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Proof step 6 from proof bertrand-small-primes (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 7:
all_prime_divisors(binom_coef(2 * i, i)) = ints + ints0, 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, ∀ j. j ∈ ints → j ≤ sqrt(2 * i) ⊦

Rule argument

Additional information

Used simplifier rules: forward-le: ⊦ i ≤ j ∧ 0 < i → 0 < j; (forward) from specification int-abs%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%prime-not-neg-01: ⊦ j ≤ 0 → ¬ prime(j); from specification int-gcd%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%all_prime_divisors-in-01: ⊦ j ∈ all_prime_divisors(i) ↔ prime(j) ∧ j | i ∧ 2 ≤ i; from specification intlist