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Proof step 23 from proof longin-mult-01 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

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: forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%lels-02: ⊦ i ≠ j → (i ≤ j ↔ i < j); from specification int-abs%abs-16: ⊦ 0 < j → abs(j) = j; 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%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%Abs-pos: ⊦ ¬ j < 0 → abs(j) = j; from specification int-abs%zero-gr-01: ⊦ i < 0 → (0 < i * j ↔ j < 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%zero-gels-03: ⊦ i < 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%abs-08: ⊦ i < 0 → abs(i) = ~ i; from specification int-abs%neg-06: ⊦ 0 - i = ~ i; from specification int-abs%minus-left: ⊦ ~ i * j = ~(i * j); from specification int-mult%neg-02: ⊦ ~ ~ i = i; from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs Used terms: i