Generic Adder from Idris to Scala? - scala

Question

Type Driven Development with Idris presents the following generic adder approach:
AdderType : (numArgs : Nat) -> Type
AdderType Z = Int
AdderType (S k) = (next : Int) -> AdderType k
adder : (n : Nat) -> (acc : Int) -> AdderType n
adder Z acc = acc
adder (S k) acc = \x => (adder k (x+acc))
Example:
-- expects 3 Int's to add, with a starting value of 0
*Work> :t (adder 3 0)
adder 3 0 : Int -> Int -> Int -> Int
-- 0 (initial) + 3 + 3 + 3 == 9
*Work> (adder 3 0) 3 3 3
9 : Int
I'm guessing that shapeless can handle the above generic adder function.
How can it be written in Scala with or without shapeless?

Answer 1

Update: I'll leave my original implementation below, but here's one that's a little more direct:
import shapeless._
trait AdderType[N <: Nat] extends DepFn1[Int]
object AdderType {
type Aux[N <: Nat, Out0] = AdderType[N] { type Out = Out0 }
def apply[N <: Nat](base: Int)(implicit at: AdderType[N]): at.Out = at(base)
implicit val adderTypeZero: Aux[Nat._0, Int] = new AdderType[Nat._0] {
type Out = Int
def apply(x: Int): Int = x
}
implicit def adderTypeSucc[N <: Nat](implicit
atN: AdderType[N]
): Aux[Succ[N], Int => atN.Out] = new AdderType[Succ[N]] {
type Out = Int => atN.Out
def apply(x: Int): Int => atN.Out = i => atN(x + i)
}
}
And then:
scala> val at3 = AdderType[Nat._3](0)
at3: Int => (Int => (Int => Int)) = <function1>
scala> at3(3)(3)(3)
res8: Int = 9
Original answer below.
Here's an off-the-cuff Scala translation:
import shapeless._
trait AdderType[N <: Nat] extends DepFn1[Int] {
protected def plus(x: Int): AdderType.Aux[N, Out]
}
object AdderType {
type Aux[N <: Nat, Out0] = AdderType[N] { type Out = Out0 }
def apply[N <: Nat](base: Int)(implicit at: AdderType[N]): Aux[N, at.Out] =
at.plus(base)
private[this] case class AdderTypeZero(acc: Int) extends AdderType[Nat._1] {
type Out = Int
def apply(x: Int): Int = acc + x
protected def plus(x: Int): Aux[Nat._1, Int] = copy(acc = acc + x)
}
private[this] case class AdderTypeSucc[N <: Nat, Out0](
atN: Aux[N, Out0],
acc: Int
) extends AdderType[Succ[N]] {
type Out = Aux[N, Out0]
def apply(x: Int): Aux[N, Out0] = atN.plus(acc + x)
protected def plus(x: Int): Aux[Succ[N], Aux[N, Out0]] = copy(acc = acc + x)
}
implicit val adderTypeZero: Aux[Nat._1, Int] = AdderTypeZero(0)
implicit def adderTypeSucc[N <: Nat](implicit
atN: AdderType[N]
): Aux[Succ[N], Aux[N, atN.Out]] = AdderTypeSucc(atN, 0)
}
And then:
scala> val at3 = AdderType[Nat._3](0)
at3: AdderType[shapeless.Succ[shapeless.Succ[shapeless.Succ[shapeless._0]]]] { ...
scala> at3(3)(3)(3)
res0: Int = 9
It's more verbose and the representation is a little different to get the Scala syntax to work out—our "base case" is essentially an Int => Int instead of an Int because otherwise I don't see a way to avoid needing to write apply or () everywhere—but the basic ideas are exactly the same.

Answer 2

In case you're on a long trip and don't have your shapeless on hands, here is how you can do this in pure Scala. It can be useful for those who are not familiar with shapeless and those who don't use it for some reason.
First of all, we'll need some way to iterate on types, i.e. represent natural numbers in types. You can use any nested type or just define a new one with some aliases for numbers:
sealed trait Nat
trait Zero extends Nat
trait Succ[N <: Nat] extends Nat
// enough for examples:
type _0 = Zero
type _1 = Succ[_0]
type _2 = Succ[_1]
type _3 = Succ[_2]
type _4 = Succ[_3]
// etc...
Of course, if you will often use types like _42 and _342923, it would be more convenient to take an existing Nat type with some macro-magic for constructing those from values, but for our examples it's enough.
Now, the AdderType dependent function type is quite straight forward:
// first we define the type which take a Nat type argument
trait AdderType[N <: Nat] {
type Out
def apply(i: Int): Out
}
// then we inductively construct its values using implicits
case object AdderType {
// base case: N = _0
implicit def zero:
AdderType[_0] { type Out = Int } =
new AdderType[_0] {
type Out = Int
def apply(i: Int): Out = i
}
// induction step: N -> Succ[N]
implicit def succ[N <: Nat, NOut](
implicit prev: AdderType[N] { type Out = NOut }
): AdderType[Succ[N]] { type Out = Int => NOut } =
new AdderType[Succ[N]] {
type Out = Int => NOut
def apply(i: Int): Out = k => prev(i + k)
}
}
Now, to construct an instance of AdderType and apply it, we write a function, which takes a N <: Nat as a type argument and implicitly constructs AdderType[N]:
def adder[N <: Nat](initial: Int)(
implicit adderFunction: AdderType[N]
): adderFunction.Out = adderFunction(initial)
That's it:
scala> val add3Numbers = adder_[_3](0)
add3Numbers: Int => (Int => (Int => Int)) = <function1>
scala> add3Numbers(1)(2)(3)
res0: Int = 6
You can see that the pure solution is not much bigger or more complicated than the one using shapeless (although the latter provides us ready-to-use Nat and DepFn types).
A little addition: if (in some more general case) you don't want to use adderFunction.Out, which sometimes leads to problems, I also have a solution without it. In this particular case it's not any better, but I'll show it anyway.
The key point is to add another type parameter for the out type: adder[N <: Nat, NOut], but then can't pass N as a type to adder, because we will need to write NOut, which want to be inferred (otherwise, what's the point). So we can pass an additional value argument, which will help to derive N type:
def adder[N <: Nat, NOut](n: NatVal[N])(initial: Int)(
implicit adderFunction: AdderType[N] { type Out = NOut }
): NOut = adderFunction(initial)
To construct NatVal[N] we don't need to create an instance of each Nat type, we can use a little trick:
// constructing "values" to derive its type arg
case class NatVal[N <: Nat]()
// just a convenience function
def nat[N <: Nat]: NatVal[N] = NatVal[N]()
Now here is how you use it:
scala> val add3Numbers = adder(nat[_3])(0)
add3Numbers: this.Out = <function1>
scala> add3Numbers(1)(2)(3)
res1: this.Out = 6
You can see that it works, but doesn't show us the actual types. Nevertheless, this approach can work better in cases when you have several implicits that depend others' type members. I mean
def foo[AOut]()(implicit
a: A { type Out = AOut},
b: B { type In = AOut }
) ...
instead of
def foo()(implicit
a: A,
b: B { type In = a.Out }
) ...
Because you cannot reffer to a.Out in the same argument list.
You can find full code in my repo on Github.