# HG changeset patch
# User Christian Urban <christian dot urban at kcl dot ac dot uk>
# Date 1414510574 0
# Node ID 33065bde3bbdbb5132433c0d88f01f9240868309
# Parent  fa92e8f089a2d25ab65dea854228b26358fea5b9
added a file for calculating all answers...still incomplete

diff -r fa92e8f089a2 -r 33065bde3bbd progs/scala/re.scala
--- a/progs/scala/re.scala	Sun Oct 26 16:42:28 2014 +0000
+++ b/progs/scala/re.scala	Tue Oct 28 15:36:14 2014 +0000
@@ -54,6 +54,7 @@
   case RECD(_, r) => 1 + size(r)
 }
 
+
 // nullable function: tests whether the regular 
 // expression can recognise the empty string
 def nullable (r: Rexp) : Boolean = r match {
@@ -107,7 +108,6 @@
   case Rec(x, v) => (x, flatten(v))::env(v)
 }
 
-
 def mkeps(r: Rexp) : Val = r match {
   case EMPTY => Void
   case ALT(r1, r2) => 
@@ -117,6 +117,7 @@
   case RECD(x, r) => Rec(x, mkeps(r))
 }
 
+
 def inj(r: Rexp, c: Char, v: Val) : Val = (r, v) match {
   case (STAR(r), Sequ(v1, Stars(vs))) => Stars(inj(r, c, v1)::vs)
   case (SEQ(r1, r2), Sequ(v1, v2)) => Sequ(inj(r1, c, v1), v2)
@@ -124,11 +125,10 @@
   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), Void) => Chr(d) 
+  case (CHAR(d), Void) => Chr(c) 
   case (RECD(x, r1), _) => Rec(x, inj(r1, c, v))
 }
 
-
 // main lexing function (produces a value)
 def lex(r: Rexp, s: List[Char]) : Val = s match {
   case Nil => if (nullable(r)) mkeps(r) else throw new Exception("Not matched")
@@ -138,11 +138,66 @@
 def lexing(r: Rexp, s: String) : Val = lex(r, s.toList)
 
 
-val r = (("1" $ "a") | (("2" $ "b") | ("3" $ "ab"))).% 
-env(lexing(r, "ba"))
+
+// 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_Void1(f1: Val => Val, f2: Val => Val) = (v:Val) => Sequ(f1(Void), f2(v))
+def F_SEQ_Void2(f1: Val => Val, f2: Val => Val) = (v:Val) => Sequ(f1(v), f2(Void))
+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")
 
-val r = "a" | "b"
-lexing(r,"a")
+// simplification of regular expressions returning also an
+// rectification function; no simplification under STAR 
+def simp(r: Rexp): (Rexp, Val => Val) = r match {
+  case ALT(r1, r2) => {
+    val (r1s, f1s) = simp(r1)
+    val (r2s, f2s) = simp(r2)
+    (r1s, r2s) match {
+      case (NULL, _) => (r2s, F_RIGHT(f2s))
+      case (_, NULL) => (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 (NULL, _) => (NULL, F_ERROR)
+      case (_, NULL) => (NULL, F_ERROR)
+      case (EMPTY, _) => (r2s, F_SEQ_Void1(f1s, f2s))
+      case (_, EMPTY) => (r1s, F_SEQ_Void2(f1s, f2s))
+      case _ => (SEQ(r1s,r2s), F_SEQ(f1s, f2s))
+    }
+  }
+  case RECD(x, r1) => {
+    val (r1s, f1s) = simp(r1)
+    (RECD(x, r1s), F_RECD(f1s))
+  }
+  case r => (r, F_ID)
+}
+
+def lex_simp(r: Rexp, s: List[Char]) : Val = s match {
+  case Nil => if (nullable(r)) mkeps(r) else throw new Exception("Not matched")
+  case c::cs => {
+    val (r_simp, f_simp) = simp(der(c, r))
+    inj(r, c, f_simp(lex_simp(r_simp, cs)))
+  }
+}
+
+def lexing_simp(r: Rexp, s: String) : Val = lex_simp(r, s.toList)
+
 
 // Lexing Rules for a Small While Language
 
@@ -193,12 +248,12 @@
 }
 
 val prog0 = """read n"""
-env (lexing_simp(WHILE_REGS, prog0))
+env (lexing(WHILE_REGS, prog0))
 
 println("Next test")
 /*
 val prog1 = """read  n; write (n)"""
-env (lexing_simp(WHILE_REGS, prog1))
+env (lexing(WHILE_REGS, prog1))
 
