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Proof step 9 from proof coef-mult-rec (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ j | i ∧ j ≠ 0 → i / j * j0 = (i * j0) / j
Proof step 11:
¬(fac(i - j) * j * fac(j - 1) | fac(i) ∧ fac(i - j) * j * fac(j - 1) ≠ 0), 0 < j, 0 < i, 0 < j, j ≤ i, 0 ≤ j, 0 ≤ i ⊦ i * fac(i - 1) / (fac(i - j) * fac(j - 1)) = j * fac(i) / (fac(i - j) * j * fac(j - 1))
Proof step 18:
fac(i - j) * j * fac(j - 1) | fac(i) ∧ fac(i - j) * j * fac(j - 1) ≠ 0, 0 < j, 0 < i, 0 < j, j ≤ i, 0 ≤ j, 0 ≤ i ⊦ i * fac(i - 1) / (fac(i - j) * fac(j - 1)) = (fac(i) * j) / (fac(i - j) * j * fac(j - 1))

Rule argument

spec: int-gcd inst: name: div-mult-shift seq: ⊦ j | i ∧ j ≠ 0 → i / j * j0 = (i * j0) / j subst: [j0, j, i] → [j, fac(i - j) * j * fac(j - 1), fac(i)] flags: rewrite

Additional information

Used simplifier rules: associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: j, fac(i - j) * j * fac(j - 1), fac(i)