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Proof step 20 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(j)) / fac(j - 1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 21:
⊦ j | i ∧ j ≠ 0 → (i / j = i0 ↔ i = i0 * j)
Proof step 22:
¬(fac(i - j) | fac(i) * fac(i + 1 - j) ∧ fac(i - j) ≠ 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(j - 1)
Proof step 27:
fac(i - j) | fac(i) * fac(i + 1 - j) ∧ fac(i - j) ≠ 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) * (i + 1 - 2 * j) + (fac(i) * fac(j)) / fac(j - 1)) * fac(i - j)

Rule argument

spec: int-gcd inst: name: div-eq seq: ⊦ j | i ∧ j ≠ 0 → (i / j = i0 ↔ i = i0 * j) subst: [i0, j, i] → [fac(i) * (i + 1 - 2 * j) + (fac(i) * fac(j)) / fac(j - 1), fac(i - j), fac(i) * fac(i + 1 - j)] flags: rewrite

Additional information

Used terms: fac(i) * (i + 1 - 2 * j) + (fac(i) * fac(j)) / fac(j - 1), fac(i - j), fac(i) * fac(i + 1 - j)