(* Title: HOL/Prolog/Func.thy
Author: David von Oheimb (based on a lecture on Lambda Prolog by Nadathur)
*)
section {* Untyped functional language, with call by value semantics *}
theory Func
imports HOHH
begin
typedecl tm
axiomatization
abs :: "(tm \<Rightarrow> tm) \<Rightarrow> tm" and
app :: "tm \<Rightarrow> tm \<Rightarrow> tm" and
cond :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm" and
"fix" :: "(tm \<Rightarrow> tm) \<Rightarrow> tm" and
true :: tm and
false :: tm and
"and" :: "tm \<Rightarrow> tm \<Rightarrow> tm" (infixr "and" 999) and
eq :: "tm \<Rightarrow> tm \<Rightarrow> tm" (infixr "eq" 999) and
Z :: tm ("Z") and
S :: "tm \<Rightarrow> tm" and
plus :: "tm \<Rightarrow> tm \<Rightarrow> tm" (infixl "+" 65) and
minus :: "tm \<Rightarrow> tm \<Rightarrow> tm" (infixl "-" 65) and
times :: "tm \<Rightarrow> tm \<Rightarrow> tm" (infixl "*" 70) and
eval :: "tm \<Rightarrow> tm \<Rightarrow> bool" where
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
schematic_goal "eval ((S (S Z)) + (S Z)) ?X"
apply (prolog prog_Func)
done
schematic_goal "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