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Proof step 20 from proof div-nearer-zero (previous step)

~ i < j * ~ i / j, i / j + 1 < j, i + 1 < j, i < 0, i / j * j < i, i / j * j < ~1, 0 < j, i / j < 0, i / j < j, i < j ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: inst: name: ls-same-pos-02 seq: ⊦ 0 ≤ i ∧ 0 < j → ¬ i < i / j * j subst: [j, i] → [j, ~ i] flags: rewrite, current

Additional information

Used simplifier rules: neg-01: j ≠ 0 ⊦ ~ i / j = ~(i / j);%numls-03: ⊦ i + (* j) < (* j0) ↔ i < j0 - j; 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%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%zero-left: ⊦ 0 * i = 0; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1%minus-left: ⊦ ~ i * j = ~(i * j); from specification int-mult%neg-01: ⊦ ~ j < ~ i ↔ i < j; from specification int-abs%negle-07: ⊦ i ≤ ~ j ↔ i + j ≤ 0; from specification int-abs%add-zero-rev: ⊦ 0 + i = i; from specification int-basic2%zero-gels-03: ⊦ i < 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: j, ~ i