src/HOL/Isar_Examples/Hoare.thy
author haftmann
Wed Jun 30 16:46:44 2010 +0200 (2010-06-30)
changeset 37659 14cabf5fa710
parent 35417 47ee18b6ae32
child 37671 fa53d267dab3
permissions -rw-r--r--
more speaking names
wenzelm@33026
     1
(*  Title:      HOL/Isar_Examples/Hoare.thy
wenzelm@10148
     2
    Author:     Markus Wenzel, TU Muenchen
wenzelm@10148
     3
wenzelm@10148
     4
A formulation of Hoare logic suitable for Isar.
wenzelm@10148
     5
*)
wenzelm@10148
     6
wenzelm@10148
     7
header {* Hoare Logic *}
wenzelm@10148
     8
wenzelm@31758
     9
theory Hoare
wenzelm@31758
    10
imports Main
wenzelm@31758
    11
uses ("~~/src/HOL/Hoare/hoare_tac.ML")
wenzelm@31758
    12
begin
wenzelm@10148
    13
wenzelm@10148
    14
subsection {* Abstract syntax and semantics *}
wenzelm@10148
    15
wenzelm@10148
    16
text {*
wenzelm@10148
    17
 The following abstract syntax and semantics of Hoare Logic over
wenzelm@10148
    18
 \texttt{WHILE} programs closely follows the existing tradition in
wenzelm@10148
    19
 Isabelle/HOL of formalizing the presentation given in
wenzelm@10148
    20
 \cite[\S6]{Winskel:1993}.  See also
wenzelm@10148
    21
 \url{http://isabelle.in.tum.de/library/Hoare/} and
wenzelm@10148
    22
 \cite{Nipkow:1998:Winskel}.
wenzelm@10148
    23
*}
wenzelm@10148
    24
wenzelm@10148
    25
types
wenzelm@10148
    26
  'a bexp = "'a set"
wenzelm@10148
    27
  'a assn = "'a set"
wenzelm@10148
    28
wenzelm@10148
    29
datatype 'a com =
wenzelm@10148
    30
    Basic "'a => 'a"
wenzelm@10148
    31
  | Seq "'a com" "'a com"    ("(_;/ _)" [60, 61] 60)
wenzelm@10148
    32
  | Cond "'a bexp" "'a com" "'a com"
wenzelm@10148
    33
  | While "'a bexp" "'a assn" "'a com"
wenzelm@10148
    34
wenzelm@19363
    35
abbreviation
wenzelm@21404
    36
  Skip  ("SKIP") where
wenzelm@19122
    37
  "SKIP == Basic id"
wenzelm@10148
    38
wenzelm@10148
    39
types
wenzelm@10148
    40
  'a sem = "'a => 'a => bool"
wenzelm@10148
    41
wenzelm@10148
    42
consts
wenzelm@10148
    43
  iter :: "nat => 'a bexp => 'a sem => 'a sem"
wenzelm@10148
    44
primrec
wenzelm@10148
    45
  "iter 0 b S s s' = (s ~: b & s = s')"
wenzelm@10148
    46
  "iter (Suc n) b S s s' =
wenzelm@10148
    47
    (s : b & (EX s''. S s s'' & iter n b S s'' s'))"
wenzelm@10148
    48
wenzelm@10148
    49
consts
wenzelm@10148
    50
  Sem :: "'a com => 'a sem"
wenzelm@10148
    51
primrec
wenzelm@10148
    52
  "Sem (Basic f) s s' = (s' = f s)"
wenzelm@10148
    53
  "Sem (c1; c2) s s' = (EX s''. Sem c1 s s'' & Sem c2 s'' s')"
wenzelm@10148
    54
  "Sem (Cond b c1 c2) s s' =
wenzelm@10148
    55
    (if s : b then Sem c1 s s' else Sem c2 s s')"
wenzelm@10148
    56
  "Sem (While b x c) s s' = (EX n. iter n b (Sem c) s s')"
wenzelm@10148
    57
haftmann@35416
    58
definition Valid :: "'a bexp => 'a com => 'a bexp => bool" ("(3|- _/ (2_)/ _)" [100, 55, 100] 50) where
