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Proof step 16 from proof coef-mult-rec-lower (previous step)

(fac(i - j) * fac(j)) * i0 = fac(i), j ≤ i, 0 < i, 0 ≤ j, 0 ≤ i, fac(i + 1 - j) * fac(j - 1) | (fac(i - j) * fac(j)) * i0, 0 < j ⊦ ((fac(i - j) * fac(j)) * i0) / (fac(i - j) * fac(j)) * j = ((fac(i - j) * fac(j)) * i0) / (fac(i + 1 - j) * fac(j - 1)) * (i + 1 - j)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 17:
j ≠ 0 ⊦ (j * i) / j = i
Proof step 18:
fac(i - j) * fac(j) ≠ 0, (fac(i - j) * fac(j)) * i0 = fac(i), j ≤ i, 0 < i, 0 ≤ j, 0 ≤ i, fac(i + 1 - j) * fac(j - 1) | (fac(i - j) * fac(j)) * i0, 0 < j ⊦ i0 * j = ((fac(i - j) * fac(j)) * i0) / (fac(i + 1 - j) * fac(j - 1)) * (i + 1 - j)

Rule argument

spec: int-div inst: name: div-mult seq: j ≠ 0 ⊦ (j * i) / j = i subst: [i, j] → [i0, fac(i - j) * fac(j)] 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: i0, fac(i - j) * fac(j)