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Proof step 5 from proof zero-ge-add-02 (previous step)

(i + j) * j0 + 1 < j0, 0 < j0, (i + j) * j0 < 0, (i + j) * j0 < i + j, (i + j) * j0 < j0, 0 ≤ i + j ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ 0 < i → (i * j < 0 ↔ j < 0)
Proof step 7:
(i + j) * j0 < j0, (i + j) * j0 < i + j, (i + j) * j0 < 0, 0 < j0, (i + j) * j0 + 1 < j0, 0 ≤ i + j ⊦ 0 < i + j

Rule argument

spec: <toplevel> name: zero-ls seq: ⊦ 0 < i → (i * j < 0 ↔ j < 0) subst: [j, i] → [j0, i + j] flags: <all paths>

Additional information

Used simplifier rules: dist-lsplit: ⊦ (i + i0) * j = i * j + i0 * j;%associativity: ⊦ i * j * j0 = (i * j) * j0;%commutativity: ⊦ i * j = j * i;%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ i < j → ¬ j < i; (s) Used terms: j0, i + j