diff -r 06aa99b54423 -r fedc16924b76 progs/scala/re-bit2.scala
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/progs/scala/re-bit2.scala	Sat Oct 24 12:13:39 2020 +0100
@@ -0,0 +1,691 @@
+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
+case class Position(i: Int, v: Val) extends Val  // for testing purposes
+case object Undefined extends Val                // for testing purposes
+   
+// 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)
+}
+
+
+// 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
+}
+
+// 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))
+}
+
+// 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))
+}
+
+// mkeps and 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)
+}
+
+
+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(c) 
+}
+
+// main lexing function (produces a value)
+// - no 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)
+
+
+
+// Bitcoded + Annotation
+//=======================
+
+//erase function: extracts the regx from Aregex
+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))
+}
+
+// translation into ARexps
+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))
+}
+
+
+// nullable 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
+}
+
+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))
+}
+
+// derivative w.r.t. a string (iterates der)
+@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
+val reg = ((("a".%)) ~ ("b" | "c")).%
+
+println(blexing(reg, "aab"))
+
+
+//=======================
+// 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 AALTS(bs1, rs) => distinctBy(flts(rs.map(bsimp)), erase) match {
+      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))
+
+println(blexing_simp(reg, "aab"))
+
+// bsimp2 by Chengsong
+
+def pos_i(rs: List[ARexp], v: Val): Int = (rs, v) match {
+    case (r::Nil,        v1) => 0
+    case ( r::rs1, Right(v)) => pos_i(rs1, v) + 1
+    case ( r::rs1, Left(v) ) => 0
+}
+
+def pos_v(rs: List[ARexp], v: Val): Val = (rs, v) match {
+    case (r::Nil,       v1) => v1
+    case (r::rs1, Right(v)) => pos_v(rs1, v)
+    case (r::rs1, Left(v) ) => v
+}
+
+def unify(rs: List[ARexp], v: Val): Val = {
+  Position(pos_i(rs, v), pos_v(rs, v))
+}
+
+// coat does the job of "coating" a value
+// given the number of right outside it
+def coat(v: Val, i: Int) : Val = i match {
+  case 0 => v
+  case i => if (i > 0) coat(Right(v), i - 1) else { println(v, i); throw new Exception("coat minus")}
+}
+
+def distinctBy2[B, C](v: Val, xs: List[B], f: B => C, acc: List[C] = Nil, res: List[B] = Nil): (List[B], Val) = xs match {
+    case Nil => (res, v)
+    case (x::xs) => {
+      val re = f(x)
+      if (acc.contains(re)) v match { 
+        case Position(i, v0) => distinctBy2(Position(i - 1, v0), xs, f, acc, res)  
+        case _ => throw new Exception("dB2")
+      }
+      else distinctBy2(v, xs, f, re::acc, x::res)
+    }
+  }
+
+
+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 flats2(front: List[ARexp], i: Int, rs: List[ARexp], v: Val): (List[ARexp], Val) = v match {
+    case Position(j, v0) => {
+      if (i < 0) (front ::: flats(rs), Position(j, v0) ) 
+      else if(i == 0){
+        rs match {
+          case AALTS(bs, rs1) :: rs2 => ( (front ::: rs1.map(fuse(bs, _))):::flats(rs2), Position(j + rs1.length - 1, pos_v(rs1, v0)))
+          case r::rs2 => (front ::: List(r) ::: flats(rs2), Position(j, v0))
+          case _ => throw new Exception("flats2 i = 0")
+        }
+      }
+      else{
+        rs match {
+          case AZERO::rs1 => flats2(front, i - 1, rs1, Position(j - 1, v0))
+          case AALTS(bs, rs1) ::rs2 => flats2(front:::rs1.map(fuse(bs, _)), i - 1, rs2, Position(j + rs1.length - 1, v0))
+          case r::rs1 => flats2(front:::List(r), i - 1, rs1, Position(j, v0)) 
+          case _ => throw new Exception("flats2 i>0")
+        }
+      }
+    }
+    case _ => throw new Exception("flats2 error")
+  }
+
+def deunify(rs_length: Int, v: Val): Val = v match{
+  case Position(i, v0) => {
+      if (i <0 ) { println(rs_length, v); throw new Exception("deunify minus") } 
+      else if (rs_length == 1) {println(v); v}
+      else if (rs_length - 1 == i) coat(v0, i) 
+      else coat(Left(v0), i)
+  }
+  case _ => throw new Exception("deunify error")
+}
+
+
+def bsimp2(r: ARexp, v: Val): (ARexp, Val) = (r, v) match {
+    case (ASEQ(bs1, r1, r2), Sequ(v1, v2)) => (bsimp2(r1, v1), bsimp2(r2, v2)) match {
+        case ((AZERO, _), (_, _)) => (AZERO, Undefined)
+        case ((_, _), (AZERO, _)) => (AZERO, Undefined)
+        case ((AONE(bs2), v1s) , (r2s, v2s)) => (fuse(bs1 ++ bs2, r2s), v2s)
+                 // v2 tells how to retrieve bits in r2s, which is enough as the bits 
+                 // of the first part of the sequence has already been integrated to 
+                 // the top level of the second regx.
+        case ((r1s, v1s), (r2s, v2s)) => (ASEQ(bs1, r1s, r2s),  Sequ(v1s, v2s))
+    }
+    case (AALTS(bs1, rs), v) => {
+      val vlist = unify(rs, v)
+      vlist match {
+        case Position(i, v0) => {
+          val v_simp = bsimp2(rs(i), v0)._2
+          val rs_simp = rs.map(bsimp)
+          val flat_res = flats2( Nil, i, rs_simp, Position(i, v_simp) )
+          val dist_res = distinctBy2(flat_res._2, flat_res._1, erase)
+          val rs_new = dist_res._1
+          val v_new = deunify(rs_new.length, dist_res._2)
+          rs_new match {
+            case Nil => (AZERO, Undefined)
+            case s :: Nil => (fuse(bs1, s), v_new)
+            case rs => (AALTS(bs1, rs), v_new)
+          }
+        }
+        case _ => throw new Exception("Funny vlist bsimp2")
+      }
+    }
+    // STAR case
+    // case ASTAR(bs, r) => ASTAR(bs, bsimp(r))
+    case (r, v) => (r, v)  
+  }
+
+
+
+val dr = ASEQ(List(),AALTS(List(S),List(AONE(List(Z)), AONE(List(S)))),ASTAR(List(),AALTS(List(),List(ACHAR(List(Z),'a'), ACHAR(List(S),'a')))))
+val dv = Sequ(Left(Empty),Stars(List()))
+println(bsimp2(dr, dv))
+  
+
+/*
+def blex_simp2(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(bsimp2(bder(c, r)), cs)
+}
+
+def blexing_simp2(r: Rexp, s: String) : Val = 
+  decode(r, blex_simp2(internalise(r), s.toList))
+
+println(blexing_simp2(reg, "aab"))
+*/
+
+// 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]
+val res2 = retrieve(bder('a', internalise(dr)), dr_val) // => [false, true]
+decode(dr_der, res1) // gives the value for derivative
+decode(dr, res2)     // gives the value for original value
+
+encode(inj(dr, 'a', decode(dr_der, res1)))
+
+*/