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Proof step 138 from proof all_prime_divisors-div (previous step)

j0 * i0 = j2 * i2, all_prime_divisors(i0) = all_prime_divisors(i * i0) -1l i, 0 ≤ j2 * i2, 0 < i, prime(i), prime(j0), j0 < j0 * i * i0, j2 < j2 * i2, prime(j2), 2 ≤ j2 * i2, 2 ≤ j0 * i * i0, 0 < j2 * i2, j0 < j2, j0 ≤ j2, j0 < j2 * i2, ∀ j1. j0 < j1 ∧ j1 < j0 * i * i0 ∧ prime(j1) → ¬ j1 | j0 * i * i0, ∀ j. j2 < j ∧ j < j2 * i2 ∧ prime(j) → ¬ j | j2 * i2, j2 < j0 * i * i0 → ¬ j2 | j0 * i * i0, ∀ 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 ⊦ all_prime_divisors(i2) + j2 ' = all_prime_divisors(i0) + j0 ', j0 = 0, prime(j0 * i * i0), prime(j2 * i2)

User applied rule cut formula

And obtained the following new goals

Proof step 139:
j0 * i * i0 = (j0 * i0) * i, j0 * i0 = j2 * i2, all_prime_divisors(i0) = all_prime_divisors(i * i0) -1l i, 0 ≤ j2 * i2, 0 < i, prime(i), prime(j0), j0 < j0 * i * i0, j2 < j2 * i2, prime(j2), 2 ≤ j2 * i2, 2 ≤ j0 * i * i0, 0 < j2 * i2, j0 < j2, j0 ≤ j2, j0 < j2 * i2, ∀ j1. j0 < j1 ∧ j1 < j0 * i * i0 ∧ prime(j1) → ¬ j1 | j0 * i * i0, ∀ j. j2 < j ∧ j < j2 * i2 ∧ prime(j) → ¬ j | j2 * i2, j2 < j0 * i * i0 → ¬ j2 | j0 * i * i0, ∀ 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 ⊦ all_prime_divisors(i2) + j2 ' = all_prime_divisors(i0) + j0 ', j0 = 0, prime(j0 * i * i0), prime(j2 * i2)

Rule argument

fma: j0 * i * i0 = (j0 * i0) * i

Additional information

Used simplifier rules: associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used formulas: j0 * i * i0 = (j0 * i0) * i