diff -r db0ff630bbb7 -r 43e070803c1c exps/bit-test.scala
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/exps/bit-test.scala	Fri May 10 11:56:37 2019 +0100
@@ -0,0 +1,758 @@
+
+import scala.language.implicitConversions    
+import scala.language.reflectiveCalls
+import scala.annotation.tailrec   
+import scala.util.Try
+
+// for escaping strings
+def escape(raw: String) : String = {
+  import scala.reflect.runtime.universe._
+  Literal(Constant(raw)).toString
+}
+
+def esc2(r: (String, String)) = (escape(r._1), escape(r._2))
+
+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)
+  }
+} 
+
+abstract class Bit
+case object Z extends Bit
+case object S extends Bit
+case class C(c: Char) extends Bit
+
+type Bits = List[Bit]
+
+// usual regular expressions with predicates
+abstract class Rexp 
+case object ZERO extends Rexp
+case object ONE extends Rexp
+case class PRED(f: Char => Boolean, s: String = "_") extends Rexp {
+  override def toString = s"PRED(${s})"
+}
+case class ALTS(rs: List[Rexp]) extends Rexp 
+case class SEQ(r1: Rexp, r2: Rexp) extends Rexp 
+case class STAR(r: Rexp) extends Rexp 
+case class RECD(x: String, r: Rexp) extends Rexp
+
+
+// abbreviations
+def CHAR(c: Char) = PRED(_ == c, c.toString)
+def ALT(r1: Rexp, r2: Rexp) = ALTS(List(r1, r2))
+def PLUS(r: Rexp) = SEQ(r, STAR(r))
+val ANYCHAR = PRED(_ => true, ".")
+
+// annotated regular expressions
+abstract class ARexp 
+case object AZERO extends ARexp
+case class AONE(bs: Bits) extends ARexp
+case class APRED(bs: Bits, f: Char => Boolean, s: String = "_") extends ARexp {
+  override def toString = s"APRED(${bs}, ${s})"
+}
+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 
+
+// abbreviations
+def AALT(bs: Bits, r1: ARexp, r2: ARexp) = AALTS(bs, List(r1, r2))
+
+// 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
+
+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)
+}
+
+// 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)
+}
+
+   
+// 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)
+}
+
+
+// string of a regular expression - for testing purposes
+def string(r: Rexp): String = r match {
+  case ZERO => "0"
+  case ONE => "1"
+  case PRED(_, s) => s
+  case ALTS(rs) => rs.map(string).mkString("[", "|", "]")
+  case SEQ(r1, r2) => s"(${string(r1)} ~ ${string(r2)})"
+  case STAR(r) => s"{${string(r)}}*"
+  case RECD(x, r) => s"(${x}! ${string(r)})"
+}
+
+// string of an annotated regular expression - for testing purposes
+def astring(a: ARexp): String = a match {
+  case AZERO => "0"
+  case AONE(_) => "1"
+  case APRED(_, _, s) => s
+  case AALTS(_, rs) => rs.map(astring).mkString("[", "|", "]")
+  case ASEQ(_, r1, r2) => s"(${astring(r1)} ~ ${astring(r2)})"
+  case ASTAR(_, r) => s"{${astring(r)}}*"
+}
+ 
+//--------------------------------------------------------------------
+// BITCODED PART
+
+def retrieve(r: ARexp, v: Val) : Bits = (r, v) match {
+  case (AONE(bs), Empty) => bs
+  case (APRED(bs, _, _), Chr(d)) => bs
+  case (AALTS(bs, r::Nil), v) => bs ++ retrieve(r, v)
+  case (AALTS(bs, r::rs), Left(v)) => bs ++ retrieve(r, v)
+  case (AALTS(bs, r::rs), Right(v)) => bs ++ retrieve(AALTS(Nil, rs), v)
+  case (ASEQ(bs, r1, r2), Sequ(v1, v2)) => 
+    bs ++ retrieve(r1, v1) ++ retrieve(r2, v2)
+  case (ASTAR(bs, r), Stars(Nil)) => bs ++ List(S)
+  case (ASTAR(bs, r), Stars(v::vs)) => 
+     bs ++ List(Z) ++ retrieve(r, v) ++ retrieve(ASTAR(Nil, r), Stars(vs))
+}
+
+def fuse(bs: Bits, r: ARexp) : ARexp = r match {
