# HG changeset patch
# User Christian Urban <urbanc@in.tum.de>
# Date 1564326903 -3600
# Node ID 6709fa87410b01b7788ea1cdad240225a38e7734
# Parent  8d0af38389bcda6e49a18a0d5e53ac9f382d9a59
updated to 2.13

diff -r 8d0af38389bc -r 6709fa87410b progs/compile.scala
--- a/progs/compile.scala	Sun Jul 28 14:24:46 2019 +0100
+++ b/progs/compile.scala	Sun Jul 28 16:15:03 2019 +0100
@@ -257,7 +257,7 @@
        Assign("minus2",Num(1)),         //  minus2 := 1;
        Assign("temp",Num(0)),           //  temp := 0;
        While(Bop("<",Num(0),Var("n")),  //  while n > 0 do  {
-          List(Assign("temp",Var("minus2")),    //  temp := minus2;
+          List(Assign("temp",Var("minus2")), //  temp := minus2;
                Assign("minus2",Aop("+",Var("minus1"),Var("minus2"))), 
                                         //  minus2 := minus1 + minus2;
                Assign("minus1",Var("temp")), //  minus1 := temp;
diff -r 8d0af38389bc -r 6709fa87410b progs/fun-bare.scala
--- a/progs/fun-bare.scala	Sun Jul 28 14:24:46 2019 +0100
+++ b/progs/fun-bare.scala	Sun Jul 28 16:15:03 2019 +0100
@@ -153,7 +153,7 @@
   }
 }
 
-// main compilation function
+// the main compilation function
 def compile(prog: List[Decl], class_name: String) : String = {
   val instructions = prog.map(compile_decl).mkString
   (library + instructions).replaceAllLiterally("XXX", class_name)
@@ -162,7 +162,7 @@
 
 
 
-// example program (factorials)
+// An example program (factorials)
 
 val test_prog = 
   List(Def("fact", List("n"),
diff -r 8d0af38389bc -r 6709fa87410b progs/fun.scala
--- a/progs/fun.scala	Sun Jul 28 14:24:46 2019 +0100
+++ b/progs/fun.scala	Sun Jul 28 16:15:03 2019 +0100
@@ -1,3 +1,6 @@
+// A Small Compiler for a Simple Functional Language
+// (includes a lexer and a parser)
+
 import scala.language.implicitConversions    
 import scala.language.reflectiveCalls 
 import scala.util._
@@ -7,196 +10,245 @@
 def fromFile(name: String) : String = 
   io.Source.fromFile(name).mkString
 
+
 abstract class Rexp 
-case object NULL extends Rexp
-case object EMPTY extends 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 RANGE(cs: List[Char]) extends Rexp 
 case class SEQ(r1: Rexp, r2: Rexp) extends Rexp 
-case class PLUS(r: Rexp) extends Rexp 
 case class STAR(r: Rexp) extends Rexp 
-case class NTIMES(r: Rexp, n: Int) extends Rexp 
-case class NUPTOM(r: Rexp, n: Int, m: Int) extends Rexp
-
-object RANGE {
-  def apply(s: String) : RANGE = RANGE(s.toList)
-}
-def NMTIMES(r: Rexp, n: Int, m: Int) = {
-  if (m < n) throw new IllegalArgumentException("the number m cannot be smaller than n.")
-  else NUPTOM(r, n, m - n)
-}
-
-case class NOT(r: Rexp) extends Rexp 
-case class OPT(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 => EMPTY
+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 string2rexp(s : String) : Rexp = 
+  charlist2rexp(s.toList)
 
-implicit def RexpOps (r: Rexp) = new {
+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 {
+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)
 }
 
