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Proof step 12 from proof coef-mult-rec-lower (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 13:
⊦ i | j ↔ (∃ i0. i * i0 = j)
Proof step 14:
0 ≤ j - 1 ∧ j - 1 ≤ i, 0 ≤ j ∧ j ≤ i, j ≤ i, 0 < j, 0 < i, ∃ i0. (fac(i - j) * fac(j)) * i0 = fac(i), 0 ≤ j, 0 ≤ i, fac(i + 1 - j) * fac(j - 1) | fac(i) ⊦ fac(i) / (fac(i - j) * fac(j)) * j = fac(i) / (fac(i - (j - 1)) * 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(i - j) * fac(j)] flags: rewrite

Additional information

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