awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.NumberTheory.Pell
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
refine : a₁.y * (a.x * a₁.x) < a₁.y * (d * (a.y * a₁.y) + a.x)
⊢ a.x * a₁.x < d * (a.y * a₁.y) + a.x

Tactic:
refine lt_of_mul_lt_mul_left ?_ h.2.1.le

Premises used: lt_of_mul_lt_mul_left, LT.lt.le, And.right, And.left
-- Module: Mathlib.NumberTheory.Pell
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) = a.x * a₁.y * a₁.x

Tactic:
ring
-- Module: Mathlib.NumberTheory.Pell
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
⊢ 1 < a₁.x

Tactic:
have := h.1
-- Module: Mathlib.NumberTheory.Pell
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 ≤ a.y * a₁.x

Tactic:
have := x_mul_y_le_y_mul_x h hax hay
-- Module: Mathlib.NumberTheory.Pell
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
ha : 0 ≤ a₁.x
⊢ a.x * a₁.y * a₁.x ≤ a.y * a₁.x * a₁.x

Tactic:
gcongr
-- Module: Mathlib.NumberTheory.Pell
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
calc : a.y * a₁.x * a₁.x < a₁.y * (d * (a.y * a₁.y) + a.x)
⊢ a₁.y * (a.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

Premises used: HMul.hMul, Pell.Solution₁.x, Pell.Solution₁.y
-- Module: Mathlib.NumberTheory.Pell
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
calc : a.y * (1 + d * HPow.hPow (β := ℕ) a₁.y 2) - a₁.y * (d * (a.y * a₁.y) + a.x) < 0
⊢ a.y * a₁.x * a₁.x < a₁.y * (d * (a.y * a₁.y) + a.x)

Tactic:
rw [mul_assoc, ← sq, a₁.prop_x, ← sub_neg]

Premises used: sub_neg, mul_assoc, sq, Pell.Solution₁.prop_x
-- Module: Mathlib.NumberTheory.Pell
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 - a.x * a₁.y < 0

Tactic:
this
-- Module: Mathlib.NumberTheory.Pell
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
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
⊢ a.y * (1 + d * HPow.hPow (β := ℕ) a₁.y 2) - a₁.y * (d * (a.y * a₁.y) + a.x) < 0

Tactic:
convert! this using 1
-- Module: Mathlib.NumberTheory.Pell
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) = a.y - a.x * a₁.y

Tactic:
ring
-- Module: Mathlib.NumberTheory.Pell
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
calc : 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

Tactic:
suffices a.y - a.x * a₁.y < 0 by convert! this using 1; ring
-- Module: Mathlib.NumberTheory.Pell
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
calc : HPow.hPow (β := ℕ) a.y 2 < (1 + d * HPow.hPow (β := ℕ) a.y 2) * HPow.hPow (β := ℕ) a₁.y 2
⊢ a.y - a.x * a₁.y < 0

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]

Premises used: sub_neg, abs_mul, zero_lt_one, sq_lt_sq, Pell.Solution₁.prop_x, And.right, And.left, LT.lt.trans, abs_of_pos, mul_pow
-- Module: Mathlib.NumberTheory.Pell
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 * HPow.hPow (β := ℕ) a.y 2

Tactic:
(one_mul _).symm

Premises used: Eq.symm, one_mul
-- Module: Mathlib.NumberTheory.Pell
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
⊢ 1 * HPow.hPow (β := ℕ) a.y 2 ≤ d * HPow.hPow (β := ℕ) a.y 2

Tactic:
(mul_le_mul_iff_left₀ <| sq_pos_of_pos hay).mpr h.d_pos

Premises used: Iff.mpr, sq_pos_of_pos, Pell.IsFundamental.d_pos, mul_le_mul_iff_left₀
-- Module: Mathlib.NumberTheory.Pell
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 < d * HPow.hPow (β := ℕ) a.y 2 + 1

Tactic:
lt_add_one _

Premises used: lt_add_one
-- Module: Mathlib.NumberTheory.Pell
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
rw₁ rw₂ : 1 + d * HPow.hPow (β := ℕ) a.y 2 = 1 + d * HPow.hPow (β := ℕ) a.y 2
⊢ d * HPow.hPow (β := ℕ) a.y 2 + 1 = (1 + d * HPow.hPow (β := ℕ) a.y 2) * 1