-
-// nullable function: tests whether the regular 
-// expression can recognise the empty string
 def nullable (r: Rexp) : Boolean = r match {
-  case NULL => false
-  case EMPTY => true
+  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 PLUS(r) => nullable(r)
-  case NTIMES(r, i) => if (i == 0) true else nullable(r)
-  case NUPTOM(r, i, j) => if (i == 0) true else nullable(r)
-  case RANGE(_) => false
-  case NOT(r) => !(nullable(r))
-  case OPT(_) => 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 NULL => NULL
-  case EMPTY => NULL
-  case CHAR(d) => if (c == d) EMPTY else NULL
+  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 PLUS(r) => SEQ(der(c, r), STAR(r))
-  case NTIMES(r, i) => 
-    if (i == 0) NULL else der(c, SEQ(r, NTIMES(r, i - 1)))
-  case NUPTOM(r, i, j) =>
-    if (i == 0 && j == 0) NULL else 
-    if (i == 0) ALT(der(c, NTIMES(r, j)), der(c, NUPTOM(r, 0, j - 1)))
-    else der(c, SEQ(r, NUPTOM(r, i - 1, j)))
-  case RANGE(cs) => if (cs contains c) EMPTY else NULL
-  case NOT(r) => NOT(der (c, r))
-  case OPT(r) => der(c, r)
+  case RECD(_, r1) => der(c, r1)
+}
+
+
+// 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 zeroable (r: Rexp) : Boolean = r match {
-  case NULL => true
-  case EMPTY => false
-  case CHAR(_) => false
-  case ALT(r1, r2) => zeroable(r1) && zeroable(r2)
-  case SEQ(r1, r2) => zeroable(r1) || zeroable(r2)
-  case STAR(_) => false
-  case PLUS(r) => zeroable(r)
-  case NTIMES(r, i) => if (i == 0) false else zeroable(r)
-  case NUPTOM(r, i, j) => if (i == 0) false else zeroable(r)
-  case RANGE(_) => false
-  case NOT(r) => !(zeroable(r))     // bug: incorrect definition for NOT
-  case OPT(_) => false
+// extracts an environment from a value;
+// used for tokenise 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 Rec(x, v) => (x, flatten(v))::env(v)
 }
 
-// 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))
+// The Injection Part of the lexer
+
+// calculates a value for how a nullable regex 
+// matches the empty string 
+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))
+}
+
+// injects back a character into a value
+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) 
+  case (RECD(x, r1), _) => Rec(x, inj(r1, c, v))
 }
 
 
-// regular expressions for the While language
-val SYM = RANGE("ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz_".toList)
-val DIGIT = RANGE("0123456789".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 returns now 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)
+}
+
+// lexing functions including simplification
+def lex_simp(r: Rexp, s: List[Char]) : Val = s match {
+  case Nil => if (nullable(r)) mkeps(r) else { println ("Lexing Error") ; sys.exit(-1) } 
+  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 Fun 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" | "T" | "_"
+val DIGIT = "0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9"
 val ID = SYM ~ (SYM | DIGIT).% 
 val NUM = PLUS(DIGIT)
 val KEYWORD : Rexp = "if" | "then" | "else" | "write" | "def"
 val SEMI: Rexp = ";"
-val COMMA: Rexp = ","
-val OP: Rexp = ":=" | "==" | "-" | "+" | "*" | "!=" | "<=" | "=>" | "<" | ">" | "%" | "=" | "/"
+val OP: Rexp = "=" | "==" | "-" | "+" | "*" | "!=" | "<" | ">" | "<=" | ">=" | "%" | "/"
 val WHITESPACE = PLUS(" " | "\n" | "\t")
 val RPAREN: Rexp = ")"
 val LPAREN: Rexp = "("
+val COMMA: Rexp = ","
 val ALL = SYM | DIGIT | OP | " " | ":" | ";" | "\"" | "=" | "," | "(" | ")"
 val ALL2 = ALL | "\n"
 val COMMENT = ("/*" ~ ALL2.% ~ "*/") | ("//" ~ ALL.% ~ "\n")
 
