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Proof step 127 from proof all_prime_divisors-div (previous step)

j0 * i1 = i * j2 * i2, all_prime_divisors(i1 / i) = all_prime_divisors(i1) -1l i, 0 < i, prime(i), i | i1, prime(j0), j0 < j0 * i1, j2 < j2 * i2, prime(j2), 2 ≤ j2 * i2, 2 ≤ j0 * i1, 0 < j2 * i2, 0 < j0 * i1, 0 ≤ j0 * i1, 0 ≤ j2 * i2, ∀ j1. j0 < j1 ∧ j1 < j0 * i1 ∧ prime(j1) → ¬ j1 | j0 * i1, ∀ j. j2 < j ∧ j < j2 * i2 ∧ prime(j) → ¬ j | j2 * i2, ∀ j0, i1. i→n(j0) < i→n(i * j2 * i2) → i1 | j0 ∧ prime(i1) ∧ 0 < j0 → all_prime_divisors(j0 / i1) = all_prime_divisors(j0) -1l i1 ⊦ j0 = 0, j2 = j0, all_prime_divisors(i2) + j2 ' = (all_prime_divisors(i1) -1l i) + j0 ', prime(j0 * i1), prime(j2 * i2)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 128:
⊦ i | j ↔ (∃ i0. i * i0 = j)
Proof step 129:
j0 * i1 = i * j2 * i2, all_prime_divisors(i1 / i) = all_prime_divisors(i1) -1l i, 0 < i, prime(i), ∃ i0. i * i0 = i1, prime(j0), j0 < j0 * i1, j2 < j2 * i2, prime(j2), 2 ≤ j2 * i2, 2 ≤ j0 * i1, 0 < j2 * i2, 0 < j0 * i1, 0 ≤ j0 * i1, 0 ≤ j2 * i2, ∀ j1. j0 < j1 ∧ j1 < j0 * i1 ∧ prime(j1) → ¬ j1 | j0 * i1, ∀ j. j2 < j ∧ j < j2 * i2 ∧ prime(j) → ¬ j | j2 * i2, ∀ j0, i1. i→n(j0) < i→n(i * j2 * i2) → i1 | j0 ∧ prime(i1) ∧ 0 < j0 → all_prime_divisors(j0 / i1) = all_prime_divisors(j0) -1l i1 ⊦ j0 = 0, j2 = j0, all_prime_divisors(i2) + j2 ' = (all_prime_divisors(i1) -1l i) + j0 ', prime(j0 * i1), prime(j2 * i2)

Rule argument

spec: int-gcd inst: name: divides seq: ⊦ i | j ↔ (∃ i0. i * i0 = j) subst: [j, i] → [i1, i] flags: rewrite

Additional information

Used terms: i1, i