 val prog2 = """
 i := 2;
@@ -222,6 +277,6 @@
 
 for (i <- 1 to 100 by 10) {
   print(i.toString + ":  ")
-  time(lexing_simp(WHILE_REGS, prog2 * i))
+  time(lexing(WHILE_REGS, prog2 * i))
 }
 */
diff -r fa92e8f089a2 -r 33065bde3bbd progs/scala/re2.scala
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/progs/scala/re2.scala	Tue Oct 28 15:36:14 2014 +0000
@@ -0,0 +1,172 @@
+import scala.language.implicitConversions    
+import scala.language.reflectiveCalls
+import scala.annotation.tailrec   
+
+abstract class Rexp 
+case object NULL extends Rexp
+case object EMPTY 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 Void 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
+  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)
+}
+
+// size of a regular expressions - for testing purposes 
+def size(r: Rexp) : Int = r match {
+  case NULL => 1
+  case EMPTY => 1
+  case CHAR(_) => 1
+  case ALT(r1, r2) => 1 + size(r1) + size(r2)
+  case SEQ(r1, r2) => 1 + size(r1) + size(r2)
+  case STAR(r) => 1 + size(r)
+  case RECD(_, r) => 1 + size(r)
+}
+
+
+// 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 CHAR(_) => false
+  case ALT(r1, r2) => nullable(r1) || nullable(r2)
+  case SEQ(r1, r2) => nullable(r1) && nullable(r2)
+  case STAR(_) => true
+  case RECD(_, r1) => nullable(r1)
+}
+
+// derivative of a regular expression w.r.t. a character
+def der (c: Char, r: Rexp) : Rexp = r match {
+  case NULL => NULL
+  case EMPTY => NULL
+  case CHAR(d) => if (c == d) EMPTY else NULL
+  case ALT(r1, r2) => ALT(der(c, r1), der(c, r2))
+  case SEQ(r1, r2) => 
+    if (nullable(r1)) ALT(SEQ(der(c, r1), r2), der(c, r2))
+    else SEQ(der(c, r1), r2)
+  case STAR(r) => SEQ(der(c, r), STAR(r))
+  case RECD(_, r1) => der(c, r1)
+}
+
+// derivative w.r.t. a string (iterates der)
+def ders (s: List[Char], r: Rexp) : Rexp = s match {
+  case Nil => r
+  case c::s => ders(s, der(c, r))
+}
+
+// extracts a string from value
+def flatten(v: Val) : String = v match {
+  case Void => ""
+  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 Void => 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)
+}
+
+def mkeps_all(r: Rexp) : Set[Val] = r match {
+  case EMPTY => Set(Void)
+  case ALT(r1, r2) => (nullable(r1), nullable(r2)) match {
+    case (true, true) => mkeps_all(r1).map(Left) ++ mkeps_all(r2).map(Right)
+    case (true, false) => mkeps_all(r1).map(Left)
+    case (false, true) => mkeps_all(r2).map(Right)
+  }
+  case SEQ(r1, r2) => for (v1 <- mkeps_all(r1);
+                           v2 <- mkeps_all(r2)) yield Sequ(v1, v2)
+  case STAR(r) => Set(Stars(Nil), Stars(List(mkeps(r))))
+  case RECD(x, r) => for (v <-  mkeps_all(r)) yield Rec(x, v)
+}
+
+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), Void) => Chr(c) 
+  case (RECD(x, r1), _) => Rec(x, inj(r1, c, v))
+}
+
+def inj_all(r: Rexp, c: Char, vs: Set[Val]) : Set[Val] =
+  for (v <- vs) yield inj(r, c, v)
+
+// main lexing function (produces a value)
+def lex(r: Rexp, s: List[Char]) : Val = s match {
+  case Nil => if (nullable(r)) mkeps(r) else throw new Exception("Not matched")
+  case c::cs => inj(r, c, lex(der(c, r), cs))
+}
+
+def lexing(r: Rexp, s: String) : Val = lex(r, s.toList)
+
+// Examples
+val K: Rexp = "a" | "b"
+val I: Rexp = "ab" | "ba"  
+
+val R0 = (K | I).%
+
+lexing(R0, "abab")
+
+val K: Rexp = ("key" $ "a" | "b")
+val I: Rexp = ("id" $ ("ab" | "ba"))  
+
+val R0 = (K | I).%
+lexing(R0, "abaa")
+env(lexing(R0, "abaa"))
+
+val r0: Rexp = (K | I).%
+val r1 = der('a', r0)
+val r1_simp = simp2(r1)
+val r2 = der('b', r1)
+val r2_simp = simp2(r2)
+nullable(r2)
+val v2 = mkeps(r2)
+val v1 = inj(r1, 'b', v2)
+val v0 = inj(r0, 'a', v1)
+env(v0)
+env(lexing(r0, "abab"))
+
diff -r fa92e8f089a2 -r 33065bde3bbd thys/CountSnoc.thy
--- a/thys/CountSnoc.thy	Sun Oct 26 16:42:28 2014 +0000
+++ b/thys/CountSnoc.thy	Tue Oct 28 15:36:14 2014 +0000
@@ -23,11 +23,15 @@
 value "count_list (1::nat) [2,2,2]"
 value "count_list (2::nat) [2,2,1]"
 
-lemma count1: "count_list n(xs @ ys) = count_list n xs + count_list n ys"
+lemma count1: "count_list n (xs @ ys) = count_list n xs + count_list n ys"
 apply(induct xs)
 apply(auto)
 done
  
+thm count1
+thm refl
+thm conjI[OF refl[of "a"] refl[of "b"]]
+thm conjI[OF conjI]