--- a/scala/ex.scala Thu Jun 06 17:27:45 2013 +0100
+++ b/scala/ex.scala Wed Jun 26 14:35:43 2013 +0100
@@ -60,51 +60,75 @@
println("Compiled Expo 3 ^ 4: " + toTM(Expo(0, 1, 2, 3, 4, 10000).p).run(Tape(3,4,0,0,0)))
// Recursive function examples
+println("Const 8: " + Const(8).eval(67))
println("Add 3 4: " + Add.eval(3, 4))
println("Mult 3 4: " + recs.Mult.eval(3, 4))
-println("Twice 4: " + Twice.eval(4))
-println("FourTm 4: " + Fourtimes.eval(4))
+println("Power 2 3: " + Power.eval(2, 3))
+println("Fact 5: " + Fact.eval(5))
println("Pred 9: " + Pred.eval(9))
println("Pred 0: " + Pred.eval(0))
println("Minus 6 2: " + Minus.eval(6, 2))
println("Minus 6 8: " + Minus.eval(6, 8))
-println("Const 8: " + Const(8).eval(67))
-println("Power 2 3: " + Power.eval(2, 3))
+
+
println("Sign 8: " + Sign.eval(8))
println("Sign 0: " + Sign.eval(0))
+println("Not 0: " + Not.eval(0))
+println("Not 6: " + Not.eval(6))
+println("Eq 4 4: " + Eq.eval(4, 4))
+println("Eq 4 6: " + Eq.eval(4, 6))
+println("Eq 6 4: " + Eq.eval(6, 4))
+println("NotEq 4 4: " + Noteq.eval(4, 4))
+println("NotEq 4 6: " + Noteq.eval(4, 6))
+println("NotEq 6 4: " + Noteq.eval(6, 4))
+println("Conj 0 6: " + Conj.eval(0, 6))
+println("Conj 6 4: " + Conj.eval(6, 4))
+println("Conj 0 0: " + Conj.eval(0, 0))
+println("Disj 0 6: " + Disj.eval(0, 6))
+println("Disj 6 4: " + Disj.eval(6, 4))
+println("Disj 0 0: " + Disj.eval(0, 0))
+println("Imp 6 1: " + Imp.eval(6, 1))
+println("Imp 6 4: " + Imp.eval(6, 4))
+println("Imp 1 0: " + Imp.eval(1, 0))
+println("Ifz 0 1 2:" + Ifz.eval(0, 1, 2))
+println("Ifz 1 1 2:" + Ifz.eval(1, 1, 2))
+println("If 0 1 2:" + If.eval(0, 1, 2))
+println("If 1 1 2:" + If.eval(1, 1, 2))
+
+
println("Less 4 4: " + Less.eval(4, 4))
println("Less 4 6: " + Less.eval(4, 6))
println("Less 6 4: " + Less.eval(6, 4))
println("Le 4 4: " + recs.Le.eval(4, 4))
println("Le 4 6: " + recs.Le.eval(4, 6))
println("Le 6 4: " + recs.Le.eval(6, 4))
-println("Not 0: " + Not.eval(0))
-println("Not 6: " + Not.eval(6))
-println("Eq 4 4: " + Eq.eval(4, 4))
-println("Eq 4 6: " + Eq.eval(4, 6))
-println("Eq 6 4: " + Eq.eval(6, 4))
-println("Conj 0 6: " + Conj.eval(0, 6))
-println("Conj 6 4: " + Conj.eval(6, 4))
-println("Conj 0 0: " + Conj.eval(0, 0))
-println("Disj 0 6: " + Disj.eval(0, 6))
-println("Disj 6 4: " + Disj.eval(6, 4))
-println("Disj 0 0: " + Disj.eval(0, 0))
-println("Sigma Add 2 3 -> 14: " + Sigma(Add).eval(2,3))
-println("Accum Add 2 3 -> 120: " + Accum(Add).eval(2,3))
-println("Accum Mult 2 3 -> 0: " + Accum(recs.Mult).eval(2,3))
-println("Fact 5: " + Fact.eval(5))
-println("Prime 0..15: " + (0 to 15).map(n => (n, Prime.eval(n))))
