src/HOL/Hoare/HoareAbort.thy

 
 Like Hoare.thy, but with an Abort statement for modelling run time errors.
 *)
 
 theory HoareAbort  imports Main
-uses ("hoareAbort.ML") begin
+begin
 
 types
     'a bexp = "'a set"
     'a assn = "'a set"
 

 done
 
 lemma AbortRule: "p \<subseteq> {s. False} \<Longrightarrow> Valid p Abort q"
 by(auto simp:Valid_def)
 
-use "hoareAbort.ML"
+
+subsection {* Derivation of the proof rules and, most importantly, the VCG tactic *}
+
+ML {*
+(*** The tactics ***)
+
+(*****************************************************************************)
+(** The function Mset makes the theorem                                     **)
+(** "?Mset <= {(x1,...,xn). ?P (x1,...,xn)} ==> ?Mset <= {s. ?P s}",        **)
+(** where (x1,...,xn) are the variables of the particular program we are    **)
+(** working on at the moment of the call                                    **)
+(*****************************************************************************)
+
+local open HOLogic in
+
+(** maps (%x1 ... xn. t) to [x1,...,xn] **)
+fun abs2list (Const ("split",_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t
+  | abs2list (Abs(x,T,t)) = [Free (x, T)]
+  | abs2list _ = [];
+
+(** maps {(x1,...,xn). t} to [x1,...,xn] **)
+fun mk_vars (Const ("Collect",_) $ T) = abs2list T
+  | mk_vars _ = [];
+
+(** abstraction of body over a tuple formed from a list of free variables. 
+Types are also built **)
+fun mk_abstupleC []     body = absfree ("x", unitT, body)
+  | mk_abstupleC (v::w) body = let val (n,T) = dest_Free v
+                               in if w=[] then absfree (n, T, body)
+        else let val z  = mk_abstupleC w body;
+                 val T2 = case z of Abs(_,T,_) => T
+                        | Const (_, Type (_,[_, Type (_,[T,_])])) $ _ => T;
+       in Const ("split", (T --> T2 --> boolT) --> mk_prodT (T,T2) --> boolT) 
+          $ absfree (n, T, z) end end;
+
+(** maps [x1,...,xn] to (x1,...,xn) and types**)
+fun mk_bodyC []      = HOLogic.unit
+  | mk_bodyC (x::xs) = if xs=[] then x 
+               else let val (n, T) = dest_Free x ;
+                        val z = mk_bodyC xs;
+                        val T2 = case z of Free(_, T) => T
+                                         | Const ("Pair", Type ("fun", [_, Type
+                                            ("fun", [_, T])])) $ _ $ _ => T;
+                 in Const ("Pair", [T, T2] ---> mk_prodT (T, T2)) $ x $ z end;
+
+(** maps a goal of the form:
+        1. [| P |] ==> VARS x1 ... xn {._.} _ {._.} or to [x1,...,xn]**) 
+fun get_vars thm = let  val c = Logic.unprotect (concl_of (thm));
+                        val d = Logic.strip_assums_concl c;
+                        val Const _ $ pre $ _ $ _ = dest_Trueprop d;
+      in mk_vars pre end;
+
+
+(** Makes Collect with type **)
+fun mk_CollectC trm = let val T as Type ("fun",[t,_]) = fastype_of trm 
+                      in Collect_const t $ trm end;
+
+fun inclt ty = Const (@{const_name HOL.less_eq}, [ty,ty] ---> boolT);
+
+(** Makes "Mset <= t" **)
+fun Mset_incl t = let val MsetT = fastype_of t 
+                 in mk_Trueprop ((inclt MsetT) $ Free ("Mset", MsetT) $ t) end;
+
+
+fun Mset thm = let val vars = get_vars(thm);
+                   val varsT = fastype_of (mk_bodyC vars);
+                   val big_Collect = mk_CollectC (mk_abstupleC vars 
+                         (Free ("P",varsT --> boolT) $ mk_bodyC vars));
+                   val small_Collect = mk_CollectC (Abs("x",varsT,
+                           Free ("P",varsT --> boolT) $ Bound 0));
+                   val impl = implies $ (Mset_incl big_Collect) $ 
+                                          (Mset_incl small_Collect);
+   in Goal.prove (ProofContext.init (Thm.theory_of_thm thm)) ["Mset", "P"] [] impl (K (CLASET' blast_tac 1)) end;
+
+end;
+*}
+
+(*****************************************************************************)
+(** Simplifying:                                                            **)
