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Joined 2026-09-21 Message
Incoming links: Lower type bound
Covariant persistent multiset 2026-10-31
An immutable multiset can expose a covariant type parameter because reading an element only produces that type. Addition accepts a new type parameter with a lower type bound,
add[U >: T](x: U), and returns a new ISet[U]. The old ISet[T] retains its contents and original element guarantee. Deletion returns both the removed value and a new remaining multiset; it does not mutate the old one. Duplicate occurrences are preserved.An immutable object can widen its element type when adding a value by returning a new object. Introduce a lower type bound on the method's fresh type parameter.As before, the representation invariant excludes duplicates, and the abstraction function returns the unordered set of entries. The constructor normalization makes the representation invariant hold even for repeated initial entries. Both operations construct a fresh set object and leave the original object unchanged, including when addition finds an existing member.
class ISet[+T](contents: List[T]) {
private val elems: List[T] = contents.distinct
def add[U >: T](x: U): ISet[U] = {
val widened: List[U] = elems
if (widened.contains(x)) new ISet[U](widened)
else new ISet[U](x :: widened)
}
def delete(): (T, ISet[T]) = {
require(elems.nonEmpty)
(elems.head, new ISet[T](elems.tail))
}
}For a receiver representing , the new object returned by
add represents . Deletion returns a pair with and representing . Crucially,The widened list is valid because immutable List itself has a covariant type parameter. A client can view an ISet[Int] as ISet[Any], but adding a string creates a separate widened result; it cannot put a string into the old integer set. This explains how using immutable objects makes the desired type variance sound. A direct add(x: T) on a covariant software class would still violate the variance-position rule; immutability alone does not repair an incorrectly typed method signature.A covariant type parameter may be produced but not consumed directly. Widen the output type of
add through a lower type bound. In compose, the first parameter of CoPartialFn is invariant, so replace its occurrence of U by a fresh type parameter W with lower type bound U.class CoPartialFn[T,+U](val data: List[(T,U)]) {
require(data.map(_._1).distinct.size == data.size)
def get(x: T): Option[U] =
data.find(p => p._1 == x).map(_._2)
def add[V >: U](x: T, y: V): CoPartialFn[T,V] = {
val old: List[(T,V)] = data.filterNot(p => p._1 == x)
new CoPartialFn[T,V]((x,y) :: old)
}
def remove(x: T): CoPartialFn[T,U] =
new CoPartialFn[T,U](data.filterNot(p => p._1 == x))
def compose[V, W >: U](that: CoPartialFn[W,V]): CoPartialFn[T,V] = {
val pairs: List[(T,V)] = data.flatMap { case (x,y) =>
that.get(y).toList.map(z => (x,z))
}
new CoPartialFn[T,V](pairs)
}
}List, Tuple2, and Option are covariant in their element/output parameters, so data, get, and remove are valid producers. The variance-position rule permits U in the lower type bounds; every value of U can be passed to that.get, since . No cast or mutation through a widened reference is needed.[ref]Scala 2.11 basic declarations and definitions specification. Variance Annotations: variance positions and lower-bound widening
The input-key representation invariant and the partial function semantics are unchanged.Because The lower type bound gives , so stored output values can be widened into
Set is invariant in the specified Scala library, even returning Set[U] would place the class's covariant type parameter in an invariant position. Introduce a fresh widened type for each lifting.class CoPartialFnExt[T,+U](data: List[(T,U)])
extends CoPartialFn[T,U](data) {
def forwardLift[V >: U](s: Set[T]): Set[V] = {
val ys: List[V] = this.data.collect {
case (x,y) if s.contains(x) => y
}
ys.toSet
}
def backwardLift[V >: U](s: Set[V]): Set[T] =
this.data.collect {
case (x,y) if s.contains(y) => x
}.toSet
}List[V] or passed to s.contains. Both exposed Set types use invariant method type parameters, not the class's U. Calling forwardLift[U] or backwardLift[U] when the concrete output type is known recovers the exact output universe; a broader admissible universe is also safe.For a partial function, the forward formula is understood only for inputs in its domain of a function. The implementations therefore computeThe conversion to a set removes repeated images. Inputs not in the domain of a function contribute nothing to either lifting.
