l3lab/ntp-mathlib huggingface dataset:
theorem isPell_pellZd : ∀ n : ℕ, IsPell (pellZd a1 n)
  | 0 => rfl
  | n + 1 => by
    let o := isPell_one a1
    

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
o : IsPell { re := ↑a, im := 1 } := isPell_one a1
⊢ IsPell (pellZd a1 (n + 1))

Next tactic:
simp
theorem isPell_pellZd : ∀ n : ℕ, IsPell (pellZd a1 n)
  | 0 => rfl
  | n + 1 => by
    let o := isPell_one a1
    simp; 

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
o : IsPell { re := ↑a, im := 1 } := isPell_one a1
⊢ IsPell (pellZd a1 n * { re := ↑a, im := 1 })

Next tactic:
exact Pell.isPell_mul (isPell_pellZd n) o
@[simp]
theorem pell_eq (n : ℕ) : xn a1 n * xn a1 n - d a1 * yn a1 n * yn a1 n = 1 :=
  let pn := pell_eqz a1 n
  have h : (↑(xn a1 n * xn a1 n) : ℤ) - ↑(d a1 * yn a1 n * yn a1 n) = 1 := by
    

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
pn : xz a1 n * xz a1 n - ↑(Pell.d a1) * yz a1 n * yz a1 n = 1 := pell_eqz a1 n
⊢ ↑(xn a1 n * xn a1 n) - ↑(Pell.d a1 * yn a1 n * yn a1 n) = 1

Next tactic:
repeat' rw [Int.ofNat_mul]; exact pn
@[simp]
theorem pell_eq (n : ℕ) : xn a1 n * xn a1 n - d a1 * yn a1 n * yn a1 n = 1 :=
  let pn := pell_eqz a1 n
  have h : (↑(xn a1 n * xn a1 n) : ℤ) - ↑(d a1 * yn a1 n * yn a1 n) = 1 := by
    repeat' 

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
pn : xz a1 n * xz a1 n - ↑(Pell.d a1) * yz a1 n * yz a1 n = 1 := pell_eqz a1 n
⊢ ↑(xn a1 n * xn a1 n) - ↑(Pell.d a1 * yn a1 n * yn a1 n) = 1

Next tactic:
rw [Int.ofNat_mul]
@[simp]
theorem pell_eq (n : ℕ) : xn a1 n * xn a1 n - d a1 * yn a1 n * yn a1 n = 1 :=
  let pn := pell_eqz a1 n
  have h : (↑(xn a1 n * xn a1 n) : ℤ) - ↑(d a1 * yn a1 n * yn a1 n) = 1 := by
    repeat' rw [Int.ofNat_mul]; 

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
pn : xz a1 n * xz a1 n - ↑(Pell.d a1) * yz a1 n * yz a1 n = 1 := pell_eqz a1 n
⊢ ↑(xn a1 n) * ↑(xn a1 n) - ↑(Pell.d a1 * yn a1 n * yn a1 n) = 1

Next tactic:
exact pn
@[simp]
theorem pell_eq (n : ℕ) : xn a1 n * xn a1 n - d a1 * yn a1 n * yn a1 n = 1 :=
  let pn := pell_eqz a1 n
  have h : (↑(xn a1 n * xn a1 n) : ℤ) - ↑(d a1 * yn a1 n * yn a1 n) = 1 := by
    repeat' rw [Int.ofNat_mul]; exact pn
  have hl : d a1 * yn a1 n * yn a1 n ≤ xn a1 n * xn a1 n :=
    Nat.cast_le.1 <| Int.le.intro _ <| add_eq_of_eq_sub' <| Eq.symm h
  Nat.cast_inj.1 (by 

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
pn : xz a1 n * xz a1 n - ↑(Pell.d a1) * yz a1 n * yz a1 n = 1 := pell_eqz a1 n
h : ↑(xn a1 n * xn a1 n) - ↑(Pell.d a1 * yn a1 n * yn a1 n) = 1
hl : Pell.d a1 * yn a1 n * yn a1 n ≤ xn a1 n * xn a1 n
⊢ ↑(xn a1 n * xn a1 n - Pell.d a1 * yn a1 n * yn a1 n) = ↑1

