import scala.language.implicitConversions    

import scala.language.reflectiveCalls

import scala.annotation.tailrec   

abstract class Rexp 

case object ZERO extends Rexp

case object ONE 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 Bit

case object Z extends Bit

case object S extends Bit

type Bits = List[Bit]

abstract class ARexp 

case object AZERO extends ARexp

case class AONE(bs: Bits) extends ARexp

case class ACHAR(bs: Bits, c: Char) 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 

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

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

// 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)

}

// Bitcoded + Annotation

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

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

  case AZERO => ZERO

  case AONE(_) => ONE

  case ACHAR(bs, c) => CHAR(c)

  case AALTS(bs, Nil) => ZERO

  case AALTS(bs, r::Nil) => erase(r)

  case AALTS(bs, r::rs) => ALT(erase(r), erase(AALTS(bs, rs)))

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

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

}

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

  case AZERO => AZERO

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

  case ACHAR(cs, c) => ACHAR(bs ++ cs, c)

  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)

}

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

  case ZERO => AZERO

  case ONE => AONE(Nil)

  case CHAR(c) => ACHAR(Nil, c)

  case ALT(r1, r2) => AALT(Nil, fuse(List(Z), internalise(r1)), fuse(List(S), internalise(r2)))

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

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

}

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

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

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

  case (CHAR(c), bs) => (Chr(c), bs)

  case (ALT(r1, r2), Z::bs) => {

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

    (Left(v), bs1)

  }

  case (ALT(r1, r2), S::bs) => {

    val (v, bs1) = decode_aux(r2, bs)

    (Right(v), bs1)

  }

  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), Z::bs) => {

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

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

    (Stars(v::vs), bs2)

  }

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

}

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

  case (v, Nil) => v

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

}

def encode(v: Val) : Bits = v match {

  case Empty => Nil

  case Chr(c) => Nil

  case Left(v) => Z :: encode(v)

  case Right(v) => S :: encode(v)

  case Sequ(v1, v2) => encode(v1) ::: encode(v2)

  case Stars(Nil) => List(S)

  case Stars(v::vs) =>  Z :: encode(v) ::: encode(Stars(vs))

}

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

  case AZERO => false

  case AONE(_) => true

  case ACHAR(_,_) => 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, r::Nil) => bs ++ bmkeps(r) 

  case AALTS(bs, r::rs) => 

    if (bnullable(r)) bs ++ bmkeps(r) else bmkeps(AALTS(bs, rs))  

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

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

}

// 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 ACHAR(bs, d) => if (c == d) AONE(bs) 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(Z), bder(c, r)), ASTAR(Nil, r))

}

@tailrec

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

  case Nil => r

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

}

// main unsimplified lexing function (produces a value)

def blex(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(bder(c, r), cs)

}

def pre_blexing(r: ARexp, s: String)  : Bits = blex(r, s.toList)

def blexing(r: Rexp, s: String) : Val = decode(r, pre_blexing(internalise(r), s))

// example by Tudor

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

// simplification 

//

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

  case Nil => Nil

  case AZERO :: rs => flts(rs)

  case AALTS(bs, rs1) :: rs => rs1.map(fuse(bs, _)) ++ flts(rs)

  case r1 :: rs => r1 :: flts(rs)

}

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)

  }

} 

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 (AALTS(bs, rs), r2) => AALTS(bs, rs.map(ASEQ(Nil, _, r2)))

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

  }

      case Nil => AZERO

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

      case rs => AALTS(bs1, rs)

  }

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

}

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

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

// 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 Sequ(v1, v2) => flatten(v1) + flatten(v2)

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

}

// 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)

}

// Some Tests

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

/*

def time_needed[T](i: Int, code: => T) = {

  val start = System.nanoTime()

  for (j <- 1 to i) code

  val end = System.nanoTime()

  (end - start)/(i * 1.0e9)

