FizzBuzz in the type system (faster)

fizzbuzz-faster.scala

import shapeless._, Nat._

trait FizzBuzz[N <: Nat] {
  type R3 <: Nat
  type R5 <: Nat
 
  def ti: ToInt[N]
  def prev: List[String]
  def d3: R3 =:= _0
  def d5: R5 =:= _0
  
  def step = if (d3 != null) if (d5 != null) "FizzBuzz" else "Fizz" else
    if (d5 != null) "Buzz" else ti().toString
  def steps = step :: prev
  def show = println(steps.reverse.mkString("\n"))
}

case class F[N <: Nat, N3 <: Nat, N5 <: Nat](prev: List[String])(implicit
  val ti: ToInt[N],
  val d3: N3 =:= _0 = null,
  val d5: N5 =:= _0 = null
) extends FizzBuzz[N] {
  type R3 = N3
  type R5 = N5
}

implicit val f1 = F[_1, _1, _1](Nil)

implicit def fN[N <: Nat, N3 <: Nat, N5 <: Nat, R3 <: Nat, R5 <: Nat](implicit
  nf: FizzBuzz[N] { type R3 = N3; type R5 = N5 },
  ti: ToInt[Succ[N]],
  m3: ModAux[Succ[N3], _3, R3],
  m5: ModAux[Succ[N5], _5, R5],
  d3: R3 =:= _0 = null,
  d5: R5 =:= _0 = null
) = F[Succ[N], R3, R5](nf.steps)

type _100 =
  Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[
  Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[
  Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[
  Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[
  Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[
  Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[
  Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[
  Succ[Succ[Succ[Succ[Succ[Succ[Succ[Succ[_22]]]]]]]]
  ]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

implicitly[FizzBuzz[_100]].show