// A simple lexer inspired by work of Sulzmann & Lu
//==================================================
//
// Call the test cases with 
//
//   amm lexer.sc small
//   amm lexer.sc fib
//   amm lexer.sc loops
//   amm lexer.sc email
//
//   amm lexer.sc all


// regular expressions including records
abstract class Rexp 
case object ZERO extends Rexp
case object ONE extends Rexp
case object ANYCHAR extends Rexp
case class CHAR(c: Char) extends Rexp
case class ALTS(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 NTIMES(n: Int, r: Rexp) extends Rexp
case class OPTIONAL(r: Rexp) extends Rexp
case class NOT(r: Rexp) extends Rexp
                // records for extracting strings or tokens
  
// 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
case class Ntime(vs: List[Val]) extends Val
case class Optionall(v: Val) extends Val
case class Nots(s: String) extends Val



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 
case class ANOT(bs: Bits, r: ARexp) extends ARexp
case class AANYCHAR(bs: Bits) extends ARexp


   
// 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) = ALTS(r, s)
  def % = STAR(r)
  def ~ (s: Rexp) = SEQ(r, s)
}

implicit def stringOps(s: String) = new {
  def | (r: Rexp) = ALTS(s, r)
  def | (r: String) = ALTS(s, r)
  def % = STAR(s)
  def ~ (r: Rexp) = SEQ(s, r)
  def ~ (r: String) = SEQ(s, r)
  def $ (r: Rexp) = RECD(s, r)
}

def nullable(r: Rexp) : Boolean = r match {
  case ZERO => false
  case ONE => true
  case CHAR(_) => false
  case ANYCHAR => false
  case ALTS(r1, r2) => nullable(r1) || nullable(r2)
  case SEQ(r1, r2) => nullable(r1) && nullable(r2)
  case STAR(_) => true
  case RECD(_, r1) => nullable(r1)
  case NTIMES(n, r) => if (n == 0) true else nullable(r)
  case OPTIONAL(r) => true
  case NOT(r) => !nullable(r)
}

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 ANYCHAR => ONE 
  case ALTS(r1, r2) => ALTS(der(c, r1), der(c, r2))
  case SEQ(r1, r2) => 
    if (nullable(r1)) ALTS(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)
  case NTIMES(n, r) => if(n > 0) SEQ(der(c, r), NTIMES(n - 1, r)) else ZERO
  case OPTIONAL(r) => der(c, r)
  case NOT(r) =>  NOT(der(c, r))
}


// extracts a string from a 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 Ntime(vs) => vs.map(flatten).mkString
  case Optionall(v) => flatten(v)
  case Rec(_, v) => flatten(v)
}


// extracts an environment from a value;
// used for tokenising a string
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 Ntime(vs) => vs.flatMap(env)
  case Rec(x, v) => (x, flatten(v))::env(v)
  case Optionall(v) => env(v)
  case Nots(s) => ("Negative", s) :: Nil
}


// The injection and mkeps part of the lexer
//===========================================

def mkeps(r: Rexp) : Val = r match {
  case ONE => Empty
  case ALTS(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))
  case NTIMES(n, r) => Ntime(List.fill(n)(mkeps(r)))
  case OPTIONAL(r) => Optionall(Empty)
  case NOT(rInner) => if(nullable(rInner)) throw new Exception("error")  
                         else Nots("")//Nots(s.reverse.toString)
//   case NOT(ZERO) => Empty
//   case NOT(CHAR(c)) => Empty
//   case NOT(SEQ(r1, r2)) => Sequ(mkeps(NOT(r1)), mkeps(NOT(r2)))
//   case NOT(ALTS(r1, r2)) => if(!nullable(r1)) Left(mkeps(NOT(r1))) else Right(mkeps(NOT(r2)))
//   case NOT(STAR(r)) => Stars(Nil) 

