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Proof step 16 from proof bertrand-small-primes (previous step)

all_prime_divisors(binom_coef(2 * i, i)) = ints + ints0, Π ints ≤ (2 * i) ^ n→i(# rmdup(ints)), 0 < i, (2 * i) ^ (sqrt(2 * i) - 1) < Π ints, 2 < i, (2 * i) ^ (sqrt(2 * i) - 1) < (2 * i) ^ n→i(# rmdup(ints)), ∀ j. j ∈ ints → j ≤ sqrt(2 * i) ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-pot inst: name: ls-pos seq: ⊦ 1 < i ∧ 0 ≤ j → (i ^ j < i ^ j0 ↔ j < j0) subst: [j0, j, i] → [n→i(# rmdup(ints)), sqrt(2 * i) - 1, 2 * i] flags: rewrite

Additional information

Used simplifier rules: nummult-ls-02: ⊦ 0 < i ∧ 0 ≤ j → ((* j) < (* i) * i0 ↔ j / i < i0); from specification int-div%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%sub-le-01: ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%leone: ⊦ 1 ≤ i ↔ 0 < i; from specification int-abs%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%square-not-neg-03: ⊦ 0 < i → ¬ sqrt(2 * i) ≤ 0; from specification int-square Used terms: n→i(# rmdup(ints)), sqrt(2 * i) - 1, 2 * i