Product type, only based on support for sum type, functions and pattern matching (Product Type as Function over sum Type) ...

Product Type as Function over sum Type

// a very simple kind of union type - or you might say 'another Either'
object Sum{
  abstract trait OR[+A,+B]
  case class FST[A,B]( a:A ) extends OR[A,B]
  case class SND[A,B]( b:B ) extends OR[A,B]
}

// a simple Product type, based on sum types
object Product{
  import Sum._

  // Product, implemented as a Function
  type x[A,B]  =  Int => A OR B

  // constructor - unfortunately we must adhere to 'A x B' and therefore the function can't simply return A xor B 
  // (but something of type 'A OR B' ), which is kind of ugly
  def pair[A,B]( a :A, b :B ) : A x B  =   i => if( i == 0 ) FST(a) else SND(b)

  // selectors - if we don't want to use 'OO-style message passing by using dot-operator' we're forced to do pattern matching
  // since our type function always returns something of type 'A OR B' (see constructor), which is again kind of ugly
  def fst[A,B]( p :A x B ) :A = p(0) match{ case FST(a) => a }
  def snd[A,B]( p :A x B ) :B = p(1) match{ case SND(b) => b }
}