diff -r fc2a5e9fbb97 -r f06aa4e1c25b scala/ex.scala
--- a/scala/ex.scala	Fri Feb 22 14:31:34 2013 +0000
+++ b/scala/ex.scala	Mon Feb 25 15:12:48 2013 +0000
@@ -1,7 +1,9 @@
 import lib._
 import turing._
 import abacus._
+import recs._
 import comp1._
+//import comp2._
 
 // Turing machine examples
 val TMCopy = TM((WBk, 5), (R, 2), (R, 3), (R, 2), (WOc, 3), 
@@ -21,7 +23,7 @@
   Abacus(List(Dec(in, jump), Inc(out), Goto(0))) 
 
 def Plus(m: Int, n: Int, tmp: Int, jump: Int) =
-  Abacus(List(Dec(m, 4), Inc(n), Inc(tmp), Goto(0), Dec(tmp, jump), Inc(m), Goto(4))) 
+  Abacus(List(Dec(m, 4), Inc(n), Inc(tmp), Goto(0), Dec(tmp, jump), Inc(m), Goto(4)))
 
 def Mult(in1: Int, in2: Int, out: Int, tmp: Int, jump: Int) = 
   Abacus(List(Dec(in1, jump))) ++ Plus(in2, out, tmp, -1).shift(1, -1).adjust(-1, 0)
@@ -44,4 +46,76 @@
 println("Compiled Mult 3 * 5: " + toTM(Mult(0, 1, 2, 3, Int.MaxValue).p).run(Tape(3,5,0,0)))
 println("Compiled Expo 3 ^ 4: " + toTM(Expo(0, 1, 2, 3, 4, 10000).p).run(Tape(3,4,0,0,0)))
 
+// Recursive Function examples
+def arity(f: Rec) = f match {
+  case Z => 1
+  case S => 1
+  case Id(n, _) => n
+  case Cn(n, _, _) => n
+  case Pr(n, _, _) => n + 1
+  case Mn(n, _) => n 
+}
 
+val Add = Pr(1, Id(1, 0), Cn(3, S, List(Id(3, 2))))
+val Mult = Pr(1, Z, Cn(3, Add, List(Id(3, 0), Id(3, 2))))
+val Pred = Cn(1, Pr(1, Z, Id(3, 1)), List(Id(1, 0), Id(1, 0)))
+val Minus = Pr(1, Id(1, 0), Cn(3, Pred, List(Id(3, 2))))
+
+def Const(n: Int) : Rec = n match {
+  case 0 => Z
+  case n => Cn(1, S, List(Const(n - 1)))
+}
+
+val Sign = Cn(1, Minus, List(Const(1), Cn(1, Minus, List(Const(1), Id(1, 0)))))
+val Less = Cn(2, Sign, List(Cn(2, Minus, List(Id(2, 1), Id(2, 0)))))
+val Not = Cn(1, Minus, List(Const(1), Id(1, 0)))
+val Eq = Cn(2, Minus, List(Cn(2, Const(1), List(Id(2, 0))), 
+           Cn(2, Add, List(Cn(2, Minus, List(Id(2, 0), Id(2, 1))), 
+             Cn(2, Minus, List(Id(2, 1), Id(2, 0)))))))
+val Conj = Cn(2, Sign, List(Cn(2, Mult, List(Id(2, 0), Id(2, 1)))))
+val Disj = Cn(2, Sign, List(Cn(2, Add, List(Id(2, 0), Id(2, 1)))))
+
+def Nargs(n: Int, m: Int) : List[Rec] = m match {
+  case 0 => Nil
+  case m => Nargs(n, m - 1) ::: List(Id(n, m - 1))
+}
+
+def Sigma(f: Rec) = {
+  val ar = arity(f)  
+  Pr(ar - 1, Cn(ar - 1, f, Nargs(ar - 1, ar - 1) :::
+                    List(Cn(ar - 1, Const(0), List(Id(ar - 1, 0))))), 
+             Cn(ar + 1, Add, List(Id(ar + 1, ar), 
+                    Cn(ar + 1, f, Nargs(ar + 1, ar - 1) :::
+                        List(Cn(ar + 1, S, List(Id(ar + 1, ar - 1))))))))
+}
+
+println("Add 3 4:   " + Add.eval(List(3, 4)))
+println("Mult 3 4:  " + Mult.eval(List(3, 4)))
+println("Pred 9:    " + Pred.eval(List(9)))
+println("Pred 0:    " + Pred.eval(List(0)))
+println("Minus 6 2: " + Minus.eval(List(6, 2)))
+println("Minus 6 8: " + Minus.eval(List(6, 8)))
+println("Const 8:   " + Const(8).eval(List(67)))
+println("Sign 8:    " + Sign.eval(List(8)))
+println("Sign 0:    " + Sign.eval(List(0)))
+println("Less 4 4:  " + Less.eval(List(4, 4)))
+println("Less 4 6:  " + Less.eval(List(4, 6)))
+println("Less 6 4:  " + Less.eval(List(6, 4)))
+println("Not 0:     " + Not.eval(List(0)))
+println("Not 6:     " + Not.eval(List(6)))
+println("Eq 4 4:    " + Eq.eval(List(4, 4)))
+println("Eq 4 6:    " + Eq.eval(List(4, 6)))
+println("Eq 6 4:    " + Eq.eval(List(6, 4)))
+println("Conj 0 6:  " + Conj.eval(List(0, 6)))
+println("Conj 6 4:  " + Conj.eval(List(6, 4)))
+println("Conj 0 0:  " + Conj.eval(List(0, 0)))
+println("Disj 0 6:  " + Disj.eval(List(0, 6)))
+println("Disj 6 4:  " + Disj.eval(List(6, 4)))
+println("Disj 0 0:  " + Disj.eval(List(0, 0)))
+//println("Sigma:     " + Sigma(S).eval(List(0,1,2)))
+//println("Sigma:     " + Sigma(S).eval(List(0,1,2,3,4,5)))
+
+
+val ABCZero = Abacus(List(Goto(1)))
+val ABCSucc = Plus(0, 1, 2, 7) ++ Abacus(List(Inc(1))).shift(Plus(0, 1, 2, 7).p.length, -1)
+def ABCId(n: Int, m: Int) = Plus(m, n, n + 1, 7)