import scala.language.implicitConversions
import scala.language.reflectiveCalls

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 

abstract class Pat

case class VAR(x: String, w: String, r: Rexp) extends Pat
case class GRP(x: String, w: String, p: Pat) extends Pat
case class PSEQ(p1: Pat, p2: Pat) extends Pat
case class PALT(p1: Pat, p2: Pat) extends Pat
case class PSTAR(p: Pat) extends Pat

object VAR { def apply(x: String, r: Rexp) : VAR = VAR(x, "", r) }
object GRP { def apply(x: String, p: Pat) : GRP = GRP(x, "", p) }


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
}

// 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))
}

def down (p: Pat) : Rexp = p match {
  case VAR(x: String, w: String, r: Rexp) => r
  case GRP(x: String, w: String, p: Pat) => down(p)
  case PSEQ(p1: Pat, p2: Pat) => SEQ(down(p1), down(p2))
  case PALT(p1: Pat, p2: Pat) => ALT(down(p1), down(p2))
  case PSTAR(p: Pat) => STAR(down(p))
}

def empty (p: Pat) : Pat = p match {
  case VAR(x: String, w: String, r: Rexp) => 
    if (nullable(r)) VAR(x, w, EMPTY) 
    else VAR(x, w, NULL)
  case GRP(x: String, w: String, p: Pat) => GRP(x, w, empty(p))
  case PSEQ(p1: Pat, p2: Pat) => PSEQ(empty(p1), empty(p2))
  case PALT(p1: Pat, p2: Pat) => PALT(empty(p1), empty(p2))
  case PSTAR(p: Pat) => PSTAR(empty(p))
}

def patder (c: Char, p: Pat) : Pat = p match {
  case VAR(x: String, w: String, r: Rexp) => VAR(x, w + c, der(c, r))
  case GRP(x: String, w: String, p: Pat) => GRP(x, w + c, patder(c, p))
  case PSEQ(p1: Pat, p2: Pat) => 
    if (nullable(down(p1))) PALT(PSEQ(patder(c, p1), p2), PSEQ(empty(p1), patder(c, p2)))
    else PSEQ(patder(c, p1), p2)
  case PALT(p1: Pat, p2: Pat) => PALT(patder(c, p1), patder(c, p2))
  case PSTAR(p: Pat) => PSEQ(patder(c, p), PSTAR(p))
}

def patders (s: List[Char], p: Pat) : Pat = s match {
  case Nil => p
  case c::s => patders(s, patder(c, p))
}

type Env = Set[List[(String, String)]]

def special (p: Pat, env: Env) : Env = 
  if (env == Set() && nullable(down(p))) Set(Nil) else env

def env (p: Pat) : Env = p match {
  case VAR(x: String, w: String, r: Rexp) => 
    if (nullable(r)) Set(List((x, w))) else Set()
  case GRP(x: String, w: String, p1: Pat) => 
    special(p, for (e <- env(p1)) yield List((x, w)) ++ e)
  case PSEQ(p1: Pat, p2: Pat) => 
    special(p, for (e1 <- env(p1); e2 <- env(p2)) yield (e1 ++ e2))
  case PALT(p1: Pat, p2: Pat) => special(p, env(p1) ++ env(p2))
  case PSTAR(p1: Pat) => special(p, env(p1))
}

def matcher (p: Pat, s: String) = env(empty(patders(s.toList, p)))


// 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)

implicit def RexpOps (r: Rexp) = new {
  def | (s: Rexp) = ALT(r, s)
  def % = STAR(r)
  def ~ (s: Rexp) = SEQ(r, s)
}

implicit def stringOps (s: String) = new {
  def | (r: Rexp) = ALT(s, r)
  def | (r: String) = ALT(s, r)
  def % = STAR(s)
  def ~ (r: Rexp) = SEQ(s, r)
  def ~ (r: String) = SEQ(s, r)
}

implicit def PatOps (p: Pat) = new {
  def | (q: Pat) = PALT(p, q)
  def % = PSTAR(p)
  def ~ (q: Pat) = PSEQ(p, q)
}


val p0 = VAR("x", "A")

patders("A".toList, p0)
matcher(p0, "A")

val p1 = VAR("x", "A") | VAR("y", "A") | VAR("z", "B")

patders("A".toList, p1)
matcher(p1, "A")
matcher(p1, "B")

val p2 = VAR("x", "AB")
matcher(p2, "AB")
matcher(p2, "AA")

val p3 = VAR("x", "A" ~ "B")
matcher(p3, "AB")
matcher(p3, "AA")

val p4 = VAR("x", "A") ~ VAR("y", "B")
matcher(p4, "AB")
matcher(p4, "AA")

val p5 = VAR("x", "A".%)
matcher(p5, "A")
matcher(p5, "AA")
matcher(p5, "")
matcher(p5, "AAA")

val p6 = VAR("x", "A").%
matcher(p6, "A")
matcher(p6, "AA")
matcher(p6, "")
matcher(p6, "AAA")

val p7 = (VAR("x", "A") | VAR("y", "AB") | VAR("z", "B")).%
matcher(p7, "AB")