src/HOL/Hoare/Hoare_Logic_Abort.thy
author haftmann
Tue Feb 23 10:11:49 2010 +0100 (2010-02-23)
changeset 35320 f80aee1ed475
parent 35113 src/HOL/Hoare/HoareAbort.thy@1a0c129bb2e0
child 35352 fa051b504c3f
child 35416 d8d7d1b785af
permissions -rw-r--r--
dropped axclass; dropped Id; session theory Hoare.thy
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
nipkow@13857
    28
consts iter :: "nat => 'a bexp => 'a sem => 'a sem"
nipkow@13857
    29
primrec
nipkow@13875
    30
"iter 0 b S = (\<lambda>s s'. s \<notin> Some ` b \<and> s=s')"
nipkow@13875
    31
"iter (Suc n) b S =
nipkow@13875
    32
  (\<lambda>s s'. s \<in> Some ` b \<and> (\<exists>s''. S s s'' \<and> iter n b S s'' s'))"
nipkow@13857
    33
nipkow@13857
    34
consts Sem :: "'a com => 'a sem"
nipkow@13857
    35
primrec
nipkow@13857
    36
"Sem(Basic f) s s' = (case s of None \<Rightarrow> s' = None | Some t \<Rightarrow> s' = Some(f t))"
nipkow@13857
    37
"Sem Abort s s' = (s' = None)"
nipkow@13875
    38
"Sem(c1;c2) s s' = (\<exists>s''. Sem c1 s s'' \<and> Sem c2 s'' s')"
nipkow@13857
    39
"Sem(IF b THEN c1 ELSE c2 FI) s s' =
nipkow@13857
    40
 (case s of None \<Rightarrow> s' = None
nipkow@13875
    41
  | Some t \<Rightarrow> ((t \<in> b \<longrightarrow> Sem c1 s s') \<and> (t \<notin> b \<longrightarrow> Sem c2 s s')))"
nipkow@13857
    42
"Sem(While b x c) s s' =
nipkow@13875
    43
 (if s = None then s' = None else \<exists>n. iter n b (Sem c) s s')"
nipkow@13857
    44
nipkow@13857
    45
constdefs Valid :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool"
nipkow@13857
    46
  "Valid p c q == \<forall>s s'. Sem c s s' \<longrightarrow> s : Some ` p \<longrightarrow> s' : Some ` q"
nipkow@13857
    47
nipkow@13857
    48
nipkow@13857
    49
nipkow@13857
    50
(** parse translations **)
nipkow@13857
    51
wenzelm@35054
    52
syntax
wenzelm@35054
    53
  "_assign"  :: "id => 'b => 'a com"        ("(2_ :=/ _)" [70,65] 61)
wenzelm@35054
    54
wenzelm@35054
    55
syntax
haftmann@35320
    56
  "_hoare_abort_vars" :: "[idts, 'a assn,'a com,'a assn] => bool"
wenzelm@35054
    57
                 ("VARS _// {_} // _ // {_}" [0,0,55,0] 50)
wenzelm@35054
    58
syntax ("" output)
haftmann@35320
    59
  "_hoare_abort"      :: "['a assn,'a com,'a assn] => bool"
wenzelm@35054
    60
                 ("{_} // _ // {_}" [0,55,0] 50)
wenzelm@35054
    61
ML {*
nipkow@13857
    62
nipkow@13857
    63
local
nipkow@13857
    64
fun free a = Free(a,dummyT)
nipkow@13857
    65
fun abs((a,T),body) =
nipkow@13857
    66
  let val a = absfree(a, dummyT, body)
nipkow@13857
    67
  in if T = Bound 0 then a else Const(Syntax.constrainAbsC,dummyT) $ a $ T end
nipkow@13857
    68
in
nipkow@13857
    69
nipkow@13857
    70
fun mk_abstuple [x] body = abs (x, body)
nipkow@13857
    71
  | mk_abstuple (x::xs) body =
wenzelm@35113
    72
      Syntax.const @{const_syntax split} $ abs (x, mk_abstuple xs body);
nipkow@13857
    73
nipkow@13857
    74
fun mk_fbody a e [x as (b,_)] = if a=b then e else free b
nipkow@13857
    75
  | mk_fbody a e ((b,_)::xs) =
wenzelm@35113
    76
      Syntax.const @{const_syntax Pair} $ (if a=b then e else free b) $ mk_fbody a e xs;
nipkow@13857
    77
nipkow@13857
    78
fun mk_fexp a e xs = mk_abstuple xs (mk_fbody a e xs)
nipkow@13857
    79
end
nipkow@13857
    80
*}
nipkow@13857
    81
nipkow@13857
    82