Next tactic:
rw [Int.ofNat_sub hl]
@[simp]
theorem pell_eq (n : ℕ) : xn a1 n * xn a1 n - d a1 * yn a1 n * yn a1 n = 1 :=
  let pn := pell_eqz a1 n
  have h : (↑(xn a1 n * xn a1 n) : ℤ) - ↑(d a1 * yn a1 n * yn a1 n) = 1 := by
    repeat' rw [Int.ofNat_mul]; exact pn
  have hl : d a1 * yn a1 n * yn a1 n ≤ xn a1 n * xn a1 n :=
    Nat.cast_le.1 <| Int.le.intro _ <| add_eq_of_eq_sub' <| Eq.symm h
  Nat.cast_inj.1 (by rw [Int.ofNat_sub hl]; 

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
pn : xz a1 n * xz a1 n - ↑(Pell.d a1) * yz a1 n * yz a1 n = 1 := pell_eqz a1 n
h : ↑(xn a1 n * xn a1 n) - ↑(Pell.d a1 * yn a1 n * yn a1 n) = 1
hl : Pell.d a1 * yn a1 n * yn a1 n ≤ xn a1 n * xn a1 n
⊢ ↑(xn a1 n * xn a1 n) - ↑(Pell.d a1 * yn a1 n * yn a1 n) = ↑1

Next tactic:
exact h
instance dnsq : Zsqrtd.Nonsquare (d a1) :=
  ⟨fun n h =>
    have : n * n + 1 = a * a := by 

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
h : Pell.d a1 = n * n
⊢ n * n + 1 = a * a

Next tactic:
rw [← h]
instance dnsq : Zsqrtd.Nonsquare (d a1) :=
  ⟨fun n h =>
    have : n * n + 1 = a * a := by rw [← h]; 

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
h : Pell.d a1 = n * n
⊢ Pell.d a1 + 1 = a * a

Next tactic:
exact Nat.succ_pred_eq_of_pos (asq_pos a1)
instance dnsq : Zsqrtd.Nonsquare (d a1) :=
  ⟨fun n h =>
    have : n * n + 1 = a * a := by rw [← h]; exact Nat.succ_pred_eq_of_pos (asq_pos a1)
    have na : n < a := Nat.mul_self_lt_mul_self_iff.2 (by 

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
h : Pell.d a1 = n * n
this : n * n + 1 = a * a
⊢ n * n < a * a

Next tactic:
rw [← this]
instance dnsq : Zsqrtd.Nonsquare (d a1) :=
  ⟨fun n h =>
    have : n * n + 1 = a * a := by rw [← h]; exact Nat.succ_pred_eq_of_pos (asq_pos a1)
    have na : n < a := Nat.mul_self_lt_mul_self_iff.2 (by rw [← this]; 

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
h : Pell.d a1 = n * n
this : n * n + 1 = a * a
⊢ n * n < n * n + 1

Next tactic:
exact Nat.lt_succ_self _
instance dnsq : Zsqrtd.Nonsquare (d a1) :=
  ⟨fun n h =>
    have : n * n + 1 = a * a := by rw [← h]; exact Nat.succ_pred_eq_of_pos (asq_pos a1)
    have na : n < a := Nat.mul_self_lt_mul_self_iff.2 (by rw [← this]; exact Nat.lt_succ_self _)
    have : (n + 1) * (n + 1) ≤ n * n + 1 := by 

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
h : Pell.d a1 = n * n
this : n * n + 1 = a * a
na : n < a
⊢ (n + 1) * (n + 1) ≤ n * n + 1

Next tactic:
rw [this]
instance dnsq : Zsqrtd.Nonsquare (d a1) :=
  ⟨fun n h =>
    have : n * n + 1 = a * a := by rw [← h]; exact Nat.succ_pred_eq_of_pos (asq_pos a1)
    have na : n < a := Nat.mul_self_lt_mul_self_iff.2 (by rw [← this]; exact Nat.lt_succ_self _)
    have : (n + 1) * (n + 1) ≤ n * n + 1 := by rw [this]; 

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
h : Pell.d a1 = n * n
this : n * n + 1 = a * a
na : n < a
⊢ (n + 1) * (n + 1) ≤ a * a

