src/HOL/Hoare/Hoare_Logic_Abort.thy
author berghofe
Tue May 04 14:11:28 2010 +0200 (2010-05-04)
changeset 36643 f36588af1ba1
parent 35417 47ee18b6ae32
child 37591 d3daea901123
permissions -rw-r--r--
Turned Sem into an inductive definition.
nipkow@13857
     1
(*  Title:      HOL/Hoare/HoareAbort.thy
nipkow@13857
     2
    Author:     Leonor Prensa Nieto & Tobias Nipkow
nipkow@13857
     3
    Copyright   2003 TUM
nipkow@13857
     4
nipkow@13857
     5
Like Hoare.thy, but with an Abort statement for modelling run time errors.
nipkow@13857
     6
*)
nipkow@13857
     7
haftmann@35320
     8
theory Hoare_Logic_Abort
wenzelm@28457
     9
imports Main
wenzelm@28457
    10
uses ("hoare_tac.ML")
wenzelm@24470
    11
begin
nipkow@13857
    12
nipkow@13857
    13
types
nipkow@13857
    14
    'a bexp = "'a set"
nipkow@13857
    15
    'a assn = "'a set"
nipkow@13857
    16
nipkow@13857
    17
datatype
nipkow@13857
    18
 'a com = Basic "'a \<Rightarrow> 'a"
nipkow@13857
    19
   | Abort
nipkow@13857
    20
   | Seq "'a com" "'a com"               ("(_;/ _)"      [61,60] 60)
nipkow@13857
    21
   | Cond "'a bexp" "'a com" "'a com"    ("(1IF _/ THEN _ / ELSE _/ FI)"  [0,0,0] 61)
nipkow@13857
    22
   | While "'a bexp" "'a assn" "'a com"  ("(1WHILE _/ INV {_} //DO _ /OD)"  [0,0,0] 61)
wenzelm@35113
    23
wenzelm@35054
    24
abbreviation annskip ("SKIP") where "SKIP == Basic id"
nipkow@13857
    25
nipkow@13857
    26
types 'a sem = "'a option => 'a option => bool"
nipkow@13857
    27
berghofe@36643
    28
inductive Sem :: "'a com \<Rightarrow> 'a sem"
berghofe@36643
    29
where
berghofe@36643
    30
  "Sem (Basic f) None None"
berghofe@36643
    31
| "Sem (Basic f) (Some s) (Some (f s))"
berghofe@36643
    32
| "Sem Abort s None"
berghofe@36643
    33
| "Sem c1 s s'' \<Longrightarrow> Sem c2 s'' s' \<Longrightarrow> Sem (c1;c2) s s'"
berghofe@36643
    34
| "Sem (IF b THEN c1 ELSE c2 FI) None None"
berghofe@36643
    35
| "s \<in> b \<Longrightarrow> Sem c1 (Some s) s' \<Longrightarrow> Sem (IF b THEN c1 ELSE c2 FI) (Some s) s'"
berghofe@36643
    36
| "s \<notin> b \<Longrightarrow> Sem c2 (Some s) s' \<Longrightarrow> Sem (IF b THEN c1 ELSE c2 FI) (Some s) s'"
berghofe@36643
    37
| "Sem (While b x c) None None"
berghofe@36643
    38
| "s \<notin> b \<Longrightarrow> Sem (While b x c) (Some s) (Some s)"
berghofe@36643
    39
| "s \<in> b \<Longrightarrow> Sem c (Some s) s'' \<Longrightarrow> Sem (While b x c) s'' s' \<Longrightarrow>
berghofe@36643
    40
   Sem (While b x c) (Some s) s'"
nipkow@13857
    41
berghofe@36643
    42
inductive_cases [elim!]:
berghofe@36643
    43
  "Sem (Basic f) s s'" "Sem (c1;c2) s s'"
berghofe@36643
    44
  "Sem (IF b THEN c1 ELSE c2 FI) s s'"
nipkow@13857
    45
haftmann@35416
    46
definition Valid :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool" where
nipkow@13857
    47
  "Valid p c q == \<forall>s s'. Sem c s s' \<longrightarrow> s : Some ` p \<longrightarrow> s' : Some ` q"
nipkow@13857
    48
nipkow@13857
    49
nipkow@13857
    50
