awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
calc : a.x * (d * (a.y * a₁.y)) = d * a.y * (a.x * a₁.y)
calc₁ : d * a.y * (a.x * a₁.y) ≤ d * a.y * (a.y * a₁.x)
⊢ a.x * (d * (a.y * a₁.y)) < a.x * (a.x * a₁.x)
have calc₂ : d * a.y * (a.y * a₁.x) = (HPow.hPow (β := ℕ) a.x 2 - 1) * a₁.x
Tactic: calc a.x * (d * (a.y * a₁.y))
    _ = d * a.y * (a.x * a₁.y) := by ring
    _ ≤ d * a.y * (a.y * a₁.x) := by have := x_mul_y_le_y_mul_x h hax hay; have := h.d_pos; gcongr
    _ = (a.x ^ 2 - 1) * a₁.x := by rw [← a.prop_y]; ring
    _ < a.x * (a.x * a₁.x) := by linarith [h.1]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
calc : a.x * (d * (a.y * a₁.y)) = d * a.y * (a.x * a₁.y)
calc₁ : d * a.y * (a.x * a₁.y) ≤ d * a.y * (a.y * a₁.x)
calc₂ : d * a.y * (a.y * a₁.x) = (HPow.hPow (β := ℕ) a.x 2 - 1) * a₁.x
⊢ a.x * (d * (a.y * a₁.y)) < a.x * (a.x * a₁.x)
have calc₃ : (HPow.hPow (β := ℕ) a.x 2 - 1) * a₁.x < a.x * (a.x * a₁.x)
Tactic: calc a.x * (d * (a.y * a₁.y))
    _ = d * a.y * (a.x * a₁.y) := by ring
    _ ≤ d * a.y * (a.y * a₁.x) := by have := x_mul_y_le_y_mul_x h hax hay; have := h.d_pos; gcongr
    _ = (a.x ^ 2 - 1) * a₁.x := by rw [← a.prop_y]; ring
    _ < a.x * (a.x * a₁.x) := by linarith [h.1]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
⊢ (a * a₁⁻¹).x < a.x
have simp : a.x * a₁.x < d * (a.y * a₁.y) + a.x
Tactic: simp only [x_mul, x_inv, y_inv, mul_neg, add_neg_lt_iff_le_add']
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
⊢ a.x * a₁.x < d * (a.y * a₁.y) + a.x
have refine : a₁.y * (a.x * a₁.x) < a₁.y * (d * (a.y * a₁.y) + a.x)
Tactic: refine lt_of_mul_lt_mul_left ?_ h.2.1.le
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
⊢ a.x * a₁.y * a₁.x ≤ a.y * a₁.x * a₁.x
have this : 1 < a₁.x
Tactic: have := h.1
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
this : 1 < a₁.x
⊢ a.x * a₁.y * a₁.x ≤ a.y * a₁.x * a₁.x
have this₁ : a.x * a₁.y ≤ a.y * a₁.x
Tactic: have := x_mul_y_le_y_mul_x h hax hay
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
this_1 : 1 < a₁.x
this : a.x * a₁.y ≤ a.y * a₁.x
⊢ a.x * a₁.y * a₁.x ≤ a.y * a₁.x * a₁.x
have ha : 0 ≤ a₁.x
Tactic: gcongr
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
⊢ a₁.y * (a.x * a₁.x) < a₁.y * (d * (a.y * a₁.y) + a.x)
have calc : a.y * a₁.x * a₁.x < a₁.y * (d * (a.y * a₁.y) + a.x)
Tactic: calc a₁.y * (a.x * a₁.x)
    _ = a.x * a₁.y * a₁.x := by ring
    _ ≤ a.y * a₁.x * a₁.x := by have := h.1; have := x_mul_y_le_y_mul_x h hax hay; gcongr
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
⊢ a.y * a₁.x * a₁.x < a₁.y * (d * (a.y * a₁.y) + a.x)
have calc : a.y * (1 + d * HPow.hPow (β := ℕ) a₁.y 2) - a₁.y * (d * (a.y * a₁.y) + a.x) < 0
