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Proof step 19 from proof equality-pos (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 20:
⊦ i ≠ 0 → i / gcd(i, j) * j0 = (i * j0) / gcd(i, j)
Proof step 21:
j0 ≠ 0, i * j0 = i0 * j, 0 < i0, 0 < i, 0 < j0, 0 < j ⊦ i0 = 0, i = 0, j0 = 0, j = 0, i = (i0 * gcd(i, j)) / gcd(i0, j0) ∧ j = (j0 * gcd(j, i)) / gcd(j0, i0)

Rule argument

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

Additional information

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