src/HOL/Hoare/HoareAbort.thy
author nipkow
Tue Mar 11 15:04:24 2003 +0100 (2003-03-11)
changeset 13857 11d7c5a8dbb7
child 13875 12997e3ddd8d
permissions -rw-r--r--
*** empty log message ***
     1 (*  Title:      HOL/Hoare/HoareAbort.thy
     2     ID:         $Id$
     3     Author:     Leonor Prensa Nieto & Tobias Nipkow
     4     Copyright   2003 TUM
     5 
     6 Like Hoare.thy, but with an Abort statement for modelling run time errors.
     7 *)
     8 
     9 theory HoareAbort  = Main
    10 files ("hoareAbort.ML"):
    11 
    12 types
    13     'a bexp = "'a set"
    14     'a assn = "'a set"
    15 
    16 datatype
    17  'a com = Basic "'a \<Rightarrow> 'a"
    18    | Abort
    19    | Seq "'a com" "'a com"               ("(_;/ _)"      [61,60] 60)
    20    | Cond "'a bexp" "'a com" "'a com"    ("(1IF _/ THEN _ / ELSE _/ FI)"  [0,0,0] 61)
    21    | While "'a bexp" "'a assn" "'a com"  ("(1WHILE _/ INV {_} //DO _ /OD)"  [0,0,0] 61)
    22   
    23 syntax
    24   "@assign"  :: "id => 'b => 'a com"        ("(2_ :=/ _)" [70,65] 61)
    25   "@annskip" :: "'a com"                    ("SKIP")
    26 
    27 translations
    28             "SKIP" == "Basic id"
    29 
    30 types 'a sem = "'a option => 'a option => bool"
    31 
    32 consts iter :: "nat => 'a bexp => 'a sem => 'a sem"
    33 primrec
    34 "iter 0 b S = (%s s'. s ~: Some ` b & (s=s'))"
    35 "iter (Suc n) b S = (%s s'. s : Some ` b & (? s''. S s s'' & iter n b S s'' s'))"
    36 
    37 consts Sem :: "'a com => 'a sem"
    38 primrec
    39 "Sem(Basic f) s s' = (case s of None \<Rightarrow> s' = None | Some t \<Rightarrow> s' = Some(f t))"
    40 "Sem Abort s s' = (s' = None)"
    41 "Sem(c1;c2) s s' = (? s''. Sem c1 s s'' & Sem c2 s'' s')"
    42 "Sem(IF b THEN c1 ELSE c2 FI) s s' =
    43  (case s of None \<Rightarrow> s' = None
    44   | Some t \<Rightarrow> ((t : b --> Sem c1 s s') & (t ~: b --> Sem c2 s s')))"
    45 "Sem(While b x c) s s' =
    46  (if s = None then s' = None
    47   else EX n. iter n b (Sem c) s s')"
    48 
    49 constdefs Valid :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool"
    50   "Valid p c q == \<forall>s s'. Sem c s s' \<longrightarrow> s : Some ` p \<longrightarrow> s' : Some ` q"
    51 
    52 
    53 syntax
    54  "@hoare_vars" :: "[idts, 'a assn,'a com,'a assn] => bool"
    55                  ("VARS _// {_} // _ // {_}" [0,0,55,0] 50)
    56 syntax ("" output)
    57  "@hoare"      :: "['a assn,'a com,'a assn] => bool"
    58                  ("{_} // _ // {_}" [0,55,0] 50)
    59 
    60 (** parse translations **)
    61 
    62 ML{*
    63 
    64 local
    65 fun free a = Free(a,dummyT)
    66 fun abs((a,T),body) =
    67   let val a = absfree(a, dummyT, body)
    68   in if T = Bound 0 then a else Const(Syntax.constrainAbsC,dummyT) $ a $ T end
    69 in
    70 
    71 fun mk_abstuple [x] body = abs (x, body)
    72   | mk_abstuple (x::xs) body =
    73       Syntax.const "split" $ abs (x, mk_abstuple xs body);
    74 
    75 fun mk_fbody a e [x as (b,_)] = if a=b then e else free b
    76   | mk_fbody a e ((b,_)::xs) =
    77       Syntax.const "Pair" $ (if a=b then e else free b) $ mk_fbody a e xs;
    78 
    79 fun mk_fexp a e xs = mk_abstuple xs (mk_fbody a e xs)
    80 end
    81 *}
    82 
    83 (* bexp_tr & assn_tr *)
    84 (*all meta-variables for bexp except for TRUE are translated as if they
