src/HOL/Hoare/Hoare.thy
author webertj
Mon Mar 07 19:30:53 2005 +0100 (2005-03-07)
changeset 15584 3478bb4f93ff
parent 15032 02aed07e01bf
child 16417 9bc16273c2d4
permissions -rw-r--r--
refute_params: default value itself=1 added (for type classes)
clasohm@1476
     1
(*  Title:      HOL/Hoare/Hoare.thy
nipkow@1335
     2
    ID:         $Id$
nipkow@5646
     3
    Author:     Leonor Prensa Nieto & Tobias Nipkow
nipkow@5646
     4
    Copyright   1998 TUM
nipkow@1335
     5
nipkow@1335
     6
Sugared semantic embedding of Hoare logic.
nipkow@5646
     7
Strictly speaking a shallow embedding (as implemented by Norbert Galm
nipkow@5646
     8
following Mike Gordon) would suffice. Maybe the datatype com comes in useful
nipkow@5646
     9
later.
nipkow@1335
    10
*)
nipkow@1335
    11
nipkow@13682
    12
theory Hoare  = Main
nipkow@13696
    13
files ("hoare.ML"):
nipkow@1335
    14
nipkow@1335
    15
types
nipkow@13682
    16
    'a bexp = "'a set"
nipkow@13682
    17
    'a assn = "'a set"
nipkow@5646
    18
nipkow@5646
    19
datatype
nipkow@13682
    20
 'a com = Basic "'a \<Rightarrow> 'a"         
nipkow@13682
    21
   | Seq "'a com" "'a com"               ("(_;/ _)"      [61,60] 60)
nipkow@13682
    22
   | Cond "'a bexp" "'a com" "'a com"    ("(1IF _/ THEN _ / ELSE _/ FI)"  [0,0,0] 61)
nipkow@13682
    23
   | While "'a bexp" "'a assn" "'a com"  ("(1WHILE _/ INV {_} //DO _ /OD)"  [0,0,0] 61)
nipkow@5646
    24
  
nipkow@5646
    25
syntax
nipkow@13682
    26
  "@assign"  :: "id => 'b => 'a com"        ("(2_ :=/ _)" [70,65] 61)
nipkow@13682
    27
  "@annskip" :: "'a com"                    ("SKIP")
nipkow@5646
    28
nipkow@5646
    29
translations
nipkow@5646
    30
            "SKIP" == "Basic id"
nipkow@5646
    31
nipkow@13682
    32
types 'a sem = "'a => 'a => bool"
nipkow@5646
    33
nipkow@13682
    34
consts iter :: "nat => 'a bexp => 'a sem => 'a sem"
nipkow@5646
    35
primrec
nipkow@5646
    36
"iter 0 b S = (%s s'. s ~: b & (s=s'))"
nipkow@5646
    37
"iter (Suc n) b S = (%s s'. s : b & (? s''. S s s'' & iter n b S s'' s'))"
nipkow@5646
    38
nipkow@13682
    39
consts Sem :: "'a com => 'a sem"
nipkow@5646
    40
primrec
nipkow@5646
    41
"Sem(Basic f) s s' = (s' = f s)"
nipkow@5646
    42
"Sem(c1;c2) s s' = (? s''. Sem c1 s s'' & Sem c2 s'' s')"
nipkow@5646
    43
"Sem(IF b THEN c1 ELSE c2 FI) s s' = ((s  : b --> Sem c1 s s') &
nipkow@5646
    44
                                      (s ~: b --> Sem c2 s s'))"
nipkow@5646
    45
"Sem(While b x c) s s' = (? n. iter n b (Sem c) s s')"
nipkow@5646
    46
nipkow@13682
    47
constdefs Valid :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool"
nipkow@5646
    48
  "Valid p c q == !s s'. Sem c s s' --> s : p --> s' : q"
wenzelm@5007
    49
wenzelm@5007
    50
nipkow@1335
    51
syntax
nipkow@13764
    52
 "@hoare_vars" :: "[idts, 'a assn,'a com,'a assn] => bool"
nipkow@13737
    53
                 ("VARS _// {_} // _ // {_}" [0,0,55,0] 50)
nipkow@5646
    54
syntax ("" output)
nipkow@13682
    55
 "@hoare"      :: "['a assn,'a com,'a assn] => bool"
nipkow@13737
    56
                 ("{_} // _ // {_}" [0,55,0] 50)
nipkow@1335
    57
