src/HOL/Hoare/hoare_tac.ML
 author blanchet Wed Feb 12 08:35:57 2014 +0100 (2014-02-12) changeset 55414 eab03e9cee8a parent 51717 9e7d1c139569 child 55659 4089f6e98ab9 permissions -rw-r--r--
renamed '{prod,sum,bool,unit}_case' to 'case_...'
1 (*  Title:      HOL/Hoare/hoare_tac.ML
2     Author:     Leonor Prensa Nieto & Tobias Nipkow
4 Derivation of the proof rules and, most importantly, the VCG tactic.
5 *)
7 (* FIXME structure Hoare: HOARE *)
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
20 (** maps (%x1 ... xn. t) to [x1,...,xn] **)
21 fun abs2list (Const (@{const_name case_prod}, _) \$ 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 (@{const_name 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", HOLogic.unitT) body
32   | mk_abstupleC [v] body = absfree (dest_Free v) body
33   | mk_abstupleC (v :: w) body =
34       let
35         val (x, T) = dest_Free v;
36         val z = mk_abstupleC w body;
37         val T2 =
38           (case z of
39             Abs (_, T, _) => T
40           | Const (_, Type (_, [_, Type (_, [T, _])])) \$ _ => T);
41       in
42         Const (@{const_name case_prod},
43             (T --> T2 --> HOLogic.boolT) --> HOLogic.mk_prodT (T, T2) --> HOLogic.boolT) \$
44           absfree (x, T) z
45       end;
47 (** maps [x1,...,xn] to (x1,...,xn) and types**)
48 fun mk_bodyC []      = HOLogic.unit
49   | mk_bodyC (x::xs) = if xs=[] then x
50                else let val (n, T) = dest_Free x ;
51                         val z = mk_bodyC xs;
52                         val T2 = case z of Free(_, T) => T
53                                          | Const (@{const_name Pair}, Type ("fun", [_, Type
54                                             ("fun", [_, T])])) \$ _ \$ _ => T;
55                  in Const (@{const_name Pair}, [T, T2] ---> HOLogic.mk_prodT (T, T2)) \$ x \$ z end;
57 (** maps a subgoal of the form:
58         VARS x1 ... xn {._.} _ {._.} or to [x1,...,xn]**)
59 fun get_vars c =
60   let
61     val d = Logic.strip_assums_concl c;
62     val Const _ \$ pre \$ _ \$ _ = HOLogic.dest_Trueprop d;
63   in mk_vars pre end;
65 fun mk_CollectC trm =
66   let val T as Type ("fun",[t,_]) = fastype_of trm
67   in HOLogic.Collect_const t \$ trm end;
69 fun inclt ty = Const (@{const_name Orderings.less_eq}, [ty,ty] ---> HOLogic.boolT);
71 in
73 fun Mset ctxt prop =
74   let
75     val [(Mset, _), (P, _)] = Variable.variant_frees ctxt [] [("Mset", ()), ("P", ())];
77     val vars = get_vars prop;
78     val varsT = fastype_of (mk_bodyC vars);
79     val big_Collect = mk_CollectC (mk_abstupleC vars (Free (P, varsT --> HOLogic.boolT) \$ mk_bodyC vars));
80     val small_Collect = mk_CollectC (Abs ("x", varsT, Free (P, varsT --> HOLogic.boolT) \$ Bound 0));
82     val MsetT = fastype_of big_Collect;
83     fun Mset_incl t = HOLogic.mk_Trueprop (inclt MsetT \$ Free (Mset, MsetT) \$ t);
84     val impl = Logic.mk_implies (Mset_incl big_Collect, Mset_incl small_Collect);
85     val th = Goal.prove ctxt [Mset, P] [] impl (fn _ => blast_tac ctxt 1);
86  in (vars, th) end;
88 end;
91 (*****************************************************************************)
92 (** Simplifying:                                                            **)
93 (** Some useful lemmata, lists and simplification tactics to control which  **)
94 (** theorems are used to simplify at each moment, so that the original      **)
95 (** input does not suffer any unexpected transformation                     **)
96 (*****************************************************************************)
98 (**Simp_tacs**)
100 fun before_set2pred_simp_tac ctxt =
