l3lab/ntp-mathlib huggingface dataset:
/-- If `d` is a positive integer that is not a square, then there is a nontrivial solution
to the Pell equation `x^2 - d*y^2 = 1`. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0 := by
  let ξ : ℝ := sqrt d
  have hξ : Irrational ξ := by
    refine' irrational_nrt_of_notint_nrt 2 d (sq_sqrt <| Int.cast_nonneg.mpr h₀.le) _ two_pos
    rintro ⟨x, hx⟩
    refine' hd ⟨x, @Int.cast_injective ℝ _ _ d (x * x) _⟩
    rw [← sq_sqrt <| Int.cast_nonneg.mpr h₀.le, Int.cast_mul, ← hx, sq]
  

Proof state:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := sqrt ↑d
hξ : Irrational ξ
⊢ ∃ x y, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0

Next tactic:
obtain ⟨M, hM₁⟩ := exists_int_gt (2 * |ξ| + 1)
/-- If `d` is a positive integer that is not a square, then there is a nontrivial solution
to the Pell equation `x^2 - d*y^2 = 1`. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0 := by
  let ξ : ℝ := sqrt d
  have hξ : Irrational ξ := by
    refine' irrational_nrt_of_notint_nrt 2 d (sq_sqrt <| Int.cast_nonneg.mpr h₀.le) _ two_pos
    rintro ⟨x, hx⟩
    refine' hd ⟨x, @Int.cast_injective ℝ _ _ d (x * x) _⟩
    rw [← sq_sqrt <| Int.cast_nonneg.mpr h₀.le, Int.cast_mul, ← hx, sq]
  obtain ⟨M, hM₁⟩ := exists_int_gt (2 * |ξ| + 1)
  

Proof state:
case intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := sqrt ↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
⊢ ∃ x y, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0

Next tactic:
have hM : {q : ℚ | |q.1 ^ 2 - d * (q.2 : ℤ) ^ 2| < M}.Infinite := by
    refine' Infinite.mono (fun q h => _) (infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational hξ)
    have h0 : 0 < (q.2 : ℝ) ^ 2 := pow_pos (Nat.cast_pos.mpr q.pos) 2
    have h1 : (q.num : ℝ) / (q.den : ℝ) = q := mod_cast q.num_div_den
    rw [mem_setOf, abs_sub_comm, ← @Int.cast_lt ℝ, ← div_lt_div_right (abs_pos_of_pos h0)]
    push_cast
    rw [← abs_div, abs_sq, sub_div, mul_div_cancel _ h0.ne', ← div_pow, h1, ←
      sq_sqrt (Int.cast_pos.mpr h₀).le, sq_sub_sq, abs_mul, ← mul_one_div]
    refine' mul_lt_mul'' (((abs_add ξ q).trans _).trans_lt hM₁) h (abs_nonneg _) (abs_nonneg _)
    rw [two_mul, add_assoc, add_le_add_iff_left, ← sub_le_iff_le_add']
    rw [mem_setOf, abs_sub_comm] at h
    refine' (abs_sub_abs_le_abs_sub (q : ℝ) ξ).trans (h.le.trans _)
    rw [div_le_one h0, one_le_sq_iff_one_le_abs, Nat.abs_cast, Nat.one_le_cast]
    exact q.pos
/-- If `d` is a positive integer that is not a square, then there is a nontrivial solution
to the Pell equation `x^2 - d*y^2 = 1`. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0 := by
  let ξ : ℝ := sqrt d
  have hξ : Irrational ξ := by
    refine' irrational_nrt_of_notint_nrt 2 d (sq_sqrt <| Int.cast_nonneg.mpr h₀.le) _ two_pos
    rintro ⟨x, hx⟩
    refine' hd ⟨x, @Int.cast_injective ℝ _ _ d (x * x) _⟩
    rw [← sq_sqrt <| Int.cast_nonneg.mpr h₀.le, Int.cast_mul, ← hx, sq]
  obtain ⟨M, hM₁⟩ := exists_int_gt (2 * |ξ| + 1)
  have hM : {q : ℚ | |q.1 ^ 2 - d * (q.2 : ℤ) ^ 2| < M}.Infinite := by
    