}

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(r1, r2), Left(v1)) => Left(inj(r1, c, v1))
  case (ALTS(r1, r2), Right(v2)) => Right(inj(r2, c, v2))
  case (CHAR(d), Empty) => Chr(c) 
  case (RECD(x, r1), _) => Rec(x, inj(r1, c, v))
  case (NTIMES(n, r), Sequ(v1, Ntime(vs))) => Ntime(inj(r, c, v1)::vs)
  case (OPTIONAL(r), v) => Optionall(inj(r, c, v))
  case (NOT(r), Nots(s)) => Nots(c.toString ++ s)
  case (ANYCHAR, Empty) => Chr(c)
}

// 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_ERROR(v: Val): Val = throw new Exception("error")

// simplification
def simp(r: Rexp): (Rexp, Val => Val) = r match {
  case ALTS(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 (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 r => (r, F_ID)
}

// lexing functions including simplification
def lex_simp(r: Rexp, s: List[Char]) : Val = s match {
  case Nil => if (nullable(r)) mkeps(r) else 
    { throw new Exception(s"lexing error $r not nullable") } 
  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) = 
  env(lex_simp(r, s.toList))

// The Lexing Rules for the WHILE Language

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

def Range(s : List[Char]) : Rexp = s match {
  case Nil => ZERO
  case c::Nil => CHAR(c)
  case c::s => ALTS(CHAR(c), Range(s))
}
def RANGE(s: String) = Range(s.toList)

val SYM = RANGE("ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz_")
val DIGIT = RANGE("0123456789")
val ID = SYM ~ (SYM | DIGIT).% 
val NUM = PLUS(DIGIT)
val KEYWORD : Rexp = "skip" | "while" | "do" | "if" | "then" | "else" | "read" | "write" 
val SEMI: Rexp = ";"
val OP: Rexp = ":=" | "=" | "-" | "+" | "*" | "!=" | "<" | ">"
val WHITESPACE = PLUS(" " | "\n" | "\t" | "\r")
val RPAREN: Rexp = "{"
val LPAREN: Rexp = "}"
val STRING: Rexp = "\"" ~ SYM.% ~ "\""


//ab \ a --> 1b
//
val WHILE_REGS = (("k" $ KEYWORD) | 
                  ("i" $ ID) | 
                  ("o" $ OP) | 
                  ("n" $ NUM) | 
                  ("s" $ SEMI) | 
                  ("str" $ STRING) |
                  ("p" $ (LPAREN | RPAREN)) | 
                  ("w" $ WHITESPACE)).%

val NREGS = NTIMES(5, OPTIONAL(SYM))
val NREGS1 = ("test" $ NREGS)
// Two Simple While Tests
//========================
val NOTREG = "hehe" ~ NOT((ANYCHAR.%) ~ "haha" ~ (ANYCHAR.%))


  // 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 ACHAR(_,_) => false
    case AALTS(_, rs) => rs.exists(bnullable)
    case ASEQ(_, r1, r2) => bnullable(r1) && bnullable(r2)
    case ASTAR(_, _) => true
    case ANOT(_, rn) => !bnullable(rn)
  }

def mkepsBC(r: ARexp) : Bits = r match {
    case AONE(bs) => bs
    case AALTS(bs, rs) => {
      val n = rs.indexWhere(bnullable)
      bs ++ mkepsBC(rs(n))
    }
    case ASEQ(bs, r1, r2) => bs ++ mkepsBC(r1) ++ mkepsBC(r2)
    case ASTAR(bs, r) => bs ++ List(Z)
    case ANOT(bs, rn) => bs
  }


def bder(c: Char, r: ARexp) : ARexp = r match {
    case AZERO => AZERO
    case AONE(_) => AZERO
    case ACHAR(bs, f) => if (c == f) AONE(bs) else AZERO
    case AALTS(bs, rs) => AALTS(bs, rs.map(bder(c, _)))
    case ASEQ(bs, r1, r2) => 
      if (bnullable(r1)) AALTS(bs, ASEQ(Nil, bder(c, r1), r2) :: fuse(mkepsBC(r1), bder(c, r2)) :: Nil )
      else ASEQ(bs, bder(c, r1), r2)
    case ASTAR(bs, r) => ASEQ(bs, fuse(List(S), bder(c, r)), ASTAR(Nil, r))
    case ANOT(bs, rn) => ANOT(bs, bder(c, rn))
    case AANYCHAR(bs) => AONE(bs)
  } 

