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Proof step 23 from proof bertrand-middle-coef-smaller (previous step)

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 = []

User applied rule insert elim lemma

And obtained the following new goals

Proof step 24:
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 = []

Rule argument

spec: list-data inst: intlista name: elim-1 seq: ⊦ ints ≠ [] → (i = ints.first ∧ ints0 = ints.rest ↔ ints = i + ints0) subst: [ints, i, ints0] → [ints1, i0, ints3] flags: []

Additional information

Used simplifier rules: ⊦ ints ≠ [] → (i = ints.first ∧ ints0 = ints.rest ↔ ints = i + ints0); (elim) Used terms: ints1, i0, ints3