l3lab/ntp-mathlib huggingface dataset:
theorem eq_pell_lem : ∀ (n) (b : ℤ√(d a1)), 1 ≤ b → IsPell b →
    b ≤ pellZd a1 n → ∃ n, b = pellZd a1 n
  | 0, b => fun h1 _ hl => ⟨0, @Zsqrtd.le_antisymm _ (dnsq a1) _ _ hl h1⟩
  | n + 1, b => fun h1 hp h =>
    have a1p : (0 : ℤ√(d a1)) ≤ ⟨a, 1⟩ := trivial
    have am1p : (0 : ℤ√(d a1)) ≤ ⟨a, -1⟩ := show (_ : Nat) ≤ _ by simp; exact Nat.pred_le _
    have a1m : (⟨a, 1⟩ * ⟨a, -1⟩ : ℤ√(d a1)) = 1 := isPell_norm.1 (isPell_one a1)
    if ha : (⟨↑a, 1⟩ : ℤ√(d a1)) ≤ b then
      let ⟨m, e⟩ :=
        eq_pell_lem n (b * ⟨a, -1⟩) (by rw [← a1m]; exact mul_le_mul_of_nonneg_right ha am1p)
          (isPell_mul hp (isPell_star.1 (isPell_one a1)))
          (by
            have t := mul_le_mul_of_nonneg_right h am1p
            rwa [pellZd_succ, mul_assoc, a1m, mul_one] at t)
      ⟨m + 1, by
        rw [show b = b * ⟨a, -1⟩ * ⟨a, 1⟩ by rw [mul_assoc, Eq.trans (mul_comm _ _) a1m]; simp,
          pellZd_succ, e]⟩
    else
      suffices ¬1 < b from ⟨0, show b = 1 from (Or.resolve_left (lt_or_eq_of_le h1) this).symm⟩
      fun h1l => by
      cases' b with x y
      exact by
        have bm : (_ * ⟨_, _⟩ : ℤ√d a1) = 1 := Pell.isPell_norm.1 hp
        have y0l : (0 : ℤ√d a1) < ⟨x - x, y - -y⟩ :=
          sub_lt_sub h1l fun hn : (1 : ℤ√d a1) ≤ ⟨x, -y⟩ => by
            have t := mul_le_mul_of_nonneg_left hn (le_trans zero_le_one h1)
            erw [bm, mul_one] at t
            exact h1l t
        have yl2 : (⟨_, _⟩ : ℤ√_) < ⟨_, _⟩ :=
          show (⟨x, y⟩ - ⟨x, -y⟩ : ℤ√d a1) < ⟨a, 1⟩ - ⟨a, -1⟩ from
            sub_lt_sub ha fun hn : (⟨x, -y⟩ : ℤ√d a1) ≤ ⟨a, -1⟩ => by
              have t := mul_le_mul_of_nonneg_right
                      (mul_le_mul_of_nonneg_left hn (le_trans zero_le_one h1)) a1p
              erw [bm, one_mul, mul_assoc, Eq.trans (mul_comm _ _) a1m, mul_one] at t
              exact ha t
        simp only [sub_self, sub_neg_eq_add] at y0l; simp only [Zsqrtd.neg_re, add_right_neg,
          Zsqrtd.neg_im, neg_neg] at yl2
        

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
a1p : 0 ≤ { re := ↑a, im := 1 }
am1p : 0 ≤ { re := ↑a, im := -1 }
a1m : { re := ↑a, im := 1 } * { re := ↑a, im := -1 } = 1
x y : ℤ
h1 : 1 ≤ { re := x, im := y }
hp : IsPell { re := x, im := y }
h : { re := x, im := y } ≤ pellZd a1 (n + 1)
ha : ¬{ re := ↑a, im := 1 } ≤ { re := x, im := y }
h1l : 1 < { re := x, im := y }
bm : { re := x, im := y } * { re := { re := x, im := y }.re, im := -{ re := x, im := y }.im } = 1
y0l : 0 < { re := 0, im := y + y }
yl2 : { re := 0, im := y + y } < { re := 0, im := 1 + 1 }
⊢ False

