(* Title: HOL/Hoare/Hoare.thy
ID: $Id$
Author: Norbert Galm & Tobias Nipkow
Copyright 1995 TUM
Sugared semantic embedding of Hoare logic.
*)
Hoare = Arith +
types
'a prog (* program syntax *)
pvar = nat (* encoding of program variables ( < 26! ) *)
'a state = pvar => 'a (* program state *)
'a exp = 'a state => 'a (* denotation of expressions *)
'a bexp = 'a state => bool (* denotation of boolean expressions *)
'a com = 'a state => 'a state => bool (* denotation of programs *)
syntax
"@Skip" :: 'a prog ("SKIP")
"@Assign" :: [id, 'a] => 'a prog ("_ := _")
"@Seq" :: ['a prog, 'a prog] => 'a prog ("_;//_" [0,1000] 999)
"@If" :: [bool, 'a prog, 'a prog] => 'a prog
("IF _//THEN// (_)//ELSE// (_)//END")
"@While" :: [bool, bool, 'a prog] => 'a prog
("WHILE _//DO {_}// (_)//END")
"@Spec" :: [bool, 'a prog, bool] => bool ("{_}//_//{_}")
constdefs
(* denotational semantics *)
Skip :: 'a com
"Skip s s' == (s=s')"
Assign :: [pvar, 'a exp] => 'a com
"Assign v e s s' == (s' = (%x.if x=v then e(s) else s(x)))"
Seq :: ['a com, 'a com] => 'a com
"Seq c c' s s' == ? s''. c s s'' & c' s'' s'"
Cond :: ['a bexp, 'a com, 'a com] => 'a com
"Cond b c c' s s' == (b(s) --> c s s') & (~b s --> c' s s')"
While :: ['a bexp, 'a bexp, 'a com] => 'a com
"While b inv c s s' == ? n. Iter n b c s s'"
Iter :: [nat, 'a bexp, 'a com] => 'a com
"Iter n b c == nat_rec (%s s'.~b(s) & (s=s'))
(%n_rec iter_rec.%s s'.b(s) & Seq c iter_rec s s') n"
Spec :: ['a bexp, 'a com, 'a bexp] => bool
"Spec p c q == !s s'. c s s' --> p s --> q s'"
end
ML
(*** term manipulation ***)
(* name_in_term:bool (name:string,trm:term) bestimmt, ob in trm der Name name als Konstante,
freie Var., scheme-Variable oder als Name fuer eine Lambda-Abstraktion vorkommt *)
fun name_in_term (name,Const (s,t)) =(name=s)
| name_in_term (name,Free (s,t)) =(name=s)
| name_in_term (name,Var ((s,i),t)) =(name=s ^ makestring i)
| name_in_term (name,Abs (s,t,trm)) =(name=s) orelse (name_in_term (name,trm))
| name_in_term (name,trm1 $ trm2) =(name_in_term (name,trm1)) orelse (name_in_term (name,trm2))
| name_in_term (_,_) =false;
(* variant_name:string (name:string,trm:term) liefert einen von name abgewandelten Namen (durch Anhaengen
von genuegend vielen "_"), der nicht in trm vorkommt. Im Gegensatz zu variant_abs beruecktsichtigt es
auch Namen von scheme-Variablen und von Lambda-Abstraktionen in trm *)
fun variant_name (name,trm) =if name_in_term (name,trm)
then variant_name (name ^ "_",trm)
else name;
(* subst_term:term (von:term,nach:term,trm:term) liefert den Term, der aus
trm entsteht, wenn alle Teilterme, die gleich von sind, durch nach ersetzt
wurden *)
fun subst_term (von,nach,Abs (s,t,trm)) =if von=Abs (s,t,trm)
then nach
else Abs (s,t,subst_term (von,nach,trm))
| subst_term (von,nach,trm1 $ trm2) =if von=trm1 $ trm2
then nach
else subst_term (von,nach,trm1) $ subst_term (von,nach,trm2)
| subst_term (von,nach,trm) =if von=trm
then nach
else trm;
(* Translation between program vars ("a" - "z") and their encoding as
natural numbers: "a" <==> 0, "b" <==> Suc(0), ..., "z" <==> 25 *)
fun is_pvarID s = size s=1 andalso "a"<=s andalso s<="z";
fun pvarID2pvar s =
let fun rest2pvar (i,arg) =
if i=0 then arg else rest2pvar(i-1, Syntax.const "Suc" $ arg)
in rest2pvar(ord s - ord "a", Syntax.const "0") end;
fun pvar2pvarID trm =
let
fun rest2pvarID (Const("0",_),i) =chr (i + ord "a")
| rest2pvarID (Const("Suc",_) $ trm,i) =rest2pvarID (trm,i+1)
in
rest2pvarID (trm,0)
end;
(*** parse translations: syntax -> semantics ***)
(* term_tr:term (name:string,trm:term) ersetzt in trm alle freien Variablen, die eine pvarID
darstellen, durch name $ pvarID2pvar(pvarID). Beispiel:
bei name="s" und dem Term "a=b & a=X" wird der Term "s(0)=s(Suc(0)) & s(0)=X" geliefert *)
fun term_tr (name,Free (s,t)) = if is_pvarID s
then Syntax.free name $ pvarID2pvar s
else Free (s,t)
| term_tr (name,Abs (s,t,trm)) = Abs (s,t,term_tr (name,trm))
| term_tr (name,trm1 $ trm2) = term_tr (name,trm1) $ term_tr (name,trm2)
| term_tr (name,trm) = trm;
fun bool_tr B =
let val name = variant_name("s",B)
in Abs (name,dummyT,abstract_over (Syntax.free name,term_tr (name,B))) end;
fun exp_tr E =
let val name = variant_name("s",E)
in Abs (name,dummyT,abstract_over (Syntax.free name,term_tr (name,E))) end;
fun prog_tr (Const ("@Skip",_)) = Syntax.const "Skip"
| prog_tr (Const ("@Assign",_) $ Free (V,_) $ E) =
if is_pvarID V
then Syntax.const "Assign" $ pvarID2pvar V $ exp_tr E
else error("Not a valid program variable: " ^ quote V)
| prog_tr (Const ("@Seq",_) $ C $ C') =
Syntax.const "Seq" $ prog_tr C $ prog_tr C'
| prog_tr (Const ("@If",_) $ B $ C $ C') =
Syntax.const "Cond" $ bool_tr B $ prog_tr C $ prog_tr C'
| prog_tr (Const ("@While",_) $ B $ INV $ C) =
Syntax.const "While" $ bool_tr B $ bool_tr INV $ prog_tr C;
fun spec_tr [P,C,Q] = Syntax.const "Spec" $ bool_tr P $ prog_tr C $ bool_tr Q;
val parse_translation = [("@Spec",spec_tr)];
(*** print translations: semantics -> syntax ***)
(* term_tr':term (name:string,trm:term) ersetzt in trm alle Vorkommen von name $ pvar durch
entsprechende freie Variablen, welche die pvarID zu pvar darstellen. Beispiel:
bei name="s" und dem Term "s(0)=s(Suc(0)) & s(0)=X" wird der Term "a=b & a=X" geliefert *)
fun term_tr' (name,Free (s,t) $ trm) =if name=s
then Syntax.free (pvar2pvarID trm)
else Free (s,t) $ term_tr' (name,trm)
| term_tr' (name,Abs (s,t,trm)) =Abs (s,t,term_tr' (name,trm))
| term_tr' (name,trm1 $ trm2) =term_tr' (name,trm1) $ term_tr' (name,trm2)
| term_tr' (name,trm) =trm;
fun bexp_tr' (Abs(_,_,b)) =term_tr' (variant_abs ("s",dummyT,b));
fun exp_tr' (Abs(_,_,e)) =term_tr' (variant_abs ("s",dummyT,e));
fun com_tr' (Const ("Skip",_)) =Syntax.const "@Skip"
| com_tr' (Const ("Assign",_) $ v $ e) =Syntax.const "@Assign" $ Syntax.free (pvar2pvarID v) $ exp_tr' e
| com_tr' (Const ("Seq",_) $ c $ c') =Syntax.const "@Seq" $ com_tr' c $ com_tr' c'
| com_tr' (Const ("Cond",_) $ b $ c $ c') =Syntax.const "@If" $ bexp_tr' b $ com_tr' c $ com_tr' c'
| com_tr' (Const ("While",_) $ b $ inv $ c) =Syntax.const "@While" $ bexp_tr' b $ bexp_tr' inv $ com_tr' c;
fun spec_tr' [p,c,q] =Syntax.const "@Spec" $ bexp_tr' p $ com_tr' c $ bexp_tr' q;
val print_translation =[("Spec",spec_tr')];