-println("NextPrime 3: " + NextPrime.eval(3))
-println("NthPrime 1: " + NthPrime.eval(1))
-println("Listsum [2, 3, 4 , 5, 6]: " + Listsum(5, 4).eval(2, 3, 4, 5, 6))
-println("Strt: " + Strt(2).eval(2, 3))
-println("(<=5) 1: " + (Less o (Id(1, 0), Const(5))).eval(1))
-println("(<=5) 5: " + (Less o (Id(1, 0), Const(5))).eval(5))
-println("(<=5) 6: " + (Less o (Id(1, 0), Const(5))).eval(6))
-println("Max (<=9): " + Maxr(Le o (Id(1, 0), Const(9))).eval(10))
-println("Max (>=9): " + Maxr(Le o (Const(9), Id(1, 0))).eval(8))
-println("test")
-println("Lg 4 2: " + Lg.eval(4, 2))
+
+println("Sigma1 Add 2 3 -> 12: " + Sigma1(Add).eval(2, 3))
+println("Accum1 Add 2 3 -> 60: " + Accum1(Add).eval(2,3))
+println("Accum1 Mult 2 3 -> 0: " + Accum1(Mult).eval(2,3))
+println("Accum1 (Id(2, 1)) 2 3 -> 27: " + Accum1(Id(2, 1)).eval(2,3))
+println("Accum2 (Id(3, 1)) 2 3 3 -> 27: " + Accum2(Id(3, 1)).eval(2,3,3))
+println("Accum3 (Id(4, 1)) 2 3 3 3 -> 27: " + Accum3(Id(4, 1)).eval(2,3,3,3))
+println("All1 Add 2 0 -> 0: " + All1(Add).eval(2,0))
+println("All1 Add 2 1 -> 1: " + All1(Add).eval(2,1))
+println("All1_less Add 3 0 -> 0: " + All1_less(Add).eval(2,0))
+println("All1_less Add 3 1 -> 1: " + All1_less(Add).eval(2,1))
+println("All2_less (Id 3 1) 2 3 0 -> 1: " + All2_less(Id(3, 1)).eval(2,3,0))
+println("All2_less (Id 3 0) 2 3 0 -> 0: " + All2_less(Id(3, 0)).eval(2,3,0))
+println("Ex1 Add 2 0 -> 1: " + Ex1(Add).eval(2,0))
+println("Ex2 Id(3,1) 2 1 0 -> 1: " + Ex2(Id(3, 1)).eval(2,1,0))
+println("Ex2 Id(3,2) 2 1 0 -> 0: " + Ex2(Id(3, 2)).eval(2,1,0))
+
+println("Quo 6 4 -> 1: " + Quo.eval(6, 4))
+println("Quo 13 4 -> 3: " + Quo.eval(13, 4))
+
+println("Triangle 0 - 5: " + (0 until 5).map(Triangle.eval(_)).mkString(","))
+println("MaxTriangle 10 -> 4 " + MaxTriangle.eval(10))
+
+println("Penc 1 2 -> 7: " + Penc.eval(1, 2))
+println("Pdec1 7 -> 1: " + Pdec1.eval(7))
+println("Pdec2 7 -> 2: " + Pdec2.eval(7))
+println("Enclen 0 .. 10: " + (0 until 10).map(EncLen.eval(_)))
// compilation of rec to abacus tests
def test_comp2(f: Rec, ns: Int*) = {
@@ -125,9 +149,6 @@
println("Power(3, 4) " + test_comp2(Power, 3, 4))
println("Minus(30, 4) " + test_comp2(Minus, 30, 4))
println("Fact(5) " + test_comp2(Fact, 5))
-println("Prime(5) " + test_comp2(Prime, 5))
-//println("Prime(4) " + test_comp2(Prime, 4))
-println("Strt(1)(2) " + test_comp2(Strt(1), 2))
def test_comp1(f: Rec, ns: Int*) = {
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/scala/recs2.scala Wed Jun 26 14:35:43 2013 +0100
@@ -0,0 +1,349 @@
+package object recs2 {
+
+//Recursive Functions
+
+abstract class Rec {
+ def eval(ns: List[Int]) : Int
