awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p' : ℕ
h : Eq (α := ZMod (HPow.hPow (α := ℕ) 2 (p' + 2) - 1)) (↑(s (p' + 2 - 2)) : ZMod (2 ^ (p' + 2) - 1)) 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 : HPow.hPow (α := ℤ) 2 (p' + 2) - 1 ∣ s p'
Tactic: replace h : 2 ^ (p' + 2) - 1 ∣ s p' := by simpa [ZMod.intCast_zmod_eq_zero_iff_dvd] using h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p' : ℕ
h : HPow.hPow (α := ℤ) 2 (p' + 2) - 1 ∣ s p'
⊢ ∃ (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 obtain : ∀ (k : ℤ) (h : s p' = (HPow.hPow (α := ℤ) 2 (p' + 2) - 1) * k),
  ∃ (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
Tactic: obtain ⟨k, h⟩ := h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p' : ℕ
k : ℤ
h : s p' = (HPow.hPow (α := ℤ) 2 (p' + 2) - 1) * k
⊢ ∃ (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 : 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
Tactic: use k
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p' : ℕ
k : ℤ
h : s p' = (HPow.hPow (α := ℤ) 2 (p' + 2) - 1) * k
⊢ 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 := X (↑(q (p' + 2)) : ℕ)) ((fun (n : ℤ) ↦ ((↑n : X (↑(q (p' + 2)) : ℕ)) : X (↑(q (p' + 2)) : ℕ))) (s p'))
  ((fun (n : ℤ) ↦ ((↑n : X (↑(q (p' + 2)) : ℕ)) : X (↑(q (p' + 2)) : ℕ))) ((HPow.hPow (α := ℤ) 2 (p' + 2) - 1) * k))
Tactic: replace h := congr_arg (fun n : ℤ => (n : X (q (p' + 2)))) h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p' : ℕ
k : ℤ
h :
  Eq (α := X (↑(q (p' + 2)) : ℕ)) (↑(s p') : X (↑(q (p' + 2)) : ℕ))
    (↑((HPow.hPow (α := ℤ) 2 (p' + 2) - 1) * k) : X (↑(q (p' + 2)) : ℕ))
⊢ 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 : HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p') +
    HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ωb (HPow.hPow (α := ℕ) 2 p') =
  (↑((HPow.hPow (α := ℤ) 2 (p' + 2) - 1) * k) : X (↑(q (p' + 2)) : ℕ))
Tactic: rw [closed_form] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p' : ℕ
k : ℤ
h :
  HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p') +
      HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ωb (HPow.hPow (α := ℕ) 2 p') =
    (↑((HPow.hPow (α := ℤ) 2 (p' + 2) - 1) * k) : X (↑(q (p' + 2)) : ℕ))
⊢ 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 := X (↑(q (p' + 2)) : ℕ))
  ((fun (x : X (↑(q (p' + 2)) : ℕ)) ↦ HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p') * x)
    (HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p') +
      HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ωb (HPow.hPow (α := ℕ) 2 p')))
  ((fun (x : X (↑(q (p' + 2)) : ℕ)) ↦ HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p') * x)
    (↑((HPow.hPow (α := ℤ) 2 (p' + 2) - 1) * k) : X (↑(q (p' + 2)) : ℕ)))
Tactic: replace h := congr_arg (fun x => ω ^ 2 ^ p' * x) h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p' : ℕ
k : ℤ
h :
  HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p') *
      (HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p') +
        HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ωb (HPow.hPow (α := ℕ) 2 p')) =
    HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p') *