+  case AZERO => AZERO
+  case AONE(cs) => AONE(bs ++ cs)
+  case APRED(cs, f, s) => APRED(bs ++ cs, f, s)
+  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)
+}
+
+// translation into ARexps
+def internalise(r: Rexp) : ARexp = r match {
+  case ZERO => AZERO
+  case ONE => AONE(Nil)
+  case PRED(f, s) => APRED(Nil, f, s)
+  case ALTS(List(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)
+}
+
+internalise(("a" | "ab") ~ ("b" | ""))
+
+// decoding of values from bit sequences
+def decode_aux(r: Rexp, bs: Bits) : (Val, Bits) = (r, bs) match {
+  case (ONE, bs) => (Empty, bs)
+  case (PRED(f, _), C(c)::bs) => (Chr(c), bs)
+  case (ALTS(r::Nil), bs) => decode_aux(r, bs)
+  case (ALTS(rs), Z::bs1) => {
+      val (v, bs2) = decode_aux(rs.head, bs1)
+      (Left(v), bs2)
+    }
+  case (ALTS(rs), S::bs1) => {
+      val (v, bs2) = decode_aux(ALTS(rs.tail), 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)
+    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)
+  }
+}
+
+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))
+}
+
+
+//erase function: extracts a Rexp from Arexp
+def erase(r: ARexp) : Rexp = r match{
+  case AZERO => ZERO
+  case AONE(_) => ONE
+  case APRED(bs, f, s) => PRED(f, s)
+  case AALTS(bs, rs) => ALTS(rs.map(erase(_)))
+  case ASEQ(bs, r1, r2) => SEQ (erase(r1), erase(r2))
+  case ASTAR(cs, r)=> STAR(erase(r))
+}
+
+
+// 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 APRED(_,_,_) => 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, rs) => {
+    val n = rs.indexWhere(bnullable)
+    bs ++ bmkeps(rs(n))
+  }
+  case ASEQ(bs, r1, r2) => bs ++ bmkeps(r1) ++ bmkeps(r2)
+  case ASTAR(bs, r) => bs ++ List(Z)
+}
+
+// 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 APRED(bs, f, _) => if (f(c)) AONE(bs:::List(C(c))) 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(S), bder(c, r)), ASTAR(Nil, r))
+}
+
+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 preblexing(r: ARexp, s: String) : Val = 
+ decode(erase(r), blex(r, s.toList))
+
+def blexing(r: Rexp, s: String) : Val = 
+ decode(r, blex(internalise(r), s.toList))
+
+
+// derivative w.r.t. a string (iterates bder)
+@tailrec
+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 vsimp(r: ARexp, v: Val): Val = (r, v) match {
+  case (ASEQ(bs1, r1, r2), Sequ(v1, v2)) => 
+    (bsimp(r1), bsimp(r2), vsimp(r1, v1), vsimp(r2, v2)) match {
+      case (AZERO, _, _, _) => throw new Exception("error")
+      case (_, AZERO, _, _) => throw new Exception("error")
+      case (AONE(_), _, _, vp2) => vp2
+      case (r1s, r2s, vp1, vp2) => Sequ(vp1, vp2) 
+  }
+  case (AALTS(bs1, rs), _) => distinctBy(flats(rs.map(bsimp)), erase) match {
+    case Nil => throw new Exception("error")
+    case r :: Nil => throw new Exception("error")
+    case rs => throw new Exception("error")
+  }
+  case _ => v
+}
+*/
+def vsimp(v: Val, a: ARexp): Val = (v, bsimp(a)) match {
+  case (Sequ(v1, v2), ASEQ(_, a1, a2)) => 
+    (vsimp(v1, a1), vsimp(v2, a2)) match {
+        case (Empty, vp2) => vp2
+        case (vp1, vp2) => Sequ(vp1, vp2) 
+    }
+  case (Left(Left(v1)), AALTS(_, r::rs)) => Left(vsimp(v1, r)) 
+  case (Left(v1), AALTS(_, rs)) => 
+    if (rs.length == 1) vsimp(v1, rs.head) else Left(vsimp(v1, rs.head)) 
+  case (Right(v1), AALTS(bs, rs)) => 
+    if (rs.length == 1) vsimp(v1, rs.head) else Right(vsimp(v1, AALTS(bs, rs.tail)))
+  case _ => v
+}
+
+
+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) => distinctBy(flats(rs.map(bsimp)), erase) match {