Proof state:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := sqrt ↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
⊢ Set.Infinite {q | |q.num ^ 2 - d * ↑q.den ^ 2| < M}

Next tactic:
refine' Infinite.mono (fun q h => _) (infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational hξ)
/-- If `d` is a positive integer that is not a square, then there is a nontrivial solution
to the Pell equation `x^2 - d*y^2 = 1`. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0 := by
  let ξ : ℝ := sqrt d
  have hξ : Irrational ξ := by
    refine' irrational_nrt_of_notint_nrt 2 d (sq_sqrt <| Int.cast_nonneg.mpr h₀.le) _ two_pos
    rintro ⟨x, hx⟩
    refine' hd ⟨x, @Int.cast_injective ℝ _ _ d (x * x) _⟩
    rw [← sq_sqrt <| Int.cast_nonneg.mpr h₀.le, Int.cast_mul, ← hx, sq]
  obtain ⟨M, hM₁⟩ := exists_int_gt (2 * |ξ| + 1)
  have hM : {q : ℚ | |q.1 ^ 2 - d * (q.2 : ℤ) ^ 2| < M}.Infinite := by
    refine' Infinite.mono (fun q h => _) (infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational hξ)
    

Proof state:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := sqrt ↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
q : ℚ
h : q ∈ {q | |ξ - ↑q| < 1 / ↑q.den ^ 2}
⊢ q ∈ {q | |q.num ^ 2 - d * ↑q.den ^ 2| < M}

Next tactic:
have h0 : 0 < (q.2 : ℝ) ^ 2 := pow_pos (Nat.cast_pos.mpr q.pos) 2
/-- If `d` is a positive integer that is not a square, then there is a nontrivial solution
to the Pell equation `x^2 - d*y^2 = 1`. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0 := by
  let ξ : ℝ := sqrt d
  have hξ : Irrational ξ := by
    refine' irrational_nrt_of_notint_nrt 2 d (sq_sqrt <| Int.cast_nonneg.mpr h₀.le) _ two_pos
    rintro ⟨x, hx⟩
    refine' hd ⟨x, @Int.cast_injective ℝ _ _ d (x * x) _⟩
    rw [← sq_sqrt <| Int.cast_nonneg.mpr h₀.le, Int.cast_mul, ← hx, sq]
  obtain ⟨M, hM₁⟩ := exists_int_gt (2 * |ξ| + 1)
  have hM : {q : ℚ | |q.1 ^ 2 - d * (q.2 : ℤ) ^ 2| < M}.Infinite := by
    refine' Infinite.mono (fun q h => _) (infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational hξ)
    have h0 : 0 < (q.2 : ℝ) ^ 2 := pow_pos (Nat.cast_pos.mpr q.pos) 2
    

Proof state:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := sqrt ↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
q : ℚ
h : q ∈ {q | |ξ - ↑q| < 1 / ↑q.den ^ 2}
h0 : 0 < ↑q.den ^ 2
⊢ q ∈ {q | |q.num ^ 2 - d * ↑q.den ^ 2| < M}

Next tactic:
have h1 : (q.num : ℝ) / (q.den : ℝ) = q := mod_cast q.num_div_den
/-- If `d` is a positive integer that is not a square, then there is a nontrivial solution
to the Pell equation `x^2 - d*y^2 = 1`. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0 := by
  let ξ : ℝ := sqrt d
  have hξ : Irrational ξ := by
    refine' irrational_nrt_of_notint_nrt 2 d (sq_sqrt <| Int.cast_nonneg.mpr h₀.le) _ two_pos
    rintro ⟨x, hx⟩
    refine' hd ⟨x, @Int.cast_injective ℝ _ _ d (x * x) _⟩
    rw [← sq_sqrt <| Int.cast_nonneg.mpr h₀.le, Int.cast_mul, ← hx, sq]
  obtain ⟨M, hM₁⟩ := exists_int_gt (2 * |ξ| + 1)
  have hM : {q : ℚ | |q.1 ^ 2 - d * (q.2 : ℤ) ^ 2| < M}.Infinite := by
    refine' Infinite.mono (fun q h => _) (infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational hξ)
    have h0 : 0 < (q.2 : ℝ) ^ 2 := pow_pos (Nat.cast_pos.mpr q.pos) 2
    have h1 : (q.num : ℝ) / (q.den : ℝ) = q := mod_cast q.num_div_den
    

