src/HOL/Hoare/hoare_tac.ML
author wenzelm
Fri Jan 07 18:32:19 2011 +0100 (2011-01-07)
changeset 41449 7339f0e7c513
parent 38012 3ca193a6ae5a
child 42793 88bee9f6eec7
permissions -rw-r--r--
do not open ML structures;
wenzelm@24475
     1
(*  Title:      HOL/Hoare/hoare_tac.ML
wenzelm@24475
     2
    Author:     Leonor Prensa Nieto & Tobias Nipkow
wenzelm@24475
     3
wenzelm@24475
     4
Derivation of the proof rules and, most importantly, the VCG tactic.
wenzelm@24475
     5
*)
wenzelm@24475
     6
wenzelm@41449
     7
(* FIXME structure Hoare: HOARE *)
wenzelm@41449
     8
wenzelm@24475
     9
(*** The tactics ***)
wenzelm@24475
    10
wenzelm@24475
    11
(*****************************************************************************)
wenzelm@24475
    12
(** The function Mset makes the theorem                                     **)
wenzelm@24475
    13
(** "?Mset <= {(x1,...,xn). ?P (x1,...,xn)} ==> ?Mset <= {s. ?P s}",        **)
wenzelm@24475
    14
(** where (x1,...,xn) are the variables of the particular program we are    **)
wenzelm@24475
    15
(** working on at the moment of the call                                    **)
wenzelm@24475
    16
(*****************************************************************************)
wenzelm@24475
    17
wenzelm@41449
    18
local
wenzelm@24475
    19
wenzelm@24475
    20
(** maps (%x1 ... xn. t) to [x1,...,xn] **)
haftmann@37591
    21
fun abs2list (Const (@{const_name prod_case}, _) $ Abs (x, T, t)) = Free (x, T) :: abs2list t
haftmann@37135
    22
  | abs2list (Abs (x, T, t)) = [Free (x, T)]
wenzelm@24475
    23
  | abs2list _ = [];
wenzelm@24475
    24
wenzelm@24475
    25
(** maps {(x1,...,xn). t} to [x1,...,xn] **)
haftmann@37677
    26
fun mk_vars (Const (@{const_name Collect},_) $ T) = abs2list T
wenzelm@24475
    27
  | mk_vars _ = [];
wenzelm@24475
    28
wenzelm@28457
    29
(** abstraction of body over a tuple formed from a list of free variables.
wenzelm@24475
    30
Types are also built **)
wenzelm@41449
    31
fun mk_abstupleC []     body = absfree ("x", HOLogic.unitT, body)
wenzelm@24475
    32
  | mk_abstupleC (v::w) body = let val (n,T) = dest_Free v
wenzelm@24475
    33
                               in if w=[] then absfree (n, T, body)
wenzelm@24475
    34
        else let val z  = mk_abstupleC w body;
wenzelm@24475
    35
                 val T2 = case z of Abs(_,T,_) => T
wenzelm@24475
    36
                        | Const (_, Type (_,[_, Type (_,[T,_])])) $ _ => T;
wenzelm@41449
    37
       in
wenzelm@41449
    38
        Const (@{const_name prod_case},
wenzelm@41449
    39
          (T --> T2 --> HOLogic.boolT) --> HOLogic.mk_prodT (T,T2) --> HOLogic.boolT)
wenzelm@41449
    40
            $ absfree (n, T, z)
wenzelm@41449
    41
       end end;
wenzelm@24475
    42
wenzelm@24475
    43
(** maps [x1,...,xn] to (x1,...,xn) and types**)
wenzelm@24475
    44
fun mk_bodyC []      = HOLogic.unit
wenzelm@28457
    45
  | mk_bodyC (x::xs) = if xs=[] then x
wenzelm@24475
    46
               else let val (n, T) = dest_Free x ;
wenzelm@24475
    47
                        val z = mk_bodyC xs;
wenzelm@24475
    48
                        val T2 = case z of Free(_, T) => T
haftmann@37391
    49
                                         | Const (@{const_name Pair}, Type ("fun", [_, Type
wenzelm@24475
    50
                                            ("fun", [_, T])])) $ _ $ _ => T;
wenzelm@41449
    51
                 in Const (@{const_name Pair}, [T, T2] ---> HOLogic.mk_prodT (T, T2)) $ x $ z end;
wenzelm@24475
    52
wenzelm@28457
    53
(** maps a subgoal of the form:
wenzelm@28457
    54
        VARS x1 ... xn {._.} _ {._.} or to [x1,...,xn]**)
wenzelm@28457
