import scala.language.implicitConversions    

import scala.language.reflectiveCalls

import scala.annotation.tailrec   

import scala.util.Try

def escape(raw: String) : String = {

  import scala.reflect.runtime.universe._

  Literal(Constant(raw)).toString

}

def esc2(r: (String, String)) = (escape(r._1), escape(r._2))

def distinctBy[B, C](xs: List[B], f: B => C, acc: List[C] = Nil): List[B] = xs match {

  case Nil => Nil

  case (x::xs) => {

    val res = f(x)

    if (acc.contains(res)) distinctBy(xs, f, acc)  

    else x::distinctBy(xs, f, res::acc)

  }

} 

abstract class Bit

case object Z extends Bit

case object S extends Bit

case class C(c: Char) extends Bit

type Bits = List[Bit]

// usual regular expressions

abstract class Rexp 

case object ZERO extends Rexp

case object ONE extends Rexp

case class PRED(f: Char => Boolean, s: String = "_") extends Rexp 

case class ALTS(rs: List[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

// abbreviations

def CHAR(c: Char) = PRED(_ == c, c.toString)

def ALT(r1: Rexp, r2: Rexp) = ALTS(List(r1, r2))

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

// annotated regular expressions

abstract class ARexp 

case object AZERO extends ARexp

case class AONE(bs: Bits) extends ARexp

case class APRED(bs: Bits, f: Char => Boolean, s: String = "_") extends ARexp

case class AALTS(bs: Bits, rs: List[ARexp]) extends ARexp 

case class ASEQ(bs: Bits, r1: ARexp, r2: ARexp) extends ARexp 

case class ASTAR(bs: Bits, r: ARexp) extends ARexp 

// abbreviations

def AALT(bs: Bits, r1: ARexp, r2: ARexp) = AALTS(bs, List(r1, r2))

// values

abstract class Val

case object Empty extends Val

case class Chr(c: Char) extends Val

case class Sequ(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 {

  case Nil => ONE

  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)

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

}

// string of a regular expressions - for testing purposes

def string(r: Rexp): String = r match {

  case ZERO => "0"

  case ONE => "1"

  case PRED(_, s) => s

  case ALTS(rs) => rs.map(string).mkString("[", "|", "]")

  case SEQ(r1, r2) => s"(${string(r1)} ~ ${string(r2)})"

  case STAR(r) => s"{${string(r)}}*"

  case RECD(x, r) => s"(${x}! ${string(r)})"

}

// string of an annotated regular expressions - for testing purposes

def astring(a: ARexp): String = a match {

  case AZERO => "0"

  case AONE(_) => "1"

  case APRED(_, _, s) => s

  case AALTS(_, rs) => rs.map(astring).mkString("[", "|", "]")

  case ASEQ(_, r1, r2) => s"(${astring(r1)} ~ ${astring(r2)})"

  case ASTAR(_, r) => s"{${astring(r)}}*"

}

//--------------------------------------------------------------

// START OF NON-BITCODE PART

//

// 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 PRED(_, _) => false

  case ALTS(rs) => rs.exists(nullable)

  case SEQ(r1, r2) => nullable(r1) && nullable(r2)

  case STAR(_) => true

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

}

// 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 PRED(f, _) => if (f(c)) ONE else ZERO

  case ALTS(List(r1, r2)) => ALTS(List(der(c, r1), der(c, r2)))

  case SEQ(r1, r2) => 

    if (nullable(r1)) ALTS(List(SEQ(der(c, r1), r2), der(c, r2)))

    else SEQ(der(c, r1), r2)

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

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

}

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 Sequ(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 Sequ(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 ONE => Empty

  case ALTS(List(r1, r2)) => 

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

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

  case STAR(r) => Stars(Nil)

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

}

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

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

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

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

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

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

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

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

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

}

// lexing without simplification

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)

//println(lexing(("ab" | "ab") ~ ("b" | ONE), "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 Sequ(v1, v2) => Sequ(f1(v1), f2(v2))

}

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

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

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

  (v:Val) => Sequ(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 ALTS(List(r1, r2)) => {

    val (r1s, f1s) = simp(r1)

    val (r2s, f2s) = simp(r2)

    (r1s, r2s) match {

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

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

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

                else (ALTS(List(r1s, r2s)), F_ALT(f1s, f2s)) 

    }

  }

  case SEQ(r1, r2) => {

    val (r1s, f1s) = simp(r1)

    val (r2s, f2s) = simp(r2)

    (r1s, r2s) match {

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

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

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

      case (_, ONE) => (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 ders_simp(s: List[Char], r: Rexp) : Rexp = s match {

  case Nil => r

  case c::s => ders_simp(s, simp(der(c, r))._1)

}

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)

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

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

  env(lexing_simp(r, s)).map(esc2)

