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Proof step 18 from proof bertrand-middle-coef-smaller (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 19:
Π ints * Π ints0 * Π ints2 = binom_coef(2 * i, i), all_prime_divisors(binom_coef(2 * i, i)) = ints + ints0 + ints2, 0 < i, 0 < Π ints, 0 < Π ints0, 0 < Π ints2, 2 < i, (2 * i) ^ (sqrt(2 * i) - 1) * 4 ^ ((2 * i) / 3) < Π ints * Π ints0, 0 ≤ Π ints2, 0 ≤ Π ints, 4 ^ ((2 * i) / 3) < Π ints0, ∀ j. j ∈ ints → j ≤ sqrt(2 * i), ∀ j. j ∈ ints0 → sqrt(2 * i) < j ∧ j ≤ (2 * i) / 3, ∀ j. j ∈ ints2 → i < j ∧ j ≤ 2 * i ⊦

Rule argument

Additional information

Used simplifier rules: in-empty: ⊦ ¬ i ∈ []; from specification list[intlista]%a: ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1; from specification list[intlista]%Nil: ⊦ [] + ints = ints; 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