def fuse(bs: Bits, r: ARexp) : ARexp = r match {
    case AZERO => AZERO
    case AONE(cs) => AONE(bs ++ cs)
    case ACHAR(cs, f) => ACHAR(bs ++ cs, f)
    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)
    case ANOT(cs, r) => ANOT(bs ++ cs, r)
  }


def internalise(r: Rexp) : ARexp = r match {
    case ZERO => AZERO
    case ONE => AONE(Nil)
    case CHAR(c) => ACHAR(Nil, c)
    //case PRED(f) => APRED(Nil, f)
    case ALTS(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)
    case NOT(r) => ANOT(Nil, internalise(r))
    case ANYCHAR => AANYCHAR(Nil)
  }


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) => {
            val rs_simp = rs.map(bsimp(_))
            val flat_res = flats(rs_simp)
            val dist_res = distinctBy(flat_res, erase)//strongDB(flat_res)//distinctBy(flat_res, erase)
            dist_res match {
              case Nil => AZERO
              case s :: Nil => fuse(bs1, s)
              case rs => AALTS(bs1, rs)  
            }
          
      }
      case r => r
    }
  }
  def strongBsimp(r: ARexp): ARexp =
  {
    r match {
      case ASEQ(bs1, r1, r2) => (strongBsimp(r1), strongBsimp(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) => {
            val rs_simp = rs.map(strongBsimp(_))
            val flat_res = flats(rs_simp)
            val dist_res = distinctBy2(flat_res)//distinctBy(flat_res, erase)
            dist_res match {
              case Nil => AZERO
              case s :: Nil => fuse(bs1, s)
              case rs => AALTS(bs1, rs)  
            }
          
      }
      case r => r
    }
  }

  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 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 projectFirstChild(rs: List[ARexp]) : List[ARexp] = rs match {
    case (ASEQ(bs, r1, r2p)::rs1) => r1::projectFirstChild(rs1)
    case Nil => Nil
  }


  def distinctBy2(xs: List[ARexp], acc: List[Rexp] = Nil): List[ARexp] = xs match {
    case Nil => Nil
    case (x::xs) => {
      val res = erase(x)
      if(acc.contains(res))
        distinctBy2(xs, acc)
      else
        x match {
          case ASEQ(bs0, AALTS(bs1, rs), r2) => 
            val newTerms =  distinctBy2(rs.map(r1 => ASEQ(Nil, r1, r2)), acc)
            val rsPrime = projectFirstChild(newTerms)
            newTerms match {
              case Nil => distinctBy2(xs, acc)
              case t::Nil => ASEQ(bs0, fuse(bs1, rsPrime.head), r2)::distinctBy2(xs, erase(t)::acc)
              case ts => ASEQ(bs0, AALTS(bs1, rsPrime), r2)::distinctBy2(xs, newTerms.map(erase(_)):::acc)
            }
          case x => x::distinctBy2(xs, res::acc)
        }
    }
  }



  def distinctBy3(xs: List[ARexp], acc: List[Rexp] = Nil): List[ARexp] = xs match {
    case Nil => Nil
    case (x::xs) => {
      val res = erase(x)
      if(acc.contains(res))
        distinctBy3(xs, acc)
      else
        x match {
          case ASEQ(bs0, AALTS(bs1, rs), r2) => 
            val newTerms =  distinctBy3(rs.map(r1 => ASEQ(Nil, r1, r2)), acc)
            val rsPrime = projectFirstChild(newTerms)
            newTerms match {
              case Nil => distinctBy3(xs, acc)
              case t::Nil => ASEQ(bs0, fuse(bs1, rsPrime.head), r2)::distinctBy3(xs, erase(t)::acc)
              case ts => ASEQ(bs0, AALTS(bs1, rsPrime), r2)::distinctBy3(xs, newTerms.map(erase(_)):::acc)
            }
          case x => x::distinctBy3(xs, res::acc)
        }
    }
  }

  def breakIntoTerms(r: Rexp) : List[Rexp] = r match {
    case SEQ(r1, r2)  => breakIntoTerms(r1).map(r11 => SEQ(r11, r2))
    case ALTS(r1, r2) => breakIntoTerms(r1) ::: breakIntoTerms(r2)
    case _ => r::Nil
  } 

