= Solution
Nonnegative count functions form a valid subtype of the immutable value interface, but they are not closed under its exact subtraction. Preserve the inherited result type rather than silently changing the subtraction contract:
``
trait MultiSet[T] extends CompleteMultiSet[T] {
// Additional invariant: count(t) >= 0 for every t.
// Inherited arithmetic still returns CompleteMultiSet[T].
}
class MultiSetImp[T](initial: Map[T, Int])
extends CompleteMultiSetImp[T](initial) with MultiSet[T] {
require(initial.values.forall(_ >= 0), "negative multiplicity")
}
def ms[T](ts: T*): MultiSet[T] =
new MultiSetImp[T](ts.distinct.map(t => (t, 1)).toMap)
``
\b[The invariant guarantees nonnegative multiplicities in every `MultiSet` object], and the factory gives each supplied element exactly one occurrence. Since the objects are immutable, inherited operations cannot invalidate that invariant in the receiver. Their declared result is a <complete multiset>, which can legitimately contain negative counts; for example, subtracting a positive count from zero produces a signed result.
If an ordinary <multiset>-only difference is desired, introduce a separate operation with count $\max(0,a(t)-b(t))$, or an interface whose subtraction explicitly has that contract. Replacing the parent's exact difference by truncated difference under the same signature would violate <behavioral subtyping>; strengthening its <precondition> to forbid negative results would also fail to implement the parent's unrestricted mathematical contract. Retaining the complete result type above reuses every correct default without that substitution error.
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