      (↑((HPow.hPow (α := ℤ) 2 (p' + 2) - 1) * k) : X (↑(q (p' + 2)) : ℕ))
⊢ 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 t : HPow.hPow (α := ℕ) 2 p' + HPow.hPow (α := ℕ) 2 p' = HPow.hPow (α := ℕ) 2 (p' + 1)
Tactic: have t : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1) := by ring
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p' : ℕ
k : ℤ
h :
  HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p') *
      (HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p') +
        HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ωb (HPow.hPow (α := ℕ) 2 p')) =
    HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p') *
      (↑((HPow.hPow (α := ℤ) 2 (p' + 2) - 1) * k) : X (↑(q (p' + 2)) : ℕ))
t : HPow.hPow (α := ℕ) 2 p' + HPow.hPow (α := ℕ) 2 p' = HPow.hPow (α := ℕ) 2 (p' + 1)
⊢ 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 : HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 1)) + 1 =
  HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p') *
    (↑((HPow.hPow (α := ℤ) 2 (p' + 2) - 1) * k) : X (↑(q (p' + 2)) : ℕ))
Tactic: rw [mul_add, ← pow_add ω, t, ← mul_pow ω ωb (2 ^ p'), ω_mul_ωb, one_pow] at h
Cleared hypotheses: h, t
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p' : ℕ
k : ℤ
h :
  HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 1)) + 1 =
    HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p') *
      (↑((HPow.hPow (α := ℤ) 2 (p' + 2) - 1) * k) : X (↑(q (p' + 2)) : ℕ))
t : HPow.hPow (α := ℕ) 2 p' + HPow.hPow (α := ℕ) 2 p' = HPow.hPow (α := ℕ) 2 (p' + 1)
⊢ 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 : HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 1)) + 1 =
  HMul.hMul (α := X (↑(q (p' + 2)) : ℕ)) (↑(HPow.hPow (α := ℤ) 2 (p' + 2) - 1) : X (↑(q (p' + 2)) : ℕ))
      (↑k : X (↑(q (p' + 2)) : ℕ)) *
    HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p')
Tactic: rw [mul_comm, coe_mul] at h
Cleared hypotheses: h, t
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p' : ℕ
k : ℤ
h :
  HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 1)) + 1 =
    HMul.hMul (α := X (↑(q (p' + 2)) : ℕ)) (↑(HPow.hPow (α := ℤ) 2 (p' + 2) - 1) : X (↑(q (p' + 2)) : ℕ))
        (↑k : X (↑(q (p' + 2)) : ℕ)) *
      HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p')
t : HPow.hPow (α := ℕ) 2 p' + HPow.hPow (α := ℕ) 2 p' = HPow.hPow (α := ℕ) 2 (p' + 1)
⊢ 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 : HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 1)) + 1 =
  HMul.hMul (α := X (↑(q (p' + 2)) : ℕ)) (↑k : X (↑(q (p' + 2)) : ℕ))
      (↑(HPow.hPow (α := ℤ) 2 (p' + 2) - 1) : X (↑(q (p' + 2)) : ℕ)) *
    HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p')
Tactic: rw [mul_comm _ (k : X (q (p' + 2)))] at h
Cleared hypotheses: h, t
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p' : ℕ
k : ℤ
h :
  HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 1)) + 1 =
    HMul.hMul (α := X (↑(q (p' + 2)) : ℕ)) (↑k : X (↑(q (p' + 2)) : ℕ))
        (↑(HPow.hPow (α := ℤ) 2 (p' + 2) - 1) : X (↑(q (p' + 2)) : ℕ)) *
      HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p')