+ def eval(ns: Int*) : Int = eval(ns.toList)
+
+ //syntactic convenience for composition
+ def o(r: Rec) = Cn(r.arity, this, List(r))
+ def o(r: Rec, f: Rec) = Cn(r.arity, this, List(r, f))
+ def o(r: Rec, f: Rec, g: Rec) = Cn(r.arity, this, List(r, f, g))
+ def o(r: Rec, f: Rec, g: Rec, h: Rec) = Cn(r.arity, this, List(r, f, g, h))
+ def arity : Int
+}
+
+case object Z extends Rec {
+ override def eval(ns: List[Int]) = ns match {
+ case n::Nil => 0
+ case _ => throw new IllegalArgumentException("Z args: " + ns)
+ }
+ override def arity = 1
+}
+
+case object S extends Rec {
+ override def eval(ns: List[Int]) = ns match {
+ case n::Nil => n + 1
+ case _ => throw new IllegalArgumentException("S args: " + ns)
+ }
+ override def arity = 1
+}
+
+case class Id(n: Int, m: Int) extends Rec {
+ require(m < n, println("Id m < n:" + m + " " + n))
+
+ override def eval(ns: List[Int]) =
+ if (ns.length == n && m < n) ns(m)
+ else throw new IllegalArgumentException("Id args: " + ns + "," + n + "," + m)
+
+ override def arity = n
+}
+
+case class Cn(n: Int, f: Rec, gs: List[Rec]) extends Rec {
+ require(f.arity == gs.length,
+ println("CN " + f + " " + gs.mkString(",") + "\n" +
+ "f.arity gs.length:" + f.arity + " " + gs.length))
+
+ override def eval(ns: List[Int]) =
+ if (ns.length == n && gs.forall(_.arity == n) && f.arity == gs.length) f.eval(gs.map(_.eval(ns)))
+ else {
+ val msg = List("Cn f: " + f,
+ "n: " + n,
+ "f arity: " + f.arity,
+ "ns-args: " + ns,
+ "gs arities: " + gs.map(_.arity).mkString(", "),
+ "gs: " + gs.mkString(", "),
+ "ns.length == n " + (ns.length == n).toString,
+ "gs.forall(_.arity == n) " + (gs.forall(_.arity == n)).toString,
+ "f.arity == gs.length " + (f.arity == gs.length).toString
+ )
+ throw new IllegalArgumentException(msg.mkString("\n"))
+ }
+
+ override def arity = n
+ override def toString = f.toString + gs.map(_.toString).mkString ("(",", ", ")")
+}
+
+// syntactic convenience
+object Cn {
+ def apply(n: Int, f: Rec, g: Rec) : Rec = new Cn(n, f, List(g))
+}
+
+case class Pr(n: Int, f: Rec, g: Rec) extends Rec {
+ override def eval(ns: List[Int]) =
+ if (ns.length == n + 1) {
+ if (ns.head == 0) f.eval(ns.tail)
+ else {
+ val r = Pr(n, f, g).eval(ns.head - 1 :: ns.tail)
+ g.eval(ns.head - 1 :: r :: ns.tail)
+ }
+ }
+ else {
+ val msg = List("Pr f: " + f,
+ "g: " + g,
+ "n: " + n,
+ "f arity: " + f.arity,
+ "g arity: " + g.arity,
+ "ns-args: " + ns)
+ throw new IllegalArgumentException(msg.mkString("\n"))
+ }
+
+ override def arity = n + 1
+ override def toString = "Pr(" + f.toString + ", " + g.toString + ")"
+}
+
+// syntactic convenience
