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Proof step 8 from proof middle-coef-largest-01 (previous step)

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

Rule argument

Additional information

Used simplifier rules: 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%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%ls-same: ⊦ 0 < i → (i * j < i ↔ j < 1); from specification int-mult%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%sub-eq: ⊦ i - j = i0 ↔ i = i0 + j; from specification int-abs%sub-add-06: ⊦ i + j - i0 + j = i - i0; from specification int-abs%ax-04: ⊦ i + 0 = i; from specification int-basic2%numsub: ⊦ (* i) + i0 - (* j) = (i - j) + i0; from specification int-abs%succ-01: ⊦ i + 1 ≤ j ↔ i < j; from specification int-abs%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%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%leone: ⊦ 1 ≤ i ↔ 0 < i; from specification int-abs