src/HOL/Hoare/Hoare.thy
author nipkow
Wed Oct 14 15:26:31 1998 +0200 (1998-10-14)
changeset 5646 7c2ddbaf8b8c
parent 5565 301a3a4d3dc7
child 6055 fdf4638bf726
permissions -rw-r--r--
New many-sorted version.
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(*  Title:      HOL/Hoare/Hoare.thy
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    ID:         $Id$
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    Author:     Leonor Prensa Nieto & Tobias Nipkow
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    Copyright   1998 TUM
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Sugared semantic embedding of Hoare logic.
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Strictly speaking a shallow embedding (as implemented by Norbert Galm
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following Mike Gordon) would suffice. Maybe the datatype com comes in useful
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later.
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*)
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Hoare  = Main +
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types
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    'a bexp = 'a set
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    'a assn = 'a set
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    'a fexp = 'a =>'a
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datatype
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 'a com = Basic ('a fexp)         
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   | Seq ('a com) ('a com)               ("(_;/_)"      [61,60] 60)
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   | Cond ('a bexp) ('a com) ('a com)    ("(1IF _/ THEN _ / ELSE _/ FI)"  [0,0,0] 61)
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   | While ('a bexp) ('a assn) ('a com)  ("(1WHILE _/ INV {_} //DO _ /OD)"  [0,0,0] 61)
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syntax
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  "@assign"  :: id => 'b => 'a com        ("(2_ :=/ _ )" [70,65] 61)
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  "@annskip" :: 'a com                    ("SKIP")
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translations
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            "SKIP" == "Basic id"
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types 'a sem = 'a => 'a => bool
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consts iter :: nat => 'a bexp => 'a sem => 'a sem
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primrec
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"iter 0 b S = (%s s'. s ~: b & (s=s'))"
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"iter (Suc n) b S = (%s s'. s : b & (? s''. S s s'' & iter n b S s'' s'))"
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consts Sem :: 'a com => 'a sem
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primrec
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"Sem(Basic f) s s' = (s' = f s)"
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"Sem(c1;c2) s s' = (? s''. Sem c1 s s'' & Sem c2 s'' s')"
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"Sem(IF b THEN c1 ELSE c2 FI) s s' = ((s  : b --> Sem c1 s s') &
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                                      (s ~: b --> Sem c2 s s'))"
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"Sem(While b x c) s s' = (? n. iter n b (Sem c) s s')"
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constdefs Valid :: ['a bexp, 'a com, 'a bexp] => bool
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  "Valid p c q == !s s'. Sem c s s' --> s : p --> s' : q"
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nonterminals
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  vars
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syntax
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  ""		     :: "id => vars"		       ("_")
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  "_vars" 	     :: "[id, vars] => vars"	       ("_ _")
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syntax
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 "@hoare_vars" :: [vars, 'a assn,'a com,'a assn] => bool
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                 ("|- VARS _.// {_} // _ // {_}" [0,0,55,0] 50)
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syntax ("" output)
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 "@hoare"      :: ['a assn,'a com,'a assn] => bool
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                 ("|- {_} // _ // {_}" [0,55,0] 50)
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end
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ML
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(** parse translations **)
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fun mk_abstuple []     body = absfree ("x", dummyT, body)
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  | mk_abstuple [v]    body = absfree ((fst o dest_Free) v, dummyT, body)
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  | mk_abstuple (v::w) body = Syntax.const "split" $
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                              absfree ((fst o dest_Free) v, dummyT, mk_abstuple w body);
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fun mk_fbody v e []      = Syntax.const "()"
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  | mk_fbody v e [x]     = if v=x then e else x
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  | mk_fbody v e (x::xs) = Syntax.const "Pair" $ (if v=x then e else x) $
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                           mk_fbody v e xs;
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fun mk_fexp v e xs = mk_abstuple xs (mk_fbody v e xs);
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(* bexp_tr & assn_tr *)
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(*all meta-variables for bexp except for TRUE and FALSE are translated as if they
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  were boolean expressions*)
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fun bexp_tr (Const ("TRUE", _)) xs = Syntax.const "TRUE"
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  | bexp_tr b xs = Syntax.const "Collect" $ mk_abstuple xs b;
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fun assn_tr r xs = Syntax.const "Collect" $ mk_abstuple xs r;
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(* com_tr *)
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fun assign_tr [Free (V,_),E] xs = Syntax.const "Basic" $
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                                      mk_fexp (Free(V,dummyT)) E xs
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  | assign_tr ts _ = raise TERM ("assign_tr", ts);
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fun com_tr (Const("@assign",_) $ Free (V,_) $ E) xs =
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               assign_tr [Free (V,dummyT),E] xs