+object Pr {
+ def apply(r: Rec, f: Rec) : Rec = Pr(r.arity, r, f)
+}
+
+case class Mn(n: Int, f: Rec) extends Rec {
+ def evaln(ns: List[Int], n: Int) : Int =
+ if (f.eval(n :: ns) == 0) n else evaln(ns, n + 1)
+
+ override def eval(ns: List[Int]) =
+ if (ns.length == n) evaln(ns, 0)
+ else throw new IllegalArgumentException("Mn: args")
+
+ override def arity = n
+}
+
+object Mn {
+ def apply(f: Rec) : Rec = Mn(f.arity - 1, f)
+}
+
+
+
+
+// Recursive Function examples
+def Const(n: Int) : Rec = n match {
+ case 0 => Z
+ case n => S o Const(n - 1)
+}
+
+def Swap(f: Rec) = f o (Id(2, 1), Id(2, 0))
+val Add = Pr(Id(1, 0), S o Id(3, 1))
+val Mult = Pr(Z, Add o (Id(3, 1), Id(3, 2)))
+val Power = Swap(Pr(Const(1), Mult o (Id(3, 1), Id(3, 2))))
+val Fact = (Pr (Const(1), Mult o (S o Id(3, 0), Id(3, 1)))) o (Id(1, 0), Id(1, 0))
+val Pred = Pr(Z, Id(3, 0)) o (Id(1, 0), Id(1, 0))
+val Minus = Swap(Pr(Id(1, 0), Pred o Id(3, 1)))
+
+val Sign = Minus o (Const(1), Minus o (Const(1), Id(1, 0)))
+val Not = Minus o (Const(1), Id(1, 0))
+val Eq = Minus o (Const(1) o Id(2, 0), Add o (Minus, Swap(Minus)))
+val Noteq = Not o Eq
+val Conj = Sign o Mult
+val Disj = Sign o Add
+val Imp = Disj o (Not o Id(2, 0), Id(2, 1))
+val Ifz = Pr(Id(2, 0), Id(4, 3))
+val If = Ifz o (Not o Id(3, 0), Id(3, 1), Id (3, 2))
+
+val Less = Sign o (Swap(Minus))
+val Le = Disj o (Less, Eq)
+
+def Sigma1(f: Rec) = Pr(f o (Z o (Id(1, 0)), Id(1, 0)),
+ Add o (Id(3, 1), f o (S o Id(3, 0), Id(3, 2))))
+def Sigma2(f: Rec) = Pr(f o (Z o (Id(2, 0)), Id(2, 0), Id(2, 1)),
+ Add o (Id(4, 1), f o (S o Id(4, 0), Id(4, 2), Id(4, 3))))
+
+def Accum1(f: Rec) = Pr(f o (Z o (Id(1, 0)), Id(1, 0)),
+ Mult o (Id(3, 1), f o (S o Id(3, 0), Id(3, 2))))
+def Accum2(f: Rec) = Pr(f o (Z o (Id(2, 0)), Id(2, 0), Id(2, 1)),
+ Mult o (Id(4, 1), f o (S o Id(4, 0), Id(4, 2), Id(4, 3))))
+def Accum3(f: Rec) = Pr(f o (Z o (Id(3, 0)), Id(3, 0), Id(3, 1), Id(3, 2)),
+ Mult o (Id(5, 1), f o (S o Id(5, 0), Id(5, 2), Id(5, 3), Id(5, 4))))
+
+def All1(f: Rec) = Sign o (Accum1(f))
+def All2(f: Rec) = Sign o (Accum2(f))
+def All3(f: Rec) = Sign o (Accum3(f))
+
+def All1_less(f: Rec) = {
+ val cond1 = Eq o (Id(3, 0), Id(3, 1))
+ val cond2 = f o (Id(3, 0), Id(3, 2))
+ All2(Disj o (cond1, cond2)) o (Id(2, 0), Id(2, 0), Id(2, 1))
+}
+
+def All2_less(f: Rec) = {
+ val cond1 = Eq o (Id(4, 0), Id(4, 1))
+ val cond2 = f o (Id(4, 0), Id(4, 2), Id(4, 3))
+ All3(Disj o (cond1, cond2)) o (Id(3, 0), Id(3, 0), Id(3, 1), Id(3, 2))
+}
+
+def Ex1(f: Rec) = Sign o (Sigma1(f))
+def Ex2(f: Rec) = Sign o (Sigma2(f))
+
+val Quo = {
+ val lhs = S o (Id(3, 0))
+ val rhs = Mult o (Id(3, 2), S o (Id(3, 1)))
