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Proof step 23 from proof coef-mult-rec-lower (previous step)

(fac(j - 1) * fac(i + 1 - j)) * i1 = fac(i), (fac(i - j) * fac(j)) * i0 = (fac(j - 1) * fac(i + 1 - j)) * i1, 0 < j, 0 ≤ i, 0 ≤ j, 0 < i, j ≤ i ⊦ i0 * j = ((fac(j - 1) * fac(i + 1 - j)) * i1) / (fac(i + 1 - j) * fac(j - 1)) * (i + 1 - j)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 24:
j ≠ 0 ⊦ (j * i) / j = i
Proof step 25:
fac(j - 1) * fac(i + 1 - j) ≠ 0, (fac(j - 1) * fac(i + 1 - j)) * i1 = fac(i), (fac(i - j) * fac(j)) * i0 = (fac(j - 1) * fac(i + 1 - j)) * i1, 0 < j, 0 ≤ i, 0 ≤ j, 0 < i, j ≤ i ⊦ i0 * j = i1 * (i + 1 - j)

Rule argument

spec: int-div inst: name: div-mult seq: j ≠ 0 ⊦ (j * i) / j = i subst: [i, j] → [i1, fac(j - 1) * fac(i + 1 - 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: i1, fac(j - 1) * fac(i + 1 - j)