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Proof step 21 from proof middle-coef-largest-01 (previous step)

binom_coef(2 * (j + 1), j + 1) = fac(2 * (j + 1)) / (fac(j + 1) * fac(2 * (j + 1) - j)) + binom_coef(2 * (j + 1), j), fac(2 * (j + 1)) / (fac(j + 1) * fac(2 * (j + 1) - j)) < 0, 0 ≤ j, fac(2 * (j + 1)) / (fac(2 * (j + 1) - j) * fac(j + 1)) < 0, j < 2 * (j + 1), ∀ i0, j0. i→n(i0 - j0) < i→n(j + 1 - j) → binom_coef(2 * i0, i0) < binom_coef(2 * i0, j0) ∧ j0 < i0 ∧ 0 ≤ j0 ∧ 0 < i0 → false ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 22:
⊦ 0 ≤ i ∧ 0 < j → ¬ i / j < 0

Rule argument

spec: int-div inst: name: div-zero-nls seq: ⊦ 0 ≤ i ∧ 0 < j → ¬ i / j < 0 subst: [j, i] → [fac(2 * (j + 1) - j) * fac(j + 1), fac(2 * (j + 1))] flags: rewrite

Additional information

Used simplifier rules: fac-not-neg: ⊦ ¬ fac(i) < 0;%fac-le-zero: ⊦ i * fac(j) ≤ 0 ↔ i ≤ 0;%fac-pos-01: ⊦ ¬ fac(i) ≤ 0;%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: fac(2 * (j + 1) - j) * fac(j + 1), fac(2 * (j + 1))