+ val cond = Eq o (lhs, rhs)
+ val if_stmt = If o (cond, S o (Id(3, 1)), Id(3, 1))
+ Pr(Z, if_stmt)
+}
+
+def Iter(f: Rec) = Pr(Id(1, 0), f o (Id(3, 1)))
+
+def Max1(f: Rec) = Pr(Z, Ifz o (f o (S o (Id(3, 0)), Id(3, 2)), S o (Id(3, 0)), Id(3, 1)))
+def Max2(f: Rec) = Pr(Z, Ifz o (f o (S o (Id(4, 0)), Id(4, 2), Id(4, 3)), S o (Id(4, 0)), Id(4, 1)))
+
+val Triangle = Quo o (Mult o (Id(1, 0), S), Const(2))
+
+val MaxTriangle = {
+ val cond = Not o (Le o (Triangle o (Id(2, 0)), Id(2, 1)))
+ Max1(cond) o (Id(1, 0), Id(1, 0))
+}
+
+val MaxTriangle2 = {
+ Pred o Mn(Le o (Triangle o (Id(2, 0)), Id(2, 1)))
+}
+
+case object MaxTriangleFast extends Rec {
+ def triangle(n: Int) : Int = (n * (n + 1)) / 2
+
+ def search(m: Int, n: Int) : Int = {
+ if (triangle(n) > m) n - 1 else search(m, n + 1)
+ }
+
+ override def eval(ns: List[Int]) = ns match {
+ case n::Nil => search(n, 0)
+ case _ => throw new IllegalArgumentException("MT args: " + ns)
+ }
+
+ override def arity = 1
+}
+
+case object TriangleFast extends Rec {
+ def triangle(n: Int) : Int = (n * (n + 1)) / 2
+
+ override def eval(ns: List[Int]) = ns match {
+ case n::Nil => triangle(n)
+ case _ => throw new IllegalArgumentException("Tr args: " + ns)
+ }
+
+ override def arity = 1
+}
+
+//(0 until 200).map(MaxTriangleFast.eval(_))
+
+
+val Penc = Add o (TriangleFast o (Add o (Id(2, 0), Id(2, 1))), Id(2, 0))
+val Pdec1 = Minus o (Id(1, 0), Triangle o (MaxTriangle o (Id(1, 0))))
+val Pdec2 = Minus o (MaxTriangle o (Id(1, 0)), Pdec1 o (Id(1, 0)))
+
+def Lenc(fs: List[Rec]) : Rec = fs match {
+ case Nil => Z
+ case f::fs => Penc o (S o (f), Lenc(fs))
+}
+
+val Ldec = Pred o (Pdec1 o (Swap (Iter(Pdec2))))
+
+val Within = Less o (Z o (Id(2, 0)), Swap (Iter(Pdec2)))
+val Enclen = (Max1(Not o (Within o (Id(2, 1), Pred o (Id(2, 0)))))) o (Id(1, 0), Id(1, 0))
+
+
+val Read = Ldec o (Id(1, 0), Const(0))
+val Write = Penc o (S o (Id(2, 0)), Pdec2 o (Id(2, 1)))
+
+val Newleft = {
+ val cond0 = Eq o (Id(3, 2), Const(0))
+ val cond1 = Eq o (Id(3, 2), Const(1))
+ val cond2 = Eq o (Id(3, 2), Const(2))
+ val cond3 = Eq o (Id(3, 2), Const(3))
+ val case3 = Penc o (S o (Read o (Id(3, 1))), Id(3, 0))
+ If o (cond0,
+ Id(3, 0),
+ If o (cond1,
+ Id(3, 0),
+ If o (cond2,
+ Pdec2 o (Id(3, 0)),
+ If o (cond3, case3, Id(3, 0)))))
+}
+
+val Newright = {
+ val cond0 = Eq o (Id(3, 2), Const(0))
+ val cond1 = Eq o (Id(3, 2), Const(1))
+ val cond2 = Eq o (Id(3, 2), Const(2))
+ val cond3 = Eq o (Id(3, 2), Const(3))
+ val case2 = Penc o (S o (Read o (Id(3, 0))), Id(3, 1))
+ If o (cond0,
+ Write o (Const(0), Id(3, 1)),
+ If o (cond1,
+ Write o (Const(1), Id(3, 1)),