wenzelm@10148
    59
  "|- P c Q == ALL s s'. Sem c s s' --> s : P --> s' : Q"
wenzelm@10148
    60
haftmann@35416
    61
notation (xsymbols) Valid ("(3\<turnstile> _/ (2_)/ _)" [100, 55, 100] 50)
wenzelm@10148
    62
wenzelm@10148
    63
lemma ValidI [intro?]:
wenzelm@10148
    64
    "(!!s s'. Sem c s s' ==> s : P ==> s' : Q) ==> |- P c Q"
wenzelm@10148
    65
  by (simp add: Valid_def)
wenzelm@10148
    66
wenzelm@10148
    67
lemma ValidD [dest?]:
wenzelm@10148
    68
    "|- P c Q ==> Sem c s s' ==> s : P ==> s' : Q"
wenzelm@10148
    69
  by (simp add: Valid_def)
wenzelm@10148
    70
wenzelm@10148
    71
wenzelm@10148
    72
subsection {* Primitive Hoare rules *}
wenzelm@10148
    73
wenzelm@10148
    74
text {*
wenzelm@10148
    75
 From the semantics defined above, we derive the standard set of
wenzelm@10148
    76
 primitive Hoare rules; e.g.\ see \cite[\S6]{Winskel:1993}.  Usually,
wenzelm@10148
    77
 variant forms of these rules are applied in actual proof, see also
wenzelm@10148
    78
 \S\ref{sec:hoare-isar} and \S\ref{sec:hoare-vcg}.
wenzelm@10148
    79
wenzelm@10148
    80
 \medskip The \name{basic} rule represents any kind of atomic access
wenzelm@10148
    81
 to the state space.  This subsumes the common rules of \name{skip}
wenzelm@10148
    82
 and \name{assign}, as formulated in \S\ref{sec:hoare-isar}.
wenzelm@10148
    83
*}
wenzelm@10148
    84
wenzelm@10148
    85
theorem basic: "|- {s. f s : P} (Basic f) P"
wenzelm@10148
    86
proof
wenzelm@10148
    87
  fix s s' assume s: "s : {s. f s : P}"
wenzelm@10148
    88
  assume "Sem (Basic f) s s'"
wenzelm@10148
    89
  hence "s' = f s" by simp
wenzelm@10148
    90
  with s show "s' : P" by simp
wenzelm@10148
    91
qed
wenzelm@10148
    92
wenzelm@10148
    93
text {*
wenzelm@10148
    94
 The rules for sequential commands and semantic consequences are
wenzelm@10148
    95
 established in a straight forward manner as follows.
wenzelm@10148
    96
*}
wenzelm@10148
    97
wenzelm@10148
    98
theorem seq: "|- P c1 Q ==> |- Q c2 R ==> |- P (c1; c2) R"
wenzelm@10148
    99
proof
wenzelm@10148
   100
  assume cmd1: "|- P c1 Q" and cmd2: "|- Q c2 R"
wenzelm@10148
   101
  fix s s' assume s: "s : P"
wenzelm@10148
   102
  assume "Sem (c1; c2) s s'"
wenzelm@10148
   103
  then obtain s'' where sem1: "Sem c1 s s''" and sem2: "Sem c2 s'' s'"
wenzelm@10148
   104
    by auto
wenzelm@10148
   105
  from cmd1 sem1 s have "s'' : Q" ..
wenzelm@10148
   106
  with cmd2 sem2 show "s' : R" ..
wenzelm@10148
   107
qed
wenzelm@10148
   108
wenzelm@10148
   109
theorem conseq: "P' <= P ==> |- P c Q ==> Q <= Q' ==> |- P' c Q'"
wenzelm@10148
   110
proof
wenzelm@10148
   111
  assume P'P: "P' <= P" and QQ': "Q <= Q'"
wenzelm@10148
   112
  assume cmd: "|- P c Q"
wenzelm@10148
   113
  fix s s' :: 'a
wenzelm@10148
   114
  assume sem: "Sem c s s'"
wenzelm@10148
   115
  assume "s : P'" with P'P have "s : P" ..
