Compiler plugin for making type lambdas (type projections) easier to write
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Kind Projector

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"But I don't want to go among mad people," Alice remarked.

"Oh, you can't help that," said the Cat: "we're all mad here. I'm mad. You're mad."

"How do you know I'm mad?" said Alice.

"You must be," said the Cat, "or you wouldn't have come here."

--Lewis Carroll, "Alice's Adventures in Wonderland"


One piece of Scala syntactic noise that often trips people up is the use of type projections to implement anonymous, partially-applied types. For example:

// partially-applied type named "IntOrA"
type IntOrA[A] = Either[Int, A]

// type projection implementing the same type anonymously (without a name).
({type L[A] = Either[Int, A]})#L

Many people have wished for a better way to do this.

The goal of this plugin is to add a syntax for type lambdas. We do this by rewriting syntactically valid programs into new programs, letting us seem to add new keywords to the language. This is achieved through a compiler plugin performing an (un-typed) tree transformation.

One problem with this approach is that it changes the meaning of (potentially) valid programs. In practice this means that you must avoid defining the following identifiers:

  1. Lambda and λ
  2. ?, +?, and -?
  3. Λ$
  4. α$, β$, ...

If you find yourself using lots of type lambdas, and you don't mind reserving those identifiers, then this compiler plugin is for you!

Using the plugin

Kind-projector supports Scala 2.10, 2.11, 2.12, and 2.13.0-M4.

To use this plugin in your own projects, add the following lines to your build.sbt file:

resolvers += Resolver.sonatypeRepo("releases")

addCompilerPlugin("org.spire-math" %% "kind-projector" % "0.9.7")

// if your project uses multiple Scala versions, use this for cross building
addCompilerPlugin("org.spire-math" % "kind-projector" % "0.9.7" cross CrossVersion.binary)

// if your project uses both 2.10 and polymorphic lambdas
libraryDependencies ++= (scalaBinaryVersion.value match {
  case "2.10" =>
    compilerPlugin("org.scalamacros" % "paradise" % "2.1.0" cross CrossVersion.full) :: Nil
  case _ =>

For maven projects, add the plugin to the configuration of the maven-scala-plugin (remember to use _2.10, _2.11 or _2.12 as appropriate):


That's it!

Versions of the plugin earlier than 0.6.2 require a different resolver. For these earlier releases, use this:

resolvers += "bintray/non" at ""

Inline Syntax

The simplest syntax to use is the inline syntax. This syntax resembles Scala's use of underscores to define anonymous functions like _ + _.

Since underscore is used for existential types in Scala (and it is probably too late to change this syntax), we use ? for the same purpose. We also use +? and -? to handle covariant and contravariant types parameters.

Here are a few examples:

Tuple2[?, Double]        // equivalent to: type R[A] = Tuple2[A, Double]
Either[Int, +?]          // equivalent to: type R[+A] = Either[Int, A]
Function2[-?, Long, +?]  // equivalent to: type R[-A, +B] = Function2[A, Long, B]
EitherT[?[_], Int, ?]    // equivalent to: type R[F[_], B] = EitherT[F, Int, B]

As you can see, this syntax works when each type parameter in the type lambda is only used in the body once, and in the same order. For more complex type lambda expressions, you will need to use the function syntax.

Function Syntax

The more powerful syntax to use is the function syntax. This syntax resembles anonymous functions like x => x + 1 or (x, y) => x + y. In the case of type lambdas, we wrap the entire function type in a Lambda or λ type. Both names are equivalent: the former may be easier to type or say, and the latter is less verbose.

Here are some examples:

Lambda[A => (A, A)]              // equivalent to: type R[A] = (A, A)
Lambda[(A, B) => Either[B, A]]   // equivalent to: type R[A, B] = Either[B, A]
Lambda[A => Either[A, List[A]]]  // equivalent to: type R[A] = Either[A, List[A]]

Since types like (+A, +B) => Either[A, B] are not syntactically valid, we provide two alternate methods to specify variance when using function syntax:

  • Plus/minus: (+[A], +[B]) => Either[A, B]
  • Backticks: (`+A`, `+B`) => Either[A, B]

(Note that unlike names like ?, + and - do not have to be reserved. They will only be interpreted this way when used in parameters to Lambda[...] types, which should never conflict with other usage.)

