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
a : Solution₁ d
ha₁ : 1 < a.x
ha₂ : 0 < a.y
P : ∃ (x' : ℕ), 1 < x' ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑x' : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1
x₁ : ℕ :=
  Nat.find (p := fun (n : ℕ) ↦
    1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1) P
hx :
  1 <
    Nat.find (p := fun (n : ℕ) ↦
      1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1) P
y₁ : ℤ
hy₀ : 0 < y₁
hy₁ :
  HPow.hPow (α := ℤ) (β := ℕ)
        (↑(Nat.find (p := fun (n : ℕ) ↦
              1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1)
              P) :
          ℤ)
        2 -
      d * HPow.hPow (β := ℕ) y₁ 2 =
    1
b : Solution₁ d
hb : 1 < b.x
⊢ (↑x₁ : ℤ) ≤ b.x
have hb' : b.x = (↑b.x.toNat : ℤ)
Tactic: have hb' := (Int.toNat_of_nonneg <| zero_le_one.trans hb.le).symm
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < a.x
ha₂ : 0 < a.y
P : ∃ (x' : ℕ), 1 < x' ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑x' : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1
x₁ : ℕ :=
  Nat.find (p := fun (n : ℕ) ↦
    1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1) P
hx :
  1 <
    Nat.find (p := fun (n : ℕ) ↦
      1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1) P
y₁ : ℤ
hy₀ : 0 < y₁
hy₁ :
  HPow.hPow (α := ℤ) (β := ℕ)
        (↑(Nat.find (p := fun (n : ℕ) ↦
              1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1)
              P) :
          ℤ)
        2 -
      d * HPow.hPow (β := ℕ) y₁ 2 =
    1
b : Solution₁ d
hb : 1 < b.x
hb' : b.x = (↑b.x.toNat : ℤ)
hb'' : 1 < b.x
⊢ (↑x₁ : ℤ) ≤ b.x
have rw : ∀ (hb : LT.lt (α := ℤ) 1 (↑b.x.toNat : ℤ)) (hb' : b.x = (↑b.x.toNat : ℤ)) (hb'' : 1 < b.x),
  LE.le (α := ℤ) (↑x₁ : ℤ) (↑b.x.toNat : ℤ)
Tactic: rw [hb'] at hb ⊢
Cleared hypotheses: hb, hb', hb''
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < a.x
ha₂ : 0 < a.y
P : ∃ (x' : ℕ), 1 < x' ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑x' : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1
x₁ : ℕ :=
  Nat.find (p := fun (n : ℕ) ↦
    1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1) P
hx :
  1 <
    Nat.find (p := fun (n : ℕ) ↦
      1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1) P
y₁ : ℤ
hy₀ : 0 < y₁
hy₁ :
  HPow.hPow (α := ℤ) (β := ℕ)
        (↑(Nat.find (p := fun (n : ℕ) ↦
              1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1)
              P) :
          ℤ)
        2 -
      d * HPow.hPow (β := ℕ) y₁ 2 =
    1
b : Solution₁ d
hb : LT.lt (α := ℤ) 1 (↑b.x.toNat : ℤ)
hb' : b.x = (↑b.x.toNat : ℤ)
hb'' : 1 < b.x
⊢ LE.le (α := ℤ) (↑x₁ : ℤ) (↑b.x.toNat : ℤ)
have norm_cast : ∀ (hb : 1 < b.x.toNat), x₁ ≤ b.x.toNat
Tactic: norm_cast at hb ⊢
Cleared hypotheses: hb
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < a.x
ha₂ : 0 < a.y
P : ∃ (x' : ℕ), 1 < x' ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑x' : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1
x₁ : ℕ :=
  Nat.find (p := fun (n : ℕ) ↦
    1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1) P
hx :
  1 <
    Nat.find (p := fun (n : ℕ) ↦
      1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1) P
y₁ : ℤ
hy₀ : 0 < y₁
hy₁ :
  HPow.hPow (α := ℤ) (β := ℕ)
