Scala Shapeless Code for Project Euler #1

Question

I am new to shapeless and have been trying to practice some type level programming. I took Problem #1 from Project Euler as my first challenge.

I started by writing regular scala code:

object ProjectEuler1 {
  def e1(limit: Int) = (1 until limit).foldLeft(0) {
    case (acc, x) if x % 3 * x % 5 == 0 => acc + x
    case (acc, _)                       => acc
  }
  val out = e1(10)
  assert(out == 23)
}

Then, I came up with this working shapeless implementation using poly:

object ProjectEuler1Shapeless extends App {
  import shapeless._
  import nat._
  import ops.nat._
  import poly._
  import test.typed

  trait eLP extends Poly1 {
    implicit def default[A <: Nat] = at[A] { _ => _0 }
  }
  object e extends eLP {
    implicit def match3[A <: Nat](implicit ev: Mod.Aux[A, _3, _0]) = at[A](identity)
    implicit def match5[A <: Nat](implicit ev: Mod.Aux[A, _5, _0]) = at[A](identity)
  }

  object sum extends Poly2 {
    implicit def sum[A <: Nat, B <: Nat, Z <: Nat](implicit s: Sum.Aux[A, B, Z],
                                                   z: Witness.Aux[Z]) =
      at[A, B] { (_, _) => z.value }
  }

  type _23 = Succ[_22]
  val l = _1 :: _2 :: _3 :: _4 :: _5 :: _6 :: _7 :: _8 :: _9 :: HNil
  val out = l.map(e).foldLeft(_0)(sum)
  typed[_23](out)
}

Next, I wanted to change the function so that I don't need to manually create a list. Instead it accepts a "limit" as an argument like the regular scala code. I came up with this:

object ProjectEuler1Shapeless2 extends App {
  import shapeless._
  import nat._
  import ops.nat._
  import test.typed

  class E1[I <: Nat, N <: Nat]
  trait ELP0 {
    implicit def default[I <: Nat, M <: Nat] = new E1[I, _0]
  }
  trait ELP1 extends E1LP0 {
    implicit def match3[A <: Nat](implicit ev: Mod.Aux[A, _3, _0]) = new E1[A, A]
    implicit def match5[A <: Nat](implicit ev: Mod.Aux[A, _5, _0]) = new E1[A, A]
  }
  object E1 extends E1LP1 {
    implicit def combine[I <: Nat, L <: Nat, M <: Nat](implicit e1: E1[I, L],
                                                       m: E1[Succ[I], M],
                                                       sum: Sum[L, M]) =
      new E1[Succ[Succ[I]], sum.Out]
  }
  def e1[N <: Nat](limit: Nat)(implicit e: E1[limit.N, N], w: Witness.Aux[N]): N = w.value

  val f1 = e1(1)
  typed[_0](f1)

  val f2 = e1(2)
  typed[_0](f2)

  val f3 = e1(3)
  typed[_3](f3) // Does not compile!
}

I've gotten stuck here. The compiler is telling me it found _0. I guess it's picking up the instance from def default.

Any tips on how I can fix this? I have a feeling my strategy for solving this problem might be a little weird also. Any pointers on how I can make this shapeless code more idiomatic are greatly appreciated.

My original strategy was to create a hylomorphism. I noticed there is an unfold example in the shapeless git but its complexity escapes me at the moment.

Answer 1

I find it a little easier to think about this problem inductively (at the type level, at least). First we can define a helper type class that returns N if N is a multiple of one of the numbers in M, and _0 otherwise:

import shapeless._, nat._0, ops.nat.Mod

trait IfMultiple[N <: Nat, M <: HList] { type Out <: Nat }

trait LowPriorityIfMultiple {
  type Aux[N <: Nat, M <: HList, Out0 <: Nat] = IfMultiple[N, M] {
    type Out = Out0
  }

  implicit def isMultiple1[N <: Nat, H <: Nat, T <: HList](implicit
    ifMultiple: IfMultiple[N, T]
  ): Aux[N, H :: T, ifMultiple.Out] = new IfMultiple[N, H :: T] {
    type Out = ifMultiple.Out
  }
}

object IfMultiple extends LowPriorityIfMultiple {
  implicit def ifMultiple0[N <: Nat]: Aux[N, HNil, _0] =
    new IfMultiple[N, HNil] {
      type Out = _0
    }

  implicit def ifMultiple2[N <: Nat, H <: Nat, T <: HList](implicit
    mod: Mod.Aux[N, H, _0]
  ): Aux[N, H :: T, N] = new IfMultiple[N, H :: T] {
    type Out = N
  }
}

And now we just need a type class to add up all these values from _0 to N - _1:

import nat._1, ops.nat.Sum

trait SumOfMultiples[N <: Nat, M <: HList] extends DepFn0 { type Out <: Nat }

object SumOfMultiples {
  type Aux[N <: Nat, M <: HList, Out0 <: Nat] = SumOfMultiples[N, M] {
    type Out = Out0
  }

  def apply[N <: Nat, M <: HList](implicit
    som: SumOfMultiples[N, M]
  ): Aux[N, M, som.Out] = som

  implicit def sum0[M <: HList]: Aux[_1, M, _0] =
    new SumOfMultiples[_1, M] {
      type Out = _0
      def apply(): _0 = _0
    }

  implicit def sumN[P <: Nat, M <: HList, NV <: Nat, PT <: Nat, NT <: Nat](implicit
    ifMultiple: IfMultiple.Aux[P, M, NV],
    som: Aux[P, M, PT],
    sum: Sum.Aux[NV, PT, NT],
    wit: Witness.Aux[NT]
  ): Aux[Succ[P], M, NT] = new SumOfMultiples[Succ[P], M] {
    type Out = NT
    def apply(): NT = wit.value
  }
}

And then we're done:

import nat._, test.typed

val result = SumOfMultiples[_10, _3 :: _5 :: HNil]

typed[Succ[_22]](result())

Which compiles as expected.

It's worth noting that there are other ways you could solve this problem. You could create a type class that would provide Nat ranges and then fold over that with a Poly2 using IfMultiple. You could also define an IsMultiple type class that just witnesses that N is a multiple of one of the numbers in M—my first quick attempt did this, but I ran into ambiguity issues, so I went with the similar version above. The implementation here is fairly straightforward, though, and unless you have other applications for e.g. Nat ranges, I think it's a pretty reasonable solution.