nipkow@13857
    51
(** parse translations **)
nipkow@13857
    52
wenzelm@35054
    53
syntax
wenzelm@35054
    54
  "_assign"  :: "id => 'b => 'a com"        ("(2_ :=/ _)" [70,65] 61)
wenzelm@35054
    55
wenzelm@35054
    56
syntax
haftmann@35320
    57
  "_hoare_abort_vars" :: "[idts, 'a assn,'a com,'a assn] => bool"
wenzelm@35054
    58
                 ("VARS _// {_} // _ // {_}" [0,0,55,0] 50)
wenzelm@35054
    59
syntax ("" output)
haftmann@35320
    60
  "_hoare_abort"      :: "['a assn,'a com,'a assn] => bool"
wenzelm@35054
    61
                 ("{_} // _ // {_}" [0,55,0] 50)
wenzelm@35054
    62
ML {*
nipkow@13857
    63
nipkow@13857
    64
local
nipkow@13857
    65
fun free a = Free(a,dummyT)
nipkow@13857
    66
fun abs((a,T),body) =
nipkow@13857
    67
  let val a = absfree(a, dummyT, body)
nipkow@13857
    68
  in if T = Bound 0 then a else Const(Syntax.constrainAbsC,dummyT) $ a $ T end
nipkow@13857
    69
in
nipkow@13857
    70
nipkow@13857
    71
fun mk_abstuple [x] body = abs (x, body)
nipkow@13857
    72
  | mk_abstuple (x::xs) body =
wenzelm@35113
    73
      Syntax.const @{const_syntax split} $ abs (x, mk_abstuple xs body);
nipkow@13857
    74
nipkow@13857
    75
fun mk_fbody a e [x as (b,_)] = if a=b then e else free b
nipkow@13857
    76
  | mk_fbody a e ((b,_)::xs) =
wenzelm@35113
    77
      Syntax.const @{const_syntax Pair} $ (if a=b then e else free b) $ mk_fbody a e xs;
nipkow@13857
    78
nipkow@13857
    79
fun mk_fexp a e xs = mk_abstuple xs (mk_fbody a e xs)
nipkow@13857
    80
end
nipkow@13857
    81
*}
nipkow@13857
    82
nipkow@13857
    83
(* bexp_tr & assn_tr *)
nipkow@13857
    84
(*all meta-variables for bexp except for TRUE are translated as if they
nipkow@13857
    85
  were boolean expressions*)
nipkow@13857
    86
ML{*
wenzelm@35113
    87
fun bexp_tr (Const ("TRUE", _)) xs = Syntax.const "TRUE"   (* FIXME !? *)
wenzelm@35113
    88
  | bexp_tr b xs = Syntax.const @{const_syntax Collect} $ mk_abstuple xs b;
wenzelm@35113
    89
wenzelm@35113
    90
fun assn_tr r xs = Syntax.const @{const_syntax Collect} $ mk_abstuple xs r;
nipkow@13857
    91
*}
nipkow@13857
    92
(* com_tr *)
nipkow@13857
    93
ML{*
wenzelm@35113
    94
fun com_tr (Const (@{syntax_const "_assign"},_) $ Free (a,_) $ e) xs =
wenzelm@35113
    95
      Syntax.const @{const_syntax Basic} $ mk_fexp a e xs
wenzelm@35113
    96
  | com_tr (Const (@{const_syntax Basic},_) $ f) xs = Syntax.const @{const_syntax Basic} $ f
wenzelm@35113
    97
  | com_tr (Const (@{const_syntax Seq},_) $ c1 $ c2) xs =
wenzelm@35113
    98
      Syntax.const @{const_syntax Seq} $ com_tr c1 xs $ com_tr c2 xs
wenzelm@35113
    99
  | com_tr (Const (@{const_syntax Cond},_) $ b $ c1 $ c2) xs =
wenzelm@35113
   100
      Syntax.const @{const_syntax Cond} $ bexp_tr b xs $ com_tr c1 xs $ com_tr c2 xs
wenzelm@35113
   101
  | com_tr (Const (@{const_syntax While},_) $ b $ I $ c) xs =
wenzelm@35113
   102
      Syntax.const @{const_syntax While} $ bexp_tr b xs $ assn_tr I xs $ com_tr c xs
nipkow@13857
   103