Proof state:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := sqrt ↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
q : ℚ
h : q ∈ {q | |ξ - ↑q| < 1 / ↑q.den ^ 2}
h0 : 0 < ↑q.den ^ 2
h1 : ↑q.num / ↑q.den = ↑q
⊢ q ∈ {q | |q.num ^ 2 - d * ↑q.den ^ 2| < M}

Next tactic:
rw [mem_setOf, abs_sub_comm, ← @Int.cast_lt ℝ, ← div_lt_div_right (abs_pos_of_pos h0)]
/-- If `d` is a positive integer that is not a square, then there is a nontrivial solution
to the Pell equation `x^2 - d*y^2 = 1`. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0 := by
  let ξ : ℝ := sqrt d
  have hξ : Irrational ξ := by
    refine' irrational_nrt_of_notint_nrt 2 d (sq_sqrt <| Int.cast_nonneg.mpr h₀.le) _ two_pos
    rintro ⟨x, hx⟩
    refine' hd ⟨x, @Int.cast_injective ℝ _ _ d (x * x) _⟩
    rw [← sq_sqrt <| Int.cast_nonneg.mpr h₀.le, Int.cast_mul, ← hx, sq]
  obtain ⟨M, hM₁⟩ := exists_int_gt (2 * |ξ| + 1)
  have hM : {q : ℚ | |q.1 ^ 2 - d * (q.2 : ℤ) ^ 2| < M}.Infinite := by
    refine' Infinite.mono (fun q h => _) (infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational hξ)
    have h0 : 0 < (q.2 : ℝ) ^ 2 := pow_pos (Nat.cast_pos.mpr q.pos) 2
    have h1 : (q.num : ℝ) / (q.den : ℝ) = q := mod_cast q.num_div_den
    rw [mem_setOf, abs_sub_comm, ← @Int.cast_lt ℝ, ← div_lt_div_right (abs_pos_of_pos h0)]
    

Proof state:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := sqrt ↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
q : ℚ
h : q ∈ {q | |ξ - ↑q| < 1 / ↑q.den ^ 2}
h0 : 0 < ↑q.den ^ 2
h1 : ↑q.num / ↑q.den = ↑q
⊢ ↑|d * ↑q.den ^ 2 - q.num ^ 2| / |↑q.den ^ 2| < ↑M / |↑q.den ^ 2|

Next tactic:
push_cast
/-- If `d` is a positive integer that is not a square, then there is a nontrivial solution
to the Pell equation `x^2 - d*y^2 = 1`. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0 := by
  let ξ : ℝ := sqrt d
  have hξ : Irrational ξ := by
    refine' irrational_nrt_of_notint_nrt 2 d (sq_sqrt <| Int.cast_nonneg.mpr h₀.le) _ two_pos
    rintro ⟨x, hx⟩
    refine' hd ⟨x, @Int.cast_injective ℝ _ _ d (x * x) _⟩
    rw [← sq_sqrt <| Int.cast_nonneg.mpr h₀.le, Int.cast_mul, ← hx, sq]
  obtain ⟨M, hM₁⟩ := exists_int_gt (2 * |ξ| + 1)
  have hM : {q : ℚ | |q.1 ^ 2 - d * (q.2 : ℤ) ^ 2| < M}.Infinite := by
    refine' Infinite.mono (fun q h => _) (infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational hξ)
    have h0 : 0 < (q.2 : ℝ) ^ 2 := pow_pos (Nat.cast_pos.mpr q.pos) 2
    have h1 : (q.num : ℝ) / (q.den : ℝ) = q := mod_cast q.num_div_den
    rw [mem_setOf, abs_sub_comm, ← @Int.cast_lt ℝ, ← div_lt_div_right (abs_pos_of_pos h0)]
    push_cast
    

