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Proof step 28 from proof longin-mult-01 (previous step)

0 < abs(i) * abs(j), 4611686014132420609 < abs(i) * abs(j), 9223372036854775807 < i * j, ~2147483648 ≤ i, i ≤ 2147483647, ~2147483648 ≤ j, j ≤ 2147483647, 0 < i * j ⊦ ¬ 2147483647 < abs(i) ∧ ¬ 2147483647 < abs(j)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 29:
⊦ j < 0 → abs(j) = 0 - j
Proof step 30:
i < 0, 0 < (0 - i) * abs(j), 4611686014132420609 < (0 - i) * abs(j), 9223372036854775807 < i * j, ~2147483648 ≤ i, i ≤ 2147483647, ~2147483648 ≤ j, j ≤ 2147483647, 0 < i * j ⊦ ¬ 2147483647 < 0 - i ∧ ¬ 2147483647 < abs(j)

Rule argument

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

Additional information

Used simplifier rules: Abs-pos: ⊦ ¬ j < 0 → abs(j) = j; from specification int-abs%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%numleledel-01: i0 ≠ 0, i0 ≤ i ⊦ (* i) ≤ j → (* i0) ≤ j; from specification int-abs%zero-gr-02: ⊦ 0 ≤ i → (0 < i * j ↔ ¬ ¬(0 < j ∧ i ≠ 0)); from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%abs-16: ⊦ 0 < j → abs(j) = j; from specification int-abs%zero-gr: ⊦ 0 < i → (0 < i * j ↔ 0 < j); from specification int-mult%numlslsdel-01: i0 ≠ 0, i0 < i ⊦ (* i) < j → (* i0) < j; from specification int-basic1%lels-02: ⊦ i ≠ j → (i ≤ j ↔ i < j); from specification int-abs%lels-04: ⊦ i ≤ j → (¬ ¬ j < i ↔ false); from specification int-abs Used terms: i