awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
a : Solution₁ d
H : 1 < a.x ∧ 0 < a.y ∧ ∀ (a_1 : Solution₁ d), ∃ (n : ℤ), a_1 = a ^ n ∨ a_1 = -a ^ n
Hx : 1 < a.x
Hy : 0 < a.y
⊢ a = a₁
have H₁ : ∀ (a_1 : Solution₁ d), ∃ (n : ℤ), a_1 = a ^ n ∨ a_1 = -a ^ n
Tactic: obtain ⟨Hx, Hy, H⟩ := H
Cleared hypotheses: H
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
a : Solution₁ d
Hx : 1 < a.x
Hy : 0 < a.y
H : ∀ (a_1 : Solution₁ d), ∃ (n : ℤ), a_1 = a ^ n ∨ a_1 = -a ^ n
⊢ a = a₁
have obtain : ∀ (n₁ : ℤ) (hn₁ : a₁ = a ^ n₁ ∨ a₁ = -a ^ n₁), a = a₁
Tactic: obtain ⟨n₁, hn₁⟩ := H a₁
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
a : Solution₁ d
Hx : 1 < a.x
Hy : 0 < a.y
H : ∀ (a_1 : Solution₁ d), ∃ (n : ℤ), a_1 = a ^ n ∨ a_1 = -a ^ n
n₁ : ℤ
hn₁ : a₁ = a ^ n₁ ∨ a₁ = -a ^ n₁
⊢ a = a₁
have obtain : ∀ (n₂ : ℤ) (hn₂ : a = a₁ ^ n₂ ∨ a = -a₁ ^ n₂), a = a₁
Tactic: obtain ⟨n₂, hn₂⟩ := ha₁.eq_zpow_or_neg_zpow a
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
a : Solution₁ d
Hx : 1 < a.x
Hy : 0 < a.y
H : ∀ (a_1 : Solution₁ d), ∃ (n : ℤ), 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
have inl : ∀ (Hx : 1 < (a₁ ^ n₂).x) (Hy : 0 < (a₁ ^ n₂).y)
  (H : ∀ (a : Solution₁ d), ∃ (n : ℤ), a = (a₁ ^ n₂) ^ n ∨ a = -(a₁ ^ n₂) ^ n)
  (hn₁ : a₁ = (a₁ ^ n₂) ^ n₁ ∨ a₁ = -(a₁ ^ n₂) ^ n₁), a₁ ^ n₂ = a₁
Tactic: rcases hn₂ with (rfl | rfl)
Cleared hypotheses: a, Hx, Hy, H, hn₁, hn₂
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
a : Solution₁ d
Hx : 1 < a.x
Hy : 0 < a.y
H : ∀ (a_1 : Solution₁ d), ∃ (n : ℤ), 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
inl :
  ∀ (Hx : 1 < (a₁ ^ n₂).x) (Hy : 0 < (a₁ ^ n₂).y)
    (H : ∀ (a : Solution₁ d), ∃ (n : ℤ), a = (a₁ ^ n₂) ^ n ∨ a = -(a₁ ^ n₂) ^ n)
    (hn₁ : a₁ = (a₁ ^ n₂) ^ n₁ ∨ a₁ = -(a₁ ^ n₂) ^ n₁), a₁ ^ n₂ = a₁
⊢ a = a₁
have inr : ∀ (Hx : 1 < (-a₁ ^ n₂).x) (Hy : 0 < (-a₁ ^ n₂).y)
  (H : ∀ (a : Solution₁ d), ∃ (n : ℤ), a = (-a₁ ^ n₂) ^ n ∨ a = -(-a₁ ^ n₂) ^ n)
  (hn₁ : a₁ = (-a₁ ^ n₂) ^ n₁ ∨ a₁ = -(-a₁ ^ n₂) ^ n₁), -a₁ ^ n₂ = a₁
Tactic: rcases hn₂ with (rfl | rfl)
Cleared hypotheses: a, Hx, Hy, H, hn₁, hn₂
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ n₂ : ℤ
Hx : 1 < (a₁ ^ n₂).x
Hy : 0 < (a₁ ^ n₂).y
H : ∀ (a : Solution₁ d), ∃ (n : ℤ), a = (a₁ ^ n₂) ^ n ∨ a = -(a₁ ^ n₂) ^ n
hn₁ : a₁ = (a₁ ^ n₂) ^ n₁ ∨ a₁ = -(a₁ ^ n₂) ^ n₁
⊢ a₁ ^ n₂ = a₁
have hn₂ : a₁ ^ (n₂ * n₁ - 1) = 1 ∨ -a₁ ^ (n₂ * n₁ - 1) = 1
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
Cleared hypotheses: hn₁
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ n₂ : ℤ
Hx : 1 < (a₁ ^ n₂).x
Hy : 0 < (a₁ ^ n₂).y
H : ∀ (a : Solution₁ d), ∃ (n : ℤ), a = (a₁ ^ n₂) ^ n ∨ a = -(a₁ ^ n₂) ^ n
hn₁ : a₁ ^ (n₂ * n₁ - 1) = 1 ∨ -a₁ ^ (n₂ * n₁ - 1) = 1
⊢ a₁ ^ n₂ = a₁
