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Proof step 32 from proof coef-mult-rec-lower (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 33:
⊦ i ≠ 0 → (i * j = i * j0 ↔ j = j0)
Proof step 34:
fac(i - j) ≠ 0, (fac(i - j) * j * fac(j - 1)) * i0 = fac(i), j * fac(j - 1) * i0 = i1 * fac(j - 1) * (i + 1 - j), 0 < j, 0 ≤ i, 0 ≤ j, 0 < i, j ≤ i ⊦ i0 * j = i1 * (i + 1 - j)

Rule argument

spec: int-mult inst: name: eq-nonzero seq: ⊦ i ≠ 0 → (i * j = i * j0 ↔ j = j0) subst: [j0, j, i] → [i1 * fac(j - 1) * (i + 1 - j), j * fac(j - 1) * i0, fac(i - j)] flags: rewrite

Additional information

Used simplifier rules: fac-zero-02: ⊦ fac(i) ≠ 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) * (i + 1 - j), j * fac(j - 1) * i0, fac(i - j)