[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 28 from proof coef-rec-add-01 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 29:
⊦ (i + i0) * j = i * j + i0 * j
Proof step 30:
(j * fac(i - 1)) % (fac(i - j) * fac(j)) = 0 ∧ 0 ≤ j * fac(i - 1) ∧ 0 ≤ (i - j) * fac(i - 1) ∧ 0 < fac(i - j) * fac(j), 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, 0 < j, j ≤ i, j < i, 1 < i, 0 < i ⊦ fac(i) / (fac(i - j) * fac(j)) = ((j + (i - j)) * fac(i - 1)) / (fac(i - j) * fac(j))

Rule argument

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

Additional information

Used terms: i - j, fac(i - 1), j