src/HOL/Hoare/Hoare_Logic.thy
author wenzelm
Wed Aug 11 18:41:06 2010 +0200 (2010-08-11)
changeset 38353 d98baa2cf589
parent 37591 d3daea901123
child 41959 b460124855b8
permissions -rw-r--r--
modernized specifications;
tuned headers;
clasohm@1476
     1
(*  Title:      HOL/Hoare/Hoare.thy
nipkow@5646
     2
    Author:     Leonor Prensa Nieto & Tobias Nipkow
nipkow@5646
     3
    Copyright   1998 TUM
nipkow@1335
     4
nipkow@1335
     5
Sugared semantic embedding of Hoare logic.
nipkow@5646
     6
Strictly speaking a shallow embedding (as implemented by Norbert Galm
nipkow@5646
     7
following Mike Gordon) would suffice. Maybe the datatype com comes in useful
nipkow@5646
     8
later.
nipkow@1335
     9
*)
nipkow@1335
    10
haftmann@35316
    11
theory Hoare_Logic
wenzelm@28457
    12
imports Main
wenzelm@24472
    13
uses ("hoare_tac.ML")
wenzelm@24470
    14
begin
nipkow@1335
    15
nipkow@1335
    16
types
nipkow@13682
    17
    'a bexp = "'a set"
nipkow@13682
    18
    'a assn = "'a set"
nipkow@5646
    19
nipkow@5646
    20
datatype
wenzelm@35113
    21
 'a com = Basic "'a \<Rightarrow> 'a"
nipkow@13682
    22
   | Seq "'a com" "'a com"               ("(_;/ _)"      [61,60] 60)
nipkow@13682
    23
   | Cond "'a bexp" "'a com" "'a com"    ("(1IF _/ THEN _ / ELSE _/ FI)"  [0,0,0] 61)
nipkow@13682
    24
   | While "'a bexp" "'a assn" "'a com"  ("(1WHILE _/ INV {_} //DO _ /OD)"  [0,0,0] 61)
wenzelm@35113
    25
wenzelm@35054
    26
abbreviation annskip ("SKIP") where "SKIP == Basic id"
nipkow@5646
    27
nipkow@13682
    28
types 'a sem = "'a => 'a => bool"
nipkow@5646
    29
berghofe@36643
    30
inductive Sem :: "'a com \<Rightarrow> 'a sem"
berghofe@36643
    31
where
berghofe@36643
    32
  "Sem (Basic f) s (f s)"
berghofe@36643
    33
| "Sem c1 s s'' \<Longrightarrow> Sem c2 s'' s' \<Longrightarrow> Sem (c1;c2) s s'"
berghofe@36643
    34
| "s \<in> b \<Longrightarrow> Sem c1 s s' \<Longrightarrow> Sem (IF b THEN c1 ELSE c2 FI) s s'"
berghofe@36643
    35
| "s \<notin> b \<Longrightarrow> Sem c2 s s' \<Longrightarrow> Sem (IF b THEN c1 ELSE c2 FI) s s'"
berghofe@36643
    36
| "s \<notin> b \<Longrightarrow> Sem (While b x c) s s"
berghofe@36643
    37
| "s \<in> b \<Longrightarrow> Sem c s s'' \<Longrightarrow> Sem (While b x c) s'' s' \<Longrightarrow>
berghofe@36643
    38
   Sem (While b x c) s s'"
nipkow@5646
    39
berghofe@36643
    40
inductive_cases [elim!]:
berghofe@36643
    41
  "Sem (Basic f) s s'" "Sem (c1;c2) s s'"
berghofe@36643
    42
  "Sem (IF b THEN c1 ELSE c2 FI) s s'"
nipkow@5646
    43
wenzelm@38353
    44
definition Valid :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool"
wenzelm@38353
    45
  where "Valid p c q \<longleftrightarrow> (!s s'. Sem c s s' --> s : p --> s' : q)"
wenzelm@5007
    46
wenzelm@5007
    47
nipkow@1335
    48
nipkow@5646
    49
(** parse translations **)
nipkow@1335
    50
wenzelm@35054
    51
syntax
wenzelm@35054
    52
  "_assign"  :: "id => 'b => 'a com"        ("(2_ :=/ _)" [70,65] 61)
wenzelm@35054
    53
wenzelm@35054
    54
syntax
wenzelm@35054
    55
 "_hoare_vars" :: "[idts, 'a assn,'a com,'a assn] => bool"
wenzelm@35054
    56
                 ("VARS _// {_} // _ // {_}" [0,0,55,0] 50)
wenzelm@35054
    57
syntax ("" output)
wenzelm@35054
    58
 "_hoare"      :: "['a assn,'a com,'a assn] => bool"
wenzelm@35054
    59
                 ("{_} // _ // {_}" [0,55,0] 50)
wenzelm@35054
    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 =
haftmann@37591
    71
      Syntax.const @{const_syntax prod_case} $ 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) =
wenzelm@35113
    75
      Syntax.const @{const_syntax 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{*
wenzelm@35113
    85
fun bexp_tr (Const ("TRUE", _)) xs = Syntax.const "TRUE"   (* FIXME !? *)
wenzelm@35113
