src/HOL/Hoare/Hoare.thy
author wenzelm
Wed Aug 29 11:10:28 2007 +0200 (2007-08-29)
changeset 24470 41c81e23c08d
parent 21588 cd0dc678a205
child 24472 943ef707396c
permissions -rw-r--r--
removed Hoare/hoare.ML, Hoare/hoareAbort.ML, ex/svc_oracle.ML (which can be mistaken as attached ML script on case-insensitive file-system);
     1 (*  Title:      HOL/Hoare/Hoare.thy
     2     ID:         $Id$
     3     Author:     Leonor Prensa Nieto & Tobias Nipkow
     4     Copyright   1998 TUM
     5 
     6 Sugared semantic embedding of Hoare logic.
     7 Strictly speaking a shallow embedding (as implemented by Norbert Galm
     8 following Mike Gordon) would suffice. Maybe the datatype com comes in useful
     9 later.
    10 *)
    11 
    12 theory Hoare  imports Main
    13 begin
    14 
    15 types
    16     'a bexp = "'a set"
    17     'a assn = "'a set"
    18 
    19 datatype
    20  'a com = Basic "'a \<Rightarrow> 'a"         
    21    | Seq "'a com" "'a com"               ("(_;/ _)"      [61,60] 60)
    22    | Cond "'a bexp" "'a com" "'a com"    ("(1IF _/ THEN _ / ELSE _/ FI)"  [0,0,0] 61)
    23    | While "'a bexp" "'a assn" "'a com"  ("(1WHILE _/ INV {_} //DO _ /OD)"  [0,0,0] 61)
    24   
    25 syntax
    26   "@assign"  :: "id => 'b => 'a com"        ("(2_ :=/ _)" [70,65] 61)
    27   "@annskip" :: "'a com"                    ("SKIP")
    28 
    29 translations
    30             "SKIP" == "Basic id"
    31 
    32 types 'a sem = "'a => 'a => bool"
    33 
    34 consts iter :: "nat => 'a bexp => 'a sem => 'a sem"
    35 primrec
    36 "iter 0 b S = (%s s'. s ~: b & (s=s'))"
    37 "iter (Suc n) b S = (%s s'. s : b & (? s''. S s s'' & iter n b S s'' s'))"
    38 
    39 consts Sem :: "'a com => 'a sem"
    40 primrec
    41 "Sem(Basic f) s s' = (s' = f s)"
    42 "Sem(c1;c2) s s' = (? s''. Sem c1 s s'' & Sem c2 s'' s')"
    43 "Sem(IF b THEN c1 ELSE c2 FI) s s' = ((s  : b --> Sem c1 s s') &
    44                                       (s ~: b --> Sem c2 s s'))"
    45 "Sem(While b x c) s s' = (? n. iter n b (Sem c) s s')"
    46 
    47 constdefs Valid :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool"
    48   "Valid p c q == !s s'. Sem c s s' --> s : p --> s' : q"
    49 
    50 
    51 syntax
    52  "@hoare_vars" :: "[idts, 'a assn,'a com,'a assn] => bool"
    53                  ("VARS _// {_} // _ // {_}" [0,0,55,0] 50)
    54 syntax ("" output)
    55  "@hoare"      :: "['a assn,'a com,'a assn] => bool"
    56                  ("{_} // _ // {_}" [0,55,0] 50)
    57 
    58 (** parse translations **)
    59 
    60 ML{*
    61 
    62 local
    63 
    64 fun abs((a,T),body) =
    65   let val a = absfree(a, dummyT, body)
    66   in if T = Bound 0 then a else Const(Syntax.constrainAbsC,dummyT) $ a $ T end
    67 in
    68 
    69 fun mk_abstuple [x] body = abs (x, body)
    70   | mk_abstuple (x::xs) body =
    71       Syntax.const "split" $ abs (x, mk_abstuple xs body);
    72 
    73 fun mk_fbody a e [x as (b,_)] = if a=b then e else Syntax.free b
    74   | mk_fbody a e ((b,_)::xs) =
    75       Syntax.const "Pair" $ (if a=b then e else Syntax.free b) $ mk_fbody a e xs;
    76 
    77 fun mk_fexp a e xs = mk_abstuple xs (mk_fbody a e xs)
    78 end
    79 *}
    80 
    81 (* bexp_tr & assn_tr *)
    82 (*all meta-variables for bexp except for TRUE are translated as if they
    83   were boolean expressions*)
    84 ML{*
    85 fun bexp_tr (Const ("TRUE", _)) xs = Syntax.const "TRUE"
    86   | bexp_tr b xs = Syntax.const "Collect" $ mk_abstuple xs b;
