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Proof step 114 from proof all_prime_divisors-div (previous step)

all_prime_divisors((j / j0) / i) = all_prime_divisors(j / j0) -1l i, j2 < j / i, j2 | j / i, prime(j2), 2 ≤ j / i, 2 ≤ j, 0 < j, prime(i), i | j, 0 ≤ j, 0 < j / i, 0 ≤ j / i, i | j / j0, prime(j0), j0 | j, j0 < j, ∀ j1. j2 < j1 ∧ j1 < j / i ∧ prime(j1) → ¬ j1 | j / i, ∀ j1. j0 < j1 ∧ j1 < j ∧ prime(j1) → ¬ j1 | j, ∀ j0. j2 < j0 ∧ j0 < j / i ∧ prime(j0) → ¬ j0 | j / i, ∀ 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 ⊦ j2 = j0, all_prime_divisors((j / i) / j2) + j2 ' = (all_prime_divisors(j / j0) -1l i) + j0 ', prime(j), prime(j / i)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 115:
⊦ i | j ↔ (∃ i0. i * i0 = j)
Proof step 116:
all_prime_divisors((j / j0) / i) = all_prime_divisors(j / j0) -1l i, j2 < j / i, j2 | j / i, prime(j2), 2 ≤ j / i, 2 ≤ j, 0 < j, prime(i), ∃ i0. i * i0 = j, 0 ≤ j, 0 < j / i, 0 ≤ j / i, i | j / j0, prime(j0), j0 | j, j0 < j, ∀ j1. j2 < j1 ∧ j1 < j / i ∧ prime(j1) → ¬ j1 | j / i, ∀ j1. j0 < j1 ∧ j1 < j ∧ prime(j1) → ¬ j1 | j, ∀ j0. j2 < j0 ∧ j0 < j / i ∧ prime(j0) → ¬ j0 | j / i, ∀ 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 ⊦ j2 = j0, all_prime_divisors((j / i) / j2) + j2 ' = (all_prime_divisors(j / j0) -1l i) + j0 ', prime(j), prime(j / i)

Rule argument

spec: int-gcd inst: name: divides seq: ⊦ i | j ↔ (∃ i0. i * i0 = j) subst: [j, i] → [j, i] flags: rewrite

Additional information

Used terms: j, i