--- a/src/HOL/Hoare/Hoare.thy Wed Oct 14 11:51:11 1998 +0200
+++ b/src/HOL/Hoare/Hoare.thy Wed Oct 14 15:26:31 1998 +0200
@@ -1,196 +1,193 @@
(* Title: HOL/Hoare/Hoare.thy
ID: $Id$
- Author: Norbert Galm & Tobias Nipkow
- Copyright 1995 TUM
+ Author: Leonor Prensa Nieto & Tobias Nipkow
+ Copyright 1998 TUM
Sugared semantic embedding of Hoare logic.
+Strictly speaking a shallow embedding (as implemented by Norbert Galm
+following Mike Gordon) would suffice. Maybe the datatype com comes in useful
+later.
*)
-Hoare = Arith +
+Hoare = Main +
types
- 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 *)
+ 'a bexp = 'a set
+ 'a assn = 'a set
+ 'a fexp = 'a =>'a
+
+datatype
+ 'a com = Basic ('a fexp)
+ | Seq ('a com) ('a com) ("(_;/_)" [61,60] 60)
+ | Cond ('a bexp) ('a com) ('a com) ("(1IF _/ THEN _ / ELSE _/ FI)" [0,0,0] 61)
+ | While ('a bexp) ('a assn) ('a com) ("(1WHILE _/ INV {_} //DO _ /OD)" [0,0,0] 61)
+
+syntax
+ "@assign" :: id => 'b => 'a com ("(2_ :=/ _ )" [70,65] 61)
+ "@annskip" :: 'a com ("SKIP")
+
+translations
+ "SKIP" == "Basic id"
+
+types 'a sem = 'a => 'a => bool
+
+consts iter :: nat => 'a bexp => 'a sem => 'a sem
+primrec
+"iter 0 b S = (%s s'. s ~: b & (s=s'))"
+"iter (Suc n) b S = (%s s'. s : b & (? s''. S s s'' & iter n b S s'' s'))"
+
+consts Sem :: 'a com => 'a sem
+primrec
+"Sem(Basic f) s s' = (s' = f s)"
+"Sem(c1;c2) s s' = (? s''. Sem c1 s s'' & Sem c2 s'' s')"
+"Sem(IF b THEN c1 ELSE c2 FI) s s' = ((s : b --> Sem c1 s s') &
+ (s ~: b --> Sem c2 s s'))"
+"Sem(While b x c) s s' = (? n. iter n b (Sem c) s s')"
+
+constdefs Valid :: ['a bexp, 'a com, 'a bexp] => bool
+ "Valid p c q == !s s'. Sem c s s' --> s : p --> s' : q"
-(* program syntax *)
-
nonterminals
- prog
+ vars
syntax
- "@Skip" :: prog ("SKIP")
- "@Assign" :: [id, 'a] => prog ("_ := _")
- "@Seq" :: [prog, prog] => prog ("_;//_" [0,1000] 999)
- "@If" :: [bool, prog, prog] => prog ("IF _//THEN// (_)//ELSE// (_)//END")
- "@While" :: [bool, bool, prog] => prog ("WHILE _//DO {_}// (_)//END")
- "@Spec" :: [bool, prog, bool] => bool ("{_}//_//{_}")
-
-
-(* denotational semantics *)
-
-constdefs
- 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)))"
+ "" :: "id => vars" ("_")
+ "_vars" :: "[id, vars] => vars" ("_ _")
- 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')"
-
-consts
- Iter :: [nat, 'a bexp, 'a com] => 'a com
-
-primrec
- "Iter 0 b c = (%s s'.~b(s) & (s=s'))"
- "Iter (Suc n) b c = (%s s'. b(s) & Seq c (Iter n b c) s s')"
-
-constdefs
- While :: ['a bexp, 'a bexp, 'a com] => 'a com
- "While b I c s s' == ? n. Iter n b c s s'"
-
- Spec :: ['a bexp, 'a com, 'a bexp] => bool
- "Spec p c q == !s s'. c s s' --> p s --> q s'"
+syntax
+ "@hoare_vars" :: [vars, 'a assn,'a com,'a assn] => bool
+ ("|- VARS _.// {_} // _ // {_}" [0,0,55,0] 50)
+syntax ("" output)