wenzelm@10148
   116
  with cmd sem have "s' : Q" ..
wenzelm@10148
   117
  with QQ' show "s' : Q'" ..
wenzelm@10148
   118
qed
wenzelm@10148
   119
wenzelm@10148
   120
text {*
wenzelm@10148
   121
 The rule for conditional commands is directly reflected by the
wenzelm@10148
   122
 corresponding semantics; in the proof we just have to look closely
wenzelm@10148
   123
 which cases apply.
wenzelm@10148
   124
*}
wenzelm@10148
   125
wenzelm@10148
   126
theorem cond:
wenzelm@10148
   127
  "|- (P Int b) c1 Q ==> |- (P Int -b) c2 Q ==> |- P (Cond b c1 c2) Q"
wenzelm@10148
   128
proof
wenzelm@10148
   129
  assume case_b: "|- (P Int b) c1 Q" and case_nb: "|- (P Int -b) c2 Q"
wenzelm@10148
   130
  fix s s' assume s: "s : P"
wenzelm@10148
   131
  assume sem: "Sem (Cond b c1 c2) s s'"
wenzelm@10148
   132
  show "s' : Q"
wenzelm@10148
   133
  proof cases
wenzelm@10148
   134
    assume b: "s : b"
wenzelm@10148
   135
    from case_b show ?thesis
wenzelm@10148
   136
    proof
wenzelm@10148
   137
      from sem b show "Sem c1 s s'" by simp
wenzelm@10148
   138
      from s b show "s : P Int b" by simp
wenzelm@10148
   139
    qed
wenzelm@10148
   140
  next
wenzelm@10148
   141
    assume nb: "s ~: b"
wenzelm@10148
   142
    from case_nb show ?thesis
wenzelm@10148
   143
    proof
wenzelm@10148
   144
      from sem nb show "Sem c2 s s'" by simp
wenzelm@10148
   145
      from s nb show "s : P Int -b" by simp
wenzelm@10148
   146
    qed
wenzelm@10148
   147
  qed
wenzelm@10148
   148
qed
wenzelm@10148
   149
wenzelm@10148
   150
text {*
wenzelm@10148
   151
 The \name{while} rule is slightly less trivial --- it is the only one
wenzelm@10148
   152
 based on recursion, which is expressed in the semantics by a
wenzelm@10148
   153
 Kleene-style least fixed-point construction.  The auxiliary statement
wenzelm@10148
   154
 below, which is by induction on the number of iterations is the main
wenzelm@10148
   155
 point to be proven; the rest is by routine application of the
wenzelm@10148
   156
 semantics of \texttt{WHILE}.
wenzelm@10148
   157
*}
wenzelm@10148
   158
wenzelm@18241
   159
theorem while:
wenzelm@18241
   160
  assumes body: "|- (P Int b) c P"
wenzelm@18241
   161
  shows "|- P (While b X c) (P Int -b)"
wenzelm@10148
   162
proof
wenzelm@10148
   163
  fix s s' assume s: "s : P"
wenzelm@10148
   164
  assume "Sem (While b X c) s s'"
wenzelm@18241
   165
  then obtain n where "iter n b (Sem c) s s'" by auto
wenzelm@18241
   166
  from this and s show "s' : P Int -b"
wenzelm@20503
   167
  proof (induct n arbitrary: s)
wenzelm@19122
   168
    case 0
wenzelm@11987
   169
    thus ?case by auto
wenzelm@11987
   170
  next
wenzelm@19122
   171
    case (Suc n)
wenzelm@11987
   172
    then obtain s'' where b: "s : b" and sem: "Sem c s s''"
wenzelm@11987
   173
      and iter: "iter n b (Sem c) s'' s'"
wenzelm@11987
   174
      by auto
wenzelm@11987
   175
    from Suc and b have "s : P Int b" by simp
wenzelm@11987
   176
    with body sem have "s'' : P" ..