    55
fun get_vars c =
wenzelm@28457
    56
  let
wenzelm@28457
    57
    val d = Logic.strip_assums_concl c;
wenzelm@41449
    58
    val Const _ $ pre $ _ $ _ = HOLogic.dest_Trueprop d;
wenzelm@28457
    59
  in mk_vars pre end;
wenzelm@24475
    60
wenzelm@28457
    61
fun mk_CollectC trm =
wenzelm@28457
    62
  let val T as Type ("fun",[t,_]) = fastype_of trm
wenzelm@41449
    63
  in HOLogic.Collect_const t $ trm end;
wenzelm@24475
    64
wenzelm@41449
    65
fun inclt ty = Const (@{const_name Orderings.less_eq}, [ty,ty] ---> HOLogic.boolT);
wenzelm@24475
    66
wenzelm@41449
    67
in
wenzelm@24475
    68
wenzelm@28457
    69
fun Mset ctxt prop =
wenzelm@28457
    70
  let
wenzelm@28457
    71
    val [(Mset, _), (P, _)] = Variable.variant_frees ctxt [] [("Mset", ()), ("P", ())];
wenzelm@24475
    72
wenzelm@28457
    73
    val vars = get_vars prop;
wenzelm@28457
    74
    val varsT = fastype_of (mk_bodyC vars);
wenzelm@41449
    75
    val big_Collect = mk_CollectC (mk_abstupleC vars (Free (P, varsT --> HOLogic.boolT) $ mk_bodyC vars));
wenzelm@41449
    76
    val small_Collect = mk_CollectC (Abs ("x", varsT, Free (P, varsT --> HOLogic.boolT) $ Bound 0));
wenzelm@28457
    77
wenzelm@28457
    78
    val MsetT = fastype_of big_Collect;
wenzelm@41449
    79
    fun Mset_incl t = HOLogic.mk_Trueprop (inclt MsetT $ Free (Mset, MsetT) $ t);
wenzelm@28457
    80
    val impl = Logic.mk_implies (Mset_incl big_Collect, Mset_incl small_Collect);
wenzelm@32149
    81
    val th = Goal.prove ctxt [Mset, P] [] impl (fn _ => blast_tac (claset_of ctxt) 1);
wenzelm@28457
    82
 in (vars, th) end;
wenzelm@24475
    83
wenzelm@24475
    84
end;
wenzelm@24475
    85
wenzelm@24475
    86
wenzelm@24475
    87
(*****************************************************************************)
wenzelm@24475
    88
(** Simplifying:                                                            **)
wenzelm@24475
    89
(** Some useful lemmata, lists and simplification tactics to control which  **)
wenzelm@24475
    90
(** theorems are used to simplify at each moment, so that the original      **)
wenzelm@24475
    91
(** input does not suffer any unexpected transformation                     **)
wenzelm@24475
    92
(*****************************************************************************)
wenzelm@24475
    93
wenzelm@24475
    94
(**Simp_tacs**)
wenzelm@24475
    95
wenzelm@24475
    96
val before_set2pred_simp_tac =
wenzelm@26300
    97
  (simp_tac (HOL_basic_ss addsimps [Collect_conj_eq RS sym, @{thm Compl_Collect}]));
wenzelm@24475
    98
haftmann@37138
    99
val split_simp_tac = (simp_tac (HOL_basic_ss addsimps [@{thm split_conv}]));
wenzelm@24475
   100
wenzelm@24475
   101
(*****************************************************************************)
wenzelm@28457
   102
(** set2pred_tac transforms sets inclusion into predicates implication,     **)
wenzelm@24475
   103
(** maintaining the original variable names.                                **)
wenzelm@24475
   104
(** Ex. "{x. x=0} <= {x. x <= 1}" -set2pred-> "x=0 --> x <= 1"              **)
wenzelm@24475
   105
(** Subgoals containing intersections (A Int B) or complement sets (-A)     **)
wenzelm@24475
   106
(** are first simplified by "before_set2pred_simp_tac", that returns only   **)
wenzelm@24475
   107
(** subgoals of the form "{x. P x} <= {x. Q x}", which are easily           **)
wenzelm@24475
   108
(** transformed.                                                            **)
wenzelm@24475
   109
(** This transformation may solve very easy subgoals due to a ligth         **)
wenzelm@24475
   110
(** simplification done by (split_all_tac)                                  **)
