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Proof step 7 from proof bertrand-prime-factors (previous step)

all_prime_divisors(binom_coef(2 * i, i)) = ints0 + ints, ordered≤(ints0 + ints), 1 < i, 0 < i, ∀ i1. i1 ∈ ints0 → i1 < sqrt(2 * i) ∨ i1 = sqrt(2 * i), ∀ i1. i1 ∈ ints → sqrt(2 * i) < i1 ⊦ ∃ ints, ints0, ints1, ints2. all_prime_divisors(binom_coef(2 * i, i)) = ints + ints0 + ints1 + ints2 ∧ (∀ 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 insert spec-lemma

And obtained the following new goals

Proof step 8:
ordered≤(ints) ⊦ ∃ ints0, ints1. ints = ints0 + ints1 ∧ (∀ i. i ∈ ints0 → i < i0 ∨ i = i0) ∧ (∀ i. i ∈ ints1 → i0 < i)
Proof step 9:
∃ ints0, ints1. ints = ints0 + ints1 ∧ (∀ i1. i1 ∈ ints0 → i1 < (2 * i) / 3 ∨ i1 = (2 * i) / 3) ∧ (∀ i1. i1 ∈ ints1 → (2 * i) / 3 < i1), all_prime_divisors(binom_coef(2 * i, i)) = ints0 + ints, ordered≤(ints0 + ints), 1 < i, 0 < i, ∀ i1. i1 ∈ ints0 → i1 < sqrt(2 * i) ∨ i1 = sqrt(2 * i), ∀ i1. i1 ∈ ints → sqrt(2 * i) < i1 ⊦ ∃ ints, ints0, ints1, ints2. all_prime_divisors(binom_coef(2 * i, i)) = ints + ints0 + ints1 + ints2 ∧ (∀ 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)

Rule argument

spec: olist inst: intlista name: ordered-ex-divide-lels seq: ordered≤(ints) ⊦ ∃ ints0, ints1. ints = ints0 + ints1 ∧ (∀ i. i ∈ ints0 → i < i0 ∨ i = i0) ∧ (∀ i. i ∈ ints1 → i0 < i) subst: [ints, i0] → [ints, (2 * i) / 3] flags: []

Additional information

Used simplifier rules: a: ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1; from specification list[intlista]%app-01: ⊦ ordered≤(ints + ints0) → ordered≤(ints0); from specification olist[intlista] Used terms: ints, (2 * i) / 3