    85   were boolean expressions*)
    86 ML{*
    87 fun bexp_tr (Const ("TRUE", _)) xs = Syntax.const "TRUE"
    88   | bexp_tr b xs = Syntax.const "Collect" $ mk_abstuple xs b;
    89   
    90 fun assn_tr r xs = Syntax.const "Collect" $ mk_abstuple xs r;
    91 *}
    92 (* com_tr *)
    93 ML{*
    94 fun com_tr (Const("@assign",_) $ Free (a,_) $ e) xs =
    95       Syntax.const "Basic" $ mk_fexp a e xs
    96   | com_tr (Const ("Basic",_) $ f) xs = Syntax.const "Basic" $ f
    97   | com_tr (Const ("Seq",_) $ c1 $ c2) xs =
    98       Syntax.const "Seq" $ com_tr c1 xs $ com_tr c2 xs
    99   | com_tr (Const ("Cond",_) $ b $ c1 $ c2) xs =
   100       Syntax.const "Cond" $ bexp_tr b xs $ com_tr c1 xs $ com_tr c2 xs
   101   | com_tr (Const ("While",_) $ b $ I $ c) xs =
   102       Syntax.const "While" $ bexp_tr b xs $ assn_tr I xs $ com_tr c xs
   103   | com_tr t _ = t (* if t is just a Free/Var *)
   104 *}
   105 
   106 (* triple_tr *)
   107 ML{*
   108 local
   109 
   110 fun var_tr(Free(a,_)) = (a,Bound 0) (* Bound 0 = dummy term *)
   111   | var_tr(Const ("_constrain", _) $ (Free (a,_)) $ T) = (a,T);
   112 
   113 fun vars_tr (Const ("_idts", _) $ idt $ vars) = var_tr idt :: vars_tr vars
   114   | vars_tr t = [var_tr t]
   115 
   116 in
   117 fun hoare_vars_tr [vars, pre, prg, post] =
   118       let val xs = vars_tr vars
   119       in Syntax.const "Valid" $
   120          assn_tr pre xs $ com_tr prg xs $ assn_tr post xs
   121       end
   122   | hoare_vars_tr ts = raise TERM ("hoare_vars_tr", ts);
   123 end
   124 *}
   125 
   126 parse_translation {* [("@hoare_vars", hoare_vars_tr)] *}
   127 
   128 
   129 (*****************************************************************************)
   130 
   131 (*** print translations ***)
   132 ML{*
   133 fun dest_abstuple (Const ("split",_) $ (Abs(v,_, body))) =
   134                             subst_bound (Syntax.free v, dest_abstuple body)
   135   | dest_abstuple (Abs(v,_, body)) = subst_bound (Syntax.free v, body)
   136   | dest_abstuple trm = trm;
   137 
   138 fun abs2list (Const ("split",_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t
   139   | abs2list (Abs(x,T,t)) = [Free (x, T)]
   140   | abs2list _ = [];
   141 
   142 fun mk_ts (Const ("split",_) $ (Abs(x,_,t))) = mk_ts t
   143   | mk_ts (Abs(x,_,t)) = mk_ts t
   144   | mk_ts (Const ("Pair",_) $ a $ b) = a::(mk_ts b)
   145   | mk_ts t = [t];
   146 
   147 fun mk_vts (Const ("split",_) $ (Abs(x,_,t))) = 
   148            ((Syntax.free x)::(abs2list t), mk_ts t)
   149   | mk_vts (Abs(x,_,t)) = ([Syntax.free x], [t])
   150   | mk_vts t = raise Match;
   151   
   152 fun find_ch [] i xs = (false, (Syntax.free "not_ch",Syntax.free "not_ch" ))
   153   | find_ch ((v,t)::vts) i xs = if t=(Bound i) then find_ch vts (i-1) xs
   154               else (true, (v, subst_bounds (xs,t)));
   155   
   156 fun is_f (Const ("split",_) $ (Abs(x,_,t))) = true
   157   | is_f (Abs(x,_,t)) = true
   158   | is_f t = false;
   159 *}
   160 
   161 (* assn_tr' & bexp_tr'*)
   162 ML{*  
   163 fun assn_tr' (Const ("Collect",_) $ T) = dest_abstuple T
   164   | assn_tr' (Const ("op Int",_) $ (Const ("Collect",_) $ T1) $ 
   165                                    (Const ("Collect",_) $ T2)) =  
   166             Syntax.const "op Int" $ dest_abstuple T1 $ dest_abstuple T2
   167   | assn_tr' t = t;