Proof state:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := sqrt ↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
q : ℚ
h : q ∈ {q | |ξ - ↑q| < 1 / ↑q.den ^ 2}
h0 : 0 < ↑q.den ^ 2
h1 : ↑q.num / ↑q.den = ↑q
⊢ |↑d * ↑q.den ^ 2 - ↑q.num ^ 2| / |↑q.den ^ 2| < ↑M / |↑q.den ^ 2|

Next tactic:
rw [← abs_div, abs_sq, sub_div, mul_div_cancel _ h0.ne', ← div_pow, h1, ←
      sq_sqrt (Int.cast_pos.mpr h₀).le, sq_sub_sq, abs_mul, ← mul_one_div]
/-- If `d` is a positive integer that is not a square, then there is a nontrivial solution
to the Pell equation `x^2 - d*y^2 = 1`. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0 := by
  let ξ : ℝ := sqrt d
  have hξ : Irrational ξ := by
    refine' irrational_nrt_of_notint_nrt 2 d (sq_sqrt <| Int.cast_nonneg.mpr h₀.le) _ two_pos
    rintro ⟨x, hx⟩
    refine' hd ⟨x, @Int.cast_injective ℝ _ _ d (x * x) _⟩
    rw [← sq_sqrt <| Int.cast_nonneg.mpr h₀.le, Int.cast_mul, ← hx, sq]
  obtain ⟨M, hM₁⟩ := exists_int_gt (2 * |ξ| + 1)
  have hM : {q : ℚ | |q.1 ^ 2 - d * (q.2 : ℤ) ^ 2| < M}.Infinite := by
    refine' Infinite.mono (fun q h => _) (infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational hξ)
    have h0 : 0 < (q.2 : ℝ) ^ 2 := pow_pos (Nat.cast_pos.mpr q.pos) 2
    have h1 : (q.num : ℝ) / (q.den : ℝ) = q := mod_cast q.num_div_den
    rw [mem_setOf, abs_sub_comm, ← @Int.cast_lt ℝ, ← div_lt_div_right (abs_pos_of_pos h0)]
    push_cast
    rw [← abs_div, abs_sq, sub_div, mul_div_cancel _ h0.ne', ← div_pow, h1, ←
      sq_sqrt (Int.cast_pos.mpr h₀).le, sq_sub_sq, abs_mul, ← mul_one_div]
    

Proof state:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := sqrt ↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
q : ℚ
h : q ∈ {q | |ξ - ↑q| < 1 / ↑q.den ^ 2}
h0 : 0 < ↑q.den ^ 2
h1 : ↑q.num / ↑q.den = ↑q
⊢ |sqrt ↑d + ↑q| * |sqrt ↑d - ↑q| < ↑M * (1 / ↑q.den ^ 2)

Next tactic:
refine' mul_lt_mul'' (((abs_add ξ q).trans _).trans_lt hM₁) h (abs_nonneg _) (abs_nonneg _)
/-- If `d` is a positive integer that is not a square, then there is a nontrivial solution
to the Pell equation `x^2 - d*y^2 = 1`. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0 := by
  let ξ : ℝ := sqrt d
  have hξ : Irrational ξ := by
    refine' irrational_nrt_of_notint_nrt 2 d (sq_sqrt <| Int.cast_nonneg.mpr h₀.le) _ two_pos
    rintro ⟨x, hx⟩
    refine' hd ⟨x, @Int.cast_injective ℝ _ _ d (x * x) _⟩
    rw [← sq_sqrt <| Int.cast_nonneg.mpr h₀.le, Int.cast_mul, ← hx, sq]
  obtain ⟨M, hM₁⟩ := exists_int_gt (2 * |ξ| + 1)
  have hM : {q : ℚ | |q.1 ^ 2 - d * (q.2 : ℤ) ^ 2| < M}.Infinite := by
    refine' Infinite.mono (fun q h => _) (infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational hξ)
    have h0 : 0 < (q.2 : ℝ) ^ 2 := pow_pos (Nat.cast_pos.mpr q.pos) 2
    have h1 : (q.num : ℝ) / (q.den : ℝ) = q := mod_cast q.num_div_den
    rw [mem_setOf, abs_sub_comm, ← @Int.cast_lt ℝ, ← div_lt_div_right (abs_pos_of_pos h0)]
    push_cast
    rw [← abs_div, abs_sq, sub_div, mul_div_cancel _ h0.ne', ← div_pow, h1, ←
      sq_sqrt (Int.cast_pos.mpr h₀).le, sq_sub_sq, abs_mul, ← mul_one_div]
    refine' mul_lt_mul'' (((abs_add ξ q).trans _).trans_lt hM₁) h (abs_nonneg _) (abs_nonneg _)
    

