awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (hypothesis)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
⊢ (A * B).ProbablePrime b
have ha₂ : 2 ∣ b ^ p + b
Tactic: have ha₂ : 2 ∣ b ^ p + b := by
    rw [← even_iff_two_dvd, Nat.even_add, Nat.even_pow' p_prime.ne_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (hypothesis)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
⊢ p ∣ b ^ (p - 1) - 1
have this : ¬p ∣ b
Tactic: have : ¬p ∣ b := mt (fun h : p ∣ b => dvd_mul_of_dvd_left h _) not_dvd
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (hypothesis)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
this : ¬p ∣ b
⊢ p ∣ b ^ (p - 1) - 1
have this₁ : p.Coprime b
Tactic: have : p.Coprime b := Or.resolve_right (Nat.coprime_or_dvd_of_prime p_prime b) this
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (hypothesis)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
this_1 : ¬p ∣ b
this : p.Coprime b
⊢ p ∣ b ^ (p - 1) - 1
have this₁ : IsCoprime (R := ℤ) (↑b : ℤ) (↑p : ℤ)
Tactic: have : IsCoprime (b : ℤ) ↑p := this.symm.isCoprime
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (hypothesis)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
this_1 : ¬p ∣ b
this_2 : p.Coprime b
this : IsCoprime (R := ℤ) (↑b : ℤ) (↑p : ℤ)
⊢ p ∣ b ^ (p - 1) - 1
have this₁ : HPow.hPow (α := ℤ) (↑b : ℤ) (p - 1) ≡ 1 [ZMOD (↑p : ℤ)]
Tactic: have : ↑b ^ (p - 1) ≡ 1 [ZMOD ↑p] := Int.ModEq.pow_card_sub_one_eq_one p_prime this
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (hypothesis)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
⊢ (A * B).ProbablePrime b
have ha₃ : p ∣ b ^ (p - 1) - 1
Tactic: have ha₃ : p ∣ b ^ (p - 1) - 1 := by
    have : ¬p ∣ b := mt (fun h : p ∣ b => dvd_mul_of_dvd_left h _) not_dvd
    have : p.Coprime b := Or.resolve_right (Nat.coprime_or_dvd_of_prime p_prime b) this
    have : IsCoprime (b : ℤ) ↑p := this.symm.isCoprime
    have : ↑b ^ (p - 1) ≡ 1 [ZMOD ↑p] := Int.ModEq.pow_card_sub_one_eq_one p_prime this
    have : ↑p ∣ ↑b ^ (p - 1) - ↑1 := mod_cast Int.ModEq.dvd (Int.ModEq.symm this)
    exact mod_cast this
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (goal)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
⊢ HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (p - 1) - 1
have obtain : ∀ (k : ℕ) (hk : p = 2 * k + 1), HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (p - 1) - 1
Tactic: obtain ⟨k, hk⟩ := p_odd
Cleared hypotheses: p_odd
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (hypothesis)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
k : ℕ
hk : p = 2 * k + 1
⊢ HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (p - 1) - 1
have this : 2 ∣ p - 1
Tactic: have : 2 ∣ p - 1 := ⟨k, by simp [hk]⟩
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (goal)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
k : ℕ
hk : p = 2 * k + 1
this : 2 ∣ p - 1
⊢ HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (p - 1) - 1
have obtain : ∀ (c : ℕ) (hc : p - 1 = 2 * c), HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (p - 1) - 1
Tactic: obtain ⟨c, hc⟩ := this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (hypothesis)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
k : ℕ
hk : p = 2 * k + 1
c : ℕ
hc : p - 1 = 2 * c
⊢ HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (p - 1) - 1
have this : HPow.hPow (β := ℕ) b 2 - 1 ∣ HPow.hPow (β := ℕ) b 2 ^ c - 1
Tactic: have : b ^ 2 - 1 ∣ (b ^ 2) ^ c - 1 := by
      simpa only [one_pow] using Nat.sub_dvd_pow_sub_pow _ 1 c
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (goal)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
k : ℕ
hk : p = 2 * k + 1
c : ℕ
hc : p - 1 = 2 * c
this : HPow.hPow (β := ℕ) b 2 - 1 ∣ HPow.hPow (β := ℕ) b 2 ^ c - 1
⊢ HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * c) - 1
