awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q i : ℕ
ih :
  (↑(s i) : LucasLehmer.X q) =
    HPow.hPow (α := LucasLehmer.X q) ω (HPow.hPow (α := ℕ) 2 i) +
      HPow.hPow (α := LucasLehmer.X q) ωb (HPow.hPow (α := ℕ) 2 i)
X :
  (↑(HPow.hPow (β := ℕ) (s i) 2 - 2) : LucasLehmer.X q) =
    HPow.hPow (α := LucasLehmer.X q) (β := ℕ) (↑(s i) : LucasLehmer.X q) 2 - 2
X₁ :
  HPow.hPow (α := LucasLehmer.X q) (β := ℕ) (↑(s i) : LucasLehmer.X q) 2 - 2 =
    HPow.hPow (β := ℕ)
        (HPow.hPow (α := LucasLehmer.X q) ω (HPow.hPow (α := ℕ) 2 i) +
          HPow.hPow (α := LucasLehmer.X q) ωb (HPow.hPow (α := ℕ) 2 i))
        2 -
      2
X₂ :
  HPow.hPow (β := ℕ)
        (HPow.hPow (α := LucasLehmer.X q) ω (HPow.hPow (α := ℕ) 2 i) +
          HPow.hPow (α := LucasLehmer.X q) ωb (HPow.hPow (α := ℕ) 2 i))
        2 -
      2 =
    HPow.hPow (β := ℕ) (HPow.hPow (α := LucasLehmer.X q) ω (HPow.hPow (α := ℕ) 2 i)) 2 +
          HPow.hPow (β := ℕ) (HPow.hPow (α := LucasLehmer.X q) ωb (HPow.hPow (α := ℕ) 2 i)) 2 +
        2 *
          (HPow.hPow (α := LucasLehmer.X q) ωb (HPow.hPow (α := ℕ) 2 i) *
            HPow.hPow (α := LucasLehmer.X q) ω (HPow.hPow (α := ℕ) 2 i)) -
      2
X₃ :
  HPow.hPow (β := ℕ) (HPow.hPow (α := LucasLehmer.X q) ω (HPow.hPow (α := ℕ) 2 i)) 2 +
          HPow.hPow (β := ℕ) (HPow.hPow (α := LucasLehmer.X q) ωb (HPow.hPow (α := ℕ) 2 i)) 2 +
        2 *
          (HPow.hPow (α := LucasLehmer.X q) ωb (HPow.hPow (α := ℕ) 2 i) *
            HPow.hPow (α := LucasLehmer.X q) ω (HPow.hPow (α := ℕ) 2 i)) -
      2 =
    HPow.hPow (β := ℕ) (HPow.hPow (α := LucasLehmer.X q) ω (HPow.hPow (α := ℕ) 2 i)) 2 +
      HPow.hPow (β := ℕ) (HPow.hPow (α := LucasLehmer.X q) ωb (HPow.hPow (α := ℕ) 2 i)) 2
⊢ (↑(s (i + 1)) : LucasLehmer.X q) =
    HPow.hPow (α := LucasLehmer.X q) ω (HPow.hPow (α := ℕ) 2 (i + 1)) +
      HPow.hPow (α := LucasLehmer.X q) ωb (HPow.hPow (α := ℕ) 2 (i + 1))
have X₄ : HPow.hPow (β := ℕ) (HPow.hPow (α := LucasLehmer.X q) ω (HPow.hPow (α := ℕ) 2 i)) 2 +
    HPow.hPow (β := ℕ) (HPow.hPow (α := LucasLehmer.X q) ωb (HPow.hPow (α := ℕ) 2 i)) 2 =
  HPow.hPow (α := LucasLehmer.X q) ω (HPow.hPow (α := ℕ) 2 (i + 1)) +
    HPow.hPow (α := LucasLehmer.X q) ωb (HPow.hPow (α := ℕ) 2 (i + 1))
Tactic: calc
      (s (i + 1) : X q) = (s i ^ 2 - 2 : ℤ) := rfl
      _ = (s i : X q) ^ 2 - 2 := by push_cast; rfl
      _ = (ω ^ 2 ^ i + ωb ^ 2 ^ i) ^ 2 - 2 := by rw [ih]
      _ = (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 + 2 * (ωb ^ 2 ^ i * ω ^ 2 ^ i) - 2 := by ring
      _ = (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 := by