have inl : ∀ (hn₁ : a₁ ^ (n₂ * n₁ - 1) = 1), a₁ ^ n₂ = a₁
Tactic: rcases hn₁ with hn₁ | hn₁
Cleared hypotheses: hn₁
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ n₂ : ℤ
Hx : 1 < (a₁ ^ n₂).x
Hy : 0 < (a₁ ^ n₂).y
H : ∀ (a : Solution₁ d), ∃ (n : ℤ), a = (a₁ ^ n₂) ^ n ∨ a = -(a₁ ^ n₂) ^ n
hn₁ : a₁ ^ (n₂ * n₁ - 1) = 1 ∨ -a₁ ^ (n₂ * n₁ - 1) = 1
inl : ∀ (hn₁ : a₁ ^ (n₂ * n₁ - 1) = 1), a₁ ^ n₂ = a₁
⊢ a₁ ^ n₂ = a₁
have inl₁ : ∀ (hn₁ : -a₁ ^ (n₂ * n₁ - 1) = 1), a₁ ^ n₂ = a₁
Tactic: rcases hn₁ with hn₁ | hn₁
Cleared hypotheses: hn₁
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ n₂ : ℤ
Hx : 1 < (a₁ ^ n₂).x
Hy : 0 < (a₁ ^ n₂).y
H : ∀ (a : Solution₁ d), ∃ (n : ℤ), a = (a₁ ^ n₂) ^ n ∨ a = -(a₁ ^ n₂) ^ n
hn₁ : a₁ ^ (n₂ * n₁ - 1) = 1
⊢ a₁ ^ n₂ = a₁
have inl : ∀ (Hx : 1 < (HPow.hPow (β := ℤ) a₁ 1).x) (Hy : 0 < (HPow.hPow (β := ℤ) a₁ 1).y)
  (H : ∀ (a : Solution₁ d), ∃ (n : ℤ), a = HPow.hPow (β := ℤ) a₁ 1 ^ n ∨ a = -HPow.hPow (β := ℤ) a₁ 1 ^ n)
  (hn₁ : a₁ ^ (1 * n₁ - 1) = 1), HPow.hPow (β := ℤ) a₁ 1 = a₁
Tactic: rcases Int.isUnit_iff.mp
          (.of_mul_eq_one _ <|
            sub_eq_zero.mp <| (ha₁.zpow_eq_one_iff (n₂ * n₁ - 1)).mp hn₁) with
        (rfl | rfl)
Cleared hypotheses: n₂, Hx, Hy, H, hn₁
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ n₂ : ℤ
Hx : 1 < (a₁ ^ n₂).x
Hy : 0 < (a₁ ^ n₂).y
H : ∀ (a : Solution₁ d), ∃ (n : ℤ), a = (a₁ ^ n₂) ^ n ∨ a = -(a₁ ^ n₂) ^ n
hn₁ : a₁ ^ (n₂ * n₁ - 1) = 1
inl :
  ∀ (Hx : 1 < (HPow.hPow (β := ℤ) a₁ 1).x) (Hy : 0 < (HPow.hPow (β := ℤ) a₁ 1).y)
    (H : ∀ (a : Solution₁ d), ∃ (n : ℤ), a = HPow.hPow (β := ℤ) a₁ 1 ^ n ∨ a = -HPow.hPow (β := ℤ) a₁ 1 ^ n)
    (hn₁ : a₁ ^ (1 * n₁ - 1) = 1), HPow.hPow (β := ℤ) a₁ 1 = a₁
⊢ a₁ ^ n₂ = a₁
have inl₁ : ∀ (Hx : 1 < (HPow.hPow (β := ℤ) a₁ (-1)).x) (Hy : 0 < (HPow.hPow (β := ℤ) a₁ (-1)).y)
  (H : ∀ (a : Solution₁ d), ∃ (n : ℤ), a = HPow.hPow (β := ℤ) a₁ (-1) ^ n ∨ a = -HPow.hPow (β := ℤ) a₁ (-1) ^ n)
  (hn₁ : a₁ ^ (-1 * n₁ - 1) = 1), HPow.hPow (β := ℤ) a₁ (-1) = a₁
Tactic: rcases Int.isUnit_iff.mp
          (.of_mul_eq_one _ <|
            sub_eq_zero.mp <| (ha₁.zpow_eq_one_iff (n₂ * n₁ - 1)).mp hn₁) with
        (rfl | rfl)
Cleared hypotheses: n₂, Hx, Hy, H, hn₁
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ : ℤ
Hx : 1 < (HPow.hPow (β := ℤ) a₁ 1).x
Hy : 0 < (HPow.hPow (β := ℤ) a₁ 1).y
H : ∀ (a : Solution₁ d), ∃ (n : ℤ), a = HPow.hPow (β := ℤ) a₁ 1 ^ n ∨ a = -HPow.hPow (β := ℤ) a₁ 1 ^ n
hn₁ : a₁ ^ (1 * n₁ - 1) = 1
⊢ HPow.hPow (β := ℤ) a₁ 1 = a₁
have inl : a₁ = a₁
Tactic: rw [zpow_one]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ : ℤ
Hx : 1 < (HPow.hPow (β := ℤ) a₁ (-1)).x
Hy : 0 < (HPow.hPow (β := ℤ) a₁ (-1)).y
H : ∀ (a : Solution₁ d), ∃ (n : ℤ), a = HPow.hPow (β := ℤ) a₁ (-1) ^ n ∨ a = -HPow.hPow (β := ℤ) a₁ (-1) ^ n