}

val evil1 = (("a").%) ~ "b"

val evil2 = (((("a").%).%).%) ~ "b"

val evil3 = (("a"~"a") | ("a")).%

for(i <- 1 to 10000 by 1000) {

    println(time_needed(1, blex_simp(internalise(evil1), ("a"*i + "b").toList)))

}

for(i <- 1 to 10000 by 1000) {

    println(time_needed(1, blexing_simp(evil1, "a"*i + "b")))

}

for(i <- 1 to 10000 by 1000) {

    println(time_needed(1, blexing_simp(evil2, "a"*i + "b")))

}

for(i <- 1 to 10000 by 1000) {

    println(time_needed(1, blexing_simp(evil3, "a"*i)))

}

*/

/////////////////////////

/////////////////////////

///// Below not relevant

/*

val rf = ("a" | "ab") ~ ("ab" | "")

println(pre_blexing(internalise(rf), "ab"))

println(blexing(rf, "ab"))

println(blexing_simp(rf, "ab"))

val r0 = ("a" | "ab") ~ ("b" | "")

println(pre_blexing(internalise(r0), "ab"))

println(blexing(r0, "ab"))

println(blexing_simp(r0, "ab"))

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

println(blexing(r1, "abcd"))

println(blexing_simp(r1, "abcd"))

println(blexing((("" | "a") ~ ("ab" | "b")), "ab"))

println(blexing_simp((("" | "a") ~ ("ab" | "b")), "ab"))

println(blexing((("" | "a") ~ ("b" | "ab")), "ab"))

println(blexing_simp((("" | "a") ~ ("b" | "ab")), "ab"))

println(blexing((("" | "a") ~ ("c" | "ab")), "ab"))

println(blexing_simp((("" | "a") ~ ("c" | "ab")), "ab"))

// Sulzmann's tests

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

val sulzmann = ("a" | "b" | "ab").%

println(blexing(sulzmann, "a" * 10))

println(blexing_simp(sulzmann, "a" * 10))

for (i <- 0 to 4000 by 500) {

  println(i + ": " + "%.5f".format(time_needed(1, blexing_simp(sulzmann, "a" * i))))

}

for (i <- 0 to 15 by 5) {

  println(i + ": " + "%.5f".format(time_needed(1, blexing_simp(sulzmann, "ab" * i))))

}

*/

// some automatic testing

/*

def clear() = {

  print("")

  //print("\33[H\33[2J")

}

def merge[A](l1: LazyList[A], l2: LazyList[A], l3: LazyList[A]) : LazyList[A] =

  l1.head #:: l2.head #:: l3.head #:: merge(l1.tail, l2.tail, l3.tail)

// enumerates regular expressions until a certain depth

def enum(rs: LazyList[Rexp]) : LazyList[Rexp] = {

  rs #::: enum( (for (r1 <- rs; r2 <- rs) yield ALT(r1, r2)) ++ 

                (for (r1 <- rs; r2 <- rs) yield SEQ(r1, r2)) ++

                (for (r1 <- rs) yield STAR(r1)))

}

enum(LazyList(ZERO, ONE, CHAR('a'), CHAR('b'))).take(200).force.mkString("\n")

enum(LazyList(ZERO, ONE, CHAR('a'), CHAR('b'))).take(200_000).force

import scala.util.Try

def test_mkeps(r: Rexp) = {

  val res1 = Try(Some(mkeps(r))).getOrElse(None)

  val res2 = Try(Some(decode(r, bmkeps(internalise(r))))).getOrElse(None) 

  if (res1 != res2) println(s"Mkeps disagrees on ${r}")

  if (res1 != res2) Some(r) else (None)

}

println("Testing mkeps")

enum(LazyList(ZERO, ONE, CHAR('a'), CHAR('b'))).take(100).exists(test_mkeps(_).isDefined)