(* bexp_tr & assn_tr *)
nipkow@13857
    83
(*all meta-variables for bexp except for TRUE are translated as if they
nipkow@13857
    84
  were boolean expressions*)
nipkow@13857
    85
ML{*
wenzelm@35113
    86
fun bexp_tr (Const ("TRUE", _)) xs = Syntax.const "TRUE"   (* FIXME !? *)
wenzelm@35113
    87
  | bexp_tr b xs = Syntax.const @{const_syntax Collect} $ mk_abstuple xs b;
wenzelm@35113
    88
wenzelm@35113
    89
fun assn_tr r xs = Syntax.const @{const_syntax Collect} $ mk_abstuple xs r;
nipkow@13857
    90
*}
nipkow@13857
    91
(* com_tr *)
nipkow@13857
    92
ML{*
wenzelm@35113
    93
fun com_tr (Const (@{syntax_const "_assign"},_) $ Free (a,_) $ e) xs =
wenzelm@35113
    94
      Syntax.const @{const_syntax Basic} $ mk_fexp a e xs
wenzelm@35113
    95
  | com_tr (Const (@{const_syntax Basic},_) $ f) xs = Syntax.const @{const_syntax Basic} $ f
wenzelm@35113
    96
  | com_tr (Const (@{const_syntax Seq},_) $ c1 $ c2) xs =
wenzelm@35113
    97
      Syntax.const @{const_syntax Seq} $ com_tr c1 xs $ com_tr c2 xs
wenzelm@35113
    98
  | com_tr (Const (@{const_syntax Cond},_) $ b $ c1 $ c2) xs =
wenzelm@35113
    99
      Syntax.const @{const_syntax Cond} $ bexp_tr b xs $ com_tr c1 xs $ com_tr c2 xs
wenzelm@35113
   100
  | com_tr (Const (@{const_syntax While},_) $ b $ I $ c) xs =
wenzelm@35113
   101
      Syntax.const @{const_syntax While} $ bexp_tr b xs $ assn_tr I xs $ com_tr c xs
nipkow@13857
   102
  | com_tr t _ = t (* if t is just a Free/Var *)
nipkow@13857
   103
*}
nipkow@13857
   104
wenzelm@17781
   105
(* triple_tr *)  (* FIXME does not handle "_idtdummy" *)
nipkow@13857
   106
ML{*
nipkow@13857
   107
local
nipkow@13857
   108
wenzelm@35113
   109
fun var_tr (Free (a, _)) = (a, Bound 0) (* Bound 0 = dummy term *)
wenzelm@35113
   110
  | var_tr (Const (@{syntax_const "_constrain"}, _) $ Free (a, _) $ T) = (a, T);
nipkow@13857
   111
wenzelm@35113
   112
fun vars_tr (Const (@{syntax_const "_idts"}, _) $ idt $ vars) = var_tr idt :: vars_tr vars
nipkow@13857
   113
  | vars_tr t = [var_tr t]
nipkow@13857
   114
nipkow@13857
   115
in
nipkow@13857
   116
fun hoare_vars_tr [vars, pre, prg, post] =
nipkow@13857
   117
      let val xs = vars_tr vars
wenzelm@35113
   118
      in Syntax.const @{const_syntax Valid} $
nipkow@13857
   119
         assn_tr pre xs $ com_tr prg xs $ assn_tr post xs
nipkow@13857
   120
      end
nipkow@13857
   121
  | hoare_vars_tr ts = raise TERM ("hoare_vars_tr", ts);
nipkow@13857
   122
end
nipkow@13857
   123
*}
nipkow@13857
   124
haftmann@35320
   125
parse_translation {* [(@{syntax_const "_hoare_abort_vars"}, hoare_vars_tr)] *}
nipkow@13857
   126
nipkow@13857
   127
nipkow@13857
   128
(*****************************************************************************)
nipkow@13857
   129
nipkow@13857
   130
(*** print translations ***)
nipkow@13857
   131
ML{*
wenzelm@35113
   132
fun dest_abstuple (Const (@{const_syntax split},_) $ (Abs(v,_, body))) =
wenzelm@35113
   133
      subst_bound (Syntax.free v, dest_abstuple body)
nipkow@13857
   134
  | dest_abstuple (Abs(v,_, body)) = subst_bound (Syntax.free v, body)
nipkow@13857
   135
  | dest_abstuple trm = trm;
nipkow@13857
   136
wenzelm@35113
   137
fun abs2list (Const (@{const_syntax split},_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t
nipkow@13857
   138
  | abs2list (Abs(x,T,t)) = [Free (x, T)]