nipkow@5646
    58
(** parse translations **)
nipkow@1335
    59
nipkow@13682
    60
ML{*
nipkow@13764
    61
nipkow@13764
    62
local
nipkow@13857
    63
nipkow@13764
    64
fun abs((a,T),body) =
nipkow@13764
    65
  let val a = absfree(a, dummyT, body)
nipkow@13764
    66
  in if T = Bound 0 then a else Const(Syntax.constrainAbsC,dummyT) $ a $ T end
nipkow@13764
    67
in
nipkow@1335
    68
nipkow@13764
    69
fun mk_abstuple [x] body = abs (x, body)
nipkow@13764
    70
  | mk_abstuple (x::xs) body =
nipkow@13764
    71
      Syntax.const "split" $ abs (x, mk_abstuple xs body);
nipkow@1335
    72
nipkow@13857
    73
fun mk_fbody a e [x as (b,_)] = if a=b then e else Syntax.free b
nipkow@13764
    74
  | mk_fbody a e ((b,_)::xs) =
nipkow@13857
    75
      Syntax.const "Pair" $ (if a=b then e else Syntax.free b) $ mk_fbody a e xs;
nipkow@13764
    76
nipkow@13764
    77
fun mk_fexp a e xs = mk_abstuple xs (mk_fbody a e xs)
nipkow@13764
    78
end
nipkow@13682
    79
*}
nipkow@1335
    80
nipkow@5646
    81
(* bexp_tr & assn_tr *)
prensani@9397
    82
(*all meta-variables for bexp except for TRUE are translated as if they
nipkow@5646
    83
  were boolean expressions*)
nipkow@13682
    84
ML{*
nipkow@5646
    85
fun bexp_tr (Const ("TRUE", _)) xs = Syntax.const "TRUE"
nipkow@5646
    86
  | bexp_tr b xs = Syntax.const "Collect" $ mk_abstuple xs b;
nipkow@5646
    87
  
nipkow@5646
    88
fun assn_tr r xs = Syntax.const "Collect" $ mk_abstuple xs r;
nipkow@13682
    89
*}
nipkow@5646
    90
(* com_tr *)
nipkow@13682
    91
ML{*
nipkow@13764
    92
fun com_tr (Const("@assign",_) $ Free (a,_) $ e) xs =
nipkow@13764
    93
      Syntax.const "Basic" $ mk_fexp a e xs
nipkow@5646
    94
  | com_tr (Const ("Basic",_) $ f) xs = Syntax.const "Basic" $ f
nipkow@13764
    95
  | com_tr (Const ("Seq",_) $ c1 $ c2) xs =
nipkow@13764
    96
      Syntax.const "Seq" $ com_tr c1 xs $ com_tr c2 xs
nipkow@13764
    97
  | com_tr (Const ("Cond",_) $ b $ c1 $ c2) xs =
nipkow@13764
    98
      Syntax.const "Cond" $ bexp_tr b xs $ com_tr c1 xs $ com_tr c2 xs
nipkow@13764
    99
  | com_tr (Const ("While",_) $ b $ I $ c) xs =
nipkow@13764
   100
      Syntax.const "While" $ bexp_tr b xs $ assn_tr I xs $ com_tr c xs
nipkow@13764
   101
  | com_tr t _ = t (* if t is just a Free/Var *)
nipkow@13682
   102
*}
nipkow@5646
   103
nipkow@5646
   104
(* triple_tr *)
nipkow@13682
   105
ML{*
nipkow@13764
   106
local
nipkow@13764
   107
nipkow@13764
   108
fun var_tr(Free(a,_)) = (a,Bound 0) (* Bound 0 = dummy term *)
nipkow@13764
   109
  | var_tr(Const ("_constrain", _) $ (Free (a,_)) $ T) = (a,T);
nipkow@5646
   110
nipkow@13764
   111
fun vars_tr (Const ("_idts", _) $ idt $ vars) = var_tr idt :: vars_tr vars
nipkow@13764
   112
  | vars_tr t = [var_tr t]
nipkow@13764
   113
nipkow@13764
   114
in
nipkow@5646
   115
fun hoare_vars_tr [vars, pre, prg, post] =
nipkow@5646
   116
      let val xs = vars_tr vars
nipkow@5646
   117
      in Syntax.const "Valid" $
nipkow@13764
   118
         assn_tr pre xs $ com_tr prg xs $ assn_tr post xs
nipkow@5646
   119
      end
nipkow@5646
   120
  | hoare_vars_tr ts = raise TERM ("hoare_vars_tr", ts);