        (↑(Nat.find (p := fun (n : ℕ) ↦
              1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1)
              P) :
          ℤ)
        2 -
      d * HPow.hPow (β := ℕ) y₁ 2 =
    1
b : Solution₁ d
hb' : b.x = (↑b.x.toNat : ℤ)
hb'' : 1 < b.x
hb : 1 < b.x.toNat
⊢ x₁ ≤ b.x.toNat
have refine : HPow.hPow (α := ℤ) (β := ℕ) (↑b.x.toNat : ℤ) 2 - d * HPow.hPow (β := ℕ) |b.y| 2 = 1
Tactic: refine Nat.find_min' P ⟨hb, |b.y|, abs_pos.mpr <| y_ne_zero_of_one_lt_x hb'', ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < a.x
ha₂ : 0 < a.y
P : ∃ (x' : ℕ), 1 < x' ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑x' : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1
x₁ : ℕ :=
  Nat.find (p := fun (n : ℕ) ↦
    1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1) P
hx :
  1 <
    Nat.find (p := fun (n : ℕ) ↦
      1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1) P
y₁ : ℤ
hy₀ : 0 < y₁
hy₁ :
  HPow.hPow (α := ℤ) (β := ℕ)
        (↑(Nat.find (p := fun (n : ℕ) ↦
              1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1)
              P) :
          ℤ)
        2 -
      d * HPow.hPow (β := ℕ) y₁ 2 =
    1
b : Solution₁ d
hb' : b.x = (↑b.x.toNat : ℤ)
hb'' : 1 < b.x
hb : 1 < b.x.toNat
⊢ HPow.hPow (α := ℤ) (β := ℕ) (↑b.x.toNat : ℤ) 2 - d * HPow.hPow (β := ℕ) |b.y| 2 = 1
have rw : HPow.hPow (β := ℕ) b.x 2 - d * HPow.hPow (β := ℕ) b.y 2 = 1
Tactic: rw [← hb', sq_abs]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < a.x
ha₂ : 0 < a.y
P : ∃ (x' : ℕ), 1 < x' ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑x' : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1
to to₁ to₂ : ∃ (a : Solution₁ d), IsFundamental a
⊢ ∃ (a : Solution₁ d), IsFundamental a
have to₃ : ∀
  (hx :
    1 <
      Nat.find (p := fun (n : ℕ) ↦
        1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1) P)
  (y₁ : ℤ) (hy₀ : 0 < y₁)
  (hy₁ :
    HPow.hPow (α := ℤ) (β := ℕ)
          (↑(Nat.find (p := fun (n : ℕ) ↦
                1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1)
                P) :
            ℤ)
          2 -
        d * HPow.hPow (β := ℕ) y₁ 2 =
      1),
  ∃ (a : Solution₁ d), IsFundamental a
Tactic: classical
  -- to avoid having to show that the predicate is decidable
  let x₁ := Nat.find P
  obtain ⟨hx, y₁, hy₀, hy₁⟩ := Nat.find_spec P
  refine ⟨mk x₁ y₁ hy₁, by rw [x_mk]; exact mod_cast hx, hy₀, fun {b} hb => ?_⟩
  rw [x_mk]
  have hb' := (Int.toNat_of_nonneg <| zero_le_one.trans hb.le).symm
  have hb'' := hb
  rw [hb'] at hb ⊢
  norm_cast at hb ⊢
  refine Nat.find_min' P ⟨hb, |b.y|, abs_pos.mpr <| y_ne_zero_of_one_lt_x hb'', ?_⟩
  rw [← hb', sq_abs]
  exact b.prop
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < a.x
ha₂ : 0 < a.y
P : ∃ (x' : ℕ), 1 < x' ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑x' : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1
to to₁ to₂ : ∃ (a : Solution₁ d), IsFundamental a
to₃ :
  ∀
    (hx :
      1 <
        Nat.find (p := fun (n : ℕ) ↦
          1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1) P)
    (y₁ : ℤ) (hy₀ : 0 < y₁)
    (hy₁ :
      HPow.hPow (α := ℤ) (β := ℕ)
            (↑(Nat.find (p := fun (n : ℕ) ↦
                  1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1)
                  P) :
