src/HOL/Hoare/hoare_tac.ML
 author wenzelm Mon Jun 23 23:45:39 2008 +0200 (2008-06-23) changeset 27330 1af2598b5f7d parent 27244 af0a44372d1f child 28457 25669513fd4c permissions -rw-r--r--
Logic.all/mk_equals/mk_implies;
1 (*  Title:      HOL/Hoare/hoare_tac.ML
2     ID:         \$Id\$
3     Author:     Leonor Prensa Nieto & Tobias Nipkow
6 Derivation of the proof rules and, most importantly, the VCG tactic.
7 *)
9 (*** The tactics ***)
11 (*****************************************************************************)
12 (** The function Mset makes the theorem                                     **)
13 (** "?Mset <= {(x1,...,xn). ?P (x1,...,xn)} ==> ?Mset <= {s. ?P s}",        **)
14 (** where (x1,...,xn) are the variables of the particular program we are    **)
15 (** working on at the moment of the call                                    **)
16 (*****************************************************************************)
18 local open HOLogic in
20 (** maps (%x1 ... xn. t) to [x1,...,xn] **)
21 fun abs2list (Const ("split",_) \$ (Abs(x,T,t))) = Free (x, T)::abs2list t
22   | abs2list (Abs(x,T,t)) = [Free (x, T)]
23   | abs2list _ = [];
25 (** maps {(x1,...,xn). t} to [x1,...,xn] **)
26 fun mk_vars (Const ("Collect",_) \$ T) = abs2list T
27   | mk_vars _ = [];
29 (** abstraction of body over a tuple formed from a list of free variables.
30 Types are also built **)
31 fun mk_abstupleC []     body = absfree ("x", unitT, body)
32   | mk_abstupleC (v::w) body = let val (n,T) = dest_Free v
33                                in if w=[] then absfree (n, T, body)
34         else let val z  = mk_abstupleC w body;
35                  val T2 = case z of Abs(_,T,_) => T
36                         | Const (_, Type (_,[_, Type (_,[T,_])])) \$ _ => T;
37        in Const ("split", (T --> T2 --> boolT) --> mk_prodT (T,T2) --> boolT)
38           \$ absfree (n, T, z) end end;
40 (** maps [x1,...,xn] to (x1,...,xn) and types**)
41 fun mk_bodyC []      = HOLogic.unit
42   | mk_bodyC (x::xs) = if xs=[] then x
43                else let val (n, T) = dest_Free x ;
44                         val z = mk_bodyC xs;
45                         val T2 = case z of Free(_, T) => T
46                                          | Const ("Pair", Type ("fun", [_, Type
47                                             ("fun", [_, T])])) \$ _ \$ _ => T;
48                  in Const ("Pair", [T, T2] ---> mk_prodT (T, T2)) \$ x \$ z end;
50 (** maps a goal of the form:
51         1. [| P |] ==> VARS x1 ... xn {._.} _ {._.} or to [x1,...,xn]**)
52 fun get_vars thm = let  val c = Logic.unprotect (concl_of (thm));
53                         val d = Logic.strip_assums_concl c;
54                         val Const _ \$ pre \$ _ \$ _ = dest_Trueprop d;
55       in mk_vars pre end;
58 (** Makes Collect with type **)
59 fun mk_CollectC trm = let val T as Type ("fun",[t,_]) = fastype_of trm
60                       in Collect_const t \$ trm end;
62 fun inclt ty = Const (@{const_name HOL.less_eq}, [ty,ty] ---> boolT);
64 (** Makes "Mset <= t" **)
65 fun Mset_incl t = let val MsetT = fastype_of t
66                  in mk_Trueprop ((inclt MsetT) \$ Free ("Mset", MsetT) \$ t) end;
69 fun Mset thm = let val vars = get_vars(thm);
70                    val varsT = fastype_of (mk_bodyC vars);
71                    val big_Collect = mk_CollectC (mk_abstupleC vars
72                          (Free ("P",varsT --> boolT) \$ mk_bodyC vars));
73                    val small_Collect = mk_CollectC (Abs("x",varsT,
74                            Free ("P",varsT --> boolT) \$ Bound 0));
75                    val impl = Logic.mk_implies (Mset_incl big_Collect, Mset_incl small_Collect);
76    in Goal.prove (ProofContext.init (Thm.theory_of_thm thm)) ["Mset", "P"] [] impl (K (CLASET' blast_tac 1)) end;
78 end;
81 (*****************************************************************************)