nipkow@13764
   121
end
nipkow@13682
   122
*}
nipkow@5646
   123
nipkow@13682
   124
parse_translation {* [("@hoare_vars", hoare_vars_tr)] *}
nipkow@1335
   125
nipkow@1335
   126
nipkow@5646
   127
(*****************************************************************************)
nipkow@5646
   128
nipkow@5646
   129
(*** print translations ***)
nipkow@13682
   130
ML{*
nipkow@5646
   131
fun dest_abstuple (Const ("split",_) $ (Abs(v,_, body))) =
nipkow@5646
   132
                            subst_bound (Syntax.free v, dest_abstuple body)
nipkow@5646
   133
  | dest_abstuple (Abs(v,_, body)) = subst_bound (Syntax.free v, body)
nipkow@5646
   134
  | dest_abstuple trm = trm;
nipkow@1335
   135
nipkow@5646
   136
fun abs2list (Const ("split",_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t
nipkow@5646
   137
  | abs2list (Abs(x,T,t)) = [Free (x, T)]
nipkow@5646
   138
  | abs2list _ = [];
nipkow@1335
   139
nipkow@5646
   140
fun mk_ts (Const ("split",_) $ (Abs(x,_,t))) = mk_ts t
nipkow@5646
   141
  | mk_ts (Abs(x,_,t)) = mk_ts t
nipkow@5646
   142
  | mk_ts (Const ("Pair",_) $ a $ b) = a::(mk_ts b)
nipkow@5646
   143
  | mk_ts t = [t];
nipkow@1335
   144
nipkow@5646
   145
fun mk_vts (Const ("split",_) $ (Abs(x,_,t))) = 
nipkow@5646
   146
           ((Syntax.free x)::(abs2list t), mk_ts t)
nipkow@5646
   147
  | mk_vts (Abs(x,_,t)) = ([Syntax.free x], [t])
nipkow@5646
   148
  | mk_vts t = raise Match;
nipkow@5646
   149
  
nipkow@5646
   150
fun find_ch [] i xs = (false, (Syntax.free "not_ch",Syntax.free "not_ch" ))
nipkow@5646
   151
  | find_ch ((v,t)::vts) i xs = if t=(Bound i) then find_ch vts (i-1) xs
nipkow@5646
   152
              else (true, (v, subst_bounds (xs,t)));
nipkow@5646
   153
  
nipkow@5646
   154
fun is_f (Const ("split",_) $ (Abs(x,_,t))) = true
nipkow@5646
   155
  | is_f (Abs(x,_,t)) = true
nipkow@5646
   156
  | is_f t = false;
nipkow@13682
   157
*}
nipkow@13682
   158
nipkow@5646
   159
(* assn_tr' & bexp_tr'*)
nipkow@13682
   160
ML{*  
nipkow@5646
   161
fun assn_tr' (Const ("Collect",_) $ T) = dest_abstuple T
nipkow@5646
   162
  | assn_tr' (Const ("op Int",_) $ (Const ("Collect",_) $ T1) $ 
nipkow@5646
   163
                                   (Const ("Collect",_) $ T2)) =  
nipkow@5646
   164
            Syntax.const "op Int" $ dest_abstuple T1 $ dest_abstuple T2
nipkow@5646
   165
  | assn_tr' t = t;
nipkow@1335
   166
nipkow@5646
   167
fun bexp_tr' (Const ("Collect",_) $ T) = dest_abstuple T 
nipkow@5646
   168
  | bexp_tr' t = t;
nipkow@13682
   169
*}
nipkow@5646
   170
nipkow@5646
   171
(*com_tr' *)
nipkow@13682
   172
ML{*
nipkow@5646
   173
fun mk_assign f =
nipkow@5646
   174
  let val (vs, ts) = mk_vts f;
nipkow@5646
   175
      val (ch, which) = find_ch (vs~~ts) ((length vs)-1) (rev vs)
nipkow@5646
   176
  in if ch then Syntax.const "@assign" $ fst(which) $ snd(which)
nipkow@5646
   177
     else Syntax.const "@skip" end;
nipkow@1335
   178
nipkow@5646
   179
fun com_tr' (Const ("Basic",_) $ f) = if is_f f then mk_assign f
nipkow@5646
   180
                                           else Syntax.const "Basic" $ f
nipkow@5646
   181