    87   
    88 fun assn_tr r xs = Syntax.const "Collect" $ mk_abstuple xs r;
    89 *}
    90 (* com_tr *)
    91 ML{*
    92 fun com_tr (Const("@assign",_) $ Free (a,_) $ e) xs =
    93       Syntax.const "Basic" $ mk_fexp a e xs
    94   | com_tr (Const ("Basic",_) $ f) xs = Syntax.const "Basic" $ f
    95   | com_tr (Const ("Seq",_) $ c1 $ c2) xs =
    96       Syntax.const "Seq" $ com_tr c1 xs $ com_tr c2 xs
    97   | com_tr (Const ("Cond",_) $ b $ c1 $ c2) xs =
    98       Syntax.const "Cond" $ bexp_tr b xs $ com_tr c1 xs $ com_tr c2 xs
    99   | com_tr (Const ("While",_) $ b $ I $ c) xs =
   100       Syntax.const "While" $ bexp_tr b xs $ assn_tr I xs $ com_tr c xs
   101   | com_tr t _ = t (* if t is just a Free/Var *)
   102 *}
   103 
   104 (* triple_tr *)    (* FIXME does not handle "_idtdummy" *)
   105 ML{*
   106 local
   107 
   108 fun var_tr(Free(a,_)) = (a,Bound 0) (* Bound 0 = dummy term *)
   109   | var_tr(Const ("_constrain", _) $ (Free (a,_)) $ T) = (a,T);
   110 
   111 fun vars_tr (Const ("_idts", _) $ idt $ vars) = var_tr idt :: vars_tr vars
   112   | vars_tr t = [var_tr t]
   113 
   114 in
   115 fun hoare_vars_tr [vars, pre, prg, post] =
   116       let val xs = vars_tr vars
   117       in Syntax.const "Valid" $
   118          assn_tr pre xs $ com_tr prg xs $ assn_tr post xs
   119       end
   120   | hoare_vars_tr ts = raise TERM ("hoare_vars_tr", ts);
   121 end
   122 *}
   123 
   124 parse_translation {* [("@hoare_vars", hoare_vars_tr)] *}
   125 
   126 
   127 (*****************************************************************************)
   128 
   129 (*** print translations ***)
   130 ML{*
   131 fun dest_abstuple (Const ("split",_) $ (Abs(v,_, body))) =
   132                             subst_bound (Syntax.free v, dest_abstuple body)
   133   | dest_abstuple (Abs(v,_, body)) = subst_bound (Syntax.free v, body)
   134   | dest_abstuple trm = trm;
   135 
   136 fun abs2list (Const ("split",_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t
   137   | abs2list (Abs(x,T,t)) = [Free (x, T)]
   138   | abs2list _ = [];
   139 
   140 fun mk_ts (Const ("split",_) $ (Abs(x,_,t))) = mk_ts t
   141   | mk_ts (Abs(x,_,t)) = mk_ts t
   142   | mk_ts (Const ("Pair",_) $ a $ b) = a::(mk_ts b)
   143   | mk_ts t = [t];
   144 
   145 fun mk_vts (Const ("split",_) $ (Abs(x,_,t))) = 
   146            ((Syntax.free x)::(abs2list t), mk_ts t)
   147   | mk_vts (Abs(x,_,t)) = ([Syntax.free x], [t])
   148   | mk_vts t = raise Match;
   149   
   150 fun find_ch [] i xs = (false, (Syntax.free "not_ch",Syntax.free "not_ch" ))
   151   | find_ch ((v,t)::vts) i xs = if t=(Bound i) then find_ch vts (i-1) xs
   152               else (true, (v, subst_bounds (xs,t)));
   153   
   154 fun is_f (Const ("split",_) $ (Abs(x,_,t))) = true
   155   | is_f (Abs(x,_,t)) = true
   156   | is_f t = false;
   157 *}
   158 
   159 (* assn_tr' & bexp_tr'*)
   160 ML{*  
   161 fun assn_tr' (Const ("Collect",_) $ T) = dest_abstuple T
   162   | assn_tr' (Const ("op Int",_) $ (Const ("Collect",_) $ T1) $ 
   163                                    (Const ("Collect",_) $ T2)) =  
   164             Syntax.const "op Int" $ dest_abstuple T1 $ dest_abstuple T2
   165   | assn_tr' t = t;
   166 
   167 fun bexp_tr' (Const ("Collect",_) $ T) = dest_abstuple T 
   168   | bexp_tr' t = t;
   169 *}
   170 
   171 (*com_tr' *)
   172 ML{*
   173 fun mk_assign f =
   174   let val (vs, ts) = mk_vts f;
   175       val (ch, which) = find_ch (vs~~ts) ((length vs)-1) (rev vs)
