-- 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)
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
this : ¬p ∣ b
⊢ p ∣ b ^ (p - 1) - 1

Tactic:
have : p.Coprime b := Or.resolve_right (Nat.coprime_or_dvd_of_prime p_prime b) 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
this✝ : ¬p ∣ b
this : p.Coprime b
⊢ p ∣ 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)
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)
q₂ : p * (b ^ 2 - 1) ∣ b ^ (p - 1) - 1
⊢ 2 * p * (b ^ 2 - 1) ∣ b * (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

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)
q₂ : p * (b ^ 2 - 1) ∣ b ^ (p - 1) - 1
q₃ : p * (b ^ 2 - 1) * 2 ∣ (b ^ (p - 1) - 1) * (b ^ p + b)
⊢ 2 * p * (b ^ 2 - 1) ∣ b * (b ^ (p - 1) - 1) * (b ^ p + b)

Proof state:
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)
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)
ha₇ : A * B ∣ b ^ (2 * p) - 1
q : ℕ
hq : A * B - 1 = 2 * p * q
ha₈ : b ^ (2 * p) - 1 ∣ b ^ (A * B - 1) - 1
⊢ (A * B).ProbablePrime b

Tactic:
exact dvd_trans ha₇ ha₈

Result:
no goals

Proof state:
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
this : 2 ∣ p - 1
⊢ b ^ 2 - 1 ∣ b ^ (p - 1) - 1

Tactic:
obtain ⟨c, hc⟩ := this

Result:
case intro.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
c : ℕ
hc : p - 1 = 2 * c
⊢ 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
⊢ 0 < b ^ 2 - 1

Tactic:
have := Nat.pow_le_pow_left b_ge_two 2

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
this : 2 ^ 2 ≤ b ^ 2
⊢ 0 < b ^ 2 - 1

Proof state:
case h
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)
ha₆ : 2 * p ∣ A * B - 1
this : A * B * (b ^ 2 - 1) = (b ^ (2 * p) - 1) / (b ^ 2 - 1) * (b ^ 2 - 1)
⊢ b ^ (2 * p) - 1 = A * B * (b ^ 2 - 1)

Tactic:
simpa only [add_comm, Nat.div_mul_cancel hd, Nat.sub_add_cancel hi_bpowtwop] using this.symm

Result:
no goals