hk0 : k ≠ 0) (hyk : y ^ k < a) :
    (↑(y ^ k) : ℤ) < 2 * a * y - y * y - 1 :=
  have hya : y < a := (Nat.le_self_pow hk0 _).trans_lt hyk
  calc
    (↑(y ^ k) : ℤ) < a := Nat.cast_lt.2 hyk
    _ ≤ (a : ℤ) ^ 2 - (a - 1 : ℤ) ^ 2 - 1 := by
      rw [sub_sq, mul_one, one_pow, sub_add, sub_sub_cancel, two_mul, sub_sub, ← add_sub,
        le_add_iff_nonneg_right, sub_nonneg, Int.add_one_le_iff]
      norm_cast
      exact lt_of_le_of_lt (Nat.succ_le_of_lt (Nat.pos_of_ne_zero hy0)) hya
    _ ≤ (a : ℤ) ^ 2 - (a - y : ℤ) ^ 2 - 1 := by
      have := hya.le
      mono * <;> 

Proof state:
case hab.hcd.ha
a y k : ℕ
hy0 : y ≠ 0
hk0 : k ≠ 0
hyk : y ^ k < a
hya : y < a
this : y ≤ a
hy0_symm : 0 ≠ y
hk0_symm : 0 ≠ k
⊢ 0 ≤ ↑a - ↑y

Next tactic:
norm_cast
theorem eq_pow_of_pell_lem {a y k : ℕ} (hy0 : y ≠ 0) (hk0 : k ≠ 0) (hyk : y ^ k < a) :
    (↑(y ^ k) : ℤ) < 2 * a * y - y * y - 1 :=
  have hya : y < a := (Nat.le_self_pow hk0 _).trans_lt hyk
  calc
    (↑(y ^ k) : ℤ) < a := Nat.cast_lt.2 hyk
    _ ≤ (a : ℤ) ^ 2 - (a - 1 : ℤ) ^ 2 - 1 := by
      rw [sub_sq, mul_one, one_pow, sub_add, sub_sub_cancel, two_mul, sub_sub, ← add_sub,
        le_add_iff_nonneg_right, sub_nonneg, Int.add_one_le_iff]
      norm_cast
      exact lt_of_le_of_lt (Nat.succ_le_of_lt (Nat.pos_of_ne_zero hy0)) hya
    _ ≤ (a : ℤ) ^ 2 - (a - y : ℤ) ^ 2 - 1 := by
      have := hya.le
      mono * <;> 

Proof state:
case hab.hcd.hab.hcd
a y k : ℕ
hy0 : y ≠ 0
hk0 : k ≠ 0
hyk : y ^ k < a
hya : y < a
this : y ≤ a
hy0_symm : 0 ≠ y
hk0_symm : 0 ≠ k
⊢ 1 ≤ ↑y

Next tactic:
norm_cast
theorem eq_pow_of_pell_lem {a y k : ℕ} (hy0 : y ≠ 0) (hk0 : k ≠ 0) (hyk : y ^ k < a) :
    (↑(y ^ k) : ℤ) < 2 * a * y - y * y - 1 :=
  have hya : y < a := (Nat.le_self_pow hk0 _).trans_lt hyk
  calc
    (↑(y ^ k) : ℤ) < a := Nat.cast_lt.2 hyk
    _ ≤ (a : ℤ) ^ 2 - (a - 1 : ℤ) ^ 2 - 1 := by
      rw [sub_sq, mul_one, one_pow, sub_add, sub_sub_cancel, two_mul, sub_sub, ← add_sub,
        le_add_iff_nonneg_right, sub_nonneg, Int.add_one_le_iff]
      norm_cast
      exact lt_of_le_of_lt (Nat.succ_le_of_lt (Nat.pos_of_ne_zero hy0)) hya
    _ ≤ (a : ℤ) ^ 2 - (a - y : ℤ) ^ 2 - 1 := by
      have := hya.le
      mono * <;> norm_cast <;> 

Proof state:
case hab.hab.ha
a y k : ℕ
hy0 : y ≠ 0
hk0 : k ≠ 0
hyk : y ^ k < a
hya : y < a
this : y ≤ a
hy0_symm : 0 ≠ y
hk0_symm : 0 ≠ k
⊢ 0 ≤ a