have rwa : ∀ (this : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * c) - 1), HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * c) - 1
Tactic: rwa [← pow_mul] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (hypothesis)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
k : ℕ
hk : p = 2 * k + 1
c : ℕ
hc : p - 1 = 2 * c
this : HPow.hPow (β := ℕ) b 2 - 1 ∣ HPow.hPow (β := ℕ) b 2 ^ c - 1
⊢ HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (p - 1) - 1
have this₁ : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * c) - 1
Tactic: have : b ^ 2 - 1 ∣ b ^ (2 * c) - 1 := by rwa [← pow_mul] at this
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (goal)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
k : ℕ
hk : p = 2 * k + 1
c : ℕ
hc : p - 1 = 2 * c
this_1 : HPow.hPow (β := ℕ) b 2 - 1 ∣ HPow.hPow (β := ℕ) b 2 ^ c - 1
this : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * c) - 1
⊢ HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (p - 1) - 1
have rwa : ∀ (this : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (p - 1) - 1), HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (p - 1) - 1
Tactic: rwa [← hc] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (hypothesis)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
⊢ (A * B).ProbablePrime b
have ha₄ : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (p - 1) - 1
Tactic: have ha₄ : b ^ 2 - 1 ∣ b ^ (p - 1) - 1 := by
    obtain ⟨k, hk⟩ := p_odd
    have : 2 ∣ p - 1 := ⟨k, by simp [hk]⟩
    obtain ⟨c, hc⟩ := this
    have : b ^ 2 - 1 ∣ (b ^ 2) ^ c - 1 := by
      simpa only [one_pow] using Nat.sub_dvd_pow_sub_pow _ 1 c
    have : b ^ 2 - 1 ∣ b ^ (2 * c) - 1 := by rwa [← pow_mul] at this
    rwa [← hc] at this
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (goal)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
ha₄ : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (p - 1) - 1
⊢ 2 * p * (HPow.hPow (β := ℕ) b 2 - 1) ∣ (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1)
have suffices : 2 * p * (HPow.hPow (β := ℕ) b 2 - 1) ∣ b * (b ^ (p - 1) - 1) * (b ^ p + b)
Tactic: suffices q : 2 * p * (b ^ 2 - 1) ∣ b * (b ^ (p - 1) - 1) * (b ^ p + b) by rwa [ha₁]
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (hypothesis)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
ha₄ : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (p - 1) - 1
⊢ ¬p ∣ HPow.hPow (β := ℕ) b 2 - 1
have not_dvd₁ : ¬p ∣ (HPow.hPow (β := ℕ) b 2 - 1) * b
Tactic: rw [mul_comm] at not_dvd
Cleared hypotheses: not_dvd, A, B, hA, hi_A, hi_B, hi_b, hi_bsquared, hi_bpowtwop, hi_bpowpsubone, p_odd, AB_not_prime, AB_id, hd, ha₁, ha₂, ha₃, ha₄
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (hypothesis)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
ha₄ : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (p - 1) - 1
⊢ 2 * p * (HPow.hPow (β := ℕ) b 2 - 1) ∣ b * (b ^ (p - 1) - 1) * (b ^ p + b)
have q₁ : p.Coprime (HPow.hPow (β := ℕ) b 2 - 1)
Tactic: have q₁ : Nat.Coprime p (b ^ 2 - 1) :=
      haveI q₂ : ¬p ∣ b ^ 2 - 1 := by
        rw [mul_comm] at not_dvd
        exact mt (fun h : p ∣ b ^ 2 - 1 => dvd_mul_of_dvd_left h _) not_dvd
      (Nat.Prime.coprime_iff_not_dvd p_prime).mpr q₂
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (hypothesis)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
ha₄ : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (p - 1) - 1
q₁ : p.Coprime (HPow.hPow (β := ℕ) b 2 - 1)
⊢ 2 * p * (HPow.hPow (β := ℕ) b 2 - 1) ∣ b * (b ^ (p - 1) - 1) * (b ^ p + b)
have q₂ : p * (HPow.hPow (β := ℕ) b 2 - 1) ∣ b ^ (p - 1) - 1
Tactic: have q₂ : p * (b ^ 2 - 1) ∣ b ^ (p - 1) - 1 := Nat.Coprime.mul_dvd_of_dvd_of_dvd q₁ ha₃ ha₄
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (hypothesis)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
ha₄ : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (p - 1) - 1
q₁ : p.Coprime (HPow.hPow (β := ℕ) b 2 - 1)
