l3lab/ntp-mathlib huggingface dataset:
/-- Every solution is, up to a sign, a power of a given fundamental solution. -/
theorem eq_zpow_or_neg_zpow {a₁ : Solution₁ d} (h : IsFundamental a₁) (a : Solution₁ d) :
    ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨b, hbx, hby, hb⟩ := exists_pos_variant h.d_pos a
  obtain ⟨n, hn⟩ := h.eq_pow_of_nonneg hbx hby
  rcases hb with (rfl | rfl | rfl | hb)
  · exact ⟨n, Or.inl (mod_cast hn)⟩
  · exact ⟨-n, Or.inl (by simp [hn])⟩
  · exact ⟨n, Or.inr (by simp [hn])⟩
  · 

Proof state:
case intro.intro.intro.intro.inr.inr.inr
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a b : Solution₁ d
hbx : 0 < Solution₁.x b
hby : 0 ≤ Solution₁.y b
n : ℕ
hn : b = a₁ ^ n
hb : a ∈ {-b⁻¹}
⊢ ∃ n, a = a₁ ^ n ∨ a = -a₁ ^ n

Next tactic:
rw [Set.mem_singleton_iff] at hb
/-- Every solution is, up to a sign, a power of a given fundamental solution. -/
theorem eq_zpow_or_neg_zpow {a₁ : Solution₁ d} (h : IsFundamental a₁) (a : Solution₁ d) :
    ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨b, hbx, hby, hb⟩ := exists_pos_variant h.d_pos a
  obtain ⟨n, hn⟩ := h.eq_pow_of_nonneg hbx hby
  rcases hb with (rfl | rfl | rfl | hb)
  · exact ⟨n, Or.inl (mod_cast hn)⟩
  · exact ⟨-n, Or.inl (by simp [hn])⟩
  · exact ⟨n, Or.inr (by simp [hn])⟩
  · rw [Set.mem_singleton_iff] at hb
    

Proof state:
case intro.intro.intro.intro.inr.inr.inr
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a b : Solution₁ d
hbx : 0 < Solution₁.x b
hby : 0 ≤ Solution₁.y b
n : ℕ
hn : b = a₁ ^ n
hb : a = -b⁻¹
⊢ ∃ n, a = a₁ ^ n ∨ a = -a₁ ^ n

Next tactic:
rw [hb]
/-- Every solution is, up to a sign, a power of a given fundamental solution. -/
theorem eq_zpow_or_neg_zpow {a₁ : Solution₁ d} (h : IsFundamental a₁) (a : Solution₁ d) :
    ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨b, hbx, hby, hb⟩ := exists_pos_variant h.d_pos a
  obtain ⟨n, hn⟩ := h.eq_pow_of_nonneg hbx hby
  rcases hb with (rfl | rfl | rfl | hb)
  · exact ⟨n, Or.inl (mod_cast hn)⟩
  · exact ⟨-n, Or.inl (by simp [hn])⟩
  · exact ⟨n, Or.inr (by simp [hn])⟩
  · rw [Set.mem_singleton_iff] at hb
    rw [hb]
    

Proof state:
case intro.intro.intro.intro.inr.inr.inr
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a b : Solution₁ d
hbx : 0 < Solution₁.x b
hby : 0 ≤ Solution₁.y b
n : ℕ
hn : b = a₁ ^ n
hb : a = -b⁻¹
⊢ ∃ n, -b⁻¹ = a₁ ^ n ∨ -b⁻¹ = -a₁ ^ n

Next tactic:
exact ⟨-n, Or.inr (by simp [hn])⟩
/-- Every solution is, up to a sign, a power of a given fundamental solution. -/
theorem eq_zpow_or_neg_zpow {a₁ : Solution₁ d} (h : IsFundamental a₁) (a : Solution₁ d) :
    ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨b, hbx, hby, hb⟩ := exists_pos_variant h.d_pos a
  obtain ⟨n, hn⟩ := h.eq_pow_of_nonneg hbx hby
  rcases hb with (rfl | rfl | rfl | hb)
  · exact ⟨n, Or.inl (mod_cast hn)⟩
  · exact ⟨-n, Or.inl (by simp [hn])⟩
  · exact ⟨n, Or.inr (by simp [hn])⟩
  · rw [Set.mem_singleton_iff] at hb
    rw [hb]
    exact ⟨-n, Or.inr (by 