hn₁ : a₁ ^ (-1 * n₁ - 1) = 1
⊢ HPow.hPow (β := ℤ) a₁ (-1) = a₁
have Hy₁ : a₁.y < 0
Tactic: rw [zpow_neg_one, y_inv, lt_neg, neg_zero] at Hy
Cleared hypotheses: Hy, H, hn₁
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ n₂ : ℤ
Hx : 1 < (a₁ ^ n₂).x
Hy : 0 < (a₁ ^ n₂).y
H : ∀ (a : Solution₁ d), ∃ (n : ℤ), a = (a₁ ^ n₂) ^ n ∨ a = -(a₁ ^ n₂) ^ n
hn₁ : -a₁ ^ (n₂ * n₁ - 1) = 1
⊢ a₁ ^ n₂ = a₁
have hn₂ : HPow.hPow (β := ℤ) a₁ 0 = -a₁ ^ (n₂ * n₁ - 1)
Tactic: rw [← zpow_zero a₁, eq_comm] at hn₁
Cleared hypotheses: hn₁
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ n₂ : ℤ
Hx : 1 < (-a₁ ^ n₂).x
Hy : 0 < (-a₁ ^ n₂).y
H : ∀ (a : Solution₁ d), ∃ (n : ℤ), a = (-a₁ ^ n₂) ^ n ∨ a = -(-a₁ ^ n₂) ^ n
hn₁ : a₁ = (-a₁ ^ n₂) ^ n₁ ∨ a₁ = -(-a₁ ^ n₂) ^ n₁
⊢ -a₁ ^ n₂ = a₁
have Hx₁ : (a₁ ^ n₂).x < -1
Tactic: rw [x_neg, lt_neg] at Hx
Cleared hypotheses: Hx, Hy, H, hn₁
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
n₁ n₂ : ℤ
Hx : (a₁ ^ n₂).x < -1
Hy : 0 < (-a₁ ^ n₂).y
H : ∀ (a : Solution₁ d), ∃ (n : ℤ), a = (-a₁ ^ n₂) ^ n ∨ a = -(-a₁ ^ n₂) ^ n
hn₁ : a₁ = (-a₁ ^ n₂) ^ n₁ ∨ a₁ = -(-a₁ ^ n₂) ^ n₁
⊢ -a₁ ^ n₂ = a₁
have this : LT.lt (α := ℤ) 0 (-1)
Tactic: have := (x_zpow_pos (zero_lt_one.trans ha₁.1) n₂).trans Hx
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
⊢ (1 < a.x ∧ 0 < a.y ∧ ∀ (b : Solution₁ d), ∃ (n : ℤ), b = a ^ n ∨ b = -a ^ n) ↔ IsFundamental a
have refine : ∀ (h : 1 < a.x ∧ 0 < a.y ∧ ∀ (b : Solution₁ d), ∃ (n : ℤ), b = a ^ n ∨ b = -a ^ n), IsFundamental a
Tactic: refine ⟨fun h => ?_, fun H => ⟨H.1, H.2.1, H.eq_zpow_or_neg_zpow⟩⟩
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
a : Solution₁ d
h : 1 < a.x ∧ 0 < a.y ∧ ∀ (b : Solution₁ d), ∃ (n : ℤ), b = a ^ n ∨ b = -a ^ n
⊢ IsFundamental a
have h₀ : 0 < d
Tactic: have h₀ := d_pos_of_one_lt_x h.1
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
a : Solution₁ d
h : 1 < a.x ∧ 0 < a.y ∧ ∀ (b : Solution₁ d), ∃ (n : ℤ), b = a ^ n ∨ b = -a ^ n
h₀ : 0 < d
⊢ IsFundamental a
have hd : ¬IsSquare d
Tactic: have hd := d_nonsquare_of_one_lt_x h.1
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
h : 1 < a.x ∧ 0 < a.y ∧ ∀ (b : Solution₁ d), ∃ (n : ℤ), b = a ^ n ∨ b = -a ^ n
h₀ : 0 < d
hd : ¬IsSquare d
⊢ IsFundamental a
have obtain : ∀ (a₁ : Solution₁ d) (ha₁ : IsFundamental a₁), IsFundamental a
Tactic: obtain ⟨a₁, ha₁⟩ := IsFundamental.exists_of_not_isSquare h₀ hd
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
h : 1 < a.x ∧ 0 < a.y ∧ ∀ (b : Solution₁ d), ∃ (n : ℤ), b = a ^ n ∨ b = -a ^ n
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
⊢ IsFundamental a
have obtain : ∀ (b : Solution₁ d)
  (hb₂ :
    ∀ (y : Solution₁ d),