Next tactic:
exact
          match y, y0l, (yl2 : (⟨_, _⟩ : ℤ√_) < ⟨_, _⟩) with
          | 0, y0l, _ => y0l (le_refl 0)
          | (y + 1 : ℕ), _, yl2 =>
            yl2
              (Zsqrtd.le_of_le_le (by simp [sub_eq_add_neg])
                (let t := Int.ofNat_le_ofNat_of_le (Nat.succ_pos y)
                add_le_add t t))
          | Int.negSucc _, y0l, _ => y0l trivial
theorem eq_pell_lem : ∀ (n) (b : ℤ√(d a1)), 1 ≤ b → IsPell b →
    b ≤ pellZd a1 n → ∃ n, b = pellZd a1 n
  | 0, b => fun h1 _ hl => ⟨0, @Zsqrtd.le_antisymm _ (dnsq a1) _ _ hl h1⟩
  | n + 1, b => fun h1 hp h =>
    have a1p : (0 : ℤ√(d a1)) ≤ ⟨a, 1⟩ := trivial
    have am1p : (0 : ℤ√(d a1)) ≤ ⟨a, -1⟩ := show (_ : Nat) ≤ _ by simp; exact Nat.pred_le _
    have a1m : (⟨a, 1⟩ * ⟨a, -1⟩ : ℤ√(d a1)) = 1 := isPell_norm.1 (isPell_one a1)
    if ha : (⟨↑a, 1⟩ : ℤ√(d a1)) ≤ b then
      let ⟨m, e⟩ :=
        eq_pell_lem n (b * ⟨a, -1⟩) (by rw [← a1m]; exact mul_le_mul_of_nonneg_right ha am1p)
          (isPell_mul hp (isPell_star.1 (isPell_one a1)))
          (by
            have t := mul_le_mul_of_nonneg_right h am1p
            rwa [pellZd_succ, mul_assoc, a1m, mul_one] at t)
      ⟨m + 1, by
        rw [show b = b * ⟨a, -1⟩ * ⟨a, 1⟩ by rw [mul_assoc, Eq.trans (mul_comm _ _) a1m]; simp,
          pellZd_succ, e]⟩
    else
      suffices ¬1 < b from ⟨0, show b = 1 from (Or.resolve_left (lt_or_eq_of_le h1) this).symm⟩
      fun h1l => by
      cases' b with x y
      exact by
        have bm : (_ * ⟨_, _⟩ : ℤ√d a1) = 1 := Pell.isPell_norm.1 hp
        have y0l : (0 : ℤ√d a1) < ⟨x - x, y - -y⟩ :=
          sub_lt_sub h1l fun hn : (1 : ℤ√d a1) ≤ ⟨x, -y⟩ => by
            have t := mul_le_mul_of_nonneg_left hn (le_trans zero_le_one h1)
            erw [bm, mul_one] at t
            exact h1l t
        have yl2 : (⟨_, _⟩ : ℤ√_) < ⟨_, _⟩ :=
          show (⟨x, y⟩ - ⟨x, -y⟩ : ℤ√d a1) < ⟨a, 1⟩ - ⟨a, -1⟩ from
            sub_lt_sub ha fun hn : (⟨x, -y⟩ : ℤ√d a1) ≤ ⟨a, -1⟩ => by
              have t := mul_le_mul_of_nonneg_right
                      (mul_le_mul_of_nonneg_left hn (le_trans zero_le_one h1)) a1p
              erw [bm, one_mul, mul_assoc, Eq.trans (mul_comm _ _) a1m, mul_one] at t
              exact ha t
        simp only [sub_self, sub_neg_eq_add] at y0l; simp only [Zsqrtd.neg_re, add_right_neg,
          Zsqrtd.neg_im, neg_neg] at yl2
        exact
          match y, y0l, (yl2 : (⟨_, _⟩ : ℤ√_) < ⟨_, _⟩) with
          | 0, y0l, _ => y0l (le_refl 0)
          | (y + 1 : ℕ), _, yl2 =>
            yl2
              (Zsqrtd.le_of_le_le (by 

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
a1p : 0 ≤ { re := ↑a, im := 1 }
am1p : 0 ≤ { re := ↑a, im := -1 }
a1m : { re := ↑a, im := 1 } * { re := ↑a, im := -1 } = 1
x y✝ : ℤ
y0l : 0 < { re := 0, im := y✝ + y✝ }
yl2✝ : { re := 0, im := y✝ + y✝ } < { re := 0, im := 1 + 1 }
y : ℕ
x✝ : 0 < { re := 0, im := ↑(y + 1) + ↑(y + 1) }
yl2 : { re := 0, im := ↑(y + 1) + ↑(y + 1) } < { re := 0, im := 1 + 1 }
h1 : 1 ≤ { re := x, im := ↑(y + 1) }
hp : IsPell { re := x, im := ↑(y + 1) }
h : { re := x, im := ↑(y + 1) } ≤ pellZd a1 (n + 1)
ha : ¬{ re := ↑a, im := 1 } ≤ { re := x, im := ↑(y + 1) }
h1l : 1 < { re := x, im := ↑(y + 1) }
bm : { re := x, im := ↑(y + 1) } * { re := { re := x, im := ↑(y + 1) }.re, im := -{ re := x, im := ↑(y + 1) }.im } = 1
⊢ 0 ≤ 0