 
-// token for While language
+
+val WHILE_REGS = (("k" $ KEYWORD) | 
+                  ("i" $ ID) | 
+                  ("o" $ OP) | 
+                  ("n" $ NUM) | 
+                  ("s" $ SEMI) | 
+                  ("c" $ COMMA) |
+                  ("pl" $ LPAREN) |
+                  ("pr" $ RPAREN) |
+                  ("w" $ (WHITESPACE | COMMENT))).%
+
+
+
+// The tokens for the While language
+
 abstract class Token
-case object T_WHITESPACE extends Token
 case object T_SEMI extends Token
 case object T_COMMA extends Token
 case object T_LPAREN extends Token
 case object T_RPAREN extends Token
-case object T_COMMENT extends Token
 case class T_ID(s: String) extends Token
 case class T_OP(s: String) extends Token
-case class T_NUM(s: String) extends Token
+case class T_NUM(n: Int) extends Token
 case class T_KWD(s: String) extends Token
-case class T_ERR(s: String) extends Token // special error token
-
 
-type TokenFun = String => Token
-type LexRules = List[(Rexp, TokenFun)]
-val While_lexing_rules: LexRules = 
-  List((KEYWORD, (s) => T_KWD(s)),
-       (ID, (s) => T_ID(s)),
-       (COMMENT, (s) => T_COMMENT),
-       (OP, (s) => T_OP(s)),
-       (NUM, (s) => T_NUM(s)),
-       (SEMI, (s) => T_SEMI),
-       (COMMA, (s) => T_COMMA),
-       (LPAREN, (s) => T_LPAREN),
-       (RPAREN, (s) => T_RPAREN),
-       (WHITESPACE, (s) => T_WHITESPACE))
+val token : PartialFunction[(String, String), Token] = {
+  case ("k", s) => T_KWD(s)
+  case ("i", s) => T_ID(s)
+  case ("o", s) => T_OP(s)
+  case ("n", s) => T_NUM(s.toInt)
+  case ("s", _) => T_SEMI
+  case ("c", _) => T_COMMA
+  case ("pl", _) => T_LPAREN
+  case ("pr", _) => T_RPAREN
+}
 
 
-// calculates derivatives until all of them are zeroable
-@tailrec
-def munch(s: List[Char], 
-          pos: Int, 
-          rs: LexRules, 
-          last: Option[(Int, TokenFun)]): Option[(Int, TokenFun)] = {
-  rs match {
-  case Nil => last
-  case rs if (s.length <= pos) => last
-  case rs => {
-    val ders = rs.map({case (r, tf) => (der(s(pos), r), tf)})
-    val rs_nzero = ders.filterNot({case (r, _) => zeroable(r)})
-    val rs_nulls = ders.filter({case (r, _) => nullable(r)})
-    val new_last = if (rs_nulls != Nil) Some((pos, rs_nulls.head._2)) else last
-    munch(s, 1 + pos, rs_nzero, new_last)
-  }
-}}
-
-// iterates the munching function and returns a Token list
-def tokenize(s: String, rs: LexRules) : List[Token] = munch(s.toList, 0, rs, None) match {
-  case None if (s == "") => Nil
-  case None => List(T_ERR(s"Lexing error: $s"))
-  case Some((n, tf)) => {
-    val (head, tail) = s.splitAt(n + 1)
-    tf(head)::tokenize(tail, rs)
-  }
-}
-
-def tokenizer(s:String) : List[Token] = 
-  tokenize(s, While_lexing_rules).filter {
-    case T_ERR(s) => { println(s); sys.exit(-1) }
-    case T_WHITESPACE => false
-    case T_COMMENT => false
-    case _ => true
-  } 
-
+def tokenise(s: String) : List[Token] = 
+  lexing_simp(WHILE_REGS, s).collect(token)
 