      (fun (a₁ : Solution₁ d) ↦ 1 < a₁.x ∧ 0 < a₁.y ∧ ∀ (a : Solution₁ d), ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n) y →
        y = b),
  IsFundamental a
Tactic: obtain ⟨b, -, hb₂⟩ := existsUnique_pos_generator h₀ hd
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
h : 1 < a.x ∧ 0 < a.y ∧ ∀ (b : Solution₁ d), ∃ (n : ℤ), b = a ^ n ∨ b = -a ^ n
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
b : Solution₁ d
hb₂ :
  ∀ (y : Solution₁ d),
    (fun (a₁ : Solution₁ d) ↦ 1 < a₁.x ∧ 0 < a₁.y ∧ ∀ (a : Solution₁ d), ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n) y → y = b
⊢ IsFundamental a
have rwa : IsFundamental b
Tactic: rwa [hb₂ a h, ← hb₂ a₁ ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
d x y : ℤ
⊢ IsPell (d := d) { re := x, im := y } ↔
    HMul.hMul (α := ℤ√d) { re := x, im := y : ℤ√d } (star { re := x, im := y : ℤ√d }) = 1
have simp : x * x - y * (d * y) = 1 ↔ x * x + -(y * (d * y)) = 1
Tactic: simp [Zsqrtd.ext_iff, IsPell, mul_comm]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
d x y : ℤ
⊢ IsPell (d := d) { re := x, im := y } ↔ Membership.mem (γ := Submonoid (ℤ√d)) (unitary (ℤ√d)) { re := x, im := y }
have rw : IsPell (d := d) { re := x, im := y } ↔
  HMul.hMul (α := ℤ√d) (star { re := x, im := y : ℤ√d }) { re := x, im := y : ℤ√d } = 1 ∧
    HMul.hMul (α := ℤ√d) { re := x, im := y : ℤ√d } (star { re := x, im := y : ℤ√d }) = 1
Tactic: rw [Unitary.mem_iff, isPell_norm, mul_comm (star _), and_self_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
d x y : ℤ
rw :
  IsPell (d := d) { re := x, im := y } ↔
    HMul.hMul (α := ℤ√d) (star { re := x, im := y : ℤ√d }) { re := x, im := y : ℤ√d } = 1 ∧
      HMul.hMul (α := ℤ√d) { re := x, im := y : ℤ√d } (star { re := x, im := y : ℤ√d }) = 1
⊢ IsPell (d := d) { re := x, im := y } ↔ Membership.mem (γ := Submonoid (ℤ√d)) (unitary (ℤ√d)) { re := x, im := y }
have rw₁ : HMul.hMul (α := ℤ√d) { re := x, im := y : ℤ√d } (star { re := x, im := y : ℤ√d }) = 1 ↔
  HMul.hMul (α := ℤ√d) (star { re := x, im := y : ℤ√d }) { re := x, im := y : ℤ√d } = 1 ∧
    HMul.hMul (α := ℤ√d) { re := x, im := y : ℤ√d } (star { re := x, im := y : ℤ√d }) = 1
Tactic: rw [Unitary.mem_iff, isPell_norm, mul_comm (star _), and_self_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
d x y : ℤ
rw :
  IsPell (d := d) { re := x, im := y } ↔
    HMul.hMul (α := ℤ√d) (star { re := x, im := y : ℤ√d }) { re := x, im := y : ℤ√d } = 1 ∧
      HMul.hMul (α := ℤ√d) { re := x, im := y : ℤ√d } (star { re := x, im := y : ℤ√d }) = 1
rw₁ :
  HMul.hMul (α := ℤ√d) { re := x, im := y : ℤ√d } (star { re := x, im := y : ℤ√d }) = 1 ↔
    HMul.hMul (α := ℤ√d) (star { re := x, im := y : ℤ√d }) { re := x, im := y : ℤ√d } = 1 ∧
      HMul.hMul (α := ℤ√d) { re := x, im := y : ℤ√d } (star { re := x, im := y : ℤ√d }) = 1