              ℤ)
            2 -
          d * HPow.hPow (β := ℕ) y₁ 2 =
        1),
    ∃ (a : Solution₁ d), IsFundamental a
⊢ ∃ (a : Solution₁ d), IsFundamental a
have to₄ : let x₁ :=
  Nat.find (p := fun (n : ℕ) ↦
    1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1) P;
∀
  (hx :
    1 <
      Nat.find (p := fun (n : ℕ) ↦
        1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1) P)
  (y₁ : ℤ) (hy₀ : 0 < y₁)
  (hy₁ :
    HPow.hPow (α := ℤ) (β := ℕ)
          (↑(Nat.find (p := fun (n : ℕ) ↦
                1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1)
                P) :
            ℤ)
          2 -
        d * HPow.hPow (β := ℕ) y₁ 2 =
      1)
  {b : Solution₁ d} (hb : 1 < b.x), (Solution₁.mk (↑x₁ : ℤ) y₁ hy₁).x ≤ b.x
Tactic: classical
  -- to avoid having to show that the predicate is decidable
  let x₁ := Nat.find P
  obtain ⟨hx, y₁, hy₀, hy₁⟩ := Nat.find_spec P
  refine ⟨mk x₁ y₁ hy₁, by rw [x_mk]; exact mod_cast hx, hy₀, fun {b} hb => ?_⟩
  rw [x_mk]
  have hb' := (Int.toNat_of_nonneg <| zero_le_one.trans hb.le).symm
  have hb'' := hb
  rw [hb'] at hb ⊢
  norm_cast at hb ⊢
  refine Nat.find_min' P ⟨hb, |b.y|, abs_pos.mpr <| y_ne_zero_of_one_lt_x hb'', ?_⟩
  rw [← hb', sq_abs]
  exact b.prop
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
a : Solution₁ d
ha₁ : 1 < a.x
ha₂ : 0 < a.y
P : ∃ (x' : ℕ), 1 < x' ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑x' : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1
to to₁ to₂ : ∃ (a : Solution₁ d), IsFundamental a
to₃ :
  ∀
    (hx :
      1 <
        Nat.find (p := fun (n : ℕ) ↦
          1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1) P)
    (y₁ : ℤ) (hy₀ : 0 < y₁)
    (hy₁ :
      HPow.hPow (α := ℤ) (β := ℕ)
            (↑(Nat.find (p := fun (n : ℕ) ↦
                  1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1)
                  P) :
              ℤ)
            2 -
          d * HPow.hPow (β := ℕ) y₁ 2 =
        1),
    ∃ (a : Solution₁ d), IsFundamental a
to₄ :
  let x₁ :=
    Nat.find (p := fun (n : ℕ) ↦
      1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1) P;
  ∀
    (hx :
      1 <
        Nat.find (p := fun (n : ℕ) ↦
          1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1) P)
    (y₁ : ℤ) (hy₀ : 0 < y₁)
    (hy₁ :
      HPow.hPow (α := ℤ) (β := ℕ)
            (↑(Nat.find (p := fun (n : ℕ) ↦
                  1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1)
                  P) :
              ℤ)
            2 -
          d * HPow.hPow (β := ℕ) y₁ 2 =
        1)
    {b : Solution₁ d} (hb : 1 < b.x), (Solution₁.mk (↑x₁ : ℤ) y₁ hy₁).x ≤ b.x
⊢ ∃ (a : Solution₁ d), IsFundamental a
have to₅ : let x₁ :=
  Nat.find (p := fun (n : ℕ) ↦
    1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1) P;
∀
  (hx :
    1 <
      Nat.find (p := fun (n : ℕ) ↦
        1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1) P)
  (y₁ : ℤ) (hy₀ : 0 < y₁)
  (hy₁ :
    HPow.hPow (α := ℤ) (β := ℕ)
          (↑(Nat.find (p := fun (n : ℕ) ↦
                1 < n ∧ ∃ (y' : ℤ), 0 < y' ∧ HPow.hPow (α := ℤ) (β := ℕ) (↑n : ℤ) 2 - d * HPow.hPow (β := ℕ) y' 2 = 1)
                P) :
            ℤ)
          2 -
        d * HPow.hPow (β := ℕ) y₁ 2 =
      1)