wenzelm@11987
   177
    with iter show ?case by (rule Suc)
wenzelm@10148
   178
  qed
wenzelm@10148
   179
qed
wenzelm@10148
   180
wenzelm@10148
   181
wenzelm@10148
   182
subsection {* Concrete syntax for assertions *}
wenzelm@10148
   183
wenzelm@10148
   184
text {*
wenzelm@10148
   185
 We now introduce concrete syntax for describing commands (with
wenzelm@10148
   186
 embedded expressions) and assertions. The basic technique is that of
wenzelm@10148
   187
 semantic ``quote-antiquote''.  A \emph{quotation} is a syntactic
wenzelm@10148
   188
 entity delimited by an implicit abstraction, say over the state
wenzelm@10148
   189
 space.  An \emph{antiquotation} is a marked expression within a
wenzelm@10148
   190
 quotation that refers the implicit argument; a typical antiquotation
wenzelm@10148
   191
 would select (or even update) components from the state.
wenzelm@10148
   192
wenzelm@10148
   193
 We will see some examples later in the concrete rules and
wenzelm@10148
   194
 applications.
wenzelm@10148
   195
*}
wenzelm@10148
   196
wenzelm@10148
   197
text {*
wenzelm@10148
   198
 The following specification of syntax and translations is for
wenzelm@10148
   199
 Isabelle experts only; feel free to ignore it.
wenzelm@10148
   200
wenzelm@10148
   201
 While the first part is still a somewhat intelligible specification
wenzelm@10148
   202
 of the concrete syntactic representation of our Hoare language, the
wenzelm@10148
   203
 actual ``ML drivers'' is quite involved.  Just note that the we
wenzelm@10148
   204
 re-use the basic quote/antiquote translations as already defined in
wenzelm@10148
   205
 Isabelle/Pure (see \verb,Syntax.quote_tr, and
wenzelm@10148
   206
 \verb,Syntax.quote_tr',).
wenzelm@10148
   207
*}
wenzelm@10148
   208
wenzelm@10148
   209
syntax
wenzelm@10874
   210
  "_quote"       :: "'b => ('a => 'b)"       ("(.'(_').)" [0] 1000)
wenzelm@10874
   211
  "_antiquote"   :: "('a => 'b) => 'b"       ("\<acute>_" [1000] 1000)
wenzelm@10874
   212
  "_Subst"       :: "'a bexp \<Rightarrow> 'b \<Rightarrow> idt \<Rightarrow> 'a bexp"
wenzelm@10874
   213
        ("_[_'/\<acute>_]" [1000] 999)
wenzelm@10874
   214
  "_Assert"      :: "'a => 'a set"           ("(.{_}.)" [0] 1000)
wenzelm@10874
   215
  "_Assign"      :: "idt => 'b => 'a com"    ("(\<acute>_ :=/ _)" [70, 65] 61)
wenzelm@10148
   216
  "_Cond"        :: "'a bexp => 'a com => 'a com => 'a com"
wenzelm@10148
   217
        ("(0IF _/ THEN _/ ELSE _/ FI)" [0, 0, 0] 61)
wenzelm@10148
   218
  "_While_inv"   :: "'a bexp => 'a assn => 'a com => 'a com"
wenzelm@10148
   219
        ("(0WHILE _/ INV _ //DO _ /OD)"  [0, 0, 0] 61)
wenzelm@10148
   220
  "_While"       :: "'a bexp => 'a com => 'a com"
wenzelm@10148
   221
        ("(0WHILE _ //DO _ /OD)"  [0, 0] 61)
wenzelm@10148
   222
wenzelm@10148
   223
syntax (xsymbols)
wenzelm@10148
   224
  "_Assert"      :: "'a => 'a set"            ("(\<lbrace>_\<rbrace>)" [0] 1000)
wenzelm@10148
   225
wenzelm@10148
   226
translations
wenzelm@35054
   227
  ".{b}."                   => "CONST Collect .(b)."
schirmer@25706
   228
  "B [a/\<acute>x]"                => ".{\<acute>(_update_name x (\<lambda>_. a)) \<in> B}."
wenzelm@35054
   229
  "\<acute>x := a"                 => "CONST Basic .(\<acute>(_update_name x (\<lambda>_. a)))."
