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Proof step 12 from proof equality-01 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 13:
⊦ i ≠ 0 → (i / gcd(i, j) = i0 ↔ i = i0 * gcd(i, j))
Proof step 14:
j ≠ 0, j = j0 / gcd(i0, j0) * gcd(j, i), i = i0 / gcd(i0, j0) * gcd(i, j), 0 < j0, 0 < j ⊦ i = 0, i0 = 0, j0 = 0, j = 0, i * j0 = i0 * j

Rule argument

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

Additional information

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