    86
  | bexp_tr b xs = Syntax.const @{const_syntax Collect} $ mk_abstuple xs b;
wenzelm@35113
    87
wenzelm@35113
    88
fun assn_tr r xs = Syntax.const @{const_syntax Collect} $ mk_abstuple xs r;
nipkow@13682
    89
*}
nipkow@5646
    90
(* com_tr *)
nipkow@13682
    91
ML{*
wenzelm@35113
    92
fun com_tr (Const(@{syntax_const "_assign"},_) $ Free (a,_) $ e) xs =
wenzelm@35113
    93
      Syntax.const @{const_syntax Basic} $ mk_fexp a e xs
wenzelm@35113
    94
  | com_tr (Const (@{const_syntax Basic},_) $ f) xs = Syntax.const @{const_syntax Basic} $ f
wenzelm@35113
    95
  | com_tr (Const (@{const_syntax Seq},_) $ c1 $ c2) xs =
wenzelm@35113
    96
      Syntax.const @{const_syntax Seq} $ com_tr c1 xs $ com_tr c2 xs
wenzelm@35113
    97
  | com_tr (Const (@{const_syntax Cond},_) $ b $ c1 $ c2) xs =
wenzelm@35113
    98
      Syntax.const @{const_syntax Cond} $ bexp_tr b xs $ com_tr c1 xs $ com_tr c2 xs
wenzelm@35113
    99
  | com_tr (Const (@{const_syntax While},_) $ b $ I $ c) xs =
wenzelm@35113
   100
      Syntax.const @{const_syntax 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
wenzelm@17781
   104
(* triple_tr *)    (* FIXME does not handle "_idtdummy" *)
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 *)
wenzelm@35113
   109
  | var_tr(Const (@{syntax_const "_constrain"}, _) $ (Free (a,_)) $ T) = (a,T);
nipkow@5646
   110
wenzelm@35113
   111
fun vars_tr (Const (@{syntax_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
wenzelm@35113
   117
      in Syntax.const @{const_syntax 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
wenzelm@35113
   124
parse_translation {* [(@{syntax_const "_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{*
haftmann@37591
   131
fun dest_abstuple (Const (@{const_syntax prod_case},_) $ (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
haftmann@37591
   136
fun abs2list (Const (@{const_syntax prod_case},_) $ (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
haftmann@37591
   140
fun mk_ts (Const (@{const_syntax prod_case},_) $ (Abs(x,_,t))) = mk_ts t
nipkow@5646
   141
  | mk_ts (Abs(x,_,t)) = mk_ts t
wenzelm@35113
   142
  | mk_ts (Const (@{const_syntax Pair},_) $ a $ b) = a::(mk_ts b)
nipkow@5646
   143
  | mk_ts t = [t];
nipkow@1335
   144
haftmann@37591
   145
fun mk_vts (Const (@{const_syntax prod_case},_) $ (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;
wenzelm@35113
   149
wenzelm@35113
   150
fun find_ch [] i xs = (false, (Syntax.free "not_ch", Syntax.free "not_ch"))
wenzelm@35113
   151
  | find_ch ((v,t)::vts) i xs =
wenzelm@35113
   152
      if t = Bound i then find_ch vts (i-1) xs
wenzelm@35113
   153
      else (true, (v, subst_bounds (xs, t)));
wenzelm@35113
   154
haftmann@37591
   155
fun is_f (Const (@{const_syntax prod_case},_) $ (Abs(x,_,t))) = true
nipkow@5646
   156
  | is_f (Abs(x,_,t)) = true
nipkow@5646
   157
  | is_f t = false;
nipkow@13682
   158
*}
nipkow@13682
   159
nipkow@5646
   160
(* assn_tr' & bexp_tr'*)
wenzelm@35113
   161
ML{*
wenzelm@35113
   162
fun assn_tr' (Const (@{const_syntax Collect},_) $ T) = dest_abstuple T
wenzelm@35113
   163
  | assn_tr' (Const (@{const_syntax inter}, _) $
wenzelm@35113
   164
        (Const (@{const_syntax Collect},_) $ T1) $ (Const (@{const_syntax Collect},_) $ T2)) =
wenzelm@35113
   165
      Syntax.const @{const_syntax inter} $ dest_abstuple T1 $ dest_abstuple T2
nipkow@5646
   166
  | assn_tr' t = t;
nipkow@1335
   167
wenzelm@35113
   168
fun bexp_tr' (Const (@{const_syntax Collect},_) $ T) = dest_abstuple T
nipkow@5646
   169
  | bexp_tr' t = t;
nipkow@13682
   170
*}
nipkow@5646
   171
nipkow@5646
   172
(*com_tr' *)
nipkow@13682
   173
ML{*
nipkow@5646
   174
fun mk_assign f =
nipkow@5646
   175
  let val (vs, ts) = mk_vts f;
nipkow@5646
   176
      val (ch, which) = find_ch (vs~~ts) ((length vs)-1) (rev vs)
