abstract class Rexp {
  def simp : Rexp = this
}

case object NULL extends Rexp
case object EMPTY extends Rexp
case class CHAR(c: Char) extends Rexp
case class ALT(r1: Rexp, r2: Rexp) extends Rexp {
  override def simp = (r1.simp, r2.simp) match {
    case (NULL, r) => r
    case (r, NULL) => r
    case (r, EMPTY) => if (nullable(r)) r else ALT(r, EMPTY)
    case (EMPTY, r) => if (nullable(r)) r else ALT(r, EMPTY)
    case (r1, r2) => if (r1 == r2) r1 else ALT(r1, r2)
  }
}
case class SEQ(r1: Rexp, r2: Rexp) extends Rexp {
  override def simp = (r1.simp, r2.simp) match {
    case (NULL, _) => NULL
    case (_, NULL) => NULL
    case (EMPTY, r) => r
    case (r, EMPTY) => r
    case (r1, r2) => SEQ(r1, r2)
  }
}
case class STAR(r: Rexp) extends Rexp {
  override def simp = r.simp match {
    case NULL => EMPTY
    case EMPTY => EMPTY
    case r => STAR(r)
  }
}
case class NTIMES(r: Rexp, n: Int) extends Rexp {
  override def simp = if (n == 0) EMPTY else 
    r.simp match {
      case NULL => NULL
      case EMPTY => EMPTY
      case r => NTIMES(r, n)
    }
}

// some convenience for typing in regular expressions
def charlist2rexp(s : List[Char]) : Rexp = s match {
  case Nil => EMPTY
  case c::Nil => CHAR(c)
  case c::s => SEQ(CHAR(c), charlist2rexp(s))
}
implicit def string2rexp(s : String) : Rexp = charlist2rexp(s.toList)


// nullable function: tests whether the regular 
// expression can recognise the empty string
def nullable (r: Rexp) : Boolean = r match {
  case NULL => false
  case EMPTY => true
  case CHAR(_) => false
  case ALT(r1, r2) => nullable(r1) || nullable(r2)
  case SEQ(r1, r2) => nullable(r1) && nullable(r2)
  case STAR(_) => true
  case NTIMES(r, i) => if (i == 0) true else nullable(r)
}

// derivative of a regular expression w.r.t. a character
def der (c: Char, r: Rexp) : Rexp = r match {
  case NULL => NULL
  case EMPTY => NULL
  case CHAR(d) => if (c == d) EMPTY else NULL
  case ALT(r1, r2) => ALT(der(c, r1), der(c, r2))
  case SEQ(r1, r2) => 
    if (nullable(r1)) ALT(SEQ(der(c, r1), r2), der(c, r2))
    else SEQ(der(c, r1), r2)
  case STAR(r) => SEQ(der(c, r), STAR(r))
  case NTIMES(r, i) => 
    if (i == 0) NULL else SEQ(der(c, r), NTIMES(r, i - 1))
}

// derivative w.r.t. a string (iterates der)
def ders (s: List[Char], r: Rexp) : Rexp = s match {
  case Nil => r
  case c::s => ders(s, der(c, r).simp)
}

// main matcher function
def matcher(r: Rexp, s: String) : Boolean = nullable(ders(s.toList, r))



//one or zero
def OPT(r: Rexp) = ALT(r, EMPTY)

def RTEST(n: Int) = SEQ(NTIMES(OPT("a"), n), NTIMES("a", n))

def time_needed[T](i: Int, code: => T) = {
  val start = System.nanoTime()
  for (j <- 1 to i) code
  val end = System.nanoTime()
  (end - start)/(i * 1.0e9)
}


for (i <- 1 to 11001 by 500) {
  println(i + " " + + " " + time_needed(1, matcher(RTEST(i), "a" * i)))
}