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Proof step 28 from proof coef-sub-mult (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 29:
⊦ (i + i0) * j = i * j + i0 * j
Proof step 30:
fac(i - j) | fac(i) * fac(i + 1 - j), j ≤ i, 0 < i, 0 < j, fac(i - j) * fac(j) | 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) * fac(j)) / fac(j - 1) * fac(i - j)

Rule argument

spec: int-mult inst: name: dist-lsplit seq: ⊦ (i + i0) * j = i * j + i0 * j subst: [j, i0, i] → [fac(i - j), (fac(i) * fac(j)) / fac(j - 1), fac(i) * (i + 1 - 2 * j)] flags: rewrite

Additional information

Used terms: fac(i - j), (fac(i) * fac(j)) / fac(j - 1), fac(i) * (i + 1 - 2 * j)