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 

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

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)

}

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

  case ZERO => "0"

  case ONE => "e"

  case CHAR(c) => c.toString 

  case ALT(r1, r2) => "(" ++ pretty(r1) ++ " | " + pretty(r2) ++ ")"

  case SEQ(r1, r2) => pretty(r1) ++ pretty(r2)

  case STAR(r) => "(" ++ pretty(r) ++ ")*"

  case RECD(x, r) => "(" ++ x ++ " : " ++ pretty(r) ++ ")"

}

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

  case Empty => "()"

  case Chr(c) => c.toString

  case Left(v) => "Left(" ++ vpretty(v) ++ ")"

  case Right(v) => "Right(" ++ vpretty(v) ++ ")"

  case Sequ(v1, v2) => vpretty(v1) ++ " ~ " ++ vpretty(v2)

  case Stars(vs) => vs.flatMap(vpretty).mkString("[", ",", "]")

  case Rec(x, v) => "(" ++ x ++ ":" ++ vpretty(v) ++ ")"

}

// size of a regular expressions - for testing purposes 

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

  case ZERO => 1

  case ONE => 1

  case CHAR(_) => 1

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

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

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

  case RECD(_, r) => 1 + size(r)

}

// extracts a set of candidate values from a "non-starred" regular expression

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(Empty) ++ values(r)   // to do more would cause the set to be infinite

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

}

// zeroable function: tests whether the regular 

// expression cannot regognise any string

def zeroable (r: Rexp) : Boolean = r match {

  case ZERO => true

  case ONE => false

  case CHAR(_) => false

  case ALT(r1, r2) => zeroable(r1) && zeroable(r2)

  case SEQ(r1, r2) => zeroable(r1) || zeroable(r2)

  case STAR(_) => false

  case RECD(_, r1) => zeroable(r1)

}

// 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 CHAR(_) => false

  case ALT(r1, r2) => nullable(r1) || nullable(r2)

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

  case STAR(_) => true

  case RECD(_, r1) => nullable(r1)

}

// 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 CHAR(d) => if (c == d) ONE else ZERO

  case ALT(r1, r2) => ALT(der(c, r1), der(c, r2))

  case SEQ(r1, r2) => 

    if (nullable(r1)) ALT(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)

}

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

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

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

}

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

  case Empty => List("")

  case Chr(c) => List(c.toString)

  case Left(v) => flattens(v)

  case Right(v) => flattens(v)

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

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

  case Rec(_, v) => flattens(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 ALT(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 (ALT(r1, r2), Left(v1)) => Left(inj(r1, c, v1))

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

  case (CHAR(d), Empty) => Chr(d) 

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

}

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

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

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

    if (flatten(v1) == "") Right(prj(r2, c, v2))      

    else { if (nullable(r1)) Left(Sequ(prj(r1, c, v1), v2))

           else Sequ(prj(r1, c, v1), v2)

    }

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

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

  case (CHAR(d), _) => Empty 

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

}

// 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)

// 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 ALT(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 (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 (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 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(("a" | "ab") ~ ("b" | ""), "ab")

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

// brute force search infrastructure

// enumerates regular expressions until a certain depth

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))

  }

}

def ordr(v1: Val, r: Rexp, v2: Val) : Boolean = (v1, r, v2) match {

  case (Empty, ONE, Empty) => true

  case (Chr(c1), CHAR(c2), Chr(c3)) if (c1 == c2 && c1 == c3) => true

  case (Left(v1), ALT(r1, r2), Left(v2)) => ordr(v1, r1, v2)

  case (Right(v1), ALT(r1, r2), Right(v2)) => ordr(v1, r2, v2)

  case (Left(v1), ALT(r1, r2), Right(v2)) => flatten(v2).length <= flatten(v1).length

  case (Right(v1), ALT(r1, r2), Left(v2)) => flatten(v2).length < flatten(v1).length