q₂ : p * (HPow.hPow (β := ℕ) b 2 - 1) ∣ b ^ (p - 1) - 1
⊢ 2 * p * (HPow.hPow (β := ℕ) b 2 - 1) ∣ b * (b ^ (p - 1) - 1) * (b ^ p + b)
have q₃ : p * (HPow.hPow (β := ℕ) b 2 - 1) * 2 ∣ (b ^ (p - 1) - 1) * (b ^ p + b)
Tactic: have q₃ : p * (b ^ 2 - 1) * 2 ∣ (b ^ (p - 1) - 1) * (b ^ p + b) := mul_dvd_mul q₂ ha₂
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (hypothesis)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
ha₄ : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (p - 1) - 1
q₁ : p.Coprime (HPow.hPow (β := ℕ) b 2 - 1)
q₂ : p * (HPow.hPow (β := ℕ) b 2 - 1) ∣ b ^ (p - 1) - 1
q₃ : p * (HPow.hPow (β := ℕ) b 2 - 1) * 2 ∣ (b ^ (p - 1) - 1) * (b ^ p + b)
⊢ 2 * p * (HPow.hPow (β := ℕ) b 2 - 1) ∣ b * (b ^ (p - 1) - 1) * (b ^ p + b)
have q₄ : p * (HPow.hPow (β := ℕ) b 2 - 1) * 2 ∣ b * ((b ^ (p - 1) - 1) * (b ^ p + b))
Tactic: have q₄ : p * (b ^ 2 - 1) * 2 ∣ b * ((b ^ (p - 1) - 1) * (b ^ p + b)) :=
      dvd_mul_of_dvd_right q₃ _
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (goal)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
ha₄ : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (p - 1) - 1
q₁ : p.Coprime (HPow.hPow (β := ℕ) b 2 - 1)
q₂ : p * (HPow.hPow (β := ℕ) b 2 - 1) ∣ b ^ (p - 1) - 1
q₃ : p * (HPow.hPow (β := ℕ) b 2 - 1) * 2 ∣ (b ^ (p - 1) - 1) * (b ^ p + b)
q₄ : p * (HPow.hPow (β := ℕ) b 2 - 1) * 2 ∣ b * ((b ^ (p - 1) - 1) * (b ^ p + b))
⊢ 2 * p * (HPow.hPow (β := ℕ) b 2 - 1) ∣ b * (b ^ (p - 1) - 1) * (b ^ p + b)
have rwa : 2 * (p * (HPow.hPow (β := ℕ) b 2 - 1)) ∣ b * (b ^ (p - 1) - 1) * (b ^ p + b)
Tactic: rwa [mul_assoc, mul_comm, mul_assoc b]
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (goal)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
ha₄ : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (p - 1) - 1
q₁ : p.Coprime (HPow.hPow (β := ℕ) b 2 - 1)
q₂ : p * (HPow.hPow (β := ℕ) b 2 - 1) ∣ b ^ (p - 1) - 1
q₃ : p * (HPow.hPow (β := ℕ) b 2 - 1) * 2 ∣ (b ^ (p - 1) - 1) * (b ^ p + b)
q₄ : p * (HPow.hPow (β := ℕ) b 2 - 1) * 2 ∣ b * ((b ^ (p - 1) - 1) * (b ^ p + b))
rwa : 2 * (p * (HPow.hPow (β := ℕ) b 2 - 1)) ∣ b * (b ^ (p - 1) - 1) * (b ^ p + b)
⊢ 2 * p * (HPow.hPow (β := ℕ) b 2 - 1) ∣ b * (b ^ (p - 1) - 1) * (b ^ p + b)
have rwa₁ : p * (HPow.hPow (β := ℕ) b 2 - 1) * 2 ∣ b * (b ^ (p - 1) - 1) * (b ^ p + b)
Tactic: rwa [mul_assoc, mul_comm, mul_assoc b]
-- Lean 4 proof step from Mathlib.NumberTheory.FermatPsp (hypothesis)
Goal:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (HPow.hPow (β := ℕ) b 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hA : p < A
hi_A : 1 < A
hi_B : 1 < B
hi_b : 0 < b
hi_bsquared : 0 < HPow.hPow (β := ℕ) b 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (HPow.hPow (β := ℕ) b 2 - 1)
hd : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
ha₄ : HPow.hPow (β := ℕ) b 2 - 1 ∣ b ^ (p - 1) - 1
⊢ (A * B).ProbablePrime b
have ha₅ : 2 * p * (HPow.hPow (β := ℕ) b 2 - 1) ∣ (HPow.hPow (β := ℕ) b 2 - 1) * (A * B - 1)
Tactic: have ha₅ : 2 * p * (b ^ 2 - 1) ∣ (b ^ 2 - 1) * (A * B - 1) := by
    suffices q : 2 * p * (b ^ 2 - 1) ∣ b * (b ^ (p - 1) - 1) * (b ^ p + b) by rwa [ha₁]
    -- We already proved that `b ^ 2 - 1 ∣ b ^ (p - 1) - 1`.
    -- Since `2 ∣ b ^ p + b` and `p ∣ b ^ p + b`, if we show that 2 and p are coprime, then we
    -- know that `2 * p ∣ b ^ p + b`
    have q₁ : Nat.Coprime p (b ^ 2 - 1) :=
      haveI q₂ : ¬p ∣ b ^ 2 - 1 := by
        rw [mul_comm] at not_dvd
        exact mt (fun h : p ∣ b ^ 2 - 1 => dvd_mul_of_dvd_left h _) not_dvd
      (Nat.Prime.coprime_iff_not_dvd p_prime).mpr q₂
    have q₂ : p * (b ^ 2 - 1) ∣ b ^ (p - 1) - 1 := Nat.Coprime.mul_dvd_of_dvd_of_dvd q₁ ha₃ ha₄
    have q₃ : p * (b ^ 2 - 1) * 2 ∣ (b ^ (p - 1) - 1) * (b ^ p + b) := mul_dvd_mul q₂ ha₂
    have q₄ : p * (b ^ 2 - 1) * 2 ∣ b * ((b ^ (p - 1) - 1) * (b ^ p + b)) :=
      dvd_mul_of_dvd_right q₃ _
    rwa [mul_assoc, mul_comm, mul_assoc b]