Tactic: rw [mul_assoc, ← sq, a₁.prop_x, ← sub_neg]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
this : a.y - a.x * a₁.y < 0
⊢ a.y * (1 + d * HPow.hPow (β := ℕ) a₁.y 2) - a₁.y * (d * (a.y * a₁.y) + a.x) < 0
have h₁ : a.y * (1 + d * HPow.hPow (β := ℕ) a₁.y 2) - a₁.y * (d * (a.y * a₁.y) + a.x) = a.y - a.x * a₁.y
Tactic: convert! this using 1
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
⊢ a.y * (1 + d * HPow.hPow (β := ℕ) a₁.y 2) - a₁.y * (d * (a.y * a₁.y) + a.x) < 0
have calc : a.y - a.x * a₁.y < 0
Tactic: suffices a.y - a.x * a₁.y < 0 by convert! this using 1; ring
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
⊢ a.y - a.x * a₁.y < 0
have calc : HPow.hPow (β := ℕ) a.y 2 < (1 + d * HPow.hPow (β := ℕ) a.y 2) * HPow.hPow (β := ℕ) a₁.y 2
Tactic: rw [sub_neg, ← abs_of_pos hay, ← abs_of_pos h.2.1, ← abs_of_pos <| zero_lt_one.trans hax, ←
    abs_mul, ← sq_lt_sq, mul_pow, a.prop_x]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
⊢ d * HPow.hPow (β := ℕ) a.y 2 + 1 = (1 + d * HPow.hPow (β := ℕ) a.y 2) * 1
have rw : 1 + d * HPow.hPow (β := ℕ) a.y 2 = (1 + d * HPow.hPow (β := ℕ) a.y 2) * 1
Tactic: rw [add_comm, mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
rw : 1 + d * HPow.hPow (β := ℕ) a.y 2 = (1 + d * HPow.hPow (β := ℕ) a.y 2) * 1
⊢ d * HPow.hPow (β := ℕ) a.y 2 + 1 = (1 + d * HPow.hPow (β := ℕ) a.y 2) * 1
have rw₁ : 1 + d * HPow.hPow (β := ℕ) a.y 2 = 1 + d * HPow.hPow (β := ℕ) a.y 2
Tactic: rw [add_comm, mul_one]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
⊢ 0 < 1 + d * HPow.hPow (β := ℕ) a.y 2
have this : 0 < d
Tactic: have := h.d_pos
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
⊢ HPow.hPow (β := ℕ) a.y 2 < (1 + d * HPow.hPow (β := ℕ) a.y 2) * HPow.hPow (β := ℕ) a₁.y 2
have _ : d * HPow.hPow (β := ℕ) a.y 2 + 1 = (1 + d * HPow.hPow (β := ℕ) a.y 2) * 1
Tactic: calc
    a.y ^ 2 = 1 * a.y ^ 2 := (one_mul _).symm
    _ ≤ d * a.y ^ 2 := (mul_le_mul_iff_left₀ <| sq_pos_of_pos hay).mpr h.d_pos
    _ < d * a.y ^ 2 + 1 := lt_add_one _
    _ = (1 + d * a.y ^ 2) * 1 := by rw [add_comm, mul_one]
    _ ≤ (1 + d * a.y ^ 2) * a₁.y ^ 2 :=
      (mul_le_mul_iff_right₀ (by have := h.d_pos; positivity)).mpr (sq_pos_of_pos h.2.1)
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 0 < a.x
hay : 0 ≤ a.y
⊢ ∃ (n : ℕ), a = a₁ ^ n
have lift : ∀ (ax : ℕ) (hax' : (↑ax : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑ax : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
Tactic: lift a.x to ℕ using hax.le with ax hax'
Cleared hypotheses: hax
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih :
  ∀ m < x,
