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Proof step 4 from proof modeqp-03 (previous step)

i * j + j0 = i0 * j + j1, j < j0, j < j1, j0 ≤ 0, j1 ≤ 0, j < 0 ⊦ i = i0 ∧ j0 = j1, abs(j1) + j < 0 ∧ abs(j0) + j < 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 5:
⊦ j < 0 → abs(j) = 0 - j
Proof step 6:
j1 < 0, i * j + j0 = i0 * j + j1, j < j0, j < j1, j0 ≤ 0, j1 ≤ 0, j < 0 ⊦ i = i0 ∧ j0 = j1, (0 - j1) + j < 0 ∧ abs(j0) + j < 0

Rule argument

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

Additional information

Used simplifier rules: eq-neg: ⊦ i < 0 → (i * j = i * j0 ↔ j = j0);%associativity: ⊦ i * j * j0 = (i * j) * j0;%commutativity: ⊦ i * j = j * i;%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%anti: ⊦ i ≤ j → (j ≤ i ↔ i = j); from specification int-abs%lf-02: ⊦ i < j ∧ j = i0 → i < i0; (forward) from specification int-basic1%add-zero-rev: ⊦ 0 + i = i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%lels-02: ⊦ i ≠ j → (i ≤ j ↔ i < j); from specification int-abs%abs-08: ⊦ i < 0 → abs(i) = ~ i; from specification int-abs%neg-15: ⊦ i + ~ j = i - j; from specification int-abs%sub-le-01: ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; from specification int-abs%lels-03: ⊦ i < j → (¬ ¬ j ≤ i ↔ false); from specification int-abs%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs Used terms: j1