  | com_tr t _ = t (* if t is just a Free/Var *)
nipkow@13857
   104
*}
nipkow@13857
   105
wenzelm@17781
   106
(* triple_tr *)  (* FIXME does not handle "_idtdummy" *)
nipkow@13857
   107
ML{*
nipkow@13857
   108
local
nipkow@13857
   109
wenzelm@35113
   110
fun var_tr (Free (a, _)) = (a, Bound 0) (* Bound 0 = dummy term *)
wenzelm@35113
   111
  | var_tr (Const (@{syntax_const "_constrain"}, _) $ Free (a, _) $ T) = (a, T);
nipkow@13857
   112
wenzelm@35113
   113
fun vars_tr (Const (@{syntax_const "_idts"}, _) $ idt $ vars) = var_tr idt :: vars_tr vars
nipkow@13857
   114
  | vars_tr t = [var_tr t]
nipkow@13857
   115
nipkow@13857
   116
in
nipkow@13857
   117
fun hoare_vars_tr [vars, pre, prg, post] =
nipkow@13857
   118
      let val xs = vars_tr vars
wenzelm@35113
   119
      in Syntax.const @{const_syntax Valid} $
nipkow@13857
   120
         assn_tr pre xs $ com_tr prg xs $ assn_tr post xs
nipkow@13857
   121
      end
nipkow@13857
   122
  | hoare_vars_tr ts = raise TERM ("hoare_vars_tr", ts);
nipkow@13857
   123
end
nipkow@13857
   124
*}
nipkow@13857
   125
haftmann@35320
   126
parse_translation {* [(@{syntax_const "_hoare_abort_vars"}, hoare_vars_tr)] *}
nipkow@13857
   127
nipkow@13857
   128
nipkow@13857
   129
(*****************************************************************************)
nipkow@13857
   130
nipkow@13857
   131
(*** print translations ***)
nipkow@13857
   132
ML{*
wenzelm@35113
   133
fun dest_abstuple (Const (@{const_syntax split},_) $ (Abs(v,_, body))) =
wenzelm@35113
   134
      subst_bound (Syntax.free v, dest_abstuple body)
nipkow@13857
   135
  | dest_abstuple (Abs(v,_, body)) = subst_bound (Syntax.free v, body)
nipkow@13857
   136
  | dest_abstuple trm = trm;
nipkow@13857
   137
wenzelm@35113
   138
fun abs2list (Const (@{const_syntax split},_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t
nipkow@13857
   139
  | abs2list (Abs(x,T,t)) = [Free (x, T)]
nipkow@13857
   140
  | abs2list _ = [];
nipkow@13857
   141
wenzelm@35113
   142
fun mk_ts (Const (@{const_syntax split},_) $ (Abs(x,_,t))) = mk_ts t
nipkow@13857
   143
  | mk_ts (Abs(x,_,t)) = mk_ts t
wenzelm@35113
   144
  | mk_ts (Const (@{const_syntax Pair},_) $ a $ b) = a::(mk_ts b)
nipkow@13857
   145
  | mk_ts t = [t];
nipkow@13857
   146
wenzelm@35113
   147
fun mk_vts (Const (@{const_syntax split},_) $ (Abs(x,_,t))) =
nipkow@13857
   148
           ((Syntax.free x)::(abs2list t), mk_ts t)
nipkow@13857
   149
  | mk_vts (Abs(x,_,t)) = ([Syntax.free x], [t])
nipkow@13857
   150
  | mk_vts t = raise Match;
wenzelm@35113
   151
wenzelm@35113
   152
fun find_ch [] i xs = (false, (Syntax.free "not_ch", Syntax.free "not_ch"))
wenzelm@35113
   153
  | find_ch ((v,t)::vts) i xs =
wenzelm@35113
   154
      if t = Bound i then find_ch vts (i-1) xs
wenzelm@35113
   155
      else (true, (v, subst_bounds (xs,t)));
wenzelm@35113
   156
wenzelm@35113
   157
fun is_f (Const (@{const_syntax split},_) $ (Abs(x,_,t))) = true
nipkow@13857
   158
  | is_f (Abs(x,_,t)) = true
nipkow@13857
   159
