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Proof step 17 from proof coef-sub-mult (previous step)

j ≤ i, 0 < i, 0 < j, fac(i - j) * fac(j) | fac(i), fac(i + 1 - j) * fac(j - 1) | fac(i), 0 ≤ j, 0 ≤ i ⊦ (fac(i) * fac(i + 1 - j)) / fac(i - j) = fac(i) * (i + 1 - 2 * j) + fac(i) / (fac(i + 1 - j) * fac(j - 1)) * fac(j) * fac(i + 1 - j)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 18:
⊦ j | i ∧ j ≠ 0 → i / j * j0 = (i * j0) / j
Proof step 19:
fac(i + 1 - j) * fac(j - 1) | fac(i) ∧ fac(i + 1 - j) * fac(j - 1) ≠ 0, j ≤ i, 0 < i, 0 < j, fac(i - j) * fac(j) | fac(i), fac(i + 1 - j) * fac(j - 1) | fac(i), 0 ≤ j, 0 ≤ i ⊦ (fac(i) * fac(i + 1 - j)) / fac(i - j) = fac(i) * (i + 1 - 2 * j) + (fac(i) * fac(j) * fac(i + 1 - j)) / (fac(i + 1 - 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] → [fac(j) * fac(i + 1 - j), fac(i + 1 - j) * fac(j - 1), fac(i)] flags: rewrite

Additional information

Used simplifier rules: fac-not-zero: ⊦ fac(i) * fac(j) ≠ 0;%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: fac(j) * fac(i + 1 - j), fac(i + 1 - j) * fac(j - 1), fac(i)