        rw [← mul_pow ωb ω, ωb_mul_ω, one_pow, mul_one, add_sub_cancel_right]
      _ = ω ^ 2 ^ (i + 1) + ωb ^ 2 ^ (i + 1) := by rw [← pow_mul, ← pow_mul, _root_.pow_succ]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q i : ℕ
⊢ (↑(s i) : X q) = HPow.hPow (α := X q) ω (HPow.hPow (α := ℕ) 2 i) + HPow.hPow (α := X q) ωb (HPow.hPow (α := ℕ) 2 i)
have zero : (↑(s 0) : X q) =
  HPow.hPow (α := X q) ω (HPow.hPow (α := ℕ) (β := ℕ) 2 0) + HPow.hPow (α := X q) ωb (HPow.hPow (α := ℕ) (β := ℕ) 2 0)
Tactic: induction i with
  | zero =>
    dsimp [s, ω, ωb]
    ext <;> norm_num
  | succ i ih =>
    calc
      (s (i + 1) : X q) = (s i ^ 2 - 2 : ℤ) := rfl
      _ = (s i : X q) ^ 2 - 2 := by push_cast; rfl
      _ = (ω ^ 2 ^ i + ωb ^ 2 ^ i) ^ 2 - 2 := by rw [ih]
      _ = (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 + 2 * (ωb ^ 2 ^ i * ω ^ 2 ^ i) - 2 := by ring
      _ = (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 := by
        rw [← mul_pow ωb ω, ωb_mul_ω, one_pow, mul_one, add_sub_cancel_right]
      _ = ω ^ 2 ^ (i + 1) + ωb ^ 2 ^ (i + 1) := by rw [← pow_mul, ← pow_mul, _root_.pow_succ]
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q i : ℕ
zero :
  (↑(s 0) : X q) =
    HPow.hPow (α := X q) ω (HPow.hPow (α := ℕ) (β := ℕ) 2 0) + HPow.hPow (α := X q) ωb (HPow.hPow (α := ℕ) (β := ℕ) 2 0)
⊢ (↑(s i) : X q) = HPow.hPow (α := X q) ω (HPow.hPow (α := ℕ) 2 i) + HPow.hPow (α := X q) ωb (HPow.hPow (α := ℕ) 2 i)
have succ : ∀ (i : ℕ)
  (ih :
    (↑(s i) : X q) =
      HPow.hPow (α := X q) ω (HPow.hPow (α := ℕ) 2 i) + HPow.hPow (α := X q) ωb (HPow.hPow (α := ℕ) 2 i)),
  (↑(s (i + 1)) : X q) =
    HPow.hPow (α := X q) ω (HPow.hPow (α := ℕ) 2 (i + 1)) + HPow.hPow (α := X q) ωb (HPow.hPow (α := ℕ) 2 (i + 1))
Tactic: induction i with
  | zero =>
    dsimp [s, ω, ωb]
    ext <;> norm_num
  | succ i ih =>
    calc
      (s (i + 1) : X q) = (s i ^ 2 - 2 : ℤ) := rfl
      _ = (s i : X q) ^ 2 - 2 := by push_cast; rfl
      _ = (ω ^ 2 ^ i + ωb ^ 2 ^ i) ^ 2 - 2 := by rw [ih]
      _ = (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 + 2 * (ωb ^ 2 ^ i * ω ^ 2 ^ i) - 2 := by ring
      _ = (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 := by
        rw [← mul_pow ωb ω, ωb_mul_ω, one_pow, mul_one, add_sub_cancel_right]
      _ = ω ^ 2 ^ (i + 1) + ωb ^ 2 ^ (i + 1) := by rw [← pow_mul, ← pow_mul, _root_.pow_succ]
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
⊢ HPow.hPow (α := X q) (β := ℕ) α 2 = 3
have h₁ : (HPow.hPow (α := X q) (β := ℕ) α 2).1 = 3.1
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