Next tactic:
simp [sub_eq_add_neg]
theorem eq_pellZd (b : ℤ√(d a1)) (b1 : 1 ≤ b) (hp : IsPell b) : ∃ n, b = pellZd a1 n :=
  let ⟨n, h⟩ := @Zsqrtd.le_arch (d a1) b
  eq_pell_lem a1 n b b1 hp <|
    h.trans <| by
      

Proof state:
a : ℕ
a1 : 1 < a
b : ℤ√↑(Pell.d a1)
b1 : 1 ≤ b
hp : IsPell b
n : ℕ
h : b ≤ ↑n
⊢ ↑n ≤ pellZd a1 n

Next tactic:
rw [Zsqrtd.coe_nat_val]
theorem eq_pellZd (b : ℤ√(d a1)) (b1 : 1 ≤ b) (hp : IsPell b) : ∃ n, b = pellZd a1 n :=
  let ⟨n, h⟩ := @Zsqrtd.le_arch (d a1) b
  eq_pell_lem a1 n b b1 hp <|
    h.trans <| by
      rw [Zsqrtd.coe_nat_val]
      

Proof state:
a : ℕ
a1 : 1 < a
b : ℤ√↑(Pell.d a1)
b1 : 1 ≤ b
hp : IsPell b
n : ℕ
h : b ≤ ↑n
⊢ { re := ↑n, im := 0 } ≤ pellZd a1 n

Next tactic:
exact
        Zsqrtd.le_of_le_le (Int.ofNat_le_ofNat_of_le <| le_of_lt <| n_lt_xn _ _)
          (Int.ofNat_zero_le _)
/-- Every solution to **Pell's equation** is recursively obtained from the initial solution
`(1,0)` using the recursion `pell`. -/
theorem eq_pell {x y : ℕ} (hp : x * x - d a1 * y * y = 1) : ∃ n, x = xn a1 n ∧ y = yn a1 n :=
  have : (1 : ℤ√(d a1)) ≤ ⟨x, y⟩ :=
    match x, hp with
    | 0, (hp : 0 - _ = 1) => by 

Proof state:
a : ℕ
a1 : 1 < a
x y : ℕ
hp✝ : x * x - Pell.d a1 * y * y = 1
hp : 0 - Pell.d a1 * y * y = 1
⊢ 1 ≤ { re := ↑0, im := ↑y }

Next tactic:
rw [zero_tsub] at hp
/-- Every solution to **Pell's equation** is recursively obtained from the initial solution
`(1,0)` using the recursion `pell`. -/
theorem eq_pell {x y : ℕ} (hp : x * x - d a1 * y * y = 1) : ∃ n, x = xn a1 n ∧ y = yn a1 n :=
  have : (1 : ℤ√(d a1)) ≤ ⟨x, y⟩ :=
    match x, hp with
    | 0, (hp : 0 - _ = 1) => by rw [zero_tsub] at hp; 

Proof state:
a : ℕ
a1 : 1 < a
x y : ℕ
hp✝ : x * x - Pell.d a1 * y * y = 1
hp : 0 = 1
⊢ 1 ≤ { re := ↑0, im := ↑y }

Next tactic:
contradiction
theorem pellZd_add (m) : ∀ n, pellZd a1 (m + n) = pellZd a1 m * pellZd a1 n
  | 0 => (mul_one _).symm
  | n + 1 => by 

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
⊢ pellZd a1 (m + (n + 1)) = pellZd a1 m * pellZd a1 (n + 1)

Next tactic:
rw [← add_assoc, pellZd_succ, pellZd_succ, pellZd_add _ n, ← mul_assoc]
theorem xn_add (m n) : xn a1 (m + n) = xn a1 m * xn a1 n + d a1 * yn a1 m * yn a1 n := by
  

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
⊢ xn a1 (m + n) = xn a1 m * xn a1 n + Pell.d a1 * yn a1 m * yn a1 n

Next tactic:
injection pellZd_add a1 m n with h _
theorem xn_add (m n) : xn a1 (m + n) = xn a1 m * xn a1 n + d a1 * yn a1 m * yn a1 n := by
  injection pellZd_add a1 m n with h _
  

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : ↑(xn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).re + ↑(Pell.d a1) * (pellZd a1 m).im * (pellZd a1 n).im
im_eq✝ : ↑(yn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).im + (pellZd a1 m).im * (pellZd a1 n).re
⊢ xn a1 (m + n) = xn a1 m * xn a1 n + Pell.d a1 * yn a1 m * yn a1 n