  def prettyRexp(r: Rexp) : String = r match {
      case STAR(r0) => s"${prettyRexp(r0)}*"
      case SEQ(CHAR(c), r2) => c.toString ++ prettyRexp(r2)
      case SEQ(r1, r2) => s"${prettyRexp(r1)}~${prettyRexp(r2)}"
      case CHAR(c) => c.toString
      case ANYCHAR => "."
    //   case NOT(r0) => s
  }

  def decode_aux(r: Rexp, bs: Bits) : (Val, Bits) = (r, bs) match {
    case (ONE, bs) => (Empty, bs)
    case (CHAR(f), bs) => (Chr(f), bs)
    case (ALTS(r1, r2), Z::bs1) => {
        val (v, bs2) = decode_aux(r1, bs1)
        (Left(v), bs2)
    }
    case (ALTS(r1, r2), S::bs1) => {
        val (v, bs2) = decode_aux(r2, 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)
      //println(v)
      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)
    }
    case (NOT(r), bs) => (Nots(prettyRexp(r)), bs)
  }

  def decode(r: Rexp, bs: Bits) = decode_aux(r, bs) match {
    case (v, Nil) => v
    case _ => throw new Exception("Not decodable")
  }



def blexSimp(r: Rexp, s: String) : List[Bit] = {
    blex_simp(internalise(r), s.toList)
}

def blexing_simp(r: Rexp, s: String) : Val = {
    val bit_code = blex_simp(internalise(r), s.toList)
    decode(r, bit_code)
  }

  def strong_blexing_simp(r: Rexp, s: String) : Val = {
    decode(r, strong_blex_simp(internalise(r), s.toList))
  }

  def strong_blex_simp(r: ARexp, s: List[Char]) :Bits = s match {
    case Nil => {
      if (bnullable(r)) {
        //println(asize(r))
        val bits = mkepsBC(r)
        bits
      }
      else 
        throw new Exception("Not matched")
    }
    case c::cs => {
      val der_res = bder(c,r)
      val simp_res = strongBsimp(der_res)  
      strong_blex_simp(simp_res, cs)      
    }
  }



  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 bdersSimp(s: String, r: Rexp) : ARexp = bders_simp(s.toList, internalise(r))

  def bders_simpS(s: List[Char], r: ARexp) : ARexp = s match {
    case Nil => r 
    case c::s => bders_simpS(s, strongBsimp(bder(c, r)))
  }

  def bdersSimpS(s: String, r: Rexp) : ARexp = bders_simpS(s.toList, internalise(r))

  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, a::Nil) => erase(a)
    case AALTS(bs, a::as) => ALTS(erase(a), erase(AALTS(bs, as)))
    case ASEQ(bs, r1, r2) => SEQ (erase(r1), erase(r2))
    case ASTAR(cs, r)=> STAR(erase(r))
    case ANOT(bs, r) => NOT(erase(r))
    case AANYCHAR(bs) => ANYCHAR
  }

def breakHead(r: ARexp) : List[ARexp] = r match {
    case AALTS(bs, rs) => rs
    case r => r::Nil
}

def distinctByWithAcc[B, C](xs: List[B], f: B => C, 
    acc: List[C] = Nil, accB: List[B] = Nil): (List[B], List[C]) = xs match {
    case Nil => (accB.reverse, acc)
    case (x::xs) => {
      val res = f(x)
      if (acc.contains(res)) distinctByWithAcc(xs, f, acc, accB)  
      else distinctByWithAcc(xs, f, res::acc, x::accB)
    }
  } 