+    case Nil => AZERO
+    case r :: Nil => fuse(bs1, r)
+    case rs => AALTS(bs1, rs)  
+  }
+  case r => r
+}
+
+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 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_simp(bsimp(bder(c, r)), cs)
+}
+
+
+def blexing_simp(r: Rexp, s: String) : Val = 
+ decode(r, blex_simp(internalise(r), s.toList))
+
+
+def btokenise_simp(r: Rexp, s: String) = 
+  env(blexing_simp(r, s)).map(esc2)
+
+// Quick example
+
+val r : Rexp = ZERO | "a" 
+
+lexing(r, "a")
+
+val a0 = internalise(r)
+val a1 = bder('a', a0)
+val a1s = bsimp(bder('a', a0))
+
+val a2 = bmkeps(a1)
+val a2s = bmkeps(a1s)
+
+val v  = decode(r, a2)
+val vs  = decode(r, a2s)
+
+
+
+val Rr : Rexp = ONE ~ "a" 
+
+lexing(Rr, "a")
+
+val Ra0 = internalise(Rr)
+astring(Ra0)
+val Ra1 = bder('a', Ra0)
+astring(Ra1)
+val Ra1s = bsimp(bder('a', Ra0))
+astring(Ra1s)
+
+val Ra2 = bmkeps(Ra1)
+val Ra2s = bmkeps(Ra1s)
+
+val Rv  = decode(Rr, Ra2)
+val Rvs  = decode(Rr, Ra2s)
+
+
+//   Testing
+//============
+
+def time[T](code: => T) = {
+  val start = System.nanoTime()
+  val result = code
+  val end = System.nanoTime()
+  ((end - start)/1.0e9).toString
+}
+
+def timeR[T](code: => T) = {
+  val start = System.nanoTime()
+  for (i <- 1 to 10) code
+  val result = code
+  val end = System.nanoTime()
+  (result, (end - start))
+}
+
+//size: of a Aregx for testing purposes 
+def size(r: Rexp) : Int = r match {
+  case ZERO => 1
+  case ONE => 1
+  case PRED(_,_) => 1
+  case SEQ(r1, r2) => 1 + size(r1) + size(r2)
+  case ALTS(rs) => 1 + rs.map(size).sum
+  case STAR(r) => 1 + size(r)
+  case RECD(_, r) => size(r)
+}
+
+def asize(a: ARexp) = size(erase(a))
+
+
+// Lexing Rules for a Small While Language
+
+//symbols
+val SYM = PRED("abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ".contains(_), "SYM")
+//digits
+val DIGIT = PRED("0123456789".contains(_), "NUM")
+//identifiers
+val ID = SYM ~ (SYM | DIGIT).% 
+//numbers
+val NUM = STAR(DIGIT)
+//keywords
+val KEYWORD : Rexp = "skip" | "while" | "do" | "if" | "then" | "else" | "read" | "write" | "true" | "false"
+//semicolons
+val SEMI: Rexp = ";"
+//operators
+val OP: Rexp = ":=" | "==" | "-" | "+" | "*" | "!=" | "<" | ">" | "<=" | ">=" | "%" | "/"
+//whitespaces
+val WHITESPACE = PLUS(" " | "\n" | "\t")
+//parentheses
+val RPAREN: Rexp = ")"
+val LPAREN: Rexp = "("
+val BEGIN: Rexp = "{"
+val END: Rexp = "}"
+//strings...but probably needs not
+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)).%
+
+
+// Some Small Tests
+//==================
+
+println("Small tests")
+
+val re1 = STAR("a" | "aa")
+println(astring(bders_simp("".toList, internalise(re1))))
+println(astring(bders_simp("a".toList, internalise(re1))))
+println(astring(bders_simp("aa".toList, internalise(re1))))
+println(astring(bders_simp("aaa".toList, internalise(re1))))
+println(astring(bders_simp("aaaaaa".toList, internalise(re1))))
+println(astring(bders_simp("aaaaaaaaa".toList, internalise(re1))))
+println(astring(bders_simp("aaaaaaaaaaaa".toList, internalise(re1))))
+println(astring(bders_simp("aaaaaaaaaaaaaaaaaaaaaaaaa".toList, internalise(re1))))
+println(astring(bders_simp("aaaaaabaaaabbbbbaaaaaaaaaaaaaaa".toList, internalise(re1))))
+
+
+for (i <- 0 to 100 by 5) {
+  //print("Old: " + time(tokenise_simp(re1, "a" * i)))
+  print(" Bit: " + time(btokenise_simp(re1, "a" * i)))
+  print(" Bit full simp: " + time(btokenise_simp_full(re1, "a" * i)))
