-- Mathlib/NumberTheory/FermatPsp.lean
theorem psp_from_prime_psp {b : ℕ} (b_ge_two : 2 ≤ b) {p : ℕ} (p_prime : p.Prime)
    (p_gt_two : 2 < p) (not_dvd : ¬p ∣ b * (b ^ 2 - 1)) : FermatPsp (psp_from_prime b p) b

Proof state:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
⊢ (A * B).FermatPsp b

Tactic:
have hi_A : 1 < A := a_id_helper (Nat.succ_le_iff.mp b_ge_two) (Nat.Prime.one_lt p_prime)

Result:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hi_A : 1 < A
⊢ (A * B).FermatPsp b

Proof state:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hi_A : 1 < A
hi_B : 1 < B
hi_AB : 1 < A * B
hi_b : 0 < b
hi_p : 1 ≤ p
hi_bsquared : 0 < 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) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (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
⊢ b ^ 2 - 1 ∣ b ^ (p - 1) - 1

Tactic:
obtain ⟨k, hk⟩ := p_odd

Result:
case intro
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hi_A : 1 < A
hi_B : 1 < B
hi_AB : 1 < A * B
hi_b : 0 < b
hi_p : 1 ≤ p
hi_bsquared : 0 < 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) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (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
⊢ b ^ 2 - 1 ∣ b ^ (p - 1) - 1

Proof state:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hi_A : 1 < A
hi_B : 1 < B
hi_AB : 1 < A * B
hi_b : 0 < b
hi_p : 1 ≤ p
hi_bsquared : 0 < 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) / (b ^ 2 - 1)
⊢ b ^ 2 - 1 ∣ b ^ (2 * p) - 1

Tactic:
simpa only [one_pow, pow_mul] using nat_sub_dvd_pow_sub_pow _ 1 p

Result:
no goals

Proof state:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hi_A : 1 < A
hi_B : 1 < B
hi_AB : 1 < A * B
hi_b : 0 < b
hi_p : 1 ≤ p
hi_bsquared : 0 < 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) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (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 : b ^ 2 - 1 ∣ b ^ (p - 1) - 1
ha₅ : 2 * p * (b ^ 2 - 1) ∣ (b ^ 2 - 1) * (A * B - 1)
⊢ 2 * p ∣ A * B - 1

Tactic:
rw [mul_comm] at ha₅

Result:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hi_A : 1 < A
hi_B : 1 < B
hi_AB : 1 < A * B
hi_b : 0 < b
hi_p : 1 ≤ p
hi_bsquared : 0 < 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) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (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 : b ^ 2 - 1 ∣ b ^ (p - 1) - 1
ha₅ : (b ^ 2 - 1) * (2 * p) ∣ (b ^ 2 - 1) * (A * B - 1)
⊢ 2 * p ∣ A * B - 1

Proof state:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hi_A : 1 < A
hi_B : 1 < B
hi_AB : 1 < A * B
hi_b : 0 < b
hi_p : 1 ≤ p
hi_bsquared : 0 < 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) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (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

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

Result:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hi_A : 1 < A
hi_B : 1 < B
hi_AB : 1 < A * B
hi_b : 0 < b
hi_p : 1 ≤ p
hi_bsquared : 0 < 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) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (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 : b ^ 2 - 1 ∣ b ^ (p - 1) - 1
⊢ (A * B).ProbablePrime b

Proof state:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hi_A : 1 < A
hi_B : 1 < B
hi_AB : 1 < A * B
hi_b : 0 < b
hi_p : 1 ≤ p
hi_bsquared : 0 < 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) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (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 : b ^ 2 - 1 ∣ b ^ (p - 1) - 1
⊢ 2 * p * (b ^ 2 - 1) ∣ b * (b ^ (p - 1) - 1) * (b ^ p + b)

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

Result:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hi_A : 1 < A
hi_B : 1 < B
hi_AB : 1 < A * B
hi_b : 0 < b
hi_p : 1 ≤ p
hi_bsquared : 0 < 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) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (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 : b ^ 2 - 1 ∣ b ^ (p - 1) - 1
q₁ : p.Coprime (b ^ 2 - 1)
⊢ 2 * p * (b ^ 2 - 1) ∣ b * (b ^ (p - 1) - 1) * (b ^ p + b)

Proof state:
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hi_A : 1 < A
hi_B : 1 < B
hi_AB : 1 < A * B
hi_b : 0 < b
hi_p : 1 ≤ p
hi_bsquared : 0 < 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) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (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
⊢ p - 1 = 2 * k

Tactic:
simp [hk]

Result:
no goals