Next tactic:
simp [Nat.zero_le, Nat.succ_le_of_lt (Nat.pos_of_ne_zero hy0)]
theorem eq_pow_of_pell_lem {a y k : ℕ} (hy0 : y ≠ 0) (hk0 : k ≠ 0) (hyk : y ^ k < a) :
    (↑(y ^ k) : ℤ) < 2 * a * y - y * y - 1 :=
  have hya : y < a := (Nat.le_self_pow hk0 _).trans_lt hyk
  calc
    (↑(y ^ k) : ℤ) < a := Nat.cast_lt.2 hyk
    _ ≤ (a : ℤ) ^ 2 - (a - 1 : ℤ) ^ 2 - 1 := by
      rw [sub_sq, mul_one, one_pow, sub_add, sub_sub_cancel, two_mul, sub_sub, ← add_sub,
        le_add_iff_nonneg_right, sub_nonneg, Int.add_one_le_iff]
      norm_cast
      exact lt_of_le_of_lt (Nat.succ_le_of_lt (Nat.pos_of_ne_zero hy0)) hya
    _ ≤ (a : ℤ) ^ 2 - (a - y : ℤ) ^ 2 - 1 := by
      have := hya.le
      mono * <;> norm_cast <;> 

Proof state:
case hab.hcd.ha
a y k : ℕ
hy0 : y ≠ 0
hk0 : k ≠ 0
hyk : y ^ k < a
hya : y < a
this : y ≤ a
hy0_symm : 0 ≠ y
hk0_symm : 0 ≠ k
⊢ 0 ≤ a - y

Next tactic:
simp [Nat.zero_le, Nat.succ_le_of_lt (Nat.pos_of_ne_zero hy0)]
theorem eq_pow_of_pell_lem {a y k : ℕ} (hy0 : y ≠ 0) (hk0 : k ≠ 0) (hyk : y ^ k < a) :
    (↑(y ^ k) : ℤ) < 2 * a * y - y * y - 1 :=
  have hya : y < a := (Nat.le_self_pow hk0 _).trans_lt hyk
  calc
    (↑(y ^ k) : ℤ) < a := Nat.cast_lt.2 hyk
    _ ≤ (a : ℤ) ^ 2 - (a - 1 : ℤ) ^ 2 - 1 := by
      rw [sub_sq, mul_one, one_pow, sub_add, sub_sub_cancel, two_mul, sub_sub, ← add_sub,
        le_add_iff_nonneg_right, sub_nonneg, Int.add_one_le_iff]
      norm_cast
      exact lt_of_le_of_lt (Nat.succ_le_of_lt (Nat.pos_of_ne_zero hy0)) hya
    _ ≤ (a : ℤ) ^ 2 - (a - y : ℤ) ^ 2 - 1 := by
      have := hya.le
      mono * <;> norm_cast <;> 

Proof state:
case hab.hcd.hab.hcd
a y k : ℕ
hy0 : y ≠ 0
hk0 : k ≠ 0
hyk : y ^ k < a
hya : y < a
this : y ≤ a
hy0_symm : 0 ≠ y
hk0_symm : 0 ≠ k
⊢ 1 ≤ y

Next tactic:
simp [Nat.zero_le, Nat.succ_le_of_lt (Nat.pos_of_ne_zero hy0)]
theorem eq_pow_of_pell_lem {a y k : ℕ} (hy0 : y ≠ 0) (hk0 : k ≠ 0) (hyk : y ^ k < a) :
    (↑(y ^ k) : ℤ) < 2 * a * y - y * y - 1 :=
  have hya : y < a := (Nat.le_self_pow hk0 _).trans_lt hyk
  calc
    (↑(y ^ k) : ℤ) < a := Nat.cast_lt.2 hyk
    _ ≤ (a : ℤ) ^ 2 - (a - 1 : ℤ) ^ 2 - 1 := by
      rw [sub_sq, mul_one, one_pow, sub_add, sub_sub_cancel, two_mul, sub_sub, ← add_sub,
        le_add_iff_nonneg_right, sub_nonneg, Int.add_one_le_iff]
      norm_cast
      exact lt_of_le_of_lt (Nat.succ_le_of_lt (Nat.pos_of_ne_zero hy0)) hya
    _ ≤ (a : ℤ) ^ 2 - (a - y : ℤ) ^ 2 - 1 := by
      have := hya.le
      mono * <;> norm_cast <;> simp [Nat.zero_le, Nat.succ_le_of_lt (Nat.pos_of_ne_zero hy0)]
    _ = 2 * a * y - y * y - 1 := by 

