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object Recursion { | |
case class Nat(n: Int) extends Ordered[Nat] { | |
require(n >= 0) | |
def toInt = n | |
def +(that: Nat): Nat = Nat(n + that.n) | |
def -(that: Nat): Nat = Nat(n - that.n) | |
def *(that: Nat): Nat = Nat(n * that.n) | |
override def compare(that: Nat) = n.compare(that.n) | |
} | |
object Nat { | |
import scala.language.implicitConversions | |
implicit def intToNat(n: Int): Nat = Nat(n) | |
def Zero = Nat(0) | |
def One = Nat(1) | |
} | |
import Nat.{Zero, One} | |
/** | |
* Type for a recursive function roughly like `type F[A, B] = F[A, B] => A => B` (if that were legal Scala). | |
* This can represent something like ((((...(A => B) => A => B)...) => A => B) => A => B). | |
*/ | |
case class RecFun[A, B](f: RecFun[A, B] => A => B) extends (RecFun[A, B] => A => B) { | |
override def apply(g: RecFun[A, B]) = f(g) | |
} | |
/** | |
* The Z-combinator is a fixed-point combinator, identical to the Y-combinator but adapted for strict evaluation | |
* by adding the extra function parameter `(a: A)`. I.e. the Y-combinator would be the same but with | |
* `def h = RecFun { (g: RecFun[A, B]) => f(g(g)) }`. | |
*/ | |
def Z[A, B](f: (A => B) => A => B): A => B = { | |
def h = RecFun { (g: RecFun[A, B]) => (a: A) => f(g(g))(a) } | |
h(h) | |
} | |
val pfact = { | |
(f: Nat => Nat) => | |
(n: Nat) => | |
if (n == Zero) One | |
else n * f(n - 1) | |
} | |
def fact = Z(pfact) | |
val pfib = { | |
(f: Nat => Nat) => | |
(n: Nat) => | |
if (n <= 1) n | |
else f(n - 1) + f(n - 2) | |
} | |
def fib = Z(pfib) | |
} | |
import Recursion._ | |
import Nat.intToNat |
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