Tactic:
rw [add_comm, mul_one]

Premises used: add_comm, mul_one
-- Module: Mathlib.NumberTheory.Pell
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
⊢ 0 < d

Tactic:
have := h.d_pos
-- Module: Mathlib.NumberTheory.Pell
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
this : 0 < d
⊢ 0 < 1 + d * HPow.hPow (β := ℕ) a.y 2

Tactic:
positivity
-- Module: Mathlib.NumberTheory.Pell
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
⊢ HPow.hPow (β := ℕ) a.y 2 < (1 + d * HPow.hPow (β := ℕ) a.y 2) * HPow.hPow (β := ℕ) a₁.y 2

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)

Premises used: Iff.mpr, HMul.hMul, lt_add_one, sq_pos_of_pos, Pell.Solution₁.y, Pell.IsFundamental.d_pos, HPow.hPow, HAdd.hAdd, mul_le_mul_iff_left₀, OfNat.ofNat, Eq.symm, one_mul, mul_le_mul_iff_right₀
-- Module: Mathlib.NumberTheory.Pell
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 0 < a.x
hay : 0 ≤ a.y
⊢ 0 ≤ a.x

Tactic:
hax.le

Premises used: LT.lt.le
-- Module: Mathlib.NumberTheory.Pell
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 0 < a.x
hay : 0 ≤ a.y
lift : ∀ (ax : ℕ) (hax' : (↑ax : ℤ) = a.x) (hax : LT.lt (α := ℤ) 0 (↑ax : ℤ)), ∃ (n : ℕ), a = a₁ ^ n
⊢ ∃ (n : ℕ), a = a₁ ^ n

Tactic:
lift a.x to ℕ using hax.le with ax hax'

Premises used: Pell.Solution₁.x, LT.lt.le
-- Module: Mathlib.NumberTheory.Pell
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
h₂ : ∀ (hy : 0 < a.y), ∃ (n : ℕ), a = a₁ ^ n
⊢ ∃ (n : ℕ), a = a₁ ^ n

Tactic:
rcases hay.eq_or_lt with hy | hy

Premises used: LE.le.eq_or_lt
-- Module: Mathlib.NumberTheory.Pell
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

Tactic:
⟨0, ?_⟩

Premises used: Exists.intro, OfNat.ofNat
-- Module: Mathlib.NumberTheory.Pell
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 : a = HPow.hPow (β := ℕ) a₁ 0
⊢ ∃ (n : ℕ), a = a₁ ^ n

Tactic:
refine ⟨0, ?_⟩
-- Module: Mathlib.NumberTheory.Pell
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
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
⊢ a = HPow.hPow (β := ℕ) a₁ 0

Tactic:
rcases eq_one_or_neg_one_iff_y_eq_zero.2 hy.symm with rfl | rfl

Premises used: Iff.mpr, Pell.Solution₁.eq_one_or_neg_one_iff_y_eq_zero, Eq.symm, rfl
-- Module: Mathlib.NumberTheory.Pell
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
hax : LT.lt (α := ℤ) 0 (↑x : ℤ)
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:
simp
-- Module: Mathlib.NumberTheory.Pell
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
hax : LT.lt (α := ℤ) 0 (↑x : ℤ)
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:
simp at hax'
-- Module: Mathlib.NumberTheory.Pell
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
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
⊢ 1 < a.x

Tactic:
nlinarith [a.prop, h.d_pos]

Premises used: Pell.Solution₁.prop, Pell.IsFundamental.d_pos
-- Module: Mathlib.NumberTheory.Pell
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

Tactic:
have hx₁ : 1 < a.x := by nlinarith [a.prop, h.d_pos]
-- Module: Mathlib.NumberTheory.Pell
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
⊢ 0 < (a * a₁⁻¹).x

Tactic:
have hxx₁ := h.mul_inv_x_pos hx₁ hy
-- Module: Mathlib.NumberTheory.Pell
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
⊢ (a * a₁⁻¹).x < a.x

Tactic:
have hxx₂ := h.mul_inv_x_lt_x hx₁ hy
-- Module: Mathlib.NumberTheory.Pell
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
⊢ 0 ≤ (a * a₁⁻¹).y