//(r1+r2)r + (r2'+r3)r' ~> (r1+r2)r + (r3)r' (erase r = erase r') (erase r2 = erase r2')
//s = s1@s2 \in L R  s1 \in L r2 s2 \in L r
//s = s3@s4  s3 \in L r1 s4 \in L r |s3| < |s1| \nexists s3' s4' s.t. s3'@s4' = s and s3' \in L r1 s4' \in L r
//(r1+r4)r ~> r1r +_usedToBeSeq r4r 
// | ss7:  "erase a01 = erase a02 ∧ (distinctBy as2 erase (set (map erase as1)) = as2p)  ⟹ 
//         (rsa@[ASEQ bs (AALTs bs1 as1) (ASTAR bs01 a01)]@rsb@[ASEQ bs (AALTs bs2 as2) (ASTAR bs02 a02)]@rsc)  s↝
//         (rsa@[ASEQ bs (AALTs bs1 as1) (ASTAR bs01 a01)]@rsb@[ASEQ bs (AALTs bs2 as2p) (ASTAR bs02 a02)]@rsc)"
// | ss8:  "erase a01 = erase a02 ∧ (distinctBy [a2] erase (set (map erase as1)) = [])  ⟹ 
//         (rsa@[ASEQ bs (AALTs bs1 as1) (ASTAR bs01 a01)]@rsb@[ASEQ bs a2 (ASTAR bs02 a02)]@rsc) s↝                                                
//         (rsa@[ASEQ bs (AALTs bs1 as1) (ASTAR bs01 a01)]@rsb@rsc)"
// | ss9:  "erase a01 = erase a02 ∧ (distinctBy as2 erase {erase a1} = as2p)  ⟹ 
//         (rsa@[ASEQ bs a1 (ASTAR bs01 a01)]@rsb@[ASEQ bs (AALTs bs2 as2) (ASTAR bs02 a02)]@rsc) s↝ 
//         (rsa@[ASEQ bs a1 (ASTAR bs01 a01)]@rsb@[ASEQ bs (AALTs bs2 as2p) (ASTAR bs02 a02)]@rsc)"


//# of top-level terms
//      r1r2 -> r11r2 + r12 -> r21r2 + r22 + r12' -> 4 terms -> 5 terms -> 6..........
// if string long enough --> #| r1r2 \s s | > unbounded? No ->  #| r1r2 \s s | <  #| pders UNIV r1| + #|pders UNIV r2| <= awidth r1r2
//r* -> r1r* -> r21r* + r22r* -> 4 terms -> 8 terms..............
  def strongDB(xs: List[ARexp], 
                       acc1: List[Rexp] = Nil, 
                       acc2 : List[(List[Rexp], Rexp)] = Nil): List[ARexp] = xs match {
    case Nil => Nil
    case (x::xs) => 
        if(acc1.contains(erase(x)))
            strongDB(xs, acc1, acc2)
        else{
            x match {
                case ASTAR(bs0, r0) => 
                    val headList : List[ARexp] = List[ARexp](AONE(Nil))
                    val i = acc2.indexWhere(
                        r2stl => {val (r2s, tl) = r2stl; tl == erase(r0) } 
                    )
                    if(i == -1){ 
                        x::strongDB(
                            xs, erase(x)::acc1, (ONE::Nil, erase(r0))::acc2
                        )
                    }
                    else{
                        val headListAlready = acc2(i)
                        val (newHeads, oldHeadsUpdated) = 
                                distinctByWithAcc(headList, erase, headListAlready._1)
                        newHeads match{
                            case newHead::Nil =>
                                ASTAR(bs0, r0) :: 
                                strongDB(xs, erase(x)::acc1, 
                                acc2.updated(i, (oldHeadsUpdated, headListAlready._2)) )
                            case Nil =>
                                strongDB(xs, erase(x)::acc1, 
                                acc2)
                        }
                    }                
                case ASEQ(bs, r1, ASTAR(bs0, r0)) => 
                    val headList = breakHead(r1)
                    val i = acc2.indexWhere(
                        r2stl => {val (r2s, tl) = r2stl; tl == erase(r0) } 
                    )
                    if(i == -1){ 
                        x::strongDB(
                            xs, erase(x)::acc1, (headList.map(erase(_)), erase(r0))::acc2
                        )
                    }
                    else{
                        val headListAlready = acc2(i)
                        val (newHeads, oldHeadsUpdated) = 
                                distinctByWithAcc(headList, erase, headListAlready._1)
                        newHeads match{
                            case newHead::Nil =>
                                ASEQ(bs, newHead, ASTAR(bs0, r0)) :: 
                                strongDB(xs, erase(x)::acc1, 
                                acc2.updated(i, (oldHeadsUpdated, headListAlready._2)) )//TODO: acc2 already contains headListAlready
                            case Nil =>
                                strongDB(xs, erase(x)::acc1, 
                                acc2)
                            case hds => val AALTS(bsp, rsp) = r1
                                ASEQ(bs, AALTS(bsp, hds), ASTAR(bs0, r0)) ::
                                strongDB(xs, erase(x)::acc1,
                                acc2.updated(i, (oldHeadsUpdated, headListAlready._2)))
                        }
                    }
                case rPrime => x::strongDB(xs, erase(x)::acc1, acc2)    
            }
                