Proof state:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := sqrt ↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
q : ℚ
h : q ∈ {q | |ξ - ↑q| < 1 / ↑q.den ^ 2}
h0 : 0 < ↑q.den ^ 2
h1 : ↑q.num / ↑q.den = ↑q
⊢ |ξ| + |↑q| ≤ 2 * |ξ| + 1

Next tactic:
rw [two_mul, add_assoc, add_le_add_iff_left, ← sub_le_iff_le_add']
/-- If `d` is a positive integer that is not a square, then there is a nontrivial solution
to the Pell equation `x^2 - d*y^2 = 1`. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0 := by
  let ξ : ℝ := sqrt d
  have hξ : Irrational ξ := by
    refine' irrational_nrt_of_notint_nrt 2 d (sq_sqrt <| Int.cast_nonneg.mpr h₀.le) _ two_pos
    rintro ⟨x, hx⟩
    refine' hd ⟨x, @Int.cast_injective ℝ _ _ d (x * x) _⟩
    rw [← sq_sqrt <| Int.cast_nonneg.mpr h₀.le, Int.cast_mul, ← hx, sq]
  obtain ⟨M, hM₁⟩ := exists_int_gt (2 * |ξ| + 1)
  have hM : {q : ℚ | |q.1 ^ 2 - d * (q.2 : ℤ) ^ 2| < M}.Infinite := by
    refine' Infinite.mono (fun q h => _) (infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational hξ)
    have h0 : 0 < (q.2 : ℝ) ^ 2 := pow_pos (Nat.cast_pos.mpr q.pos) 2
    have h1 : (q.num : ℝ) / (q.den : ℝ) = q := mod_cast q.num_div_den
    rw [mem_setOf, abs_sub_comm, ← @Int.cast_lt ℝ, ← div_lt_div_right (abs_pos_of_pos h0)]
    push_cast
    rw [← abs_div, abs_sq, sub_div, mul_div_cancel _ h0.ne', ← div_pow, h1, ←
      sq_sqrt (Int.cast_pos.mpr h₀).le, sq_sub_sq, abs_mul, ← mul_one_div]
    refine' mul_lt_mul'' (((abs_add ξ q).trans _).trans_lt hM₁) h (abs_nonneg _) (abs_nonneg _)
    rw [two_mul, add_assoc, add_le_add_iff_left, ← sub_le_iff_le_add']
    

Proof state:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := sqrt ↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
q : ℚ
h : q ∈ {q | |ξ - ↑q| < 1 / ↑q.den ^ 2}
h0 : 0 < ↑q.den ^ 2
h1 : ↑q.num / ↑q.den = ↑q
⊢ |↑q| - |ξ| ≤ 1

