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Proof step 19 from proof coef-mult-rec-lower (previous step)

(fac(i - j) * fac(j)) * i0 = fac(i), j ≤ i, 0 < i, 0 ≤ j, 0 ≤ i, fac(i + 1 - j) * fac(j - 1) | fac(i), 0 < j ⊦ i0 * j = fac(i) / (fac(i + 1 - j) * fac(j - 1)) * (i + 1 - j)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 20:
⊦ i | j ↔ (∃ i0. i * i0 = j)
Proof step 21:
(fac(i - j) * fac(j)) * i0 = fac(i), j ≤ i, 0 < i, 0 ≤ j, 0 ≤ i, ∃ i0. (fac(j - 1) * fac(i + 1 - j)) * i0 = fac(i), 0 < j ⊦ i0 * j = fac(i) / (fac(i + 1 - j) * fac(j - 1)) * (i + 1 - j)

Rule argument

spec: int-gcd inst: name: divides seq: ⊦ i | j ↔ (∃ i0. i * i0 = j) subst: [j, i] → [fac(i), fac(j - 1) * fac(i + 1 - j)] flags: rewrite

Additional information

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