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Proof step 19 from proof bertrand-small-primes (previous step)

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

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 20:
¬ dups(ints), ∀ i. i ∈ ints → j0 < i ∧ i ≤ j1, ints ≠ [] ⊦ n→i(# ints) ≤ j1 - j0
Proof step 21:
all_prime_divisors(binom_coef(2 * i, i)) = ints + ints0, 0 < sqrt(2 * i), sqrt(2 * i) ≤ n→i(# rmdup(ints)), 2 < i, (2 * i) ^ (sqrt(2 * i) - 1) < Π ints, Π ints ≤ (2 * i) ^ n→i(# rmdup(ints)), 0 < i, ∀ j. j ∈ ints → j ≤ sqrt(2 * i) ⊦ ¬ dups(rmdup(ints)) ∧ (∀ i0. i0 ∈ rmdup(ints) → 1 < i0 ∧ i0 ≤ sqrt(2 * i)) ∧ rmdup(ints) ≠ [], ints = []

Rule argument

spec: intlist inst: name: nodups-interval-size seq: ¬ dups(ints), ∀ i. i ∈ ints → j0 < i ∧ i ≤ j1, ints ≠ [] ⊦ n→i(# ints) ≤ j1 - j0 subst: [ints, j0, j1] → [rmdup(ints), 1, sqrt(2 * i)] flags: []

Additional information

Used simplifier rules: sub-le-01: ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; from specification int-abs%succ-01: ⊦ i + 1 ≤ j ↔ i < j; from specification int-abs%lenls: ⊦ i ≤ j → ¬ j < i; 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 Used terms: rmdup(ints), 1, sqrt(2 * i)