        }
  
}

def allCharSeq(r: Rexp) : Boolean = r match {
  case CHAR(c) => true
  case SEQ(r1, r2) => allCharSeq(r1) && allCharSeq(r2)
  case _ => false
}

def flattenSeq(r: Rexp) : String = r match {
  case CHAR(c) => c.toString
  case SEQ(r1, r2) => flattenSeq(r1) ++ flattenSeq(r2)
  case _ => throw new Error("flatten unflattenable rexp")
} 

def shortRexpOutput(r: Rexp) : String = r match {
    case CHAR(c) => c.toString
    case ONE => "1"
    case ZERO => "0"
    case SEQ(r1, r2) if(allCharSeq(r)) => flattenSeq(r)//"[" ++ shortRexpOutput(r1) ++ "~" ++ shortRexpOutput(r2) ++ "]"
    case SEQ(r1, r2) => "[" ++ shortRexpOutput(r1) ++ "~" ++ shortRexpOutput(r2) ++ "]"
    case ALTS(r1, r2) => "(" ++ shortRexpOutput(r1) ++ "+" ++ shortRexpOutput(r2) ++ ")"
    case STAR(r) => "[" ++ shortRexpOutput(r) ++ "]*"
    //case STAR(r) => "STAR(" ++ shortRexpOutput(r) ++ ")"
    //case RTOP => "RTOP"
  }


def blex_simp(r: ARexp, s: List[Char]) : Bits = s match {
    case Nil => {
      if (bnullable(r)) {
        //println(asize(r))
        val bits = mkepsBC(r)
        bits
      }
      else 
        throw new Exception("Not matched")
    }
    case c::cs => {
      val der_res = bder(c,r)
      val simp_res = bsimp(der_res)  
      blex_simp(simp_res, cs)      
    }
  }
  def size(r: Rexp) : Int = r match {
    case ZERO => 1
    case ONE => 1
    case CHAR(_) => 1
    case ANYCHAR => 1
    case NOT(r0) => 1 + size(r0)
    case SEQ(r1, r2) => 1 + size(r1) + size(r2)
    case ALTS(r1, r2) => 1 + List(r1, r2).map(size).sum
    case STAR(r) => 1 + size(r)
  }

  def asize(a: ARexp) = size(erase(a))

//pder related
type Mon = (Char, Rexp)
type Lin = Set[Mon]

def dot_prod(rs: Set[Rexp], r: Rexp): Set[Rexp] = r match {
    case ZERO => Set()
    case ONE => rs
    case r => rs.map((re) => if (re == ONE) r else SEQ(re, r)  )   
  }
  def cir_prod(l: Lin, t: Rexp): Lin = t match {//remember this Lin is different from the Lin in Antimirov's paper. Here it does not mean the set of all subsets of linear forms that does not contain zero, but rather the type a set of linear forms
    case ZERO => Set()
    case ONE => l
    case t => l.map( m => m._2 match {case ZERO => m case ONE => (m._1, t) case p => (m._1, SEQ(p, t)) }  )
  }
  def lf(r: Rexp): Lin = r match {
    case ZERO => Set()
    case ONE => Set()
    case CHAR(f) => {
      //val Some(c) = alphabet.find(f) 
      Set((f, ONE))
    }
    case ALTS(r1, r2) => {
      lf(r1 ) ++ lf(r2)
    }
    case STAR(r1) => cir_prod(lf(r1), STAR(r1)) //may try infix notation later......
    case SEQ(r1, r2) =>{
      if (nullable(r1))
        cir_prod(lf(r1), r2) ++ lf(r2)
      else
        cir_prod(lf(r1), r2) 
    }
  }
  def lfs(r: Set[Rexp]): Lin = {
    r.foldLeft(Set[Mon]())((acc, r) => acc ++ lf(r))
  }

