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Proof step 14 from proof bertrand-middle-coef-smaller (previous step)

¬(Π ints ≤ (2 * i) ^ (sqrt(2 * i) - 1) ∧ Π ints0 * Π ints1 * Π ints2 ≤ 4 ^ ((2 * i) / 3) * Π ints2 ∧ 0 ≤ Π ints ∧ 0 ≤ Π ints0 * Π ints1 * Π ints2), Π ints * Π ints0 * Π ints1 * Π ints2 = binom_coef(2 * i, i), all_prime_divisors(Π ints * Π ints0 * Π ints1 * Π ints2) = ints + ints0 + ints1 + ints2, 2 < i, (2 * i) ^ (sqrt(2 * i) - 1) * 4 ^ ((2 * i) / 3) * Π ints2 < Π ints * Π ints0 * Π ints1 * Π ints2, 0 < i, 0 < Π ints, 0 < Π ints0, 0 < Π ints1, 0 < Π ints2, ∀ j. j ∈ ints → j ≤ sqrt(2 * i), ∀ j. j ∈ ints0 → sqrt(2 * i) < j ∧ j ≤ (2 * i) / 3, ∀ j. j ∈ ints1 → (2 * i) / 3 < j ∧ j ≤ i, ∀ j. j ∈ ints2 → i < j ∧ j ≤ 2 * i ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%zero-ge: ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); from specification int-mult%le-pos: ⊦ 0 < i → (i * j ≤ i * i0 ↔ j ≤ i0); from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%ls-pos: ⊦ 0 < i → (i * j < i * j0 ↔ j < j0); from specification int-mult