Implicit Resolution Failure?

Question

I have been working on a "shapeless style" implementation of Okasaki's dense binary number system. It's simply a type-level linked-list of bits; a sort of HList of binary Digits. I have completed a first draft of my ops, which include the standard math operations you'd expect for natural numbers. Only now do I realize a big problem in my encoding. How do I fix the implicit resolution in my Induction example? Feel free to paste the entire snippet into a REPL. In this example, the only dependency on shapeless is for DepFn1, and DepFn2.

import shapeless.{ DepFn1, DepFn2 }

sealed trait Digit
case object Zero extends Digit
case object One extends Digit

sealed trait Dense { type N <: Dense }

final case class ::[+H <: Digit, +T <: Dense](digit: H, tail: T) extends Dense {
  type N = digit.type :: tail.N
}

sealed trait DNil extends Dense {
  type N = DNil
}

case object DNil extends DNil

/* ops */
trait IsDCons[N <: Dense] {
  type H <: Digit
  type T <: Dense

  def digit(n: N): H
  def tail(n: N): T
}

object IsDCons {
  type Aux[N <: Dense, H0 <: Digit, T0 <: Dense] = IsDCons[N] {
    type H = H0
    type T = T0
  }

  def apply[N <: Dense](implicit ev: IsDCons[N]): Aux[N, ev.H, ev.T] = ev

  implicit def isDCons[H0 <: Digit, T0 <: Dense]: Aux[H0 :: T0, H0, T0] =
    new IsDCons[H0 :: T0] {
      type H = H0
      type T = T0

      def digit(n: H0 :: T0): H = n.digit
      def tail(n: H0 :: T0): T = n.tail
    }
}

// Disallows Leading Zeros
trait SafeCons[H <: Digit, T <: Dense] extends DepFn2[H, T] { type Out <: Dense }

trait LowPrioritySafeCons {
  type Aux[H <: Digit, T <: Dense, Out0 <: Dense] = SafeCons[H, T] { type Out = Out0 }

  implicit def sc1[H <: Digit, T <: Dense]: Aux[H, T, H :: T] =
    new SafeCons[H, T] {
      type Out = H :: T
      def apply(h: H, t: T) = h :: t
  }
}

object SafeCons extends LowPrioritySafeCons {
  implicit val sc0: Aux[Zero.type, DNil, DNil] =
    new SafeCons[Zero.type, DNil] {
      type Out = DNil
      def apply(h: Zero.type, t: DNil) = DNil
  }
}

trait ShiftLeft[N <: Dense] extends DepFn1[N] { type Out <: Dense }

object ShiftLeft {
  type Aux[N <: Dense, Out0 <: Dense] = ShiftLeft[N] { type Out = Out0 }

  implicit def sl1[T <: Dense](implicit sc: SafeCons[Zero.type, T]): Aux[T, sc.Out] =
    new ShiftLeft[T] {
      type Out = sc.Out
      def apply(n: T) = Zero safe_:: n
    }
}

trait Succ[N <: Dense] extends DepFn1[N] { type Out <: Dense }

object Succ {
  type Aux[N <: Dense, Out0 <: Dense] = Succ[N] { type Out = Out0 }

  def apply[N <: Dense](implicit succ: Succ[N]): Aux[N, succ.Out] = succ

  implicit val succ0: Aux[DNil, One.type :: DNil] =
    new Succ[DNil] {
      type Out = One.type :: DNil
      def apply(DNil: DNil) = One :: DNil
    }

  implicit def succ1[T <: Dense]: Aux[Zero.type :: T, One.type :: T] =
    new Succ[Zero.type :: T] {
      type Out = One.type :: T
      def apply(n: Zero.type :: T) = One :: n.tail
  }

  implicit def succ2[T <: Dense, S <: Dense]
    (implicit ev: Aux[T, S], sl: ShiftLeft[S]): Aux[One.type :: T, sl.Out] =
      new Succ[One.type :: T] {
        type Out = sl.Out
        def apply(n: One.type :: T) = n.tail.succ.shiftLeft
      }
}

/* syntax */
val Cons = ::
implicit class DenseOps[N <: Dense](val n: N) extends AnyVal {
  def ::[H <: Digit](h: H): H :: N = Cons(h, n)

  def safe_::[H <: Digit](h: H)(implicit sc: SafeCons[H, N]): sc.Out = sc(h, n)

  def succ(implicit s: Succ[N]): s.Out = s(n)

  def digit(implicit c: IsDCons[N]): c.H = c.digit(n)

  def tail(implicit c: IsDCons[N]): c.T = c.tail(n)

  def shiftLeft(implicit sl: ShiftLeft[N]): sl.Out = sl(n)
}

/* aliases */
type _0 = DNil
val _0: _0 = DNil

val _1 = _0.succ
type _1 = _1.N

val _2 = _1.succ
type _2 = _2.N

/* test */
trait Induction[A <: Dense]

object Induction{
  def apply[A <: Dense](a: A)(implicit r: Induction[A]) = r
  implicit val r0 = new Induction[_0] {}
  implicit def r1[A <: Dense](implicit r: Induction[A], s: Succ[A]) = 
    new Induction[s.Out]{}
}

Induction(_0)
Induction(_1)
Induction(_2) // <- Could not find implicit value for parameter r...

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Answer 1

This is a somewhat incomplete answer, but hopefully it'll get you unstuck ...

I think your problem is the definition of r1 here,

object Induction{
  def apply[A <: Dense](a: A)(implicit r: Induction[A]) = r
  implicit val r0 = new Induction[_0] {}
  implicit def r1[A <: Dense](implicit r: Induction[A], s: Succ[A]) = 
    new Induction[s.Out]{}
}

When you ask for Induction(_2) you're hoping for r1 to be applicable and s.Out to be fixed as _2 and that that will drive the inference process from right to left in r1s implicit parameter block.

Unfortunately that won't happen. First, s.Out won't be fixed as _2 because it isn't a type variable. So you would at the very least have to rewrite this as,

implicit def r1[A <: Dense, SO <: Dense]
  (implicit r: Induction[A], s: Succ.Aux[A, SO]): Induction[SO] = 
    new Induction[SO]{}

for r1 even to be applicable. This won't get you much further, however, because SO is merely constrained to be equal to the type member Out of s ... it doesn't play a role in the selection of the Succ instance for s. And we can't make any progress from the other end, because at this point A is completely undetermined as far as the typechecker is concerned.

So I'm afraid you'll have to rethink this a little. I think your best approach would be to define a Pred operator which would allow you to define something along these lines,

implicit def r1[S <: Dense, PO <: Dense]
  (implicit p: Pred.Aux[S, PO], r: Induction[PO]): Induction[S] = 
    new Induction[S]{}

Now when you ask for Induction(_2) S will be solved immediately as _2, the Pred instance for _2 will be resolved, yielding a solution of _1 for PO which gives the typechecker what it needs to resolve the next step of the induction.

Notice that the general strategy is to start with the result type (Induction[S]) to fix the initial type variable(s), and then work from left to right across the implicit parameter list.

Also notice that I've added explicit result types to the implicit definitions: you should almost always do this (there are rare exceptions to this rule).