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Proof step 13 from proof longin-mult-01 (previous step)

~(2 ^ 31) * j ≤ i * j, i * j < ~9223372036854775808, ~(2 ^ 31) ≤ i, ~(2 ^ 31) ≤ j, j ≤ 2147483647, i < 0, 0 < j ⊦ ~(2 ^ 31) * j < ~(2 ^ 31 * 2 ^ 32)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-abs inst: name: forward-lels seq: ⊦ i ≤ j ∧ j < j0 → i < j0 subst: [i, j, j0] → [~(2 ^ 31) * j, i * j, ~(2 ^ 31 * 2 ^ 32)] flags: rewrite

Additional information

Used simplifier rules: le-pos: ⊦ 0 < i → (i * j ≤ i * i0 ↔ j ≤ i0); from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%numleledel-01: i0 ≠ 0, i0 ≤ i ⊦ (* i) ≤ j → (* i0) ≤ j; from specification int-abs%numlsneg-02: i < 0 ⊦ (* i) * i0 < j ↔ 0 < ~ i * i0 + j; from specification int-mult%minus-add: i < 0 ⊦ (* i) + j = j - ~ i; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%sub-ls-01: ⊦ i < j - i0 ↔ i + i0 < j; from specification int-abs%nummult-ls-02: ⊦ 0 < i ∧ 0 ≤ j → ((* j) < (* i) * i0 ↔ j / i < i0); from specification int-div%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%numleledel: j0 ≠ 0, j ≤ j0 ⊦ i ≤ (* j) → i ≤ (* j0); from specification int-abs%numlelsdel: j0 ≠ 0, j ≤ j0 ⊦ i < (* j) → i < (* j0); from specification int-abs Used terms: ~(2 ^ 31) * j, i * j, ~(2 ^ 31 * 2 ^ 32)