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Proof step 30 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 - 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 31:
⊦ 0 < i → fac(i) = i * fac(i - 1)
Proof step 32:
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)

Rule argument

spec: inst: name: fac-rec seq: ⊦ 0 < i → fac(i) = i * fac(i - 1) subst: [i] → [j] flags: rewrite, current

Additional information

Used terms: j