h₁ : (HPow.hPow (α := X q) (β := ℕ) α 2).1 = 3.1
⊢ HPow.hPow (α := X q) (β := ℕ) α 2 = 3
have h₂ : (HPow.hPow (α := X q) (β := ℕ) α 2).2 = 3.2
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
⊢ HPow.hPow (β := ℕ) (HAdd.hAdd (α := X q) 1 α) 2 = HMul.hMul (α := X q) 2 ω
have h₁ : (HPow.hPow (β := ℕ) (HAdd.hAdd (α := X q) 1 α) 2).1 = (HMul.hMul (α := X q) 2 ω).1
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
h₁ : (HPow.hPow (β := ℕ) (HAdd.hAdd (α := X q) 1 α) 2).1 = (HMul.hMul (α := X q) 2 ω).1
⊢ HPow.hPow (β := ℕ) (HAdd.hAdd (α := X q) 1 α) 2 = HMul.hMul (α := X q) 2 ω
have h₂ : (HPow.hPow (β := ℕ) (HAdd.hAdd (α := X q) 1 α) 2).2 = (HMul.hMul (α := X q) 2 ω).2
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
⊢ (HPow.hPow (β := ℕ) (HAdd.hAdd (α := X q) 1 α) 2).1 = (HMul.hMul (α := X q) 2 ω).1
have h₁ : HAdd.hAdd (α := ZMod q) 1 3 = HMul.hMul (α := ZMod q) 2 2
Tactic: simp [α, ω, sq]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
⊢ (HPow.hPow (β := ℕ) (HAdd.hAdd (α := X q) 1 α) 2).2 = (HMul.hMul (α := X q) 2 ω).2
have h₂ : HAdd.hAdd (α := ZMod q) 1 1 = 2
Tactic: simp [α, ω, sq]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q i : ℕ
⊢ HPow.hPow (α := X q) α (2 * i + 1) = HPow.hPow (α := X q) 3 i * α
have rw : HPow.hPow (α := X q) α (2 * i) * α = HPow.hPow (α := X q) 3 i * α
Tactic: rw [pow_succ, pow_mul, α_sq]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q i : ℕ
rw : HPow.hPow (α := X q) α (2 * i) * α = HPow.hPow (α := X q) 3 i * α
⊢ HPow.hPow (α := X q) α (2 * i + 1) = HPow.hPow (α := X q) 3 i * α
have rw₁ : HPow.hPow (α := X q) (β := ℕ) α 2 ^ i * α = HPow.hPow (α := X q) 3 i * α
Tactic: rw [pow_succ, pow_mul, α_sq]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q i : ℕ
rw : HPow.hPow (α := X q) α (2 * i) * α = HPow.hPow (α := X q) 3 i * α
rw₁ : HPow.hPow (α := X q) (β := ℕ) α 2 ^ i * α = HPow.hPow (α := X q) 3 i * α
⊢ HPow.hPow (α := X q) α (2 * i + 1) = HPow.hPow (α := X q) 3 i * α
have rw₂ : HPow.hPow (α := X q) 3 i * α = HPow.hPow (α := X q) 3 i * α
Tactic: rw [pow_succ, pow_mul, α_sq]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q x : ℕ
⊢ Eq (α := X q) (↑x : X q) 0 ↔ q ∣ x
have h : Eq (α := X q) (↑x : X q) 0 ↔ Eq (α := ZMod q) (↑x : ZMod q) 0
Tactic: convert! ZMod.natCast_eq_zero_iff _ _
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
⊢ HAdd.hAdd (α := X q) 1 α ^ q = HSub.hSub (α := X q) 1 α
have obtain : ∀ (k : ℕ) [inst : Fact (Nat.Prime (2 * k + 1))] (leg3 : legendreSym (2 * k + 1) 3 = -1),
  HAdd.hAdd (α := X (2 * k + 1)) 1 α ^ (2 * k + 1) = HSub.hSub (α := X (2 * k + 1)) 1 α