Proof state:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a b : Solution₁ d
hbx : 0 < Solution₁.x b
hby : 0 ≤ Solution₁.y b
n : ℕ
hn : b = a₁ ^ n
hb : a = -b⁻¹
⊢ -b⁻¹ = -a₁ ^ (-↑n)

Next tactic:
simp [hn]
/-- When `d` is positive and not a square, then the group of solutions to the Pell equation
`x^2 - d*y^2 = 1` has a unique positive generator (up to sign). -/
theorem existsUnique_pos_generator (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃! a₁ : Solution₁ d,
      1 < a₁.x ∧ 0 < a₁.y ∧ ∀ a : Solution₁ d, ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  

Proof state:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
⊢ ∃! a₁, 1 < Solution₁.x a₁ ∧ 0 < Solution₁.y a₁ ∧ ∀ (a : Solution₁ d), ∃ n, a = a₁ ^ n ∨ a = -a₁ ^ n

Next tactic:
obtain ⟨a₁, ha₁⟩ := IsFundamental.exists_of_not_isSquare h₀ hd
/-- When `d` is positive and not a square, then the group of solutions to the Pell equation
`x^2 - d*y^2 = 1` has a unique positive generator (up to sign). -/
theorem existsUnique_pos_generator (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃! a₁ : Solution₁ d,
      1 < a₁.x ∧ 0 < a₁.y ∧ ∀ a : Solution₁ d, ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨a₁, ha₁⟩ := IsFundamental.exists_of_not_isSquare h₀ hd
  

Proof state:
case intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
⊢ ∃! a₁, 1 < Solution₁.x a₁ ∧ 0 < Solution₁.y a₁ ∧ ∀ (a : Solution₁ d), ∃ n, a = a₁ ^ n ∨ a = -a₁ ^ n

Next tactic:
refine' ⟨a₁, ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩, fun a (H : 1 < _ ∧ _) => _⟩
/-- When `d` is positive and not a square, then the group of solutions to the Pell equation
`x^2 - d*y^2 = 1` has a unique positive generator (up to sign). -/
theorem existsUnique_pos_generator (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃! a₁ : Solution₁ d,
      1 < a₁.x ∧ 0 < a₁.y ∧ ∀ a : Solution₁ d, ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨a₁, ha₁⟩ := IsFundamental.exists_of_not_isSquare h₀ hd
  refine' ⟨a₁, ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩, fun a (H : 1 < _ ∧ _) => _⟩
  

Proof state:
case intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
a : Solution₁ d
H : 1 < Solution₁.x a ∧ 0 < Solution₁.y a ∧ ∀ (a_1 : Solution₁ d), ∃ n, a_1 = a ^ n ∨ a_1 = -a ^ n
⊢ a = a₁

Next tactic:
obtain ⟨Hx, Hy, H⟩ := H
/-- When `d` is positive and not a square, then the group of solutions to the Pell equation
`x^2 - d*y^2 = 1` has a unique positive generator (up to sign). -/
theorem existsUnique_pos_generator (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃! a₁ : Solution₁ d,
      1 < a₁.x ∧ 0 < a₁.y ∧ ∀ a : Solution₁ d, ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨a₁, ha₁⟩ := IsFundamental.exists_of_not_isSquare h₀ hd
  refine' ⟨a₁, ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩, fun a (H : 1 < _ ∧ _) => _⟩
  obtain ⟨Hx, Hy, H⟩ := H
  

Proof state:
case intro.intro.intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
a : Solution₁ d
Hx : 1 < Solution₁.x a
Hy : 0 < Solution₁.y a
H : ∀ (a_1 : Solution₁ d), ∃ n, a_1 = a ^ n ∨ a_1 = -a ^ n
⊢ a = a₁

Next tactic:
obtain ⟨n₁, hn₁⟩ := H a₁
/-- When `d` is positive and not a square, then the group of solutions to the Pell equation
`x^2 - d*y^2 = 1` has a unique positive generator (up to sign). -/
theorem existsUnique_pos_generator (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃! a₁ : Solution₁ d,
      1 < a₁.x ∧ 0 < a₁.y ∧ ∀ a : Solution₁ d, ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨a₁, ha₁⟩ := IsFundamental.exists_of_not_isSquare h₀ hd
  refine' ⟨a₁, ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩, fun a (H : 1 < _ ∧ _) => _⟩
  obtain ⟨Hx, Hy, H⟩ := H
  obtain ⟨n₁, hn₁⟩ := H a₁
  

