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Joined 2026-09-21 Message
Incoming links: Covariant type parameter
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.If the mutable set were allowed a covariant type parameter, both variables below could refer to the same mutable object:The broad view accepts any value, including a string. The narrow view still promises that deletion returns an
val s1: Set[Int] = new Set[Int](List())
val s2: Set[Any] = s1 // hypothetically permitted
s2.add("not an integer")
val n: Int = s1.delete() // cannot return an IntInt, yet the sole stored value is that string. This violates type safety:This is covariance failure for a mutable collection. The input occurrence in add(x: T) conflicts with the positive occurrence in the result of delete. Keeping a generic type invariance restriction prevents the unsafe aliasing. The example uses an initially empty set so that the problematic deletion result is forced.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.Immutable Scala
List[+A] is covariant: if Juice is a subtype of Drink, a list of juices can be used as a list of drinks. Scala Array[A] is invariant because it permits both reading and writing elements.If an array of juices could be treated as an array of drinks, writing a beer through that widened reference would violate the original juice-array contract. This illustrates covariant type parameter versus generic type invariance; immutability permits the first relationship for lists.
A covariant type parameter preserves subtyping: if , then
PartialFn[T,A] <: PartialFn[T,B] when its second parameter is covariant. In Scala the class annotation is +U. A mapping that produces a narrower output can be used where a broader output is expected. Covariance describes a type relationship; it does not convert stored values or make mutation through a widened reference safe.[ref]Scala 2.12 types specification. Relations between types: conformance of parameterized types and variance
The plus marks a covariant type parameter: if , then
Iterator[S] <: Iterator[T]. An iterator producing a subtype's values can be used where an iterator of the supertype is required. For example, a string iterator can be viewed as an iterator of AnyRef, because strings are reference values. The direction is appropriate for a producer: calling next() gives a value valid at the wider type. It would not be safe to expose an ordinary insertion method accepting arbitrary wider-type values into a subtype-only collection. This is a subtyping relationship, not a request to transform each produced value.[ref]Scala 2.12 types specification. Conformance of parameterized types with covariant parameters
The expression constructs a valid iterator object; it does not itself return the concatenated numeric or character values. The element types have a common supertype Successive calls return
AnyVal, and covariance lets Iterator[Int] and Iterator[Char] serve as the two inputs to ConcatIterator[AnyVal]. In the ordinary Scala 2 typing of this expression, the common element type is AnyVal. One can make the intended widening explicit:new ConcatIterator[AnyVal](
List(1, 2, 3).iterator,
List('a', 'b', 'c').iterator
)1, 2, 3, 'a', 'b', 'c', then exhaustion. Characters remain characters: viewing an existing iterator at a wider type does not convert its contents to integer character codes. No next() is invoked merely by constructing this adapter. The common-type argument uses subtype polymorphism and covariant type parameters, rather than requiring the two concrete input types to be equal.The maximally variant safe declarations are:
trait Rel1[+X, +Y] { val pairs: List[(X, Y)] }
trait Rel2[-X, +Y] { def relatives(x: X): List[Y] }
trait Rel3[-X, -Y] { def related(x: X, y: Y): Boolean }Leaving one or both parameters invariant is also legal but admits fewer subtyping conversions. These are type variance annotations.
For
Rel1, both parameters occur only as outputs. Scala's immutable List and Tuple2 are covariant, so a list of pairs of narrower types can safely be read as pairs of broader types. Thus both may be covariant type parameters. A contravariant declaration would permit, for example, treating a Rel1[Any, Int] containing (7, 0) as a Rel1[String, Int], then reading its first component as a String. That would be unsound; the compiler rejects the negative annotation at this output position.For
Rel2, X occurs as a method argument and therefore may be a contravariant type parameter; Y is returned in an immutable list and may be a covariant type parameter. A function that accepts every Any is safe to use where only String queries are issued. The opposite direction would be unsafe: if X were covariant, the following hypothetical conversion could call a string-only implementation with an integer:val strings: Rel2[String, Int] = new Rel2[String, Int] {
def relatives(x: String): List[Int] = List(x.length)
}
// Unsafe if Rel2 were incorrectly covariant in X:
// val arbitrary: Rel2[Any, Int] = strings
// arbitrary.relatives(7)Making
Y contravariant would instead allow an integer-valued Rel2[String, Any] to be read as a string-valued Rel2[String, String], recreating the output error.For
Rel3, both generic values are consumed and only a Boolean function result is produced, so both may be contravariant type parameters. A predicate accepting all values of broad types can answer queries restricted to narrower types. Covariance in either parameter would allow passing an unsupported value to a specialized implementation. For example, treating a string-only first-argument predicate as accepting Any would permit related(7, y) even though its body might call x.length. These are applications of the variance-position rule: method inputs reverse variance, whereas immutable outputs preserve it.[ref]Scala 2.13 variance annotations specification. Variance annotations and the change of position in method parameter types
Type variance 2026-10-24
How subtyping between parameter types affects subtyping between instances of a parameterized type. A covariant type parameter preserves the direction, a contravariant type parameter reverses it, and generic type invariance permits neither conversion solely from the parameter relationship. Safe variance depends on how a type is consumed, produced and stored.