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Proof step 30 from proof longin-mult-01 (previous step)

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)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%numlslsdel-01: i0 ≠ 0, i0 < i ⊦ (* i) < j → (* i0) < j; from specification int-basic1%sub-ls-01: ⊦ i < j - i0 ↔ i + i0 < j; from specification int-abs%minus-add: i < 0 ⊦ (* i) + j = j - ~ i; from specification int-abs%neg-13: ⊦ ~ i < j ↔ 0 < j + i; from specification int-abs%abs-08: ⊦ i < 0 → abs(i) = ~ i; from specification int-abs%zero-gr-01: ⊦ i < 0 → (0 < i * j ↔ j < 0); from specification int-mult%zero-gels-03: ⊦ i < 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%numls-03: ⊦ i + (* j) < (* j0) ↔ i < j0 - j; from specification int-abs%neg-07: ⊦ i < ~ j ↔ i + j < 0; from specification int-abs%abs-06: ⊦ 0 < abs(i) ↔ i ≠ 0; from specification int-abs%zero-ls-01: ⊦ i < 0 → (i * j < 0 ↔ 0 < j); from specification int-mult%neg: ⊦ 0 < ~ i ↔ i < 0; from specification int-abs%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%minus-left: ⊦ ~ i * j = ~(i * j); from specification int-mult%neg-06: ⊦ 0 - i = ~ i; from specification int-abs