Next tactic:
exact Nat.mul_self_le_mul_self na
instance dnsq : Zsqrtd.Nonsquare (d a1) :=
  ⟨fun n h =>
    have : n * n + 1 = a * a := by rw [← h]; exact Nat.succ_pred_eq_of_pos (asq_pos a1)
    have na : n < a := Nat.mul_self_lt_mul_self_iff.2 (by rw [← this]; exact Nat.lt_succ_self _)
    have : (n + 1) * (n + 1) ≤ n * n + 1 := by rw [this]; exact Nat.mul_self_le_mul_self na
    have : n + n ≤ 0 :=
      @Nat.le_of_add_le_add_right _ (n * n + 1) _ (by 

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
h : Pell.d a1 = n * n
this✝ : n * n + 1 = a * a
na : n < a
this : (n + 1) * (n + 1) ≤ n * n + 1
⊢ n + n + (n * n + 1) ≤ 0 + (n * n + 1)

Next tactic:
ring_nf at this ⊢
instance dnsq : Zsqrtd.Nonsquare (d a1) :=
  ⟨fun n h =>
    have : n * n + 1 = a * a := by rw [← h]; exact Nat.succ_pred_eq_of_pos (asq_pos a1)
    have na : n < a := Nat.mul_self_lt_mul_self_iff.2 (by rw [← this]; exact Nat.lt_succ_self _)
    have : (n + 1) * (n + 1) ≤ n * n + 1 := by rw [this]; exact Nat.mul_self_le_mul_self na
    have : n + n ≤ 0 :=
      @Nat.le_of_add_le_add_right _ (n * n + 1) _ (by ring_nf at this ⊢; 

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
h : Pell.d a1 = n * n
this✝ : n * n + 1 = a * a
na : n < a
this : 1 + n * 2 + n ^ 2 ≤ 1 + n ^ 2
⊢ 1 + n * 2 + n ^ 2 ≤ 1 + n ^ 2

Next tactic:
assumption
instance dnsq : Zsqrtd.Nonsquare (d a1) :=
  ⟨fun n h =>
    have : n * n + 1 = a * a := by rw [← h]; exact Nat.succ_pred_eq_of_pos (asq_pos a1)
    have na : n < a := Nat.mul_self_lt_mul_self_iff.2 (by rw [← this]; exact Nat.lt_succ_self _)
    have : (n + 1) * (n + 1) ≤ n * n + 1 := by rw [this]; exact Nat.mul_self_le_mul_self na
    have : n + n ≤ 0 :=
      @Nat.le_of_add_le_add_right _ (n * n + 1) _ (by ring_nf at this ⊢; assumption)
    Nat.ne_of_gt (d_pos a1) <| by
      

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
h : Pell.d a1 = n * n
this✝¹ : n * n + 1 = a * a
na : n < a
this✝ : (n + 1) * (n + 1) ≤ n * n + 1
this : n + n ≤ 0
⊢ Pell.d a1 = 0

Next tactic:
rwa [Nat.eq_zero_of_le_zero ((Nat.le_add_left _ _).trans this)] at h
theorem xn_ge_a_pow : ∀ n : ℕ, a ^ n ≤ xn a1 n
  | 0 => le_refl 1
  | n + 1 => by
    

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
⊢ a ^ (n + 1) ≤ xn a1 (n + 1)

Next tactic:
simp only [_root_.pow_succ', xn_succ]
theorem xn_ge_a_pow : ∀ n : ℕ, a ^ n ≤ xn a1 n
  | 0 => le_refl 1
  | n + 1 => by
    simp only [_root_.pow_succ', xn_succ]
    

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

Next tactic:
exact le_trans (Nat.mul_le_mul_right _ (xn_ge_a_pow n)) (Nat.le_add_right _ _)
theorem n_lt_a_pow : ∀ n : ℕ, n < a ^ n
  | 0 => Nat.le_refl 1
  | n + 1 => by
    

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
⊢ n + 1 < a ^ (n + 1)

Next tactic:
have IH := n_lt_a_pow n
theorem n_lt_a_pow : ∀ n : ℕ, n < a ^ n
  | 0 => Nat.le_refl 1
  | n + 1 => by
    have IH := n_lt_a_pow n
    

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
IH : n < a ^ n
⊢ n + 1 < a ^ (n + 1)

