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Proof step 116 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), ∃ 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)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 117:
all_prime_divisors(((i * i0) / j0) / i) = all_prime_divisors((i * i0) / j0) -1l i, 0 ≤ i0, 2 ≤ i * i0, 2 ≤ i0, prime(j2), j2 | i0, j2 < i0, j0 < i * i0, j0 | i * i0, prime(j0), i | (i * i0) / j0, prime(i), 0 < i, 0 < i0, ∀ j. j2 < j ∧ j < i0 ∧ prime(j) → ¬ j | i0, ∀ j1. j0 < j1 ∧ j1 < i * i0 ∧ prime(j1) → ¬ j1 | 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 ⊦ all_prime_divisors(i0 / j2) + j2 ' = (all_prime_divisors((i * i0) / j0) -1l i) + j0 ', j2 = j0, prime(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%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; 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%prime-not-neg-01: ⊦ j ≤ 0 → ¬ prime(j); from specification int-gcd%div-zero-01: ⊦ i | 0; from specification int-gcd%div-zero-zero: ⊦ 0 | i ↔ i = 0; from specification int-gcd%div-mult-03: j ≠ 0 ⊦ (i * j) / j = i; from specification int-div%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