src/HOL/Hoare/hoare.ML
author webertj
Mon Mar 07 19:30:53 2005 +0100 (2005-03-07)
changeset 15584 3478bb4f93ff
parent 15531 08c8dad8e399
child 15661 9ef583b08647
permissions -rw-r--r--
refute_params: default value itself=1 added (for type classes)
     1 (*  Title:      HOL/Hoare/Hoare.ML
     2     ID:         $Id$
     3     Author:     Leonor Prensa Nieto & Tobias Nipkow
     4     Copyright   1998 TUM
     5 
     6 Derivation of the proof rules and, most importantly, the VCG tactic.
     7 *)
     8 
     9 val SkipRule = thm"SkipRule";
    10 val BasicRule = thm"BasicRule";
    11 val SeqRule = thm"SeqRule";
    12 val CondRule = thm"CondRule";
    13 val WhileRule = thm"WhileRule";
    14 
    15 (*** The tactics ***)
    16 
    17 (*****************************************************************************)
    18 (** The function Mset makes the theorem                                     **)
    19 (** "?Mset <= {(x1,...,xn). ?P (x1,...,xn)} ==> ?Mset <= {s. ?P s}",        **)
    20 (** where (x1,...,xn) are the variables of the particular program we are    **)
    21 (** working on at the moment of the call                                    **)
    22 (*****************************************************************************)
    23 
    24 local open HOLogic in
    25 
    26 (** maps (%x1 ... xn. t) to [x1,...,xn] **)
    27 fun abs2list (Const ("split",_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t
    28   | abs2list (Abs(x,T,t)) = [Free (x, T)]
    29   | abs2list _ = [];
    30 
    31 (** maps {(x1,...,xn). t} to [x1,...,xn] **)
    32 fun mk_vars (Const ("Collect",_) $ T) = abs2list T
    33   | mk_vars _ = [];
    34 
    35 (** abstraction of body over a tuple formed from a list of free variables. 
    36 Types are also built **)
    37 fun mk_abstupleC []     body = absfree ("x", unitT, body)
    38   | mk_abstupleC (v::w) body = let val (n,T) = dest_Free v
    39                                in if w=[] then absfree (n, T, body)
    40         else let val z  = mk_abstupleC w body;
    41                  val T2 = case z of Abs(_,T,_) => T
    42                         | Const (_, Type (_,[_, Type (_,[T,_])])) $ _ => T;
    43        in Const ("split", (T --> T2 --> boolT) --> mk_prodT (T,T2) --> boolT) 
    44           $ absfree (n, T, z) end end;
    45 
    46 (** maps [x1,...,xn] to (x1,...,xn) and types**)
    47 fun mk_bodyC []      = HOLogic.unit
    48   | mk_bodyC (x::xs) = if xs=[] then x 
    49                else let val (n, T) = dest_Free x ;
    50                         val z = mk_bodyC xs;
    51                         val T2 = case z of Free(_, T) => T
    52                                          | Const ("Pair", Type ("fun", [_, Type
    53                                             ("fun", [_, T])])) $ _ $ _ => T;
    54                  in Const ("Pair", [T, T2] ---> mk_prodT (T, T2)) $ x $ z end;
    55 
    56 fun dest_Goal (Const ("Goal", _) $ P) = P;
    57 
    58 (** maps a goal of the form:
    59         1. [| P |] ==> VARS x1 ... xn {._.} _ {._.} or to [x1,...,xn]**) 
    60 fun get_vars thm = let  val c = dest_Goal (concl_of (thm));
    61                         val d = Logic.strip_assums_concl c;
    62                         val Const _ $ pre $ _ $ _ = dest_Trueprop d;
    63       in mk_vars pre end;
    64 
    65 
    66 (** Makes Collect with type **)
    67 fun mk_CollectC trm = let val T as Type ("fun",[t,_]) = fastype_of trm 
    68                       in Collect_const t $ trm end;
    69 
    70 fun inclt ty = Const ("op <=", [ty,ty] ---> boolT);
    71 
    72 (** Makes "Mset <= t" **)
    73 fun Mset_incl t = let val MsetT = fastype_of t 
    74                  in mk_Trueprop ((inclt MsetT) $ Free ("Mset", MsetT) $ t) end;
    75 
    76 
    77 fun Mset thm = let val vars = get_vars(thm);
    78                    val varsT = fastype_of (mk_bodyC vars);
    79                    val big_Collect = mk_CollectC (mk_abstupleC vars 
    80                          (Free ("P",varsT --> boolT) $ mk_bodyC vars));
    81                    val small_Collect = mk_CollectC (Abs("x",varsT,
    82                            Free ("P",varsT --> boolT) $ Bound 0));
    83                    val impl = implies $ (Mset_incl big_Collect) $ 
    84                                           (Mset_incl small_Collect);
    85    in Tactic.prove (Thm.sign_of_thm thm) ["Mset", "P"] [] impl (K (CLASET' blast_tac 1)) end;
    86 
    87 end;
    88 
    89 
    90 (*****************************************************************************)
