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Proof step 22 from proof square-neq-any-pos-01 (previous step)

j / gcd(j, j0) * j / gcd(j, j0) = ((j0 * i) * j0 / gcd(j0, j)) / gcd(j, j0), 0 < i, 0 < j, 0 < j0 ⊦ gcd(j, j0) = 1, j0 = 0, j = 0, ∃ i0. i = i0 * i0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 23:
⊦ i0 ≠ 0 → (i0 * j) / gcd(i0, j0) = i0 / gcd(i0, j0) * j
Proof step 24:
j0 ≠ 0, j / gcd(j, j0) * j / gcd(j, j0) = j0 / gcd(j0, j) * i * j0 / gcd(j0, j), 0 < i, 0 < j, 0 < j0 ⊦ gcd(j, j0) = 1, j0 = 0, j = 0, ∃ i0. i = i0 * i0

Rule argument

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

Additional information

Used simplifier rules: gcd-com: ⊦ gcd(i, j) = gcd(j, i);%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: j, i * j0 / gcd(j0, j), j0