+ If o (cond2,
+ case2,
+ If o (cond3, Pdec2 o (Id(3, 1)), Id(3, 0)))))
+}
+
+val Actn = Swap (Pr(Pdec1 o (Pdec1 o (Id(1, 0))), Pdec1 o (Pdec2 o (Id(3, 2)))))
+
+val Action = {
+ val cond1 = Noteq o (Id(3, 1), Z)
+ val cond2 = Within o (Id(3, 0), Pred o (Id(3, 1)))
+ val if_branch = Actn o (Ldec o (Id(3, 0), Pred o (Id(3, 1))), Id(3, 2))
+ If o (Conj o (cond1, cond2), if_branch, Const(4))
+}
+
+val Newstat = Swap (Pr (Pdec2 o (Pdec1 o (Id(1, 0))),
+ Pdec2 o (Pdec2 o (Id(3, 2)))))
+
+val Newstate = {
+ val cond = Noteq o (Id(3, 1), Z)
+ val if_branch = Newstat o (Ldec o (Id(3, 0), Pred o (Id(3, 1))), Id(3, 2))
+ If o (cond, if_branch, Z)
+}
+
+val Conf = Lenc (List(Id(3, 0), Id(3, 1), Id(3, 2)))
+
+val State = Ldec o (Id(1, 0), Z)
+
+val Left = Ldec o (Id(1, 0), Const(1))
+
+val Right = Ldec o (Id(1, 0), Const(2))
+
+val Step = {
+ val left = Left o (Id(2, 0))
+ val right = Right o (Id(2, 0))
+ val state = State o (Id(2, 0))
+ val read = Read o (right)
+ val action = Action o (Id(2, 1), state, read)
+ val newstate = Newstate o (Id(2, 1), state, read)
+ val newleft = Newleft o (left, right, action)
+ val newright = Newright o (left, right, action)
+ Conf o (newstate, newleft, newright)
+ }
+
+val Steps = Pr (Id(2, 0), Step o (Id(4, 1), Id(4, 3)))
+
+val Stknum = Minus o (Sigma1(Ldec o (Id(2, 1), Id(2, 0))) o (Enclen o (Id(1, 0)), Id(1, 0)),
+ Ldec o (Id(1, 0), Enclen o (Id(1, 0))))
+
+val Right_std = {
+ val bound = Enclen o (Id(1, 0))
+ val cond1 = Le o (Const(1) o (Id(2, 0)), Id(2, 0))
+ val cond2 = All1_less (Eq o (Ldec o (Id(2, 1), Id(2, 0)), Const(1)))
+ val bound2 = Minus o (Enclen o (Id(2, 1)), Id(2, 0))
+ val cond3 =
+ (All2_less (Eq o (Ldec o (Id(3, 2), Add o (Id(3, 1), Id(3, 0))), Z))) o (bound2, Id(2, 0), Id(2, 1))
+ Ex1(Conj o (Conj o (cond1, cond2), cond3)) o (bound, Id(1, 0))
+}
+
+val Left_std = {
+ val cond = Eq o (Ldec o (Id(2, 1), Id(2, 0)), Z)
+ (All1_less(cond)) o (Enclen o (Id(1, 0)), Id(1, 0))
+}
+
+val Std = Conj o (Left_std o (Left o (Id(1, 0))), Right_std o (Right o (Id(1, 0))))
+
+val Final = Eq o (State o (Id(1, 0)), Z)
+
+val Stop = {
+ val stps = Steps o (Id(3, 2), Id(3, 1), Id(3, 0))
+ Conj o (Final o (stps), Std o (stps))
+}
+
+val Halt = Mn (Not o (Stop o (Id(3, 1), Id(3, 2), Id(3, 0))))
+
+val UF = Pred o (Stknum o (Right o (Steps o (Halt o (Id(2, 0), Id(2, 1)), Id(2, 1), Id(2, 0)))))
+
+}
--- a/thys2/Recs.thy Thu Jun 06 17:27:45 2013 +0100
+++ b/thys2/Recs.thy Wed Jun 26 14:35:43 2013 +0100
@@ -213,7 +213,7 @@
| termi_pr: "\<lbrakk>\<forall> y < x. terminates g (y # (rec_eval (Pr n f g) (y # xs) # xs));
terminates f xs;
length xs = n\<rbrakk>