Next tactic:
have : a ^ n + a ^ n ≤ a ^ n * a := by
      rw [← mul_two]
      exact Nat.mul_le_mul_left _ a1
theorem n_lt_a_pow : ∀ n : ℕ, n < a ^ n
  | 0 => Nat.le_refl 1
  | n + 1 => by
    have IH := n_lt_a_pow n
    have : a ^ n + a ^ n ≤ a ^ n * a := by
      

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
IH : n < a ^ n
⊢ a ^ n + a ^ n ≤ a ^ n * a

Next tactic:
rw [← mul_two]
theorem n_lt_a_pow : ∀ n : ℕ, n < a ^ n
  | 0 => Nat.le_refl 1
  | n + 1 => by
    have IH := n_lt_a_pow n
    have : a ^ n + a ^ n ≤ a ^ n * a := by
      rw [← mul_two]
      

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
IH : n < a ^ n
⊢ a ^ n * 2 ≤ a ^ n * a

Next tactic:
exact Nat.mul_le_mul_left _ a1
theorem n_lt_a_pow : ∀ n : ℕ, n < a ^ n
  | 0 => Nat.le_refl 1
  | n + 1 => by
    have IH := n_lt_a_pow n
    have : a ^ n + a ^ n ≤ a ^ n * a := by
      rw [← mul_two]
      exact Nat.mul_le_mul_left _ a1
    

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
IH : n < a ^ n
this : a ^ n + a ^ n ≤ a ^ n * a
⊢ n + 1 < a ^ (n + 1)

Next tactic:
simp only [_root_.pow_succ', gt_iff_lt]
theorem n_lt_a_pow : ∀ n : ℕ, n < a ^ n
  | 0 => Nat.le_refl 1
  | n + 1 => by
    have IH := n_lt_a_pow n
    have : a ^ n + a ^ n ≤ a ^ n * a := by
      rw [← mul_two]
      exact Nat.mul_le_mul_left _ a1
    simp only [_root_.pow_succ', gt_iff_lt]
    

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
IH : n < a ^ n
this : a ^ n + a ^ n ≤ a ^ n * a
⊢ n + 1 < a ^ n * a

Next tactic:
refine' lt_of_lt_of_le _ this
theorem n_lt_a_pow : ∀ n : ℕ, n < a ^ n
  | 0 => Nat.le_refl 1
  | n + 1 => by
    have IH := n_lt_a_pow n
    have : a ^ n + a ^ n ≤ a ^ n * a := by
      rw [← mul_two]
      exact Nat.mul_le_mul_left _ a1
    simp only [_root_.pow_succ', gt_iff_lt]
    refine' lt_of_lt_of_le _ this
    

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
IH : n < a ^ n
this : a ^ n + a ^ n ≤ a ^ n * a
⊢ n + 1 < a ^ n + a ^ n

Next tactic:
exact add_lt_add_of_lt_of_le IH (lt_of_le_of_lt (Nat.zero_le _) IH)
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 

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
b : ℤ√↑(Pell.d a1)
h1 : 1 ≤ b
hp : IsPell b
h : b ≤ pellZd a1 (n + 1)
a1p : 0 ≤ { re := ↑a, im := 1 }
⊢ Pell.d a1 * (0 + 1) * (0 + 1) ≤ 1 * (a + 0) * (a + 0)

Next tactic:
simp
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; 

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
b : ℤ√↑(Pell.d a1)
h1 : 1 ≤ b
hp : IsPell b
h : b ≤ pellZd a1 (n + 1)
a1p : 0 ≤ { re := ↑a, im := 1 }
⊢ Pell.d a1 ≤ a * a

Next tactic:
exact Nat.pred_le _
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 

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
b : ℤ√↑(Pell.d a1)
h1 : 1 ≤ b
hp : IsPell b
h : b ≤ pellZd a1 (n + 1)
a1p : 0 ≤ { re := ↑a, im := 1 }
am1p : 0 ≤ { re := ↑a, im := -1 }
a1m : { re := ↑a, im := 1 } * { re := ↑a, im := -1 } = 1
ha : { re := ↑a, im := 1 } ≤ b
⊢ 1 ≤ b * { re := ↑a, im := -1 }

Next tactic:
rw [← a1m]
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]; 

