diff -r 3e48dcd25746 -r 631460c31a1f src/HOL/Hoare/Hoare.ML --- a/src/HOL/Hoare/Hoare.ML Tue Nov 05 15:59:17 2002 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,217 +0,0 @@ -(* Title: HOL/Hoare/Hoare.ML - ID: $Id$ - Author: Leonor Prensa Nieto & Tobias Nipkow - Copyright 1998 TUM - -Derivation of the proof rules and, most importantly, the VCG tactic. -*) - -(*** The proof rules ***) - -Goalw [thm "Valid_def"] "p <= q ==> Valid p (Basic id) q"; -by (Auto_tac); -qed "SkipRule"; - -Goalw [thm "Valid_def"] "p <= {s. (f s):q} ==> Valid p (Basic f) q"; -by (Auto_tac); -qed "BasicRule"; - -Goalw [thm "Valid_def"] "Valid P c1 Q ==> Valid Q c2 R ==> Valid P (c1;c2) R"; -by (Asm_simp_tac 1); -by (Blast_tac 1); -qed "SeqRule"; - -Goalw [thm "Valid_def"] - "p <= {s. (s:b --> s:w) & (s~:b --> s:w')} \ -\ ==> Valid w c1 q ==> Valid w' c2 q \ -\ ==> Valid p (Cond b c1 c2) q"; -by (Asm_simp_tac 1); -by (Blast_tac 1); -qed "CondRule"; - -Goal "! s s'. Sem c s s' --> s : I Int b --> s' : I ==> \ -\ ! s s'. s : I --> iter n b (Sem c) s s' --> s' : I & s' ~: b"; -by (induct_tac "n" 1); - by (Asm_simp_tac 1); -by (Simp_tac 1); -by (Blast_tac 1); -val lemma = result() RS spec RS spec RS mp RS mp; - -Goalw [thm "Valid_def"] - "p <= i ==> Valid (i Int b) c i ==> i Int (-b) <= q \ -\ ==> Valid p (While b j c) q"; -by (Asm_simp_tac 1); -by (Clarify_tac 1); -by (dtac lemma 1); -by (assume_tac 2); -by (Blast_tac 1); -by (Blast_tac 1); -qed "WhileRule'"; - -Goal - "p <= i ==> Valid (i Int b) c i ==> i Int (-b) <= q \ -\ ==> Valid p (While b i c) q"; -by (rtac WhileRule' 1); -by (ALLGOALS assume_tac); -qed "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, - 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;