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Proof step 12 from proof bertrand-small-primes (previous step)

1 < i0 ∧ 0 ≤ max_exp(i0, binom_coef(2 * i, i)), all_prime_divisors(binom_coef(2 * i, i)) = ints + ints0, i0 ^ max_exp(i0, binom_coef(2 * i, i)) ≤ 2 * i, 2 < i, (2 * i) ^ (sqrt(2 * i) - 1) < Π ints, 0 < i, i0 ∈ ints, prime(i0), i0 | binom_coef(2 * i, i), 2 ≤ binom_coef(2 * i, i), 2 * i < i0 ^ n→i(#oc(i0, ints)), 0 < binom_coef(2 * i, i), 0 ≤ binom_coef(2 * i, i), 0 < i0, max_exp(i0, binom_coef(2 * i, i)) < n→i(#oc(i0, ints)), ∀ j. j ∈ ints → j ≤ sqrt(2 * i) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
all_prime_divisors(binom_coef(2 * i, i)) = ints + ints0, i0 ^ max_exp(i0, binom_coef(2 * i, i)) ≤ 2 * i, 2 < i, (2 * i) ^ (sqrt(2 * i) - 1) < Π ints, 0 < i, i0 ∈ ints, prime(i0), i0 | binom_coef(2 * i, i), 2 ≤ binom_coef(2 * i, i), 2 * i < i0 ^ n→i(#oc(i0, ints)), 0 < binom_coef(2 * i, i), 0 ≤ binom_coef(2 * i, i), max_exp(i0, binom_coef(2 * i, i)) < n→i(#oc(i0, ints)), 1 < i0, 0 < i0, ∀ j. j ∈ ints → j ≤ sqrt(2 * i) ⊦

Rule argument

Additional information

Used simplifier rules: max_exp-pos: ⊦ 0 ≤ max_exp(i, j);%forward-pos: ⊦ i < j ∧ 0 < i → 0 < j ∧ 1 < j; (forward) from specification int-basic1