import scala.language.implicitConversions    

import scala.annotation.tailrec   

abstract class 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 RECD(x: String, r: Rexp) extends Rexp

case class CRANGE(cs: String) extends Rexp

case class PLUS(r: Rexp) extends Rexp

abstract class Val

case object Empty extends Val

case class Chr(c: Char) extends Val

case class Seq(v1: Val, v2: Val) extends Val

case class Left(v: Val) extends Val

case class Right(v: Val) extends Val

case class Stars(vs: List[Val]) extends Val

case class Rec(x: String, v: Val) extends Val

// some convenience for typing in regular expressions

def charlist2rexp(s : List[Char]): Rexp = s match {

  def $ (r: Rexp) = RECD(s, r)

// nullable function: tests whether the regular 

// expression can recognise the empty string

  case RECD(_, r) => nullable(r)

  case CRANGE(_) => false

  case PLUS(r) => nullable(r)

// derivative of a regular expression w.r.t. a character

  case EMPTY => NULL

  case RECD(_, r1) => der(c, r1)

  case CRANGE(cs) => if (cs.contains(c)) EMPTY else NULL

  case PLUS(r) => SEQ(der(c, r), STAR(r))

}

// derivative w.r.t. a string (iterates der)

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

  case Nil => r

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

}

// extracts a string from value

def flatten(v: Val) : String = v match {

  case Empty => ""

  case Chr(c) => c.toString

  case Left(v) => flatten(v)

  case Right(v) => flatten(v)

  case Seq(v1, v2) => flatten(v1) + flatten(v2)

  case Stars(vs) => vs.map(flatten).mkString

  case Rec(_, v) => flatten(v)

}

// extracts an environment from a value

def env(v: Val) : List[(String, String)] = v match {

  case Empty => Nil

  case Chr(c) => Nil

  case Left(v) => env(v)

  case Right(v) => env(v)

  case Seq(v1, v2) => env(v1) ::: env(v2)

  case Stars(vs) => vs.flatMap(env)

  case Rec(x, v) => (x, flatten(v))::env(v)

}

// injection part

def mkeps(r: Rexp) : Val = r match {

  case EMPTY => Empty

  case ALT(r1, r2) => 

    if (nullable(r1)) Left(mkeps(r1)) else Right(mkeps(r2))

  case SEQ(r1, r2) => Seq(mkeps(r1), mkeps(r2))

  case STAR(r) => Stars(Nil)

  case RECD(x, r) => Rec(x, mkeps(r))

  case PLUS(r) => Stars(List(mkeps(r)))

def inj(r: Rexp, c: Char, v: Val) : Val = (r, v) match {

  case (STAR(r), Seq(v1, Stars(vs))) => Stars(inj(r, c, v1)::vs)

  case (SEQ(r1, r2), Seq(v1, v2)) => Seq(inj(r1, c, v1), v2)

  case (SEQ(r1, r2), Left(Seq(v1, v2))) => Seq(inj(r1, c, v1), v2)

  case (SEQ(r1, r2), Right(v2)) => Seq(mkeps(r1), inj(r2, c, v2))

  case (ALT(r1, r2), Left(v1)) => Left(inj(r1, c, v1))

  case (ALT(r1, r2), Right(v2)) => Right(inj(r2, c, v2))

  case (CHAR(_), Empty) => Chr(c) 

  case (CRANGE(_), Empty) => Chr(c) 

  case (RECD(x, r1), _) => Rec(x, inj(r1, c, v))

  case (PLUS(r), Seq(v1, Stars(vs))) => Stars(inj(r, c, v1)::vs)

}

// main lexing function (produces a value)

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

  case Nil => if (nullable(r)) mkeps(r) else throw new Exception("Not matched")

  case c::cs => inj(r, c, lex(der(c, r), cs))

}

def lexing(r: Rexp, s: String) : Val = lex(r, s.toList)

lexing(("ab" | "ab") ~ ("b" | EMPTY), "ab")

// some "rectification" functions for simplification

def F_ID(v: Val): Val = v

def F_RIGHT(f: Val => Val) = (v:Val) => Right(f(v))

def F_LEFT(f: Val => Val) = (v:Val) => Left(f(v))

def F_ALT(f1: Val => Val, f2: Val => Val) = (v:Val) => v match {

  case Right(v) => Right(f2(v))

  case Left(v) => Left(f1(v))

}

def F_SEQ(f1: Val => Val, f2: Val => Val) = (v:Val) => v match {

  case Seq(v1, v2) => Seq(f1(v1), f2(v2))

}

def F_SEQ_Empty1(f1: Val => Val, f2: Val => Val) = 

  (v:Val) => Seq(f1(Empty), f2(v))

def F_SEQ_Empty2(f1: Val => Val, f2: Val => Val) = 

  (v:Val) => Seq(f1(v), f2(Empty))

def F_RECD(f: Val => Val) = (v:Val) => v match {

  case Rec(x, v) => Rec(x, f(v))