Proof state:
a y k : ℕ
hy0 : y ≠ 0
hk0 : k ≠ 0
hyk : y ^ k < a
hya : y < a
⊢ ↑a ^ 2 - (↑a - ↑y) ^ 2 - 1 = 2 * ↑a * ↑y - ↑y * ↑y - 1

Next tactic:
ring
theorem eq_pow_of_pell {m n k} :
    n ^ k = m ↔ k = 0 ∧ m = 1 ∨0 < k ∧ (n = 0 ∧ m = 0 ∨
      0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
      2 * a * n = t + (n * n + 1) ∧ m < t ∧
      n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) := by
  

Proof state:
m n k : ℕ
⊢ n ^ k = m ↔
    k = 0 ∧ m = 1 ∨
      0 < k ∧
        (n = 0 ∧ m = 0 ∨
          0 < n ∧
            ∃ w a t z,
              ∃ (a1 : 1 < a),
                xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
                  2 * a * n = t + (n * n + 1) ∧
                    m < t ∧ n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1)

Next tactic:
constructor
theorem eq_pow_of_pell {m n k} :
    n ^ k = m ↔ k = 0 ∧ m = 1 ∨0 < k ∧ (n = 0 ∧ m = 0 ∨
      0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
      2 * a * n = t + (n * n + 1) ∧ m < t ∧
      n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) := by
  constructor
  · 

Proof state:
case mp
m n k : ℕ
⊢ n ^ k = m →
    k = 0 ∧ m = 1 ∨
      0 < k ∧
        (n = 0 ∧ m = 0 ∨
          0 < n ∧
            ∃ w a t z,
              ∃ (a1 : 1 < a),
                xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
                  2 * a * n = t + (n * n + 1) ∧
                    m < t ∧ n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1)

Next tactic:
rintro rfl
theorem eq_pow_of_pell {m n k} :
    n ^ k = m ↔ k = 0 ∧ m = 1 ∨0 < k ∧ (n = 0 ∧ m = 0 ∨
      0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
      2 * a * n = t + (n * n + 1) ∧ m < t ∧
      n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) := by
  constructor
  · rintro rfl
    

Proof state:
case mp
n k : ℕ
⊢ k = 0 ∧ n ^ k = 1 ∨
    0 < k ∧
      (n = 0 ∧ n ^ k = 0 ∨
        0 < n ∧
          ∃ w a t z,
            ∃ (a1 : 1 < a),
              xn a1 k ≡ yn a1 k * (a - n) + n ^ k [MOD t] ∧
                2 * a * n = t + (n * n + 1) ∧
                  n ^ k < t ∧ n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1)

Next tactic:
refine' k.eq_zero_or_pos.imp (fun k0 : k = 0 => k0.symm ▸ ⟨rfl, rfl⟩) fun hk => ⟨hk, _⟩
theorem eq_pow_of_pell {m n k} :
    n ^ k = m ↔ k = 0 ∧ m = 1 ∨0 < k ∧ (n = 0 ∧ m = 0 ∨
      0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
      2 * a * n = t + (n * n + 1) ∧ m < t ∧
      n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) := by
  constructor
  · rintro rfl
    refine' k.eq_zero_or_pos.imp (fun k0 : k = 0 => k0.symm ▸ ⟨rfl, rfl⟩) fun hk => ⟨hk, _⟩
    

Proof state:
case mp
n k : ℕ
hk : 0 < k
⊢ n = 0 ∧ n ^ k = 0 ∨
    0 < n ∧
      ∃ w a t z,
        ∃ (a1 : 1 < a),
          xn a1 k ≡ yn a1 k * (a - n) + n ^ k [MOD t] ∧
            2 * a * n = t + (n * n + 1) ∧
              n ^ k < t ∧ n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1