wenzelm@35054
   230
  "IF b THEN c1 ELSE c2 FI" => "CONST Cond .{b}. c1 c2"
wenzelm@35054
   231
  "WHILE b INV i DO c OD"   => "CONST While .{b}. i c"
haftmann@28524
   232
  "WHILE b DO c OD"         == "WHILE b INV CONST undefined DO c OD"
wenzelm@10148
   233
wenzelm@10148
   234
parse_translation {*
wenzelm@10148
   235
  let
wenzelm@35113
   236
    fun quote_tr [t] = Syntax.quote_tr @{syntax_const "_antiquote"} t
wenzelm@10148
   237
      | quote_tr ts = raise TERM ("quote_tr", ts);
wenzelm@35113
   238
  in [(@{syntax_const "_quote"}, quote_tr)] end
wenzelm@10148
   239
*}
wenzelm@10148
   240
wenzelm@10148
   241
text {*
wenzelm@10148
   242
 As usual in Isabelle syntax translations, the part for printing is
wenzelm@10148
   243
 more complicated --- we cannot express parts as macro rules as above.
wenzelm@10148
   244
 Don't look here, unless you have to do similar things for yourself.
wenzelm@10148
   245
*}
wenzelm@10148
   246
wenzelm@10148
   247
print_translation {*
wenzelm@10148
   248
  let
wenzelm@10148
   249
    fun quote_tr' f (t :: ts) =
wenzelm@35113
   250
          Term.list_comb (f $ Syntax.quote_tr' @{syntax_const "_antiquote"} t, ts)
wenzelm@10148
   251
      | quote_tr' _ _ = raise Match;
wenzelm@10148
   252
wenzelm@35113
   253
    val assert_tr' = quote_tr' (Syntax.const @{syntax_const "_Assert"});
wenzelm@10148
   254
wenzelm@35113
   255
    fun bexp_tr' name ((Const (@{const_syntax Collect}, _) $ t) :: ts) =
wenzelm@10148
   256
          quote_tr' (Syntax.const name) (t :: ts)
wenzelm@10148
   257
      | bexp_tr' _ _ = raise Match;
wenzelm@10148
   258
wenzelm@35145
   259
    fun K_tr' (Abs (_, _, t)) =
wenzelm@35145
   260
          if null (loose_bnos t) then t else raise Match
wenzelm@35145
   261
      | K_tr' (Abs (_, _, Abs (_, _, t) $ Bound 0)) =
wenzelm@35145
   262
          if null (loose_bnos t) then t else raise Match
schirmer@25706
   263
      | K_tr' _ = raise Match;
schirmer@25706
   264
schirmer@25706
   265
    fun assign_tr' (Abs (x, _, f $ k $ Bound 0) :: ts) =
wenzelm@35145
   266
          quote_tr' (Syntax.const @{syntax_const "_Assign"} $ Syntax.update_name_tr' f)
schirmer@25706
   267
            (Abs (x, dummyT, K_tr' k) :: ts)
wenzelm@10148
   268
      | assign_tr' _ = raise Match;
wenzelm@10148
   269
  in
wenzelm@35113
   270
   [(@{const_syntax Collect}, assert_tr'),
wenzelm@35113
   271
    (@{const_syntax Basic}, assign_tr'),
wenzelm@35113
   272
    (@{const_syntax Cond}, bexp_tr' @{syntax_const "_Cond"}),
wenzelm@35113
   273
    (@{const_syntax While}, bexp_tr' @{syntax_const "_While_inv"})]
wenzelm@10148
   274
  end
wenzelm@10148
   275
*}
wenzelm@10148
   276
wenzelm@10148
   277
wenzelm@10148
   278
subsection {* Rules for single-step proof \label{sec:hoare-isar} *}
wenzelm@10148
   279
wenzelm@10148
   280
text {*
wenzelm@10148
   281
 We are now ready to introduce a set of Hoare rules to be used in
wenzelm@10148
   282
 single-step structured proofs in Isabelle/Isar.  We refer to the
wenzelm@10148
   283
 concrete syntax introduce above.