Next tactic:
rw [mem_setOf, abs_sub_comm] at h
/-- If `d` is a positive integer that is not a square, then there is a nontrivial solution
to the Pell equation `x^2 - d*y^2 = 1`. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0 := by
  let ξ : ℝ := sqrt d
  have hξ : Irrational ξ := by
    refine' irrational_nrt_of_notint_nrt 2 d (sq_sqrt <| Int.cast_nonneg.mpr h₀.le) _ two_pos
    rintro ⟨x, hx⟩
    refine' hd ⟨x, @Int.cast_injective ℝ _ _ d (x * x) _⟩
    rw [← sq_sqrt <| Int.cast_nonneg.mpr h₀.le, Int.cast_mul, ← hx, sq]
  obtain ⟨M, hM₁⟩ := exists_int_gt (2 * |ξ| + 1)
  have hM : {q : ℚ | |q.1 ^ 2 - d * (q.2 : ℤ) ^ 2| < M}.Infinite := by
    refine' Infinite.mono (fun q h => _) (infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational hξ)
    have h0 : 0 < (q.2 : ℝ) ^ 2 := pow_pos (Nat.cast_pos.mpr q.pos) 2
    have h1 : (q.num : ℝ) / (q.den : ℝ) = q := mod_cast q.num_div_den
    rw [mem_setOf, abs_sub_comm, ← @Int.cast_lt ℝ, ← div_lt_div_right (abs_pos_of_pos h0)]
    push_cast
    rw [← abs_div, abs_sq, sub_div, mul_div_cancel _ h0.ne', ← div_pow, h1, ←
      sq_sqrt (Int.cast_pos.mpr h₀).le, sq_sub_sq, abs_mul, ← mul_one_div]
    refine' mul_lt_mul'' (((abs_add ξ q).trans _).trans_lt hM₁) h (abs_nonneg _) (abs_nonneg _)
    rw [two_mul, add_assoc, add_le_add_iff_left, ← sub_le_iff_le_add']
    rw [mem_setOf, abs_sub_comm] at h
    

Proof state:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := sqrt ↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
q : ℚ
h : |↑q - ξ| < 1 / ↑q.den ^ 2
h0 : 0 < ↑q.den ^ 2
h1 : ↑q.num / ↑q.den = ↑q
⊢ |↑q| - |ξ| ≤ 1

Next tactic:
refine' (abs_sub_abs_le_abs_sub (q : ℝ) ξ).trans (h.le.trans _)
/-- If `d` is a positive integer that is not a square, then there is a nontrivial solution
to the Pell equation `x^2 - d*y^2 = 1`. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0 := by
  let ξ : ℝ := sqrt d
  have hξ : Irrational ξ := by
    refine' irrational_nrt_of_notint_nrt 2 d (sq_sqrt <| Int.cast_nonneg.mpr h₀.le) _ two_pos
    rintro ⟨x, hx⟩
    refine' hd ⟨x, @Int.cast_injective ℝ _ _ d (x * x) _⟩
    rw [← sq_sqrt <| Int.cast_nonneg.mpr h₀.le, Int.cast_mul, ← hx, sq]
  obtain ⟨M, hM₁⟩ := exists_int_gt (2 * |ξ| + 1)
  have hM : {q : ℚ | |q.1 ^ 2 - d * (q.2 : ℤ) ^ 2| < M}.Infinite := by
    refine' Infinite.mono (fun q h => _) (infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational hξ)
    have h0 : 0 < (q.2 : ℝ) ^ 2 := pow_pos (Nat.cast_pos.mpr q.pos) 2
    have h1 : (q.num : ℝ) / (q.den : ℝ) = q := mod_cast q.num_div_den
    rw [mem_setOf, abs_sub_comm, ← @Int.cast_lt ℝ, ← div_lt_div_right (abs_pos_of_pos h0)]
    push_cast
    rw [← abs_div, abs_sq, sub_div, mul_div_cancel _ h0.ne', ← div_pow, h1, ←
      sq_sqrt (Int.cast_pos.mpr h₀).le, sq_sub_sq, abs_mul, ← mul_one_div]
    refine' mul_lt_mul'' (((abs_add ξ q).trans _).trans_lt hM₁) h (abs_nonneg _) (abs_nonneg _)
    rw [two_mul, add_assoc, add_le_add_iff_left, ← sub_le_iff_le_add']
    rw [mem_setOf, abs_sub_comm] at h
    refine' (abs_sub_abs_le_abs_sub (q : ℝ) ξ).trans (h.le.trans _)
    

Proof state:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := sqrt ↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
q : ℚ
h : |↑q - ξ| < 1 / ↑q.den ^ 2
h0 : 0 < ↑q.den ^ 2
h1 : ↑q.num / ↑q.den = ↑q
⊢ 1 / ↑q.den ^ 2 ≤ 1