Next tactic:
refine' n.eq_zero_or_pos.imp (fun n0 : n = 0 => n0.symm ▸ ⟨rfl, zero_pow hk⟩) fun hn => ⟨hn, _⟩
theorem eq_pow_of_pell {m n k} :
    n ^ k = m ↔ k = 0 ∧ m = 1 ∨0 < k ∧ (n = 0 ∧ m = 0 ∨
      0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
      2 * a * n = t + (n * n + 1) ∧ m < t ∧
      n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) := by
  constructor
  · rintro rfl
    refine' k.eq_zero_or_pos.imp (fun k0 : k = 0 => k0.symm ▸ ⟨rfl, rfl⟩) fun hk => ⟨hk, _⟩
    refine' n.eq_zero_or_pos.imp (fun n0 : n = 0 => n0.symm ▸ ⟨rfl, zero_pow hk⟩) fun hn => ⟨hn, _⟩
    

Proof state:
case mp
n k : ℕ
hk : 0 < k
hn : 0 < n
⊢ ∃ w a t z,
    ∃ (a1 : 1 < a),
      xn a1 k ≡ yn a1 k * (a - n) + n ^ k [MOD t] ∧
        2 * a * n = t + (n * n + 1) ∧
          n ^ k < t ∧ n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1

Next tactic:
set w := max n k
theorem eq_pow_of_pell {m n k} :
    n ^ k = m ↔ k = 0 ∧ m = 1 ∨0 < k ∧ (n = 0 ∧ m = 0 ∨
      0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
      2 * a * n = t + (n * n + 1) ∧ m < t ∧
      n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) := by
  constructor
  · rintro rfl
    refine' k.eq_zero_or_pos.imp (fun k0 : k = 0 => k0.symm ▸ ⟨rfl, rfl⟩) fun hk => ⟨hk, _⟩
    refine' n.eq_zero_or_pos.imp (fun n0 : n = 0 => n0.symm ▸ ⟨rfl, zero_pow hk⟩) fun hn => ⟨hn, _⟩
    set w := max n k
    

Proof state:
case mp
n k : ℕ
hk : 0 < k
hn : 0 < n
w : ℕ := max n k
⊢ ∃ w a t z,
    ∃ (a1 : 1 < a),
      xn a1 k ≡ yn a1 k * (a - n) + n ^ k [MOD t] ∧
        2 * a * n = t + (n * n + 1) ∧
          n ^ k < t ∧ n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1

Next tactic:
have nw : n ≤ w := le_max_left _ _
theorem eq_pow_of_pell {m n k} :
    n ^ k = m ↔ k = 0 ∧ m = 1 ∨0 < k ∧ (n = 0 ∧ m = 0 ∨
      0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
      2 * a * n = t + (n * n + 1) ∧ m < t ∧
      n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) := by
  constructor
  · rintro rfl
    refine' k.eq_zero_or_pos.imp (fun k0 : k = 0 => k0.symm ▸ ⟨rfl, rfl⟩) fun hk => ⟨hk, _⟩
    refine' n.eq_zero_or_pos.imp (fun n0 : n = 0 => n0.symm ▸ ⟨rfl, zero_pow hk⟩) fun hn => ⟨hn, _⟩
    set w := max n k
    have nw : n ≤ w := le_max_left _ _
    

Proof state:
case mp
n k : ℕ
hk : 0 < k
hn : 0 < n
w : ℕ := max n k
nw : n ≤ w
⊢ ∃ w a t z,
    ∃ (a1 : 1 < a),
      xn a1 k ≡ yn a1 k * (a - n) + n ^ k [MOD t] ∧
        2 * a * n = t + (n * n + 1) ∧
          n ^ k < t ∧ n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1

Next tactic:
have kw : k ≤ w := le_max_right _ _
theorem eq_pow_of_pell {m n k} :
    n ^ k = m ↔ k = 0 ∧ m = 1 ∨0 < k ∧ (n = 0 ∧ m = 0 ∨
      0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
      2 * a * n = t + (n * n + 1) ∧ m < t ∧
      n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) := by
  constructor
  · rintro rfl
    refine' k.eq_zero_or_pos.imp (fun k0 : k = 0 => k0.symm ▸ ⟨rfl, rfl⟩) fun hk => ⟨hk, _⟩
    refine' n.eq_zero_or_pos.imp (fun n0 : n = 0 => n0.symm ▸ ⟨rfl, zero_pow hk⟩) fun hn => ⟨hn, _⟩
    set w := max n k
    have nw : n ≤ w := le_max_left _ _
    have kw : k ≤ w := le_max_right _ _
    

