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Proof step 38 from proof div-pos-def (previous step)

0 < gcd(j0, i0), 0 < gcd(i, j), 0 < (j * i0 / gcd(i0, j0)) / gcd(j, i), 0 < i0, 0 < j0, 0 < j, 0 < gcd(i0, j0), 0 < gcd(j, i), 0 < i ⊦ j0 = 0, i0 = 0, gcd(j, i) = 0, j = 0, (j0 * i) / (gcd(j0, i0) * gcd(i, j)) * j * i0 = (i0 * i * j0 * j) / (gcd(i0, j0) * gcd(j, i)), i = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 39:
⊦ j | i ∧ j ≠ 0 → i / j * j0 = (i * j0) / j
Proof step 40:
¬(gcd(j0, i0) * gcd(i, j) | j0 * i ∧ gcd(j0, i0) * gcd(i, j) ≠ 0), 0 < gcd(j0, i0), 0 < gcd(i, j), 0 < (j * i0 / gcd(i0, j0)) / gcd(j, i), 0 < i0, 0 < j0, 0 < j, 0 < gcd(i0, j0), 0 < gcd(j, i), 0 < i ⊦ j0 = 0, i0 = 0, gcd(j, i) = 0, j = 0, (j0 * i) / (gcd(j0, i0) * gcd(i, j)) * j * i0 = (i0 * i * j0 * j) / (gcd(i0, j0) * gcd(j, i)), i = 0

Rule argument

seq: ⊦ j | i ∧ j ≠ 0 → i / j * j0 = (i * j0) / j subst: [j0, j, i] → [j * i0, gcd(j0, i0) * gcd(i, j), j0 * i] rotate: F spec: int-gcd inst: name: div-mult-shift

Additional information

Used simplifier rules: ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i;% ⊦ gcd(i, j) = gcd(j, i); Used terms: j * i0, gcd(j0, i0) * gcd(i, j), j0 * i