⊢ IsPell (d := d) { re := x, im := y } ↔ Membership.mem (γ := Submonoid (ℤ√d)) (unitary (ℤ√d)) { re := x, im := y }
have rw₂ : HMul.hMul (α := ℤ√d) { re := x, im := y : ℤ√d } (star { re := x, im := y : ℤ√d }) = 1 ↔
  HMul.hMul (α := ℤ√d) { re := x, im := y : ℤ√d } (star { re := x, im := y : ℤ√d }) = 1 ∧
    HMul.hMul (α := ℤ√d) { re := x, im := y : ℤ√d } (star { re := x, im := y : ℤ√d }) = 1
Tactic: rw [Unitary.mem_iff, isPell_norm, mul_comm (star _), and_self_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
d x y : ℤ
rw :
  IsPell (d := d) { re := x, im := y } ↔
    HMul.hMul (α := ℤ√d) (star { re := x, im := y : ℤ√d }) { re := x, im := y : ℤ√d } = 1 ∧
      HMul.hMul (α := ℤ√d) { re := x, im := y : ℤ√d } (star { re := x, im := y : ℤ√d }) = 1
rw₁ :
  HMul.hMul (α := ℤ√d) { re := x, im := y : ℤ√d } (star { re := x, im := y : ℤ√d }) = 1 ↔
    HMul.hMul (α := ℤ√d) (star { re := x, im := y : ℤ√d }) { re := x, im := y : ℤ√d } = 1 ∧
      HMul.hMul (α := ℤ√d) { re := x, im := y : ℤ√d } (star { re := x, im := y : ℤ√d }) = 1
rw₂ :
  HMul.hMul (α := ℤ√d) { re := x, im := y : ℤ√d } (star { re := x, im := y : ℤ√d }) = 1 ↔
    HMul.hMul (α := ℤ√d) { re := x, im := y : ℤ√d } (star { re := x, im := y : ℤ√d }) = 1 ∧
      HMul.hMul (α := ℤ√d) { re := x, im := y : ℤ√d } (star { re := x, im := y : ℤ√d }) = 1
⊢ IsPell (d := d) { re := x, im := y } ↔ Membership.mem (γ := Submonoid (ℤ√d)) (unitary (ℤ√d)) { re := x, im := y }
have rw₃ : HMul.hMul (α := ℤ√d) { re := x, im := y : ℤ√d } (star { re := x, im := y : ℤ√d }) = 1 ↔
  HMul.hMul (α := ℤ√d) { re := x, im := y : ℤ√d } (star { re := x, im := y : ℤ√d }) = 1
Tactic: rw [Unitary.mem_iff, isPell_norm, mul_comm (star _), and_self_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
⊢ a ≤ a * a
have this : a * 1 ≤ a * a
Tactic: have := @Nat.mul_le_mul_left 1 a a (le_of_lt a1)
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
this : a * 1 ≤ a * a
⊢ a ≤ a * a
have rwa : ∀ (this : a ≤ a * a), a ≤ a * a
Tactic: rwa [mul_one] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
this : 1 ≤ a * a
⊢ (↑(d a1) : ℤ) = az a * az a - 1
have rw : HSub.hSub (α := ℤ) (↑(a * a) : ℤ) (↑1 : ℤ) = az a * az a - 1
Tactic: rw [Pell.d, Int.ofNat_sub this]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
x y : ℕ
h : IsPell (d := (↑(d a1) : ℤ)) { re := (↑x : ℤ), im := (↑y : ℤ) }
⊢ Eq (α := ℤ) (↑(x * x - d a1 * y * y) : ℤ) (↑1 : ℤ)
have rw : HSub.hSub (α := ℤ) (↑(x * x) : ℤ) (↑(d a1 * y * y) : ℤ) = (↑1 : ℤ)
Tactic: rw [Int.ofNat_sub (Int.le_of_ofNat_le_ofNat <| Int.le.intro_sub _ h)]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
x y : ℕ
h : x * x - d a1 * y * y = 1
⊢ HSub.hSub (α := ℤ) (↑(x * x) : ℤ) (↑(d a1 * y * y) : ℤ) = 1