Tactic: obtain ⟨k, rfl⟩ := odd
Cleared hypotheses: q, inst, odd, leg3
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
k : ℕ
inst : Fact (Nat.Prime (2 * k + 1))
leg3 : legendreSym (2 * k + 1) 3 = -1
q : ℕ := 2 * k + 1
⊢ HAdd.hAdd (α := X (2 * k + 1)) 1 α ^ (2 * k + 1) = HSub.hSub (α := X (2 * k + 1)) 1 α
have this : HPow.hPow (α := ZMod q) 3 k = -1
Tactic: have : (3 ^ k : ZMod q) = -1 := by
    simpa [leg3, mul_add_div, eq_comm] using legendreSym.eq_pow (2 * k + 1) 3
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
k : ℕ
inst : Fact (Nat.Prime (2 * k + 1))
leg3 : legendreSym (2 * k + 1) 3 = -1
q : ℕ := 2 * k + 1
this : HPow.hPow (α := ZMod q) 3 k = -1
⊢ Eq (α := X q) 3 ((ZMod.castHom ⋯ (X q) : (a : ZMod q) → X q) 3)
have rw : Eq (α := X q) 3 3
Tactic: rw [map_ofNat]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
k : ℕ
inst : Fact (Nat.Prime (2 * k + 1))
leg3 : legendreSym (2 * k + 1) 3 = -1
q : ℕ := 2 * k + 1
this : HPow.hPow (α := ZMod q) 3 k = -1
⊢ HAdd.hAdd (α := X (2 * k + 1)) 1 α ^ (2 * k + 1) = HSub.hSub (α := X (2 * k + 1)) 1 α
have rw : HPow.hPow (α := X (2 * k + 1)) 1 (2 * k + 1) +
    HMul.hMul (α := X (2 * k + 1)) (-(ZMod.castHom sorry (X q) : (a : ZMod q) → X q) 1) α =
  HSub.hSub (α := X (2 * k + 1)) 1 α
Tactic: rw [add_pow_expChar, α_pow, show (3 : X q) = (3 : ZMod q) by rw [map_ofNat], ← map_pow, this,
    map_neg]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
⊢ HAdd.hAdd (α := X q) 1 α ^ (q + 1) = -2
have rw : HPow.hPow (α := X q) (β := ℕ) 1 2 - 3 = -2
Tactic: rw [pow_succ, one_add_α_pow_q odd leg3, mul_comm, ← _root_.sq_sub_sq, α_sq]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
⊢ HMul.hMul (α := X q) 2 ω ^ ((q + 1) / 2) = -2
have rw : HAdd.hAdd (α := X q) 1 α ^ (2 * ((q + 1) / 2)) = -2
Tactic: rw [← one_add_α_sq, ← pow_mul]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
⊢ 2 * ((q + 1) / 2) = q + 1
have H : 2 ∣ q + 1
Tactic: apply Nat.mul_div_cancel'
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
⊢ 2 ∣ q + 1
have H : Even (q + 1)
Tactic: rw [← even_iff_two_dvd]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
⊢ HAdd.hAdd (α := X q) 1 α ^ (2 * ((q + 1) / 2)) = -2
have this : 2 * ((q + 1) / 2) = q + 1
Tactic: have : 2 * ((q + 1) / 2) = q + 1 := by
    apply Nat.mul_div_cancel'
    rw [← even_iff_two_dvd]
    exact Odd.add_one odd
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
this : 2 * ((q + 1) / 2) = q + 1
⊢ HAdd.hAdd (α := X q) 1 α ^ (2 * ((q + 1) / 2)) = -2
have rw : HAdd.hAdd (α := X q) 1 α ^ (q + 1) = -2
