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Proof step 39 from proof coef-sub-mult (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 40:
⊦ (i + i0) * j = i * j + i0 * j
Proof step 41:
0 ≤ i, 0 ≤ j, fac(i + 1 - j) * fac(j - 1) | fac(i), fac(i - j) * j * fac(j - 1) | fac(i), 0 < j, 0 < i, j ≤ i, fac(i - j) | fac(i) * fac(i + 1 - j) ⊦ fac(i + 1 - j) = ((i + 1 - 2 * j) + j) * fac(i - j)

Rule argument

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

Additional information

Used simplifier rules: commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: j, fac(i - j), i + 1 - 2 * j