+ "@hoare" :: ['a assn,'a com,'a assn] => bool
+ ("|- {_} // _ // {_}" [0,55,0] 50)
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 *)
+(** parse translations **)
-fun name_in_term (name,Const (s,t)) = (name=s)
- | name_in_term (name,Free (s,t)) = (name=s)
- | name_in_term (name,Var (ix,t)) = (name= string_of_indexname ix)
- | 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 mk_abstuple [] body = absfree ("x", dummyT, body)
+ | mk_abstuple [v] body = absfree ((fst o dest_Free) v, dummyT, body)
+ | mk_abstuple (v::w) body = Syntax.const "split" $
+ absfree ((fst o dest_Free) v, dummyT, mk_abstuple w body);
-(*This could be done more simply by calling Term.variant, supplying a list of
- all free, bound and scheme variables in the term.*)
-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 mk_fbody v e [] = Syntax.const "()"
+ | mk_fbody v e [x] = if v=x then e else x
+ | mk_fbody v e (x::xs) = Syntax.const "Pair" $ (if v=x then e else x) $
+ mk_fbody v e xs;
-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;
+fun mk_fexp v e xs = mk_abstuple xs (mk_fbody v e xs);
-(* Translation between program vars ("a" - "z") and their encoding as
- natural numbers: "a" <==> 0, "b" <==> Suc(0), ..., "z" <==> 25 *)
+(* bexp_tr & assn_tr *)
+(*all meta-variables for bexp except for TRUE and FALSE are translated as if they
+ were boolean expressions*)
+
+fun bexp_tr (Const ("TRUE", _)) xs = Syntax.const "TRUE"
+ | bexp_tr b xs = Syntax.const "Collect" $ mk_abstuple xs b;
+
+fun assn_tr r xs = Syntax.const "Collect" $ mk_abstuple xs r;
-fun is_pvarID s = size s=1 andalso "a"<=s andalso s<="z";
+(* com_tr *)
+
+fun assign_tr [Free (V,_),E] xs = Syntax.const "Basic" $
+ mk_fexp (Free(V,dummyT)) E xs
+ | assign_tr ts _ = raise TERM ("assign_tr", ts);
-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 com_tr (Const("@assign",_) $ Free (V,_) $ E) xs =
+ assign_tr [Free (V,dummyT),E] xs
+ | com_tr (Const ("Basic",_) $ f) xs = Syntax.const "Basic" $ f
+ | com_tr (Const ("Seq",_) $ c1 $ c2) xs = Syntax.const "Seq" $
+ com_tr c1 xs $ com_tr c2 xs
+ | com_tr (Const ("Cond",_) $ b $ c1 $ c2) xs = Syntax.const "Cond" $
+ bexp_tr b xs $ com_tr c1 xs $ com_tr c2 xs
+ | com_tr (Const ("While",_) $ b $ I $ c) xs = Syntax.const "While" $
+ bexp_tr b xs $ assn_tr I xs $ com_tr c xs
+ | com_tr trm _ = trm;
+
+(* triple_tr *)
-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;
+fun vars_tr (x as Free _) = [x]
+ | vars_tr (Const ("_vars", _) $ (x as Free _) $ vars) = x :: vars_tr vars
+ | vars_tr t = raise TERM ("vars_tr", [t]);
+
+fun hoare_vars_tr [vars, pre, prg, post] =
+ let val xs = vars_tr vars
+ in Syntax.const "Valid" $
+ assn_tr pre xs $ com_tr prg xs $ assn_tr post xs
+ end
+ | hoare_vars_tr ts = raise TERM ("hoare_vars_tr", ts);
+
+
+
+val parse_translation = [("@hoare_vars", hoare_vars_tr)];