Proof state:
case intro.intro.intro.intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
a : Solution₁ d
Hx : 1 < Solution₁.x a
Hy : 0 < Solution₁.y a
H : ∀ (a_1 : Solution₁ d), ∃ n, a_1 = a ^ n ∨ a_1 = -a ^ n
n₁ : ℤ
hn₁ : a₁ = a ^ n₁ ∨ a₁ = -a ^ n₁
⊢ a = a₁

Next tactic:
obtain ⟨n₂, hn₂⟩ := ha₁.eq_zpow_or_neg_zpow a
/-- When `d` is positive and not a square, then the group of solutions to the Pell equation
`x^2 - d*y^2 = 1` has a unique positive generator (up to sign). -/
theorem existsUnique_pos_generator (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃! a₁ : Solution₁ d,
      1 < a₁.x ∧ 0 < a₁.y ∧ ∀ a : Solution₁ d, ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨a₁, ha₁⟩ := IsFundamental.exists_of_not_isSquare h₀ hd
  refine' ⟨a₁, ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩, fun a (H : 1 < _ ∧ _) => _⟩
  obtain ⟨Hx, Hy, H⟩ := H
  obtain ⟨n₁, hn₁⟩ := H a₁
  obtain ⟨n₂, hn₂⟩ := ha₁.eq_zpow_or_neg_zpow a
  

Proof state:
case intro.intro.intro.intro.intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
a : Solution₁ d
Hx : 1 < Solution₁.x a
Hy : 0 < Solution₁.y a
H : ∀ (a_1 : Solution₁ d), ∃ n, a_1 = a ^ n ∨ a_1 = -a ^ n
n₁ : ℤ
hn₁ : a₁ = a ^ n₁ ∨ a₁ = -a ^ n₁
n₂ : ℤ
hn₂ : a = a₁ ^ n₂ ∨ a = -a₁ ^ n₂
⊢ a = a₁

Next tactic:
rcases hn₂ with (rfl | rfl)
/-- When `d` is positive and not a square, then the group of solutions to the Pell equation
`x^2 - d*y^2 = 1` has a unique positive generator (up to sign). -/
theorem existsUnique_pos_generator (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃! a₁ : Solution₁ d,
      1 < a₁.x ∧ 0 < a₁.y ∧ ∀ a : Solution₁ d, ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨a₁, ha₁⟩ := IsFundamental.exists_of_not_isSquare h₀ hd
  refine' ⟨a₁, ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩, fun a (H : 1 < _ ∧ _) => _⟩
  obtain ⟨Hx, Hy, H⟩ := H
  obtain ⟨n₁, hn₁⟩ := H a₁
  obtain ⟨n₂, hn₂⟩ := ha₁.eq_zpow_or_neg_zpow a
  rcases hn₂ with (rfl | rfl)
  · 

Proof state:
case intro.intro.intro.intro.intro.inl
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ n₂ : ℤ
Hx : 1 < Solution₁.x (a₁ ^ n₂)
Hy : 0 < Solution₁.y (a₁ ^ n₂)
H : ∀ (a : Solution₁ d), ∃ n, a = (a₁ ^ n₂) ^ n ∨ a = -(a₁ ^ n₂) ^ n
hn₁ : a₁ = (a₁ ^ n₂) ^ n₁ ∨ a₁ = -(a₁ ^ n₂) ^ n₁
⊢ a₁ ^ n₂ = a₁

Next tactic:
rw [← zpow_mul, eq_comm, @eq_comm _ a₁, ← mul_inv_eq_one, ← @mul_inv_eq_one _ _ _ a₁, ←
      zpow_neg_one, neg_mul, ← zpow_add, ← sub_eq_add_neg] at hn₁
/-- When `d` is positive and not a square, then the group of solutions to the Pell equation
`x^2 - d*y^2 = 1` has a unique positive generator (up to sign). -/
theorem existsUnique_pos_generator (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃! a₁ : Solution₁ d,
      1 < a₁.x ∧ 0 < a₁.y ∧ ∀ a : Solution₁ d, ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨a₁, ha₁⟩ := IsFundamental.exists_of_not_isSquare h₀ hd
  refine' ⟨a₁, ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩, fun a (H : 1 < _ ∧ _) => _⟩
  obtain ⟨Hx, Hy, H⟩ := H
  obtain ⟨n₁, hn₁⟩ := H a₁
  obtain ⟨n₂, hn₂⟩ := ha₁.eq_zpow_or_neg_zpow a
  rcases hn₂ with (rfl | rfl)
  · rw [← zpow_mul, eq_comm, @eq_comm _ a₁, ← mul_inv_eq_one, ← @mul_inv_eq_one _ _ _ a₁, ←
      zpow_neg_one, neg_mul, ← zpow_add, ← sub_eq_add_neg] at hn₁
    