Next tactic:
rw [div_le_one h0, one_le_sq_iff_one_le_abs, Nat.abs_cast, Nat.one_le_cast]
/-- If `d` is a positive integer that is not a square, then there is a nontrivial solution
to the Pell equation `x^2 - d*y^2 = 1`. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0 := by
  let ξ : ℝ := sqrt d
  have hξ : Irrational ξ := by
    refine' irrational_nrt_of_notint_nrt 2 d (sq_sqrt <| Int.cast_nonneg.mpr h₀.le) _ two_pos
    rintro ⟨x, hx⟩
    refine' hd ⟨x, @Int.cast_injective ℝ _ _ d (x * x) _⟩
    rw [← sq_sqrt <| Int.cast_nonneg.mpr h₀.le, Int.cast_mul, ← hx, sq]
  obtain ⟨M, hM₁⟩ := exists_int_gt (2 * |ξ| + 1)
  have hM : {q : ℚ | |q.1 ^ 2 - d * (q.2 : ℤ) ^ 2| < M}.Infinite := by
    refine' Infinite.mono (fun q h => _) (infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational hξ)
    have h0 : 0 < (q.2 : ℝ) ^ 2 := pow_pos (Nat.cast_pos.mpr q.pos) 2
    have h1 : (q.num : ℝ) / (q.den : ℝ) = q := mod_cast q.num_div_den
    rw [mem_setOf, abs_sub_comm, ← @Int.cast_lt ℝ, ← div_lt_div_right (abs_pos_of_pos h0)]
    push_cast
    rw [← abs_div, abs_sq, sub_div, mul_div_cancel _ h0.ne', ← div_pow, h1, ←
      sq_sqrt (Int.cast_pos.mpr h₀).le, sq_sub_sq, abs_mul, ← mul_one_div]
    refine' mul_lt_mul'' (((abs_add ξ q).trans _).trans_lt hM₁) h (abs_nonneg _) (abs_nonneg _)
    rw [two_mul, add_assoc, add_le_add_iff_left, ← sub_le_iff_le_add']
    rw [mem_setOf, abs_sub_comm] at h
    refine' (abs_sub_abs_le_abs_sub (q : ℝ) ξ).trans (h.le.trans _)
    rw [div_le_one h0, one_le_sq_iff_one_le_abs, Nat.abs_cast, Nat.one_le_cast]
    

Proof state:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := sqrt ↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
q : ℚ
h : |↑q - ξ| < 1 / ↑q.den ^ 2
h0 : 0 < ↑q.den ^ 2
h1 : ↑q.num / ↑q.den = ↑q
⊢ 1 ≤ q.den

Next tactic:
exact q.pos
/-- If `d` is a positive integer that is not a square, then there is a nontrivial solution
to the Pell equation `x^2 - d*y^2 = 1`. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0 := by
  let ξ : ℝ := sqrt d
  have hξ : Irrational ξ := by
    refine' irrational_nrt_of_notint_nrt 2 d (sq_sqrt <| Int.cast_nonneg.mpr h₀.le) _ two_pos
    rintro ⟨x, hx⟩
    refine' hd ⟨x, @Int.cast_injective ℝ _ _ d (x * x) _⟩
    rw [← sq_sqrt <| Int.cast_nonneg.mpr h₀.le, Int.cast_mul, ← hx, sq]
  obtain ⟨M, hM₁⟩ := exists_int_gt (2 * |ξ| + 1)
  have hM : {q : ℚ | |q.1 ^ 2 - d * (q.2 : ℤ) ^ 2| < M}.Infinite := by
    refine' Infinite.mono (fun q h => _) (infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational hξ)
    have h0 : 0 < (q.2 : ℝ) ^ 2 := pow_pos (Nat.cast_pos.mpr q.pos) 2
    have h1 : (q.num : ℝ) / (q.den : ℝ) = q := mod_cast q.num_div_den
    rw [mem_setOf, abs_sub_comm, ← @Int.cast_lt ℝ, ← div_lt_div_right (abs_pos_of_pos h0)]
    push_cast
    rw [← abs_div, abs_sq, sub_div, mul_div_cancel _ h0.ne', ← div_pow, h1, ←
      sq_sqrt (Int.cast_pos.mpr h₀).le, sq_sub_sq, abs_mul, ← mul_one_div]
    refine' mul_lt_mul'' (((abs_add ξ q).trans _).trans_lt hM₁) h (abs_nonneg _) (abs_nonneg _)
    rw [two_mul, add_assoc, add_le_add_iff_left, ← sub_le_iff_le_add']
    rw [mem_setOf, abs_sub_comm] at h
    refine' (abs_sub_abs_le_abs_sub (q : ℝ) ξ).trans (h.le.trans _)
    rw [div_le_one h0, one_le_sq_iff_one_le_abs, Nat.abs_cast, Nat.one_le_cast]
    exact q.pos
  

