(* Title: HOL/Prolog/Func.thy
ID: $Id$
Author: David von Oheimb (based on a lecture on Lambda Prolog by Nadathur)
*)
header {* Untyped functional language, with call by value semantics *}
theory Func
imports HOHH
begin
typedecl tm
consts
abs :: "(tm => tm) => tm"
app :: "tm => tm => tm"
cond :: "tm => tm => tm => tm"
"fix" :: "(tm => tm) => tm"
true :: tm
false :: tm
"and" :: "tm => tm => tm" (infixr "and" 999)
eq :: "tm => tm => tm" (infixr "eq" 999)
Z :: tm ("Z")
S :: "tm => tm"
(*
"++", "--",
"**" :: tm => tm => tm (infixr 999)
*)
eval :: "[tm, tm] => bool"
instance tm :: plus ..
instance tm :: minus ..
instance tm :: times ..
axioms eval: "
eval (abs RR) (abs RR)..
eval (app F X) V :- eval F (abs R) & eval X U & eval (R U) V..
eval (cond P L1 R1) D1 :- eval P true & eval L1 D1..
eval (cond P L2 R2) D2 :- eval P false & eval R2 D2..
eval (fix G) W :- eval (G (fix G)) W..
eval true true ..
eval false false..
eval (P and Q) true :- eval P true & eval Q true ..
eval (P and Q) false :- eval P false | eval Q false..
eval (A1 eq B1) true :- eval A1 C1 & eval B1 C1..
eval (A2 eq B2) false :- True..
eval Z Z..
eval (S N) (S M) :- eval N M..
eval ( Z + M) K :- eval M K..
eval ((S N) + M) (S K) :- eval (N + M) K..
eval (N - Z) K :- eval N K..
eval ((S N) - (S M)) K :- eval (N- M) K..
eval ( Z * M) Z..
eval ((S N) * M) K :- eval (N * M) L & eval (L + M) K"
lemmas prog_Func = eval
lemma "eval ((S (S Z)) + (S Z)) ?X"
apply (prolog prog_Func)
done
lemma "eval (app (fix (%fact. abs(%n. cond (n eq Z) (S Z)
(n * (app fact (n - (S Z))))))) (S (S (S Z)))) ?X"
apply (prolog prog_Func)
done
end