Proof state:
case intro.intro.intro.intro.intro.inl
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ n₂ : ℤ
Hx : 1 < Solution₁.x (a₁ ^ n₂)
Hy : 0 < Solution₁.y (a₁ ^ n₂)
H : ∀ (a : Solution₁ d), ∃ n, a = (a₁ ^ n₂) ^ n ∨ a = -(a₁ ^ n₂) ^ n
hn₁ : a₁ ^ (n₂ * n₁ - 1) = 1 ∨ -a₁ ^ (n₂ * n₁ - 1) = 1
⊢ a₁ ^ n₂ = a₁

Next tactic:
cases' hn₁ with hn₁ hn₁
/-- When `d` is positive and not a square, then the group of solutions to the Pell equation
`x^2 - d*y^2 = 1` has a unique positive generator (up to sign). -/
theorem existsUnique_pos_generator (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃! a₁ : Solution₁ d,
      1 < a₁.x ∧ 0 < a₁.y ∧ ∀ a : Solution₁ d, ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨a₁, ha₁⟩ := IsFundamental.exists_of_not_isSquare h₀ hd
  refine' ⟨a₁, ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩, fun a (H : 1 < _ ∧ _) => _⟩
  obtain ⟨Hx, Hy, H⟩ := H
  obtain ⟨n₁, hn₁⟩ := H a₁
  obtain ⟨n₂, hn₂⟩ := ha₁.eq_zpow_or_neg_zpow a
  rcases hn₂ with (rfl | rfl)
  · rw [← zpow_mul, eq_comm, @eq_comm _ a₁, ← mul_inv_eq_one, ← @mul_inv_eq_one _ _ _ a₁, ←
      zpow_neg_one, neg_mul, ← zpow_add, ← sub_eq_add_neg] at hn₁
    cases' hn₁ with hn₁ hn₁
    · 

Proof state:
case intro.intro.intro.intro.intro.inl.inl
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ n₂ : ℤ
Hx : 1 < Solution₁.x (a₁ ^ n₂)
Hy : 0 < Solution₁.y (a₁ ^ n₂)
H : ∀ (a : Solution₁ d), ∃ n, a = (a₁ ^ n₂) ^ n ∨ a = -(a₁ ^ n₂) ^ n
hn₁ : a₁ ^ (n₂ * n₁ - 1) = 1
⊢ a₁ ^ n₂ = a₁

Next tactic:
rcases Int.isUnit_iff.mp
          (isUnit_of_mul_eq_one _ _ <|
            sub_eq_zero.mp <| (ha₁.zpow_eq_one_iff (n₂ * n₁ - 1)).mp hn₁) with
        (rfl | rfl)
/-- When `d` is positive and not a square, then the group of solutions to the Pell equation
`x^2 - d*y^2 = 1` has a unique positive generator (up to sign). -/
theorem existsUnique_pos_generator (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃! a₁ : Solution₁ d,
      1 < a₁.x ∧ 0 < a₁.y ∧ ∀ a : Solution₁ d, ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨a₁, ha₁⟩ := IsFundamental.exists_of_not_isSquare h₀ hd
  refine' ⟨a₁, ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩, fun a (H : 1 < _ ∧ _) => _⟩
  obtain ⟨Hx, Hy, H⟩ := H
  obtain ⟨n₁, hn₁⟩ := H a₁
  obtain ⟨n₂, hn₂⟩ := ha₁.eq_zpow_or_neg_zpow a
  rcases hn₂ with (rfl | rfl)
  · rw [← zpow_mul, eq_comm, @eq_comm _ a₁, ← mul_inv_eq_one, ← @mul_inv_eq_one _ _ _ a₁, ←
      zpow_neg_one, neg_mul, ← zpow_add, ← sub_eq_add_neg] at hn₁
    cases' hn₁ with hn₁ hn₁
    · rcases Int.isUnit_iff.mp
          (isUnit_of_mul_eq_one _ _ <|
            sub_eq_zero.mp <| (ha₁.zpow_eq_one_iff (n₂ * n₁ - 1)).mp hn₁) with
        (rfl | rfl)
      · 