    91 (** Simplifying:                                                            **)
    92 (** SOME useful lemmata, lists and simplification tactics to control which  **)
    93 (** theorems are used to simplify at each moment, so that the original      **)
    94 (** input does not suffer any unexpected transformation                     **)
    95 (*****************************************************************************)
    96 
    97 Goal "-(Collect b) = {x. ~(b x)}";
    98 by (Fast_tac 1);
    99 qed "Compl_Collect";
   100 
   101 
   102 (**Simp_tacs**)
   103 
   104 val before_set2pred_simp_tac =
   105   (simp_tac (HOL_basic_ss addsimps [Collect_conj_eq RS sym,Compl_Collect]));
   106 
   107 val split_simp_tac = (simp_tac (HOL_basic_ss addsimps [split_conv]));
   108 
   109 (*****************************************************************************)
   110 (** set2pred transforms sets inclusion into predicates implication,         **)
   111 (** maintaining the original variable names.                                **)
   112 (** Ex. "{x. x=0} <= {x. x <= 1}" -set2pred-> "x=0 --> x <= 1"              **)
   113 (** Subgoals containing intersections (A Int B) or complement sets (-A)     **)
   114 (** are first simplified by "before_set2pred_simp_tac", that returns only   **)
   115 (** subgoals of the form "{x. P x} <= {x. Q x}", which are easily           **)
   116 (** transformed.                                                            **)
   117 (** This transformation may solve very easy subgoals due to a ligth         **)
   118 (** simplification done by (split_all_tac)                                  **)
   119 (*****************************************************************************)
   120 
   121 fun set2pred i thm = let fun mk_string [] = ""
   122                            | mk_string (x::xs) = x^" "^mk_string xs;
   123                          val vars=get_vars(thm);
   124                          val var_string = mk_string (map (fst o dest_Free) vars);
   125       in ((before_set2pred_simp_tac i) THEN_MAYBE
   126           (EVERY [rtac subsetI i, 
   127                   rtac CollectI i,
   128                   dtac CollectD i,
   129                   (TRY(split_all_tac i)) THEN_MAYBE
   130                   ((rename_tac var_string i) THEN
   131                    (full_simp_tac (HOL_basic_ss addsimps [split_conv]) i)) ])) thm
   132       end;
   133 
   134 (*****************************************************************************)
   135 (** BasicSimpTac is called to simplify all verification conditions. It does **)
   136 (** a light simplification by applying "mem_Collect_eq", then it calls      **)
   137 (** MaxSimpTac, which solves subgoals of the form "A <= A",                 **)
   138 (** and transforms any other into predicates, applying then                 **)
   139 (** the tactic chosen by the user, which may solve the subgoal completely.  **)
   140 (*****************************************************************************)
   141 
   142 fun MaxSimpTac tac = FIRST'[rtac subset_refl, set2pred THEN_MAYBE' tac];
   143 
   144 fun BasicSimpTac tac =
   145   simp_tac
   146     (HOL_basic_ss addsimps [mem_Collect_eq,split_conv] addsimprocs [record_simproc])
   147   THEN_MAYBE' MaxSimpTac tac;
   148 
   149 (** HoareRuleTac **)
   150 
   151 fun WlpTac Mlem tac i =
   152   rtac SeqRule i THEN  HoareRuleTac Mlem tac false (i+1)
   153 and HoareRuleTac Mlem tac pre_cond i st = st |>
   154         (*abstraction over st prevents looping*)
   155     ( (WlpTac Mlem tac i THEN HoareRuleTac Mlem tac pre_cond i)
   156       ORELSE
   157       (FIRST[rtac SkipRule i,
   158              EVERY[rtac BasicRule i,
   159                    rtac Mlem i,
   160                    split_simp_tac i],
   161              EVERY[rtac CondRule i,
   162                    HoareRuleTac Mlem tac false (i+2),
   163                    HoareRuleTac Mlem tac false (i+1)],
   164              EVERY[rtac WhileRule i,
   165                    BasicSimpTac tac (i+2),
   166                    HoareRuleTac Mlem tac true (i+1)] ] 
   167        THEN (if pre_cond then (BasicSimpTac tac i) else (rtac subset_refl i)) ));
   168 
   169 
   170 (** tac:(int -> tactic) is the tactic the user chooses to solve or simplify **)
   171 (** the final verification conditions                                       **)
   172  
   173 fun hoare_tac tac i thm =
   174   let val Mlem = Mset(thm)
   175   in SELECT_GOAL(EVERY[HoareRuleTac Mlem tac true 1]) i thm end;