  case (Sequ(v1, v2), SEQ(r1, r2), Sequ(v3, v4)) if (v1 == v3) => ordr(v2, r2, v4)

  case (Sequ(v1, v2), SEQ(r1, r2), Sequ(v3, v4)) if (v1 != v3) => ordr(v1, r1, v3)

  case _ => false

}     

def ord(v1: Val, v2: Val) : Boolean = (v1, v2) match {

  case (Empty, Empty) => true

  case (Chr(c1), Chr(c2)) if (c1 == c2) => true

  case (Left(v1), Left(v2)) => ord(v1, v2)

  case (Right(v1), Right(v2)) => ord(v1, v2)

  case (Left(v1), Right(v2)) => flatten(v2).length <= flatten(v1).length

  case (Right(v1), Left(v2)) => flatten(v2).length < flatten(v1).length

  case (Sequ(v1, v2), Sequ(v3, v4)) if (v1 == v3) => ord(v2, v4)

  case (Sequ(v1, v2), Sequ(v3, v4)) if (v1 != v3) => ord(v1, v3)

  case _ => false

}     

// exhaustively tests values of a regular expression

def vfilter1(f: Rexp => Val => Boolean)(r: Rexp) : Set[(Rexp, Val)] = {

  val vs = values(r)

  for (v <- vs; if (f(r)(v))) yield (r, v)

}

def vfilter2(f: Rexp => Val => Val => Boolean)(r: Rexp) : Set[(Rexp, Val, Val)] = {

  val vs = values(r)

  for (v1 <- vs; v2 <- vs; if (f(r)(v1)(v2))) yield (r, v1, v2)

}

def vfilter3(f: Rexp => Val => Val => Val => Boolean)(r: Rexp) : Set[(Rexp, Val, Val, Val)] = {

  val vs = values(r)

  for (v1 <- vs; v2 <- vs; v3 <- vs; if (f(r)(v1)(v2)(v3))) yield (r, v1, v2, v3)

}

// number of test cases

enum(3, "a").size

enum(2, "ab").size

enum(2, "abc").size

//enum(3, "ab").size

// test for inj/prj

def injprj_test(r:Rexp)(v:Val) : Boolean =

  if (flatten(v) != "") (inj(r, 'a', prj(r, 'a', v)) != v) else false

val injprj_tst = enum(2, "ab").flatMap(vfilter1(injprj_test))

val injprj_smallest = injprj_tst.toList.sortWith { (x,y) => size(x._1) < size(y._1) }.head

prj(CHAR('b'), 'a', Chr('b'))

inj(CHAR('b'), 'a', Empty)

prj(SEQ(ALT(ONE,ONE),CHAR('a')), 'a', Sequ(Right(Empty),Chr('a')))

inj(SEQ(ALT(ONE,ONE),CHAR('a')), 'a', Right(Empty))

// tests for totality

def totality_test(r: Rexp)(v1: Val)(v2: Val) : Boolean =

  !(ord(v1, v2) || ord(v2, v1))

enum(2, "ab").flatMap(vfilter2(totality_test))

enum(3, "a").flatMap(vfilter2(totality_test))

//tests for transitivity (simple transitivity fails)

def transitivity_test(r: Rexp)(v1: Val)(v2: Val)(v3: Val) : Boolean = 

  ord(v1, v2) && ord(v2, v3) && !ord(v1, v3)

val test0 = enum(3, "a").flatMap(vfilter3(transitivity_test))

val test0_smallest = test0.toList.sortWith { (x,y) => size(x._1) < size(y._1) }.head

pretty(test0_smallest._1)

vpretty(test0_smallest._2)

vpretty(test0_smallest._3)

vpretty(test0_smallest._4)

/* Counter example for simple transitivity:

 r = a | ((a | a)(a | e))

 v1 = Left(a)

 v2 = Right(Left(a) ~ Right(()))

 v3 = Right(Right(a) ~ Left(a))