    ∀ {a : Solution₁ d} (hay : 0 ≤ a.y) (hax' : (↑m : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑m : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : (↑x : ℤ) = a.x
hax : LT.lt (α := ℤ) 0 (↑x : ℤ)
⊢ ∃ (n : ℕ), a = a₁ ^ n
have h₁ : ∀ (hy : 0 = a.y), ∃ (n : ℕ), a = a₁ ^ n
Tactic: rcases hay.eq_or_lt with hy | hy
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih :
  ∀ m < x,
    ∀ {a : Solution₁ d} (hay : 0 ≤ a.y) (hax' : (↑m : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑m : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : (↑x : ℤ) = a.x
hax : LT.lt (α := ℤ) 0 (↑x : ℤ)
h₁ : ∀ (hy : 0 = a.y), ∃ (n : ℕ), a = a₁ ^ n
⊢ ∃ (n : ℕ), a = a₁ ^ n
have h₂ : ∀ (hy : 0 < a.y), ∃ (n : ℕ), a = a₁ ^ n
Tactic: rcases hay.eq_or_lt with hy | hy
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih :
  ∀ m < x,
    ∀ {a : Solution₁ d} (hay : 0 ≤ a.y) (hax' : (↑m : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑m : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : (↑x : ℤ) = a.x
hax : LT.lt (α := ℤ) 0 (↑x : ℤ)
hy : 0 = a.y
⊢ ∃ (n : ℕ), a = a₁ ^ n
have h₁ : a = HPow.hPow (β := ℕ) a₁ 0
Tactic: refine ⟨0, ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih :
  ∀ m < x,
    ∀ {a : Solution₁ d} (hay : 0 ≤ a.y) (hax' : (↑m : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑m : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : (↑x : ℤ) = a.x
hax : LT.lt (α := ℤ) 0 (↑x : ℤ)
hy : 0 = a.y
⊢ a = HPow.hPow (β := ℕ) a₁ 0
have h₁ : ∀ (hay : 0 ≤ Solution₁.y (d := d) 1) (hax' : (↑x : ℤ) = Solution₁.x (d := d) 1) (hy : 0 = Solution₁.y (d := d) 1),
  1 = HPow.hPow (β := ℕ) a₁ 0
Tactic: rcases eq_one_or_neg_one_iff_y_eq_zero.2 hy.symm with rfl | rfl
Cleared hypotheses: a, hay, hax', hy
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih :
  ∀ m < x,
    ∀ {a : Solution₁ d} (hay : 0 ≤ a.y) (hax' : (↑m : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑m : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : (↑x : ℤ) = a.x
hax : LT.lt (α := ℤ) 0 (↑x : ℤ)
hy : 0 = a.y
h₁ :
  ∀ (hay : 0 ≤ Solution₁.y (d := d) 1) (hax' : (↑x : ℤ) = Solution₁.x (d := d) 1) (hy : 0 = Solution₁.y (d := d) 1),
    1 = HPow.hPow (β := ℕ) a₁ 0
⊢ a = HPow.hPow (β := ℕ) a₁ 0
have h₂ : ∀ (hay : 0 ≤ Solution₁.y (d := d) (-1)) (hax' : (↑x : ℤ) = Solution₁.x (d := d) (-1))
  (hy : 0 = Solution₁.y (d := d) (-1)), -1 = HPow.hPow (β := ℕ) a₁ 0
Tactic: rcases eq_one_or_neg_one_iff_y_eq_zero.2 hy.symm with rfl | rfl
Cleared hypotheses: a, hay, hax', hy
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih :
  ∀ m < x,