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  | com_tr (Const ("Basic",_) $ f) xs = Syntax.const "Basic" $ f
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  | com_tr (Const ("Seq",_) $ c1 $ c2) xs = Syntax.const "Seq" $
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                                                 com_tr c1 xs $ com_tr c2 xs
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  | com_tr (Const ("Cond",_) $ b $ c1 $ c2) xs = Syntax.const "Cond" $
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                                  bexp_tr b xs $ com_tr c1 xs $ com_tr c2 xs
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  | com_tr (Const ("While",_) $ b $ I $ c) xs = Syntax.const "While" $
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                                         bexp_tr b xs $ assn_tr I xs $ com_tr c xs
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  | com_tr trm _ = trm;
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(* triple_tr *)
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fun vars_tr (x as Free _) = [x]
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  | vars_tr (Const ("_vars", _) $ (x as Free _) $ vars) = x :: vars_tr vars
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  | vars_tr t = raise TERM ("vars_tr", [t]);
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fun hoare_vars_tr [vars, pre, prg, post] =
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      let val xs = vars_tr vars
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      in Syntax.const "Valid" $
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           assn_tr pre xs $ com_tr prg xs $ assn_tr post xs
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      end
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  | hoare_vars_tr ts = raise TERM ("hoare_vars_tr", ts);
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val parse_translation = [("@hoare_vars", hoare_vars_tr)];
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(*****************************************************************************)
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(*** print translations ***)
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fun dest_abstuple (Const ("split",_) $ (Abs(v,_, body))) =
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                            subst_bound (Syntax.free v, dest_abstuple body)
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  | dest_abstuple (Abs(v,_, body)) = subst_bound (Syntax.free v, body)
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  | dest_abstuple trm = trm;
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fun abs2list (Const ("split",_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t
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  | abs2list (Abs(x,T,t)) = [Free (x, T)]
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  | abs2list _ = [];
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fun mk_ts (Const ("split",_) $ (Abs(x,_,t))) = mk_ts t
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  | mk_ts (Abs(x,_,t)) = mk_ts t
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  | mk_ts (Const ("Pair",_) $ a $ b) = a::(mk_ts b)
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  | mk_ts t = [t];
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fun mk_vts (Const ("split",_) $ (Abs(x,_,t))) = 
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           ((Syntax.free x)::(abs2list t), mk_ts t)
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  | mk_vts (Abs(x,_,t)) = ([Syntax.free x], [t])
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  | mk_vts t = raise Match;
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fun find_ch [] i xs = (false, (Syntax.free "not_ch",Syntax.free "not_ch" ))
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  | find_ch ((v,t)::vts) i xs = if t=(Bound i) then find_ch vts (i-1) xs
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              else (true, (v, subst_bounds (xs,t)));
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fun is_f (Const ("split",_) $ (Abs(x,_,t))) = true
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  | is_f (Abs(x,_,t)) = true
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  | is_f t = false;
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(* assn_tr' & bexp_tr'*)
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fun assn_tr' (Const ("Collect",_) $ T) = dest_abstuple T
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  | assn_tr' (Const ("op Int",_) $ (Const ("Collect",_) $ T1) $ 
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                                   (Const ("Collect",_) $ T2)) =  
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            Syntax.const "op Int" $ dest_abstuple T1 $ dest_abstuple T2
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  | assn_tr' t = t;
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fun bexp_tr' (Const ("Collect",_) $ T) = dest_abstuple T 
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  | bexp_tr' t = t;
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(*com_tr' *)
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fun mk_assign f =
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  let val (vs, ts) = mk_vts f;
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      val (ch, which) = find_ch (vs~~ts) ((length vs)-1) (rev vs)
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  in if ch then Syntax.const "@assign" $ fst(which) $ snd(which)
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     else Syntax.const "@skip" end;
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fun com_tr' (Const ("Basic",_) $ f) = if is_f f then mk_assign f
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                                           else Syntax.const "Basic" $ f
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  | com_tr' (Const ("Seq",_) $ c1 $ c2) = Syntax.const "Seq" $
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                                                 com_tr' c1 $ com_tr' c2
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  | com_tr' (Const ("Cond",_) $ b $ c1 $ c2) = Syntax.const "Cond" $
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                                           bexp_tr' b $ com_tr' c1 $ com_tr' c2
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  | com_tr' (Const ("While",_) $ b $ I $ c) = Syntax.const "While" $
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                                               bexp_tr' b $ assn_tr' I $ com_tr' c
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  | com_tr' t = t;
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fun spec_tr' [p, c, q] =
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  Syntax.const "@hoare" $ assn_tr' p $ com_tr' c $ assn_tr' q
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val print_translation = [("Valid", spec_tr')];