   176   in if ch then Syntax.const "@assign" $ fst(which) $ snd(which)
   177      else Syntax.const "@skip" end;
   178 
   179 fun com_tr' (Const ("Basic",_) $ f) = if is_f f then mk_assign f
   180                                            else Syntax.const "Basic" $ f
   181   | com_tr' (Const ("Seq",_) $ c1 $ c2) = Syntax.const "Seq" $
   182                                                  com_tr' c1 $ com_tr' c2
   183   | com_tr' (Const ("Cond",_) $ b $ c1 $ c2) = Syntax.const "Cond" $
   184                                            bexp_tr' b $ com_tr' c1 $ com_tr' c2
   185   | com_tr' (Const ("While",_) $ b $ I $ c) = Syntax.const "While" $
   186                                                bexp_tr' b $ assn_tr' I $ com_tr' c
   187   | com_tr' t = t;
   188 
   189 
   190 fun spec_tr' [p, c, q] =
   191   Syntax.const "@hoare" $ assn_tr' p $ com_tr' c $ assn_tr' q
   192 *}
   193 
   194 print_translation {* [("Valid", spec_tr')] *}
   195 
   196 lemma SkipRule: "p \<subseteq> q \<Longrightarrow> Valid p (Basic id) q"
   197 by (auto simp:Valid_def)
   198 
   199 lemma BasicRule: "p \<subseteq> {s. f s \<in> q} \<Longrightarrow> Valid p (Basic f) q"
   200 by (auto simp:Valid_def)
   201 
   202 lemma SeqRule: "Valid P c1 Q \<Longrightarrow> Valid Q c2 R \<Longrightarrow> Valid P (c1;c2) R"
   203 by (auto simp:Valid_def)
   204 
   205 lemma CondRule:
   206  "p \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> w) \<and> (s \<notin> b \<longrightarrow> s \<in> w')}
   207   \<Longrightarrow> Valid w c1 q \<Longrightarrow> Valid w' c2 q \<Longrightarrow> Valid p (Cond b c1 c2) q"
   208 by (auto simp:Valid_def)
   209 
   210 lemma iter_aux: "! s s'. Sem c s s' --> s : I & s : b --> s' : I ==>
   211        (\<And>s s'. s : I \<Longrightarrow> iter n b (Sem c) s s' \<Longrightarrow> s' : I & s' ~: b)";
   212 apply(induct n)
   213  apply clarsimp
   214 apply(simp (no_asm_use))
   215 apply blast
   216 done
   217 
   218 lemma WhileRule:
   219  "p \<subseteq> i \<Longrightarrow> Valid (i \<inter> b) c i \<Longrightarrow> i \<inter> (-b) \<subseteq> q \<Longrightarrow> Valid p (While b i c) q"
   220 apply (clarsimp simp:Valid_def)
   221 apply(drule iter_aux)
   222   prefer 2 apply assumption
   223  apply blast
   224 apply blast
   225 done
   226 
   227 
   228 subsection {* Derivation of the proof rules and, most importantly, the VCG tactic *}
   229 
   230 ML {*
   231 (*** The tactics ***)
   232 
   233 (*****************************************************************************)
   234 (** The function Mset makes the theorem                                     **)
   235 (** "?Mset <= {(x1,...,xn). ?P (x1,...,xn)} ==> ?Mset <= {s. ?P s}",        **)
   236 (** where (x1,...,xn) are the variables of the particular program we are    **)
   237 (** working on at the moment of the call                                    **)
   238 (*****************************************************************************)
   239 
   240 local open HOLogic in
   241 
   242 (** maps (%x1 ... xn. t) to [x1,...,xn] **)
   243 fun abs2list (Const ("split",_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t
   244   | abs2list (Abs(x,T,t)) = [Free (x, T)]
   245   | abs2list _ = [];
   246 
   247 (** maps {(x1,...,xn). t} to [x1,...,xn] **)
   248 fun mk_vars (Const ("Collect",_) $ T) = abs2list T
   249   | mk_vars _ = [];
   250 
   251 (** abstraction of body over a tuple formed from a list of free variables. 