  def pder(x: Char, t: Rexp): Set[Rexp] = {
    val lft = lf(t)
    (lft.filter(mon => if(mon._1 == x) true else false)).map(mon => mon._2)
  }
  def pders_single(s: List[Char], t: Rexp) : Set[Rexp] = s match {
    case x::xs => pders(xs, pder(x, t))
    case Nil => Set(t)
  }
  def pders(s: List[Char], ts: Set[Rexp]) : Set[Rexp] = s match {
    case x::xs => pders(xs, ts.foldLeft(Set[Rexp]())((acc, t) => acc ++ pder(x, t)))
    case Nil => ts 
  }
  def pderss(ss: List[List[Char]], t: Rexp): Set[Rexp] = ss.foldLeft( Set[Rexp]() )( (acc, s) => pders_single(s, t) ++ acc )
  def pdera(t: Rexp): Set[Rexp] = lf(t).map(mon => mon._2)
  //all implementation of partial derivatives that involve set union are potentially buggy
  //because they don't include the original regular term before they are pdered.
  //now only pderas is fixed.
  def pderas(t: Set[Rexp], d: Int): Set[Rexp] = if(d > 0) pderas(lfs(t).map(mon => mon._2), d - 1) ++ t else lfs(t).map(mon => mon._2) ++ t//repeated application of pderas over the newest set of pders.
  def pderUNIV(r: Rexp) : Set[Rexp] = pderas(Set(r), awidth(r) + 1)
  def awidth(r: Rexp) : Int = r match {
    case CHAR(c) => 1
    case SEQ(r1, r2) => awidth(r1) + awidth(r2)
    case ALTS(r1, r2) => awidth(r1) + awidth(r2)
    case ONE => 0
    case ZERO => 0
    case STAR(r) => awidth(r)
  }
  //def sigma_lf(l: Set[Mon]) : Rexp = ALTS(l.map(mon => SEQ(CHAR(mon._1),mon._2)).toList)
  //def sigma(rs: Set[Rexp]) : Rexp = ALTS(rs.toList)
  def o(r: Rexp) = if (nullable(r)) ONE else ZERO
  //def nlf(t: Rexp) : Rexp = ALTS(List( o(t), sigma_lf(lf(t)) ))
  def pdp(x: Char, r: Rexp) : Set[Rexp] = r match {
    case ZERO => Set[Rexp]()
    case ONE => Set[Rexp]()
    case CHAR(f) => if(x == f) Set(ONE) else Set[Rexp]()
    case ALTS(r1, r2) => pdp(x, r1) ++ pdp(x, r2)
    case STAR(r1) => pdp(x, r).map(a => SEQ(a, STAR(r1)))
    case SEQ(a0, b) => if(nullable(a0)) pdp(x, a0).map(a => SEQ(a, b)) ++ pdp(x, b) else pdp(x, a0).map(a => SEQ(a, b))
  }
  def pdps(s: List[Char], ts: Set[Rexp]): Set[Rexp] = s match {
    case x::xs => pdps(xs, ts.foldLeft(Set[Rexp]())((acc, t) => acc ++ pder(x, t)))
    case Nil => ts   
  }
  def pdpss(ss: List[List[Char]], t: Rexp): Set[Rexp] = ss.foldLeft( Set[Rexp]())((acc, s) => pdps(s, Set(t)) ++ acc)



// @arg(doc = "small tests")
val STARREG = (((STAR("a") | (STAR("aa")) | STAR("aaa") | STAR("aaaa") | STAR("aaaaa") | (STAR("aaaaaa")) | STAR("aaaaaaa") | STAR("aaaaaaaa")).%))
//(((STAR("a") | ( STAR("aaa")) | STAR("aaaaa")).%).%).%

