test.scala

import annotation.implicitNotFound

@implicitNotFound(msg = "This message can never appear!")
trait ~>[A, B] { self =>
  def apply(a: A): B
  def invert(b: B): A = inverse.apply(b)
  def inverse: B ~> A = new ~>[B,A] {
    def apply(b: B) = self.invert(b)
    override def invert(a: A) = self(a)
  }
}
 
object ~> {
  def connect[A,B](implicit prov: Provider[A,B]): A ~> B = prov.bij
}

sealed trait Provider[A,B] { def bij: A ~> B }
class Forward[A,B](override val bij: A ~> B) extends Provider[A,B]
class Reverse[A,B](invbij: B ~> A) extends Provider[A,B] { lazy val bij = invbij.inverse }

trait LowPriority {
  implicit def reverse[A,B](implicit bij: B ~> A): Provider[A,B] = new Reverse(bij)
}

object Provider extends LowPriority {
  implicit def forward[A,B](implicit bij: A ~> B): Provider[A,B] = new Forward(bij)
}

// Obviously not a bijection. Chill
implicit val toSBij = new ~>[Int,String] {
  def apply(i: Int) = i.toString
  override def invert(s: String) = s.toInt
}
/*
scala> ~>.connect[Int, String]
res7: ~>[Int,String] = $anon$1@4d90b2ca

scala> ~>.connect[String, Int]
res8: ~>[String,Int] = $tilde$greater$$anon$2@78dc17f

scala> ~>.connect[String, Double]
<console>:28: error: could not find implicit value for parameter prov: Provider[String,Double]
              ~>.connect[String, Double]
*/