   252 Types are also built **)
   253 fun mk_abstupleC []     body = absfree ("x", unitT, body)
   254   | mk_abstupleC (v::w) body = let val (n,T) = dest_Free v
   255                                in if w=[] then absfree (n, T, body)
   256         else let val z  = mk_abstupleC w body;
   257                  val T2 = case z of Abs(_,T,_) => T
   258                         | Const (_, Type (_,[_, Type (_,[T,_])])) $ _ => T;
   259        in Const ("split", (T --> T2 --> boolT) --> mk_prodT (T,T2) --> boolT) 
   260           $ absfree (n, T, z) end end;
   261 
   262 (** maps [x1,...,xn] to (x1,...,xn) and types**)
   263 fun mk_bodyC []      = HOLogic.unit
   264   | mk_bodyC (x::xs) = if xs=[] then x 
   265                else let val (n, T) = dest_Free x ;
   266                         val z = mk_bodyC xs;
   267                         val T2 = case z of Free(_, T) => T
   268                                          | Const ("Pair", Type ("fun", [_, Type
   269                                             ("fun", [_, T])])) $ _ $ _ => T;
   270                  in Const ("Pair", [T, T2] ---> mk_prodT (T, T2)) $ x $ z end;
   271 
   272 (** maps a goal of the form:
   273         1. [| P |] ==> VARS x1 ... xn {._.} _ {._.} or to [x1,...,xn]**) 
   274 fun get_vars thm = let  val c = Logic.unprotect (concl_of (thm));
   275                         val d = Logic.strip_assums_concl c;
   276                         val Const _ $ pre $ _ $ _ = dest_Trueprop d;
   277       in mk_vars pre end;
   278 
   279 
   280 (** Makes Collect with type **)
   281 fun mk_CollectC trm = let val T as Type ("fun",[t,_]) = fastype_of trm 
   282                       in Collect_const t $ trm end;
   283 
   284 fun inclt ty = Const (@{const_name HOL.less_eq}, [ty,ty] ---> boolT);
   285 
   286 (** Makes "Mset <= t" **)
   287 fun Mset_incl t = let val MsetT = fastype_of t 
   288                  in mk_Trueprop ((inclt MsetT) $ Free ("Mset", MsetT) $ t) end;
   289 
   290 
   291 fun Mset thm = let val vars = get_vars(thm);
   292                    val varsT = fastype_of (mk_bodyC vars);
   293                    val big_Collect = mk_CollectC (mk_abstupleC vars 
   294                          (Free ("P",varsT --> boolT) $ mk_bodyC vars));
   295                    val small_Collect = mk_CollectC (Abs("x",varsT,
   296                            Free ("P",varsT --> boolT) $ Bound 0));
   297                    val impl = implies $ (Mset_incl big_Collect) $ 
   298                                           (Mset_incl small_Collect);
   299    in Goal.prove (ProofContext.init (Thm.theory_of_thm thm)) ["Mset", "P"] [] impl (K (CLASET' blast_tac 1)) end;
   300 
   301 end;
   302 *}
   303 
   304 (*****************************************************************************)
   305 (** Simplifying:                                                            **)
   306 (** Some useful lemmata, lists and simplification tactics to control which  **)
   307 (** theorems are used to simplify at each moment, so that the original      **)
   308 (** input does not suffer any unexpected transformation                     **)
   309 (*****************************************************************************)
   310 
   311 lemma Compl_Collect: "-(Collect b) = {x. ~(b x)}"
   312   by blast
   313 
   314 
   315 ML {*
   316 (**Simp_tacs**)
   317 
   318 val before_set2pred_simp_tac =
   319   (simp_tac (HOL_basic_ss addsimps [@{thm Collect_conj_eq} RS sym, @{thm Compl_Collect}]));
   320 
   321 val split_simp_tac = (simp_tac (HOL_basic_ss addsimps [split_conv]));
   322 
   323 (*****************************************************************************)
   324 (** set2pred transforms sets inclusion into predicates implication,         **)