    ∀ {a : Solution₁ d} (hay : 0 ≤ a.y) (hax' : (↑m : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑m : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : (↑x : ℤ) = a.x
hax : LT.lt (α := ℤ) 0 (↑x : ℤ)
hy : 0 < a.y
⊢ 1 < a.x
have nlinarith : a.x ≤ 1 →
  -1 + ((↑x : ℤ) - a.x) + (HAdd.hAdd (α := ℤ) 0 1 - a.y) + (a.x - 1) +
                -(HPow.hPow (β := ℕ) a.x 2 - d * HPow.hPow (β := ℕ) a.y 2 - 1) +
              (0 - (0 - HPow.hPow (β := ℕ) a.y 2) * (HAdd.hAdd (α := ℤ) 0 1 - d)) +
            (HSub.hSub (α := ℤ) 0 (↑x : ℤ) * ((↑x : ℤ) - a.x) - 0) +
          (0 - HSub.hSub (α := ℤ) 0 (↑x : ℤ) * (a.x - 1)) +
        (0 - (0 - a.y) * (HAdd.hAdd (α := ℤ) 0 1 - a.y)) +
      (((↑x : ℤ) - a.x) * ((↑x : ℤ) - a.x) - 0) =
    0
Tactic: nlinarith [a.prop, h.d_pos]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih :
  ∀ m < x,
    ∀ {a : Solution₁ d} (hay : 0 ≤ a.y) (hax' : (↑m : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑m : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : (↑x : ℤ) = a.x
hax : LT.lt (α := ℤ) 0 (↑x : ℤ)
hy : 0 < a.y
⊢ ∃ (n : ℕ), a = a₁ ^ n
have hx₁ : 1 < a.x
Tactic: have hx₁ : 1 < a.x := by nlinarith [a.prop, h.d_pos]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih :
  ∀ m < x,
    ∀ {a : Solution₁ d} (hay : 0 ≤ a.y) (hax' : (↑m : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑m : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : (↑x : ℤ) = a.x
hax : LT.lt (α := ℤ) 0 (↑x : ℤ)
hy : 0 < a.y
hx₁ : 1 < a.x
⊢ ∃ (n : ℕ), a = a₁ ^ n
have hxx₁ : 0 < (a * a₁⁻¹).x
Tactic: have hxx₁ := h.mul_inv_x_pos hx₁ hy
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih :
  ∀ m < x,
    ∀ {a : Solution₁ d} (hay : 0 ≤ a.y) (hax' : (↑m : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑m : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : (↑x : ℤ) = a.x
hax : LT.lt (α := ℤ) 0 (↑x : ℤ)
hy : 0 < a.y
hx₁ : 1 < a.x
hxx₁ : 0 < (a * a₁⁻¹).x
⊢ ∃ (n : ℕ), a = a₁ ^ n
have hxx₂ : (a * a₁⁻¹).x < a.x
Tactic: have hxx₂ := h.mul_inv_x_lt_x hx₁ hy
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih :
  ∀ m < x,
    ∀ {a : Solution₁ d} (hay : 0 ≤ a.y) (hax' : (↑m : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑m : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : (↑x : ℤ) = a.x
hax : LT.lt (α := ℤ) 0 (↑x : ℤ)
hy : 0 < a.y
hx₁ : 1 < a.x
hxx₁ : 0 < (a * a₁⁻¹).x
hxx₂ : (a * a₁⁻¹).x < a.x
⊢ ∃ (n : ℕ), a = a₁ ^ n
have hyy : 0 ≤ (a * a₁⁻¹).y
Tactic: have hyy := h.mul_inv_y_nonneg hx₁ hy
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih :
  ∀ m < x,
    ∀ {a : Solution₁ d} (hay : 0 ≤ a.y) (hax' : (↑m : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑m : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : (↑x : ℤ) = a.x