nipkow@13857
   139
  | abs2list _ = [];
nipkow@13857
   140
wenzelm@35113
   141
fun mk_ts (Const (@{const_syntax split},_) $ (Abs(x,_,t))) = mk_ts t
nipkow@13857
   142
  | mk_ts (Abs(x,_,t)) = mk_ts t
wenzelm@35113
   143
  | mk_ts (Const (@{const_syntax Pair},_) $ a $ b) = a::(mk_ts b)
nipkow@13857
   144
  | mk_ts t = [t];
nipkow@13857
   145
wenzelm@35113
   146
fun mk_vts (Const (@{const_syntax split},_) $ (Abs(x,_,t))) =
nipkow@13857
   147
           ((Syntax.free x)::(abs2list t), mk_ts t)
nipkow@13857
   148
  | mk_vts (Abs(x,_,t)) = ([Syntax.free x], [t])
nipkow@13857
   149
  | mk_vts t = raise Match;
wenzelm@35113
   150
wenzelm@35113
   151
fun find_ch [] i xs = (false, (Syntax.free "not_ch", Syntax.free "not_ch"))
wenzelm@35113
   152
  | find_ch ((v,t)::vts) i xs =
wenzelm@35113
   153
      if t = Bound i then find_ch vts (i-1) xs
wenzelm@35113
   154
      else (true, (v, subst_bounds (xs,t)));
wenzelm@35113
   155
wenzelm@35113
   156
fun is_f (Const (@{const_syntax split},_) $ (Abs(x,_,t))) = true
nipkow@13857
   157
  | is_f (Abs(x,_,t)) = true
nipkow@13857
   158
  | is_f t = false;
nipkow@13857
   159
*}
nipkow@13857
   160
nipkow@13857
   161
(* assn_tr' & bexp_tr'*)
wenzelm@35113
   162
ML{*
wenzelm@35113
   163
fun assn_tr' (Const (@{const_syntax Collect},_) $ T) = dest_abstuple T
wenzelm@35113
   164
  | assn_tr' (Const (@{const_syntax inter},_) $ (Const (@{const_syntax Collect},_) $ T1) $
wenzelm@35113
   165
        (Const (@{const_syntax Collect},_) $ T2)) =
wenzelm@35113
   166
      Syntax.const @{const_syntax inter} $ dest_abstuple T1 $ dest_abstuple T2
nipkow@13857
   167
  | assn_tr' t = t;
nipkow@13857
   168
wenzelm@35113
   169
fun bexp_tr' (Const (@{const_syntax Collect},_) $ T) = dest_abstuple T
nipkow@13857
   170
  | bexp_tr' t = t;
nipkow@13857
   171
*}
nipkow@13857
   172
nipkow@13857
   173
(*com_tr' *)
nipkow@13857
   174
ML{*
nipkow@13857
   175
fun mk_assign f =
nipkow@13857
   176
  let val (vs, ts) = mk_vts f;
nipkow@13857
   177
      val (ch, which) = find_ch (vs~~ts) ((length vs)-1) (rev vs)
wenzelm@35113
   178
  in
wenzelm@35113
   179
    if ch then Syntax.const @{syntax_const "_assign"} $ fst which $ snd which
wenzelm@35113
   180
    else Syntax.const @{const_syntax annskip}
wenzelm@35113
   181
  end;
nipkow@13857
   182
wenzelm@35113
   183
fun com_tr' (Const (@{const_syntax Basic},_) $ f) =
wenzelm@35113
   184
      if is_f f then mk_assign f 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@13857
   191
  | com_tr' t = t;
nipkow@13857
   192
nipkow@13857
   193
fun spec_tr' [p, c, q] =
haftmann@35320
   194
  Syntax.const @{syntax_const "_hoare_abort"} $ assn_tr' p $ com_tr' c $ assn_tr' q
nipkow@13857
   195
*}
nipkow@13857
   196
wenzelm@35054
   197
print_translation {* [(@{const_syntax Valid}, spec_tr')] *}
nipkow@13857
   198
nipkow@13857
   199
(*** The proof rules ***)
nipkow@13857
   200
nipkow@13857
   201
lemma SkipRule: "p \<subseteq> q \<Longrightarrow> Valid p (Basic id) q"
nipkow@13857
   202
by (auto simp:Valid_def)
nipkow@13857
   203
nipkow@13857
   204
lemma BasicRule: "p \<subseteq> {s. f s \<in> q} \<Longrightarrow> Valid p (Basic f) q"
nipkow@13857
   205
by (auto simp:Valid_def)
nipkow@13857
   206
nipkow@13857
   207
lemma SeqRule: "Valid P c1 Q \<Longrightarrow> Valid Q c2 R \<Longrightarrow> Valid P (c1;c2) R"