Proof state:
case intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := sqrt ↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
hM : Set.Infinite {q | |q.num ^ 2 - d * ↑q.den ^ 2| < M}
⊢ ∃ x y, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0

Next tactic:
obtain ⟨m, hm⟩ : ∃ m : ℤ, {q : ℚ | q.1 ^ 2 - d * (q.den : ℤ) ^ 2 = m}.Infinite := by
    contrapose! hM
    simp only [not_infinite] at hM ⊢
    refine' (congr_arg _ (ext fun x => _)).mp (Finite.biUnion (finite_Ioo (-M) M) fun m _ => hM m)
    simp only [abs_lt, mem_setOf, mem_Ioo, mem_iUnion, exists_prop, exists_eq_right']
/-- If `d` is a positive integer that is not a square, then there is a nontrivial solution
to the Pell equation `x^2 - d*y^2 = 1`. -/
theorem exists_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃ x y : ℤ, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0 := by
  let ξ : ℝ := sqrt d
  have hξ : Irrational ξ := by
    refine' irrational_nrt_of_notint_nrt 2 d (sq_sqrt <| Int.cast_nonneg.mpr h₀.le) _ two_pos
    rintro ⟨x, hx⟩
    refine' hd ⟨x, @Int.cast_injective ℝ _ _ d (x * x) _⟩
    rw [← sq_sqrt <| Int.cast_nonneg.mpr h₀.le, Int.cast_mul, ← hx, sq]
  obtain ⟨M, hM₁⟩ := exists_int_gt (2 * |ξ| + 1)
  have hM : {q : ℚ | |q.1 ^ 2 - d * (q.2 : ℤ) ^ 2| < M}.Infinite := by
    refine' Infinite.mono (fun q h => _) (infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational hξ)
    have h0 : 0 < (q.2 : ℝ) ^ 2 := pow_pos (Nat.cast_pos.mpr q.pos) 2
    have h1 : (q.num : ℝ) / (q.den : ℝ) = q := mod_cast q.num_div_den
    rw [mem_setOf, abs_sub_comm, ← @Int.cast_lt ℝ, ← div_lt_div_right (abs_pos_of_pos h0)]
    push_cast
    rw [← abs_div, abs_sq, sub_div, mul_div_cancel _ h0.ne', ← div_pow, h1, ←
      sq_sqrt (Int.cast_pos.mpr h₀).le, sq_sub_sq, abs_mul, ← mul_one_div]
    refine' mul_lt_mul'' (((abs_add ξ q).trans _).trans_lt hM₁) h (abs_nonneg _) (abs_nonneg _)
    rw [two_mul, add_assoc, add_le_add_iff_left, ← sub_le_iff_le_add']
    rw [mem_setOf, abs_sub_comm] at h
    refine' (abs_sub_abs_le_abs_sub (q : ℝ) ξ).trans (h.le.trans _)
    rw [div_le_one h0, one_le_sq_iff_one_le_abs, Nat.abs_cast, Nat.one_le_cast]
    exact q.pos
  obtain ⟨m, hm⟩ : ∃ m : ℤ, {q : ℚ | q.1 ^ 2 - d * (q.den : ℤ) ^ 2 = m}.Infinite := by
    

Proof state:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
ξ : ℝ := sqrt ↑d
hξ : Irrational ξ
M : ℤ
hM₁ : 2 * |ξ| + 1 < ↑M
hM : Set.Infinite {q | |q.num ^ 2 - d * ↑q.den ^ 2| < M}
⊢ ∃ m, Set.Infinite {q | q.num ^ 2 - d * ↑q.den ^ 2 = m}

Next tactic:
contrapose! hM