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Proof step 8 from proof coef-sub-mult (previous step)

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

User applied rule insert lemma

And obtained the following new goals

Proof step 9:
⊦ i ≤ j ∧ 0 ≤ i → fac(j - i) * fac(i) | fac(j)
Proof step 10:
j ≤ i ∧ 0 ≤ j → fac(i - j) * fac(j) | fac(i), j - 1 ≤ i ∧ 0 ≤ j - 1 → fac(i - (j - 1)) * fac(j - 1) | fac(i), 0 ≤ j - 1 ∧ j - 1 ≤ i, 0 ≤ j ∧ j ≤ i, j ≤ i, 0 < j, 0 < i ⊦ (fac(i) / (fac(i - j) * fac(j)) - fac(i) / (fac(i - (j - 1)) * fac(j - 1))) * fac(j) * fac(i + 1 - j) = fac(i) * (i + 1 - 2 * j)

Rule argument

spec: inst: name: fac-divides seq: ⊦ i ≤ j ∧ 0 ≤ i → fac(j - i) * fac(i) | fac(j) subst: [i, j] → [j, i] flags: current, []

Additional information

Used terms: j, i