Next tactic:
zify
theorem xn_add (m n) : xn a1 (m + n) = xn a1 m * xn a1 n + d a1 * yn a1 m * yn a1 n := by
  injection pellZd_add a1 m n with h _
  zify
  

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : ↑(xn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).re + ↑(Pell.d a1) * (pellZd a1 m).im * (pellZd a1 n).im
im_eq✝ : ↑(yn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).im + (pellZd a1 m).im * (pellZd a1 n).re
⊢ ↑(xn a1 (m + n)) = ↑(xn a1 m) * ↑(xn a1 n) + ↑(Pell.d a1) * ↑(yn a1 m) * ↑(yn a1 n)

Next tactic:
rw [h]
theorem xn_add (m n) : xn a1 (m + n) = xn a1 m * xn a1 n + d a1 * yn a1 m * yn a1 n := by
  injection pellZd_add a1 m n with h _
  zify
  rw [h]
  

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : ↑(xn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).re + ↑(Pell.d a1) * (pellZd a1 m).im * (pellZd a1 n).im
im_eq✝ : ↑(yn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).im + (pellZd a1 m).im * (pellZd a1 n).re
⊢ (pellZd a1 m).re * (pellZd a1 n).re + ↑(Pell.d a1) * (pellZd a1 m).im * (pellZd a1 n).im =
    ↑(xn a1 m) * ↑(xn a1 n) + ↑(Pell.d a1) * ↑(yn a1 m) * ↑(yn a1 n)

Next tactic:
simp [pellZd]
theorem yn_add (m n) : yn a1 (m + n) = xn a1 m * yn a1 n + yn a1 m * xn a1 n := by
  

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
⊢ yn a1 (m + n) = xn a1 m * yn a1 n + yn a1 m * xn a1 n

Next tactic:
injection pellZd_add a1 m n with _ h
theorem yn_add (m n) : yn a1 (m + n) = xn a1 m * yn a1 n + yn a1 m * xn a1 n := by
  injection pellZd_add a1 m n with _ h
  

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
re_eq✝ : ↑(xn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).re + ↑(Pell.d a1) * (pellZd a1 m).im * (pellZd a1 n).im
h : ↑(yn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).im + (pellZd a1 m).im * (pellZd a1 n).re
⊢ yn a1 (m + n) = xn a1 m * yn a1 n + yn a1 m * xn a1 n

Next tactic:
zify
theorem yn_add (m n) : yn a1 (m + n) = xn a1 m * yn a1 n + yn a1 m * xn a1 n := by
  injection pellZd_add a1 m n with _ h
  zify
  

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
re_eq✝ : ↑(xn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).re + ↑(Pell.d a1) * (pellZd a1 m).im * (pellZd a1 n).im
h : ↑(yn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).im + (pellZd a1 m).im * (pellZd a1 n).re
⊢ ↑(yn a1 (m + n)) = ↑(xn a1 m) * ↑(yn a1 n) + ↑(yn a1 m) * ↑(xn a1 n)

Next tactic:
rw [h]
theorem yn_add (m n) : yn a1 (m + n) = xn a1 m * yn a1 n + yn a1 m * xn a1 n := by
  injection pellZd_add a1 m n with _ h
  zify
  rw [h]
  

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
re_eq✝ : ↑(xn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).re + ↑(Pell.d a1) * (pellZd a1 m).im * (pellZd a1 n).im
h : ↑(yn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).im + (pellZd a1 m).im * (pellZd a1 n).re
⊢ (pellZd a1 m).re * (pellZd a1 n).im + (pellZd a1 m).im * (pellZd a1 n).re =
    ↑(xn a1 m) * ↑(yn a1 n) + ↑(yn a1 m) * ↑(xn a1 n)

Next tactic:
simp [pellZd]
theorem pellZd_sub {m n} (h : n ≤ m) : pellZd a1 (m - n) = pellZd a1 m * star (pellZd a1 n) := by
  

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : n ≤ m
⊢ pellZd a1 (m - n) = pellZd a1 m * star (pellZd a1 n)

Next tactic:
let t := pellZd_add a1 n (m - n)
theorem pellZd_sub {m n} (h : n ≤ m) : pellZd a1 (m - n) = pellZd a1 m * star (pellZd a1 n) := by
  let t := pellZd_add a1 n (m - n)
  

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : n ≤ m
t : pellZd a1 (n + (m - n)) = pellZd a1 n * pellZd a1 (m - n) := pellZd_add a1 n (m - n)
⊢ pellZd a1 (m - n) = pellZd a1 m * star (pellZd a1 n)

