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Proof step 8 from proof modlep-04 (previous step)

j * i0 < j * i + abs(j1), i0 < i, abs(j1) < j, 0 < j, j1 < j, j1 ≤ 0, j * i ≤ j * i0 + j1 ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-abs name: Abs-neg seq: ⊦ j < 0 → abs(j) = 0 - j subst: [j] → [j1] flags: <all paths>

Additional information

Used simplifier rules: ls-pos: ⊦ 0 < i → (i * j < i * j0 ↔ j < j0);%associativity: ⊦ i * j * j0 = (i * j) * j0;%commutativity: ⊦ i * j = j * i;%le-pos: ⊦ 0 < i → (i * j ≤ i * i0 ↔ j ≤ i0);% ⊦ i < 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ 0 - i = ~ i; (s)% ⊦ i + ~ j = i - j; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ i < j - i0 ↔ i + i0 < j; (s)% ⊦ i ≤ j → (j < i ↔ false); (as)% ⊦ ¬ j < 0 → abs(j) = j; (s)% ⊦ i < j ↔ ¬ j ≤ i; (ss)% ⊦ i ≤ j → (j ≤ i ↔ i = j); (s)% ⊦ i + 0 = i; (s)% ⊦ i < j → (j ≤ i ↔ false); (as) Used terms: j1