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Proof step 4 from proof bertrand-middle-coef-smaller (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
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, ∀ 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, 0 < binom_coef(2 * i, i) → Π ints * Π ints0 * Π ints1 * Π ints2 = binom_coef(2 * i, i) ⊦

Rule argument

Additional information

Used simplifier rules: a: ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1; from specification list[intlista]%product-app: ⊦ Π(ints + ints0) = Π ints * Π ints0; from specification intlist