 
 // Parser - Abstract syntax trees
@@ -213,7 +265,7 @@
 case class Var(s: String) extends Exp
 case class Num(i: Int) extends Exp
 case class Aop(o: String, a1: Exp, a2: Exp) extends Exp
-case class Sequ(e1: Exp, e2: Exp) extends Exp
+case class Sequence(e1: Exp, e2: Exp) extends Exp
 
 case class Bop(o: String, a1: Exp, a2: Exp) extends BExp
 
@@ -225,14 +277,14 @@
   case Var(_) => 1
   case Num(_) => 1
   case Aop(_, a1, a2) => max_stack_exp(a1) + max_stack_exp(a2)
-  case Sequ(e1, e2) => List(max_stack_exp(e1), max_stack_exp(e2)).max
+  case Sequence(e1, e2) => List(max_stack_exp(e1), max_stack_exp(e2)).max
 }
 def max_stack_bexp(e: BExp): Int = e match {
   case Bop(_, a1, a2) => max_stack_exp(a1) + max_stack_exp(a2)
 }
 
 // Parser combinators
-abstract class Parser[I <% Seq[_], T] {
+abstract class Parser[I, T](implicit ev: I => Seq[_]) {
   def parse(ts: I): Set[(T, I)]
 
   def parse_all(ts: I) : Set[T] =
@@ -240,25 +292,40 @@
 
   def parse_single(ts: I) : T = parse_all(ts).toList match {
     case List(t) => t
-    case _ => { println ("Parse Error") ; sys.exit(-1) }
+    case _ => { println ("Parse Error\n") ; sys.exit(-1) }
   }
 }
 
-class SeqParser[I <% Seq[_], T, S](p: => Parser[I, T], q: => Parser[I, S]) extends Parser[I, (T, S)] {
+class SeqParser[I, T, S](p: => Parser[I, T], 
+                         q: => Parser[I, S])(implicit ev: I => Seq[_]) extends Parser[I, (T, S)] {
   def parse(sb: I) = 
     for ((head1, tail1) <- p.parse(sb); 
          (head2, tail2) <- q.parse(tail1)) yield ((head1, head2), tail2)
 }
 
-class AltParser[I <% Seq[_], T](p: => Parser[I, T], q: => Parser[I, T]) extends Parser[I, T] {
+class AltParser[I, T](p: => Parser[I, T], 
+                      q: => Parser[I, T])(implicit ev: I => Seq[_]) extends Parser[I, T] {
   def parse(sb: I) = p.parse(sb) ++ q.parse(sb)   
 }
 
-class FunParser[I <% Seq[_], T, S](p: => Parser[I, T], f: T => S) extends Parser[I, S] {
+class FunParser[I, T, S](p: => Parser[I, T], 
+                         f: T => S)(implicit ev: I => Seq[_]) extends Parser[I, S] {
   def parse(sb: I) = 
     for ((head, tail) <- p.parse(sb)) yield (f(head), tail)
 }
 
+implicit def ParserOps[I, T](p: Parser[I, T])(implicit ev: I => Seq[_]) = new {
+  def || (q : => Parser[I, T]) = new AltParser[I, T](p, q)
+  def ==>[S] (f: => T => S) = new FunParser[I, T, S](p, f)
+  def ~[S] (q : => Parser[I, S]) = new SeqParser[I, T, S](p, q)
+}
+
+def ListParser[I, T, S](p: => Parser[I, T], 
+                        q: => Parser[I, S])(implicit ev: I => Seq[_]): Parser[I, List[T]] = {
+  (p ~ q ~ ListParser(p, q)) ==> { case ((x, y), z) => x :: z : List[T] } ||
+  (p ==> ((s) => List(s)))
+}
+
 case class TokParser(tok: Token) extends Parser[List[Token], Token] {
   def parse(ts: List[Token]) = ts match {
     case t::ts if (t == tok) => Set((t, ts)) 
@@ -268,9 +335,15 @@
 
 implicit def token2tparser(t: Token) = TokParser(t)
 