wenzelm@10148
   284
wenzelm@10148
   285
 \medskip Assertions of Hoare Logic may be manipulated in
wenzelm@10148
   286
 calculational proofs, with the inclusion expressed in terms of sets
wenzelm@10148
   287
 or predicates.  Reversed order is supported as well.
wenzelm@10148
   288
*}
wenzelm@10148
   289
wenzelm@10148
   290
lemma [trans]: "|- P c Q ==> P' <= P ==> |- P' c Q"
wenzelm@10148
   291
  by (unfold Valid_def) blast
wenzelm@10148
   292
lemma [trans] : "P' <= P ==> |- P c Q ==> |- P' c Q"
wenzelm@10148
   293
  by (unfold Valid_def) blast
wenzelm@10148
   294
wenzelm@10148
   295
lemma [trans]: "Q <= Q' ==> |- P c Q ==> |- P c Q'"
wenzelm@10148
   296
  by (unfold Valid_def) blast
wenzelm@10148
   297
lemma [trans]: "|- P c Q ==> Q <= Q' ==> |- P c Q'"
wenzelm@10148
   298
  by (unfold Valid_def) blast
wenzelm@10148
   299
wenzelm@10148
   300
lemma [trans]:
wenzelm@10838
   301
    "|- .{\<acute>P}. c Q ==> (!!s. P' s --> P s) ==> |- .{\<acute>P'}. c Q"
wenzelm@10148
   302
  by (simp add: Valid_def)
wenzelm@10148
   303
lemma [trans]:
wenzelm@10838
   304
    "(!!s. P' s --> P s) ==> |- .{\<acute>P}. c Q ==> |- .{\<acute>P'}. c Q"
wenzelm@10148
   305
  by (simp add: Valid_def)
wenzelm@10148
   306
wenzelm@10148
   307
lemma [trans]:
wenzelm@10838
   308
    "|- P c .{\<acute>Q}. ==> (!!s. Q s --> Q' s) ==> |- P c .{\<acute>Q'}."
wenzelm@10148
   309
  by (simp add: Valid_def)
wenzelm@10148
   310
lemma [trans]:
wenzelm@10838
   311
    "(!!s. Q s --> Q' s) ==> |- P c .{\<acute>Q}. ==> |- P c .{\<acute>Q'}."
wenzelm@10148
   312
  by (simp add: Valid_def)
wenzelm@10148
   313
wenzelm@10148
   314
wenzelm@10148
   315
text {*
wenzelm@10148
   316
 Identity and basic assignments.\footnote{The $\idt{hoare}$ method
wenzelm@10148
   317
 introduced in \S\ref{sec:hoare-vcg} is able to provide proper
wenzelm@10148
   318
 instances for any number of basic assignments, without producing
wenzelm@10148
   319
 additional verification conditions.}
wenzelm@10148
   320
*}
wenzelm@10148
   321
wenzelm@10148
   322
lemma skip [intro?]: "|- P SKIP P"
wenzelm@10148
   323
proof -
wenzelm@10148
   324
  have "|- {s. id s : P} SKIP P" by (rule basic)
wenzelm@10148
   325
  thus ?thesis by simp
wenzelm@10148
   326
qed
wenzelm@10148
   327
wenzelm@10869
   328
lemma assign: "|- P [\<acute>a/\<acute>x] \<acute>x := \<acute>a P"
wenzelm@10148
   329
  by (rule basic)
wenzelm@10148
   330
wenzelm@10148
   331
text {*
wenzelm@10148
   332
 Note that above formulation of assignment corresponds to our
wenzelm@10148
   333
 preferred way to model state spaces, using (extensible) record types
wenzelm@10148
   334
 in HOL \cite{Naraschewski-Wenzel:1998:HOOL}.  For any record field
wenzelm@10148
   335
 $x$, Isabelle/HOL provides a functions $x$ (selector) and
wenzelm@10148
   336
 $\idt{x{\dsh}update}$ (update).  Above, there is only a place-holder
wenzelm@10148
   337
 appearing for the latter kind of function: due to concrete syntax
wenzelm@10838
   338
 \isa{\'x := \'a} also contains \isa{x\_update}.\footnote{Note that due
wenzelm@10148
   339
 to the external nature of HOL record fields, we could not even state
wenzelm@10148
   340
 a general theorem relating selector and update functions (if this
wenzelm@10148
   341
 were required here); this would only work for any particular instance
wenzelm@10148
   342
 of record fields introduced so far.}
wenzelm@10148
   343
*}
wenzelm@10148
   344
wenzelm@10148
   345
text {*
wenzelm@10148
   346
 Sequential composition --- normalizing with associativity achieves
wenzelm@10148
   347
 proper of chunks of code verified separately.
