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Proof step 129 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), ∃ 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)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: all_prime_divisors-prime: ⊦ prime(i) → all_prime_divisors(i) = i ';%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%forward-le: ⊦ i ≤ j ∧ 0 < i → 0 < j; (forward) from specification int-abs%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%zero-ge: ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); from specification int-mult%zero-gr: ⊦ 0 < i → (0 < i * j ↔ 0 < j); from specification int-mult%div-mult-03: j ≠ 0 ⊦ (i * j) / j = i; from specification int-div%a: ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1; from specification list[intlista]%Nil: ⊦ [] + ints = ints; from specification list[intlista]%one-01: ⊦ i ' -1l i = []; from specification list-del[intlista]%Mult-zero: ⊦ i * 0 = 0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%eq-pos: ⊦ 0 < i → (i * j = i * j0 ↔ j = j0); from specification int-mult