  | is_f t = false;
nipkow@13857
   160
*}
nipkow@13857
   161
nipkow@13857
   162
(* assn_tr' & bexp_tr'*)
wenzelm@35113
   163
ML{*
wenzelm@35113
   164
fun assn_tr' (Const (@{const_syntax Collect},_) $ T) = dest_abstuple T
wenzelm@35113
   165
  | assn_tr' (Const (@{const_syntax inter},_) $ (Const (@{const_syntax Collect},_) $ T1) $
wenzelm@35113
   166
        (Const (@{const_syntax Collect},_) $ T2)) =
wenzelm@35113
   167
      Syntax.const @{const_syntax inter} $ dest_abstuple T1 $ dest_abstuple T2
nipkow@13857
   168
  | assn_tr' t = t;
nipkow@13857
   169
wenzelm@35113
   170
fun bexp_tr' (Const (@{const_syntax Collect},_) $ T) = dest_abstuple T
nipkow@13857
   171
  | bexp_tr' t = t;
nipkow@13857
   172
*}
nipkow@13857
   173
nipkow@13857
   174
(*com_tr' *)
nipkow@13857
   175
ML{*
nipkow@13857
   176
fun mk_assign f =
nipkow@13857
   177
  let val (vs, ts) = mk_vts f;
nipkow@13857
   178
      val (ch, which) = find_ch (vs~~ts) ((length vs)-1) (rev vs)
wenzelm@35113
   179
  in
wenzelm@35113
   180
    if ch then Syntax.const @{syntax_const "_assign"} $ fst which $ snd which
wenzelm@35113
   181
    else Syntax.const @{const_syntax annskip}
wenzelm@35113
   182
  end;
nipkow@13857
   183
wenzelm@35113
   184
fun com_tr' (Const (@{const_syntax Basic},_) $ f) =
wenzelm@35113
   185
      if is_f f then mk_assign f else Syntax.const @{const_syntax Basic} $ f
wenzelm@35113
   186
  | com_tr' (Const (@{const_syntax Seq},_) $ c1 $ c2) =
wenzelm@35113
   187
      Syntax.const @{const_syntax Seq} $ com_tr' c1 $ com_tr' c2
wenzelm@35113
   188
  | com_tr' (Const (@{const_syntax Cond},_) $ b $ c1 $ c2) =
wenzelm@35113
   189
      Syntax.const @{const_syntax Cond} $ bexp_tr' b $ com_tr' c1 $ com_tr' c2
wenzelm@35113
   190
  | com_tr' (Const (@{const_syntax While},_) $ b $ I $ c) =
wenzelm@35113
   191
      Syntax.const @{const_syntax While} $ bexp_tr' b $ assn_tr' I $ com_tr' c
nipkow@13857
   192
  | com_tr' t = t;
nipkow@13857
   193
nipkow@13857
   194
fun spec_tr' [p, c, q] =
haftmann@35320
   195
  Syntax.const @{syntax_const "_hoare_abort"} $ assn_tr' p $ com_tr' c $ assn_tr' q
nipkow@13857
   196
*}
nipkow@13857
   197
wenzelm@35054
   198
print_translation {* [(@{const_syntax Valid}, spec_tr')] *}
nipkow@13857
   199
nipkow@13857
   200
(*** The proof rules ***)
nipkow@13857
   201
nipkow@13857
   202
lemma SkipRule: "p \<subseteq> q \<Longrightarrow> Valid p (Basic id) q"
nipkow@13857
   203
by (auto simp:Valid_def)
nipkow@13857
   204
nipkow@13857
   205
lemma BasicRule: "p \<subseteq> {s. f s \<in> q} \<Longrightarrow> Valid p (Basic f) q"
nipkow@13857
   206
by (auto simp:Valid_def)
nipkow@13857
   207
nipkow@13857
   208
lemma SeqRule: "Valid P c1 Q \<Longrightarrow> Valid Q c2 R \<Longrightarrow> Valid P (c1;c2) R"
nipkow@13857
   209
by (auto simp:Valid_def)
nipkow@13857
   210
nipkow@13857
   211
lemma CondRule:
nipkow@13857
   212
 "p \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> w) \<and> (s \<notin> b \<longrightarrow> s \<in> w')}
nipkow@13857
   213