Tactic: rw [this, one_add_α_pow_q_succ odd leg3]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
this : 2 * ((q + 1) / 2) = q + 1
rw : HAdd.hAdd (α := X q) 1 α ^ (q + 1) = -2
⊢ HAdd.hAdd (α := X q) 1 α ^ (2 * ((q + 1) / 2)) = -2
have rw₁ : Eq (α := X q) (-2) (-2)
Tactic: rw [this, one_add_α_pow_q_succ odd leg3]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
⊢ HPow.hPow (α := ZMod q) 2 ((q + 1) / 2) = 2
have obtain : ∀ (w : ℕ), q = 2 * w + 1 → HPow.hPow (α := ZMod q) 2 ((q + 1) / 2) = 2
Tactic: obtain ⟨_, _⟩ := odd
Cleared hypotheses: odd
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
w : ℕ
h : q = 2 * w + 1
⊢ HPow.hPow (α := ZMod q) 2 ((q + 1) / 2) = 2
have rw : HPow.hPow (α := ZMod q) 2 (q / 2) * 2 = 2
Tactic: rw [(by lia : (q + 1) / 2 = q / 2 + 1), pow_succ]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
w : ℕ
h : q = 2 * w + 1
⊢ HPow.hPow (α := ZMod q) 2 (q / 2) * 2 = 2
have leg : (↑(legendreSym q 2) : ZMod q) = HPow.hPow (α := ZMod q) (↑2 : ZMod q) (q / 2)
Tactic: have leg := legendreSym.eq_pow q 2
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
w : ℕ
h : q = 2 * w + 1
leg : (↑(legendreSym q 2) : ZMod q) = HPow.hPow (α := ZMod q) (↑2 : ZMod q) (q / 2)
⊢ Eq (α := ZMod q) 2 (↑2 : ZMod q)
have norm_cast : Eq (α := ZMod q) 2 2
Tactic: norm_cast
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
w : ℕ
h : q = 2 * w + 1
leg : (↑(legendreSym q 2) : ZMod q) = HPow.hPow (α := ZMod q) (↑2 : ZMod q) (q / 2)
⊢ HPow.hPow (α := ZMod q) 2 (q / 2) * 2 = 2
have this : Eq (α := ZMod q) 2 (↑2 : ZMod q)
Tactic: have : (2 : ZMod q) = ((2 : ℤ) : ZMod q) := by norm_cast
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
w : ℕ
h : q = 2 * w + 1
leg : (↑(legendreSym q 2) : ZMod q) = HPow.hPow (α := ZMod q) (↑2 : ZMod q) (q / 2)
this : Eq (α := ZMod q) 2 (↑2 : ZMod q)
⊢ HPow.hPow (α := ZMod q) 2 (q / 2) * 2 = 2
have rw : HMul.hMul (α := ZMod q) (↑1 : ZMod q) (↑2 : ZMod q) = (↑2 : ZMod q)
Tactic: rw [this, ← leg, leg2]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
⊢ HPow.hPow (α := X q) ω ((q + 1) / 2) = -1
have pow2 : HPow.hPow (α := ZMod q) 2 ((q + 1) / 2) = 2
Tactic: have pow2 : (2 : ZMod q) ^ ((q + 1) / 2) = 2 := by
    obtain ⟨_, _⟩ := odd
    rw [(by lia : (q + 1) / 2 = q / 2 + 1), pow_succ]
    have leg := legendreSym.eq_pow q 2
    have : (2 : ZMod q) = ((2 : ℤ) : ZMod q) := by norm_cast
    rw [this, ← leg, leg2]
    ring
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
pow2 : HPow.hPow (α := ZMod q) 2 ((q + 1) / 2) = 2
⊢ HPow.hPow (α := X q) ω ((q + 1) / 2) = -1