  | com_tr' (Const ("Seq",_) $ c1 $ c2) = Syntax.const "Seq" $
nipkow@5646
   182
                                                 com_tr' c1 $ com_tr' c2
nipkow@5646
   183
  | com_tr' (Const ("Cond",_) $ b $ c1 $ c2) = Syntax.const "Cond" $
nipkow@5646
   184
                                           bexp_tr' b $ com_tr' c1 $ com_tr' c2
nipkow@5646
   185
  | com_tr' (Const ("While",_) $ b $ I $ c) = Syntax.const "While" $
nipkow@5646
   186
                                               bexp_tr' b $ assn_tr' I $ com_tr' c
nipkow@5646
   187
  | com_tr' t = t;
nipkow@1335
   188
nipkow@1335
   189
nipkow@5646
   190
fun spec_tr' [p, c, q] =
nipkow@5646
   191
  Syntax.const "@hoare" $ assn_tr' p $ com_tr' c $ assn_tr' q
nipkow@13682
   192
*}
nipkow@13682
   193
nipkow@13682
   194
print_translation {* [("Valid", spec_tr')] *}
nipkow@13682
   195
nipkow@13857
   196
lemma SkipRule: "p \<subseteq> q \<Longrightarrow> Valid p (Basic id) q"
nipkow@13857
   197
by (auto simp:Valid_def)
nipkow@13857
   198
nipkow@13857
   199
lemma BasicRule: "p \<subseteq> {s. f s \<in> q} \<Longrightarrow> Valid p (Basic f) q"
nipkow@13857
   200
by (auto simp:Valid_def)
nipkow@13857
   201
nipkow@13857
   202
lemma SeqRule: "Valid P c1 Q \<Longrightarrow> Valid Q c2 R \<Longrightarrow> Valid P (c1;c2) R"
nipkow@13857
   203
by (auto simp:Valid_def)
nipkow@13857
   204
nipkow@13857
   205
lemma CondRule:
nipkow@13857
   206
 "p \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> w) \<and> (s \<notin> b \<longrightarrow> s \<in> w')}
nipkow@13857
   207
  \<Longrightarrow> Valid w c1 q \<Longrightarrow> Valid w' c2 q \<Longrightarrow> Valid p (Cond b c1 c2) q"
nipkow@13857
   208
by (auto simp:Valid_def)
nipkow@13857
   209
nipkow@13857
   210
lemma iter_aux: "! s s'. Sem c s s' --> s : I & s : b --> s' : I ==>
nipkow@13857
   211
       (\<And>s s'. s : I \<Longrightarrow> iter n b (Sem c) s s' \<Longrightarrow> s' : I & s' ~: b)";
nipkow@13857
   212
apply(induct n)
nipkow@13857
   213
 apply clarsimp
nipkow@13857
   214
apply(simp (no_asm_use))
nipkow@13857
   215
apply blast
nipkow@13857
   216
done
nipkow@13857
   217
nipkow@13857
   218
lemma WhileRule:
nipkow@13857
   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"
nipkow@13857
   220
apply (clarsimp simp:Valid_def)
nipkow@13857
   221
apply(drule iter_aux)
nipkow@13857
   222
  prefer 2 apply assumption
nipkow@13857
   223
 apply blast
nipkow@13857
   224
apply blast
nipkow@13857
   225
done
nipkow@13857
   226
nipkow@13857
   227
nipkow@13696
   228
use "hoare.ML"
nipkow@13682
   229
nipkow@13682
   230
method_setup vcg = {*
nipkow@13682
   231
  Method.no_args
nipkow@13682
   232
    (Method.SIMPLE_METHOD' HEADGOAL (hoare_tac (K all_tac))) *}
nipkow@13682
   233
  "verification condition generator"
nipkow@13682
   234
nipkow@13682
   235
method_setup vcg_simp = {*
nipkow@13682
   236
  Method.ctxt_args (fn ctxt =>
nipkow@13682
   237
    Method.METHOD (fn facts => 
wenzelm@15032
   238
      hoare_tac (asm_full_simp_tac (local_simpset_of ctxt))1)) *}
nipkow@13682
   239
  "verification condition generator plus simplification"
nipkow@13682
   240
nipkow@13682
   241
end