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Proof step 19 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(j + 1) * fac(2 * (j + 1) - j)) + binom_coef(2 * (j + 1), j) < binom_coef(2 * (j + 1), j), 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 20:
⊦ j + i < j ↔ i < 0
Proof step 21:
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 ⊦

Rule argument

spec: int-basic2 inst: name: ls-03 seq: ⊦ j + i < j ↔ i < 0 subst: [i, j] → [fac(2 * (j + 1)) / (fac(2 * (j + 1) - j) * fac(j + 1)), binom_coef(2 * (j + 1), j)] flags: rewrite

Additional information

Used simplifier rules: commutativity: ⊦ i * j = j * i; from specification int-mult%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: fac(2 * (j + 1)) / (fac(2 * (j + 1) - j) * fac(j + 1)), binom_coef(2 * (j + 1), j)