Proof state:
case intro.intro.intro.intro.intro.inl.inl.inl
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ : ℤ
Hx : 1 < Solution₁.x (a₁ ^ 1)
Hy : 0 < Solution₁.y (a₁ ^ 1)
H : ∀ (a : Solution₁ d), ∃ n, a = (a₁ ^ 1) ^ n ∨ a = -(a₁ ^ 1) ^ n
hn₁ : a₁ ^ (1 * n₁ - 1) = 1
⊢ a₁ ^ 1 = a₁

Next tactic:
rw [zpow_one]
/-- When `d` is positive and not a square, then the group of solutions to the Pell equation
`x^2 - d*y^2 = 1` has a unique positive generator (up to sign). -/
theorem existsUnique_pos_generator (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃! a₁ : Solution₁ d,
      1 < a₁.x ∧ 0 < a₁.y ∧ ∀ a : Solution₁ d, ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨a₁, ha₁⟩ := IsFundamental.exists_of_not_isSquare h₀ hd
  refine' ⟨a₁, ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩, fun a (H : 1 < _ ∧ _) => _⟩
  obtain ⟨Hx, Hy, H⟩ := H
  obtain ⟨n₁, hn₁⟩ := H a₁
  obtain ⟨n₂, hn₂⟩ := ha₁.eq_zpow_or_neg_zpow a
  rcases hn₂ with (rfl | rfl)
  · rw [← zpow_mul, eq_comm, @eq_comm _ a₁, ← mul_inv_eq_one, ← @mul_inv_eq_one _ _ _ a₁, ←
      zpow_neg_one, neg_mul, ← zpow_add, ← sub_eq_add_neg] at hn₁
    cases' hn₁ with hn₁ hn₁
    · rcases Int.isUnit_iff.mp
          (isUnit_of_mul_eq_one _ _ <|
            sub_eq_zero.mp <| (ha₁.zpow_eq_one_iff (n₂ * n₁ - 1)).mp hn₁) with
        (rfl | rfl)
      · rw [zpow_one]
      · 

Proof state:
case intro.intro.intro.intro.intro.inl.inl.inr
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ : ℤ
Hx : 1 < Solution₁.x (a₁ ^ (-1))
Hy : 0 < Solution₁.y (a₁ ^ (-1))
H : ∀ (a : Solution₁ d), ∃ n, a = (a₁ ^ (-1)) ^ n ∨ a = -(a₁ ^ (-1)) ^ n
hn₁ : a₁ ^ (-1 * n₁ - 1) = 1
⊢ a₁ ^ (-1) = a₁

Next tactic:
rw [zpow_neg_one, y_inv, lt_neg, neg_zero] at Hy
/-- When `d` is positive and not a square, then the group of solutions to the Pell equation
`x^2 - d*y^2 = 1` has a unique positive generator (up to sign). -/
theorem existsUnique_pos_generator (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃! a₁ : Solution₁ d,
      1 < a₁.x ∧ 0 < a₁.y ∧ ∀ a : Solution₁ d, ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨a₁, ha₁⟩ := IsFundamental.exists_of_not_isSquare h₀ hd
  refine' ⟨a₁, ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩, fun a (H : 1 < _ ∧ _) => _⟩
  obtain ⟨Hx, Hy, H⟩ := H
  obtain ⟨n₁, hn₁⟩ := H a₁
  obtain ⟨n₂, hn₂⟩ := ha₁.eq_zpow_or_neg_zpow a
  rcases hn₂ with (rfl | rfl)
  · rw [← zpow_mul, eq_comm, @eq_comm _ a₁, ← mul_inv_eq_one, ← @mul_inv_eq_one _ _ _ a₁, ←
      zpow_neg_one, neg_mul, ← zpow_add, ← sub_eq_add_neg] at hn₁
    cases' hn₁ with hn₁ hn₁
    · rcases Int.isUnit_iff.mp
          (isUnit_of_mul_eq_one _ _ <|
            sub_eq_zero.mp <| (ha₁.zpow_eq_one_iff (n₂ * n₁ - 1)).mp hn₁) with
        (rfl | rfl)
      · rw [zpow_one]
      · rw [zpow_neg_one, y_inv, lt_neg, neg_zero] at Hy
        

Proof state:
case intro.intro.intro.intro.intro.inl.inl.inr
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ : ℤ
Hx : 1 < Solution₁.x (a₁ ^ (-1))
Hy : Solution₁.y a₁ < 0
H : ∀ (a : Solution₁ d), ∃ n, a = (a₁ ^ (-1)) ^ n ∨ a = -(a₁ ^ (-1)) ^ n
hn₁ : a₁ ^ (-1 * n₁ - 1) = 1
⊢ a₁ ^ (-1) = a₁