  \<Longrightarrow> Valid w c1 q \<Longrightarrow> Valid w' c2 q \<Longrightarrow> Valid p (Cond b c1 c2) q"
nipkow@13857
   214
by (fastsimp simp:Valid_def image_def)
nipkow@13857
   215
berghofe@36643
   216
lemma While_aux:
berghofe@36643
   217
  assumes "Sem (WHILE b INV {i} DO c OD) s s'"
berghofe@36643
   218
  shows "\<forall>s s'. Sem c s s' \<longrightarrow> s \<in> Some ` (I \<inter> b) \<longrightarrow> s' \<in> Some ` I \<Longrightarrow>
berghofe@36643
   219
    s \<in> Some ` I \<Longrightarrow> s' \<in> Some ` (I \<inter> -b)"
berghofe@36643
   220
  using assms
berghofe@36643
   221
  by (induct "WHILE b INV {i} DO c OD" s s') auto
nipkow@13857
   222
nipkow@13857
   223
lemma WhileRule:
nipkow@13857
   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"
nipkow@13857
   225
apply(simp add:Valid_def)
nipkow@13857
   226
apply(simp (no_asm) add:image_def)
nipkow@13857
   227
apply clarify
berghofe@36643
   228
apply(drule While_aux)
berghofe@36643
   229
  apply assumption
nipkow@13857
   230
 apply blast
nipkow@13857
   231
apply blast
nipkow@13857
   232
done
nipkow@13857
   233
nipkow@13857
   234
lemma AbortRule: "p \<subseteq> {s. False} \<Longrightarrow> Valid p Abort q"
nipkow@13857
   235
by(auto simp:Valid_def)
nipkow@13857
   236
wenzelm@24470
   237
wenzelm@24470
   238
subsection {* Derivation of the proof rules and, most importantly, the VCG tactic *}
wenzelm@24470
   239
wenzelm@24470
   240
lemma Compl_Collect: "-(Collect b) = {x. ~(b x)}"
wenzelm@24470
   241
  by blast
wenzelm@24470
   242
wenzelm@28457
   243
use "hoare_tac.ML"
nipkow@13857
   244
nipkow@13857
   245
method_setup vcg = {*
wenzelm@30549
   246
  Scan.succeed (fn ctxt => SIMPLE_METHOD' (hoare_tac ctxt (K all_tac))) *}
nipkow@13857
   247
  "verification condition generator"
nipkow@13857
   248
nipkow@13857
   249
method_setup vcg_simp = {*
wenzelm@30549
   250
  Scan.succeed (fn ctxt =>
wenzelm@32149
   251
    SIMPLE_METHOD' (hoare_tac ctxt (asm_full_simp_tac (simpset_of ctxt)))) *}
nipkow@13857
   252
  "verification condition generator plus simplification"
nipkow@13857
   253
nipkow@13875
   254
(* Special syntax for guarded statements and guarded array updates: *)
nipkow@13875
   255
nipkow@13875
   256
syntax
wenzelm@35352
   257
  "_guarded_com" :: "bool \<Rightarrow> 'a com \<Rightarrow> 'a com"  ("(2_ \<rightarrow>/ _)" 71)
wenzelm@35352
   258
  "_array_update" :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a com"  ("(2_[_] :=/ _)" [70, 65] 61)
nipkow@13875
   259
translations
wenzelm@35101
   260
  "P \<rightarrow> c" == "IF P THEN c ELSE CONST Abort FI"
haftmann@34940
   261
  "a[i] := v" => "(i < CONST length a) \<rightarrow> (a := CONST list_update a i v)"
nipkow@13875
   262
  (* reverse translation not possible because of duplicate "a" *)
nipkow@13875
   263
nipkow@13875
   264
text{* Note: there is no special syntax for guarded array access. Thus
nipkow@13875
   265
you must write @{text"j < length a \<rightarrow> a[i] := a!j"}. *}
nipkow@13875
   266
nipkow@13857
   267
end