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Proof step 9 from proof ls-def (previous step)

i * j0 = i1 + i0 * j, 0 < j0, 0 < j ⊦ j0 = 0, j = 0, rational(i1, j * j0).num < 0 ↔ ¬ i0 * j ≤ i * j0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ 0 < j → (rational(i, j).num < 0 ↔ i < 0)
Proof step 11:
0 < j * j0, i * j0 = i1 + i0 * j, 0 < j0, 0 < j ⊦ j0 = 0, j = 0, i1 < 0 ↔ ¬ i0 * j ≤ i * j0

Rule argument

name: num-neg-02 rotate: F subst: [j, i] → [j * j0, i1]

Additional information

Used simplifier rules: ⊦ i * j = j * i;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ 0 < i → (0 < i * j ↔ 0 < j);% ⊦ i ≠ j → (i ≤ j ↔ i < j);% ⊦ i < j → ¬ j < i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ ¬ i ≤ j ↔ j < i; Used terms: j * j0, i1