t : HPow.hPow (α := ℕ) 2 p' + HPow.hPow (α := ℕ) 2 p' = HPow.hPow (α := ℕ) 2 (p' + 1)
⊢ 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 : 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))
        (↑(HPow.hPow (α := ℤ) 2 (p' + 2) - 1) : X (↑(q (p' + 2)) : ℕ)) *
      HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p') -
    1
Tactic: replace h := eq_sub_of_add_eq h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p' : ℕ
k : ℤ
t : HPow.hPow (α := ℕ) 2 p' + HPow.hPow (α := ℕ) 2 p' = HPow.hPow (α := ℕ) 2 (p' + 1)
h :
  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))
          (↑(HPow.hPow (α := ℤ) 2 (p' + 2) - 1) : X (↑(q (p' + 2)) : ℕ)) *
        HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p') -
      1
⊢ 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 this : 1 ≤ HPow.hPow (α := ℕ) 2 (p' + 2)
Tactic: have : 1 ≤ 2 ^ (p' + 2) := Nat.one_le_pow _ _ (by decide)
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p : ℕ
⊢ Eq (α := X (↑(q p) : ℕ)) (↑(mersenne p) : X (↑(q p) : ℕ)) 0
have h₁ : Eq (α := ZMod (↑(q p) : ℕ)) (↑(mersenne p) : X (↑(q p) : ℕ)).1 0.1
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p : ℕ
h₁ : Eq (α := ZMod (↑(q p) : ℕ)) (↑(mersenne p) : X (↑(q p) : ℕ)).1 0.1
⊢ Eq (α := X (↑(q p) : ℕ)) (↑(mersenne p) : X (↑(q p) : ℕ)) 0
have h₂ : Eq (α := ZMod (↑(q p) : ℕ)) (↑(mersenne p) : X (↑(q p) : ℕ)).2 0.2
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p : ℕ
⊢ Eq (α := ZMod (↑(q p) : ℕ)) (↑(mersenne p) : X (↑(q p) : ℕ)).1 0.1
have h₁ : (HPow.hPow (α := ℕ) 2 p - 1).minFac ∣ HPow.hPow (α := ℕ) 2 p - 1
Tactic: simp [mersenne, q, ZMod.natCast_eq_zero_iff, -pow_pos]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p' : ℕ
h : Eq (α := ZMod (HPow.hPow (α := ℕ) 2 (p' + 2) - 1)) (lucasLehmerResidue (p' + 2)) 0
⊢ HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 1)) = -1
have obtain : ∀ (k : ℤ)
  (w :
    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),
  HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 1)) = -1
Tactic: obtain ⟨k, w⟩ := ω_pow_formula p' h
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p' : ℕ
h : Eq (α := ZMod (HPow.hPow (α := ℕ) 2 (p' + 2) - 1)) (lucasLehmerResidue (p' + 2)) 0
k : ℤ
w :
  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
⊢ HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 1)) = -1
have w₁ : 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)) 0 *
      HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 p') -
    1
Tactic: rw [mersenne_coe_X] at w
Cleared hypotheses: w
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p' : ℕ
h : Eq (α := ZMod (HPow.hPow (α := ℕ) 2 (p' + 2) - 1)) (lucasLehmerResidue (p' + 2)) 0
⊢ HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 2)) =
    HPow.hPow (β := ℕ) (HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 1))) 2
have rw : HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 2)) =
  HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 1) * 2)
Tactic: rw [← pow_mul, ← Nat.pow_succ]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p' : ℕ
h : Eq (α := ZMod (HPow.hPow (α := ℕ) 2 (p' + 2) - 1)) (lucasLehmerResidue (p' + 2)) 0
rw :
  HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 2)) =
    HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 1) * 2)
⊢ HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 2)) =
    HPow.hPow (β := ℕ) (HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 1))) 2
have rw₁ : HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 2)) =
  HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 1).succ)
Tactic: rw [← pow_mul, ← Nat.pow_succ]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p' : ℕ
h : Eq (α := ZMod (HPow.hPow (α := ℕ) 2 (p' + 2) - 1)) (lucasLehmerResidue (p' + 2)) 0
⊢ HPow.hPow (β := ℕ) (HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) ω (HPow.hPow (α := ℕ) 2 (p' + 1))) 2 =
    HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) (β := ℕ) (-1) 2
have rw : HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) (β := ℕ) (-1) 2 = HPow.hPow (α := X (↑(q (p' + 2)) : ℕ)) (β := ℕ) (-1) 2
Tactic: rw [ω_pow_eq_neg_one p' h]
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p' : ℕ
h : Eq (α := ZMod (HPow.hPow (α := ℕ) 2 (p' + 2) - 1)) (lucasLehmerResidue (p' + 2)) 0
⊢ orderOf (G := (X (↑(q (p' + 2)) : ℕ))ˣ) (ωUnit (p' + 2)) = HPow.hPow (α := ℕ) 2 (p' + 2)
have pp : Nat.Prime 2
Tactic: apply Nat.eq_prime_pow_of_dvd_least_prime_pow
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p' : ℕ
h : Eq (α := ZMod (HPow.hPow (α := ℕ) 2 (p' + 2) - 1)) (lucasLehmerResidue (p' + 2)) 0
pp : Nat.Prime 2
⊢ orderOf (G := (X (↑(q (p' + 2)) : ℕ))ˣ) (ωUnit (p' + 2)) = HPow.hPow (α := ℕ) 2 (p' + 2)
have h₁ : ¬orderOf (G := (X (↑(q (p' + 2)) : ℕ))ˣ) (ωUnit (p' + 2)) ∣ HPow.hPow (α := ℕ) 2 (p' + 1)
Tactic: apply Nat.eq_prime_pow_of_dvd_least_prime_pow
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p' : ℕ
h : Eq (α := ZMod (HPow.hPow (α := ℕ) 2 (p' + 2) - 1)) (lucasLehmerResidue (p' + 2)) 0
pp : Nat.Prime 2
h₁ : ¬orderOf (G := (X (↑(q (p' + 2)) : ℕ))ˣ) (ωUnit (p' + 2)) ∣ HPow.hPow (α := ℕ) 2 (p' + 1)
⊢ orderOf (G := (X (↑(q (p' + 2)) : ℕ))ˣ) (ωUnit (p' + 2)) = HPow.hPow (α := ℕ) 2 (p' + 2)
have h₂ : orderOf (G := (X (↑(q (p' + 2)) : ℕ))ˣ) (ωUnit (p' + 2)) ∣ HPow.hPow (α := ℕ) 2 (p' + 1 + 1)
Tactic: apply Nat.eq_prime_pow_of_dvd_least_prime_pow
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p' : ℕ
h : Eq (α := ZMod (HPow.hPow (α := ℕ) 2 (p' + 2) - 1)) (lucasLehmerResidue (p' + 2)) 0
o : orderOf (G := (X (↑(q (p' + 2)) : ℕ))ˣ) (ωUnit (p' + 2)) ∣ HPow.hPow (α := ℕ) 2 (p' + 1)
⊢ False
have ω_pow : Eq (α := X (↑(q (p' + 2)) : ℕ))
  ((Units.coeHom (X (↑(q (p' + 2)) : ℕ)) : (a : (X (↑(q (p' + 2)) : ℕ))ˣ) → X (↑(q (p' + 2)) : ℕ))
      (HPow.hPow (α := (X (↑(q (p' + 2)) : ℕ))ˣ) (ωUnit (p' + 2)) (HPow.hPow (α := ℕ) 2 (p' + 1))) :
    X (↑(q (p' + 2)) : ℕ))
  ((Units.coeHom (X (↑(q (p' + 2)) : ℕ)) : (a : (X (↑(q (p' + 2)) : ℕ))ˣ) → X (↑(q (p' + 2)) : ℕ)) 1 :
    X (↑(q (p' + 2)) : ℕ))
Tactic: have ω_pow :=
      congr_arg (Units.coeHom (X (q (p' + 2))) : Units (X (q (p' + 2))) → X (q (p' + 2))) <|
        orderOf_dvd_iff_pow_eq_one.1 o
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p' : ℕ
h : Eq (α := ZMod (HPow.hPow (α := ℕ) 2 (p' + 2) - 1)) (lucasLehmerResidue (p' + 2)) 0