Next tactic:
exact False.elim (lt_irrefl _ <| ha₁.2.1.trans Hy)
/-- When `d` is positive and not a square, then the group of solutions to the Pell equation
`x^2 - d*y^2 = 1` has a unique positive generator (up to sign). -/
theorem existsUnique_pos_generator (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃! a₁ : Solution₁ d,
      1 < a₁.x ∧ 0 < a₁.y ∧ ∀ a : Solution₁ d, ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨a₁, ha₁⟩ := IsFundamental.exists_of_not_isSquare h₀ hd
  refine' ⟨a₁, ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩, fun a (H : 1 < _ ∧ _) => _⟩
  obtain ⟨Hx, Hy, H⟩ := H
  obtain ⟨n₁, hn₁⟩ := H a₁
  obtain ⟨n₂, hn₂⟩ := ha₁.eq_zpow_or_neg_zpow a
  rcases hn₂ with (rfl | rfl)
  · rw [← zpow_mul, eq_comm, @eq_comm _ a₁, ← mul_inv_eq_one, ← @mul_inv_eq_one _ _ _ a₁, ←
      zpow_neg_one, neg_mul, ← zpow_add, ← sub_eq_add_neg] at hn₁
    cases' hn₁ with hn₁ hn₁
    · rcases Int.isUnit_iff.mp
          (isUnit_of_mul_eq_one _ _ <|
            sub_eq_zero.mp <| (ha₁.zpow_eq_one_iff (n₂ * n₁ - 1)).mp hn₁) with
        (rfl | rfl)
      · rw [zpow_one]
      · rw [zpow_neg_one, y_inv, lt_neg, neg_zero] at Hy
        exact False.elim (lt_irrefl _ <| ha₁.2.1.trans Hy)
    · 

Proof state:
case intro.intro.intro.intro.intro.inl.inr
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ n₂ : ℤ
Hx : 1 < Solution₁.x (a₁ ^ n₂)
Hy : 0 < Solution₁.y (a₁ ^ n₂)
H : ∀ (a : Solution₁ d), ∃ n, a = (a₁ ^ n₂) ^ n ∨ a = -(a₁ ^ n₂) ^ n
hn₁ : -a₁ ^ (n₂ * n₁ - 1) = 1
⊢ a₁ ^ n₂ = a₁

Next tactic:
rw [← zpow_zero a₁, eq_comm] at hn₁
/-- When `d` is positive and not a square, then the group of solutions to the Pell equation
`x^2 - d*y^2 = 1` has a unique positive generator (up to sign). -/
theorem existsUnique_pos_generator (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃! a₁ : Solution₁ d,
      1 < a₁.x ∧ 0 < a₁.y ∧ ∀ a : Solution₁ d, ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨a₁, ha₁⟩ := IsFundamental.exists_of_not_isSquare h₀ hd
  refine' ⟨a₁, ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩, fun a (H : 1 < _ ∧ _) => _⟩
  obtain ⟨Hx, Hy, H⟩ := H
  obtain ⟨n₁, hn₁⟩ := H a₁
  obtain ⟨n₂, hn₂⟩ := ha₁.eq_zpow_or_neg_zpow a
  rcases hn₂ with (rfl | rfl)
  · rw [← zpow_mul, eq_comm, @eq_comm _ a₁, ← mul_inv_eq_one, ← @mul_inv_eq_one _ _ _ a₁, ←
      zpow_neg_one, neg_mul, ← zpow_add, ← sub_eq_add_neg] at hn₁
    cases' hn₁ with hn₁ hn₁
    · rcases Int.isUnit_iff.mp
          (isUnit_of_mul_eq_one _ _ <|
            sub_eq_zero.mp <| (ha₁.zpow_eq_one_iff (n₂ * n₁ - 1)).mp hn₁) with
        (rfl | rfl)
      · rw [zpow_one]
      · rw [zpow_neg_one, y_inv, lt_neg, neg_zero] at Hy
        exact False.elim (lt_irrefl _ <| ha₁.2.1.trans Hy)
    · rw [← zpow_zero a₁, eq_comm] at hn₁
      

Proof state:
case intro.intro.intro.intro.intro.inl.inr
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ n₂ : ℤ
Hx : 1 < Solution₁.x (a₁ ^ n₂)
Hy : 0 < Solution₁.y (a₁ ^ n₂)
H : ∀ (a : Solution₁ d), ∃ n, a = (a₁ ^ n₂) ^ n ∨ a = -(a₁ ^ n₂) ^ n
hn₁ : a₁ ^ 0 = -a₁ ^ (n₂ * n₁ - 1)
⊢ a₁ ^ n₂ = a₁

