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Proof step 13 from proof prime-product (previous step)

0 < j, 0 ≤ j, max(ints) ≤ 2 * j + 1, 4 ^ (2 * j + 1) < Π ints, primes(ints), perm(ints, ints1 + ints0), ints0 = [] ∨ j + 1 < min(ints0), ints1 = [] ∨ max(ints1) ≤ j + 1, ∀ i0, ints. i→n(i0) < i→n(j + j + 1) → primes(ints) ∧ max(ints) ≤ i0 ∧ ¬ dups(ints) → Π ints ≤ 4 ^ i0 ⊦ ints = [], dups(ints)

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 14:
⊦ perm(ints, ints0 + ints1) → Π ints = Π ints0 * Π ints1
Proof step 15:
perm(ints, ints1 + ints0) → Π ints = Π ints1 * Π ints0, 0 < j, 0 ≤ j, max(ints) ≤ 2 * j + 1, 4 ^ (2 * j + 1) < Π ints, primes(ints), perm(ints, ints1 + ints0), ints0 = [] ∨ j + 1 < min(ints0), ints1 = [] ∨ max(ints1) ≤ j + 1, ∀ i0, ints. i→n(i0) < i→n(j + j + 1) → primes(ints) ∧ max(ints) ≤ i0 ∧ ¬ dups(ints) → Π ints ≤ 4 ^ i0 ⊦ ints = [], dups(ints)

Rule argument

spec: intlist inst: name: product-divide seq: ⊦ perm(ints, ints0 + ints1) → Π ints = Π ints0 * Π ints1 subst: [ints, ints0, ints1] → [ints, ints1, ints0] flags: []

Additional information

Used terms: ints, ints1, ints0