+implicit def TokOps(t: Token) = new {
+  def || (q : => Parser[List[Token], Token]) = new AltParser[List[Token], Token](t, q)
+  def ==>[S] (f: => Token => S) = new FunParser[List[Token], Token, S](t, f)
+  def ~[S](q : => Parser[List[Token], S]) = new SeqParser[List[Token], Token, S](t, q)
+}
+
 case object NumParser extends Parser[List[Token], Int] {
   def parse(ts: List[Token]) = ts match {
-    case T_NUM(s)::ts => Set((s.toInt, ts)) 
+    case T_NUM(n)::ts => Set((n, ts)) 
     case _ => Set ()
   }
 }
@@ -283,28 +356,13 @@
 }
 
 
-implicit def ParserOps[I<% Seq[_], T](p: Parser[I, T]) = new {
-  def || (q : => Parser[I, T]) = new AltParser[I, T](p, q)
-  def ==>[S] (f: => T => S) = new FunParser[I, T, S](p, f)
-  def ~[S] (q : => Parser[I, S]) = new SeqParser[I, T, S](p, q)
-}
-implicit def TokOps(t: Token) = new {
-  def || (q : => Parser[List[Token], Token]) = new AltParser[List[Token], Token](t, q)
-  def ==>[S] (f: => Token => S) = new FunParser[List[Token], Token, S](t, f)
-  def ~[S](q : => Parser[List[Token], S]) = new SeqParser[List[Token], Token, S](t, q)
-}
+// Grammar Rules
 
-def ListParser[I <% Seq[_], T, S](p: => Parser[I, T], q: => Parser[I, S]): Parser[I, List[T]] = {
-  (p ~ q ~ ListParser(p, q)) ==> { case ((x, y), z) => x :: z : List[T] } ||
-  (p ==> ((s) => List(s)))
-}
-
-
-// expressions
+// arithmetic expressions
 lazy val Exp: Parser[List[Token], Exp] = 
   (T_KWD("if") ~ BExp ~ T_KWD("then") ~ Exp ~ T_KWD("else") ~ Exp) ==>
     { case (((((x, y), z), u), v), w) => If(y, u, w): Exp } ||
-  (M ~ T_SEMI ~ Exp) ==> { case ((x, y), z) => Sequ(x, z): Exp } || M
+  (M ~ T_SEMI ~ Exp) ==> { case ((x, y), z) => Sequence(x, z): Exp } || M
 lazy val M: Parser[List[Token], Exp] =
   (T_KWD("write") ~ L) ==> { case (x, y) => Write(y): Exp } || L
 lazy val L: Parser[List[Token], Exp] = 
@@ -372,87 +430,100 @@
   x ++ "_" ++ counter.toString()
 }
 
-type Mem = Map[String, Int]
-type Instrs = List[String]
+// convenient string interpolations 
+// for instructions, labels and methods
+import scala.language.implicitConversions
+import scala.language.reflectiveCalls
+
+implicit def sring_inters(sc: StringContext) = new {
+    def i(args: Any*): String = "   " ++ sc.s(args:_*) ++ "\n"
+    def l(args: Any*): String = sc.s(args:_*) ++ ":\n"
+    def m(args: Any*): String = sc.s(args:_*) ++ "\n"
+}
+
 