have this : HMul.hMul (α := X q) 2 ω ^ ((q + 1) / 2) = -2
Tactic: have := two_mul_ω_pow odd leg3
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
pow2 : HPow.hPow (α := ZMod q) 2 ((q + 1) / 2) = 2
this : HMul.hMul (α := X q) 2 ω ^ ((q + 1) / 2) = -2
⊢ HPow.hPow (α := X q) ω ((q + 1) / 2) = -1
have this₁ : HPow.hPow (α := X q) 2 ((q + 1) / 2) * HPow.hPow (α := X q) ω ((q + 1) / 2) = -2
Tactic: rw [mul_pow] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
pow2 : HPow.hPow (α := ZMod q) 2 ((q + 1) / 2) = 2
this : HPow.hPow (α := X q) 2 ((q + 1) / 2) * HPow.hPow (α := X q) ω ((q + 1) / 2) = -2
⊢ Eq (α := X q) 2 ((ZMod.castHom ⋯ (X q) : (a : ZMod q) → X q) 2)
have rw : Eq (α := X q) 2 2
Tactic: rw [map_ofNat]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
pow2 : HPow.hPow (α := ZMod q) 2 ((q + 1) / 2) = 2
this : HPow.hPow (α := X q) 2 ((q + 1) / 2) * HPow.hPow (α := X q) ω ((q + 1) / 2) = -2
⊢ HPow.hPow (α := X q) ω ((q + 1) / 2) = -1
have coe : Eq (α := X q) 2 ((ZMod.castHom sorry (X q) : (a : ZMod q) → X q) 2)
Tactic: have coe : (2 : X q) = (2 : ZMod q) := by rw [map_ofNat]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
pow2 : HPow.hPow (α := ZMod q) 2 ((q + 1) / 2) = 2
this : HPow.hPow (α := X q) 2 ((q + 1) / 2) * HPow.hPow (α := X q) ω ((q + 1) / 2) = -2
coe : Eq (α := X q) 2 ((ZMod.castHom ⋯ (X q) : (a : ZMod q) → X q) 2)
⊢ HPow.hPow (α := X q) ω ((q + 1) / 2) = -1
have this₁ : 2 * HPow.hPow (α := X q) ω ((q + 1) / 2) = HMul.hMul (α := X q) 2 (-1)
Tactic: rw [coe, ← map_pow, pow2, ← coe,
    (by ring : (-2 : X q) = 2 * -1)] at this
Cleared hypotheses: this, coe
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
pow2 : HPow.hPow (α := ZMod q) 2 ((q + 1) / 2) = 2
this : 2 * HPow.hPow (α := X q) ω ((q + 1) / 2) = HMul.hMul (α := X q) 2 (-1)
coe : Eq (α := X q) 2 ((ZMod.castHom ⋯ (X q) : (a : ZMod q) → X q) 2)
⊢ HPow.hPow (α := X q) ω ((q + 1) / 2) = -1
have refine : HMul.hMul (α := X q) 2 (↑((q + 1) / 2) : X q) = 1
Tactic: refine (IsUnit.of_mul_eq_one (M := X q) ↑((q + 1) / 2) ?_).mul_left_cancel this
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
pow2 : HPow.hPow (α := ZMod q) 2 ((q + 1) / 2) = 2
this : 2 * HPow.hPow (α := X q) ω ((q + 1) / 2) = HMul.hMul (α := X q) 2 (-1)
coe : Eq (α := X q) 2 ((ZMod.castHom ⋯ (X q) : (a : ZMod q) → X q) 2)
⊢ HMul.hMul (α := X q) 2 (↑((q + 1) / 2) : X q) = 1
have norm_cast : Eq (α := X q) (↑(2 * ((q + 1) / 2)) : X q) 1
Tactic: norm_cast
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
hq4 : 4 ∣ q + 1
⊢ HPow.hPow (α := X q) ω ((q + 1) / 2) * HPow.hPow (α := X q) ωb ((q + 1) / 4) = -HPow.hPow (α := X q) ωb ((q + 1) / 4)