nipkow@13857
   208
by (auto simp:Valid_def)
nipkow@13857
   209
nipkow@13857
   210
lemma CondRule:
nipkow@13857
   211
 "p \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> w) \<and> (s \<notin> b \<longrightarrow> s \<in> w')}
nipkow@13857
   212
  \<Longrightarrow> Valid w c1 q \<Longrightarrow> Valid w' c2 q \<Longrightarrow> Valid p (Cond b c1 c2) q"
nipkow@13857
   213
by (fastsimp simp:Valid_def image_def)
nipkow@13857
   214
nipkow@13875
   215
lemma iter_aux:
nipkow@13875
   216
 "! s s'. Sem c s s' \<longrightarrow> s \<in> Some ` (I \<inter> b) \<longrightarrow> s' \<in> Some ` I \<Longrightarrow>
nipkow@13875
   217
  (\<And>s s'. s \<in> Some ` I \<Longrightarrow> iter n b (Sem c) s s' \<Longrightarrow> s' \<in> Some ` (I \<inter> -b))";
nipkow@13857
   218
apply(unfold image_def)
nipkow@13857
   219
apply(induct n)
nipkow@13857
   220
 apply clarsimp
nipkow@13857
   221
apply(simp (no_asm_use))
nipkow@13857
   222
apply blast
nipkow@13857
   223
done
nipkow@13857
   224
nipkow@13857
   225
lemma WhileRule:
nipkow@13857
   226
 "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
   227
apply(simp add:Valid_def)
nipkow@13857
   228
apply(simp (no_asm) add:image_def)
nipkow@13857
   229
apply clarify
nipkow@13857
   230
apply(drule iter_aux)
nipkow@13857
   231
  prefer 2 apply assumption
nipkow@13857
   232
 apply blast
nipkow@13857
   233
apply blast
nipkow@13857
   234
done
nipkow@13857
   235
nipkow@13857
   236
lemma AbortRule: "p \<subseteq> {s. False} \<Longrightarrow> Valid p Abort q"
nipkow@13857
   237
by(auto simp:Valid_def)
nipkow@13857
   238
wenzelm@24470
   239
wenzelm@24470
   240
subsection {* Derivation of the proof rules and, most importantly, the VCG tactic *}
wenzelm@24470
   241
wenzelm@24470
   242
lemma Compl_Collect: "-(Collect b) = {x. ~(b x)}"
wenzelm@24470
   243
  by blast
wenzelm@24470
   244
wenzelm@28457
   245
use "hoare_tac.ML"
nipkow@13857
   246
nipkow@13857
   247
method_setup vcg = {*
wenzelm@30549
   248
  Scan.succeed (fn ctxt => SIMPLE_METHOD' (hoare_tac ctxt (K all_tac))) *}
nipkow@13857
   249
  "verification condition generator"
nipkow@13857
   250
nipkow@13857
   251
method_setup vcg_simp = {*
wenzelm@30549
   252
  Scan.succeed (fn ctxt =>
wenzelm@32149
   253
    SIMPLE_METHOD' (hoare_tac ctxt (asm_full_simp_tac (simpset_of ctxt)))) *}
nipkow@13857
   254
  "verification condition generator plus simplification"
nipkow@13857
   255
nipkow@13875
   256
(* Special syntax for guarded statements and guarded array updates: *)
nipkow@13875
   257
nipkow@13875
   258
syntax
nipkow@13875
   259
  guarded_com :: "bool \<Rightarrow> 'a com \<Rightarrow> 'a com"  ("(2_ \<rightarrow>/ _)" 71)
wenzelm@35054
   260
  array_update :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a com"  ("(2_[_] :=/ _)" [70, 65] 61)
nipkow@13875
   261
translations
wenzelm@35101
   262
  "P \<rightarrow> c" == "IF P THEN c ELSE CONST Abort FI"
haftmann@34940
   263
  "a[i] := v" => "(i < CONST length a) \<rightarrow> (a := CONST list_update a i v)"
nipkow@13875
   264
  (* reverse translation not possible because of duplicate "a" *)
nipkow@13875
   265
nipkow@13875
   266
text{* Note: there is no special syntax for guarded array access. Thus
nipkow@13875
   267
you must write @{text"j < length a \<rightarrow> a[i] := a!j"}. *}
nipkow@13875
   268
nipkow@13857
   269
end