hax : LT.lt (α := ℤ) 0 (↑x : ℤ)
hy : 0 < a.y
hx₁ : 1 < a.x
hxx₁ : 0 < (a * a₁⁻¹).x
hxx₂ : (a * a₁⁻¹).x < a.x
hyy : 0 ≤ (a * a₁⁻¹).y
⊢ ∃ (n : ℕ), a = a₁ ^ n
have h₁ : ∀ (x' : ℕ) (hx' : (↑x' : ℤ) = (a * a₁⁻¹).x) (hxx₁ : LT.lt (α := ℤ) 0 (↑x' : ℤ)) (hxx₂ : (↑x' : ℤ) < a.x),
  ∃ (n : ℕ), a = a₁ ^ n
Tactic: lift (a * a₁⁻¹).x to ℕ using hxx₁.le with x' hx'
Cleared hypotheses: hxx₁, hxx₂
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih :
  ∀ m < x,
    ∀ {a : Solution₁ d} (hay : 0 ≤ a.y) (hax' : (↑m : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑m : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : (↑x : ℤ) = a.x
hax : LT.lt (α := ℤ) 0 (↑x : ℤ)
hy : 0 < a.y
hx₁ : 1 < a.x
hyy : 0 ≤ (a * a₁⁻¹).y
x' : ℕ
hx' : (↑x' : ℤ) = (a * a₁⁻¹).x
hxx₁ : LT.lt (α := ℤ) 0 (↑x' : ℤ)
hxx₂ : (↑x' : ℤ) < a.x
⊢ ∃ (n : ℕ), a = a₁ ^ n
have h₁ : ∀ (n : ℕ) (hn : a * a₁⁻¹ = a₁ ^ n), ∃ (n : ℕ), a = a₁ ^ n
Tactic: obtain ⟨n, hn⟩ := ih x' (mod_cast hxx₂.trans_eq hax'.symm) hyy hx' hxx₁
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih :
  ∀ m < x,
    ∀ {a : Solution₁ d} (hay : 0 ≤ a.y) (hax' : (↑m : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑m : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : (↑x : ℤ) = a.x
hax : LT.lt (α := ℤ) 0 (↑x : ℤ)
hy : 0 < a.y
hx₁ : 1 < a.x
hyy : 0 ≤ (a * a₁⁻¹).y
x' : ℕ
hx' : (↑x' : ℤ) = (a * a₁⁻¹).x
hxx₁ : LT.lt (α := ℤ) 0 (↑x' : ℤ)
hxx₂ : (↑x' : ℤ) < a.x
n : ℕ
hn : a * a₁⁻¹ = a₁ ^ n
⊢ a = a₁ ^ (n + 1)
have rw : a = a₁ * a₁ ^ n
Tactic: rw [pow_succ', ← hn, mul_comm a, ← mul_assoc, mul_inv_cancel, one_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih :
  ∀ m < x,
    ∀ {a : Solution₁ d} (hay : 0 ≤ a.y) (hax' : (↑m : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑m : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : (↑x : ℤ) = a.x
hax : LT.lt (α := ℤ) 0 (↑x : ℤ)
hy : 0 < a.y
hx₁ : 1 < a.x
hyy : 0 ≤ (a * a₁⁻¹).y
x' : ℕ
hx' : (↑x' : ℤ) = (a * a₁⁻¹).x
hxx₁ : LT.lt (α := ℤ) 0 (↑x' : ℤ)
hxx₂ : (↑x' : ℤ) < a.x
n : ℕ
hn : a * a₁⁻¹ = a₁ ^ n
rw : a = a₁ * a₁ ^ n
⊢ a = a₁ ^ (n + 1)
have rw₁ : a = a₁ * (a * a₁⁻¹)
Tactic: rw [pow_succ', ← hn, mul_comm a, ← mul_assoc, mul_inv_cancel, one_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih :
  ∀ m < x,
    ∀ {a : Solution₁ d} (hay : 0 ≤ a.y) (hax' : (↑m : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑m : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : (↑x : ℤ) = a.x
hax : LT.lt (α := ℤ) 0 (↑x : ℤ)
hy : 0 < a.y
hx₁ : 1 < a.x
hyy : 0 ≤ (a * a₁⁻¹).y
x' : ℕ
hx' : (↑x' : ℤ) = (a * a₁⁻¹).x
hxx₁ : LT.lt (α := ℤ) 0 (↑x' : ℤ)
hxx₂ : (↑x' : ℤ) < a.x
n : ℕ
hn : a * a₁⁻¹ = a₁ ^ n