have rw : Eq (α := ℤ) (↑1 : ℤ) 1
Tactic: rw [← Int.ofNat_sub <| le_of_lt <| Nat.lt_of_sub_eq_succ h, h]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
⊢ pellZd a1 (n + 1) = pellZd a1 n * { re := (↑a : ℤ), im := 1 }
have re : (pellZd a1 (n + 1)).re = (pellZd a1 n * { re := (↑a : ℤ), im := 1 : ℤ√(↑(d a1) : ℤ) }).re
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
re : (pellZd a1 (n + 1)).re = (pellZd a1 n * { re := (↑a : ℤ), im := 1 : ℤ√(↑(d a1) : ℤ) }).re
⊢ pellZd a1 (n + 1) = pellZd a1 n * { re := (↑a : ℤ), im := 1 }
have im : (pellZd a1 (n + 1)).im = (pellZd a1 n * { re := (↑a : ℤ), im := 1 : ℤ√(↑(d a1) : ℤ) }).im
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
pn : xz a1 n * xz a1 n - (↑(d a1) : ℤ) * yz a1 n * yz a1 n = 1 := ⋯
⊢ HSub.hSub (α := ℤ) (↑(xn a1 n * xn a1 n) : ℤ) (↑(d a1 * yn a1 n * yn a1 n) : ℤ) = 1
have repeat' : HMul.hMul (α := ℤ) (↑(xn a1 n) : ℤ) (↑(xn a1 n) : ℤ) - (↑(d a1 * yn a1 n * yn a1 n) : ℤ) = 1
Tactic: repeat' rw [Int.natCast_mul]; exact pn
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
pn : xz a1 n * xz a1 n - (↑(d a1) : ℤ) * yz a1 n * yz a1 n = 1 := ⋯
h : HSub.hSub (α := ℤ) (↑(xn a1 n * xn a1 n) : ℤ) (↑(d a1 * yn a1 n * yn a1 n) : ℤ) = 1
hl : d a1 * yn a1 n * yn a1 n ≤ xn a1 n * xn a1 n
⊢ Eq (α := ℤ) (↑(xn a1 n * xn a1 n - d a1 * yn a1 n * yn a1 n) : ℤ) (↑1 : ℤ)
have rw : HSub.hSub (α := ℤ) (↑(xn a1 n * xn a1 n) : ℤ) (↑(d a1 * yn a1 n * yn a1 n) : ℤ) = (↑1 : ℤ)
Tactic: rw [Int.ofNat_sub hl]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
h : d a1 = n * n
⊢ n * n + 1 = a * a
have rw : d a1 + 1 = a * a
Tactic: rw [← h]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
h : d a1 = n * n
this : n * n + 1 = a * a
⊢ n * n < a * a
have rw : n * n < n * n + 1
Tactic: rw [← this]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
h : d a1 = n * n
this : n * n + 1 = a * a
na : n < a
⊢ (n + 1) * (n + 1) ≤ n * n + 1
have rw : (n + 1) * (n + 1) ≤ a * a
Tactic: rw [this]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
h : d a1 = n * n
this_1 : 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)
have ring_nf : ∀ (this : 1 + n * 2 + HPow.hPow (β := ℕ) n 2 ≤ 1 + HPow.hPow (β := ℕ) n 2),
  1 + n * 2 + HPow.hPow (β := ℕ) n 2 ≤ 1 + HPow.hPow (β := ℕ) n 2
Tactic: ring_nf at this ⊢
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
h : d a1 = n * n
this_1 : n * n + 1 = a * a
na : n < a
this_2 : (n + 1) * (n + 1) ≤ n * n + 1
this : n + n ≤ 0
⊢ d a1 = 0
have rwa : ∀ (h : d a1 = HMul.hMul (α := ℕ) 0 0) (this : n * n + 1 = a * a) (na : n < a) (this : (n + 1) * (n + 1) ≤ n * n + 1)
  (this : n + n ≤ 0), d a1 = 0
Tactic: rwa [Nat.eq_zero_of_le_zero ((Nat.le_add_left _ _).trans this)] at h