//--------------------------------------------------------------------

// Partial Derivatives

def pder(c: Char, r: Rexp): Set[Rexp] = r match {

  case ZERO => Set()

  case ONE => Set()

  case PRED(f, _) => if (f(c)) Set(ONE) else Set()

  case ALTS(rs) => rs.toSet.flatMap(pder(c, _))

  case SEQ(r1, r2) =>

    (for (pr1 <- pder(c, r1)) yield SEQ(pr1, r2)) ++

    (if (nullable(r1)) pder(c, r2) else Set())

  case STAR(r1) =>

    for (pr1 <- pder(c, r1)) yield SEQ(pr1, STAR(r1))

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

}

def pders(cs: List[Char], r: Rexp): Set[Rexp] = cs match {

  case Nil => Set(r)

  case c::cs => pder(c, r).flatMap(pders(cs, _))

}

def pders_simp(cs: List[Char], r: Rexp): Set[Rexp] = cs match {

  case Nil => Set(r)

  case c::cs => pder(c, r).flatMap(pders_simp(cs, _)).map(simp(_)._1)

}

def psize(rs: Set[Rexp])  = 

  rs.map(size).sum

//--------------------------------------------------------------------

// BITCODED PART

def fuse(bs: Bits, r: ARexp) : ARexp = r match {

  case AZERO => AZERO

  case AONE(cs) => AONE(bs ++ cs)

  case APRED(cs, f, s) => APRED(bs ++ cs, f, s)

  case AALTS(cs, rs) => AALTS(bs ++ cs, rs)

  case ASEQ(cs, r1, r2) => ASEQ(bs ++ cs, r1, r2)

  case ASTAR(cs, r) => ASTAR(bs ++ cs, r)

}

// translation into ARexps

def internalise(r: Rexp) : ARexp = r match {

  case ZERO => AZERO

  case ONE => AONE(Nil)

  case PRED(f, s) => APRED(Nil, f, s)

  case ALTS(List(r1, r2)) => 

    AALTS(Nil, List(fuse(List(Z), internalise(r1)), fuse(List(S), internalise(r2))))

  case ALTS(r1::rs) => {

     val AALTS(Nil, rs2) = internalise(ALTS(rs))

     AALTS(Nil, fuse(List(Z), internalise(r1)) :: rs2.map(fuse(List(S), _)))

  }

  case SEQ(r1, r2) => ASEQ(Nil, internalise(r1), internalise(r2))

  case STAR(r) => ASTAR(Nil, internalise(r))

  case RECD(x, r) => internalise(r)

}

internalise(("a" | "ab") ~ ("b" | ""))

// decoding of values from bit sequences

def decode_aux(r: Rexp, bs: Bits) : (Val, Bits) = (r, bs) match {

  case (ONE, bs) => (Empty, bs)

  case (PRED(f, _), C(c)::bs) => (Chr(c), bs)

  case (ALTS(r::Nil), bs) => decode_aux(r, bs)

  case (ALTS(rs), bs) => bs match {

    case Z::bs1 => {

      val (v, bs2) = decode_aux(rs.head, bs1)

      (Left(v), bs2)

    }

    case S::bs1 => {

      val (v, bs2) = decode_aux(ALTS(rs.tail), bs1)

      (Right(v), bs2)			

    }

  }

  case (SEQ(r1, r2), bs) => {

    val (v1, bs1) = decode_aux(r1, bs)

    val (v2, bs2) = decode_aux(r2, bs1)

    (Sequ(v1, v2), bs2)

  }

  case (STAR(r1), S::bs) => {

    val (v, bs1) = decode_aux(r1, bs)

    val (Stars(vs), bs2) = decode_aux(STAR(r1), bs1)

    (Stars(v::vs), bs2)

  }

  case (STAR(_), Z::bs) => (Stars(Nil), bs)

  case (RECD(x, r1), bs) => {

    val (v, bs1) = decode_aux(r1, bs)

    (Rec(x, v), bs1)

  }

}

def decode(r: Rexp, bs: Bits) = decode_aux(r, bs) match {

  case (v, Nil) => v

  case _ => throw new Exception("Not decodable")

}

//erase function: extracts a Rexp from Arexp

def erase(r: ARexp) : Rexp = r match{

  case AZERO => ZERO

  case AONE(_) => ONE

  case APRED(bs, f, s) => PRED(f, s)

  case AALTS(bs, rs) => ALTS(rs.map(erase(_)))

  case ASEQ(bs, r1, r2) => SEQ (erase(r1), erase(r2))

  case ASTAR(cs, r)=> STAR(erase(r))

}

// bnullable function: tests whether the aregular 

// expression can recognise the empty string

def bnullable (r: ARexp) : Boolean = r match {

  case AZERO => false

  case AONE(_) => true

  case APRED(_,_,_) => false

  case AALTS(_, rs) => rs.exists(bnullable)

  case ASEQ(_, r1, r2) => bnullable(r1) && bnullable(r2)

  case ASTAR(_, _) => true

}

def bmkeps(r: ARexp) : Bits = r match {

  case AONE(bs) => bs

  case AALTS(bs, rs) => {

    val n = rs.indexWhere(bnullable)

    bs ++ bmkeps(rs(n))