Next tactic:
rw [add_tsub_cancel_of_le h] at t
theorem pellZd_sub {m n} (h : n ≤ m) : pellZd a1 (m - n) = pellZd a1 m * star (pellZd a1 n) := by
  let t := pellZd_add a1 n (m - n)
  rw [add_tsub_cancel_of_le h] at t
  

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : n ≤ m
t : pellZd a1 m = pellZd a1 n * pellZd a1 (m - n)
⊢ pellZd a1 (m - n) = pellZd a1 m * star (pellZd a1 n)

Next tactic:
rw [t, mul_comm (pellZd _ n) _, mul_assoc, isPell_norm.1 (isPell_pellZd _ _), mul_one]
theorem xz_sub {m n} (h : n ≤ m) :
    xz a1 (m - n) = xz a1 m * xz a1 n - d a1 * yz a1 m * yz a1 n := by
  

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : n ≤ m
⊢ xz a1 (m - n) = xz a1 m * xz a1 n - ↑(Pell.d a1) * yz a1 m * yz a1 n

Next tactic:
rw [sub_eq_add_neg, ← mul_neg]
theorem xz_sub {m n} (h : n ≤ m) :
    xz a1 (m - n) = xz a1 m * xz a1 n - d a1 * yz a1 m * yz a1 n := by
  rw [sub_eq_add_neg, ← mul_neg]
  

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : n ≤ m
⊢ xz a1 (m - n) = xz a1 m * xz a1 n + ↑(Pell.d a1) * yz a1 m * -yz a1 n

Next tactic:
exact congr_arg Zsqrtd.re (pellZd_sub a1 h)
theorem yz_sub {m n} (h : n ≤ m) : yz a1 (m - n) = xz a1 n * yz a1 m - xz a1 m * yz a1 n := by
  

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : n ≤ m
⊢ yz a1 (m - n) = xz a1 n * yz a1 m - xz a1 m * yz a1 n

Next tactic:
rw [sub_eq_add_neg, ← mul_neg, mul_comm, add_comm]
theorem yz_sub {m n} (h : n ≤ m) : yz a1 (m - n) = xz a1 n * yz a1 m - xz a1 m * yz a1 n := by
  rw [sub_eq_add_neg, ← mul_neg, mul_comm, add_comm]
  

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : n ≤ m
⊢ yz a1 (m - n) = xz a1 m * -yz a1 n + yz a1 m * xz a1 n

Next tactic:
exact congr_arg Zsqrtd.im (pellZd_sub a1 h)
theorem xy_coprime (n) : (xn a1 n).Coprime (yn a1 n) :=
  Nat.coprime_of_dvd' fun k _ kx ky => by
    

Proof state:
a : ℕ
a1 : 1 < a
n k : ℕ
x✝ : Nat.Prime k
kx : k ∣ xn a1 n
ky : k ∣ yn a1 n
⊢ k ∣ 1

Next tactic:
let p := pell_eq a1 n
theorem xy_coprime (n) : (xn a1 n).Coprime (yn a1 n) :=
  Nat.coprime_of_dvd' fun k _ kx ky => by
    let p := pell_eq a1 n
    

Proof state:
a : ℕ
a1 : 1 < a
n k : ℕ
x✝ : Nat.Prime k
kx : k ∣ xn a1 n
ky : k ∣ yn a1 n
p : xn a1 n * xn a1 n - Pell.d a1 * yn a1 n * yn a1 n = 1 := pell_eq a1 n
⊢ k ∣ 1

Next tactic:
rw [← p]
theorem xy_coprime (n) : (xn a1 n).Coprime (yn a1 n) :=
  Nat.coprime_of_dvd' fun k _ kx ky => by
    let p := pell_eq a1 n
    rw [← p]
    

Proof state:
a : ℕ
a1 : 1 < a
n k : ℕ
x✝ : Nat.Prime k
kx : k ∣ xn a1 n
ky : k ∣ yn a1 n
p : xn a1 n * xn a1 n - Pell.d a1 * yn a1 n * yn a1 n = 1 := pell_eq a1 n
⊢ k ∣ xn a1 n * xn a1 n - Pell.d a1 * yn a1 n * yn a1 n

Next tactic:
exact Nat.dvd_sub (le_of_lt <| Nat.lt_of_sub_eq_succ p) (kx.mul_left _) (ky.mul_left _)
theorem strictMono_y : StrictMono (yn a1)
  | m, 0, h => absurd h <| Nat.not_lt_zero _
  | m, n + 1, h => by
    