+(** Some useful lemmata, lists and simplification tactics to control which  **)
+(** theorems are used to simplify at each moment, so that the original      **)
+(** input does not suffer any unexpected transformation                     **)
+(*****************************************************************************)
+
+lemma Compl_Collect: "-(Collect b) = {x. ~(b x)}"
+  by blast
+
+
+ML {*
+(**Simp_tacs**)
+
+val before_set2pred_simp_tac =
+  (simp_tac (HOL_basic_ss addsimps [@{thm Collect_conj_eq} RS sym, @{thm Compl_Collect}]));
+
+val split_simp_tac = (simp_tac (HOL_basic_ss addsimps [split_conv]));
+
+(*****************************************************************************)
+(** set2pred transforms sets inclusion into predicates implication,         **)
+(** maintaining the original variable names.                                **)
+(** Ex. "{x. x=0} <= {x. x <= 1}" -set2pred-> "x=0 --> x <= 1"              **)
+(** Subgoals containing intersections (A Int B) or complement sets (-A)     **)
+(** are first simplified by "before_set2pred_simp_tac", that returns only   **)
+(** subgoals of the form "{x. P x} <= {x. Q x}", which are easily           **)
+(** transformed.                                                            **)
+(** This transformation may solve very easy subgoals due to a ligth         **)
+(** simplification done by (split_all_tac)                                  **)
+(*****************************************************************************)
+
+fun set2pred i thm =
+  let val var_names = map (fst o dest_Free) (get_vars thm) in
+    ((before_set2pred_simp_tac i) THEN_MAYBE
+      (EVERY [rtac subsetI i, 
+              rtac CollectI i,
+              dtac CollectD i,
+              (TRY(split_all_tac i)) THEN_MAYBE
+              ((rename_params_tac var_names i) THEN
+               (full_simp_tac (HOL_basic_ss addsimps [split_conv]) i)) ])) thm
+  end;
+
+(*****************************************************************************)
+(** BasicSimpTac is called to simplify all verification conditions. It does **)
+(** a light simplification by applying "mem_Collect_eq", then it calls      **)
+(** MaxSimpTac, which solves subgoals of the form "A <= A",                 **)
+(** and transforms any other into predicates, applying then                 **)
+(** the tactic chosen by the user, which may solve the subgoal completely.  **)
+(*****************************************************************************)
+
+fun MaxSimpTac tac = FIRST'[rtac subset_refl, set2pred THEN_MAYBE' tac];
+
+fun BasicSimpTac tac =
+  simp_tac
+    (HOL_basic_ss addsimps [mem_Collect_eq,split_conv] addsimprocs [record_simproc])
+  THEN_MAYBE' MaxSimpTac tac;
+
+(** HoareRuleTac **)
+
+fun WlpTac Mlem tac i =
+  rtac @{thm SeqRule} i THEN  HoareRuleTac Mlem tac false (i+1)
+and HoareRuleTac Mlem tac pre_cond i st = st |>
+        (*abstraction over st prevents looping*)
+    ( (WlpTac Mlem tac i THEN HoareRuleTac Mlem tac pre_cond i)
+      ORELSE
+      (FIRST[rtac @{thm SkipRule} i,
+             rtac @{thm AbortRule} i,
+             EVERY[rtac @{thm BasicRule} i,
+                   rtac Mlem i,
+                   split_simp_tac i],
+             EVERY[rtac @{thm CondRule} i,
+                   HoareRuleTac Mlem tac false (i+2),
+                   HoareRuleTac Mlem tac false (i+1)],
+             EVERY[rtac @{thm WhileRule} i,
+                   BasicSimpTac tac (i+2),
+                   HoareRuleTac Mlem tac true (i+1)] ] 
+       THEN (if pre_cond then (BasicSimpTac tac i) else rtac subset_refl i) ));
+
+
+(** tac:(int -> tactic) is the tactic the user chooses to solve or simplify **)
+(** the final verification conditions                                       **)
+ 
+fun hoare_tac tac i thm =
+  let val Mlem = Mset(thm)
+  in SELECT_GOAL(EVERY[HoareRuleTac Mlem tac true 1]) i thm end;
+*}
 
 method_setup vcg = {*
   Method.no_args (Method.SIMPLE_METHOD' (hoare_tac (K all_tac))) *}
   "verification condition generator"