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

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

User applied rule insert rewrite lemma

And obtained the following new goals

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

Rule argument

spec: <toplevel> name: mod-neg-01 seq: j ≠ 0 ⊦ ~ i % j = ~(i % j) subst: [j, i] → [j, j * ~ i + ~ i0] flags: <all paths>

Additional information

Used simplifier rules: mod-zero-05: ⊦ abs(i) < j → i % j = i;% ⊦ i * 0 = 0; (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ i + 0 = i; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ ~ ~ i = i; (s) Used terms: j, j * ~ i + ~ i0