  }

  case ASEQ(bs, r1, r2) => bs ++ bmkeps(r1) ++ bmkeps(r2)

  case ASTAR(bs, r) => bs ++ List(Z)

}

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

def bder(c: Char, r: ARexp) : ARexp = r match {

  case AZERO => AZERO

  case AONE(_) => AZERO

  case APRED(bs, f, _) => if (f(c)) AONE(bs:::List(C(c))) else AZERO

  case AALTS(bs, rs) => AALTS(bs, rs.map(bder(c, _)))

  case ASEQ(bs, r1, r2) => 

    if (bnullable(r1)) AALT(bs, ASEQ(Nil, bder(c, r1), r2), fuse(bmkeps(r1), bder(c, r2)))

    else ASEQ(bs, bder(c, r1), r2)

  case ASTAR(bs, r) => ASEQ(bs, fuse(List(S), bder(c, r)), ASTAR(Nil, r))

}

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

@tailrec

def bders (s: List[Char], r: ARexp) : ARexp = s match {

  case Nil => r

  case c::s => bders(s, bder(c, r))

}

def flats(rs: List[ARexp]): List[ARexp] = rs match {

    case Nil => Nil

    case AZERO :: rs1 => flats(rs1)

    case AALTS(bs, rs1) :: rs2 => rs1.map(fuse(bs, _)) ::: flats(rs2)

    case r1 :: rs2 => r1 :: flats(rs2)

}

def bsimp(r: ARexp): ARexp = r match {

  case ASEQ(bs1, r1, r2) => (bsimp(r1), bsimp(r2)) match {

      case (AZERO, _) => AZERO

      case (_, AZERO) => AZERO

      case (AONE(bs2), r2s) => fuse(bs1 ++ bs2, r2s)

      case (r1s, r2s) => ASEQ(bs1, r1s, r2s)

  }

  case AALTS(bs1, rs) => distinctBy(flats(rs.map(bsimp)), erase) match {

    case Nil => AZERO

    case r :: Nil => fuse(bs1, r)

    case rs => AALTS(bs1, rs)  

  }

  //case ASTAR(bs1, r1) => ASTAR(bs1, bsimp(r1))

  case r => r

}

def bders_simp (s: List[Char], r: ARexp) : ARexp = s match {

  case Nil => r

  case c::s => bders_simp(s, bsimp(bder(c, r)))

}

def blex_simp(r: ARexp, s: List[Char]) : Bits = s match {

  case Nil => if (bnullable(r)) bmkeps(r)

	      else throw new Exception("Not matched")

  case c::cs => blex_simp(bsimp(bder(c, r)), cs)

}

def blexing_simp(r: Rexp, s: String) : Val = 

 decode(r, blex_simp(internalise(r), s.toList))

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

  env(blexing_simp(r, s)).map(esc2)

// INCLUDING SIMPLIFICATION UNDER STARS

def bsimp_full(r: ARexp): ARexp = r match {

  case ASEQ(bs1, r1, r2) => (bsimp_full(r1), bsimp_full(r2)) match {

      case (AZERO, _) => AZERO

      case (_, AZERO) => AZERO

      case (AONE(bs2), r2s) => fuse(bs1 ++ bs2, r2s)

      case (r1s, r2s) => ASEQ(bs1, r1s, r2s)

  }

  case AALTS(bs1, rs) => distinctBy(flats(rs.map(bsimp_full)), erase) match {

    case Nil => AZERO

    case r :: Nil => fuse(bs1, r)

    case rs => AALTS(bs1, rs)  

  }

  case ASTAR(bs1, r1) => ASTAR(bs1, bsimp_full(r1))

  case r => r

}

def bders_simp_full(s: List[Char], r: ARexp) : ARexp = s match {

  case Nil => r

  case c::s => bders_simp_full(s, bsimp_full(bder(c, r)))

}

def blex_simp_full(r: ARexp, s: List[Char]) : Bits = s match {

  case Nil => if (bnullable(r)) bmkeps(r)

	      else throw new Exception("Not matched")

  case c::cs => blex_simp_full(bsimp_full(bder(c, r)), cs)

}

def blexing_simp_full(r: Rexp, s: String) : Val = 

 decode(r, blex_simp_full(internalise(r), s.toList))

def btokenise_simp_full(r: Rexp, s: String) = env(blexing_simp_full(r, s)).map(esc2)

// bders2 for strings in the ALTS case

def bders2_simp(s: List[Char], a: ARexp) : ARexp = {

  //println(s"s = ${s.length}   a = ${asize(a)}")

  //Console.readLine

  (s, a) match {

  case (Nil, r) => r

  case (s, AZERO) => AZERO

  case (s, AONE(_)) => AZERO

  case (s, APRED(bs, f, _)) => 

    if (f(s.head) && s.tail == Nil) AONE(bs:::List(C(s.head))) else AZERO

  case (s, AALTS(bs, rs)) => bsimp(AALTS(bs, rs.map(bders2_simp(s, _))))