o : orderOf (G := (X (↑(q (p' + 2)) : ℕ))ˣ) (ωUnit (p' + 2)) ∣ HPow.hPow (α := ℕ) 2 (p' + 1)
ω_pow :
  Eq (α := X (↑(q (p' + 2)) : ℕ))
    ((Units.coeHom (X (↑(q (p' + 2)) : ℕ)) : (a : (X (↑(q (p' + 2)) : ℕ))ˣ) → X (↑(q (p' + 2)) : ℕ))
        (HPow.hPow (α := (X (↑(q (p' + 2)) : ℕ))ˣ) (ωUnit (p' + 2)) (HPow.hPow (α := ℕ) 2 (p' + 1))) :
      X (↑(q (p' + 2)) : ℕ))
    ((Units.coeHom (X (↑(q (p' + 2)) : ℕ)) : (a : (X (↑(q (p' + 2)) : ℕ))ˣ) → X (↑(q (p' + 2)) : ℕ)) 1 :
      X (↑(q (p' + 2)) : ℕ))
⊢ False
have h₁ : Eq (α := ZMod (↑(q (p' + 2)) : ℕ)) 1 (-1)
Tactic: have h : (1 : ZMod (q (p' + 2))) = -1 :=
      congr_arg Prod.fst (ω_pow.symm.trans (ω_pow_eq_neg_one p' h))
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p' : ℕ
h_1 : Eq (α := ZMod (HPow.hPow (α := ℕ) 2 (p' + 2) - 1)) (lucasLehmerResidue (p' + 2)) 0
o : orderOf (G := (X (↑(q (p' + 2)) : ℕ))ˣ) (ωUnit (p' + 2)) ∣ HPow.hPow (α := ℕ) 2 (p' + 1)
ω_pow :
  Eq (α := X (↑(q (p' + 2)) : ℕ))
    ((Units.coeHom (X (↑(q (p' + 2)) : ℕ)) : (a : (X (↑(q (p' + 2)) : ℕ))ˣ) → X (↑(q (p' + 2)) : ℕ))
        (HPow.hPow (α := (X (↑(q (p' + 2)) : ℕ))ˣ) (ωUnit (p' + 2)) (HPow.hPow (α := ℕ) 2 (p' + 1))) :
      X (↑(q (p' + 2)) : ℕ))
    ((Units.coeHom (X (↑(q (p' + 2)) : ℕ)) : (a : (X (↑(q (p' + 2)) : ℕ))ˣ) → X (↑(q (p' + 2)) : ℕ)) 1 :
      X (↑(q (p' + 2)) : ℕ))
h : Eq (α := ZMod (↑(q (p' + 2)) : ℕ)) 1 (-1)
⊢ False
have this : Fact (2 < (↑(q (p' + 2)) : ℕ))
Tactic: haveI : Fact (2 < (q (p' + 2) : ℕ)) := ⟨two_lt_q _⟩
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p' : ℕ
h : Eq (α := ZMod (HPow.hPow (α := ℕ) 2 (p' + 2) - 1)) (lucasLehmerResidue (p' + 2)) 0
⊢ orderOf (G := (X (↑(q (p' + 2)) : ℕ))ˣ) (ωUnit (p' + 2)) ∣ HPow.hPow (α := ℕ) 2 (p' + 1 + 1)
have h₂ : HPow.hPow (α := (X (↑(q (p' + 2)) : ℕ))ˣ) (ωUnit (p' + 2)) (HPow.hPow (α := ℕ) 2 (p' + 1 + 1)) = 1
Tactic: apply orderOf_dvd_iff_pow_eq_one.2
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p' : ℕ
h : Eq (α := ZMod (HPow.hPow (α := ℕ) 2 (p' + 2) - 1)) (lucasLehmerResidue (p' + 2)) 0
⊢ HPow.hPow (α := (X (↑(q (p' + 2)) : ℕ))ˣ) (ωUnit (p' + 2)) (HPow.hPow (α := ℕ) 2 (p' + 1 + 1)) = 1
have h₂ : (↑(HPow.hPow (α := (X (↑(q (p' + 2)) : ℕ))ˣ) (ωUnit (p' + 2)) (HPow.hPow (α := ℕ) 2 (p' + 1 + 1))) :
    X (↑(q (p' + 2)) : ℕ)) =
  (↑1 : X (↑(q (p' + 2)) : ℕ))
Tactic: apply Units.ext
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p : ℕ
w : 1 < p
⊢ LucasLehmerTest p → Nat.Prime (mersenne p)
have hp' : let p' := p - 2;
p' = p - 2
Tactic: set p' := p - 2 with hp'
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p : ℕ
w : 1 < p
p' : ℕ := p - 2
hp' : p' = p - 2
⊢ LucasLehmerTest p → Nat.Prime (mersenne p)
have clear_value : ∀ (p' : ℕ) (hp' : p' = p - 2), LucasLehmerTest p → Nat.Prime (mersenne p)
Tactic: clear_value p'
Cleared hypotheses: p', hp'
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p : ℕ
w : 1 < p
p' : ℕ
hp' : p' = p - 2
⊢ LucasLehmerTest p → Nat.Prime (mersenne p)
have obtain : ∀ (w : 1 < p' + 2) (hp' : p' = p' + 2 - 2), LucasLehmerTest (p' + 2) → Nat.Prime (mersenne (p' + 2))
Tactic: obtain rfl : p = p' + 2 := by lia
Cleared hypotheses: p, w, hp'
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p' : ℕ