have rw : -1 * HPow.hPow (α := X q) ωb ((q + 1) / 4) = -HPow.hPow (α := X q) ωb ((q + 1) / 4)
Tactic: rw [pow_ω odd leg3 leg2]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
hq4 : 4 ∣ q + 1
⊢ HPow.hPow (α := X q) ω ((q + 1) / 4) + HPow.hPow (α := X q) ωb ((q + 1) / 4) = 0
have this : HPow.hPow (α := X q) ω ((q + 1) / 2) * HPow.hPow (α := X q) ωb ((q + 1) / 4) = -HPow.hPow (α := X q) ωb ((q + 1) / 4)
Tactic: have : (ω : X q) ^ ((q + 1) / 2) * ωb ^ ((q + 1) / 4) = -ωb ^ ((q + 1) / 4) := by
    rw [pow_ω odd leg3 leg2]
    ring
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
hq4 : 4 ∣ q + 1
this :
  HPow.hPow (α := X q) ω ((q + 1) / 2) * HPow.hPow (α := X q) ωb ((q + 1) / 4) = -HPow.hPow (α := X q) ωb ((q + 1) / 4)
⊢ (q + 1) / 2 = (q + 1) / 4 + (q + 1) / 4
have rcases : ∀ (k : ℕ) (hk : q + 1 = 4 * k), (q + 1) / 2 = (q + 1) / 4 + (q + 1) / 4
Tactic: rcases hq4 with ⟨k, hk⟩
Cleared hypotheses: hq4
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
hq4 : 4 ∣ q + 1
this :
  HPow.hPow (α := X q) ω ((q + 1) / 2) * HPow.hPow (α := X q) ωb ((q + 1) / 4) = -HPow.hPow (α := X q) ωb ((q + 1) / 4)
⊢ HPow.hPow (α := X q) ω ((q + 1) / 4) + HPow.hPow (α := X q) ωb ((q + 1) / 4) = 0
have div4 : (q + 1) / 2 = (q + 1) / 4 + (q + 1) / 4
Tactic: have div4 : (q + 1) / 2 = (q + 1) / 4 + (q + 1) / 4 := by rcases hq4 with ⟨k, hk⟩; lia
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
hq4 : 4 ∣ q + 1
this :
  HPow.hPow (α := X q) ω ((q + 1) / 2) * HPow.hPow (α := X q) ωb ((q + 1) / 4) = -HPow.hPow (α := X q) ωb ((q + 1) / 4)
div4 : (q + 1) / 2 = (q + 1) / 4 + (q + 1) / 4
⊢ HPow.hPow (α := X q) ω ((q + 1) / 4) + HPow.hPow (α := X q) ωb ((q + 1) / 4) = 0
have this₁ : HPow.hPow (α := X q) ω ((q + 1) / 4) = -HPow.hPow (α := X q) ωb ((q + 1) / 4)
Tactic: rw [div4, pow_add, mul_assoc, ← mul_pow, ω_mul_ωb, one_pow, mul_one] at this
Cleared hypotheses: this, div4
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : Fact (Nat.Prime q)
odd : Odd q
leg3 : legendreSym q 3 = -1
leg2 : legendreSym q 2 = 1
hq4 : 4 ∣ q + 1
this : HPow.hPow (α := X q) ω ((q + 1) / 4) = -HPow.hPow (α := X q) ωb ((q + 1) / 4)
div4 : (q + 1) / 2 = (q + 1) / 4 + (q + 1) / 4
⊢ HPow.hPow (α := X q) ω ((q + 1) / 4) + HPow.hPow (α := X q) ωb ((q + 1) / 4) = 0
have rw : -HPow.hPow (α := X q) ωb ((q + 1) / 4) + HPow.hPow (α := X q) ωb ((q + 1) / 4) = 0
Tactic: rw [this]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : NeZero q
⊢ Fintype.card (ZMod q × ZMod q) = HPow.hPow (β := ℕ) q 2
