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Proof step 42 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, i0 ∈ ints0, 2 < i, (2 * i) ^ (sqrt(2 * i) - 1) * 4 ^ ((2 * i) / 3) * Π ints2 < binom_coef(2 * i, i), 0 < i, Π ints0 ≤ 0, Π ints0 < i, i0 ≤ 0, i0 < i, ∀ 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 ⊦

User applied rule cut formula

And closed the goal!

Rule argument

fma: i0 ∈ all_prime_divisors(binom_coef(2 * i, i))

Additional information

Used simplifier rules: in-append: ⊦ i ∈ ints + ints0 ↔ ¬ ¬(i ∈ ints ∨ i ∈ ints0); from specification list[intlista]%in-11: ⊦ i ∈ ints → i ∈ ints + ints0; from specification list[intlista]%a: ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1; from specification list[intlista]%all_prime_divisors-in-01: ⊦ j ∈ all_prime_divisors(i) ↔ prime(j) ∧ j | i ∧ 2 ≤ i; from specification intlist%prime-not-neg-01: ⊦ j ≤ 0 → ¬ prime(j); from specification int-gcd Used formulas: i0 ∈ all_prime_divisors(binom_coef(2 * i, i))