Next tactic:
exact False.elim (ha₁.zpow_ne_neg_zpow hn₁)
/-- When `d` is positive and not a square, then the group of solutions to the Pell equation
`x^2 - d*y^2 = 1` has a unique positive generator (up to sign). -/
theorem existsUnique_pos_generator (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃! a₁ : Solution₁ d,
      1 < a₁.x ∧ 0 < a₁.y ∧ ∀ a : Solution₁ d, ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨a₁, ha₁⟩ := IsFundamental.exists_of_not_isSquare h₀ hd
  refine' ⟨a₁, ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩, fun a (H : 1 < _ ∧ _) => _⟩
  obtain ⟨Hx, Hy, H⟩ := H
  obtain ⟨n₁, hn₁⟩ := H a₁
  obtain ⟨n₂, hn₂⟩ := ha₁.eq_zpow_or_neg_zpow a
  rcases hn₂ with (rfl | rfl)
  · rw [← zpow_mul, eq_comm, @eq_comm _ a₁, ← mul_inv_eq_one, ← @mul_inv_eq_one _ _ _ a₁, ←
      zpow_neg_one, neg_mul, ← zpow_add, ← sub_eq_add_neg] at hn₁
    cases' hn₁ with hn₁ hn₁
    · rcases Int.isUnit_iff.mp
          (isUnit_of_mul_eq_one _ _ <|
            sub_eq_zero.mp <| (ha₁.zpow_eq_one_iff (n₂ * n₁ - 1)).mp hn₁) with
        (rfl | rfl)
      · rw [zpow_one]
      · rw [zpow_neg_one, y_inv, lt_neg, neg_zero] at Hy
        exact False.elim (lt_irrefl _ <| ha₁.2.1.trans Hy)
    · rw [← zpow_zero a₁, eq_comm] at hn₁
      exact False.elim (ha₁.zpow_ne_neg_zpow hn₁)
  · 

Proof state:
case intro.intro.intro.intro.intro.inr
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ n₂ : ℤ
Hx : 1 < Solution₁.x (-a₁ ^ n₂)
Hy : 0 < Solution₁.y (-a₁ ^ n₂)
H : ∀ (a : Solution₁ d), ∃ n, a = (-a₁ ^ n₂) ^ n ∨ a = -(-a₁ ^ n₂) ^ n
hn₁ : a₁ = (-a₁ ^ n₂) ^ n₁ ∨ a₁ = -(-a₁ ^ n₂) ^ n₁
⊢ -a₁ ^ n₂ = a₁

Next tactic:
rw [x_neg, lt_neg] at Hx
/-- When `d` is positive and not a square, then the group of solutions to the Pell equation
`x^2 - d*y^2 = 1` has a unique positive generator (up to sign). -/
theorem existsUnique_pos_generator (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃! a₁ : Solution₁ d,
      1 < a₁.x ∧ 0 < a₁.y ∧ ∀ a : Solution₁ d, ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨a₁, ha₁⟩ := IsFundamental.exists_of_not_isSquare h₀ hd
  refine' ⟨a₁, ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩, fun a (H : 1 < _ ∧ _) => _⟩
  obtain ⟨Hx, Hy, H⟩ := H
  obtain ⟨n₁, hn₁⟩ := H a₁
  obtain ⟨n₂, hn₂⟩ := ha₁.eq_zpow_or_neg_zpow a
  rcases hn₂ with (rfl | rfl)
  · rw [← zpow_mul, eq_comm, @eq_comm _ a₁, ← mul_inv_eq_one, ← @mul_inv_eq_one _ _ _ a₁, ←
      zpow_neg_one, neg_mul, ← zpow_add, ← sub_eq_add_neg] at hn₁
    cases' hn₁ with hn₁ hn₁
    · rcases Int.isUnit_iff.mp
          (isUnit_of_mul_eq_one _ _ <|
            sub_eq_zero.mp <| (ha₁.zpow_eq_one_iff (n₂ * n₁ - 1)).mp hn₁) with
        (rfl | rfl)
      · rw [zpow_one]
      · rw [zpow_neg_one, y_inv, lt_neg, neg_zero] at Hy
        exact False.elim (lt_irrefl _ <| ha₁.2.1.trans Hy)
    · rw [← zpow_zero a₁, eq_comm] at hn₁
      exact False.elim (ha₁.zpow_ne_neg_zpow hn₁)
  · rw [x_neg, lt_neg] at Hx
    

