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+++ b/src/HOL/Hoare/hoare_tac.ML Wed Aug 29 16:46:08 2007 +0200
@@ -0,0 +1,165 @@
+(* Title: HOL/Hoare/hoare_tac.ML
+ ID: $Id$
+ Author: Leonor Prensa Nieto & Tobias Nipkow
+ Copyright 1998 TUM
+
+Derivation of the proof rules and, most importantly, the VCG tactic.
+*)
+
+(*** 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 **)
+(*****************************************************************************)
+
+Goal "-(Collect b) = {x. ~(b x)}";
+by (Fast_tac 1);
+qed "Compl_Collect";
+
+
+(**Simp_tacs**)
+
+val before_set2pred_simp_tac =
+ (simp_tac (HOL_basic_ss addsimps [Collect_conj_eq RS sym,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,
+ 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;