Here are some examples with variance:

λ[`-A` => Function1[A, Double]]          // equivalent to: type R[-A] = Function1[A, Double]
λ[(-[A], +[B]) => Function2[A, Int, B]]  // equivalent to: type R[-A, +B] = Function2[A, Int, B]
λ[`+A` => Either[List[A], List[A]]]      // equivalent to: type R[+A] = Either[List[A], List[A]]

The function syntax also supports higher-kinded types as type parameters. The syntax overloads the existential syntax in this case (since the type parameters to a type lambda should never contain an existential).

Here are a few examples with higher-kinded types:

Lambda[A[_] => List[A[Int]]]  // equivalent to: type R[A[_]] = List[A[Int]]
Lambda[(A, B[_]) => B[A]]     // equivalent to: type R[A, B[_]] = B[A]

Finally, variance annotations on higher-kinded sub-parameters are supported using backticks:

Lambda[`x[+_]` => Q[x, List] // equivalent to: type R[x[+_]] = Q[x, List]
Lambda[`f[-_, +_]` => B[f]   // equivalent to: type R[f[-_, +_]] = B[f]

The function syntax with backtick type parameters is the most expressive syntax kind-projector supports. The other syntaxes are easier to read at the cost of being unable to express certain (hopefully rare) type lambdas.

Type lambda gotchas

The inline syntax is the tersest and is often preferable when possible. However, there are some type lambdas which it cannot express.

For example, imagine that we have trait Functor[F[_]].

You might want to write Functor[Future[List[?]]], expecting to get something like:

type X[a] = Future[List[a]]

However, ? always binds at the tightest level, meaning that List[?] is interpreted as type X[a] = List[a], and that Future[List[?]] is invalid.

In these cases you should prefer the lambda syntax, which would be written as:

Functor[Lambda[a => Future[List[a]]]]

Other types which cannot be written correctly using inline syntax are:

  • Lambda[a => (a, a)] (repeated use of a).
  • Lambda[(a, b) => Either[b, a]] (reverse order of type params).
  • Lambda[(a, b) => Function1[a, Option[b]]] (similar to example).

(And of course, you can use λ[...] instead of Lambda[...] in any of these expressions.)

Under The Hood

This section shows the exact code produced for a few type lambda expressions.

Either[Int, ?]
({type Λ$[β$0$] = Either[Int, β$0$]})#Λ$

Function2[-?, String, +?]
({type Λ$[-α$0$, +γ$0$] = Function2[α$0$, String, γ$0$]})#Λ$

Lambda[A => (A, A)]
({type Λ$[A] = (A, A)})#Λ$

Lambda[(`+A`, B) => Either[A, Option[B]]]
({type Λ$[+A, B] = Either[A, Option[B]]})#Λ$

Lambda[(A, B[_]) => B[A]]
({type Λ$[A, B[_]] = B[A]})#Λ$

As you can see, names like Λ$ and α$ are forbidden because they might conflict with names the plugin generates.

If you dislike these unicode names, pass -Dkp:genAsciiNames=true to scalac to use munged ASCII names. This will use L_kp in place of Λ$, X_kp0$ in place of α$, and so on.

Polymorphic lambda values

Scala does not have built-in syntax or types for anonymous function values which are polymorphic (i.e. which can be parameterized with types). To illustrate that consider both of these methods:

def firstInt(xs: List[Int]): Option[Int] = xs.headOption
def firstGeneric[A](xs: List[A]): Option[A] = xs.headOption

Having implemented these methods, we can see that the second just generalizes the first to work with any type: the function bodies are identical. We'd like to be able to rewrite each of these methods as a function value, but we can only represent the first method (firstInt) this way:

val firstInt0: List[Int] => Option[Int] = _.headOption
val firstGeneric0 <what to put here???>

(One reason to want to do this rewrite is that we might have a method like .map which we'd like to pass an anonymous function value.)

