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Proof step 14 from proof div-pos-def (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

seq: ⊦ 0 < i → (i * j ≤ 0 ↔ j ≤ 0) subst: [j, i] → [i0 / gcd(i0, j0), j / gcd(i, j)] rotate: F spec: int-mult inst: name: zero-le

Additional information

Used simplifier rules: ⊦ ¬ i < j ↔ j ≤ i;% ⊦ 0 < i → ¬ i / gcd(i, i0) ≤ 0;% ⊦ gcd(i, j) = gcd(j, i); Used terms: i0 / gcd(i0, j0), j / gcd(i, j)