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Proof step 20 from proof div-pos-def (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 21:
⊦ j | i ∧ j ≠ 0 → i / j * j0 = (i * j0) / j
Proof step 22:
gcd(j, i) | j * i0 / gcd(i0, j0) ∧ gcd(j, i) ≠ 0, 0 < i, 0 < j, 0 < j0, 0 < i0, 0 < (j * i0 / gcd(i0, j0)) / gcd(j, i) ⊦ i = 0, j0 = 0, i0 = 0, (i / gcd(i, j) * j0 / gcd(i0, j0)) * j * i0 = ((j * i0 / gcd(i0, j0)) * j0 * i) / gcd(j, i), j = 0

Rule argument

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

Additional information

Used simplifier rules: ⊦ gcd(i, j) | i * i0;% ⊦ gcd(i, j) = 0 ↔ i = 0 ∧ j = 0;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i; Used terms: j0 * i, gcd(j, i), j * i0 / gcd(i0, j0)