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Proof step 4 from proof div-mult-neg (previous step)

abs(i0) < j, 0 < j, i ≤ 0, i < j, i0 ≤ 0, i0 < j ⊦ ~(j * ~ i + ~ i0) / j = i

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 5:
⊦ j ≠ 0 → ~ i / j = ~(i / j)
Proof step 6:
j ≠ 0, abs(i0) < j, 0 < j, i ≤ 0, i < j, i0 ≤ 0, i0 < j ⊦ ~((j * ~ i + ~ i0) / j) = i

Rule argument

spec: int-div inst: name: Div-left seq: ⊦ j ≠ 0 → ~ i / j = ~(i / j) subst: [j, i] → [j, j * ~ i + ~ i0] flags: rewrite, current

Additional information

Used simplifier rules: minus-left: ⊦ ~ i * j = ~(i * j); from specification int-mult%zero-left: ⊦ 0 * 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%add-zero-rev: ⊦ 0 + i = i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%neg-02: ⊦ ~ ~ i = i; from specification int-abs%abs-08: ⊦ i < 0 → abs(i) = ~ i; from specification int-abs%neg-05: ⊦ ~ i < 0 ↔ 0 < i; from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1 Used terms: j, j * ~ i + ~ i0