have rw : Fintype.card (ZMod q) * Fintype.card (ZMod q) = HPow.hPow (β := ℕ) q 2
Tactic: rw [Fintype.card_prod, ZMod.card q, sq]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : NeZero q
rw : Fintype.card (ZMod q) * Fintype.card (ZMod q) = HPow.hPow (β := ℕ) q 2
⊢ Fintype.card (ZMod q × ZMod q) = HPow.hPow (β := ℕ) q 2
have rw₁ : q * q = HPow.hPow (β := ℕ) q 2
Tactic: rw [Fintype.card_prod, ZMod.card q, sq]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : NeZero q
rw : Fintype.card (ZMod q) * Fintype.card (ZMod q) = HPow.hPow (β := ℕ) q 2
rw₁ : q * q = HPow.hPow (β := ℕ) q 2
⊢ Fintype.card (ZMod q × ZMod q) = HPow.hPow (β := ℕ) q 2
have rw₂ : q * q = q * q
Tactic: rw [Fintype.card_prod, ZMod.card q, sq]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
q : ℕ
inst : NeZero q
w : 1 < q
⊢ Fintype.card (X q)ˣ < HPow.hPow (β := ℕ) q 2
have this : Fact (1 < q)
Tactic: have : Fact (1 < (q : ℕ)) := ⟨w⟩
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : NeZero q
w : 1 < q
this : Fact (1 < q)
⊢ Fintype.card (X q)ˣ < HPow.hPow (β := ℕ) q 2
have h : HPow.hPow (β := ℕ) q 2 = Fintype.card (X q)
Tactic: convert! card_units_lt (X q)
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
q : ℕ
inst : NeZero q
w : 1 < q
this : Fact (1 < q)
⊢ HPow.hPow (β := ℕ) q 2 = Fintype.card (X q)
have h : HPow.hPow (β := ℕ) q 2 = HPow.hPow (β := ℕ) q 2
Tactic: rw [card_eq]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p' : ℕ
⊢ 2 < q (p' + 2)
have refine_1 : 1 < p' + 2
Tactic: refine (minFac_prime (one_lt_mersenne.2 ?_).ne').two_le.lt_of_ne' ?_
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p' : ℕ
refine_1 : 1 < p' + 2
⊢ 2 < q (p' + 2)
have refine_2 : (mersenne (p' + 2)).minFac ≠ 2
Tactic: refine (minFac_prime (one_lt_mersenne.2 ?_).ne').two_le.lt_of_ne' ?_
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p' : ℕ
⊢ (mersenne (p' + 2)).minFac ≠ 2
have refine_2 : ¬2 ∣ 2 * HPow.hPow (α := ℕ) 2 (p' + 1) - 1
Tactic: rw [Ne, minFac_eq_two_iff, mersenne, Nat.pow_succ']
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p' : ℕ
h : Eq (α := ZMod (HPow.hPow (α := ℕ) 2 (p' + 2) - 1)) (sZMod (p' + 2) (p' + 2 - 2)) 0
⊢ ∃ (k : ℤ),
    HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 1)) =
      HMul.hMul (α := X (↑(q (p' + 2)) : ℕ)) (↑k : X (↑(q (p' + 2)) : ℕ))
            (↑(mersenne (p' + 2)) : X (↑(q (p' + 2)) : ℕ)) *
          HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p') -
        1
have h₁ : Eq (α := ZMod (HPow.hPow (α := ℕ) 2 (p' + 2) - 1)) (↑(s (p' + 2 - 2)) : ZMod (2 ^ (p' + 2) - 1)) 0
Tactic: rw [sZMod_eq_s p'] at h
Cleared hypotheses: h