- \<Longrightarrow> terminates (Pr n f g) (xs @ [x])"
+ \<Longrightarrow> terminates (Pr n f g) (x # xs)"
| termi_mn: "\<lbrakk>length xs = n; terminates f (r # xs);
rec_eval f (r # xs) = 0;
\<forall> i < r. terminates f (i # xs) \<and> rec_eval f (i # xs) > 0\<rbrakk> \<Longrightarrow> terminates (Mn n f) xs"
@@ -242,8 +242,11 @@
definition
"rec_power = rec_swap (PR (constn 1) (CN rec_mult [Id 3 1, Id 3 2]))"
+definition
+ "rec_fact_aux = PR (constn 1) (CN rec_mult [CN S [Id 3 0], Id 3 1])"
+
definition
- "rec_fact = PR (constn 1) (CN rec_mult [CN S [Id 3 0], Id 3 1])"
+ "rec_fact = CN rec_fact_aux [Id 1 0, Id 1 0]"
definition
"rec_pred = CN (PR Z (Id 3 0)) [Id 1 0, Id 1 0]"
@@ -271,9 +274,13 @@
"rec_eval rec_power [x, y] = x ^ y"
by (induct y) (simp_all add: rec_power_def)
+lemma fact_aux_lemma [simp]:
+ "rec_eval rec_fact_aux [x, y] = fact x"
+by (induct x) (simp_all add: rec_fact_aux_def)
+
lemma fact_lemma [simp]:
"rec_eval rec_fact [x] = fact x"
-by (induct x) (simp_all add: rec_fact_def)
+by (simp add: rec_fact_def)
lemma pred_lemma [simp]:
"rec_eval rec_pred [x] = x - 1"
@@ -300,7 +307,7 @@
@{text "rec_eq"} compares two arguments: returns @{text "1"}
if they are equal; @{text "0"} otherwise. *}
definition
- "rec_eq = CN rec_minus [constn 1, CN rec_add [rec_minus, rec_swap rec_minus]]"
+ "rec_eq = CN rec_minus [CN (constn 1) [Id 2 0], CN rec_add [rec_minus, rec_swap rec_minus]]"
definition
"rec_noteq = CN rec_not [rec_eq]"
@@ -385,36 +392,40 @@
section {* Summation and Product Functions *}
definition
- "rec_sigma1 f = PR (CN f [Z, Id 1 0]) (CN rec_add [Id 3 1, CN f [S, Id 3 2]])"
+ "rec_sigma1 f = PR (CN f [CN Z [Id 1 0], Id 1 0])
+ (CN rec_add [Id 3 1, CN f [CN S [Id 3 0], Id 3 2]])"
definition
- "rec_sigma2 f = PR (CN f [Z, Id 2 0, Id 2 1]) (CN rec_add [Id 4 1, CN f [S, Id 4 2, Id 4 3]])"
+ "rec_sigma2 f = PR (CN f [CN Z [Id 2 0], Id 2 0, Id 2 1])
+ (CN rec_add [Id 4 1, CN f [CN S [Id 4 0], Id 4 2, Id 4 3]])"
definition
- "rec_accum1 f = PR (CN f [Z, Id 1 0]) (CN rec_mult [Id 3 1, CN f [S, Id 3 2]])"
+ "rec_accum1 f = PR (CN f [CN Z [Id 1 0], Id 1 0])
+ (CN rec_mult [Id 3 1, CN f [CN S [Id 3 0], Id 3 2]])"
definition
- "rec_accum2 f = PR (CN f [Z, Id 2 0, Id 2 1]) (CN rec_mult [Id 4 1, CN f [S, Id 4 2, Id 4 3]])"
+ "rec_accum2 f = PR (CN f [CN Z [Id 2 0], Id 2 0, Id 2 1])
+ (CN rec_mult [Id 4 1, CN f [CN S [Id 4 0], Id 4 2, Id 4 3]])"
definition
- "rec_accum3 f = PR (CN f [Z, Id 3 0, Id 3 1, Id 3 2])
- (CN rec_mult [Id 5 1, CN f [S, Id 5 2, Id 5 3, Id 5 4]])"
+ "rec_accum3 f = PR (CN f [CN Z [Id 3 0], Id 3 0, Id 3 1, Id 3 2])
+ (CN rec_mult [Id 5 1, CN f [CN S [Id 5 0], Id 5 2, Id 5 3, Id 5 4]])"