+  println(" Bit2: " + time(btokenise2_simp(re1, "a" * i)))
+}
+
+Console.readLine
+
+
+// Bigger Tests
+//==============
+
+
+println("Big tests")
+
+val fib_prog = """
+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
+"""
+
+
+println("fib prog tests :")
+println(tokenise_simp(WHILE_REGS, fib_prog))
+println(btokenise_simp(WHILE_REGS, fib_prog))
+println("equal? " + (tokenise_simp(WHILE_REGS, fib_prog) == btokenise_simp(WHILE_REGS, fib_prog)))
+
+for (i <- 1 to 20) {
+  print("Old: " + time(tokenise_simp(WHILE_REGS, fib_prog * i)))
+  print(" Bit: " + time(btokenise_simp(WHILE_REGS, fib_prog * i)))
+  println(" Bit full simp: " + time(btokenise_simp_full(WHILE_REGS, fib_prog * i)))
+  //println(" Bit2: " + time(btokenise2_simp(WHILE_REGS, fib_prog * i)))
+}
+
+
+println("Original " + size(WHILE_REGS))
+println("Size Bit  " + asize(bders_simp((fib_prog * 1).toList, internalise(WHILE_REGS))))
+println("Size Bitf " + asize(bders_simp_full((fib_prog * 1).toList, internalise(WHILE_REGS))))
+println("Size Bit2 " + asize(bders2_simp((fib_prog * 1).toList, internalise(WHILE_REGS))))
+println("Size Old  " + size(ders_simp((fib_prog * 1).toList, WHILE_REGS)))
+println("Size Pder " + psize(pders_simp((fib_prog * 1).toList, WHILE_REGS)))
+
+System.exit(0)
+
+println("Internal sizes test OK or strange")
+
+def perc(p1: Double, p2: Double) : String =
+  f"${(((p1 - p2) / p2) * 100.0) }%5.0f" + "%"
+
+def ders_test(n: Int, s: List[Char], r: Rexp, a: ARexp) : (Rexp, ARexp) = s match {
+  case Nil => (r, a)
+  case c::s => {
+    // derivative 
+    val (rd1, tr1) = timeR(der(c, r))
+    val (ad1, ta1) = timeR(bder(c, a))
+    val trs1 = f"${tr1}%.5f"
+    val tas1 = f"${ta1}%.5f"
+    if (tr1 < ta1) println(s"Time strange der  (step) ${n} ${perc(ta1, tr1)} sizes  der ${size(rd1)} ${asize(ad1)}")
+    //simplification
+    val (rd, tr) = timeR(simp(rd1)._1)
+    val (ad, ta) = timeR(bsimp(ad1))
+    val trs = f"${tr}%.5f"
+    val tas = f"${ta}%.5f"
+    //full simplification
+    val (adf, taf) = timeR(bsimp_full(ad1))
+    if (tr < ta) println(s"Time strange simp (step) ${n} ${perc(ta, tr)} sizes simp ${size(rd)} ${asize(ad)}")
+    if (n == 1749 || n == 1734) {
+      println{s"Aregex before bder (size: ${asize(a)})\n ${string(erase(a))}"}
+      println{s"Aregex after bder (size: ${asize(ad1)})\n ${string(erase(ad1))}"}
+      println{s"Aregex after bsimp (size: ${asize(ad)})\n ${string(erase(ad))}"}
+      println{s"Aregex after bsimp_full (size: ${asize(adf)})\n ${string(erase(adf))}"}
+    }
+    ders_test(n + 1, s, rd, ad)
+  }
+}
+
+val prg = (fib_prog * 10).toList
+ders_test(0, prg, WHILE_REGS, internalise(WHILE_REGS))
+
+
+//testing the two lexings produce the same value
+//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)
+  }
+}
+
+def enum(n: Int, s: String) : Stream[Rexp] = n match {
+  case 0 => ZERO #:: ONE #:: s.toStream.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 benum(n: Int, s: String) = enum(n, s).map(internalise)
+
+def values(r: Rexp) : Set[Val] = r match {
+  case ZERO => Set()
+  case ONE => Set(Empty)
+  case PRED(_, s) => Set(Chr(s.head))
+  case ALTS(List(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
+}
+
+
+// tests about retrieve
+
+def tests_retrieve(r: Rexp) = {
+  val vs = values(r)
+  val a = internalise(r)
+  val as = bsimp(a)
+  for (v <- vs) {
+    println(s"Testing ${string(r)} and ${v}")
+    val bs1 = retrieve(a, v)
