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Proof step 33 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) * 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)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 34:
⊦ i * j + i * j0 = i * (j + j0)
Proof step 35:
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) * j)

Rule argument

spec: int-mult inst: name: dist-rmerge seq: ⊦ i * j + i * j0 = i * (j + j0) subst: [j0, j, i] → [fac(i - j) * j, (i + 1 - 2 * j) * fac(i - j), fac(i)] flags: rewrite

Additional information

Used simplifier rules: associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: fac(i - j) * j, (i + 1 - 2 * j) * fac(i - j), fac(i)