-def compile_exp(a: Exp, env : Mem) : Instrs = a match {
-  case Num(i) => List("ldc " + i.toString + "\n")
-  case Var(s) => List("iload " + env(s).toString + "\n")
-  case Aop("+", a1, a2) => compile_exp(a1, env) ++ compile_exp(a2, env) ++ List("iadd\n")
-  case Aop("-", a1, a2) => compile_exp(a1, env) ++ compile_exp(a2, env) ++ List("isub\n")
-  case Aop("*", a1, a2) => compile_exp(a1, env) ++ compile_exp(a2, env) ++ List("imul\n")
-  case Aop("/", a1, a2) => compile_exp(a1, env) ++ compile_exp(a2, env) ++ List("idiv\n")
-  case Aop("%", a1, a2) => compile_exp(a1, env) ++ compile_exp(a2, env) ++ List("irem\n")
+type Env = Map[String, Int]
+
+// compile expressions
+def compile_exp(a: Exp, env : Env) : String = a match {
+  case Num(i) => i"ldc $i"
+  case Var(s) => i"iload ${env(s)}"
+  case Aop("+", a1, a2) => compile_exp(a1, env) ++ compile_exp(a2, env) ++ i"iadd"
+  case Aop("-", a1, a2) => compile_exp(a1, env) ++ compile_exp(a2, env) ++ i"isub"
+  case Aop("*", a1, a2) => compile_exp(a1, env) ++ compile_exp(a2, env) ++ i"imul"
+  case Aop("/", a1, a2) => compile_exp(a1, env) ++ compile_exp(a2, env) ++ i"idiv"
+  case Aop("%", a1, a2) => compile_exp(a1, env) ++ compile_exp(a2, env) ++ i"irem"
   case If(b, a1, a2) => {
     val if_else = Fresh("If_else")
     val if_end = Fresh("If_end")
     compile_bexp(b, env, if_else) ++
     compile_exp(a1, env) ++
-    List("goto " + if_end + "\n") ++
-    List("\n" + if_else + ":\n\n") ++
+    i"goto $if_end" ++
+    l"$if_else" ++
     compile_exp(a2, env) ++
-    List("\n" + if_end + ":\n\n")
+    l"$if_end"
   }
-  case Call(n, args) => {
+  case Call(name, args) => {
     val is = "I" * args.length
-    args.flatMap(a => compile_exp(a, env)) ++
-    List ("invokestatic XXX/XXX/" + n + "(" + is + ")I\n")
+    args.map(a => compile_exp(a, env)).mkString ++
+    i"invokestatic XXX/XXX/$name($is)I"
   }
-  case Sequ(a1, a2) => {
-    compile_exp(a1, env) ++ List("pop\n") ++ compile_exp(a2, env)
+  case Sequence(a1, a2) => {
+    compile_exp(a1, env) ++ i"pop" ++ compile_exp(a2, env)
   }
   case Write(a1) => {
     compile_exp(a1, env) ++
-    List("dup\n",
-         "invokestatic XXX/XXX/write(I)V\n")
+    i"dup" ++
+    i"invokestatic XXX/XXX/write(I)V"
   }
 }
 
-def compile_bexp(b: BExp, env : Mem, jmp: String) : Instrs = b match {
+// compile boolean expressions
+def compile_bexp(b: BExp, env : Env, jmp: String) : String = b match {
   case Bop("==", a1, a2) => 
-    compile_exp(a1, env) ++ compile_exp(a2, env) ++ List("if_icmpne " + jmp + "\n")
+    compile_exp(a1, env) ++ compile_exp(a2, env) ++ i"if_icmpne $jmp"
   case Bop("!=", a1, a2) => 
-    compile_exp(a1, env) ++ compile_exp(a2, env) ++ List("if_icmpeq " + jmp + "\n")
+    compile_exp(a1, env) ++ compile_exp(a2, env) ++ i"if_icmpeq $jmp"
   case Bop("<", a1, a2) => 
-    compile_exp(a1, env) ++ compile_exp(a2, env) ++ List("if_icmpge " + jmp + "\n")
+    compile_exp(a1, env) ++ compile_exp(a2, env) ++ i"if_icmpge $jmp"
   case Bop("<=", a1, a2) => 
-    compile_exp(a1, env) ++ compile_exp(a2, env) ++ List("if_icmpgt " + jmp + "\n")
+    compile_exp(a1, env) ++ compile_exp(a2, env) ++ i"if_icmpgt $jmp"
 }
 
