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Proof step 32 from proof coef-sub-mult (previous step)

0 < j, fac(i - j) | fac(i) * fac(i + 1 - j), j ≤ i, 0 < i, 0 < j, fac(i - j) * j * fac(j - 1) | fac(i), fac(i + 1 - j) * fac(j - 1) | fac(i), 0 ≤ j, 0 ≤ i ⊦ fac(i) * fac(i + 1 - j) = (fac(i) * (i + 1 - 2 * j)) * fac(i - j) + (fac(i) * j * fac(j - 1)) / fac(j - 1) * fac(i - j)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 33:
fac(i - j) | fac(i) * fac(i + 1 - j), j ≤ i, 0 < i, 0 < j, fac(i - j) * j * fac(j - 1) | fac(i), fac(i + 1 - j) * fac(j - 1) | fac(i), 0 ≤ j, 0 ≤ i ⊦ fac(i) * fac(i + 1 - j) = (fac(i) * (i + 1 - 2 * j)) * fac(i - j) + (fac(i) * j) * fac(i - j)

Rule argument

Additional information

Used simplifier rules: fac-zero-02: ⊦ fac(i) ≠ 0;%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%div-mult-03: j ≠ 0 ⊦ (i * j) / j = i; from specification int-div