// regular expressions including NOT

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 NOT(r: Rexp) extends Rexp

// 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 NOT(r) => !(nullable(r))

}

// tests whether a regular expression 

// cannot recognise more

def no_more (r: Rexp) : Boolean = r match {

  case NULL => true

  case EMPTY => false

  case CHAR(_) => false

  case ALT(r1, r2) => no_more(r1) && no_more(r2)

  case SEQ(r1, r2) => if (nullable(r1)) (no_more(r1) && no_more(r2)) else no_more(r1)

  case STAR(_) => false

  case NOT(r) => !(no_more(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 NOT(r) => NOT(der (c, r))

}

// regular expression for specifying 

// ranges of characters

def RANGE(s : List[Char]) : Rexp = s match {

  case Nil => NULL

  case c::Nil => CHAR(c)

  case c::s => ALT(CHAR(c), RANGE(s))

}

// one or more

def PLUS(r: Rexp) = SEQ(r, STAR(r))

// some regular expressions

val DIGIT = RANGE("0123456789".toList)

val NONZERODIGIT = RANGE("123456789".toList)

val NUMBER = ALT(SEQ(NONZERODIGIT, STAR(DIGIT)), "0")

val LPAREN = CHAR('(')

val RPAREN = CHAR(')')

val WHITESPACE = PLUS(RANGE(" \n".toList))

val OPS = RANGE("+-*".toList)

// for classifying the strings that have been recognised

abstract class Token

case object T_WHITESPACE extends Token

case object T_NUM extends Token

case class T_OP(s: String) extends Token

case object T_LPAREN extends Token

case object T_RPAREN extends Token

case class NT(s: String) extends Token

type Rule = (Rexp, List[Char] => Token)

def error (s: String) = throw new IllegalArgumentException ("Cannot tokenize: " + s)

def munch(r: Rexp, action: List[Char] => Token, s: List[Char], t: List[Char]) : Option[(List[Char], Token)] = 

  s match {

    case Nil if (nullable(r)) => Some(Nil, action(t))

    case Nil => None

    case c::s if (no_more(der (c, r)) && nullable(r)) => Some(c::s, action(t))

    case c::s if (no_more(der (c, r))) => None

    case c::s => munch(der (c, r), action, s, t ::: List(c))

  }

def one_token (rs: List[Rule], s: List[Char]) : (List[Char], Token) = {

 val somes = rs.map { (r) => munch(r._1, r._2, s, Nil) } .flatten

 if (somes == Nil) error(s.mkString) else (somes sortBy (_._1.length) head)

}

def tokenize (rs: List[Rule], s: List[Char]) : List[Token] = s match {

  case Nil => Nil

  case _ => one_token(rs, s) match {

    case (rest, token) => token :: tokenize(rs, rest) 

  }

}

def tokenizer(rs: List[Rule], s: String) : List[Token] = 

  tokenize(rs, s.toList).filterNot(_ match {

    case T_WHITESPACE => true

    case _ => false

  })

// lexing rules for arithmetic expressions

val lexing_rules: List[Rule]= 

  List((NUMBER, (s) => T_NUM),

       (WHITESPACE, (s) => T_WHITESPACE),

       (LPAREN, (s) => T_LPAREN),

       (RPAREN, (s) => T_RPAREN),

       (OPS, (s) => T_OP(s.mkString)))

// examples

println(tokenizer(lexing_rules, "2 + 3 * 4 + 1"))

println(tokenizer(lexing_rules, "(2 + 3) * (4 + 1)"))

type Grammar = List[(String, List[Token])]

// grammar for arithmetic expressions

val grammar = 

        "E" -> List(NT("E"), T_OP("+"), NT("E")),

        "E" -> List(NT("E"), T_OP("-"), NT("E")),

        "E" -> List(NT("E"), T_OP("*"), NT("E")),    

        "E" -> List(T_LPAREN, NT("E"), T_RPAREN))

def chop[A](ts1: List[A], prefix: List[A], ts2: List[A]) : Option[(List[A], List[A])] = 

  ts1 match {

    case Nil => None

    case t::ts => 

      if (ts1.startsWith(prefix)) Some(ts2.reverse, ts1.drop(prefix.length))

      else chop(ts, prefix, t::ts2)

  }

// examples

chop(List(1,2,3,4,5,6,7,8,9), List(4,5), Nil)  

chop(List(1,2,3,4,5,6,7,8,9), List(3,5), Nil)  

def replace[A](ts: List[A], out: List[A], in: List [A]) = 

  chop(ts, out, Nil) match {

    case None => None

    case Some((before, after)) => Some(before ::: in ::: after)

  }  

def parse(g: Grammar, ts: List[Token]) : Boolean = {

  if (ts == List(NT("E"))) true

  else {

    val tss = for ((lhs, rhs) <- g) yield replace(ts, rhs, List(NT(lhs)))

    tss.flatten.exists(parse(g, _))

  }

}

def parser(g: Grammar, rs: List[Rule], s: String) = {

  println("\n")

  parse(g, tokenizer(rs, s))

}

parser(grammar, lexing_rules, "(2 + 3) * (4 + 1)")

parser(grammar, lexing_rules, "(2 + 3) * 4 (4 + 1)")