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Proof step 3 from proof modzero-05 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult inst: name: zero-ls-01 seq: ⊦ i < 0 → (i * j < 0 ↔ 0 < j) subst: [j, i] → [j, i + 1] flags: rewrite, current

Additional information

Used simplifier rules: dist-rsplit: ⊦ i * (j + j0) = i * j + i * j0;%one: ⊦ 1 * i = i;%associativity: ⊦ i * j * j0 = (i * j) * j0;%commutativity: ⊦ i * j = j * i;%minus-one: ⊦ i + 1 = 0 → i * j = ~ j;%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%lst-04: ⊦ i + j < i0 ∧ j0 < j → i + j0 < i0; from specification int-basic2%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%les: ⊦ i < j → (j ≤ i + 1 ↔ j = i + 1); from specification int-abs%neg-15: ⊦ i + ~ j = i - j; from specification int-abs%mone-03: ⊦ i + 1 = 0 → i < 0; (forward) from specification int-basic2%lf-02: ⊦ i < j ∧ j = i0 → i < i0; (forward) from specification int-basic1%succ-02: ⊦ i < j + 1 ↔ i ≤ j; from specification int-abs%sub-le: ⊦ i - j ≤ i0 ↔ i ≤ j + i0; from specification int-abs%ler: ⊦ i ≤ i; from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%ls-02: ⊦ i < i + j ↔ 0 < j; from specification int-basic2 Used terms: j, i + 1