Tactic:
have hyy := h.mul_inv_y_nonneg hx₁ hy
-- Module: Mathlib.NumberTheory.Pell
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
⊢ 0 ≤ (a * a₁⁻¹).x

Tactic:
hxx₁.le

Premises used: LT.lt.le
-- Module: Mathlib.NumberTheory.Pell
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
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
⊢ ∃ (n : ℕ), a = a₁ ^ n

Tactic:
lift (a * a₁⁻¹).x to ℕ using hxx₁.le with x' hx'

Premises used: HMul.hMul, Pell.Solution₁.x, LT.lt.le, Inv.inv
-- Module: Mathlib.NumberTheory.Pell
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
h₁ : ∀ (n : ℕ) (hn : a * a₁⁻¹ = a₁ ^ n), ∃ (n : ℕ), a = a₁ ^ n
⊢ ∃ (n : ℕ), a = a₁ ^ n

Tactic:
obtain ⟨n, hn⟩ := ih x' (mod_cast hxx₂.trans_eq hax'.symm) hyy hx' hxx₁

Premises used: Nat.cast_lt._simp_1, cast, LT.lt.trans_eq, Eq.symm
-- Module: Mathlib.NumberTheory.Pell
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
rw₅ rw₆ : a = a
⊢ a = a₁ ^ (n + 1)

Tactic:
rw [pow_succ', ← hn, mul_comm a, ← mul_assoc, mul_inv_cancel, one_mul]

Premises used: mul_assoc, mul_comm, mul_inv_cancel, pow_succ', one_mul
-- Module: Mathlib.NumberTheory.Pell
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
⊢ ∃ (n : ℕ), a = a₁ ^ n

Tactic:
exact ⟨n + 1, by rw [pow_succ', ← hn, mul_comm a, ← mul_assoc, mul_inv_cancel, one_mul]⟩
-- Module: Mathlib.NumberTheory.Pell
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)
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
⊢ ∃ (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]⟩

Premises used: LE.le.eq_or_lt, HMul.hMul, Pell.Solution₁.x, Inv.inv, rfl
-- Module: Mathlib.NumberTheory.Pell
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
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
⊢ ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n

Tactic:
obtain ⟨b, hbx, hby, hb⟩ := exists_pos_variant h.d_pos a

Premises used: Pell.Solution₁.exists_pos_variant, Pell.IsFundamental.d_pos
-- Module: Mathlib.NumberTheory.Pell
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
obtain : ∀ (n : ℕ) (hn : b = a₁ ^ n), ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n
⊢ ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n

Tactic:
obtain ⟨n, hn⟩ := h.eq_pow_of_nonneg hbx hby

Premises used: Pell.IsFundamental.eq_pow_of_nonneg
-- Module: Mathlib.NumberTheory.Pell
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
inr₂ : ∀ (hb : Membership.mem (γ := Set (Solution₁ d)) {-b⁻¹} a), ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n
⊢ ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n

Tactic:
rcases hb with (rfl | rfl | rfl | hb)

Premises used: rfl
-- Module: Mathlib.NumberTheory.Pell
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
n : ℕ
hbx : 0 < a.x
hby : 0 ≤ a.y
hn : a = a₁ ^ n
⊢ ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n

Tactic:
⟨n, Or.inl (mod_cast hn)⟩

Premises used: zpow_natCast, congrArg, cast, Nat.cast, Exists.intro, Or.inl, Eq.symm, Eq
-- Module: Mathlib.NumberTheory.Pell
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
n : ℕ
hbx : 0 < a.x
hby : 0 ≤ a.y
hn : a = a₁ ^ n
⊢ ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n

Tactic:
exact ⟨n, Or.inl (mod_cast hn)⟩
-- Module: Mathlib.NumberTheory.Pell
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
b : Solution₁ d
hbx : 0 < b.x
hby : 0 ≤ b.y
n : ℕ
hn : b = a₁ ^ n
⊢ b⁻¹ = HPow.hPow (β := ℤ) a₁ (-(↑n : ℤ))

