Type Class Derivation

Type class derivation is a way to automatically generate given instances for type classes which satisfy some simple conditions. A type class in this sense is any trait or class with a type parameter determining the type being operated on. Common examples are Eq, Ordering, or Show. For example, given the following Tree algebraic data type (ADT),

enum Tree[T] derives Eq, Ordering, Show {
  case Branch[T](left: Tree[T], right: Tree[T])
  case Leaf[T](elem: T)
}

The derives clause generates the following given instances for the Eq, Ordering and Show type classes in the companion object of Tree,

given [T: Eq]       : Eq[Tree[T]]    = Eq.derived
given [T: Ordering] : Ordering[Tree] = Ordering.derived
given [T: Show]     : Show[Tree]     = Show.derived

We say that Tree is the deriving type and that the Eq, Ordering and Show instances are derived instances.

Types supporting derives clauses

All data types can have a derives clause. This document focuses primarily on data types which also have an instance of the Mirror type class available. Instances of the Mirror type class are generated automatically by the compiler for,

Mirror type class instances provide information at the type level about the components and labelling of the type. They also provide minimal term level infrastructure to allow higher level libraries to provide comprehensive derivation support.

sealed trait Mirror {

  /** the type being mirrored */
  type MirroredType

  /** the type of the elements of the mirrored type */
  type MirroredElemTypes

  /** The mirrored *-type */
  type MirroredMonoType

  /** The name of the type */
  type MirroredLabel <: String

  /** The names of the elements of the type */
  type MirroredElemLabels <: Tuple
}

object Mirror {
  /** The Mirror for a product type */
  trait Product extends Mirror {

    /** Create a new instance of type `T` with elements taken from product `p`. */
    def fromProduct(p: scala.Product): MirroredMonoType
  }

  trait Sum extends Mirror { self =>
    /** The ordinal number of the case class of `x`. For enums, `ordinal(x) == x.ordinal` */
    def ordinal(x: MirroredMonoType): Int
  }
}

Product types (i.e. case classes and objects, and enum cases) have mirrors which are subtypes of Mirror.Product. Sum types (i.e. sealed class or traits with product children, and enums) have mirrors which are subtypes of Mirror.Sum.

For the Tree ADT from above the following Mirror instances will be automatically provided by the compiler,

// Mirror for Tree
Mirror.Sum {
  type MirroredType = Tree
  type MirroredElemTypes[T] = (Branch[T], Leaf[T])
  type MirroredMonoType = Tree[_]
  type MirroredLabels = "Tree"
  type MirroredElemLabels = ("Branch", "Leaf")

  def ordinal(x: MirroredMonoType): Int = x match {
    case _: Branch[_] => 0
    case _: Leaf[_] => 1
  }
}

// Mirror for Branch
Mirror.Product {
  type MirroredType = Branch
  type MirroredElemTypes[T] = (Tree[T], Tree[T])
  type MirroredMonoType = Branch[_]
  type MirroredLabels = "Branch"
  type MirroredElemLabels = ("left", "right")

  def fromProduct(p: Product): MirroredMonoType =
    new Branch(...)
}

// Mirror for Leaf
Mirror.Product {
  type MirroredType = Leaf
  type MirroredElemTypes[T] = Tuple1[T]
  type MirroredMonoType = Leaf[_]
  type MirroredLabels = "Leaf"
  type MirroredElemLabels = Tuple1["elem"]

  def fromProduct(p: Product): MirroredMonoType =
    new Leaf(...)
}

Note the following properties of Mirror types,

Type classes supporting automatic deriving

A trait or class can appear in a derives clause if its companion object defines a method named derived. The signature and implementation of a derived method for a type class TC[_] are arbitrary but it is typically of the following form,

def derived[T](given Mirror.Of[T]): TC[T] = ...

That is, the derived method takes a given parameter of (some subtype of) type Mirror which defines the shape of the deriving type T, and computes the type class implementation according to that shape. This is all that the provider of an ADT with a derives clause has to know about the derivation of a type class instance.

Note that derived methods may have given Mirror arguments indirectly (e.g. by having a given argument which in turn has a given Mirror, or not at all (e.g. they might use some completely different user-provided mechanism, for instance using Dotty macros or runtime reflection). We expect that (direct or indirect) Mirror based implementations will be the most common and that is what this document emphasises.

Type class authors will most likely use higher level derivation or generic programming libraries to implement derived methods. An example of how a derived method might be implemented using only the low level facilities described above and Dotty's general metaprogramming features is provided below. It is not anticipated that type class authors would normally implement a derived method in this way, however this walkthrough can be taken as a guide for authors of the higher level derivation libraries that we expect typical type class authors will use (for a fully worked out example of such a library, see shapeless 3).