Proof state:
case intro.intro.intro.intro.intro.inr
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ n₂ : ℤ
Hx : Solution₁.x (a₁ ^ n₂) < -1
Hy : 0 < Solution₁.y (-a₁ ^ n₂)
H : ∀ (a : Solution₁ d), ∃ n, a = (-a₁ ^ n₂) ^ n ∨ a = -(-a₁ ^ n₂) ^ n
hn₁ : a₁ = (-a₁ ^ n₂) ^ n₁ ∨ a₁ = -(-a₁ ^ n₂) ^ n₁
⊢ -a₁ ^ n₂ = a₁

Next tactic:
have := (x_zpow_pos (zero_lt_one.trans ha₁.1) n₂).trans Hx
/-- When `d` is positive and not a square, then the group of solutions to the Pell equation
`x^2 - d*y^2 = 1` has a unique positive generator (up to sign). -/
theorem existsUnique_pos_generator (h₀ : 0 < d) (hd : ¬IsSquare d) :
    ∃! a₁ : Solution₁ d,
      1 < a₁.x ∧ 0 < a₁.y ∧ ∀ a : Solution₁ d, ∃ n : ℤ, a = a₁ ^ n ∨ a = -a₁ ^ n := by
  obtain ⟨a₁, ha₁⟩ := IsFundamental.exists_of_not_isSquare h₀ hd
  refine' ⟨a₁, ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩, fun a (H : 1 < _ ∧ _) => _⟩
  obtain ⟨Hx, Hy, H⟩ := H
  obtain ⟨n₁, hn₁⟩ := H a₁
  obtain ⟨n₂, hn₂⟩ := ha₁.eq_zpow_or_neg_zpow a
  rcases hn₂ with (rfl | rfl)
  · rw [← zpow_mul, eq_comm, @eq_comm _ a₁, ← mul_inv_eq_one, ← @mul_inv_eq_one _ _ _ a₁, ←
      zpow_neg_one, neg_mul, ← zpow_add, ← sub_eq_add_neg] at hn₁
    cases' hn₁ with hn₁ hn₁
    · rcases Int.isUnit_iff.mp
          (isUnit_of_mul_eq_one _ _ <|
            sub_eq_zero.mp <| (ha₁.zpow_eq_one_iff (n₂ * n₁ - 1)).mp hn₁) with
        (rfl | rfl)
      · rw [zpow_one]
      · rw [zpow_neg_one, y_inv, lt_neg, neg_zero] at Hy
        exact False.elim (lt_irrefl _ <| ha₁.2.1.trans Hy)
    · rw [← zpow_zero a₁, eq_comm] at hn₁
      exact False.elim (ha₁.zpow_ne_neg_zpow hn₁)
  · rw [x_neg, lt_neg] at Hx
    have := (x_zpow_pos (zero_lt_one.trans ha₁.1) n₂).trans Hx
    

Proof state:
case intro.intro.intro.intro.intro.inr
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ n₂ : ℤ
Hx : Solution₁.x (a₁ ^ n₂) < -1
Hy : 0 < Solution₁.y (-a₁ ^ n₂)
H : ∀ (a : Solution₁ d), ∃ n, a = (-a₁ ^ n₂) ^ n ∨ a = -(-a₁ ^ n₂) ^ n
hn₁ : a₁ = (-a₁ ^ n₂) ^ n₁ ∨ a₁ = -(-a₁ ^ n₂) ^ n₁
this : 0 < -1
⊢ -a₁ ^ n₂ = a₁

Next tactic:
norm_num at this
/-- A positive solution is a generator (up to sign) of the group of all solutions to the
Pell equation `x^2 - d*y^2 = 1` if and only if it is a fundamental solution. -/
theorem pos_generator_iff_fundamental (a : Solution₁ d) :
    (1 < a.x ∧ 0 < a.y ∧ ∀ b : Solution₁ d, ∃ n : ℤ, b = a ^ n ∨ b = -a ^ n) ↔ IsFundamental a := by
  

Proof state:
d : ℤ
a : Solution₁ d
⊢ (1 < Solution₁.x a ∧ 0 < Solution₁.y a ∧ ∀ (b : Solution₁ d), ∃ n, b = a ^ n ∨ b = -a ^ n) ↔ IsFundamental a

Next tactic:
refine' ⟨fun h => _, fun H => ⟨H.1, H.2.1, H.eq_zpow_or_neg_zpow⟩⟩