w_1 : 1 < p' + 2
hp' : p' = p' + 2 - 2
w : 1 < p' + 2
⊢ LucasLehmerTest (p' + 2) → Nat.Prime (mersenne (p' + 2))
have contrapose : ¬Nat.Prime (mersenne (p' + 2)) → ¬LucasLehmerTest (p' + 2)
Tactic: contrapose
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p' : ℕ
w_1 : 1 < p' + 2
hp' : p' = p' + 2 - 2
w : 1 < p' + 2
a : ¬Nat.Prime (mersenne (p' + 2))
t : LucasLehmerTest (p' + 2)
⊢ False
have h₁ : HPow.hPow (α := ℕ) 2 (p' + 2) < HPow.hPow (β := ℕ) (↑(q (p' + 2)) : ℕ) 2
Tactic: have h₁ := order_ineq p' t
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p' : ℕ
w_1 : 1 < p' + 2
hp' : p' = p' + 2 - 2
w : 1 < p' + 2
a : ¬Nat.Prime (mersenne (p' + 2))
t : LucasLehmerTest (p' + 2)
h₁ : HPow.hPow (α := ℕ) 2 (p' + 2) < HPow.hPow (β := ℕ) (↑(q (p' + 2)) : ℕ) 2
⊢ False
have h₂ : HPow.hPow (β := ℕ) (mersenne (p' + 2)).minFac 2 ≤ mersenne (p' + 2)
Tactic: have h₂ := Nat.minFac_sq_le_self (mersenne_pos.2 (Nat.lt_of_succ_lt w)) a
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p' : ℕ
w_1 : 1 < p' + 2
hp' : p' = p' + 2 - 2
w : 1 < p' + 2
a : ¬Nat.Prime (mersenne (p' + 2))
t : LucasLehmerTest (p' + 2)
h₁ : HPow.hPow (α := ℕ) 2 (p' + 2) < HPow.hPow (β := ℕ) (↑(q (p' + 2)) : ℕ) 2
h₂ : HPow.hPow (β := ℕ) (mersenne (p' + 2)).minFac 2 ≤ mersenne (p' + 2)
⊢ False
have h : HPow.hPow (α := ℕ) 2 (p' + 2) < mersenne (p' + 2)
Tactic: have h := lt_of_lt_of_le h₁ h₂
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p : ℕ
w : 3 ≤ p
hp : Nat.Prime (mersenne p)
⊢ LucasLehmerTest p
have this : Fact (Nat.Prime (mersenne p))
Tactic: have : Fact (mersenne p).Prime := ⟨‹_›⟩
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (hypothesis)
Goal:
p : ℕ
w : 3 ≤ p
hp : Nat.Prime (mersenne p)
this : Fact (Nat.Prime (mersenne p))
⊢ LucasLehmerTest p
have hp' : let p' := p - 2;
p' = p - 2
Tactic: set p' := p - 2 with hp'
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p : ℕ
w : 3 ≤ p
hp : Nat.Prime (mersenne p)
this : Fact (Nat.Prime (mersenne p))
p' : ℕ := p - 2
hp' : p' = p - 2
⊢ LucasLehmerTest p
have clear_value : ∀ (p' : ℕ) (hp' : p' = p - 2), LucasLehmerTest p
Tactic: clear_value p'
Cleared hypotheses: p', hp'
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p : ℕ
w : 3 ≤ p
hp : Nat.Prime (mersenne p)
this : Fact (Nat.Prime (mersenne p))
p' : ℕ
hp' : p' = p - 2
⊢ LucasLehmerTest p
have obtain : ∀ (w : 3 ≤ p' + 2) (hp : Nat.Prime (mersenne (p' + 2))) (this : Fact (Nat.Prime (mersenne (p' + 2))))
  (hp' : p' = p' + 2 - 2), LucasLehmerTest (p' + 2)
Tactic: obtain rfl : p = p' + 2 := by lia
Cleared hypotheses: p, w, hp, this, hp'
-- Lean 4 proof step from Mathlib.NumberTheory.LucasLehmer (goal)
Goal:
p' : ℕ
w : 3 ≤ p' + 2
hp : Nat.Prime (mersenne (p' + 2))
this : Fact (Nat.Prime (mersenne (p' + 2)))
hp' : p' = p' + 2 - 2
⊢ Eq (α := ZMod (HPow.hPow (α := ℕ) 2 (p' + 2) - 1)) (sZMod (p' + 2) (p' + 2 - 2)) 0
have rw : (HPow.hPow (α := X (HPow.hPow (α := ℕ) 2 (p' + 2) - 1)) X.ω (HPow.hPow (α := ℕ) 2 p') +
      HPow.hPow (α := X (HPow.hPow (α := ℕ) 2 (p' + 2) - 1)) X.ωb (HPow.hPow (α := ℕ) 2 p')).1 =
  0
Tactic: rw [sZMod_eq_s p', ← X.fst_intCast, X.closed_form, add_tsub_cancel_right]