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Proof step 27 from proof BertrandsPostulate (previous step)

ints = i0 + ints3, binom_coef(2 * i, i) ≤ (2 * i) ^ (sqrt(2 * i) - 1) * 4 ^ ((2 * i) / 3) * Π ints, all_prime_divisors(binom_coef(2 * i, i)) = ints2 + ints1 + ints0 + ints, 0 < i, 2 < i, ∀ j. j ∈ ints → i < j ∧ j ≤ 2 * i, ∀ j. j ∈ ints0 → (2 * i) / 3 < j ∧ j ≤ i, ∀ j. j ∈ ints1 → sqrt(2 * i) < j ∧ j ≤ (2 * i) / 3, ∀ j. j ∈ ints2 → j ≤ sqrt(2 * i) ⊦ ints = [], prime(i0) ∧ i < i0 ∧ i0 ≤ 2 * i

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 28:
all_prime_divisors(binom_coef(2 * i, i)) = ints2 + ints1 + ints0 + i0 ' + ints3, 0 < i, 2 < i, binom_coef(2 * i, i) ≤ (2 * i) ^ (sqrt(2 * i) - 1) * 4 ^ ((2 * i) / 3) * i0 * Π ints3, ∀ j. j ∈ ints0 → (2 * i) / 3 < j ∧ j ≤ i, ∀ j. j ∈ ints1 → sqrt(2 * i) < j ∧ j ≤ (2 * i) / 3, ∀ j. j ∈ ints2 → j ≤ sqrt(2 * i), ∀ j. j = i0 ∨ j ∈ ints3 → i < j ∧ j ≤ 2 * i ⊦ prime(i0) ∧ i < i0 ∧ i0 ≤ 2 * i

Rule argument

Additional information

Used simplifier rules: in-17: ⊦ i ∈ i0 ' ↔ i = i0; from specification list[intlista]%in-append: ⊦ i ∈ ints + ints0 ↔ ¬ ¬(i ∈ ints ∨ i ∈ ints0); from specification list[intlista]%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%prod-rec: ⊦ Π(i ' + ints) = i * Π ints; from specification intlist%e-01: ⊦ i ' + ints ≠ []; from specification list[intlista]%a: ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1; from specification list[intlista]%c: ⊦ i + ints = i ' + ints; from specification list[intlista]