case class ALTS(rs: List[Rexp]) extends Rexp 

// ALT is now an abbreviation

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

// proj is now used instead for Left and Right

case class Proj(n: Int, v: Val) extends Val

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

// for manipulating projections  

def IncProj(m: Int, v: Val) = v match {

  case Proj(n, v) => Proj(n + m, v)

}

def DecProj(m: Int, v: Val) = v match {

  case Proj(n, v) => Proj(n - m, v)

}

// string of a regular expression - for testing purposes 

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

  case ZERO => "0"

  case ONE => "1"

  case CHAR(c) => c.toString 

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

  case SEQ(CHAR(c), CHAR(d)) => s"${c}${d}"

  case SEQ(ONE, CHAR(c)) => s"1${c}"

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

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

}

// size of a regular expression - for testing purposes 

def size(r: Rexp) : Int = r match {

  case ZERO => 1

  case ONE => 1

  case CHAR(_) => 1

  case ALTS(rs) => 1 + rs.map(size).sum

  case SEQ(r1, r2) => 1 + size(r1) + size(r2)

  case STAR(r) => 1 + size(r)

}

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

  case ALTS(rs) => ALTS(rs.map(der(c, _)))

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

// mkeps

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

  case ONE => Empty

  case ALTS(r1::rs) => 

    if (nullable(r1)) Proj(0, mkeps(r1)) 

    else IncProj(1, mkeps(ALTS(rs)))

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

  case STAR(r) => Stars(Nil)

}

// injection 

  case (SEQ(r1, r2), Proj(0, Sequ(v1, v2))) => Sequ(inj(r1, c, v1), v2)

  case (SEQ(r1, r2), Proj(1, v2)) => Sequ(mkeps(r1), inj(r2, c, v2))

  case (ALTS(rs), Proj(n, v1)) => Proj(n, inj(rs(n), c, v1))

//  - does  not simplify

  case Nil => {

    //println(s"Size of the last regex: ${size(r)}")

    if (nullable(r)) mkeps(r) else throw new Exception("Not matched")

  }

lexing("a" | ZERO, "a")

lexing(ZERO | "a", "a")

// removing duplicate regular expressions

val unit = (ZERO, F_ERROR(_))

def dups2(xs: List[(Rexp, Val => Val)], 

	 acc: List[(Rexp, Val => Val)] = Nil) : List[(Rexp, Val => Val)] = xs match {

  case Nil => acc

  case (x, y)::xs => 

	     if (acc.map(_._1).contains(x)) dups2(xs, acc :+ unit)

	     else dups2(xs, acc :+ (x, y))

}

def dups(xs: List[(Rexp, Val => Val)]) : List[(Rexp, Val => Val)] = {

  val out = dups2(xs)

  //if (out != xs) {

  //  println()

  //  println(s"Input ${string(ALTS(xs.map(_._1)))}")

  //  println(s"Ouput ${string(ALTS(out.map(_._1)))}")

  //}

  out

}

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

def F_PRINT(v: Val): Val = {

  println(s"value is ${v}")

  throw new Exception("caught error")

def flats(rs: List[Rexp], seen: Set[Rexp]) : (List[Rexp], Val => Val) = {

  //println(s"I am flats: ${string(ALTS(rs))}")

  //println(s"The size of seen is ${seen.size}, ${seen.map(string)}")

  rs match {

  case Nil => (Nil, F_ERROR)

  case r::rs1 if seen.contains(simp(r)._1) => {

    //println(s" I remove ${string(r)}")

    val (rs2, f) = flats(rs1, seen)

    (rs2, (v:Val) => IncProj(1, f(v)))

  }

  case ZERO::rs1 => {

    val (rs2, f) = flats(rs1, seen)

    (rs2, (v:Val) => IncProj(1, f(v)))

  }

  case ALTS(rs0)::rs1 => {

    val (rss, f1) = flats(rs0, seen)

    val (rs2, f2) = flats(rs1, rss.toSet ++ seen)

    (rss:::rs2, (v:Val) => v match {

      case Proj(n, vn) => 

     	if (n < rss.length) Proj(0, f1(Proj(n, vn)))

  	else IncProj(1, f2(Proj(n - rss.length, vn)))

    })

  }

  case r1::rs2 => {

    val (r1s, f1) = simp(r1)

    val (rs3, f2) = flats(rs2, seen + r1s)

    (r1s::rs3, (v:Val) => v match {

      case Proj(0, vn) => Proj(0, f1(vn))

      case Proj(n, vn) => IncProj(1, f2(Proj(n - 1, vn)))

    })

  }

}}

  case ALTS(rs) => {

    val (rs_s, fs_s) = flats(rs, Set())

    rs_s match {

      case Nil => (ZERO, F_ERROR) 

      case r::Nil => (r, (v:Val) => fs_s(Proj(0, v)))

      case rs_sd => (ALTS(rs_sd), fs_s)

      case (ALTS(rs), r2s) => (ALTS(rs.map(SEQ(_, r2s))), 

			       (v:Val) => v match {

				 case Proj(n, Sequ(v1, v2)) => Sequ(f1s(Proj(n, v1)), f2s(v2))

			       }

			      )

  case Nil => {

    //println(s"Size of the last regex: ${size(r)}")

