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Proof step 3 from proof mod-mult-neg (previous step)

⊦ 0 < j ∧ i ≤ 0 ∧ i0 ≤ 0 ∧ abs(i0) < j → ~(j * ~ i + ~ i0) % j = i0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
j ≠ 0 ⊦ ~ i % j = ~(i % j)
Proof step 5:
j ≠ 0 ⊦ 0 < j ∧ i ≤ 0 ∧ i0 ≤ 0 ∧ abs(i0) < j → ~((j * ~ i + ~ i0) % j) = i0

Rule argument

spec: int-div inst: name: mod-neg-01 seq: j ≠ 0 ⊦ ~ i % j = ~(i % j) subst: [j, i] → [j, j * ~ i + ~ i0] flags: rewrite, current

Additional information

Used simplifier rules: mod-zero-05: ⊦ abs(i) < j → i % j = i;%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 Used terms: j, j * ~ i + ~ i0