wenzelm@10148
   348
*}
wenzelm@10148
   349
wenzelm@10148
   350
lemmas [trans, intro?] = seq
wenzelm@10148
   351
wenzelm@10148
   352
lemma seq_assoc [simp]: "( |- P c1;(c2;c3) Q) = ( |- P (c1;c2);c3 Q)"
wenzelm@10148
   353
  by (auto simp add: Valid_def)
wenzelm@10148
   354
wenzelm@10148
   355
text {*
wenzelm@10148
   356
 Conditional statements.
wenzelm@10148
   357
*}
wenzelm@10148
   358
wenzelm@10148
   359
lemmas [trans, intro?] = cond
wenzelm@10148
   360
wenzelm@10148
   361
lemma [trans, intro?]:
wenzelm@10838
   362
  "|- .{\<acute>P & \<acute>b}. c1 Q
wenzelm@10838
   363
      ==> |- .{\<acute>P & ~ \<acute>b}. c2 Q
wenzelm@10838
   364
      ==> |- .{\<acute>P}. IF \<acute>b THEN c1 ELSE c2 FI Q"
wenzelm@10148
   365
    by (rule cond) (simp_all add: Valid_def)
wenzelm@10148
   366
wenzelm@10148
   367
text {*
wenzelm@10148
   368
 While statements --- with optional invariant.
wenzelm@10148
   369
*}
wenzelm@10148
   370
wenzelm@10148
   371
lemma [intro?]:
wenzelm@10148
   372
    "|- (P Int b) c P ==> |- P (While b P c) (P Int -b)"
wenzelm@10148
   373
  by (rule while)
wenzelm@10148
   374
wenzelm@10148
   375
lemma [intro?]:
haftmann@28524
   376
    "|- (P Int b) c P ==> |- P (While b undefined c) (P Int -b)"
wenzelm@10148
   377
  by (rule while)
wenzelm@10148
   378
wenzelm@10148
   379
wenzelm@10148
   380
lemma [intro?]:
wenzelm@10838
   381
  "|- .{\<acute>P & \<acute>b}. c .{\<acute>P}.
wenzelm@10838
   382
    ==> |- .{\<acute>P}. WHILE \<acute>b INV .{\<acute>P}. DO c OD .{\<acute>P & ~ \<acute>b}."
wenzelm@10148
   383
  by (simp add: while Collect_conj_eq Collect_neg_eq)
wenzelm@10148
   384
wenzelm@10148
   385
lemma [intro?]:
wenzelm@10838
   386
  "|- .{\<acute>P & \<acute>b}. c .{\<acute>P}.
wenzelm@10838
   387
    ==> |- .{\<acute>P}. WHILE \<acute>b DO c OD .{\<acute>P & ~ \<acute>b}."
wenzelm@10148
   388
  by (simp add: while Collect_conj_eq Collect_neg_eq)
wenzelm@10148
   389
wenzelm@10148
   390
wenzelm@10148
   391
subsection {* Verification conditions \label{sec:hoare-vcg} *}
wenzelm@10148
   392
wenzelm@10148
   393
text {*
wenzelm@10148
   394
 We now load the \emph{original} ML file for proof scripts and tactic
wenzelm@10148
   395
 definition for the Hoare Verification Condition Generator (see
wenzelm@10148
   396
 \url{http://isabelle.in.tum.de/library/Hoare/}).  As far as we are
wenzelm@10148
   397
 concerned here, the result is a proof method \name{hoare}, which may
wenzelm@10148
   398
 be applied to a Hoare Logic assertion to extract purely logical
wenzelm@10148
   399
 verification conditions.  It is important to note that the method
wenzelm@10148
   400
 requires \texttt{WHILE} loops to be fully annotated with invariants
wenzelm@10148
   401
 beforehand.  Furthermore, only \emph{concrete} pieces of code are
wenzelm@10148
   402
 handled --- the underlying tactic fails ungracefully if supplied with
wenzelm@10148
   403
 meta-variables or parameters, for example.