Several libraries define their own polymorphic function types, such as the following polymorphic version of Function1 (which we can use to implement firstGeneric0):

trait PolyFunction1[-F[_], +G[_]] {
  def apply[A](fa: F[A]): G[A]

val firstGeneric0: PolyFunction1[List, Option] =
  new PolyFunction1[List, Option] {
    def apply[A](xs: List[A]): Option[A] = xs.headOption

It's nice that PolyFunction1 enables us to express polymorphic function values, but at the level of syntax it's not clear that we've saved much over defining a polymorphic method (i.e. firstGeneric).

Since 0.9.0, Kind-projector provides a value-level rewrite to fix this issue and make polymorphic functions (and other types that share their general shape) easier to work with:

val firstGeneric0 = λ[PolyFunction1[List, Option]](_.headOption)

Either λ or Lambda can be used (in a value position) to trigger this rewrite. By default, the rewrite assumes that the "target method" to define is called apply (as in the previous example), but a different method can be selected via an explicit call.

In the following example we are using the polymorphic lambda syntax to define a run method on an instance of the PF trait:

trait PF[-F[_], +G[_]] {
  def run[A](fa: F[A]): G[A]

val f = Lambda[PF[List, Option]].run(_.headOption)

It's possible to nest this syntax. Here's an example taken from the wild of using nested polymorphic lambdas to remove boilerplate:

// without polymorphic lambdas, as in the slide
def injectFC[F[_], G[_]](implicit I: Inject[F, G]) =
  new (FreeC[F, ?] ~> FreeC[G, ?]) {
    def apply[A](fa: FreeC[F, A]): FreeC[G, A] =
      fa.mapSuspension[Coyoneda[G, ?]](
        new (Coyoneda[F, ?] ~> Coyoneda[G, ?]) {
          def apply[B](fb: Coyoneda[F, B]): Coyoneda[G, B] = fb.trans(I)

// with polymorphic lambdas
def injectFC[F[_], G[_]](implicit I: Inject[F, G]) =
  λ[FreeC[F, ?] ~> FreeC[G, ?]](
    _.mapSuspension(λ[Coyoneda[F, ?] ~> Coyoneda[G, ?]](_.trans(I)))

Kind-projector's support for type lambdas operates at the type level (in type positions), whereas this feature operates at the value level (in value positions). To avoid reserving too many names the λ and Lambda names were overloaded to do both (mirroring the relationship between types and their companion objects).

Here are some examples of expressions, along with whether the lambda symbol involved represents a type (traditional type lambda) or a value (polymorphic lambda):

// type lambda (type level)
val functor: Functor[λ[a => Either[Int, a]]] = implicitly

// polymorphic lambda (value level)
val f = λ[Vector ~> List](_.toList)

// type lambda (type level)
trait CF2 extends Contravariant[λ[a => Function2[a, a, Double]]] {

// polymorphic lambda (value level)
xyz.translate(λ[F ~> G](fx => fx.flatMap(g)))

One pattern you might notice is that when λ occurs immediately within [] it is referring to a type lambda (since [] signals a type application), whereas when it occurs after = or within () it usually refers to a polymorphic lambda, since those tokens usually signal a value. (The () syntax for tuple and function types is an exception to this pattern.)

The bottom line is that if you could replace a λ-expression with a type constructor, it's a type lambda, and if you could replace it with a value (e.g. new Xyz[...] { ... }) then it's a polymorphic lambda.

Polymorphic lambdas under the hood

What follows are the gory details of the polymorphic lambda rewrite.

Polymorphic lambdas are a syntactic transformation that occurs just after parsing (before name resolution or typechecking). Your code will be typechecked after the rewrite.

Written in its most explicit form, a polymorphic lambda looks like this:

λ[Op[F, G]].someMethod(<expr>)

and is rewritten into something like this:

new Op[F, G] {
  def someMethod[A](x: F[A]): G[A] = <expr>(x)

(The names A and x are used for clarity –- in practice unique names will be used for both.)

