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Proof step 40 from proof all_prime_divisors-div (previous step)

all_prime_divisors((j / max_prime_divisor(j)) / i) = all_prime_divisors(j / max_prime_divisor(j)) -1l i, prime(j / i), 0 ≤ j, ∃ i0. i * i0 = j, prime(i), 0 < j, 2 ≤ j, 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 ⊦ (j / i)' = (all_prime_divisors(j / max_prime_divisor(j)) -1l i) + max_prime_divisor(j) ', prime(j)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 41:
all_prime_divisors(((i * i0) / max_prime_divisor(i * i0)) / i) = all_prime_divisors((i * i0) / max_prime_divisor(i * i0)) -1l i, prime(i), 2 ≤ i * i0, i | (i * i0) / max_prime_divisor(i * i0), prime(i0), 0 < i * i0, 0 ≤ i * i0, ∀ j0, i1. i→n(j0) < i→n(i * i0) → i1 | j0 ∧ prime(i1) ∧ 0 < j0 → all_prime_divisors(j0 / i1) = all_prime_divisors(j0) -1l i1 ⊦ i0 ' = (all_prime_divisors((i * i0) / max_prime_divisor(i * i0)) -1l i) + max_prime_divisor(i * i0) ', prime(i * i0)

Rule argument

Additional information

Used simplifier rules: 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%div-mult-03: j ≠ 0 ⊦ (i * j) / j = i; from specification int-div%prime-not-neg-01: ⊦ j ≤ 0 → ¬ prime(j); from specification int-gcd%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%Mult-zero: ⊦ i * 0 = 0; from specification int-mult