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Proof step 14 from proof div-ls-zero (previous step)

j0 < j, i0 < j, 0 < j, j0 < 0, 0 < j + j0, i0 < 0, 0 < j * (i0 + 1) + j0 ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 15:
⊦ i < 0 ∧ j < 0 → ¬ 0 < i + j

Rule argument

spec: int-basic2 name: ls-zero seq: ⊦ i < 0 ∧ j < 0 → ¬ 0 < i + j subst: [j, i] → [j0, j * (i0 + 1)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ 0 < i → (i * j < 0 ↔ j < 0); (s)% ⊦ i + (* j) < (* j0) ↔ i < j0 - j; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ i < j ↔ ¬ j ≤ i; (ss)% ⊦ i < 0 → (~1 ≤ i ↔ i = ~1); (s)% ⊦ i * 0 = 0; (s)% ⊦ i + 0 = i; (s)% ⊦ i < j → ¬ j < i; (s) Used terms: j0, j * (i0 + 1)