src/HOL/Hoare/Hoare_Logic.thy
author wenzelm
Mon Mar 22 20:58:52 2010 +0100 (2010-03-22)
changeset 35898 c890a3835d15
parent 35416 d8d7d1b785af
child 36643 f36588af1ba1
permissions -rw-r--r--
recovered header;
     1 (*  Title:      HOL/Hoare/Hoare.thy
     2     Author:     Leonor Prensa Nieto & Tobias Nipkow
     3     Copyright   1998 TUM
     4 
     5 Sugared semantic embedding of Hoare logic.
     6 Strictly speaking a shallow embedding (as implemented by Norbert Galm
     7 following Mike Gordon) would suffice. Maybe the datatype com comes in useful
     8 later.
     9 *)
    10 
    11 theory Hoare_Logic
    12 imports Main
    13 uses ("hoare_tac.ML")
    14 begin
    15 
    16 types
    17     'a bexp = "'a set"
    18     'a assn = "'a set"
    19 
    20 datatype
    21  'a com = Basic "'a \<Rightarrow> 'a"
    22    | Seq "'a com" "'a com"               ("(_;/ _)"      [61,60] 60)
    23    | Cond "'a bexp" "'a com" "'a com"    ("(1IF _/ THEN _ / ELSE _/ FI)"  [0,0,0] 61)
    24    | While "'a bexp" "'a assn" "'a com"  ("(1WHILE _/ INV {_} //DO _ /OD)"  [0,0,0] 61)
    25 
    26 abbreviation annskip ("SKIP") where "SKIP == Basic id"
    27 
    28 types 'a sem = "'a => 'a => bool"
    29 
    30 consts iter :: "nat => 'a bexp => 'a sem => 'a sem"
    31 primrec
    32 "iter 0 b S = (%s s'. s ~: b & (s=s'))"
    33 "iter (Suc n) b S = (%s s'. s : b & (? s''. S s s'' & iter n b S s'' s'))"
    34 
    35 consts Sem :: "'a com => 'a sem"
    36 primrec
    37 "Sem(Basic f) s s' = (s' = f s)"
    38 "Sem(c1;c2) s s' = (? s''. Sem c1 s s'' & Sem c2 s'' s')"
    39 "Sem(IF b THEN c1 ELSE c2 FI) s s' = ((s  : b --> Sem c1 s s') &
    40                                       (s ~: b --> Sem c2 s s'))"
    41 "Sem(While b x c) s s' = (? n. iter n b (Sem c) s s')"
    42 
    43 definition Valid :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool" where
    44   "Valid p c q == !s s'. Sem c s s' --> s : p --> s' : q"
    45 
    46 
    47 
    48 (** parse translations **)
    49 
    50 syntax
    51   "_assign"  :: "id => 'b => 'a com"        ("(2_ :=/ _)" [70,65] 61)
    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 ML {*
    60 
    61 local
    62 
    63 fun abs((a,T),body) =
    64   let val a = absfree(a, dummyT, body)
    65   in if T = Bound 0 then a else Const(Syntax.constrainAbsC,dummyT) $ a $ T end
    66 in
    67 
    68 fun mk_abstuple [x] body = abs (x, body)
    69   | mk_abstuple (x::xs) body =
    70       Syntax.const @{const_syntax split} $ abs (x, mk_abstuple xs body);
    71 
    72 fun mk_fbody a e [x as (b,_)] = if a=b then e else Syntax.free b
    73   | mk_fbody a e ((b,_)::xs) =
    74       Syntax.const @{const_syntax Pair} $ (if a=b then e else Syntax.free b) $ mk_fbody a e xs;
    75 
    76 fun mk_fexp a e xs = mk_abstuple xs (mk_fbody a e xs)
    77 end
    78 *}
    79 
    80 (* bexp_tr & assn_tr *)
    81 (*all meta-variables for bexp except for TRUE are translated as if they
    82   were boolean expressions*)
    83 ML{*
    84 fun bexp_tr (Const ("TRUE", _)) xs = Syntax.const "TRUE"   (* FIXME !? *)
    85   | bexp_tr b xs = Syntax.const @{const_syntax Collect} $ mk_abstuple xs b;
    86 