Proof state:
case mp
n k : ℕ
hk : 0 < k
hn : 0 < n
w : ℕ := max n k
nw : n ≤ w
kw : k ≤ w
⊢ ∃ w a t z,
    ∃ (a1 : 1 < a),
      xn a1 k ≡ yn a1 k * (a - n) + n ^ k [MOD t] ∧
        2 * a * n = t + (n * n + 1) ∧
          n ^ k < t ∧ n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1

Next tactic:
have wpos : 0 < w := hn.trans_le nw
theorem eq_pow_of_pell {m n k} :
    n ^ k = m ↔ k = 0 ∧ m = 1 ∨0 < k ∧ (n = 0 ∧ m = 0 ∨
      0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
      2 * a * n = t + (n * n + 1) ∧ m < t ∧
      n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) := by
  constructor
  · rintro rfl
    refine' k.eq_zero_or_pos.imp (fun k0 : k = 0 => k0.symm ▸ ⟨rfl, rfl⟩) fun hk => ⟨hk, _⟩
    refine' n.eq_zero_or_pos.imp (fun n0 : n = 0 => n0.symm ▸ ⟨rfl, zero_pow hk⟩) fun hn => ⟨hn, _⟩
    set w := max n k
    have nw : n ≤ w := le_max_left _ _
    have kw : k ≤ w := le_max_right _ _
    have wpos : 0 < w := hn.trans_le nw
    

Proof state:
case mp
n k : ℕ
hk : 0 < k
hn : 0 < n
w : ℕ := max n k
nw : n ≤ w
kw : k ≤ w
wpos : 0 < w
⊢ ∃ w a t z,
    ∃ (a1 : 1 < a),
      xn a1 k ≡ yn a1 k * (a - n) + n ^ k [MOD t] ∧
        2 * a * n = t + (n * n + 1) ∧
          n ^ k < t ∧ n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1

Next tactic:
have w1 : 1 < w + 1 := Nat.succ_lt_succ wpos
theorem eq_pow_of_pell {m n k} :
    n ^ k = m ↔ k = 0 ∧ m = 1 ∨0 < k ∧ (n = 0 ∧ m = 0 ∨
      0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
      2 * a * n = t + (n * n + 1) ∧ m < t ∧
      n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) := by
  constructor
  · rintro rfl
    refine' k.eq_zero_or_pos.imp (fun k0 : k = 0 => k0.symm ▸ ⟨rfl, rfl⟩) fun hk => ⟨hk, _⟩
    refine' n.eq_zero_or_pos.imp (fun n0 : n = 0 => n0.symm ▸ ⟨rfl, zero_pow hk⟩) fun hn => ⟨hn, _⟩
    set w := max n k
    have nw : n ≤ w := le_max_left _ _
    have kw : k ≤ w := le_max_right _ _
    have wpos : 0 < w := hn.trans_le nw
    have w1 : 1 < w + 1 := Nat.succ_lt_succ wpos
    

Proof state:
case mp
n k : ℕ
hk : 0 < k
hn : 0 < n
w : ℕ := max n k
nw : n ≤ w
kw : k ≤ w
wpos : 0 < w
w1 : 1 < w + 1
⊢ ∃ w a t z,
    ∃ (a1 : 1 < a),
      xn a1 k ≡ yn a1 k * (a - n) + n ^ k [MOD t] ∧
        2 * a * n = t + (n * n + 1) ∧
          n ^ k < t ∧ n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1

Next tactic:
set a := xn w1 w
theorem eq_pow_of_pell {m n k} :
    n ^ k = m ↔ k = 0 ∧ m = 1 ∨0 < k ∧ (n = 0 ∧ m = 0 ∨
      0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
      2 * a * n = t + (n * n + 1) ∧ m < t ∧
      n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) := by
  constructor
  · rintro rfl
    refine' k.eq_zero_or_pos.imp (fun k0 : k = 0 => k0.symm ▸ ⟨rfl, rfl⟩) fun hk => ⟨hk, _⟩
    refine' n.eq_zero_or_pos.imp (fun n0 : n = 0 => n0.symm ▸ ⟨rfl, zero_pow hk⟩) fun hn => ⟨hn, _⟩
    set w := max n k
    have nw : n ≤ w := le_max_left _ _
    have kw : k ≤ w := le_max_right _ _
    have wpos : 0 < w := hn.trans_le nw
    have w1 : 1 < w + 1 := Nat.succ_lt_succ wpos
    set a := xn w1 w
    