lemma sigma1_lemma [simp]:
- shows "rec_eval (rec_sigma1 f) [x, y] = (\<Sum> z \<le> x. (rec_eval f) [z, y])"
+ shows "rec_eval (rec_sigma1 f) [x, y] = (\<Sum> z \<le> x. rec_eval f [z, y])"
by (induct x) (simp_all add: rec_sigma1_def)
lemma sigma2_lemma [simp]:
- shows "rec_eval (rec_sigma2 f) [x, y1, y2] = (\<Sum> z \<le> x. (rec_eval f) [z, y1, y2])"
+ shows "rec_eval (rec_sigma2 f) [x, y1, y2] = (\<Sum> z \<le> x. rec_eval f [z, y1, y2])"
by (induct x) (simp_all add: rec_sigma2_def)
lemma accum1_lemma [simp]:
- shows "rec_eval (rec_accum1 f) [x, y] = (\<Prod> z \<le> x. (rec_eval f) [z, y])"
+ shows "rec_eval (rec_accum1 f) [x, y] = (\<Prod> z \<le> x. rec_eval f [z, y])"
by (induct x) (simp_all add: rec_accum1_def)
lemma accum2_lemma [simp]:
- shows "rec_eval (rec_accum2 f) [x, y1, y2] = (\<Prod> z \<le> x. (rec_eval f) [z, y1, y2])"
+ shows "rec_eval (rec_accum2 f) [x, y1, y2] = (\<Prod> z \<le> x. rec_eval f [z, y1, y2])"
by (induct x) (simp_all add: rec_accum2_def)
lemma accum3_lemma [simp]:
@@ -434,8 +445,9 @@
"rec_all3 f = CN rec_sign [rec_accum3 f]"
definition
- "rec_all1_less f = (let f' = CN rec_disj [CN rec_eq [Id 3 0, Id 3 1], CN f [Id 3 0, Id 3 2]]
- in CN (rec_all2 f') [Id 2 0, Id 2 0, Id 2 1])"
+ "rec_all1_less f = (let cond1 = CN rec_eq [Id 3 0, Id 3 1] in
+ let cond2 = CN f [Id 3 0, Id 3 2]
+ in CN (rec_all2 (CN rec_disj [cond1, cond2])) [Id 2 0, Id 2 0, Id 2 1])"
definition
"rec_all2_less f = (let cond1 = CN rec_eq [Id 4 0, Id 4 1] in
@@ -540,7 +552,7 @@
section {* Iteration *}
definition
- "rec_iter f = PR (Id 2 0) (CN f [Id 3 1])"
+ "rec_iter f = PR (Id 1 0) (CN f [Id 3 1])"
fun Iter where
"Iter f 0 = id"
@@ -585,14 +597,14 @@
by (simp add: BMax_rec_eq2 Set.filter_def)
definition
- "rec_max1 f = PR (constn 0) (CN rec_ifz [CN f [CN S [Id 3 0], Id 3 2], CN S [Id 4 0], Id 4 1])"
+ "rec_max1 f = PR Z (CN rec_ifz [CN f [CN S [Id 3 0], Id 3 2], CN S [Id 3 0], Id 3 1])"
lemma max1_lemma [simp]:
"rec_eval (rec_max1 f) [x, y] = BMax_rec (\<lambda>u. rec_eval f [u, y] = 0) x"
by (induct x) (simp_all add: rec_max1_def)
definition
- "rec_max2 f = PR (constn 0) (CN rec_ifz [CN f [CN S [Id 4 0], Id 4 2, Id 4 3], CN S [Id 4 0], Id 4 1])"
+ "rec_max2 f = PR Z (CN rec_ifz [CN f [CN S [Id 4 0], Id 4 2, Id 4 3], CN S [Id 4 0], Id 4 1])"
lemma max2_lemma [simp]:
"rec_eval (rec_max2 f) [x, y1, y2] = BMax_rec (\<lambda>u. rec_eval f [u, y1, y2] = 0) x"
@@ -801,7 +813,7 @@
by (simp add: enclen_def)
-text {* Recursive De3finitions for List Encodings *}
+text {* Recursive Definitions for List Encodings *}
fun
rec_lenc :: "recf list \<Rightarrow> recf"