   168 
   169 fun bexp_tr' (Const ("Collect",_) $ T) = dest_abstuple T 
   170   | bexp_tr' t = t;
   171 *}
   172 
   173 (*com_tr' *)
   174 ML{*
   175 fun mk_assign f =
   176   let val (vs, ts) = mk_vts f;
   177       val (ch, which) = find_ch (vs~~ts) ((length vs)-1) (rev vs)
   178   in if ch then Syntax.const "@assign" $ fst(which) $ snd(which)
   179      else Syntax.const "@skip" end;
   180 
   181 fun com_tr' (Const ("Basic",_) $ f) = if is_f f then mk_assign f
   182                                            else Syntax.const "Basic" $ f
   183   | com_tr' (Const ("Seq",_) $ c1 $ c2) = Syntax.const "Seq" $
   184                                                  com_tr' c1 $ com_tr' c2
   185   | com_tr' (Const ("Cond",_) $ b $ c1 $ c2) = Syntax.const "Cond" $
   186                                            bexp_tr' b $ com_tr' c1 $ com_tr' c2
   187   | com_tr' (Const ("While",_) $ b $ I $ c) = Syntax.const "While" $
   188                                                bexp_tr' b $ assn_tr' I $ com_tr' c
   189   | com_tr' t = t;
   190 
   191 
   192 fun spec_tr' [p, c, q] =
   193   Syntax.const "@hoare" $ assn_tr' p $ com_tr' c $ assn_tr' q
   194 *}
   195 
   196 print_translation {* [("Valid", spec_tr')] *}
   197 
   198 (*** The proof rules ***)
   199 
   200 lemma SkipRule: "p \<subseteq> q \<Longrightarrow> Valid p (Basic id) q"
   201 by (auto simp:Valid_def)
   202 
   203 lemma BasicRule: "p \<subseteq> {s. f s \<in> q} \<Longrightarrow> Valid p (Basic f) q"
   204 by (auto simp:Valid_def)
   205 
   206 lemma SeqRule: "Valid P c1 Q \<Longrightarrow> Valid Q c2 R \<Longrightarrow> Valid P (c1;c2) R"
   207 by (auto simp:Valid_def)
   208 
   209 lemma CondRule:
   210  "p \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> w) \<and> (s \<notin> b \<longrightarrow> s \<in> w')}
   211   \<Longrightarrow> Valid w c1 q \<Longrightarrow> Valid w' c2 q \<Longrightarrow> Valid p (Cond b c1 c2) q"
   212 by (fastsimp simp:Valid_def image_def)
   213 
   214 lemma iter_aux: "! s s'. Sem c s s' --> s : Some ` (I \<inter> b) --> s' : Some ` I ==>
   215        (\<And>s s'. s : Some ` I \<Longrightarrow> iter n b (Sem c) s s' \<Longrightarrow> s' : Some ` (I \<inter> -b))";
   216 apply(unfold image_def)
   217 apply(induct n)
   218  apply clarsimp
   219 apply(simp (no_asm_use))
   220 apply blast
   221 done
   222 
   223 lemma WhileRule:
   224  "p \<subseteq> i \<Longrightarrow> Valid (i \<inter> b) c i \<Longrightarrow> i \<inter> (-b) \<subseteq> q \<Longrightarrow> Valid p (While b i c) q"
   225 apply(simp add:Valid_def)
   226 apply(simp (no_asm) add:image_def)
   227 apply clarify
   228 apply(drule iter_aux)
   229   prefer 2 apply assumption
   230  apply blast
   231 apply blast
   232 done
   233 
   234 lemma AbortRule: "p \<subseteq> {s. False} \<Longrightarrow> Valid p Abort q"
   235 by(auto simp:Valid_def)
   236 
   237 use "hoareAbort.ML"
   238 
   239 method_setup vcg = {*
   240   Method.no_args
   241     (Method.SIMPLE_METHOD' HEADGOAL (hoare_tac (K all_tac))) *}
   242   "verification condition generator"
   243 
   244 method_setup vcg_simp = {*
   245   Method.ctxt_args (fn ctxt =>
   246     Method.METHOD (fn facts => 
   247       hoare_tac (asm_full_simp_tac (Simplifier.get_local_simpset ctxt))1)) *}
   248   "verification condition generator plus simplification"
   249 
   250 end