*/ 

def is_seq(v: Val) : Boolean = v match {

  case Seq(_, _) => true

  case _ => false

}

def transitivity_test_extended(r: Rexp)(v1: Val)(v2: Val)(v3: Val) : Boolean = {

  //flatten(v1).size >= flatten(v2).size && 

  //flatten(v2).size >= flatten(v3).size && 

  is_seq(v1) &&

  ord(v1, v2) && ord(v2, v3) && !ord(v1, v3)

}

// smallest(?) example where simple transitivity fails 

val test1 = enum(3, "a").flatMap(vfilter3(transitivity_test_extended))

val test1_smallest = test1.toList.sortWith { (x,y) => size(x._1) < size(y._1) }.head

pretty(test1_smallest._1)

vpretty(test1_smallest._2)

vpretty(test1_smallest._3)

vpretty(test1_smallest._4)

ord(test1_smallest._2, test1_smallest._3)

ord(test1_smallest._3, test1_smallest._4)

ord(test1_smallest._2, test1_smallest._4)

ordr(test1_smallest._3, test1_smallest._1, test1_smallest._4)

/* Counter example for extended transitivity:

 r = ((e | e)(e | a)) | a

 v1 = Left(Right(()) ~ Right(a))

 v2 = Right(a)

 v3 = Left(Left(()) ~ Left(()))

*/ 

// with flatten test

enum(3, "a").flatMap(vfilter3(transitivity_test_extended))

//injection tests

def injection_test(r: Rexp)(c: Char)(v1: Val)(v2: Val) : Boolean = {

  v1 != v2 && 

  ord(v1, v2) && 

  ord(inj(r, c, v2), inj(r, c, v1)) && 

  flatten(v1) == flatten(v2)

}

def injection_testA(r: Rexp)(c: Char)(v1: Val)(v2: Val) : Boolean = {

  v1 != v2 && 

  ord(v1, v2) && 

  ord(inj(r, c, v2), inj(r, c, v1)) && 

  flatten(v1).length == flatten(v2).length

}

def injection_test2(r: Rexp)(c: Char)(v1: Val)(v2: Val) : Boolean = {

  v1 != v2 && 

  ord(v1, v2) && 

  ord(inj(r, c, v2), inj(r, c, v1))

}

def vfilter4(f: Rexp => Char => Val => Val => Boolean)(c: Char)(r: Rexp) : Set[(Rexp, Rexp, Val, Val)] = {

  val r_der = der(c, r)

  val vs = values(r_der)

  for (v1 <- vs; v2 <- vs; if (f(r)(c)(v1)(v2))) yield (r, r_der, v1, v2)

} 

enum(3, "a").flatMap(vfilter4(injection_test)('a'))

enum(3, "a").flatMap(vfilter4(injection_testA)('a'))

val test2 = enum(3, "a").flatMap(vfilter4(injection_test2)('a'))

val test2_smallest = test2.toList.sortWith { (x,y) => size(x._1) < size(y._1) }.head

pretty(test2_smallest._1)

pretty(test2_smallest._2)

vpretty(test2_smallest._3)

vpretty(test2_smallest._4)

vpretty(inj(test2_smallest._1, 'a', test2_smallest._3))

vpretty(inj(test2_smallest._1, 'a', test2_smallest._4))

// Lexing Rules for a Small While Language

def PLUS(r: Rexp) = r ~ r.%

val SYM = "a" | "b" | "c" | "d" | "e" | "f" | "g" | "h" | "i" | "j" | "k" | "l" | "m" | "n" | "o" | "p" | "q" | "r" | "s" | "t" | "u" | "v" | "w" | "x" | "y" | "z"

val DIGIT = "0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9"

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)).%

/*

val WHILE_REGS = (KEYWORD | 

                  ID | 

                  OP | 

                  NUM | 

                  SEMI | 

                  LPAREN | RPAREN | 

                  BEGIN | END | 

                  WHITESPACE).%

*/