Tactic:
simp [hn]
-- Module: Mathlib.NumberTheory.Pell
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
b : Solution₁ d
hbx : 0 < b.x
hby : 0 ≤ b.y
n : ℕ
hn : b = a₁ ^ n
⊢ ∃ (n : ℤ), b⁻¹ = a₁ ^ n ∨ b⁻¹ = -a₁ ^ n

Tactic:
exact ⟨-n, Or.inl (by simp [hn])⟩
-- Module: Mathlib.NumberTheory.Pell
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
b : Solution₁ d
hbx : 0 < b.x
hby : 0 ≤ b.y
n : ℕ
hn : b = a₁ ^ n
⊢ -b = -HPow.hPow (β := ℤ) a₁ (↑n : ℤ)

Tactic:
simp [hn]
-- Module: Mathlib.NumberTheory.Pell
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
b : Solution₁ d
hbx : 0 < b.x
hby : 0 ≤ b.y
n : ℕ
hn : b = a₁ ^ n
⊢ ∃ (n : ℤ), -b = a₁ ^ n ∨ -b = -a₁ ^ n

Tactic:
exact ⟨n, Or.inr (by simp [hn])⟩
-- Module: Mathlib.NumberTheory.Pell
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
⊢ a = -b⁻¹

Tactic:
rw [Set.mem_singleton_iff] at hb

Premises used: Set.mem_singleton_iff
-- Module: Mathlib.NumberTheory.Pell
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
inr : ∃ (n : ℤ), -b⁻¹ = a₁ ^ n ∨ -b⁻¹ = -a₁ ^ n
⊢ ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n

Tactic:
rw [hb]
-- Module: Mathlib.NumberTheory.Pell
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
⊢ -b⁻¹ = -HPow.hPow (β := ℤ) a₁ (-(↑n : ℤ))

Tactic:
simp [hn]
-- Module: Mathlib.NumberTheory.Pell
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), -b⁻¹ = a₁ ^ n ∨ -b⁻¹ = -a₁ ^ n

Tactic:
exact ⟨-n, Or.inr (by simp [hn])⟩
-- Module: Mathlib.NumberTheory.Pell
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
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
⊢ ∃! 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

Premises used: Pell.IsFundamental.exists_of_not_isSquare
-- Module: Mathlib.NumberTheory.Pell
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

Tactic:
⟨a₁, ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩, fun a (H : 1 < _ ∧ _) => ?_⟩

Premises used: Pell.IsFundamental.eq_zpow_or_neg_zpow, And.right, And.left, ExistsUnique, And.intro, Exists.intro
-- Module: Mathlib.NumberTheory.Pell
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a₁ : Solution₁ d
ha₁ : IsFundamental a₁
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
⊢ ∃! a₁ : Solution₁ d, 1 < a₁.x ∧ 0 < a₁.y ∧ ∀ (a : Solution₁ d), ∃ (n : ℤ), a = a₁ ^ n ∨ a = -a₁ ^ n

Tactic:
refine ⟨a₁, ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩, fun a (H : 1 < _ ∧ _) => ?_⟩
-- Module: Mathlib.NumberTheory.Pell
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
⊢ 1 < a.x

Tactic:
obtain ⟨Hx, Hy, H⟩ := H
-- Module: Mathlib.NumberTheory.Pell
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
⊢ 0 < a.y

Tactic:
obtain ⟨Hx, Hy, H⟩ := H
-- Module: Mathlib.NumberTheory.Pell
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_1 : Solution₁ d), ∃ (n : ℤ), a_1 = a ^ n ∨ a_1 = -a ^ n

Tactic:
obtain ⟨Hx, Hy, H⟩ := H
-- Module: Mathlib.NumberTheory.Pell
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
obtain : ∀ (n₁ : ℤ) (hn₁ : a₁ = a ^ n₁ ∨ a₁ = -a ^ n₁), a = a₁
⊢ a = a₁

Tactic:
obtain ⟨n₁, hn₁⟩ := H a₁
-- Module: Mathlib.NumberTheory.Pell
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
obtain : ∀ (n₂ : ℤ) (hn₂ : a = a₁ ^ n₂ ∨ a = -a₁ ^ n₂), a = a₁
⊢ a = a₁

Tactic:
obtain ⟨n₂, hn₂⟩ := ha₁.eq_zpow_or_neg_zpow a

Premises used: Pell.IsFundamental.eq_zpow_or_neg_zpow