:load matcher.scala

// some regular expressions

val KEYWORDS =  ALTS(List("#", "(", ")", ",", "->", "...", ":", ":>", ";", "=",

  "=>", "[", "]", "_", "{", "|", "}", "abstype", "and", "andalso", "as",

  "case", "datatype", "do", "else", "end", "eqtype", "exception", "fn",

  "fun", "functor", "handle", "if", "in", "include", "infix", "infixr",

  "let", "local", "nonfix", "of", "op", "open", "orelse", "raise", "rec",

  "sharing", "sig", "signature", "struct", "structure", "then", "type",

  "val", "where", "while", "with", "withtype"))

val DIGITS = RANGE("0123456789")

val NONZERODIGITS = RANGE("123456789")

val POSITIVES = ALT(SEQ(NONZERODIGITS, STAR(DIGITS)), "0")

val INTEGERS = ALT(SEQ("~", POSITIVES), POSITIVES)

val ALL = ALTS(KEYWORDS, INTEGERS)

val COMMENT = SEQS("/*", NOT(SEGS(STAR(ALL), "*/", STAR(ALL))), "*/")

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 T_NT(s: String, rhs: List[Token]) extends Token

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

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

       (WHITESPACE, (s) => T_WHITESPACE),

       (LPAREN, (s) => T_LPAREN),

       (RPAREN, (s) => T_RPAREN),

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

tokenize_file(Nil, "nominal_library.ML")

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

// grammar for arithmetic expressions

val grammar = 

  List ("E" -> List(T_NUM),

        "E" -> List(T_NT("E", Nil), T_OP("+"), T_NT("E", Nil)),

        "E" -> List(T_NT("E", Nil), T_OP("-"), T_NT("E", Nil)),

        "E" -> List(T_NT("E", Nil), T_OP("*"), T_NT("E", Nil)),    

        "E" -> List(T_LPAREN, T_NT("E", Nil), T_RPAREN))

def startsWith[A](ts1: List[A], ts2: List[A]) : Boolean = (ts1, ts2) match {

  case (_, Nil) => true

  case (T_NT(e, _)::ts1,T_NT(f, _)::ts2) => (e == f) && startsWith(ts1, ts2)

  case (t1::ts1, t2::ts2) => (t1 == t2) && startsWith(ts1, ts2)

  case _ => false

}

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 (startsWith(ts1, 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 parse1(g: Grammar, ts: List[Token]) : Boolean = ts match {

  case List(T_NT("E", tree)) => { println(tree); true }

  case _ => {

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

    tss.flatten.exists(parse1(g, _))

  }

}

println() ; parse1(grammar, tokenizer(lexing_rules, "2 + 3 * 4 + 1"))

println() ; parse1(grammar, tokenizer(lexing_rules, "(2 + 3) * (4 + 1)"))

println() ; parse1(grammar, tokenizer(lexing_rules, "(2 + 3) * 4 (4 + 1)"))