rw : a = a₁ * a₁ ^ n
rw₁ : a = a₁ * (a * a₁⁻¹)
⊢ a = a₁ ^ (n + 1)
have rw₂ : a = a₁ * (a₁⁻¹ * a)
Tactic: rw [pow_succ', ← hn, mul_comm a, ← mul_assoc, mul_inv_cancel, one_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih :
  ∀ m < x,
    ∀ {a : Solution₁ d} (hay : 0 ≤ a.y) (hax' : (↑m : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑m : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : (↑x : ℤ) = a.x
hax : LT.lt (α := ℤ) 0 (↑x : ℤ)
hy : 0 < a.y
hx₁ : 1 < a.x
hyy : 0 ≤ (a * a₁⁻¹).y
x' : ℕ
hx' : (↑x' : ℤ) = (a * a₁⁻¹).x
hxx₁ : LT.lt (α := ℤ) 0 (↑x' : ℤ)
hxx₂ : (↑x' : ℤ) < a.x
n : ℕ
hn : a * a₁⁻¹ = a₁ ^ n
rw : a = a₁ * a₁ ^ n
rw₁ : a = a₁ * (a * a₁⁻¹)
rw₂ : a = a₁ * (a₁⁻¹ * a)
⊢ a = a₁ ^ (n + 1)
have rw₃ : a = a₁ * a₁⁻¹ * a
Tactic: rw [pow_succ', ← hn, mul_comm a, ← mul_assoc, mul_inv_cancel, one_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih :
  ∀ m < x,
    ∀ {a : Solution₁ d} (hay : 0 ≤ a.y) (hax' : (↑m : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑m : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : (↑x : ℤ) = a.x
hax : LT.lt (α := ℤ) 0 (↑x : ℤ)
hy : 0 < a.y
hx₁ : 1 < a.x
hyy : 0 ≤ (a * a₁⁻¹).y
x' : ℕ
hx' : (↑x' : ℤ) = (a * a₁⁻¹).x
hxx₁ : LT.lt (α := ℤ) 0 (↑x' : ℤ)
hxx₂ : (↑x' : ℤ) < a.x
n : ℕ
hn : a * a₁⁻¹ = a₁ ^ n
rw : a = a₁ * a₁ ^ n
rw₁ : a = a₁ * (a * a₁⁻¹)
rw₂ : a = a₁ * (a₁⁻¹ * a)
rw₃ : a = a₁ * a₁⁻¹ * a
⊢ a = a₁ ^ (n + 1)
have rw₄ : a = 1 * a
Tactic: rw [pow_succ', ← hn, mul_comm a, ← mul_assoc, mul_inv_cancel, one_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih :
  ∀ m < x,
    ∀ {a : Solution₁ d} (hay : 0 ≤ a.y) (hax' : (↑m : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑m : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : (↑x : ℤ) = a.x
hax : LT.lt (α := ℤ) 0 (↑x : ℤ)
hy : 0 < a.y
hx₁ : 1 < a.x
hyy : 0 ≤ (a * a₁⁻¹).y
x' : ℕ
hx' : (↑x' : ℤ) = (a * a₁⁻¹).x
hxx₁ : LT.lt (α := ℤ) 0 (↑x' : ℤ)
hxx₂ : (↑x' : ℤ) < a.x
n : ℕ
hn : a * a₁⁻¹ = a₁ ^ n
rw : a = a₁ * a₁ ^ n
rw₁ : a = a₁ * (a * a₁⁻¹)
rw₂ : a = a₁ * (a₁⁻¹ * a)
rw₃ : a = a₁ * a₁⁻¹ * a
rw₄ : a = 1 * a
⊢ a = a₁ ^ (n + 1)
have rw₅ : a = a
Tactic: rw [pow_succ', ← hn, mul_comm a, ← mul_assoc, mul_inv_cancel, one_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hay : 0 ≤ a.y
ax : ℕ
hax' : (↑ax : ℤ) = a.x
hax : LT.lt (α := ℤ) 0 (↑ax : ℤ)
⊢ ∃ (n : ℕ), a = a₁ ^ n
have h₁ : ∀ (x : ℕ)
  (ih :
    ∀ m < x,
      ∀ {a : Solution₁ d} (hay : 0 ≤ a.y) (hax' : (↑m : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑m : ℤ)),
        ∃ (n : ℕ), a = a₁ ^ n)
  {a : Solution₁ d} (hay : 0 ≤ a.y) (hax' : (↑x : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑x : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
