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Proof step 34 from proof bertrand-middle-coef-smaller (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 35:
∀ i. i ∈ ints → 0 < i ⊦ ¬ Π ints ≤ 0
Proof step 36:
¬(∀ i. i ∈ ints → 0 < i), Π ints * Π ints0 * Π ints1 * Π ints2 = binom_coef(2 * i, i), all_prime_divisors(binom_coef(2 * i, i)) = ints + ints0 + ints1 + ints2, 2 < i, (2 * i) ^ (sqrt(2 * i) - 1) * 4 ^ ((2 * i) / 3) * Π ints2 < binom_coef(2 * i, i), 0 < i, Π ints ≤ 0, Π ints < i, ∀ 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 ⊦

Rule argument

spec: intlist inst: name: product-positive seq: ∀ i. i ∈ ints → 0 < i ⊦ ¬ Π ints ≤ 0 subst: [ints] → [ints] flags: rewrite

Additional information

Used terms: ints