wenzelm@10148
   404
*}
wenzelm@10148
   405
kleing@13862
   406
lemma SkipRule: "p \<subseteq> q \<Longrightarrow> Valid p (Basic id) q"
wenzelm@18193
   407
  by (auto simp add: Valid_def)
kleing@13862
   408
kleing@13862
   409
lemma BasicRule: "p \<subseteq> {s. f s \<in> q} \<Longrightarrow> Valid p (Basic f) q"
wenzelm@18193
   410
  by (auto simp: Valid_def)
kleing@13862
   411
kleing@13862
   412
lemma SeqRule: "Valid P c1 Q \<Longrightarrow> Valid Q c2 R \<Longrightarrow> Valid P (c1;c2) R"
wenzelm@18193
   413
  by (auto simp: Valid_def)
kleing@13862
   414
kleing@13862
   415
lemma CondRule:
wenzelm@18193
   416
  "p \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> w) \<and> (s \<notin> b \<longrightarrow> s \<in> w')}
wenzelm@18193
   417
    \<Longrightarrow> Valid w c1 q \<Longrightarrow> Valid w' c2 q \<Longrightarrow> Valid p (Cond b c1 c2) q"
wenzelm@18193
   418
  by (auto simp: Valid_def)
kleing@13862
   419
wenzelm@18241
   420
lemma iter_aux:
wenzelm@18241
   421
  "\<forall>s s'. Sem c s s' --> s : I & s : b --> s' : I ==>
wenzelm@18193
   422
       (\<And>s s'. s : I \<Longrightarrow> iter n b (Sem c) s s' \<Longrightarrow> s' : I & s' ~: b)"
wenzelm@18193
   423
  apply(induct n)
wenzelm@18193
   424
   apply clarsimp
wenzelm@18193
   425
   apply (simp (no_asm_use))
wenzelm@18193
   426
   apply blast
wenzelm@18193
   427
  done
kleing@13862
   428
kleing@13862
   429
lemma WhileRule:
wenzelm@18193
   430
    "p \<subseteq> i \<Longrightarrow> Valid (i \<inter> b) c i \<Longrightarrow> i \<inter> (-b) \<subseteq> q \<Longrightarrow> Valid p (While b i c) q"
wenzelm@18193
   431
  apply (clarsimp simp: Valid_def)
wenzelm@18193
   432
  apply (drule iter_aux)
wenzelm@18193
   433
    prefer 2
wenzelm@18193
   434
    apply assumption
wenzelm@18193
   435
   apply blast
wenzelm@18193
   436
  apply blast
wenzelm@18193
   437
  done
kleing@13862
   438
wenzelm@26303
   439
lemma Compl_Collect: "- Collect b = {x. \<not> b x}"
wenzelm@26303
   440
  by blast
wenzelm@26303
   441
wenzelm@28457
   442
lemmas AbortRule = SkipRule  -- "dummy version"
wenzelm@28457
   443
wenzelm@24472
   444
use "~~/src/HOL/Hoare/hoare_tac.ML"
wenzelm@10148
   445
wenzelm@10148
   446
method_setup hoare = {*
wenzelm@30549
   447
  Scan.succeed (fn ctxt =>
wenzelm@30510
   448
    (SIMPLE_METHOD'
wenzelm@28457
   449
       (hoare_tac ctxt (simp_tac (HOL_basic_ss addsimps [@{thm "Record.K_record_comp"}] ))))) *}
wenzelm@10148
   450
  "verification condition generator for Hoare logic"
wenzelm@10148
   451
kleing@13703
   452
end