Tactic: induction ax using Nat.strong_induction_on generalizing a with | h x ih =>
  rcases hay.eq_or_lt with hy | hy
  · -- case 1: `a = 1`
    refine ⟨0, ?_⟩
    rcases eq_one_or_neg_one_iff_y_eq_zero.2 hy.symm with rfl | rfl
    · simp
    · simp at hax'
  · -- case 2: `a ≥ a₁`
    have hx₁ : 1 < a.x := by nlinarith [a.prop, h.d_pos]
    have hxx₁ := h.mul_inv_x_pos hx₁ hy
    have hxx₂ := h.mul_inv_x_lt_x hx₁ hy
    have hyy := h.mul_inv_y_nonneg hx₁ hy
    lift (a * a₁⁻¹).x to ℕ using hxx₁.le with x' hx'
    obtain ⟨n, hn⟩ := ih x' (mod_cast hxx₂.trans_eq hax'.symm) hyy hx' hxx₁
    exact ⟨n + 1, by rw [pow_succ', ← hn, mul_comm a, ← mul_assoc, mul_inv_cancel, one_mul]⟩
Cleared hypotheses: a, hay, ax, hax', hax
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
⊢ ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n
have obtain : ∀ (b : Solution₁ d) (hbx : 0 < b.x) (hby : 0 ≤ b.y) (hb : Membership.mem (γ := Set (Solution₁ d)) {b, b⁻¹, -b, -b⁻¹} a),
  ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n
Tactic: obtain ⟨b, hbx, hby, hb⟩ := exists_pos_variant h.d_pos a
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a b : Solution₁ d
hbx : 0 < b.x
hby : 0 ≤ b.y
hb : Membership.mem (γ := Set (Solution₁ d)) {b, b⁻¹, -b, -b⁻¹} a
⊢ ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n
have obtain : ∀ (n : ℕ) (hn : b = a₁ ^ n), ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n
Tactic: obtain ⟨n, hn⟩ := h.eq_pow_of_nonneg hbx hby
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a b : Solution₁ d
hbx : 0 < b.x
hby : 0 ≤ b.y
hb : Membership.mem (γ := Set (Solution₁ d)) {b, b⁻¹, -b, -b⁻¹} a
n : ℕ
hn : b = a₁ ^ n
⊢ ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n
have inl : ∀ (hbx : 0 < a.x) (hby : 0 ≤ a.y) (hn : a = a₁ ^ n), ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n
Tactic: rcases hb with (rfl | rfl | rfl | hb)
Cleared hypotheses: b, hbx, hby, hb, hn
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a b : Solution₁ d
hbx : 0 < b.x
hby : 0 ≤ b.y
hb : Membership.mem (γ := Set (Solution₁ d)) {b, b⁻¹, -b, -b⁻¹} a
n : ℕ
hn : b = a₁ ^ n
inl : ∀ (hbx : 0 < a.x) (hby : 0 ≤ a.y) (hn : a = a₁ ^ n), ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n
⊢ ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n
have inr : ∃ (n : ℤ), b⁻¹ = a₁ ^ n ∨ b⁻¹ = -a₁ ^ n
Tactic: rcases hb with (rfl | rfl | rfl | hb)
Cleared hypotheses: a, hb
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a b : Solution₁ d
hbx : 0 < b.x
hby : 0 ≤ b.y
hb : Membership.mem (γ := Set (Solution₁ d)) {b, b⁻¹, -b, -b⁻¹} a
n : ℕ