wenzelm@35113
   177
  in
wenzelm@35113
   178
    if ch then Syntax.const @{syntax_const "_assign"} $ fst which $ snd which
wenzelm@35113
   179
    else Syntax.const @{const_syntax annskip}
wenzelm@35113
   180
  end;
nipkow@1335
   181
wenzelm@35113
   182
fun com_tr' (Const (@{const_syntax Basic},_) $ f) =
wenzelm@35113
   183
      if is_f f then mk_assign f
wenzelm@35113
   184
      else Syntax.const @{const_syntax Basic} $ f
wenzelm@35113
   185
  | com_tr' (Const (@{const_syntax Seq},_) $ c1 $ c2) =
wenzelm@35113
   186
      Syntax.const @{const_syntax Seq} $ com_tr' c1 $ com_tr' c2
wenzelm@35113
   187
  | com_tr' (Const (@{const_syntax Cond},_) $ b $ c1 $ c2) =
wenzelm@35113
   188
      Syntax.const @{const_syntax Cond} $ bexp_tr' b $ com_tr' c1 $ com_tr' c2
wenzelm@35113
   189
  | com_tr' (Const (@{const_syntax While},_) $ b $ I $ c) =
wenzelm@35113
   190
      Syntax.const @{const_syntax While} $ bexp_tr' b $ assn_tr' I $ com_tr' c
nipkow@5646
   191
  | com_tr' t = t;
nipkow@1335
   192
nipkow@5646
   193
fun spec_tr' [p, c, q] =
wenzelm@35113
   194
  Syntax.const @{syntax_const "_hoare"} $ assn_tr' p $ com_tr' c $ assn_tr' q
nipkow@13682
   195
*}
nipkow@13682
   196
wenzelm@35054
   197
print_translation {* [(@{const_syntax Valid}, spec_tr')] *}
nipkow@13682
   198
nipkow@13857
   199
lemma SkipRule: "p \<subseteq> q \<Longrightarrow> Valid p (Basic id) q"
nipkow@13857
   200
by (auto simp:Valid_def)
nipkow@13857
   201
nipkow@13857
   202
lemma BasicRule: "p \<subseteq> {s. f s \<in> q} \<Longrightarrow> Valid p (Basic f) q"
nipkow@13857
   203
by (auto simp:Valid_def)
nipkow@13857
   204
nipkow@13857
   205
lemma SeqRule: "Valid P c1 Q \<Longrightarrow> Valid Q c2 R \<Longrightarrow> Valid P (c1;c2) R"
nipkow@13857
   206
by (auto simp:Valid_def)
nipkow@13857
   207
nipkow@13857
   208
lemma CondRule:
nipkow@13857
   209
 "p \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> w) \<and> (s \<notin> b \<longrightarrow> s \<in> w')}
nipkow@13857
   210
  \<Longrightarrow> Valid w c1 q \<Longrightarrow> Valid w' c2 q \<Longrightarrow> Valid p (Cond b c1 c2) q"
nipkow@13857
   211
by (auto simp:Valid_def)
nipkow@13857
   212
berghofe@36643
   213
lemma While_aux:
berghofe@36643
   214
  assumes "Sem (WHILE b INV {i} DO c OD) s s'"
berghofe@36643
   215
  shows "\<forall>s s'. Sem c s s' \<longrightarrow> s \<in> I \<and> s \<in> b \<longrightarrow> s' \<in> I \<Longrightarrow>
berghofe@36643
   216
    s \<in> I \<Longrightarrow> s' \<in> I \<and> s' \<notin> b"
berghofe@36643
   217
  using assms
berghofe@36643
   218
  by (induct "WHILE b INV {i} DO c OD" s s') auto
nipkow@13857
   219
nipkow@13857
   220
lemma WhileRule:
nipkow@13857
   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"
nipkow@13857
   222
apply (clarsimp simp:Valid_def)
berghofe@36643
   223
apply(drule While_aux)
berghofe@36643
   224
  apply assumption
nipkow@13857
   225
 apply blast
nipkow@13857
   226
apply blast
nipkow@13857
   227
done
nipkow@13857
   228
nipkow@13857
   229
wenzelm@24470
   230
lemma Compl_Collect: "-(Collect b) = {x. ~(b x)}"
wenzelm@24470
   231
  by blast
wenzelm@24470
   232
wenzelm@28457
   233
lemmas AbortRule = SkipRule  -- "dummy version"
wenzelm@24472
   234
use "hoare_tac.ML"
nipkow@13682
   235
nipkow@13682
   236
method_setup vcg = {*
wenzelm@30549
   237
  Scan.succeed (fn ctxt => SIMPLE_METHOD' (hoare_tac ctxt (K all_tac))) *}
nipkow@13682
   238
  "verification condition generator"
nipkow@13682
   239
nipkow@13682
   240
method_setup vcg_simp = {*
wenzelm@30549
   241
  Scan.succeed (fn ctxt =>
wenzelm@32149
   242
    SIMPLE_METHOD' (hoare_tac ctxt (asm_full_simp_tac (simpset_of ctxt)))) *}
nipkow@13682
   243
  "verification condition generator plus simplification"
nipkow@13682
   244
nipkow@13682
   245
end