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : m < n + 1
⊢ yn a1 m < yn a1 (n + 1)

Next tactic:
have : yn a1 m ≤ yn a1 n :=
      Or.elim (lt_or_eq_of_le <| Nat.le_of_succ_le_succ h) (fun hl => le_of_lt <| strictMono_y hl)
        fun e => by rw [e]
theorem strictMono_y : StrictMono (yn a1)
  | m, 0, h => absurd h <| Nat.not_lt_zero _
  | m, n + 1, h => by
    have : yn a1 m ≤ yn a1 n :=
      Or.elim (lt_or_eq_of_le <| Nat.le_of_succ_le_succ h) (fun hl => le_of_lt <| strictMono_y hl)
        fun e => by 

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : m < n + 1
e : m = n
⊢ yn a1 m ≤ yn a1 n

Next tactic:
rw [e]
theorem strictMono_y : StrictMono (yn a1)
  | m, 0, h => absurd h <| Nat.not_lt_zero _
  | m, n + 1, h => by
    have : yn a1 m ≤ yn a1 n :=
      Or.elim (lt_or_eq_of_le <| Nat.le_of_succ_le_succ h) (fun hl => le_of_lt <| strictMono_y hl)
        fun e => by rw [e]
    

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : m < n + 1
this : yn a1 m ≤ yn a1 n
⊢ yn a1 m < yn a1 (n + 1)

Next tactic:
simp
theorem strictMono_y : StrictMono (yn a1)
  | m, 0, h => absurd h <| Nat.not_lt_zero _
  | m, n + 1, h => by
    have : yn a1 m ≤ yn a1 n :=
      Or.elim (lt_or_eq_of_le <| Nat.le_of_succ_le_succ h) (fun hl => le_of_lt <| strictMono_y hl)
        fun e => by rw [e]
    simp; 

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : m < n + 1
this : yn a1 m ≤ yn a1 n
⊢ yn a1 m < xn a1 n + yn a1 n * a

Next tactic:
refine' lt_of_le_of_lt _ (Nat.lt_add_of_pos_left <| x_pos a1 n)
theorem strictMono_y : StrictMono (yn a1)
  | m, 0, h => absurd h <| Nat.not_lt_zero _
  | m, n + 1, h => by
    have : yn a1 m ≤ yn a1 n :=
      Or.elim (lt_or_eq_of_le <| Nat.le_of_succ_le_succ h) (fun hl => le_of_lt <| strictMono_y hl)
        fun e => by rw [e]
    simp; refine' lt_of_le_of_lt _ (Nat.lt_add_of_pos_left <| x_pos a1 n)
    

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : m < n + 1
this : yn a1 m ≤ yn a1 n
⊢ yn a1 m ≤ yn a1 n * a

Next tactic:
rw [← mul_one (yn a1 m)]
theorem strictMono_y : StrictMono (yn a1)
  | m, 0, h => absurd h <| Nat.not_lt_zero _
  | m, n + 1, h => by
    have : yn a1 m ≤ yn a1 n :=
      Or.elim (lt_or_eq_of_le <| Nat.le_of_succ_le_succ h) (fun hl => le_of_lt <| strictMono_y hl)
        fun e => by rw [e]
    simp; refine' lt_of_le_of_lt _ (Nat.lt_add_of_pos_left <| x_pos a1 n)
    rw [← mul_one (yn a1 m)]
    

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : m < n + 1
this : yn a1 m ≤ yn a1 n
⊢ yn a1 m * 1 ≤ yn a1 n * a

Next tactic:
exact mul_le_mul this (le_of_lt a1) (Nat.zero_le _) (Nat.zero_le _)
theorem strictMono_x : StrictMono (xn a1)
  | m, 0, h => absurd h <| Nat.not_lt_zero _
  | m, n + 1, h => by
    

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : m < n + 1
⊢ xn a1 m < xn a1 (n + 1)

Next tactic:
have : xn a1 m ≤ xn a1 n :=
      Or.elim (lt_or_eq_of_le <| Nat.le_of_succ_le_succ h) (fun hl => le_of_lt <| strictMono_x hl)
        fun e => by rw [e]
theorem strictMono_x : StrictMono (xn a1)
  | m, 0, h => absurd h <| Nat.not_lt_zero _
  | m, n + 1, h => by
    have : xn a1 m ≤ xn a1 n :=
      Or.elim (lt_or_eq_of_le <| Nat.le_of_succ_le_succ h) (fun hl => le_of_lt <| strictMono_x hl)
        fun e => by 

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : m < n + 1
e : m = n
⊢ xn a1 m ≤ xn a1 n