Proof state:
case mp
n k : ℕ
hk : 0 < k
hn : 0 < n
w : ℕ := max n k
nw : n ≤ w
kw : k ≤ w
wpos : 0 < w
w1 : 1 < w + 1
a : ℕ := xn w1 w
⊢ ∃ w a t z,
    ∃ (a1 : 1 < a),
      xn a1 k ≡ yn a1 k * (a - n) + n ^ k [MOD t] ∧
        2 * a * n = t + (n * n + 1) ∧
          n ^ k < t ∧ n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1

Next tactic:
have a1 : 1 < a := strictMono_x w1 wpos
theorem eq_pow_of_pell {m n k} :
    n ^ k = m ↔ k = 0 ∧ m = 1 ∨0 < k ∧ (n = 0 ∧ m = 0 ∨
      0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
      2 * a * n = t + (n * n + 1) ∧ m < t ∧
      n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) := by
  constructor
  · rintro rfl
    refine' k.eq_zero_or_pos.imp (fun k0 : k = 0 => k0.symm ▸ ⟨rfl, rfl⟩) fun hk => ⟨hk, _⟩
    refine' n.eq_zero_or_pos.imp (fun n0 : n = 0 => n0.symm ▸ ⟨rfl, zero_pow hk⟩) fun hn => ⟨hn, _⟩
    set w := max n k
    have nw : n ≤ w := le_max_left _ _
    have kw : k ≤ w := le_max_right _ _
    have wpos : 0 < w := hn.trans_le nw
    have w1 : 1 < w + 1 := Nat.succ_lt_succ wpos
    set a := xn w1 w
    have a1 : 1 < a := strictMono_x w1 wpos
    

Proof state:
case mp
n k : ℕ
hk : 0 < k
hn : 0 < n
w : ℕ := max n k
nw : n ≤ w
kw : k ≤ w
wpos : 0 < w
w1 : 1 < w + 1
a : ℕ := xn w1 w
a1 : 1 < a
⊢ ∃ w a t z,
    ∃ (a1 : 1 < a),
      xn a1 k ≡ yn a1 k * (a - n) + n ^ k [MOD t] ∧
        2 * a * n = t + (n * n + 1) ∧
          n ^ k < t ∧ n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1

Next tactic:
have na : n ≤ a := nw.trans (n_lt_xn w1 w).le
theorem eq_pow_of_pell {m n k} :
    n ^ k = m ↔ k = 0 ∧ m = 1 ∨0 < k ∧ (n = 0 ∧ m = 0 ∨
      0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
      2 * a * n = t + (n * n + 1) ∧ m < t ∧
      n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) := by
  constructor
  · rintro rfl
    refine' k.eq_zero_or_pos.imp (fun k0 : k = 0 => k0.symm ▸ ⟨rfl, rfl⟩) fun hk => ⟨hk, _⟩
    refine' n.eq_zero_or_pos.imp (fun n0 : n = 0 => n0.symm ▸ ⟨rfl, zero_pow hk⟩) fun hn => ⟨hn, _⟩
    set w := max n k
    have nw : n ≤ w := le_max_left _ _
    have kw : k ≤ w := le_max_right _ _
    have wpos : 0 < w := hn.trans_le nw
    have w1 : 1 < w + 1 := Nat.succ_lt_succ wpos
    set a := xn w1 w
    have a1 : 1 < a := strictMono_x w1 wpos
    have na : n ≤ a := nw.trans (n_lt_xn w1 w).le
    

Proof state:
case mp
n k : ℕ
hk : 0 < k
hn : 0 < n
w : ℕ := max n k
nw : n ≤ w
kw : k ≤ w
wpos : 0 < w
w1 : 1 < w + 1
a : ℕ := xn w1 w
a1 : 1 < a
na : n ≤ a
⊢ ∃ w a t z,
    ∃ (a1 : 1 < a),
      xn a1 k ≡ yn a1 k * (a - n) + n ^ k [MOD t] ∧
        2 * a * n = t + (n * n + 1) ∧
          n ^ k < t ∧ n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1