-def compile_decl(d: Decl) : Instrs = d match {
+// compile function for declarations and main
+def compile_decl(d: Decl) : String = d match {
   case Def(name, args, a) => { 
     val env = args.zipWithIndex.toMap
     val is = "I" * args.length
-    List(".method public static " + name + "(" + is + ")I \n",
-         ".limit locals " + args.length.toString + "\n",
-         ".limit stack " + (1 + max_stack_exp(a)).toString + "\n",
-         name + "_Start:\n") ++   
+    m".method public static $name($is)I" ++
+    m".limit locals ${args.length.toString}" ++
+    m".limit stack ${1 + max_stack_exp(a)}" ++
+    l"${name}_Start" ++   
     compile_exp(a, env) ++
-    List("ireturn\n",
-         ".end method \n\n")
+    i"ireturn" ++
+    m".end method\n"
   }
   case Main(a) => {
-    List(".method public static main([Ljava/lang/String;)V\n",
-         ".limit locals 200\n",
-         ".limit stack 200\n") ++
+    m".method public static main([Ljava/lang/String;)V" ++
+    m".limit locals 200" ++
+    m".limit stack 200" ++
     compile_exp(a, Map()) ++
-    List("invokestatic XXX/XXX/write(I)V\n",
-         "return\n",
-         ".end method\n")
+    i"invokestatic XXX/XXX/write(I)V" ++
+    i"return" ++
+    m".end method\n"
   }
 }
 
+
 def compile(class_name: String, input: String) : String = {
-  val tks = tokenizer(input)
+  val tks = tokenise(input)
   println(Prog.parse_single(tks).mkString("\n"))
   val ast = Prog.parse_single(tks)
-  val instructions = ast.flatMap(compile_decl).mkString
+  val instructions = ast.map(compile_decl).mkString
   (library + instructions).replaceAllLiterally("XXX", class_name)
 }
 
 
-
-
 def compile_file(file_name: String) = {
   val class_name = file_name.split('.')(0)
   val output = compile(class_name, fromFile(file_name))
@@ -471,12 +542,12 @@
 def compile_run(file_name: String) : Unit = {
   val class_name = file_name.split('.')(0)
   compile_file(file_name)
-  val test = ("java -jar jvm/jasmin-2.4/jasmin.jar " + class_name + ".j").!!
-  println("Time: " + time_needed(2, ("java " + class_name + "/" + class_name).!))
+  val test = (s"java -jar jvm/jasmin-2.4/jasmin.jar ${class_name}.j").!!
+  println("Time: " + time_needed(2, (s"java ${class_name}/${class_name}").!))
 }
 
