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Proof step 11 from proof middle-coef-largest-01 (previous step)

binom_coef(2 * i, j + 1) = (fac(2 * i) * (2 * i + 1 - 2 * (j + 1))) / (fac(j + 1) * fac(2 * i - j)) + binom_coef(2 * i, j), (fac(2 * i) * (2 * i + 1 - 2 * (j + 1))) / (fac(j + 1) * fac(2 * i - j)) < i, (fac(2 * i) * (2 * i + 1 - 2 * (j + 1))) / (fac(j + 1) * fac(2 * i - j)) < 0, binom_coef(2 * i, j + 1) < binom_coef(2 * i, j), binom_coef(2 * i, j + 1) ≤ binom_coef(2 * i, i), 0 < i, 0 ≤ j, j < i, binom_coef(2 * i, i) < binom_coef(2 * i, j), j < 2 * i, j + 1 < i, ∀ i0, j0. i→n(i0 - j0) < i→n(i - 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 12:
⊦ 0 ≤ i ∧ 0 < j → ¬ i / j < 0
Proof step 13:
¬(0 ≤ fac(2 * i) * (2 * i + 1 - 2 * (j + 1)) ∧ 0 < fac(2 * i - j) * fac(j + 1)), binom_coef(2 * i, j + 1) = (fac(2 * i) * (2 * i + 1 - 2 * (j + 1))) / (fac(j + 1) * fac(2 * i - j)) + binom_coef(2 * i, j), (fac(2 * i) * (2 * i + 1 - 2 * (j + 1))) / (fac(j + 1) * fac(2 * i - j)) < i, (fac(2 * i) * (2 * i + 1 - 2 * (j + 1))) / (fac(j + 1) * fac(2 * i - j)) < 0, binom_coef(2 * i, j + 1) < binom_coef(2 * i, j), binom_coef(2 * i, j + 1) ≤ binom_coef(2 * i, i), 0 < i, 0 ≤ j, j < i, binom_coef(2 * i, i) < binom_coef(2 * i, j), j < 2 * i, j + 1 < i, ∀ i0, j0. i→n(i0 - j0) < i→n(i - j) → binom_coef(2 * i0, i0) < binom_coef(2 * i0, j0) ∧ j0 < i0 ∧ 0 ≤ j0 ∧ 0 < i0 → false ⊦

Rule argument

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

Additional information

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