+    val bs2 = Try(Some(retrieve(as, decode(erase(as), bs1)))).getOrElse(None)
+    if (Some(bs1) != bs2) println(s"Disagree on ${string(r)}, ${v}")
+    if (Some(bs1) != bs2) Some((r, v)) else None
+  }
+}
+
+println("Testing retrieve 1")
+println(enum(1, "ab").map(tests_retrieve).toList)
+
+// an example where the property fails
+val r = (ZERO ~ "b") | "a"
+val a = internalise(r)
+val as = bsimp(a)
+val v = Right(Chr('a'))
+
+println("arexp      " ++ astring(a))
+println("simplified " ++ astring(as))
+
+val bs1 = retrieve(a, v)
+encode(v)
+retrieve(as, decode(erase(as), bs1))
+
+//tests retrieve and vsimp
+
+def tests_retrieve_vsimp(ss: Set[String])(r: Rexp) = {
+  val a = internalise(r)
+  val as = bsimp(a)
+  for (s <- ss.par) yield {  
+    val v = Try(Some(preblexing(a, s))).getOrElse(None)
+    if  (v.isDefined) {
+      val bs1 = retrieve(a, v.get)
+      val bs2 = Try(retrieve(as, vsimp(v.get, as))).getOrElse(Nil)
+      if  (bs1 != bs2) {
+        println(s"Disagree on ${astring(a)}, ${astring(as)}, ${s}")
+        println(s"  ${v.get}  and  ${vsimp(v.get)}")
+        println(s"  ${bs1}  and  ${bs2}")
+        Some(a, as, s, v.get, vsimp(v.get, as), bs1, bs2)  
+      } else None 
+    } else None
+  }
+}
+
+println("Partial searching: ")
+enum(2, "abc").map(tests_retrieve_vsimp(strs(3, "abc"))).
+  flatten.toSet.flatten.minBy(a => asize(a._1))
+
+//tests derivatives and bsimp
+
+def tests_ders_bsimp(ss: Set[String])(r: Rexp) = {
+  val a = internalise(r)
+  for (s <- ss.par) yield {  
+    val d1 = bsimp(bders(s.toList, bsimp(a)))
+    val d2 = bsimp(bders(s.toList, a))
+    if  (d1 != d2) {
+        println(s"Disagree on ${astring(a)}")
+        println(s"  ${astring(d1)}  and  ${astring(d2)}")
+        Some(a, d1, d2)  
+      } else None 
+    }
+}
+
+println("Partial searching: ")
+enum(2, "abc").map(tests_ders_bsimp(strs(1, "abc"))).
+  flatten.toSet.flatten.minBy(a => asize(a._1))
+
+
+
+//tests retrieve and lexing
+
+def tests_retrieve_lex(ss: Set[String])(r: Rexp) = {
+  val a = internalise(r)
+  val as = bsimp(a)
+  for (s <- ss.par) yield {  
+    val bs1 = Try(Some(blex(a, s.toList))).getOrElse(None)
+    val bs2 = Try(Some(blex(as, s.toList))).getOrElse(None)
+    if  (bs1 != bs2) {
+        println(s"Disagree on ${astring(a)}, ${astring(as)}, ${s}")
+        println(s"  ${bs1}  and  ${bs2}")
+        Some(a, as, s)  
+      } else None 
+    }
+}
+
+println("Partial searching: ")
+enum(2, "abc").map(tests_retrieve_lex(strs(3, "abc"))).flatten.toSet
+
+//Disagree on [[c|b]|[a|c]], [c|b|a], a
+//Right(Left(Chr(a)))  and  Right(Left(Chr(a)))
+//List(S, Z)  and  List(Z, S)
+
+val s = "c"
+val ar : Rexp = "a"
+val br : Rexp = "b"
+val cr : Rexp = "c"
+val r1 : Rexp = ALT(ALT(cr, br), ALT(ar,cr))
+val a1 = internalise(r1)
+val a2 = bsimp(a1)
+val a2a = internalise(erase(a2))
+
+astring(a1)
+astring(a2)
+astring(a2a)
+
+blexing(r1 ,s)
+blexing_simp(r1 ,s)
+val v1 = preblexing(a1, s)
+val v2 = preblexing(a2a, s)
+retrieve(a1, v1)
+retrieve(a2, v2)
+
+
+//tests blexing and lexing
+def tests(ss: Set[String])(r: Rexp) = {
+  //println(s"Testing ${r}")
+  for (s <- ss.par) yield {
+    val res1 = Try(Some(lexing_simp(r, s))).getOrElse(None)
+    val res2 = Try(Some(blexing_simp(r, s))).getOrElse(None)
+    if (res1 != res2) 
+      { println(s"Disagree on ${r} and ${s}")
+	println(s"   ${res1} !=  ${res2}")
+	Some((r, s)) } else None
+  }
+}
+
+
+println("Partial searching: ")
+enum(2, "abc").map(tests(strs(3, "abc"))).toSet
+
+
+
+