 
-//examples
+// some examples
 compile_file("fact.rec")
-//compile_run("defs.rec")
-//compile_run("fact.rec")
+compile_run("defs.rec")
+compile_run("fact.rec")
diff -r 8d0af38389bc -r 6709fa87410b progs/lexer.scala
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/progs/lexer.scala	Sun Jul 28 16:15:03 2019 +0100
@@ -0,0 +1,313 @@
+// A Simple Lexer according to Sulzmann & Lu
+
+import scala.language.implicitConversions    
+import scala.language.reflectiveCalls
+
+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)
+}
+
+// A test for more conveninet syntax
+val re : Rexp = ("ab" | "a") ~ ("b" | ONE)
+
+// the 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)
+}
+
+// the 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)
+}
+
+// the 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)
+}
+
+// extracts an environment from a value;
+// used for lexing 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 Rec(x, v) => (x, flatten(v))::env(v)
+}
+
+// The Injection Part of the Lexer
+
+// calculates a value for how a nullable regex 
+// matches the empty string 
+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))
+}
+
+// injects back a character into a value
+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) 
+  case (RECD(x, r1), _) => Rec(x, inj(r1, c, v))
+}
+
+// the 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)
+
+// a simple test for extracting an environment
+val re1 : Rexp = ("first" $ ("a" | "ab")) ~ ("second" $ ("b" | ONE))
+env(lexing(re1, "ab"))
+
+// 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 returns now 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)
+}
+
+// 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("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_simp(("a" | "ab") ~ ("b" | ""), "ab")
+
+// The 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)).%
+
+//   Testing
+//============
+
+def time[T](code: => T) = {
+  val start = System.nanoTime()
+  val result = code
+  val end = System.nanoTime()
+  println((end - start)/1.0e9)
+  result
+}
+
+val r1 = ("a" | "ab") ~ ("bcd" | "c")
+println(lexing(r1, "abcd"))
+
+val r2 = ("" | "a") ~ ("ab" | "b")
+println(lexing(r2, "ab"))
+
+
+// Two Simple While Tests
+//========================
+println("prog0 test")
+
+val prog0 = """read if"""
+println(env(lexing_simp(WHILE_REGS, prog0)))
+
+println("prog1 test")
+
+val prog1 = """read  n; write (n)"""
+println(env(lexing_simp(WHILE_REGS, prog1)))
+
+
+// Bigger Test
+//=============
+
+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
+"""
+
+println("Tokens")
+println(env(lexing_simp(WHILE_REGS, prog2)))
+println(env(lexing_simp(WHILE_REGS, prog2)).filterNot{_._1 == "w"}.mkString("\n"))
+
+// some more timing tests with
+// i copies of the program
+
+for (i <- 0 to 20 by 10) {
+  print(i.toString + ":  ")
+  time(lexing_simp(WHILE_REGS, prog2 * i))
+}
+
+
+val fib = """
+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(env(lexing_simp(WHILE_REGS, prog2)).filterNot{_._1 == "w"})
diff -r 8d0af38389bc -r 6709fa87410b progs/token.scala
--- a/progs/token.scala	Sun Jul 28 14:24:46 2019 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,313 +0,0 @@
-// A Simple Tokenizer according to Sulzmann & Lu
-
-import scala.language.implicitConversions    
-import scala.language.reflectiveCalls
-
-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)
-}
-
-// A test for more conveninet syntax
-val re : Rexp = ("ab" | "a") ~ ("b" | ONE)
-
-// the 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)
-}
-
-// the 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)
-}
-
-// the 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)
-}
-
-// extracts an environment from a value;
-// used for tokenise 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 Rec(x, v) => (x, flatten(v))::env(v)
-}
-
-// The Injection Part of the Tokeniser
-
-// calculates a value for how a nullable regex 
-// matches the empty string 
-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))
-}
-
-// injects back a character into a value
-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) 
-  case (RECD(x, r1), _) => Rec(x, inj(r1, c, v))
-}
-
-// the 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)
-
-// a simple test for extracting an environment
-val re1 : Rexp = ("first" $ ("a" | "ab")) ~ ("second" $ ("b" | ONE))
-env(lexing(re1, "ab"))
-
-// 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 returns now 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)
-}
-
-// 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("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_simp(("a" | "ab") ~ ("b" | ""), "ab")
-
-// The 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)).%
-
-//   Testing
-//============
-
-def time[T](code: => T) = {
-  val start = System.nanoTime()
-  val result = code
-  val end = System.nanoTime()
-  println((end - start)/1.0e9)
-  result
-}
-
-val r1 = ("a" | "ab") ~ ("bcd" | "c")
-println(lexing(r1, "abcd"))
-
-val r2 = ("" | "a") ~ ("ab" | "b")
-println(lexing(r2, "ab"))
-
-
-// Two Simple While Tests
-//========================
-println("prog0 test")
-
-val prog0 = """read if"""
-println(env(lexing_simp(WHILE_REGS, prog0)))
-
-println("prog1 test")
-
-val prog1 = """read  n; write (n)"""
-println(env(lexing_simp(WHILE_REGS, prog1)))
-
-
-// Bigger Test
-//=============
-
-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
-"""
-
-println("Tokens")
-println(env(lexing_simp(WHILE_REGS, prog2)))
-println(env(lexing_simp(WHILE_REGS, prog2)).filterNot{_._1 == "w"}.mkString("\n"))
-
-// some more timing tests with
-// i copies of the program
-
-for (i <- 0 to 20 by 10) {
-  print(i.toString + ":  ")
-  time(lexing_simp(WHILE_REGS, prog2 * i))
-}
-
-
-val fib = """
-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(env(lexing_simp(WHILE_REGS, prog2)).filterNot{_._1 == "w"})