    87 fun assn_tr r xs = Syntax.const @{const_syntax Collect} $ mk_abstuple xs r;
    88 *}
    89 (* com_tr *)
    90 ML{*
    91 fun com_tr (Const(@{syntax_const "_assign"},_) $ Free (a,_) $ e) xs =
    92       Syntax.const @{const_syntax Basic} $ mk_fexp a e xs
    93   | com_tr (Const (@{const_syntax Basic},_) $ f) xs = Syntax.const @{const_syntax Basic} $ f
    94   | com_tr (Const (@{const_syntax Seq},_) $ c1 $ c2) xs =
    95       Syntax.const @{const_syntax Seq} $ com_tr c1 xs $ com_tr c2 xs
    96   | com_tr (Const (@{const_syntax Cond},_) $ b $ c1 $ c2) xs =
    97       Syntax.const @{const_syntax Cond} $ bexp_tr b xs $ com_tr c1 xs $ com_tr c2 xs
    98   | com_tr (Const (@{const_syntax While},_) $ b $ I $ c) xs =
    99       Syntax.const @{const_syntax While} $ bexp_tr b xs $ assn_tr I xs $ com_tr c xs
   100   | com_tr t _ = t (* if t is just a Free/Var *)
   101 *}
   102 
   103 (* triple_tr *)    (* FIXME does not handle "_idtdummy" *)
   104 ML{*
   105 local
   106 
   107 fun var_tr(Free(a,_)) = (a,Bound 0) (* Bound 0 = dummy term *)
   108   | var_tr(Const (@{syntax_const "_constrain"}, _) $ (Free (a,_)) $ T) = (a,T);
   109 
   110 fun vars_tr (Const (@{syntax_const "_idts"}, _) $ idt $ vars) = var_tr idt :: vars_tr vars
   111   | vars_tr t = [var_tr t]
   112 
   113 in
   114 fun hoare_vars_tr [vars, pre, prg, post] =
   115       let val xs = vars_tr vars
   116       in Syntax.const @{const_syntax Valid} $
   117          assn_tr pre xs $ com_tr prg xs $ assn_tr post xs
   118       end
   119   | hoare_vars_tr ts = raise TERM ("hoare_vars_tr", ts);
   120 end
   121 *}
   122 
   123 parse_translation {* [(@{syntax_const "_hoare_vars"}, hoare_vars_tr)] *}
   124 
   125 
   126 (*****************************************************************************)
   127 
   128 (*** print translations ***)
   129 ML{*
   130 fun dest_abstuple (Const (@{const_syntax split},_) $ (Abs(v,_, body))) =
   131                             subst_bound (Syntax.free v, dest_abstuple body)
   132   | dest_abstuple (Abs(v,_, body)) = subst_bound (Syntax.free v, body)
   133   | dest_abstuple trm = trm;
   134 
   135 fun abs2list (Const (@{const_syntax split},_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t
   136   | abs2list (Abs(x,T,t)) = [Free (x, T)]
   137   | abs2list _ = [];
   138 
   139 fun mk_ts (Const (@{const_syntax split},_) $ (Abs(x,_,t))) = mk_ts t
   140   | mk_ts (Abs(x,_,t)) = mk_ts t
   141   | mk_ts (Const (@{const_syntax Pair},_) $ a $ b) = a::(mk_ts b)
   142   | mk_ts t = [t];
   143 
   144 fun mk_vts (Const (@{const_syntax split},_) $ (Abs(x,_,t))) =
   145            ((Syntax.free x)::(abs2list t), mk_ts t)
   146   | mk_vts (Abs(x,_,t)) = ([Syntax.free x], [t])
   147   | mk_vts t = raise Match;
   148 
   149 fun find_ch [] i xs = (false, (Syntax.free "not_ch", Syntax.free "not_ch"))
   150   | find_ch ((v,t)::vts) i xs =
   151       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 (@{const_syntax 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 (@{const_syntax Collect},_) $ T) = dest_abstuple T
   162   | assn_tr' (Const (@{const_syntax inter}, _) $
