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Proof step 83 from proof all_prime_divisors-div (previous step)

max_prime_divisor(j / i) = max_prime_divisor(j), all_prime_divisors((j / max_prime_divisor(j)) / i) = all_prime_divisors(j / max_prime_divisor(j)) -1l i, 2 ≤ j / i, 2 ≤ j, 0 < j, prime(i), i | j, 0 ≤ j, 0 < j / i, 0 ≤ j / i, i | j / max_prime_divisor(j), ∀ j0, i. i→n(j0) < i→n(j) → i | j0 ∧ prime(i) ∧ 0 < j0 → all_prime_divisors(j0 / i) = all_prime_divisors(j0) -1l i ⊦ all_prime_divisors((j / i) / max_prime_divisor(j)) = all_prime_divisors((j / max_prime_divisor(j)) / i), prime(j), prime(j / i)

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 84:
⊦ ¬ prime(i) ∧ 2 ≤ i → (∃ j. prime(j) ∧ j | i ∧ j < i ∧ (∀ j0. j < j0 ∧ j0 < i ∧ prime(j0) → ¬ j0 | i))
Proof step 85:
¬ prime(j) ∧ 2 ≤ j → (∃ j0. prime(j0) ∧ j0 | j ∧ j0 < j ∧ (∀ j1. j0 < j1 ∧ j1 < j ∧ prime(j1) → ¬ j1 | j)), max_prime_divisor(j / i) = max_prime_divisor(j), all_prime_divisors((j / max_prime_divisor(j)) / i) = all_prime_divisors(j / max_prime_divisor(j)) -1l i, 2 ≤ j / i, 2 ≤ j, 0 < j, prime(i), i | j, 0 ≤ j, 0 < j / i, 0 ≤ j / i, i | j / max_prime_divisor(j), ∀ j0, i. i→n(j0) < i→n(j) → i | j0 ∧ prime(i) ∧ 0 < j0 → all_prime_divisors(j0 / i) = all_prime_divisors(j0) -1l i ⊦ all_prime_divisors((j / i) / max_prime_divisor(j)) = all_prime_divisors((j / max_prime_divisor(j)) / i), prime(j), prime(j / i)

Rule argument

spec: int-gcd inst: name: prime-ex-max seq: ⊦ ¬ prime(i) ∧ 2 ≤ i → (∃ j. prime(j) ∧ j | i ∧ j < i ∧ (∀ j0. j < j0 ∧ j0 < i ∧ prime(j0) → ¬ j0 | i)) subst: [i] → [j] flags: []

Additional information

Used terms: j