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Proof step 5 from proof ls-def (previous step)

0 < j0, 0 < j ⊦ (rational(i, j) + rational(~ i0, j0)).num < 0 ↔ ¬ i0 * j ≤ i * j0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ j1 ≠ 0 ∧ j2 ≠ 0 → rational(i1, j1) + rational(i2, j2) = rational(i1 * j2 + j1 * i2, j1 * j2)
Proof step 7:
j ≠ 0 ∧ j0 ≠ 0, 0 < j0, 0 < j ⊦ rational(i * j0 + j * ~ i0, j * j0).num < 0 ↔ ¬ i0 * j ≤ i * j0

Rule argument

name: add-def-03 rotate: F subst: [j2, i2, j1, i1] → [j0, ~ i0, j, i]

Additional information

Used simplifier rules: ⊦ i * 0 = 0;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i;% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j);% ⊦ 0 < i → (i * j ≤ 0 ↔ j ≤ 0); Used terms: j0, ~ i0, j, i