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Proof step 34 from proof prime-product (previous step)

Π ints = Π ints1 * Π ints0, min(ints0) < j + 2, max(ints0) ≤ j * 2 + 1, j + 1 < min(ints0), 4 ^ j < Π ints0, primes(ints1), primes(ints0), 0 < j, 0 ≤ j, max(ints) ≤ 2 * j + 1, 4 ^ (2 * j + 1) < Π ints, primes(ints), perm(ints, ints1 + ints0), Π ints1 ≤ 4 ^ (j + 1), 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 ⊦ ints0 = [], ints = [], dups(ints), dups(ints0), dups(ints1)

User applied rule cut formula

And obtained the following new goals

Proof step 35:
Π ints = Π ints1 * Π ints0, min(ints0) < j + 2, max(ints0) ≤ j * 2 + 1, j + 1 < min(ints0), 4 ^ j < Π ints0, primes(ints1), primes(ints0), 0 < j, 0 ≤ j, max(ints) ≤ 2 * j + 1, 4 ^ (2 * j + 1) < Π ints, primes(ints), perm(ints, ints1 + ints0), Π ints1 ≤ 4 ^ (j + 1), 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 ⊦ ints0 = [], ints0 = [], ints = [], dups(ints), dups(ints0), dups(ints1)

Rule argument

fma: ints0 = []

Additional information

Used formulas: ints0 = []