  {b : Solution₁ d} (hb : 1 < b.x), (↑x₁ : ℤ) ≤ b.x
Tactic: classical
  -- to avoid having to show that the predicate is decidable
  let x₁ := Nat.find P
  obtain ⟨hx, y₁, hy₀, hy₁⟩ := Nat.find_spec P
  refine ⟨mk x₁ y₁ hy₁, by rw [x_mk]; exact mod_cast hx, hy₀, fun {b} hb => ?_⟩
  rw [x_mk]
  have hb' := (Int.toNat_of_nonneg <| zero_le_one.trans hb.le).symm
  have hb'' := hb
  rw [hb'] at hb ⊢
  norm_cast at hb ⊢
  refine Nat.find_min' P ⟨hb, |b.y|, abs_pos.mpr <| y_ne_zero_of_one_lt_x hb'', ?_⟩
  rw [← hb', sq_abs]
  exact b.prop
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
h : IsFundamental a
n : ℤ
hn : 0 ≤ n
⊢ (a ^ n).y < (a ^ (n + 1)).y
have rw : 0 < (a ^ n).x * a.y + ((a ^ n).y * a.x - (a ^ n).y)
Tactic: rw [← sub_pos, zpow_add, zpow_one, y_mul, add_sub_assoc]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
h : IsFundamental a
n : ℤ
hn : 0 ≤ n
⊢ 0 < (a ^ n).x * a.y + ((a ^ n).y * a.x - (a ^ n).y)
have rw : 0 < (a ^ n).x * a.y + (a ^ n).y * (a.x - 1)
Tactic: rw [show (a ^ n).y * a.x - (a ^ n).y = (a ^ n).y * (a.x - 1) by ring]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
h : IsFundamental a
n : ℤ
hn : 0 ≤ n
⊢ 0 ≤ a.x - 1
have rw : 1 ≤ a.x
Tactic: rw [sub_nonneg]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
h : IsFundamental a
n : ℤ
hn : 0 ≤ n
⊢ 0 < (a ^ n).x * a.y + (a ^ n).y * (a.x - 1)
have add_pos_of_pos_of_nonneg : 0 ≤ (a ^ n).y
Tactic: refine
      add_pos_of_pos_of_nonneg (mul_pos (x_zpow_pos h.x_pos _) h.2.1)
        (mul_nonneg ?_ (by rw [sub_nonneg]; exact h.1.le))
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
h : IsFundamental a
n : ℤ
hn : 0 ≤ n
⊢ 0 ≤ (a ^ n).y
have inl : ∀ (hn : LE.le (α := ℤ) 0 0), 0 ≤ (HPow.hPow (β := ℤ) a 0).y
Tactic: rcases hn.eq_or_lt with (rfl | hn)
Cleared hypotheses: n, hn
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
h : IsFundamental a
n : ℤ
hn : 0 ≤ n
inl : ∀ (hn : LE.le (α := ℤ) 0 0), 0 ≤ (HPow.hPow (β := ℤ) a 0).y
⊢ 0 ≤ (a ^ n).y
have inr : ∀ (hn : 0 < n), 0 ≤ (a ^ n).y
Tactic: rcases hn.eq_or_lt with (rfl | hn)
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
a : Solution₁ d
h : IsFundamental a
⊢ StrictMono (β := ℤ) fun (n : ℤ) ↦ (a ^ n).y
have H : ∀ (n : ℤ), 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y
Tactic: have H : ∀ n : ℤ, 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y := by
    intro n hn
    rw [← sub_pos, zpow_add, zpow_one, y_mul, add_sub_assoc]
    rw [show (a ^ n).y * a.x - (a ^ n).y = (a ^ n).y * (a.x - 1) by ring]
    refine
      add_pos_of_pos_of_nonneg (mul_pos (x_zpow_pos h.x_pos _) h.2.1)
        (mul_nonneg ?_ (by rw [sub_nonneg]; exact h.1.le))
    rcases hn.eq_or_lt with (rfl | hn)
    · simp only [zpow_zero, y_one, le_refl]
    · exact (y_zpow_pos h.x_pos h.2.1 hn).le
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
h : IsFundamental a
H : ∀ (n : ℤ), 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y
⊢ StrictMono (β := ℤ) fun (n : ℤ) ↦ (a ^ n).y
have refine : ∀ (n : ℤ), (a ^ n).y < (a ^ (n + 1)).y
Tactic: refine strictMono_int_of_lt_succ fun n => ?_
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
h : IsFundamental a
H : ∀ (n : ℤ), 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y
n : ℤ
⊢ (a ^ n).y < (a ^ (n + 1)).y
