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Proof step 23 from proof coef-rec-add-01 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 24:
⊦ 0 < i → fac(i) = i * fac(i - 1)
Proof step 25:
0 < j, fac(i - 1) / (fac(i - 1 + j) * j * fac(j - 1)) = ((i - j) * fac(i - 1)) / (fac(i - j) * j * fac(j - 1)), fac(i - 1) / (fac(i - j) * fac(j - 1)) = (j * fac(i - 1)) / (fac(i - j) * j * fac(j - 1)), 0 ≤ j, 0 ≤ i, j ≤ i, j < i, 0 < i, 0 < fac(i - j) * j * fac(j - 1), 0 ≤ fac(i - 1), fac(j - 1) * fac(i - j) | fac(i - 1), 1 < j, 0 < j, 1 < i ⊦ fac(i) / (fac(i - j) * j * fac(j - 1)) = (j * fac(i - 1)) / (fac(i - j) * j * fac(j - 1)) + ((i - j) * fac(i - 1)) / (fac(i - j) * j * fac(j - 1)), (j * fac(i - 1)) % (fac(i - j) * j * fac(j - 1)) = 0

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