}

def F_ERROR(v: Val): Val = throw new Exception("error")

// simplification of regular expressions returning also an

// rectification function; no simplification under STAR 

def simp(r: Rexp): (Rexp, Val => Val) = r match {

  case ALT(r1, r2) => {

    val (r1s, f1s) = simp(r1)

    val (r2s, f2s) = simp(r2)

    (r1s, r2s) match {

      case (NULL, _) => (r2s, F_RIGHT(f2s))

      case (_, NULL) => (r1s, F_LEFT(f1s))

      case _ => if (r1s == r2s) (r1s, F_LEFT(f1s))

                else (ALT (r1s, r2s), F_ALT(f1s, f2s)) 

    }

  }

  case SEQ(r1, r2) => {

    val (r1s, f1s) = simp(r1)

    val (r2s, f2s) = simp(r2)

    (r1s, r2s) match {

      case (NULL, _) => (NULL, F_ERROR)

      case (_, NULL) => (NULL, F_ERROR)

      case (EMPTY, _) => (r2s, F_SEQ_Empty1(f1s, f2s))

      case (_, EMPTY) => (r1s, F_SEQ_Empty2(f1s, f2s))

      case _ => (SEQ(r1s,r2s), F_SEQ(f1s, f2s))

    }

  }

  case RECD(x, r1) => {

    val (r1s, f1s) = simp(r1)

    (RECD(x, r1s), F_RECD(f1s))

  }

  case r => (r, F_ID)

}

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

  case Nil => if (nullable(r)) mkeps(r) else throw new Exception("Not matched")

  case c::cs => {

    val (r_simp, f_simp) = simp(der(c, r))

    inj(r, c, f_simp(lex_simp(r_simp, cs)))

def lexing_simp(r: Rexp, s: String) : Val = lex_simp(r, s.toList)

lexing_simp(("a" | "ab") ~ ("b" | ""), "ab")

// Lexing Rules for a Small While Language

val SYM = CRANGE("abcdefghijklmnopqrstuvwxyz")

val DIGIT = CRANGE("0123456789")

val ID = SYM ~ (SYM | DIGIT).% 

val NUM = PLUS(DIGIT)

val KEYWORD : Rexp = "skip" | "while" | "do" | "if" | "then" | "else" | "read" | "write" | "true" | "false"

val SEMI: Rexp = ";"

val OP: Rexp = ":=" | "==" | "-" | "+" | "*" | "!=" | "<" | ">" | "<=" | ">=" | "%" | "/"

val WHITESPACE = PLUS(" " | "\n" | "\t")

val RPAREN: Rexp = ")"

val LPAREN: Rexp = "("

val BEGIN: Rexp = "{"

val END: Rexp = "}"

val STRING: Rexp = "\"" ~ SYM.% ~ "\""

val WHILE_REGS = (("k" $ KEYWORD) | 

                  ("i" $ ID) | 

                  ("o" $ OP) | 

                  ("n" $ NUM) | 

                  ("s" $ SEMI) | 

                  ("str" $ STRING) |

                  ("p" $ (LPAREN | RPAREN)) | 

                  ("b" $ (BEGIN | END)) | 

                  ("w" $ WHITESPACE)).%

// filters out all white spaces

def tokenise(r: Rexp, s: String) = 

  env(lexing_simp(r, s)).filterNot { (s) => s._1 == "w"}.mkString("\n")

//   Testing

//============

def time[T](code: => T) = {

  val start = System.nanoTime()

  val result = code

  val end = System.nanoTime()

  println((end - start)/1.0e9)

  result

val r1 = ("a" | "ab") ~ ("bcd" | "c")

println(lexing(r1, "abcd"))

val r2 = ("" | "a") ~ ("ab" | "b")

println(lexing(r2, "ab"))

// Two Simple While Tests

//========================

println("prog0 test")

val prog0 = """read n"""

println(env(lexing_simp(WHILE_REGS, prog0)))

println("prog1 test")

val prog1 = """read  n; write (n)"""

println(env(lexing_simp(WHILE_REGS, prog1)))

// Big Test

//==========

val prog2 = """

write "fib";

read n;

minus1 := 0;

minus2 := 1;

while n > 0 do {

  temp := minus2;

  minus2 := minus1 + minus2;

  minus1 := temp;

  n := n - 1

write "result";

write minus2

println("Tokens")

println(tokenise(WHILE_REGS, prog2))

for (i <- 1 to 120 by 10) {

  print(i.toString + ":  ")

  time(lexing_simp(WHILE_REGS, prog2 * i))

}