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Proof step 24 from proof bertrand-middle-coef-smaller (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 25:
all_prime_divisors(binom_coef(2 * i, i)) = ints + ints0 + (i0 ' + ints3) + ints2, Π ints * Π ints0 * (i0 * Π ints3) * Π ints2 = binom_coef(2 * i, i), 0 < Π ints0, 0 < Π ints, 0 < i, i0 ≤ i, (2 * i) / 3 < i0, 4 ^ ((2 * i) / 3) < Π ints0 * i0 * Π ints3, 0 ≤ Π ints, 0 ≤ Π ints2, (2 * i) ^ (sqrt(2 * i) - 1) * 4 ^ ((2 * i) / 3) < Π ints * Π ints0 * i0 * Π ints3, 2 < i, 0 < Π ints2, 0 < i0 * Π ints3, ∀ j. j ∈ ints0 → sqrt(2 * i) < j ∧ j ≤ (2 * i) / 3, ∀ j. j ∈ ints → j ≤ sqrt(2 * i), ∀ j. j ∈ ints2 → i < j ∧ j ≤ 2 * i, ∀ j. j = i0 ∨ j ∈ ints3 → (2 * i) / 3 < j ∧ j ≤ i ⊦

Rule argument

Additional information

Used simplifier rules: forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%in-17: ⊦ i ∈ i0 ' ↔ i = i0; from specification list[intlista]%in-append: ⊦ i ∈ ints + ints0 ↔ ¬ ¬(i ∈ ints ∨ i ∈ ints0); from specification list[intlista]%e-01: ⊦ i ' + ints ≠ []; 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%a: ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1; from specification list[intlista]%c: ⊦ i + ints = i ' + ints; from specification list[intlista]