Next tactic:
rw [e]
theorem strictMono_x : StrictMono (xn a1)
  | m, 0, h => absurd h <| Nat.not_lt_zero _
  | m, n + 1, h => by
    have : xn a1 m ≤ xn a1 n :=
      Or.elim (lt_or_eq_of_le <| Nat.le_of_succ_le_succ h) (fun hl => le_of_lt <| strictMono_x hl)
        fun e => by rw [e]
    

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : m < n + 1
this : xn a1 m ≤ xn a1 n
⊢ xn a1 m < xn a1 (n + 1)

Next tactic:
simp
theorem strictMono_x : StrictMono (xn a1)
  | m, 0, h => absurd h <| Nat.not_lt_zero _
  | m, n + 1, h => by
    have : xn a1 m ≤ xn a1 n :=
      Or.elim (lt_or_eq_of_le <| Nat.le_of_succ_le_succ h) (fun hl => le_of_lt <| strictMono_x hl)
        fun e => by rw [e]
    simp; 

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : m < n + 1
this : xn a1 m ≤ xn a1 n
⊢ xn a1 m < xn a1 n * a + Pell.d a1 * yn a1 n

Next tactic:
refine' lt_of_lt_of_le (lt_of_le_of_lt this _) (Nat.le_add_right _ _)
theorem strictMono_x : StrictMono (xn a1)
  | m, 0, h => absurd h <| Nat.not_lt_zero _
  | m, n + 1, h => by
    have : xn a1 m ≤ xn a1 n :=
      Or.elim (lt_or_eq_of_le <| Nat.le_of_succ_le_succ h) (fun hl => le_of_lt <| strictMono_x hl)
        fun e => by rw [e]
    simp; refine' lt_of_lt_of_le (lt_of_le_of_lt this _) (Nat.le_add_right _ _)
    

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : m < n + 1
this : xn a1 m ≤ xn a1 n
⊢ xn a1 n < xn a1 n * a

Next tactic:
have t := Nat.mul_lt_mul_of_pos_left a1 (x_pos a1 n)
theorem strictMono_x : StrictMono (xn a1)
  | m, 0, h => absurd h <| Nat.not_lt_zero _
  | m, n + 1, h => by
    have : xn a1 m ≤ xn a1 n :=
      Or.elim (lt_or_eq_of_le <| Nat.le_of_succ_le_succ h) (fun hl => le_of_lt <| strictMono_x hl)
        fun e => by rw [e]
    simp; refine' lt_of_lt_of_le (lt_of_le_of_lt this _) (Nat.le_add_right _ _)
    have t := Nat.mul_lt_mul_of_pos_left a1 (x_pos a1 n)
    

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : m < n + 1
this : xn a1 m ≤ xn a1 n
t : xn a1 n * 1 < xn a1 n * a
⊢ xn a1 n < xn a1 n * a

Next tactic:
rwa [mul_one] at t
theorem y_mul_dvd (n) : ∀ k, yn a1 n ∣ yn a1 (n * k)
  | 0 => dvd_zero _
  | k + 1 => by
    

Proof state:
a : ℕ
a1 : 1 < a
n k : ℕ
⊢ yn a1 n ∣ yn a1 (n * (k + 1))

Next tactic:
rw [Nat.mul_succ, yn_add]
theorem y_mul_dvd (n) : ∀ k, yn a1 n ∣ yn a1 (n * k)
  | 0 => dvd_zero _
  | k + 1 => by
    rw [Nat.mul_succ, yn_add]; 

Proof state:
a : ℕ
a1 : 1 < a
n k : ℕ
⊢ yn a1 n ∣ xn a1 (n * k) * yn a1 n + yn a1 (n * k) * xn a1 n

Next tactic:
exact dvd_add (dvd_mul_left _ _) ((y_mul_dvd _ k).mul_right _)
theorem y_dvd_iff (m n) : yn a1 m ∣ yn a1 n ↔ m ∣ n :=
  ⟨fun h =>
    Nat.dvd_of_mod_eq_zero <|
      (Nat.eq_zero_or_pos _).resolve_right fun hp => by
        

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : yn a1 m ∣ yn a1 n
hp : n % m > 0
⊢ False

Next tactic:
have co : Nat.Coprime (yn a1 m) (xn a1 (m * (n / m))) :=
          Nat.Coprime.symm <| (xy_coprime a1 _).coprime_dvd_right (y_mul_dvd a1 m (n / m))
theorem y_dvd_iff (m n) : yn a1 m ∣ yn a1 n ↔ m ∣ n :=
  ⟨fun h =>
    Nat.dvd_of_mod_eq_zero <|
      (Nat.eq_zero_or_pos _).resolve_right fun hp => by
        have co : Nat.Coprime (yn a1 m) (xn a1 (m * (n / m))) :=
          Nat.Coprime.symm <| (xy_coprime a1 _).coprime_dvd_right (y_mul_dvd a1 m (n / m))
        

