theorem matiyasevic {a k x y} :
    (∃ a1 : 1 < a, xn a1 k = x ∧ yn a1 k = y) ↔
      1 < a ∧ k ≤ y ∧ (x = 1 ∧ y = 0 ∨
        ∃ u v s t b : ℕ,
          x * x - (a * a - 1) * y * y = 1 ∧ u * u - (a * a - 1) * v * v = 1 ∧
          s * s - (b * b - 1) * t * t = 1 ∧ 1 < b ∧ b ≡ 1 [MOD 4 * y] ∧
          b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y]) :=
  ⟨fun ⟨a1, hx, hy⟩ => by
    rw [← hx, ← hy]
    refine' ⟨a1,
        (Nat.eq_zero_or_pos k).elim (fun k0 => by rw [k0]; exact ⟨le_rfl, Or.inl ⟨rfl, rfl⟩⟩)
          fun kpos => _⟩
    exact
      let x := xn a1 k
      let y := yn a1 k
      let m := 2 * (k * y)
      let u := xn a1 m
      let v := yn a1 m
      have ky : k ≤ y := yn_ge_n a1 k
      have yv : y * y ∣ v := (ysq_dvd_yy a1 k).trans <| (y_dvd_iff _ _ _).2 <| dvd_mul_left _ _
      have uco : Nat.Coprime u (4 * y) :=
        have : 2 ∣ v :=
          modEq_zero_iff_dvd.1 <| (yn_modEq_two _ _).trans (dvd_mul_right _ _).modEq_zero_nat
        have : Nat.Coprime u 2 := (xy_coprime a1 m).coprime_dvd_right this
        (this.mul_right this).mul_right <|
          (xy_coprime _ _).coprime_dvd_right (dvd_of_mul_left_dvd yv)
      let ⟨b, ba, bm1⟩ := chineseRemainder uco a 1
      have m1 : 1 < m :=
        have : 0 < k * y := mul_pos kpos (strictMono_y a1 kpos)
        Nat.mul_le_mul_left 2 this
      have vp : 0 < v := strictMono_y a1 (lt_trans zero_lt_one m1)
      have b1 : 1 < b :=
        have : xn a1 1 < u := strictMono_x a1 m1
        have : a < u := by simp at this; exact this
        lt_of_lt_of_le a1 <| by
          delta ModEq at ba; rw [Nat.mod_eq_of_lt this] at ba; rw [← ba]
          apply Nat.mod_le
      let s := xn b1 k
      let t := yn b1 k
      have sx : s ≡ x [MOD u] := (xy_modEq_of_modEq b1 a1 ba k).left
      have tk : t ≡ k [MOD 4 * y] :=
        have : 4 * y ∣ b - 1 :=
          Int.coe_nat_dvd.1 <| by rw [Int.ofNat_sub (le_of_lt b1)]; exact bm1.symm.dvd
        (yn_modEq_a_sub_one _ _).of_dvd this
      ⟨ky,
        Or.inr
          ⟨u, v, s, t, b, pell_eq _ _, pell_eq _ _, pell_eq _ _, b1, bm1, ba, vp, yv, sx, tk⟩⟩,
    fun ⟨a1, ky, o⟩ =>
    ⟨a1,
      match o with
      | Or.inl ⟨x1, y0⟩ => by
        rw [y0] at ky; rw [Nat.eq_zero_of_le_zero ky, x1, y0]; exact ⟨rfl, rfl⟩
      | Or.inr ⟨u, v, s, t, b, xy, uv, st, b1, rem⟩ =>
        match x, y, eq_pell a1 xy, u, v, eq_pell a1 uv, s, t, eq_pell b1 st, rem, ky with
        | _, _, ⟨i, rfl, rfl⟩, _, _, ⟨n, rfl, rfl⟩, _, _, ⟨j, rfl, rfl⟩,
          ⟨(bm1 : b ≡ 1 [MOD 4 * yn a1 i]), (ba : b ≡ a [MOD xn a1 n]), (vp : 0 < yn a1 n),
            (yv : yn a1 i * yn a1 i ∣ yn a1 n), (sx : xn b1 j ≡ xn a1 i [MOD xn a1 n]),
            (tk : yn b1 j ≡ k [MOD 4 * yn a1 i])⟩,
          (ky : k ≤ yn a1 i) =>
          (Nat.eq_zero_or_pos i).elim
            (fun i0 => by simp [i0] at ky; rw [i0, ky]; exact ⟨rfl, rfl⟩) fun ipos => by
            suffices i = k by rw [this]; exact ⟨rfl, rfl⟩
            clear o rem xy uv st
            have iln : i ≤ n :=
              le_of_not_gt fun hin =>
                not_lt_of_ge (Nat.le_of_dvd vp (dvd_of_mul_left_dvd yv)) (strictMono_y a1 hin)
            have yd : 4 * yn a1 i ∣ 4 * n := mul_dvd_mul_left _ <| dvd_of_ysq_dvd a1 yv
            have jk : j ≡ k [MOD 4 * yn a1 i] :=
              have : 4 * yn a1 i ∣ b - 1 :=
                Int.coe_nat_dvd.1 <| by rw [Int.ofNat_sub (le_of_lt b1)]; exact bm1.symm.dvd
              ((yn_modEq_a_sub_one b1 _).of_dvd this).symm.trans tk
            have ki : k + i < 4 * yn a1 i :=
              lt_of_le_of_lt (_root_.add_le_add ky (yn_ge_n a1 i)) <| by
                rw [← two_mul]
                

Proof state:
a k x y : ℕ
a1 : 1 < a
ky✝ : k ≤ y
u v s t b : ℕ
b1 : 1 < b
i n j : ℕ
bm1 : b ≡ 1 [MOD 4 * yn a1 i]
ba : b ≡ a [MOD xn a1 n]
vp : 0 < yn a1 n
yv : yn a1 i * yn a1 i ∣ yn a1 n
sx : xn b1 j ≡ xn a1 i [MOD xn a1 n]
tk : yn b1 j ≡ k [MOD 4 * yn a1 i]
ky : k ≤ yn a1 i
x✝ :
  1 < a ∧
    k ≤ yn a1 i ∧
      (xn a1 i = 1 ∧ yn a1 i = 0 ∨
        ∃ u v s t b,
          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧
            u * u - (a * a - 1) * v * v = 1 ∧
              s * s - (b * b - 1) * t * t = 1 ∧
                1 < b ∧
                  b ≡ 1 [MOD 4 * yn a1 i] ∧
                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])
ipos : i > 0
iln : i ≤ n
yd : 4 * yn a1 i ∣ 4 * n
jk : j ≡ k [MOD 4 * yn a1 i]
⊢ 2 * yn a1 i < 4 * yn a1 i

Next tactic:
exact Nat.mul_lt_mul_of_pos_right (by decide) (strictMono_y a1 ipos)