/-- 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
    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']
  have hm₀ : m ≠ 0 := by
    rintro rfl
    obtain ⟨q, hq⟩ := hm.nonempty
    rw [mem_setOf, sub_eq_zero, mul_comm] at hq
    obtain ⟨a, ha⟩ := (Int.pow_dvd_pow_iff two_pos).mp ⟨d, hq⟩
    rw [ha, mul_pow, mul_right_inj' (pow_pos (Int.coe_nat_pos.mpr q.pos) 2).ne'] at hq
    exact hd ⟨a, sq a ▸ hq.symm⟩
  haveI := neZero_iff.mpr (Int.natAbs_ne_zero.mpr hm₀)
  let f : ℚ → ZMod m.natAbs × ZMod m.natAbs := fun q => (q.num, q.den)
  obtain ⟨q₁, h₁ : q₁.num ^ 2 - d * (q₁.den : ℤ) ^ 2 = m,
      q₂, h₂ : q₂.num ^ 2 - d * (q₂.den : ℤ) ^ 2 = m, hne, hqf⟩ :=
    hm.exists_ne_map_eq_of_mapsTo (mapsTo_univ f _) finite_univ
  obtain ⟨hq1 : (q₁.num : ZMod m.natAbs) = q₂.num, hq2 : (q₁.den : ZMod m.natAbs) = q₂.den⟩ :=
    Prod.ext_iff.mp hqf
  have hd₁ : m ∣ q₁.num * q₂.num - d * (q₁.den * q₂.den) := by
    rw [← Int.natAbs_dvd, ← ZMod.int_cast_zmod_eq_zero_iff_dvd]
    push_cast
    rw [hq1, hq2, ← sq, ← sq]
    norm_cast
    rw [ZMod.int_cast_zmod_eq_zero_iff_dvd, Int.natAbs_dvd, Nat.cast_pow, ← h₂]
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.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}
m : ℤ
hm : Set.Infinite {q | q.num ^ 2 - d * ↑q.den ^ 2 = m}
hm₀ : m ≠ 0
this : NeZero (Int.natAbs m)
f : ℚ → ZMod (Int.natAbs m) × ZMod (Int.natAbs m) := fun q => (↑q.num, ↑q.den)
q₁ : ℚ
h₁ : q₁.num ^ 2 - d * ↑q₁.den ^ 2 = m
q₂ : ℚ
h₂ : q₂.num ^ 2 - d * ↑q₂.den ^ 2 = m
hne : q₁ ≠ q₂
hqf : f q₁ = f q₂
hq1 : ↑q₁.num = ↑q₂.num
hq2 : ↑q₁.den = ↑q₂.den
hd₁ : m ∣ q₁.num * q₂.num - d * (↑q₁.den * ↑q₂.den)
⊢ ∃ x y, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0

Next tactic:
have hd₂ : m ∣ q₁.num * q₂.den - q₂.num * q₁.den := by
    rw [← Int.natAbs_dvd, ← ZMod.int_cast_eq_int_cast_iff_dvd_sub]
    push_cast
    rw [hq1, hq2]