have inl : ∀ (hn : 0 ≤ n), (a ^ n).y < (a ^ (n + 1)).y
Tactic: rcases le_or_gt 0 n with hn | hn
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
h : IsFundamental a
H : ∀ (n : ℤ), 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y
n : ℤ
inl : ∀ (hn : 0 ≤ n), (a ^ n).y < (a ^ (n + 1)).y
⊢ (a ^ n).y < (a ^ (n + 1)).y
have inr : ∀ (hn : n < 0), (a ^ n).y < (a ^ (n + 1)).y
Tactic: rcases le_or_gt 0 n with hn | hn
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
a : Solution₁ d
h : IsFundamental a
H : ∀ (n : ℤ), 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y
n : ℤ
hn : n < 0
m : ℤ := -n - 1
⊢ (a ^ n).y < (a ^ (n + 1)).y
have hm : n = -m - 1
Tactic: have hm : n = -m - 1 := by simp only [m, neg_sub, sub_neg_eq_add, add_tsub_cancel_left]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
h : IsFundamental a
H : ∀ (n : ℤ), 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y
n : ℤ
hn : n < 0
m : ℤ := -n - 1
hm : n = -m - 1
⊢ (a ^ n).y < (a ^ (n + 1)).y
have inr : (a ^ m).y < (a ^ (m + 1)).y
Tactic: rw [hm, sub_add_cancel, ← neg_add', zpow_neg, zpow_neg, y_inv, y_inv, neg_lt_neg_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
h : IsFundamental a
m n : ℤ
⊢ (a ^ m).y < (a ^ n).y ↔ m < n
have refine : ∀ (H : (a ^ m).y < (a ^ n).y), m < n
Tactic: refine ⟨fun H => ?_, fun H => h.y_strictMono H⟩
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (goal)
Goal:
d : ℤ
a : Solution₁ d
h : IsFundamental a
m n : ℤ
H : (a ^ m).y < (a ^ n).y
⊢ m < n
have contrapose! : ∀ (H : n ≤ m), (a ^ n).y ≤ (a ^ m).y
Tactic: contrapose! H
Cleared hypotheses: H
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
a : Solution₁ d
h : IsFundamental a
n n' : ℤ
hf : a ^ n = -a ^ n'
⊢ False
have hf₁ : (a ^ n).x = (-a ^ n').x
Tactic: apply_fun Solution₁.x at hf
Cleared hypotheses: hf
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
a : Solution₁ d
h : IsFundamental a
n n' : ℤ
hf : (a ^ n).x = (-a ^ n').x
⊢ False
have H : 0 < (a ^ n).x
Tactic: have H := x_zpow_pos h.x_pos n
-- Lean 4 proof step from Mathlib.NumberTheory.Pell (hypothesis)
Goal:
d : ℤ
a : Solution₁ d
h : IsFundamental a
n n' : ℤ
hf : (a ^ n).x = (-a ^ n').x
H : 0 < (a ^ n).x
⊢ False
have H₁ : (a ^ n').x < 0
Tactic: rw [hf, x_neg, lt_neg, neg_zero] at H
Cleared hypotheses: H
-- 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) a₁.y 2 - HPow.hPow (β := ℕ) a.y 2) = HPow.hPow (β := ℕ) a₁.x 2 - HPow.hPow (β := ℕ) a.x 2
have rw : d * (HPow.hPow (β := ℕ) a₁.y 2 - HPow.hPow (β := ℕ) a.y 2) =
  1 + d * HPow.hPow (β := ℕ) a₁.y 2 - (1 + d * HPow.hPow (β := ℕ) a.y 2)
Tactic: rw [a.prop_x, a₁.prop_x]
-- 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
⊢ a₁.y ≤ a.y
have H : d * (HPow.hPow (β := ℕ) a₁.y 2 - HPow.hPow (β := ℕ) a.y 2) = HPow.hPow (β := ℕ) a₁.x 2 - HPow.hPow (β := ℕ) a.x 2
Tactic: have H : d * (a₁.y ^ 2 - a.y ^ 2) = a₁.x ^ 2 - a.x ^ 2 := by rw [a.prop_x, a₁.prop_x]; 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
H : d * (HPow.hPow (β := ℕ) a₁.y 2 - HPow.hPow (β := ℕ) a.y 2) = HPow.hPow (β := ℕ) a₁.x 2 - HPow.hPow (β := ℕ) a.x 2
⊢ a₁.y ≤ a.y
have rw : a₁.x ≤ a.x
Tactic: rw [← abs_of_pos hay, ← abs_of_pos h.2.1, ← sq_le_sq, ← mul_le_mul_iff_right₀ h.d_pos,
    ← sub_nonpos, ← mul_sub, H, sub_nonpos, sq_le_sq, abs_of_pos (zero_lt_one.trans h.1),