Next tactic:
set x := xn a1 k
theorem eq_pow_of_pell {m n k} :
    n ^ k = m ↔ k = 0 ∧ m = 1 ∨0 < k ∧ (n = 0 ∧ m = 0 ∨
      0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
      2 * a * n = t + (n * n + 1) ∧ m < t ∧
      n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) := by
  constructor
  · rintro rfl
    refine' k.eq_zero_or_pos.imp (fun k0 : k = 0 => k0.symm ▸ ⟨rfl, rfl⟩) fun hk => ⟨hk, _⟩
    refine' n.eq_zero_or_pos.imp (fun n0 : n = 0 => n0.symm ▸ ⟨rfl, zero_pow hk⟩) fun hn => ⟨hn, _⟩
    set w := max n k
    have nw : n ≤ w := le_max_left _ _
    have kw : k ≤ w := le_max_right _ _
    have wpos : 0 < w := hn.trans_le nw
    have w1 : 1 < w + 1 := Nat.succ_lt_succ wpos
    set a := xn w1 w
    have a1 : 1 < a := strictMono_x w1 wpos
    have na : n ≤ a := nw.trans (n_lt_xn w1 w).le
    set x := xn a1 k
    

Proof state:
case mp
n k : ℕ
hk : 0 < k
hn : 0 < n
w : ℕ := max n k
nw : n ≤ w
kw : k ≤ w
wpos : 0 < w
w1 : 1 < w + 1
a : ℕ := xn w1 w
a1 : 1 < a
na : n ≤ a
x : ℕ := xn a1 k
⊢ ∃ w a t z,
    ∃ (a1 : 1 < a),
      xn a1 k ≡ yn a1 k * (a - n) + n ^ k [MOD t] ∧
        2 * a * n = t + (n * n + 1) ∧
          n ^ k < t ∧ n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1

Next tactic:
set y := yn a1 k
theorem eq_pow_of_pell {m n k} :
    n ^ k = m ↔ k = 0 ∧ m = 1 ∨0 < k ∧ (n = 0 ∧ m = 0 ∨
      0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
      2 * a * n = t + (n * n + 1) ∧ m < t ∧
      n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) := by
  constructor
  · rintro rfl
    refine' k.eq_zero_or_pos.imp (fun k0 : k = 0 => k0.symm ▸ ⟨rfl, rfl⟩) fun hk => ⟨hk, _⟩
    refine' n.eq_zero_or_pos.imp (fun n0 : n = 0 => n0.symm ▸ ⟨rfl, zero_pow hk⟩) fun hn => ⟨hn, _⟩
    set w := max n k
    have nw : n ≤ w := le_max_left _ _
    have kw : k ≤ w := le_max_right _ _
    have wpos : 0 < w := hn.trans_le nw
    have w1 : 1 < w + 1 := Nat.succ_lt_succ wpos
    set a := xn w1 w
    have a1 : 1 < a := strictMono_x w1 wpos
    have na : n ≤ a := nw.trans (n_lt_xn w1 w).le
    set x := xn a1 k
    set y := yn a1 k
    

Proof state:
case mp
n k : ℕ
hk : 0 < k
hn : 0 < n
w : ℕ := max n k
nw : n ≤ w
kw : k ≤ w
wpos : 0 < w
w1 : 1 < w + 1
a : ℕ := xn w1 w
a1 : 1 < a
na : n ≤ a
x : ℕ := xn a1 k
y : ℕ := yn a1 k
⊢ ∃ w a t z,
    ∃ (a1 : 1 < a),
      xn a1 k ≡ yn a1 k * (a - n) + n ^ k [MOD t] ∧
        2 * a * n = t + (n * n + 1) ∧
          n ^ k < t ∧ n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1

Next tactic:
obtain ⟨z, ze⟩ : w ∣ yn w1 w
theorem eq_pow_of_pell {m n k} :
    n ^ k = m ↔ k = 0 ∧ m = 1 ∨0 < k ∧ (n = 0 ∧ m = 0 ∨
      0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
      2 * a * n = t + (n * n + 1) ∧ m < t ∧
      n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) := by
  constructor
  · rintro rfl
    refine' k.eq_zero_or_pos.imp (fun k0 : k = 0 => k0.symm ▸ ⟨rfl, rfl⟩) fun hk => ⟨hk, _⟩
    refine' n.eq_zero_or_pos.imp (fun n0 : n = 0 => n0.symm ▸ ⟨rfl, zero_pow hk⟩) fun hn => ⟨hn, _⟩
    set w := max n k
    have nw : n ≤ w := le_max_left _ _
    have kw : k ≤ w := le_max_right _ _
    have wpos : 0 < w := hn.trans_le nw
    have w1 : 1 < w + 1 := Nat.succ_lt_succ wpos
    set a := xn w1 w
    have a1 : 1 < a := strictMono_x w1 wpos
    have na : n ≤ a := nw.trans (n_lt_xn w1 w).le
    set x := xn a1 k
    set y := yn a1 k
    obtain ⟨z, ze⟩ : w ∣ yn w1 w
    