wenzelm@24475
   111
(*****************************************************************************)
wenzelm@24475
   112
wenzelm@28457
   113
fun set2pred_tac var_names = SUBGOAL (fn (goal, i) =>
wenzelm@28457
   114
  before_set2pred_simp_tac i THEN_MAYBE
wenzelm@28457
   115
  EVERY [
wenzelm@28457
   116
    rtac subsetI i,
wenzelm@28457
   117
    rtac CollectI i,
wenzelm@28457
   118
    dtac CollectD i,
wenzelm@28457
   119
    TRY (split_all_tac i) THEN_MAYBE
haftmann@37138
   120
     (rename_tac var_names i THEN full_simp_tac (HOL_basic_ss addsimps [@{thm split_conv}]) i)]);
wenzelm@24475
   121
wenzelm@24475
   122
(*****************************************************************************)
wenzelm@24475
   123
(** BasicSimpTac is called to simplify all verification conditions. It does **)
wenzelm@24475
   124
(** a light simplification by applying "mem_Collect_eq", then it calls      **)
wenzelm@24475
   125
(** MaxSimpTac, which solves subgoals of the form "A <= A",                 **)
wenzelm@24475
   126
(** and transforms any other into predicates, applying then                 **)
wenzelm@24475
   127
(** the tactic chosen by the user, which may solve the subgoal completely.  **)
wenzelm@24475
   128
(*****************************************************************************)
wenzelm@24475
   129
wenzelm@28457
   130
fun MaxSimpTac var_names tac = FIRST'[rtac subset_refl, set2pred_tac var_names THEN_MAYBE' tac];
wenzelm@24475
   131
wenzelm@28457
   132
fun BasicSimpTac var_names tac =
wenzelm@28457
   133
  simp_tac
haftmann@38012
   134
    (HOL_basic_ss addsimps [mem_Collect_eq, @{thm split_conv}] addsimprocs [Record.simproc])
wenzelm@28457
   135
  THEN_MAYBE' MaxSimpTac var_names tac;
wenzelm@24475
   136
wenzelm@24475
   137
wenzelm@28457
   138
(** hoare_rule_tac **)
wenzelm@28457
   139
wenzelm@28457
   140
fun hoare_rule_tac (vars, Mlem) tac =
wenzelm@28457
   141
  let
wenzelm@28457
   142
    val var_names = map (fst o dest_Free) vars;
wenzelm@28457
   143
    fun wlp_tac i =
wenzelm@28457
   144
      rtac @{thm SeqRule} i THEN rule_tac false (i + 1)
wenzelm@28457
   145
    and rule_tac pre_cond i st = st |> (*abstraction over st prevents looping*)
wenzelm@28457
   146
      ((wlp_tac i THEN rule_tac pre_cond i)
wenzelm@28457
   147
        ORELSE
wenzelm@28457
   148
        (FIRST [
wenzelm@28457
   149
          rtac @{thm SkipRule} i,
wenzelm@28457
   150
          rtac @{thm AbortRule} i,
wenzelm@28457
   151
          EVERY [
wenzelm@28457
   152
            rtac @{thm BasicRule} i,
wenzelm@28457
   153
            rtac Mlem i,
wenzelm@28457
   154
            split_simp_tac i],
wenzelm@28457
   155
          EVERY [
wenzelm@28457
   156
            rtac @{thm CondRule} i,
wenzelm@28457
   157
            rule_tac false (i + 2),
wenzelm@28457
   158
            rule_tac false (i + 1)],
wenzelm@28457
   159
          EVERY [
wenzelm@28457
   160
            rtac @{thm WhileRule} i,
wenzelm@28457
   161
            BasicSimpTac var_names tac (i + 2),
wenzelm@28457
   162
            rule_tac true (i + 1)]]
wenzelm@28457
   163
         THEN (if pre_cond then BasicSimpTac var_names tac i else rtac subset_refl i)));
wenzelm@28457
   164
  in rule_tac end;
wenzelm@28457
   165
wenzelm@28457
   166
wenzelm@28457
   167
(** tac is the tactic the user chooses to solve or simplify **)
wenzelm@28457
   168
(** the final verification conditions                       **)
wenzelm@28457
   169
wenzelm@28457
   170
fun hoare_tac ctxt (tac: int -> tactic) = SUBGOAL (fn (goal, i) =>
wenzelm@28457
   171
  SELECT_GOAL (hoare_rule_tac (Mset ctxt goal) tac true 1) i);
wenzelm@28457
   172