hn : b = a₁ ^ n
inl : ∀ (hbx : 0 < a.x) (hby : 0 ≤ a.y) (hn : a = a₁ ^ n), ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n
inr : ∃ (n : ℤ), b⁻¹ = a₁ ^ n ∨ b⁻¹ = -a₁ ^ n
⊢ ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n
have inr₁ : ∃ (n : ℤ), -b = a₁ ^ n ∨ -b = -a₁ ^ n
Tactic: rcases hb with (rfl | rfl | rfl | hb)
Cleared hypotheses: a, hb
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a b : Solution₁ d
hbx : 0 < b.x
hby : 0 ≤ b.y
hb : Membership.mem (γ := Set (Solution₁ d)) {b, b⁻¹, -b, -b⁻¹} a
n : ℕ
hn : b = a₁ ^ n
inl : ∀ (hbx : 0 < a.x) (hby : 0 ≤ a.y) (hn : a = a₁ ^ n), ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n
inr : ∃ (n : ℤ), b⁻¹ = a₁ ^ n ∨ b⁻¹ = -a₁ ^ n
inr₁ : ∃ (n : ℤ), -b = a₁ ^ n ∨ -b = -a₁ ^ n
⊢ ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n
have inr₂ : ∀ (hb : Membership.mem (γ := Set (Solution₁ d)) {-b⁻¹} a), ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n
Tactic: rcases hb with (rfl | rfl | rfl | hb)
Cleared hypotheses: hb
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a b : Solution₁ d
hbx : 0 < b.x
hby : 0 ≤ b.y
n : ℕ
hn : b = a₁ ^ n
hb : Membership.mem (γ := Set (Solution₁ d)) {-b⁻¹} a
⊢ ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n
have hb₁ : a = -b⁻¹
Tactic: rw [Set.mem_singleton_iff] at hb
Cleared hypotheses: hb
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a b : Solution₁ d
hbx : 0 < b.x
hby : 0 ≤ b.y
n : ℕ
hn : b = a₁ ^ n
hb : a = -b⁻¹
⊢ ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n
have inr : ∃ (n : ℤ), -b⁻¹ = a₁ ^ n ∨ -b⁻¹ = -a₁ ^ n
Tactic: rw [hb]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
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
have obtain : ∀ (a₁ : Solution₁ d) (ha₁ : IsFundamental a₁),
  ∃! a₁ : Solution₁ d, 1 < a₁.x ∧ 0 < a₁.y ∧ ∀ (a : Solution₁ d), ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n
Tactic: obtain ⟨a₁, ha₁⟩ := IsFundamental.exists_of_not_isSquare h₀ hd
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
⊢ ∃! a₁ : Solution₁ d, 1 < a₁.x ∧ 0 < a₁.y ∧ ∀ (a : Solution₁ d), ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n
have refine : ∀ (a : Solution₁ d) (H : 1 < a.x ∧ 0 < a.y ∧ ∀ (a_1 : Solution₁ d), ∃ (n : ℤ), a_1 = a ^ n ∨ a_1 = -a ^ n), a = a₁
Tactic: refine ⟨a₁, ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩, fun a (H : 1 < _ ∧ _) => ?_⟩
-- 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
⊢ a = a₁
have Hx : 1 < a.x
Tactic: obtain ⟨Hx, Hy, H⟩ := H
Cleared hypotheses: H
-- 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
⊢ a = a₁
have Hy : 0 < a.y
Tactic: obtain ⟨Hx, Hy, H⟩ := H
Cleared hypotheses: H