Proof state:
n k : ℕ
hk : 0 < k
hn : 0 < n
w : ℕ := max n k
nw : n ≤ w
kw : k ≤ w
wpos : 0 < w
w1 : 1 < w + 1
a : ℕ := xn w1 w
a1 : 1 < a
na : n ≤ a
x : ℕ := xn a1 k
y : ℕ := yn a1 k
⊢ w ∣ yn w1 w
case mp.intro
n k : ℕ
hk : 0 < k
hn : 0 < n
w : ℕ := max n k
nw : n ≤ w
kw : k ≤ w
wpos : 0 < w
w1 : 1 < w + 1
a : ℕ := xn w1 w
a1 : 1 < a
na : n ≤ a
x : ℕ := xn a1 k
y : ℕ := yn a1 k
z : ℕ
ze : yn w1 w = w * z
⊢ ∃ w a t z,
    ∃ (a1 : 1 < a),
      xn a1 k ≡ yn a1 k * (a - n) + n ^ k [MOD t] ∧
        2 * a * n = t + (n * n + 1) ∧
          n ^ k < t ∧ n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1

Next tactic:
exact modEq_zero_iff_dvd.1 ((yn_modEq_a_sub_one w1 w).trans dvd_rfl.modEq_zero_nat)
theorem eq_pow_of_pell {m n k} :
    n ^ k = m ↔ k = 0 ∧ m = 1 ∨0 < k ∧ (n = 0 ∧ m = 0 ∨
      0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
      2 * a * n = t + (n * n + 1) ∧ m < t ∧
      n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) := by
  constructor
  · rintro rfl
    refine' k.eq_zero_or_pos.imp (fun k0 : k = 0 => k0.symm ▸ ⟨rfl, rfl⟩) fun hk => ⟨hk, _⟩
    refine' n.eq_zero_or_pos.imp (fun n0 : n = 0 => n0.symm ▸ ⟨rfl, zero_pow hk⟩) fun hn => ⟨hn, _⟩
    set w := max n k
    have nw : n ≤ w := le_max_left _ _
    have kw : k ≤ w := le_max_right _ _
    have wpos : 0 < w := hn.trans_le nw
    have w1 : 1 < w + 1 := Nat.succ_lt_succ wpos
    set a := xn w1 w
    have a1 : 1 < a := strictMono_x w1 wpos
    have na : n ≤ a := nw.trans (n_lt_xn w1 w).le
    set x := xn a1 k
    set y := yn a1 k
    obtain ⟨z, ze⟩ : w ∣ yn w1 w
    exact modEq_zero_iff_dvd.1 ((yn_modEq_a_sub_one w1 w).trans dvd_rfl.modEq_zero_nat)
    

Proof state:
case mp.intro
n k : ℕ
hk : 0 < k
hn : 0 < n
w : ℕ := max n k
nw : n ≤ w
kw : k ≤ w
wpos : 0 < w
w1 : 1 < w + 1
a : ℕ := xn w1 w
a1 : 1 < a
na : n ≤ a
x : ℕ := xn a1 k
y : ℕ := yn a1 k
z : ℕ
ze : yn w1 w = w * z
⊢ ∃ w a t z,
    ∃ (a1 : 1 < a),
      xn a1 k ≡ yn a1 k * (a - n) + n ^ k [MOD t] ∧
        2 * a * n = t + (n * n + 1) ∧
          n ^ k < t ∧ n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1

Next tactic:
have nt : (↑(n ^ k) : ℤ) < 2 * a * n - n * n - 1 := by
      refine' eq_pow_of_pell_lem hn.ne' hk.ne' _
      calc
        n ^ k ≤ n ^ w := Nat.pow_le_pow_of_le_right hn kw
        _ < (w + 1) ^ w := (Nat.pow_lt_pow_left (Nat.lt_succ_of_le nw) wpos.ne')
        _ ≤ a := xn_ge_a_pow w1 w