How to write a type class derived method using low level mechanisms

The low-level method we will use to implement a type class derived method in this example exploits three new type-level constructs in Dotty: inline methods, inline matches, and implicit searches via summonFrom. Given this definition of the Eq type class,

trait Eq[T] {
  def eqv(x: T, y: T): Boolean
}

we need to implement a method Eq.derived on the companion object of Eq that produces an instance for Eq[T] given a Mirror[T]. Here is a possible implementation,

inline given derived[T](given m: Mirror.Of[T]): Eq[T] = {
  val elemInstances = summonAll[m.MirroredElemTypes]           // (1)
  inline m match {                                             // (2)
    case s: Mirror.SumOf[T]     => eqSum(s, elemInstances)
    case p: Mirror.ProductOf[T] => eqProduct(p, elemInstances)
  }
}

Note that the derived method is defined as both inline and given. This means that the method will be expanded at call sites (for instance the compiler generated instance definitions in the companion objects of ADTs which have a derived Eq clause), and also that it can be used recursively if necessary, to compute instance for children.

The body of this method (1) first materializes the Eq instances for all the child types of type the instance is being derived for. This is either all the branches of a sum type or all the fields of a product type. The implementation of summonAll is inline and uses Dotty's given match construct to collect the instances as a List,

inline def summon[T]: T = given match {
  case t: T => t
}

inline def summonAll[T <: Tuple]: List[Eq[_]] = inline erasedValue[T] match {
  case _: Unit => Nil
  case _: (t *: ts) => summon[Eq[t]] :: summonAll[ts]
}

with the instances for children in hand the derived method uses an inline match to dispatch to methods which can construct instances for either sums or products (2). Note that because derived is inline the match will be resolved at compile-time and only the left-hand side of the matching case will be inlined into the generated code with types refined as revealed by the match.

In the sum case, eqSum, we use the runtime ordinal values of the arguments to eqv to first check if the two values are of the same subtype of the ADT (3) and then, if they are, to further test for equality based on the Eq instance for the appropriate ADT subtype using the auxiliary method check (4).

def eqSum[T](s: Mirror.SumOf[T], elems: List[Eq[_]]): Eq[T] =
  new Eq[T] {
    def eqv(x: T, y: T): Boolean = {
      val ordx = s.ordinal(x)                            // (3)
      (s.ordinal(y) == ordx) && check(elems(ordx))(x, y) // (4)
    }
  }

In the product case, eqProduct we test the runtime values of the arguments to eqv for equality as products based on the Eq instances for the fields of the data type (5),

def eqProduct[T](p: Mirror.ProductOf[T], elems: List[Eq[_]]): Eq[T] =
  new Eq[T] {
    def eqv(x: T, y: T): Boolean =
      iterator(x).zip(iterator(y)).zip(elems.iterator).forall {  // (5)
        case ((x, y), elem) => check(elem)(x, y)
      }
  }

Pulling this all together we have the following complete implementation,

import scala.deriving._
import scala.compiletime.erasedValue

inline def summon[T]: T = given match {
  case t: T => t
}

inline def summonAll[T <: Tuple]: List[Eq[_]] = inline erasedValue[T] match {
  case _: Unit => Nil
  case _: (t *: ts) => summon[Eq[t]] :: summonAll[ts]
}

trait Eq[T] {
  def eqv(x: T, y: T): Boolean
}

object Eq {
  given Eq[Int] {
    def eqv(x: Int, y: Int) = x == y
  }

  def check(elem: Eq[_])(x: Any, y: Any): Boolean =
    elem.asInstanceOf[Eq[Any]].eqv(x, y)

  def iterator[T](p: T) = p.asInstanceOf[Product].productIterator

  def eqSum[T](s: Mirror.SumOf[T], elems: List[Eq[_]]): Eq[T] =
    new Eq[T] {
      def eqv(x: T, y: T): Boolean = {
        val ordx = s.ordinal(x)
        (s.ordinal(y) == ordx) && check(elems(ordx))(x, y)
      }
    }

  def eqProduct[T](p: Mirror.ProductOf[T], elems: List[Eq[_]]): Eq[T] =
    new Eq[T] {
      def eqv(x: T, y: T): Boolean =
        iterator(x).zip(iterator(y)).zip(elems.iterator).forall {
          case ((x, y), elem) => check(elem)(x, y)
        }
    }

  inline given derived[T](given m: Mirror.Of[T]): Eq[T] = {
    val elemInstances = summonAll[m.MirroredElemTypes]
    inline m match {
      case s: Mirror.SumOf[T]     => eqSum(s, elemInstances)
      case p: Mirror.ProductOf[T] => eqProduct(p, elemInstances)
    }
  }
}

we can test this relative to a simple ADT like so,

enum Opt[+T] derives Eq {
  case Sm(t: T)
  case Nn
}

object Test extends App {
  import Opt._
  val eqoi = summon[Eq[Opt[Int]]]
  assert(eqoi.eqv(Sm(23), Sm(23)))
  assert(!eqoi.eqv(Sm(23), Sm(13)))
  assert(!eqoi.eqv(Sm(23), Nn))
}