//enum(3, "ab").map(test_mkeps).toSet

//enum(3, "abc").map(test_mkeps).toSet

//enumerates strings of length n over alphabet cs

def strs(n: Int, cs: String) : Set[String] = {

  if (n == 0) Set("")

  else {

    val ss = strs(n - 1, cs)

    ss ++

    (for (s <- ss; c <- cs.toList) yield c + s)

  }

}

//tests lexing and lexingB

def tests_inj(ss: Set[String])(r: Rexp) = {

  clear()

  println(s"Testing ${r}")

  for (s <- ss.par) yield {

    val res1 = Try(Some(alexing(r, s))).getOrElse(None)

    val res2 = Try(Some(alexing_simp(r, s))).getOrElse(None)

    if (res1 != res2) println(s"Disagree on ${r} and ${s}")

    if (res1 != res2) println(s"   ${res1} !=  ${res2}")

    if (res1 != res2) Some((r, s)) else None

  }

}

//println("Testing lexing 1")

//enum(2, "ab").map(tests_inj(strs(2, "ab"))).toSet

//println("Testing lexing 2")

//enum(2, "ab").map(tests_inj(strs(3, "abc"))).toSet

//println("Testing lexing 3")

//enum(3, "ab").map(tests_inj(strs(3, "abc"))).toSet

def tests_alexer(ss: Set[String])(r: Rexp) = {

  clear()

  println(s"Testing ${r}")

  for (s <- ss.par) yield {

    val d = der('b', r)

    val ad = bder('b', internalise(r))

    val res1 = Try(Some(encode(inj(r, 'a', alexing(d, s))))).getOrElse(None)

    val res2 = Try(Some(pre_alexing(ad, s))).getOrElse(None)

    if (res1 != res2) println(s"Disagree on ${r} and 'a'::${s}")

    if (res1 != res2) println(s"   ${res1} !=  ${res2}")

    if (res1 != res2) Some((r, s)) else None

  }

}

println("Testing alexing 1")

println(enum(2, "ab").map(tests_alexer(strs(2, "ab"))).toSet)

def values(r: Rexp) : Set[Val] = r match {

  case ZERO => Set()

  case ONE => Set(Empty)

  case CHAR(c) => Set(Chr(c))

  case ALT(r1, r2) => (for (v1 <- values(r1)) yield Left(v1)) ++ 

                      (for (v2 <- values(r2)) yield Right(v2))

  case SEQ(r1, r2) => for (v1 <- values(r1); v2 <- values(r2)) yield Sequ(v1, v2)

  case STAR(r) => (Set(Stars(Nil)) ++ 

                  (for (v <- values(r)) yield Stars(List(v)))) 

    // to do more would cause the set to be infinite

}

def tests_bder(c: Char)(r: Rexp) = {

  val d = der(c, r)

  val vals = values(d)

  for (v <- vals) {

    println(s"Testing ${r} and ${v}")

    val res1 = retrieve(bder(c, internalise(r)), v)

    val res2 = encode(inj(r, c, decode(d, retrieve(internalise(der(c, r)), v))))

    if (res1 != res2) println(s"Disagree on ${r}, ${v} and der = ${d}")

    if (res1 != res2) println(s"   ${res1} !=  ${res2}")

    if (res1 != res2) Some((r, v)) else None

  }

}

println("Testing bder/der")

println(enum(2, "ab").map(tests_bder('a')).toSet)

val er = SEQ(ONE,CHAR('a')) 

val ev = Right(Empty) 

val ed = ALT(SEQ(ZERO,CHAR('a')),ONE)

retrieve(internalise(ed), ev) // => [true]

internalise(er)

bder('a', internalise(er))

retrieve(bder('a', internalise(er)), ev) // => []

decode(ed, List(true)) // gives the value for derivative

decode(er, List())     // gives the value for original value

val dr = STAR(CHAR('a'))

val dr_der = SEQ(ONE,STAR(CHAR('a'))) // derivative of dr

val dr_val = Sequ(Empty,Stars(List())) // value of dr_def

val res1 = retrieve(internalise(der('a', dr)), dr_val) // => [true]