(* Title: HOL/Hoare/Hoare.ML ID: $Id: hoareAbort.ML,v 1.3 2005/04/07 07:25:35 wenzelm Exp $ Author: Leonor Prensa Nieto & Tobias Nipkow Copyright 1998 TUM Derivation of the proof rules and, most importantly, the VCG tactic. *) val SkipRule = thm"SkipRule"; val BasicRule = thm"BasicRule"; val AbortRule = thm"AbortRule"; val SeqRule = thm"SeqRule"; val CondRule = thm"CondRule"; val WhileRule = thm"WhileRule"; (*** 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; fun dest_Goal (Const ("Goal", _) $ P) = P; (** maps a goal of the form: 1. [| P |] ==> VARS x1 ... xn {._.} _ {._.} or to [x1,...,xn]**) fun get_vars thm = let val c = dest_Goal (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 ("op <=", [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 Tactic.prove (Thm.sign_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 fun mk_string [] = "" | mk_string (x::xs) = x^" "^mk_string xs; val vars=get_vars(thm); val var_string = mk_string (map (fst o dest_Free) vars); 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_tac var_string 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 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 SkipRule i, rtac AbortRule i, EVERY[rtac BasicRule i, rtac Mlem i, split_simp_tac i], EVERY[rtac CondRule i, HoareRuleTac Mlem tac false (i+2), HoareRuleTac Mlem tac false (i+1)], EVERY[rtac 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;