   163         (Const (@{const_syntax Collect},_) $ T1) $ (Const (@{const_syntax Collect},_) $ T2)) =
   164       Syntax.const @{const_syntax inter} $ dest_abstuple T1 $ dest_abstuple T2
   165   | assn_tr' t = t;
   166 
   167 fun bexp_tr' (Const (@{const_syntax 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
   177     if ch then Syntax.const @{syntax_const "_assign"} $ fst which $ snd which
   178     else Syntax.const @{const_syntax annskip}
   179   end;
   180 
   181 fun com_tr' (Const (@{const_syntax Basic},_) $ f) =
   182       if is_f f then mk_assign f
   183       else Syntax.const @{const_syntax Basic} $ f
   184   | com_tr' (Const (@{const_syntax Seq},_) $ c1 $ c2) =
   185       Syntax.const @{const_syntax Seq} $ com_tr' c1 $ com_tr' c2
   186   | com_tr' (Const (@{const_syntax Cond},_) $ b $ c1 $ c2) =
   187       Syntax.const @{const_syntax Cond} $ bexp_tr' b $ com_tr' c1 $ com_tr' c2
   188   | com_tr' (Const (@{const_syntax While},_) $ b $ I $ c) =
   189       Syntax.const @{const_syntax While} $ bexp_tr' b $ assn_tr' I $ com_tr' c
   190   | com_tr' t = t;
   191 
   192 fun spec_tr' [p, c, q] =
   193   Syntax.const @{syntax_const "_hoare"} $ assn_tr' p $ com_tr' c $ assn_tr' q
   194 *}
   195 
   196 print_translation {* [(@{const_syntax Valid}, spec_tr')] *}
   197 
   198 lemma SkipRule: "p \<subseteq> q \<Longrightarrow> Valid p (Basic id) q"
   199 by (auto simp:Valid_def)
   200 
   201 lemma BasicRule: "p \<subseteq> {s. f s \<in> q} \<Longrightarrow> Valid p (Basic f) q"
   202 by (auto simp:Valid_def)
   203 
   204 lemma SeqRule: "Valid P c1 Q \<Longrightarrow> Valid Q c2 R \<Longrightarrow> Valid P (c1;c2) R"
   205 by (auto simp:Valid_def)
   206 
   207 lemma CondRule:
   208  "p \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> w) \<and> (s \<notin> b \<longrightarrow> s \<in> w')}
   209   \<Longrightarrow> Valid w c1 q \<Longrightarrow> Valid w' c2 q \<Longrightarrow> Valid p (Cond b c1 c2) q"
   210 by (auto simp:Valid_def)
   211 
   212 lemma iter_aux: "! s s'. Sem c s s' --> s : I & s : b --> s' : I ==>
   213        (\<And>s s'. s : I \<Longrightarrow> iter n b (Sem c) s s' \<Longrightarrow> s' : I & s' ~: b)";
   214 apply(induct n)
   215  apply clarsimp
   216 apply(simp (no_asm_use))
   217 apply blast
   218 done
   219 
   220 lemma WhileRule:
   221  "p \<subseteq> i \<Longrightarrow> Valid (i \<inter> b) c i \<Longrightarrow> i \<inter> (-b) \<subseteq> q \<Longrightarrow> Valid p (While b i c) q"
   222 apply (clarsimp simp:Valid_def)
   223 apply(drule iter_aux)
   224   prefer 2 apply assumption
   225  apply blast
   226 apply blast
   227 done
   228 
   229 
   230 lemma Compl_Collect: "-(Collect b) = {x. ~(b x)}"
   231   by blast
   232 
   233 lemmas AbortRule = SkipRule  -- "dummy version"
   234 use "hoare_tac.ML"
   235 
   236 method_setup vcg = {*
   237   Scan.succeed (fn ctxt => SIMPLE_METHOD' (hoare_tac ctxt (K all_tac))) *}
   238   "verification condition generator"
   239 
   240 method_setup vcg_simp = {*
   241   Scan.succeed (fn ctxt =>
   242     SIMPLE_METHOD' (hoare_tac ctxt (asm_full_simp_tac (simpset_of ctxt)))) *}
   243   "verification condition generator plus simplification"
   244 
   245 end