In this case the code that is generated by the inline expansion for the derived Eq instance for Opt looks like the following, after a little polishing,

given derived$Eq[T](given eqT: Eq[T]): Eq[Opt[T]] =
  eqSum(summon[Mirror[Opt[T]]],
    List(
      eqProduct(summon[Mirror[Sm[T]]], List(summon[Eq[T]]))
      eqProduct(summon[Mirror[Nn.type]], Nil)
    )
  )

Alternative approaches can be taken to the way that derived methods can be defined. For example, more aggressively inlined variants using Dotty macros, whilst being more involved for type class authors to write than the example above, can produce code for type classes like Eq which eliminate all the abstraction artefacts (eg. the Lists of child instances in the above) and generate code which is indistinguishable from what a programmer might write by hand. As a third example, using a higher level library such as shapeless the type class author could define an equivalent derived method as,

given eqSum[A](given inst: => K0.CoproductInstances[Eq, A]): Eq[A] {
  def eqv(x: A, y: A): Boolean = inst.fold2(x, y)(false)(
    [t] => (eqt: Eq[t], t0: t, t1: t) => eqt.eqv(t0, t1)
  )
}

given eqProduct[A](given inst: K0.ProductInstances[Eq, A]): Eq[A] {
  def eqv(x: A, y: A): Boolean = inst.foldLeft2(x, y)(true: Boolean)(
    [t] => (acc: Boolean, eqt: Eq[t], t0: t, t1: t) => Complete(!eqt.eqv(t0, t1))(false)(true)
  )
}


inline def derived[A](given gen: K0.Generic[A]): Eq[A] = gen.derive(eqSum, eqProduct)

The framework described here enables all three of these approaches without mandating any of them.

Deriving instances elsewhere

Sometimes one would like to derive a type class instance for an ADT after the ADT is defined, without being able to change the code of the ADT itself. To do this, simply define an instance using the derived method of the type class as right-hand side. E.g, to implement Ordering for Option define,

given [T: Ordering] : Ordering[Option[T]] = Ordering.derived

Assuming the Ordering.derived method has a given parameter of type Mirror[T] it will be satisfied by the compiler generated Mirror instance for Option and the derivation of the instance will be expanded on the right hand side of this definition in the same way as an instance defined in ADT companion objects.

Syntax

Template          ::=  InheritClauses [TemplateBody]
EnumDef           ::=  id ClassConstr InheritClauses EnumBody
InheritClauses    ::=  [‘extends’ ConstrApps] [‘derives’ QualId {‘,’ QualId}]
ConstrApps        ::=  ConstrApp {‘with’ ConstrApp}
                    |  ConstrApp {‘,’ ConstrApp}

Discussion

This type class derivation framework is intentionally very small and low-level. There are essentially two pieces of infrastructure in compiler-generated Mirror instances,

The Mirror infrastructure can be seen as an extension of the existing Product infrastructure for case classes: typically Mirror types will be implemented by the ADTs companion object, hence the type members and the ordinal or fromProduct methods will be members of that object. The primary motivation for this design decision, and the decision to encode properties via types rather than terms was to keep the bytecode and runtime footprint of the feature small enough to make it possible to provide Mirror instances unconditionally.

Whilst Mirrors encode properties precisely via type members, the value level ordinal and fromProduct are somewhat weakly typed (because they are defined in terms of MirroredMonoType) just like the members of Product. This means that code for generic type classes has to ensure that type exploration and value selection proceed in lockstep and it has to assert this conformance in some places using casts. If generic type classes are correctly written these casts will never fail.

As mentioned, however, the compiler-provided mechansim is intentionally very low level and it is anticipated that higher level type class derivation and generic programming libraries will build on this and Dotty's other metaprogramming facilities to hide these low-level details from type class authors and general users. Type class derivation in the style of both shapeless and Magnolia are possible (a prototype of shapeless 3, which combines aspects of both shapeless 2 and Magnolia has been developed alongside this language feature) as is a more aggressively inlined style, supported by Dotty's new quote/splice macro and inlining facilities.