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Proof step 6 from proof bertrand-middle-coef-smaller (previous step)

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)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: coef-le-zero: ⊦ binom_coef(i, j) ≤ 0 ↔ binom_coef(i, j) = 0;%in-empty: ⊦ ¬ i ∈ []; from specification list[intlista]%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%one: ⊦ 1 * i = i; from specification int-mult%prod-empty: ⊦ Π [] = 1; from specification intlist%a: ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1; from specification list[intlista]%app-02-ant: ⊦ ¬ ints + ints0 ≠ [] ↔ ints = [] ∧ ints0 = []; from specification list[intlista]%all_prime_divisors-base: ⊦ i < 2 → all_prime_divisors(i) = []; from specification intlist%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%transitivity: ⊦ i < i0 ∧ i0 < i1 → i < i1; (forward) from specification oelem[intlista]%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs