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Proof step 22 from proof bertrand-middle-coef-smaller (previous step)

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, ints1.first ∈ ints1 → (2 * i) / 3 < ints1.first ∧ ints1.first ≤ i, ∀ j. j ∈ ints1 → (2 * i) / 3 < j ∧ j ≤ i, ∀ j. j ∈ ints2 → i < j ∧ j ≤ 2 * i ⊦ ints1 = []

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 23:
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), ints1.first ≤ i, (2 * i) / 3 < ints1.first, 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 = []

Rule argument

Additional information

Used simplifier rules: commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%less-mult-div-01: ⊦ 0 < i ∧ 0 < j ∧ 0 < j1 ∧ i1 < j → (i * i1) / j < i; from specification int-div%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%in-08: ⊦ ints.first ∈ ints ↔ ints ≠ []; from specification list[intlista]