Cleared hypotheses: h, this_1, na, this_2, this
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
⊢ a ^ (n + 1) ≤ xn a1 (n + 1)
have simp : a ^ n * a ≤ xn a1 n * a + d a1 * yn a1 n
Tactic: simp only [_root_.pow_succ, xn_succ]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
b : ℤ√(↑(d a1) : ℤ)
h1 : 1 ≤ b
hp : IsPell b
h : b ≤ pellZd a1 (n + 1)
a1p : LE.le (α := ℤ√(↑(d a1) : ℤ)) 0 { re := (↑a : ℤ), im := 1 }
am1p : LE.le (α := ℤ√(↑(d a1) : ℤ)) 0 { re := (↑a : ℤ), im := -1 }
a1m :
  HMul.hMul (α := ℤ√(↑(d a1) : ℤ)) { re := (↑a : ℤ), im := 1 : ℤ√(↑(d a1) : ℤ) }
      { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } =
    1
ha : { re := (↑a : ℤ), im := 1 } ≤ b
⊢ 1 ≤ b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) }
have rw : HMul.hMul (α := ℤ√(↑(d a1) : ℤ)) { re := (↑a : ℤ), im := 1 : ℤ√(↑(d a1) : ℤ) }
    { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤
  b * { re := (↑a : ℤ), im := -1 }
Tactic: rw [← a1m]
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (hypothesis)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
b : ℤ√(↑(d a1) : ℤ)
h1 : 1 ≤ b
hp : IsPell b
h : b ≤ pellZd a1 (n + 1)
a1p : LE.le (α := ℤ√(↑(d a1) : ℤ)) 0 { re := (↑a : ℤ), im := 1 }
am1p : LE.le (α := ℤ√(↑(d a1) : ℤ)) 0 { re := (↑a : ℤ), im := -1 }
a1m :
  HMul.hMul (α := ℤ√(↑(d a1) : ℤ)) { re := (↑a : ℤ), im := 1 : ℤ√(↑(d a1) : ℤ) }
      { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } =
    1
ha : { re := (↑a : ℤ), im := 1 } ≤ b
⊢ b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤ pellZd a1 n
have t : b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤ pellZd a1 (n + 1) * { re := (↑a : ℤ), im := -1 }
Tactic: have t := mul_le_mul_of_nonneg_right h am1p
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
b : ℤ√(↑(d a1) : ℤ)
h1 : 1 ≤ b
hp : IsPell b
h : b ≤ pellZd a1 (n + 1)
a1p : LE.le (α := ℤ√(↑(d a1) : ℤ)) 0 { re := (↑a : ℤ), im := 1 }
am1p : LE.le (α := ℤ√(↑(d a1) : ℤ)) 0 { re := (↑a : ℤ), im := -1 }
a1m :
  HMul.hMul (α := ℤ√(↑(d a1) : ℤ)) { re := (↑a : ℤ), im := 1 : ℤ√(↑(d a1) : ℤ) }
      { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } =
    1
ha : { re := (↑a : ℤ), im := 1 } ≤ b
t : b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤ pellZd a1 (n + 1) * { re := (↑a : ℤ), im := -1 }
⊢ b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤ pellZd a1 n
have rwa : ∀
  (t :
    b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤
      pellZd a1 n * { re := (↑a : ℤ), im := 1 : ℤ√(↑(d a1) : ℤ) } * { re := (↑a : ℤ), im := -1 }),
  b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤ pellZd a1 n
Tactic: rwa [pellZd_succ, mul_assoc, a1m, mul_one] at t
Cleared hypotheses: t
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
b : ℤ√(↑(d a1) : ℤ)
