abstract class Rexp 

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

case class SEQ(r1: Rexp, r2: Rexp) extends Rexp 

case class STAR(r: Rexp) extends Rexp 

case class NTIMES(r: Rexp, n: Int) extends Rexp 

def simp(r: Rexp): Rexp = r match {

  case ALT(r1, r2) => {

    }

  }

  case SEQ(r1, r2) => {

      case (NULL, _) => NULL

      case (_, NULL) => NULL

    }

  }

  case NTIMES(r, n) => NTIMES(simp(r), n)    

  case r => r

}

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)

}

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))

}

def ders (s: List[Char], r: Rexp) : Rexp = s match {

  case Nil => r

  case c::s => ders(s, simp(der(c, r)))

}

def matches(r: Rexp, s: String) : Boolean = nullable(ders(s.toList, r))

//one or zero

def OPT(r: Rexp) = ALT(r, EMPTY)

def EVIL(n: Int) = SEQ(NTIMES(OPT(CHAR('a')), n), NTIMES(CHAR('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 12001 by 500) {

  println(i + " " + "%.5f".format(time_needed(1, matches(EVIL(i), "a" * i))))

}