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Proof step 2 from proof bertrand-middle-coef-smaller (previous step)

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) ⊦

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 3:
⊦ 0 < i → Π all_prime_divisors(i) = i
Proof step 4:
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) ⊦

Rule argument

spec: intlist inst: name: all_prime_divisors-product seq: ⊦ 0 < i → Π all_prime_divisors(i) = i subst: [i] → [binom_coef(2 * i, i)] flags: []

Additional information

Used terms: binom_coef(2 * i, i)