82 (** Simplifying:                                                            **)
83 (** Some useful lemmata, lists and simplification tactics to control which  **)
84 (** theorems are used to simplify at each moment, so that the original      **)
85 (** input does not suffer any unexpected transformation                     **)
86 (*****************************************************************************)
88 (**Simp_tacs**)
90 val before_set2pred_simp_tac =
91   (simp_tac (HOL_basic_ss addsimps [Collect_conj_eq RS sym, @{thm Compl_Collect}]));
93 val split_simp_tac = (simp_tac (HOL_basic_ss addsimps [split_conv]));
95 (*****************************************************************************)
96 (** set2pred transforms sets inclusion into predicates implication,         **)
97 (** maintaining the original variable names.                                **)
98 (** Ex. "{x. x=0} <= {x. x <= 1}" -set2pred-> "x=0 --> x <= 1"              **)
99 (** Subgoals containing intersections (A Int B) or complement sets (-A)     **)
100 (** are first simplified by "before_set2pred_simp_tac", that returns only   **)
101 (** subgoals of the form "{x. P x} <= {x. Q x}", which are easily           **)
102 (** transformed.                                                            **)
103 (** This transformation may solve very easy subgoals due to a ligth         **)
104 (** simplification done by (split_all_tac)                                  **)
105 (*****************************************************************************)
107 fun set2pred i thm =
108   let val var_names = map (fst o dest_Free) (get_vars thm) in
109     ((before_set2pred_simp_tac i) THEN_MAYBE
110      (EVERY [rtac subsetI i,
111              rtac CollectI i,
112              dtac CollectD i,
113              (TRY(split_all_tac i)) THEN_MAYBE
114              ((rename_tac var_names i) THEN
115               (full_simp_tac (HOL_basic_ss addsimps [split_conv]) i)) ])) thm
116   end;
118 (*****************************************************************************)
119 (** BasicSimpTac is called to simplify all verification conditions. It does **)
120 (** a light simplification by applying "mem_Collect_eq", then it calls      **)
121 (** MaxSimpTac, which solves subgoals of the form "A <= A",                 **)
122 (** and transforms any other into predicates, applying then                 **)
123 (** the tactic chosen by the user, which may solve the subgoal completely.  **)
124 (*****************************************************************************)
126 fun MaxSimpTac tac = FIRST'[rtac subset_refl, set2pred THEN_MAYBE' tac];
128 fun BasicSimpTac tac =
129   simp_tac
131   THEN_MAYBE' MaxSimpTac tac;
133 (** HoareRuleTac **)
135 fun WlpTac Mlem tac i =
136   rtac @{thm SeqRule} i THEN  HoareRuleTac Mlem tac false (i+1)
137 and HoareRuleTac Mlem tac pre_cond i st = st |>
138         (*abstraction over st prevents looping*)
139     ( (WlpTac Mlem tac i THEN HoareRuleTac Mlem tac pre_cond i)
140       ORELSE
141       (FIRST[rtac @{thm SkipRule} i,
142              EVERY[rtac @{thm BasicRule} i,
143                    rtac Mlem i,
144                    split_simp_tac i],
145              EVERY[rtac @{thm CondRule} i,
146                    HoareRuleTac Mlem tac false (i+2),
147                    HoareRuleTac Mlem tac false (i+1)],
148              EVERY[rtac @{thm WhileRule} i,
149                    BasicSimpTac tac (i+2),
150                    HoareRuleTac Mlem tac true (i+1)] ]
151        THEN (if pre_cond then (BasicSimpTac tac i) else (rtac subset_refl i)) ));
154 (** tac:(int -> tactic) is the tactic the user chooses to solve or simplify **)
155 (** the final verification conditions                                       **)
157 fun hoare_tac tac i thm =
158   let val Mlem = Mset(thm)
159   in SELECT_GOAL(EVERY[HoareRuleTac Mlem tac true 1]) i thm end;