This rewrite requires that the following are true:

  • F and G are unary type constructors (i.e. of shape F[_] and G[_]).
  • <expr> is an expression of type Function1[_, _].
  • Op is parameterized on two unary type constructors.
  • someMethod is parametric (for any type A it takes F[A] and returns G[A]).

For example, Op might be defined like this:

trait Op[M[_], N[_]] {
  def someMethod[A](x: M[A]): N[A]

The entire λ-expression will be rewritten immediately after parsing (and before name resolution or typechecking). If any of these constraints are not met, then a compiler error will occur during a later phase (likely type-checking).

Here are some polymorphic lambdas along with the corresponding code after the rewrite:

val f = Lambda[NaturalTransformation[Stream, List]](_.toList)
val f = new NaturalTransformation[Stream, List] {
  def apply[A](x: Stream[A]): List[A] = x.toList

type Id[A] = A
val g = λ[Id ~> Option].run(x => Some(x))
val g = new (Id ~> Option) {
  def run[A](x: Id[A]): Option[A] = Some(x)

val h = λ[Either[Unit, ?] Convert Option](_.fold(_ => None, a => Some(a)))
val h = new Convert[Either[Unit, ?], Option] {
  def apply[A](x: Either[Unit, A]): Option[A] =
    x.fold(_ => None, a => Some(a))

// that last example also includes a type lambda.
// the full expansion would be:
val h = new Convert[({type Λ$[β$0$] = Either[Unit, β$0$]})#Λ$, Option] {
  def apply[A](x: ({type Λ$[β$0$] = Either[Unit, β$0$]})#Λ$): Option[A] =
    x.fold(_ => None, a => Some(a))

Unfortunately the type errors produced by invalid polymorphic lambdas are likely to be difficult to read. This is an unavoidable consequence of doing this transformation at the syntactic level.

Building the plugin

You can build kind-projector using SBT 0.13.0 or newer.

Here are some useful targets:

  • compile: compile the code
  • package: build the plugin jar
  • test: compile the test files (no tests run; compilation is the test)
  • console: launch a REPL with the plugin loaded so you can play around

You can use the plugin with scalac by specifying it on the command-line. For instance:

scalac -Xplugin:kind-projector_2.10-0.6.0.jar test.scala

Known issues & errata

When dealing with type parameters that take covariant or contravariant type parameters, only the function syntax is supported. Huh???

Here's an example that highlights this issue:

def xyz[F[_[+_]]] = 12345
trait Q[A[+_], B[+_]]

// we can use kind-projector to adapt Q for xyz
xyz[λ[`x[+_]` => Q[x, List]] // ok

// but these don't work (although support for the second form
// could be added in a future release).
xyz[Q[?[+_], List]]          // invalid syntax
xyz[Q[?[`+_`], List]]        // unsupported

There have been suggestions for better syntax, like [A, B]Either[B, A] or [A, B] => Either[B, A] instead of Lambda[(A, B) => Either[B, A]]. Unfortunately this would actually require modifying the parser (i.e. the language itself) which is outside the scope of this project (at least, until there is an earlier compiler phase to plug into).

Others have noted that it would be nicer to be able to use _ for types the way we do for values, so that we could use Either[Int, _] to define a type lambda the way we use 3 + _ to define a function. Unfortunately, it's probably too late to modify the meaning of _, which is why we chose to use ? instead.

Future Work

As of 0.5.3, kind-projector should be able to support any type lambda that can be expressed via type projections, at least using the function syntax. If you come across a type for which kind-projector lacks a syntax, please report it.


Kind projector is an unusual compiler plugin in that it runs before the typer phase. This means that the rewrites and renaming we are doing are relatively fragile, and the author disclaims all warranty or liability of any kind.

(That said, there are currently no known bugs.)

If you are using kind-projector in one of your projects, please feel free to get in touch to report problems (or a lack of problems)!


The project's current maintainers are:

Copyright and License

All code is available to you under the MIT license, available at and also in the COPYING file.

Copyright Erik Osheim, 2011-2017.