101   simp_tac (put_simpset HOL_basic_ss ctxt addsimps [Collect_conj_eq RS sym, @{thm Compl_Collect}]);
103 fun split_simp_tac ctxt =
104   simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm split_conv}]);
106 (*****************************************************************************)
107 (** set2pred_tac transforms sets inclusion into predicates implication,     **)
108 (** maintaining the original variable names.                                **)
109 (** Ex. "{x. x=0} <= {x. x <= 1}" -set2pred-> "x=0 --> x <= 1"              **)
110 (** Subgoals containing intersections (A Int B) or complement sets (-A)     **)
111 (** are first simplified by "before_set2pred_simp_tac", that returns only   **)
112 (** subgoals of the form "{x. P x} <= {x. Q x}", which are easily           **)
113 (** transformed.                                                            **)
114 (** This transformation may solve very easy subgoals due to a ligth         **)
115 (** simplification done by (split_all_tac)                                  **)
116 (*****************************************************************************)
118 fun set2pred_tac ctxt var_names = SUBGOAL (fn (goal, i) =>
119   before_set2pred_simp_tac ctxt i THEN_MAYBE
120   EVERY [
121     rtac subsetI i,
122     rtac CollectI i,
123     dtac CollectD i,
124     TRY (split_all_tac ctxt i) THEN_MAYBE
125      (rename_tac var_names i THEN
126       full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm split_conv}]) i)]);
128 (*****************************************************************************)
129 (** BasicSimpTac is called to simplify all verification conditions. It does **)
130 (** a light simplification by applying "mem_Collect_eq", then it calls      **)
131 (** MaxSimpTac, which solves subgoals of the form "A <= A",                 **)
132 (** and transforms any other into predicates, applying then                 **)
133 (** the tactic chosen by the user, which may solve the subgoal completely.  **)
134 (*****************************************************************************)
136 fun MaxSimpTac ctxt var_names tac =
137   FIRST'[rtac subset_refl, set2pred_tac ctxt var_names THEN_MAYBE' tac];
139 fun BasicSimpTac ctxt var_names tac =
140   simp_tac
141     (put_simpset HOL_basic_ss ctxt
143   THEN_MAYBE' MaxSimpTac ctxt var_names tac;
146 (** hoare_rule_tac **)
148 fun hoare_rule_tac ctxt (vars, Mlem) tac =
149   let
150     val var_names = map (fst o dest_Free) vars;
151     fun wlp_tac i =
152       rtac @{thm SeqRule} i THEN rule_tac false (i + 1)
153     and rule_tac pre_cond i st = st |> (*abstraction over st prevents looping*)
154       ((wlp_tac i THEN rule_tac pre_cond i)
155         ORELSE
156         (FIRST [
157           rtac @{thm SkipRule} i,
158           rtac @{thm AbortRule} i,
159           EVERY [
160             rtac @{thm BasicRule} i,
161             rtac Mlem i,
162             split_simp_tac ctxt i],
163           EVERY [
164             rtac @{thm CondRule} i,
165             rule_tac false (i + 2),
166             rule_tac false (i + 1)],
167           EVERY [
168             rtac @{thm WhileRule} i,
169             BasicSimpTac ctxt var_names tac (i + 2),
170             rule_tac true (i + 1)]]
171          THEN (if pre_cond then BasicSimpTac ctxt var_names tac i else rtac subset_refl i)));
172   in rule_tac end;
175 (** tac is the tactic the user chooses to solve or simplify **)
176 (** the final verification conditions                       **)
178 fun hoare_tac ctxt (tac: int -> tactic) = SUBGOAL (fn (goal, i) =>
179   SELECT_GOAL (hoare_rule_tac ctxt (Mset ctxt goal) tac true 1) i);