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Proof step 11 from proof coef-sub-mult (previous step)

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 - j) * fac(j)) - fac(i) / (fac(i + 1 - j) * fac(j - 1))) * fac(j) * fac(i + 1 - j) = fac(i) * (i + 1 - 2 * j)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 12:
⊦ (i - j) * j0 = i * j0 - j * j0
Proof step 13:
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 - j) * fac(j)) * fac(j) * fac(i + 1 - j) - fac(i) / (fac(i + 1 - j) * fac(j - 1)) * fac(j) * fac(i + 1 - j) = fac(i) * (i + 1 - 2 * j)

Rule argument

spec: int-mult inst: name: dist-sub-split seq: ⊦ (i - j) * j0 = i * j0 - j * j0 subst: [j0, j, i] → [fac(j) * fac(i + 1 - j), fac(i) / (fac(i + 1 - j) * fac(j - 1)), fac(i) / (fac(i - j) * fac(j))] flags: rewrite

Additional information

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