// @main
def small() = {


//   println(lexing_simp(NOTREG, prog0))
//   val v = lex_simp(NOTREG, prog0.toList)
//   println(v)

//   val d =  (lex_simp(NOTREG, prog0.toList))
//   println(d)
  val pderSTAR = pderUNIV(STARREG)

  val refSize = pderSTAR.map(size(_)).sum
  println("different partial derivative terms:")
  pderSTAR.foreach(r => r match {
      
        case SEQ(head, rstar) =>
          println(shortRexpOutput(head) ++ "~STARREG")
        case STAR(rstar) =>
          println("STARREG")
      
    }
    )
  println("the total number of terms is")
  //println(refSize)
  println(pderSTAR.size)
  for(i <- List(1, 10, 100, 400, 800, 840, 841, 900) ){
    val prog0 = "a" * i
    //println(s"test: $prog0")
    println(s"testing with $i a's" )
    val bd = bdersSimp(prog0, STARREG)//DB
    val sbd = bdersSimpS(prog0, STARREG)//strongDB
    val subTerms = breakIntoTerms(erase(sbd))
    val subTermsLarge = breakIntoTerms(erase(bd))
    
    println(s"subterms of regex with strongDB: ${subTerms.length}, standard DB: ${subTermsLarge.length}")
    println("the number of distinct subterms for bsimp with strongDB and standardDB")
    println(subTerms.distinct.size)
    println(subTermsLarge.distinct.size)
    println("which coincides with the number of PDER terms")


    // println(shortRexpOutput(erase(sbd)))
    // println(shortRexpOutput(erase(bd)))
    
    println("pdersize, original, strongSimp, bsimp")
    println(refSize, size(STARREG),  asize(sbd), asize(bd))

    // val vres = strong_blexing_simp( STARREG, prog0)
    // println(vres)
  }
//   println(vs.length)
//   println(vs)
   

  // val prog1 = """read  n; write n"""  
  // println(s"test: $prog1")
  // println(lexing_simp(WHILE_REGS, prog1))
}

small()

// // Bigger Tests
// //==============

// // escapes strings and prints them out as "", "\n" and so on
// def esc(raw: String): String = {
//   import scala.reflect.runtime.universe._
//   Literal(Constant(raw)).toString
// }

// def escape(tks: List[(String, String)]) =
//   tks.map{ case (s1, s2) => (s1, esc(s2))}


// 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
// """

// @arg(doc = "Fibonacci test")
// @main
// def fib() = {
//   println("lexing fib program")
//   println(escape(lexing_simp(WHILE_REGS, prog2)).mkString("\n"))
// }


// val prog3 = """
// start := 1000;
// x := start;
// y := start;
// z := start;
// while 0 < x do {
//  while 0 < y do {
//   while 0 < z do {
//     z := z - 1
//   };
//   z := start;
//   y := y - 1
//  };     
//  y := start;
//  x := x - 1
// }
// """

// @arg(doc = "Loops test")
// @main
// def loops() = {
//   println("lexing Loops")
//   println(escape(lexing_simp(WHILE_REGS, prog3)).mkString("\n"))
// }

// @arg(doc = "Email Test")
// @main
// def email() = {
//   val lower = "abcdefghijklmnopqrstuvwxyz"

//   val NAME = RECD("name", PLUS(RANGE(lower ++ "_.-")))
//   val DOMAIN = RECD("domain", PLUS(RANGE(lower ++ "-")))
//   val RE = RANGE(lower ++ ".")
//   val TOPLEVEL = RECD("top", (RE ~ RE) |
//                              (RE ~ RE ~ RE) | 
//                              (RE ~ RE ~ RE ~ RE) | 
//                              (RE ~ RE ~ RE ~ RE ~ RE) |
//                              (RE ~ RE ~ RE ~ RE ~ RE ~ RE))

//   val EMAIL = NAME ~ "@" ~ DOMAIN ~ "." ~ TOPLEVEL

//   println(lexing_simp(EMAIL, "christian.urban@kcl.ac.uk"))
// }


// @arg(doc = "All tests.")
// @main
// def all() = { small(); fib() ; loops() ; email() } 




// runs with amm2 and amm3