Store one entry per occurrence in an immutable list. Use lower type bounds for arguments that might widen the element type, and remove only the first matching occurrence:A covariant type parameter cannot appear directly as an ordinary object method argument type. The fresh with permits insertion, deletion queries and count queries without imposing such a negative occurrence of .
class CoMSet[+T](c: List[T]) {
private val elems: List[T] = c
def insert[U >: T](x: U): CoMSet[U] =
new CoMSet[U](x :: elems)
def delete[U >: T](x: U): CoMSet[T] = {
def removeOne(xs: List[T]): List[T] = xs match {
case Nil => Nil
case h :: tail =>
if (h == x) tail else h :: removeOne(tail)
}
new CoMSet[T](removeOne(elems))
}
def count[U >: T](x: U): Int =
elems.count(y => y == x)
def union[U >: T](that: CoMSet[U]): CoMSet[U] =
new CoMSet[U](elems ::: that.elems)
}delete can retain result type CoMSet[T] because it introduces no new elements; widening its result to CoMSet[U] would also be valid. union must widen because the other multiset may contain the broader type. The class-private immutable elems is accessible on another instance of the same software class; it is not private[this].[ref]Scala 2.11 basic declarations and definitions specification. Type Parameters, Variance Annotations; variable and method parameter positions
Let be a multiset multiplicity. The implementations satisfyThe brackets denote an indicator function. Structural induction on the list proves
removeOne removes precisely one equal entry if present, otherwise none. filterNot would incorrectly remove every occurrence. List concatenation adds multiplicities, implementing the paper's additive additive multiset union, rather than the maximum-multiplicity convention sometimes used for that name.All methods preserve the original immutable object.
insert takes constant time through structural sharing; count and delete take time, and union takes time to copy the receiver's list spine. Equality is ordinary stable value equality; finite representable counts are assumed.Use a lower type bound to widen the element type only in the new value:
class ISet[+T](contents: List[T]) {
private val data: List[T] = contents
def add[U >: T](x: U): ISet[U] =
new ISet[U](x :: data)
def delete(): (T, ISet[T]) = data match {
case x :: xs => (x, new ISet[T](xs))
case Nil => throw new NoSuchElementException("ISet.delete")
}
}The list is immutable and neither method changes this instance. A new wider value cannot be accepted directly as the old narrow
T; instead add[U >: T] returns a fresh ISet[U]. The original ISet[T] retains its element guarantee, and deletion returns both its old head and a fresh bag for the tail. This covariant persistent multiset safely shares unchanged list tails and preserves duplicate multiplicities. The historical name ISet does not require silently changing the question's bag semantics into duplicate-removing set semantics.[ref]Scala 2.13 variance annotations specification. Variance Annotations: lower-bound widening of Sequence.append
class VendingMachine[+A](val currentItem: Option[A], items: List[A]) {
def this(items: List[A]) = this(None,items)
def dispenseNext(): VendingMachine[A] = items match {
case Nil =>
if (currentItem.isDefined) new VendingMachine[A](None,Nil)
else this
case t :: ts => new VendingMachine[A](Some(t),ts)
}
def addAll[B >: A](newItems: List[B]): VendingMachine[B] =
new VendingMachine[B](items ++ newItems)
}The lower type bound permits the old pending elements of type
A to be used as type B, so concatenation is type-correct. The resulting machine is immutable and can return the wider type without changing the existing machine. With the given client expressions, VM1 has type VendingMachine[Juice] and VM2 can have type VendingMachine[Drink]. The supplied addAll body constructs a new pending-item machine; its constructor sets currentItem to None.Declare
+U, keep its stored linked list immutable, and generalize operations that need to accept output values by a fresh lower type bound:class PartialFn[T, +U](val data: List[(T, U)]) {
def get(x: T): Option[U] =
data.find { case (key, _) => key == x }.map(_._2)
def remove(x: T): PartialFn[T, U] =
new PartialFn[T, U](data.filterNot { case (key, _) => key == x })
def add[V >: U](x: T, y: V): PartialFn[T, V] =
new PartialFn[T, V]((x, y) ::
data.filterNot { case (key, _) => key == x })
def compose[V >: U, W](that: PartialFn[V, W]): PartialFn[T, W] = {
val pairs: List[(T, W)] = data.flatMap {
case (x, y) => that.get(y).map(z => (x, z)).toList
}
new PartialFn[T, W](pairs)
}
}get, remove, and the immutable val data expose U in covariant result positions: Scala's immutable List, Tuple2 and Option preserve that position. The old add(x:T,y:U) consumed U directly, violating the variance-position rule. The new add accepts a value of a chosen supertype V and returns a new mapping with that widened output type; it never inserts the wider value into the old narrower mapping.The original
compose also mentions U as the first, invariant parameter of PartialFn[U,V]. Changing only add is therefore insufficient. Its new first type parameter V >: U is independent of the class's covariant parameter except for the allowed lower bound. Any stored y:U is acceptable to that.get as a V, while the result has output type W. This lower type bound solution avoids unchecked variance annotations or casts. The official variance rules illustrate the same lower-bound widening technique for an immutable collection's consuming method; the later 2.13 text corroborates the rule, without identifying an installed 2018 compiler version.[ref]Scala 2.13 variance annotations specification. Variance Annotations: method-parameter positions, invariant type arguments, and the lower-bound widening example for Sequence.append
Supertype 2026-10-31
A type is a supertype of when is a subtype of . A lower type bound requires an admissible type argument to lie above a specified type in this subtyping relation.