-(*** parse translations: syntax -> semantics ***)
+(*****************************************************************************)
+
+(*** print translations ***)
-(* 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 dest_abstuple (Const ("split",_) $ (Abs(v,_, body))) =
+ subst_bound (Syntax.free v, dest_abstuple body)
+ | dest_abstuple (Abs(v,_, body)) = subst_bound (Syntax.free v, body)
+ | dest_abstuple trm = trm;
-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 abs2list (Const ("split",_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t
+ | abs2list (Abs(x,T,t)) = [Free (x, T)]
+ | abs2list _ = [];
-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 mk_ts (Const ("split",_) $ (Abs(x,_,t))) = mk_ts t
+ | mk_ts (Abs(x,_,t)) = mk_ts t
+ | mk_ts (Const ("Pair",_) $ a $ b) = a::(mk_ts b)
+ | mk_ts t = [t];
-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 mk_vts (Const ("split",_) $ (Abs(x,_,t))) =
+ ((Syntax.free x)::(abs2list t), mk_ts t)
+ | mk_vts (Abs(x,_,t)) = ([Syntax.free x], [t])
+ | mk_vts t = raise Match;
+
+fun find_ch [] i xs = (false, (Syntax.free "not_ch",Syntax.free "not_ch" ))
+ | find_ch ((v,t)::vts) i xs = if t=(Bound i) then find_ch vts (i-1) xs
+ else (true, (v, subst_bounds (xs,t)));
+
+fun is_f (Const ("split",_) $ (Abs(x,_,t))) = true
+ | is_f (Abs(x,_,t)) = true
+ | is_f t = false;
+
+(* assn_tr' & bexp_tr'*)
+
+fun assn_tr' (Const ("Collect",_) $ T) = dest_abstuple T
+ | assn_tr' (Const ("op Int",_) $ (Const ("Collect",_) $ T1) $
+ (Const ("Collect",_) $ T2)) =
+ Syntax.const "op Int" $ dest_abstuple T1 $ dest_abstuple T2
+ | assn_tr' t = t;
-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 bexp_tr' (Const ("Collect",_) $ T) = dest_abstuple T
+ | bexp_tr' t = t;
+
+(*com_tr' *)
-fun spec_tr [P,C,Q] = Syntax.const "Spec" $ bool_tr P $ prog_tr C $ bool_tr Q;
+fun mk_assign f =
+ let val (vs, ts) = mk_vts f;
+ val (ch, which) = find_ch (vs~~ts) ((length vs)-1) (rev vs)
+ in if ch then Syntax.const "@assign" $ fst(which) $ snd(which)
+ else Syntax.const "@skip" end;
-val parse_translation = [("@Spec",spec_tr)];
+fun com_tr' (Const ("Basic",_) $ f) = if is_f f then mk_assign f
+ else Syntax.const "Basic" $ f
+ | com_tr' (Const ("Seq",_) $ c1 $ c2) = Syntax.const "Seq" $
+ com_tr' c1 $ com_tr' c2
+ | com_tr' (Const ("Cond",_) $ b $ c1 $ c2) = Syntax.const "Cond" $
+ bexp_tr' b $ com_tr' c1 $ com_tr' c2
+ | com_tr' (Const ("While",_) $ b $ I $ c) = Syntax.const "While" $
+ bexp_tr' b $ assn_tr' I $ com_tr' c
+ | com_tr' t = t;
-(*** print translations: semantics -> syntax ***)
-
-(* Note: does not mark tokens *)
-
-(* 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')];
+fun spec_tr' [p, c, q] =
+ Syntax.const "@hoare" $ assn_tr' p $ com_tr' c $ assn_tr' q
+
+val print_translation = [("Valid", spec_tr')];