// A template of the simple matcher with simplification 
// of derivatives to be used in CW1


abstract class Rexp
case object ZERO extends Rexp
case object ONE 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 RANGE(cs: Set[Char]) extends Rexp             // set of characters
case class PLUS(r: Rexp) extends Rexp                    // plus
case class OPTIONAL(r: Rexp) extends Rexp                // optional
case class INTER(r1: Rexp, r2: Rexp) extends Rexp        // intersection
case class NTIMES(r: Rexp, n: Int) extends Rexp          // n-times
case class UPTO(r: Rexp, n: Int) extends Rexp            // up n-times
case class FROM(r: Rexp, n: Int) extends Rexp            // from n-times
case class BETWEEN(r: Rexp, n: Int, m: Int) extends Rexp // between nm-times
case class NOT(r: Rexp) extends Rexp                     // not
case class CFUN(f: Char => Boolean) extends Rexp  
 


// the nullable function: tests whether the regular 
// expression can recognise the empty string
def nullable (r: Rexp) : Boolean = r match {
  case ZERO => false
  case ONE => 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)
  // ??? other cases
}

// the derivative of a regular expression w.r.t. a character
def der(c: Char, r: Rexp) : Rexp = r match {
  case ZERO => ZERO
  case ONE => ZERO
  case CHAR(d) => if (c == d) ONE else ZERO
  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(r1) => SEQ(der(c, r1), STAR(r1))
  case NTIMES(r, i) => 
    if (i == 0) ZERO else SEQ(der(c, r), NTIMES(r, i - 1))
  // ??? other cases  
}

// simplification
def simp(r: Rexp) : Rexp = r match {
  case ALT(r1, r2) => (simp(r1), simp(r2)) match {
    case (ZERO, r2s) => r2s
    case (r1s, ZERO) => r1s
    case (r1s, r2s) => if (r1s == r2s) r1s else ALT (r1s, r2s)
  }
  case SEQ(r1, r2) =>  (simp(r1), simp(r2)) match {
    case (ZERO, _) => ZERO
    case (_, ZERO) => ZERO
    case (ONE, r2s) => r2s
    case (r1s, ONE) => r1s
    case (r1s, r2s) => SEQ(r1s, r2s)
  }
  case r => r
}

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

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