   325 (** maintaining the original variable names.                                **)
   326 (** Ex. "{x. x=0} <= {x. x <= 1}" -set2pred-> "x=0 --> x <= 1"              **)
   327 (** Subgoals containing intersections (A Int B) or complement sets (-A)     **)
   328 (** are first simplified by "before_set2pred_simp_tac", that returns only   **)
   329 (** subgoals of the form "{x. P x} <= {x. Q x}", which are easily           **)
   330 (** transformed.                                                            **)
   331 (** This transformation may solve very easy subgoals due to a ligth         **)
   332 (** simplification done by (split_all_tac)                                  **)
   333 (*****************************************************************************)
   334 
   335 fun set2pred i thm =
   336   let val var_names = map (fst o dest_Free) (get_vars thm) in
   337     ((before_set2pred_simp_tac i) THEN_MAYBE
   338      (EVERY [rtac subsetI i, 
   339              rtac CollectI i,
   340              dtac CollectD i,
   341              (TRY(split_all_tac i)) THEN_MAYBE
   342              ((rename_params_tac var_names i) THEN
   343               (full_simp_tac (HOL_basic_ss addsimps [split_conv]) i)) ])) thm
   344   end;
   345 
   346 (*****************************************************************************)
   347 (** BasicSimpTac is called to simplify all verification conditions. It does **)
   348 (** a light simplification by applying "mem_Collect_eq", then it calls      **)
   349 (** MaxSimpTac, which solves subgoals of the form "A <= A",                 **)
   350 (** and transforms any other into predicates, applying then                 **)
   351 (** the tactic chosen by the user, which may solve the subgoal completely.  **)
   352 (*****************************************************************************)
   353 
   354 fun MaxSimpTac tac = FIRST'[rtac subset_refl, set2pred THEN_MAYBE' tac];
   355 
   356 fun BasicSimpTac tac =
   357   simp_tac
   358     (HOL_basic_ss addsimps [mem_Collect_eq,split_conv] addsimprocs [record_simproc])
   359   THEN_MAYBE' MaxSimpTac tac;
   360 
   361 (** HoareRuleTac **)
   362 
   363 fun WlpTac Mlem tac i =
   364   rtac @{thm SeqRule} i THEN  HoareRuleTac Mlem tac false (i+1)
   365 and HoareRuleTac Mlem tac pre_cond i st = st |>
   366         (*abstraction over st prevents looping*)
   367     ( (WlpTac Mlem tac i THEN HoareRuleTac Mlem tac pre_cond i)
   368       ORELSE
   369       (FIRST[rtac @{thm SkipRule} i,
   370              EVERY[rtac @{thm BasicRule} i,
   371                    rtac Mlem i,
   372                    split_simp_tac i],
   373              EVERY[rtac @{thm CondRule} i,
   374                    HoareRuleTac Mlem tac false (i+2),
   375                    HoareRuleTac Mlem tac false (i+1)],
   376              EVERY[rtac @{thm WhileRule} i,
   377                    BasicSimpTac tac (i+2),
   378                    HoareRuleTac Mlem tac true (i+1)] ] 
   379        THEN (if pre_cond then (BasicSimpTac tac i) else (rtac subset_refl i)) ));
   380 
   381 
   382 (** tac:(int -> tactic) is the tactic the user chooses to solve or simplify **)
   383 (** the final verification conditions                                       **)
   384  
   385 fun hoare_tac tac i thm =
   386   let val Mlem = Mset(thm)
   387   in SELECT_GOAL(EVERY[HoareRuleTac Mlem tac true 1]) i thm end;
   388 *}
   389 
   390 method_setup vcg = {*
   391   Method.no_args (Method.SIMPLE_METHOD' (hoare_tac (K all_tac))) *}
   392   "verification condition generator"
   393 
   394 method_setup vcg_simp = {*
   395   Method.ctxt_args (fn ctxt =>
   396     Method.SIMPLE_METHOD' (hoare_tac (asm_full_simp_tac (local_simpset_of ctxt)))) *}
   397   "verification condition generator plus simplification"
   398 
   399 end