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
b : ℤ√↑(Pell.d a1)
h1 : 1 ≤ b
hp : IsPell b
h : b ≤ pellZd a1 (n + 1)
a1p : 0 ≤ { re := ↑a, im := 1 }
am1p : 0 ≤ { re := ↑a, im := -1 }
a1m : { re := ↑a, im := 1 } * { re := ↑a, im := -1 } = 1
ha : { re := ↑a, im := 1 } ≤ b
⊢ { re := ↑a, im := 1 } * { re := ↑a, im := -1 } ≤ b * { re := ↑a, im := -1 }

Next tactic:
exact mul_le_mul_of_nonneg_right ha am1p
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
            

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
b : ℤ√↑(Pell.d a1)
h1 : 1 ≤ b
hp : IsPell b
h : b ≤ pellZd a1 (n + 1)
a1p : 0 ≤ { re := ↑a, im := 1 }
am1p : 0 ≤ { re := ↑a, im := -1 }
a1m : { re := ↑a, im := 1 } * { re := ↑a, im := -1 } = 1
ha : { re := ↑a, im := 1 } ≤ b
⊢ b * { re := ↑a, im := -1 } ≤ pellZd a1 n

Next tactic:
have t := mul_le_mul_of_nonneg_right h am1p
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
            

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
b : ℤ√↑(Pell.d a1)
h1 : 1 ≤ b
hp : IsPell b
h : b ≤ pellZd a1 (n + 1)
a1p : 0 ≤ { re := ↑a, im := 1 }
am1p : 0 ≤ { re := ↑a, im := -1 }
a1m : { re := ↑a, im := 1 } * { re := ↑a, im := -1 } = 1
ha : { re := ↑a, im := 1 } ≤ b
t : b * { re := ↑a, im := -1 } ≤ pellZd a1 (n + 1) * { re := ↑a, im := -1 }
⊢ b * { re := ↑a, im := -1 } ≤ pellZd a1 n

Next tactic:
rwa [pellZd_succ, mul_assoc, a1m, mul_one] at t
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
        

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
b : ℤ√↑(Pell.d a1)
h1 : 1 ≤ b
hp : IsPell b
h : b ≤ pellZd a1 (n + 1)
a1p : 0 ≤ { re := ↑a, im := 1 }
am1p : 0 ≤ { re := ↑a, im := -1 }
a1m : { re := ↑a, im := 1 } * { re := ↑a, im := -1 } = 1
ha : { re := ↑a, im := 1 } ≤ b
m : ℕ
e : b * { re := ↑a, im := -1 } = pellZd a1 m
⊢ b = pellZd a1 (m + 1)

Next tactic:
rw [show b = b * ⟨a, -1⟩ * ⟨a, 1⟩ by rw [mul_assoc, Eq.trans (mul_comm _ _) a1m]; simp,
          pellZd_succ, e]
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 

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
b : ℤ√↑(Pell.d a1)
h1 : 1 ≤ b
hp : IsPell b
h : b ≤ pellZd a1 (n + 1)
a1p : 0 ≤ { re := ↑a, im := 1 }
am1p : 0 ≤ { re := ↑a, im := -1 }
a1m : { re := ↑a, im := 1 } * { re := ↑a, im := -1 } = 1
ha : { re := ↑a, im := 1 } ≤ b
m : ℕ
e : b * { re := ↑a, im := -1 } = pellZd a1 m
⊢ b = b * { re := ↑a, im := -1 } * { re := ↑a, im := 1 }

Next tactic:
rw [mul_assoc, Eq.trans (mul_comm _ _) a1m]
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]; 

Proof state:
a : ℕ
a1 : 1 < a
n : ℕ
b : ℤ√↑(Pell.d a1)
h1 : 1 ≤ b
hp : IsPell b
h : b ≤ pellZd a1 (n + 1)
a1p : 0 ≤ { re := ↑a, im := 1 }
am1p : 0 ≤ { re := ↑a, im := -1 }
a1m : { re := ↑a, im := 1 } * { re := ↑a, im := -1 } = 1
ha : { re := ↑a, im := 1 } ≤ b
m : ℕ
e : b * { re := ↑a, im := -1 } = pellZd a1 m
⊢ b = b * 1

Next tactic:
simp
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
      

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

Next tactic:
cases' b with x y
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
      

Proof state:
case mk
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 }
⊢ False

Next tactic:
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 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