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : yn a1 m ∣ yn a1 n
hp : n % m > 0
co : Coprime (yn a1 m) (xn a1 (m * (n / m)))
⊢ False

Next tactic:
have m0 : 0 < m :=
          m.eq_zero_or_pos.resolve_left fun e => by
            rw [e, Nat.mod_zero] at hp;rw [e] at h
            exact _root_.ne_of_lt (strictMono_y a1 hp) (eq_zero_of_zero_dvd h).symm
theorem y_dvd_iff (m n) : yn a1 m ∣ yn a1 n ↔ m ∣ n :=
  ⟨fun h =>
    Nat.dvd_of_mod_eq_zero <|
      (Nat.eq_zero_or_pos _).resolve_right fun hp => by
        have co : Nat.Coprime (yn a1 m) (xn a1 (m * (n / m))) :=
          Nat.Coprime.symm <| (xy_coprime a1 _).coprime_dvd_right (y_mul_dvd a1 m (n / m))
        have m0 : 0 < m :=
          m.eq_zero_or_pos.resolve_left fun e => by
            

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : yn a1 m ∣ yn a1 n
hp : n % m > 0
co : Coprime (yn a1 m) (xn a1 (m * (n / m)))
e : m = 0
⊢ False

Next tactic:
rw [e, Nat.mod_zero] at hp
theorem y_dvd_iff (m n) : yn a1 m ∣ yn a1 n ↔ m ∣ n :=
  ⟨fun h =>
    Nat.dvd_of_mod_eq_zero <|
      (Nat.eq_zero_or_pos _).resolve_right fun hp => by
        have co : Nat.Coprime (yn a1 m) (xn a1 (m * (n / m))) :=
          Nat.Coprime.symm <| (xy_coprime a1 _).coprime_dvd_right (y_mul_dvd a1 m (n / m))
        have m0 : 0 < m :=
          m.eq_zero_or_pos.resolve_left fun e => by
            rw [e, Nat.mod_zero] at hp;

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : yn a1 m ∣ yn a1 n
hp : n > 0
co : Coprime (yn a1 m) (xn a1 (m * (n / m)))
e : m = 0
⊢ False

Next tactic:
rw [e] at h
theorem y_dvd_iff (m n) : yn a1 m ∣ yn a1 n ↔ m ∣ n :=
  ⟨fun h =>
    Nat.dvd_of_mod_eq_zero <|
      (Nat.eq_zero_or_pos _).resolve_right fun hp => by
        have co : Nat.Coprime (yn a1 m) (xn a1 (m * (n / m))) :=
          Nat.Coprime.symm <| (xy_coprime a1 _).coprime_dvd_right (y_mul_dvd a1 m (n / m))
        have m0 : 0 < m :=
          m.eq_zero_or_pos.resolve_left fun e => by
            rw [e, Nat.mod_zero] at hp;rw [e] at h
            

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : yn a1 0 ∣ yn a1 n
hp : n > 0
co : Coprime (yn a1 m) (xn a1 (m * (n / m)))
e : m = 0
⊢ False

Next tactic:
exact _root_.ne_of_lt (strictMono_y a1 hp) (eq_zero_of_zero_dvd h).symm
theorem y_dvd_iff (m n) : yn a1 m ∣ yn a1 n ↔ m ∣ n :=
  ⟨fun h =>
    Nat.dvd_of_mod_eq_zero <|
      (Nat.eq_zero_or_pos _).resolve_right fun hp => by
        have co : Nat.Coprime (yn a1 m) (xn a1 (m * (n / m))) :=
          Nat.Coprime.symm <| (xy_coprime a1 _).coprime_dvd_right (y_mul_dvd a1 m (n / m))
        have m0 : 0 < m :=
          m.eq_zero_or_pos.resolve_left fun e => by
            rw [e, Nat.mod_zero] at hp;rw [e] at h
            exact _root_.ne_of_lt (strictMono_y a1 hp) (eq_zero_of_zero_dvd h).symm
        

Proof state:
a : ℕ
a1 : 1 < a
m n : ℕ
h : yn a1 m ∣ yn a1 n
hp : n % m > 0
co : Coprime (yn a1 m) (xn a1 (m * (n / m)))
m0 : 0 < m
⊢ False

Next tactic:
rw [← Nat.mod_add_div n m, yn_add] at h