h1 : 1 ≤ b
hp : IsPell b
h : b ≤ pellZd a1 (n + 1)
a1p : LE.le (α := ℤ√(↑(d a1) : ℤ)) 0 { re := (↑a : ℤ), im := 1 }
am1p : LE.le (α := ℤ√(↑(d a1) : ℤ)) 0 { re := (↑a : ℤ), im := -1 }
a1m :
  HMul.hMul (α := ℤ√(↑(d a1) : ℤ)) { re := (↑a : ℤ), im := 1 : ℤ√(↑(d a1) : ℤ) }
      { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } =
    1
ha : { re := (↑a : ℤ), im := 1 } ≤ b
t : b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤ pellZd a1 (n + 1) * { re := (↑a : ℤ), im := -1 }
rwa :
  ∀
    (t :
      b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤
        pellZd a1 n * { re := (↑a : ℤ), im := 1 : ℤ√(↑(d a1) : ℤ) } * { re := (↑a : ℤ), im := -1 }),
    b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤ pellZd a1 n
⊢ b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤ pellZd a1 n
have rwa₁ : ∀
  (t :
    b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤
      pellZd a1 n *
        HMul.hMul (α := ℤ√(↑(d a1) : ℤ)) { re := (↑a : ℤ), im := 1 : ℤ√(↑(d a1) : ℤ) }
          { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) }),
  b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤ pellZd a1 n
Tactic: rwa [pellZd_succ, mul_assoc, a1m, mul_one] at t
Cleared hypotheses: t
-- Lean 4 proof step from Mathlib.NumberTheory.PellMatiyasevic (goal)
Goal:
a : ℕ
a1 : 1 < a
n : ℕ
b : ℤ√(↑(d a1) : ℤ)
h1 : 1 ≤ b
hp : IsPell b
h : b ≤ pellZd a1 (n + 1)
a1p : LE.le (α := ℤ√(↑(d a1) : ℤ)) 0 { re := (↑a : ℤ), im := 1 }
am1p : LE.le (α := ℤ√(↑(d a1) : ℤ)) 0 { re := (↑a : ℤ), im := -1 }
a1m :
  HMul.hMul (α := ℤ√(↑(d a1) : ℤ)) { re := (↑a : ℤ), im := 1 : ℤ√(↑(d a1) : ℤ) }
      { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } =
    1
ha : { re := (↑a : ℤ), im := 1 } ≤ b
t : b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤ pellZd a1 (n + 1) * { re := (↑a : ℤ), im := -1 }
rwa :
  ∀
    (t :
      b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤
        pellZd a1 n * { re := (↑a : ℤ), im := 1 : ℤ√(↑(d a1) : ℤ) } * { re := (↑a : ℤ), im := -1 }),
    b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤ pellZd a1 n
rwa₁ :
  ∀
    (t :
      b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤
        pellZd a1 n *
          HMul.hMul (α := ℤ√(↑(d a1) : ℤ)) { re := (↑a : ℤ), im := 1 : ℤ√(↑(d a1) : ℤ) }
            { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) }),
    b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤ pellZd a1 n
⊢ b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤ pellZd a1 n
have rwa₂ : ∀ (t : b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤ pellZd a1 n * 1),
  b * { re := (↑a : ℤ), im := -1 : ℤ√(↑(d a1) : ℤ) } ≤ pellZd a1 n
Tactic: rwa [pellZd_succ, mul_assoc, a1m, mul_one] at t
Cleared hypotheses: t