    //println(s"${string(r)}")

    if (nullable(r)) mkeps(r) else throw new Exception("Not matched")

  }

    val rd = der(c, r)     

    val (r_simp, f_simp) = simp(rd)

    //println(s"BEFORE ${string(rd)}")

    //println(s"AFTER  ${string(r_simp)}")

    val rec = lex_simp(r_simp, cs)

    inj(r, c, f_simp(rec))

lexing_simp("ab" | "aa", "ab")

lexing_simp("ab" | "aa", "aa")

lexing(STAR("a" | "aa"), "aaaaa")

lexing_simp(STAR("a" | "aa"), "aaaaa")

lexing(STAR("a" | "aa"), "aaaaaaaaaaa")

lexing_simp(STAR("a" | "aa"), "aaaaaaaaaaa")

lexing_simp(STAR("a" | "aa"), "a" * 2)

lexing_simp(STAR("a" | "aa"), "a" * 3)

lexing_simp(STAR("a" | "aa"), "a" * 4)

lexing_simp(STAR("a" | "aa"), "a" * 20)

lexing_simp(STAR("a" | "aa"), "a" * 2000)

lexing(ALTS(List("aa", "aa", "aa", "ab", "ab")), "ab")

lexing_simp(ALTS(List("aa", "aa", "aa", "ab", "ab")), "ab")

lexing(ALTS(List(("aa" | "ab" | "aa"), "aa", "ab", "ab")), "ab")

lexing_simp(ALTS(List(("aa" | "ab" | "aa"), "aa", "ab", "ab")), "ab")

lexing(ALTS(List(ZERO, ZERO, ONE, "aa", ZERO, "aa", "aa")), "aa")

lexing_simp(ALTS(List(ZERO, ZERO, ONE, "aa", ZERO, "aa", "aa")), "aa")

lexing_simp(ONE | ZERO, "")

lexing_simp(ZERO | ONE, "")

lexing("a" | ZERO, "a")

lexing_simp("a" | ZERO, "a")

lexing(ZERO | "a", "a")

lexing_simp(ZERO | "a", "a")

lexing(ALTS(List(ZERO, ZERO, ONE, "a", ZERO, "a")), "a")

lexing_simp(ALTS(List(ZERO, ZERO, ONE, "a", ZERO, "a")), "a")

lexing(ALTS(List("a")), "a")

lexing_simp(ALTS(List("a")), "a")

lexing_simp(("a" | ZERO) | ZERO, "a")

lexing_simp("a" | (ZERO | ZERO), "a")

lexing_simp(ZERO | ("a" | ZERO), "a")

lexing_simp("abc", "abc")

lexing_simp("abc" | ONE, "abc")

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

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

lexing_simp(("ba" | "c" | "ab"), "ab")

lexing(ALTS(List(ALTS(Nil), "c", "ab")), "ab")

lexing_simp(ALTS(List(ALTS(Nil), "c", "ab")), "ab")

lexing(ALTS(List(ALTS("ab"::Nil), "c", "ab")), "ab")

lexing_simp(ALTS(List(ALTS("ab"::Nil), "c", "ab")), "ab")

lexing(ALTS(List(ALTS(List("a","ab")), "c", "ab")), "ab")

lexing_simp(ALTS(List(ALTS(List("a","ab")), "c", "ab")), "ab")

lexing(ALTS(List(ALTS(List("ab","a")), "c", "ab")), "ab")

lexing_simp(ALTS(List(ALTS(List("ab","a")), "c", "ab")), "ab")

lexing(ALTS(List(ALTS(List("ab","a","a")), "c", "ab")), "ab")

lexing_simp(ALTS(List(ALTS(List("ab","a","a")), "c", "ab")), "ab")

lexing(ALTS(List(ALTS(List("a","ab","a")), "c", "ab")), "ab")

lexing_simp(ALTS(List(ALTS(List("a","ab","a")), "c", "ab")), "ab")

lexing(ALTS(List(ALTS(List("b","a","ab")), "c", "ab")), "ab")

lexing_simp(ALTS(List(ALTS(List("b","a","ab")), "c", "ab")), "ab")

lexing_simp(ALTS(List("ba", "c", "ab")), "ab")

lexing(ALTS(List("a", "ab", "a")), "ab")

lexing_simp(ALTS(List("a", "ab", "a")), "ab")

lexing(STAR("a" | "aa"), "aa")

lexing_simp(STAR("a" | "aa"), "aa")

def enum(n: Int, s: String) : Set[Rexp] = n match {

  case 0 => Set(ZERO, ONE) ++ s.toSet.map(CHAR)

  case n => {

    val rs = enum(n - 1, s)

    rs ++

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

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

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

  }

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)

  }

import scala.util.Try

def tests(n: Int, m: Int, s: String) = {

  val rs = enum(n, s)

  val ss = strs(m, s)

  println(s"cases generated: ${rs.size} regexes and ${ss.size} strings")

  for (r1 <- rs.par; s1 <- ss.par) yield {

    val res1 = Try(Some(lexing(r1, s1))).getOrElse(None)

    val res2 = Try(Some(lexing_simp(r1, s1))).getOrElse(None)

    if (res1 != res2) println(s"Disagree on ${r1} and ${s1}")

    if (res1 != res2) Some((r1, s1)) else None

  }

}

println("Testing")

println(tests(2,7,"abc"))