    abs_of_pos (zero_lt_one.trans hax)]
-- 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.y ≤ a.y * a₁.x
have rw : 0 ≤
  HPow.hPow (β := ℕ) a.y 2 * (1 + d * HPow.hPow (β := ℕ) a₁.y 2) -
    (1 + d * HPow.hPow (β := ℕ) a.y 2) * HPow.hPow (β := ℕ) a₁.y 2
Tactic: rw [← abs_of_pos <| zero_lt_one.trans hax, ← abs_of_pos hay, ← abs_of_pos h.x_pos, ←
    abs_of_pos h.2.1, ← abs_mul, ← abs_mul, ← sq_le_sq, mul_pow, mul_pow, a.prop_x, a₁.prop_x, ←
    sub_nonneg]
-- 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
⊢ 0 ≤
    HPow.hPow (β := ℕ) a.y 2 * (1 + d * HPow.hPow (β := ℕ) a₁.y 2) -
      (1 + d * HPow.hPow (β := ℕ) a.y 2) * HPow.hPow (β := ℕ) a₁.y 2
have ring_nf : 0 ≤ HPow.hPow (β := ℕ) a.y 2 - HPow.hPow (β := ℕ) a₁.y 2
Tactic: ring_nf
-- 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
⊢ 0 ≤ HPow.hPow (β := ℕ) a.y 2 - HPow.hPow (β := ℕ) a₁.y 2
have rw : a₁.y ≤ a.y
Tactic: rw [sub_nonneg, sq_le_sq, abs_of_pos hay, abs_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 : 1 < a.x
hay : 0 < a.y
⊢ 0 < (a * a₁⁻¹).x
have simp : d * (a.y * a₁.y) < a.x * a₁.x
Tactic: simp only [x_mul, x_inv, y_inv, mul_neg, lt_add_neg_iff_add_lt, zero_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
⊢ d * (a.y * a₁.y) < a.x * a₁.x
have refine : a.x * (d * (a.y * a₁.y)) < a.x * (a.x * a₁.x)
Tactic: refine lt_of_mul_lt_mul_left ?_ <| zero_le_one.trans hax.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
⊢ d * a.y * (a.x * a₁.y) ≤ d * a.y * (a.y * a₁.x)
have this : 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 (hypothesis)
Goal:
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
this : a.x * a₁.y ≤ a.y * a₁.x
⊢ d * a.y * (a.x * a₁.y) ≤ d * a.y * (a.y * a₁.x)
have this₁ : 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
this_1 : a.x * a₁.y ≤ a.y * a₁.x
this : 0 < d
⊢ d * a.y * (a.x * a₁.y) ≤ d * a.y * (a.y * a₁.x)
have ha : 0 ≤ d * a.y
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
⊢ d * a.y * (a.y * a₁.x) = (HPow.hPow (β := ℕ) a.x 2 - 1) * a₁.x
have rw : d * a.y * (a.y * a₁.x) = d * HPow.hPow (β := ℕ) a.y 2 * a₁.x
Tactic: rw [← a.prop_y]
-- 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.x 2 - 1) * a₁.x < a.x * (a.x * a₁.x)
have linarith : a.x * (a.x * a₁.x) ≤ (HPow.hPow (β := ℕ) a.x 2 - 1) * a₁.x →
  HMul.hMul (α := ℤ) 2 (-1) + (a.x * (a.x * a₁.x) - (HPow.hPow (β := ℕ) a.x 2 - 1) * a₁.x) +
      (HAdd.hAdd (α := ℤ) 1 1 - a₁.x) =
    0
Tactic: 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
linarith :
  a.x * (a.x * a₁.x) ≤ (HPow.hPow (β := ℕ) a.x 2 - 1) * a₁.x →
    HMul.hMul (α := ℤ) 2 (-1) + (a.x * (a.x * a₁.x) - (HPow.hPow (β := ℕ) a.x 2 - 1) * a₁.x) +
        (HAdd.hAdd (α := ℤ) 1 1 - a₁.x) =
      0
⊢ (HPow.hPow (β := ℕ) a.x 2 - 1) * a₁.x < a.x * (a.x * a₁.x)
have linarith₁ : a.x * (a.x * a₁.x) ≤ (HPow.hPow (β := ℕ) a.x 2 - 1) * a₁.x → GT.gt (α := ℤ) 2 0
Tactic: 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.x * (d * (a.y * a₁.y)) < a.x * (a.x * a₁.x)
have calc : a.x * (d * (a.y * a₁.y)) = d * a.y * (a.x * a₁.y)
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)
⊢ a.x * (d * (a.y * a₁.y)) < a.x * (a.x * a₁.x)
have calc₁ : d * a.y * (a.x * a₁.y) ≤ d * a.y * (a.y * 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]