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Proof step 17 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), j < 2 * i, binom_coef(2 * i, i) < binom_coef(2 * i, j), j < i, 0 ≤ j, 0 < 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 ⊦ j + 1 < i, binom_coef(2 * i, i) < binom_coef(2 * i, j + 1) ∧ j + 1 < i

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 18:
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, binom_coef(2 * (j + 1), j + 1) < 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 ⊦

Rule argument

Additional information

Used simplifier rules: zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%zero-gr: ⊦ 0 < i → (0 < i * j ↔ 0 < j); from specification int-mult%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%ler: ⊦ i ≤ i; from specification int-abs%succ-02: ⊦ i < j + 1 ↔ i ≤ j; from specification int-abs%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%mone-06: ⊦ ~1 < i ↔ 0 ≤ i; from specification int-abs%numls-02: ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0; from specification int-abs%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%one: ⊦ 1 * i = i; from specification int-mult%sub-add: ⊦ i + j - i = j; from specification int-abs%irreflexivity: ⊦ ¬ i < i; from specification oelem[intlista]%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%les: ⊦ i < j → (j ≤ i + 1 ↔ j = i + 1); from specification int-abs%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs