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Proof step 7 from proof factor-rat-03 (previous step)

0 < j, 0 < i1 ⊦ i1 = 0, j = 0, i = 0, rational(i1 * i, i1 * j).num = i / gcd(i, j)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ i ≠ 0 ∧ 0 < j → rational(i, j).num = i / gcd(i, j)

Rule argument

name: numerator-pos rotate: F subst: [j, i] → [i1 * j, i1 * i]

Additional information

Used simplifier rules: ⊦ i < j → ¬ j < i;% ⊦ i ≠ j → (i ≤ j ↔ i < j);% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ 0 < i → (0 < i * j ↔ 0 < j);% ⊦ gcd(i * i1, j * i1) = gcd(i, j) * abs(i1);% ⊦ 0 < j → abs(j) = j;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i;% ⊦ gcd(i, j) = 0 ↔ i = 0 ∧ j = 0;%j ≠ 0, i0 ≠ 0 ⊦ (i * i0) / (j * i0) = i / j;% ⊦ i * j = 0 ↔ i = 0 ∨ j = 0; Used terms: i1 * j, i1 * i