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theory HoareAbort(* Title: HOL/Hoare/HoareAbort.thy ID: $Id: HoareAbort.thy,v 1.4 2005/06/17 14:13:07 haftmann Exp $ Author: Leonor Prensa Nieto & Tobias Nipkow Copyright 2003 TUM Like Hoare.thy, but with an Abort statement for modelling run time errors. *) theory HoareAbort imports Main uses ("hoareAbort.ML") begin types 'a bexp = "'a set" 'a assn = "'a set" datatype 'a com = Basic "'a => 'a" | Abort | Seq "'a com" "'a com" ("(_;/ _)" [61,60] 60) | Cond "'a bexp" "'a com" "'a com" ("(1IF _/ THEN _ / ELSE _/ FI)" [0,0,0] 61) | While "'a bexp" "'a assn" "'a com" ("(1WHILE _/ INV {_} //DO _ /OD)" [0,0,0] 61) syntax "@assign" :: "id => 'b => 'a com" ("(2_ :=/ _)" [70,65] 61) "@annskip" :: "'a com" ("SKIP") translations "SKIP" == "Basic id" types 'a sem = "'a option => 'a option => bool" consts iter :: "nat => 'a bexp => 'a sem => 'a sem" primrec "iter 0 b S = (λs s'. s ∉ Some ` b ∧ s=s')" "iter (Suc n) b S = (λs s'. s ∈ Some ` b ∧ (∃s''. S s s'' ∧ iter n b S s'' s'))" consts Sem :: "'a com => 'a sem" primrec "Sem(Basic f) s s' = (case s of None => s' = None | Some t => s' = Some(f t))" "Sem Abort s s' = (s' = None)" "Sem(c1;c2) s s' = (∃s''. Sem c1 s s'' ∧ Sem c2 s'' s')" "Sem(IF b THEN c1 ELSE c2 FI) s s' = (case s of None => s' = None | Some t => ((t ∈ b --> Sem c1 s s') ∧ (t ∉ b --> Sem c2 s s')))" "Sem(While b x c) s s' = (if s = None then s' = None else ∃n. iter n b (Sem c) s s')" constdefs Valid :: "'a bexp => 'a com => 'a bexp => bool" "Valid p c q == ∀s s'. Sem c s s' --> s : Some ` p --> s' : Some ` q" syntax "@hoare_vars" :: "[idts, 'a assn,'a com,'a assn] => bool" ("VARS _// {_} // _ // {_}" [0,0,55,0] 50) syntax ("" output) "@hoare" :: "['a assn,'a com,'a assn] => bool" ("{_} // _ // {_}" [0,55,0] 50) (** parse translations **) ML{* local fun free a = Free(a,dummyT) fun abs((a,T),body) = let val a = absfree(a, dummyT, body) in if T = Bound 0 then a else Const(Syntax.constrainAbsC,dummyT) $ a $ T end in fun mk_abstuple [x] body = abs (x, body) | mk_abstuple (x::xs) body = Syntax.const "split" $ abs (x, mk_abstuple xs body); fun mk_fbody a e [x as (b,_)] = if a=b then e else free b | mk_fbody a e ((b,_)::xs) = Syntax.const "Pair" $ (if a=b then e else free b) $ mk_fbody a e xs; fun mk_fexp a e xs = mk_abstuple xs (mk_fbody a e xs) end *} (* bexp_tr & assn_tr *) (*all meta-variables for bexp except for TRUE are translated as if they were boolean expressions*) ML{* fun bexp_tr (Const ("TRUE", _)) xs = Syntax.const "TRUE" | bexp_tr b xs = Syntax.const "Collect" $ mk_abstuple xs b; fun assn_tr r xs = Syntax.const "Collect" $ mk_abstuple xs r; *} (* com_tr *) ML{* fun com_tr (Const("@assign",_) $ Free (a,_) $ e) xs = Syntax.const "Basic" $ mk_fexp a e xs | com_tr (Const ("Basic",_) $ f) xs = Syntax.const "Basic" $ f | com_tr (Const ("Seq",_) $ c1 $ c2) xs = Syntax.const "Seq" $ com_tr c1 xs $ com_tr c2 xs | com_tr (Const ("Cond",_) $ b $ c1 $ c2) xs = Syntax.const "Cond" $ bexp_tr b xs $ com_tr c1 xs $ com_tr c2 xs | com_tr (Const ("While",_) $ b $ I $ c) xs = Syntax.const "While" $ bexp_tr b xs $ assn_tr I xs $ com_tr c xs | com_tr t _ = t (* if t is just a Free/Var *) *} (* triple_tr *) ML{* local fun var_tr(Free(a,_)) = (a,Bound 0) (* Bound 0 = dummy term *) | var_tr(Const ("_constrain", _) $ (Free (a,_)) $ T) = (a,T); fun vars_tr (Const ("_idts", _) $ idt $ vars) = var_tr idt :: vars_tr vars | vars_tr t = [var_tr t] in fun hoare_vars_tr [vars, pre, prg, post] = let val xs = vars_tr vars in Syntax.const "Valid" $ assn_tr pre xs $ com_tr prg xs $ assn_tr post xs end | hoare_vars_tr ts = raise TERM ("hoare_vars_tr", ts); end *} parse_translation {* [("@hoare_vars", hoare_vars_tr)] *} (*****************************************************************************) (*** print translations ***) ML{* fun dest_abstuple (Const ("split",_) $ (Abs(v,_, body))) = subst_bound (Syntax.free v, dest_abstuple body) | dest_abstuple (Abs(v,_, body)) = subst_bound (Syntax.free v, body) | dest_abstuple trm = trm; fun abs2list (Const ("split",_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t | abs2list (Abs(x,T,t)) = [Free (x, T)] | abs2list _ = []; fun mk_ts (Const ("split",_) $ (Abs(x,_,t))) = mk_ts t | mk_ts (Abs(x,_,t)) = mk_ts t | mk_ts (Const ("Pair",_) $ a $ b) = a::(mk_ts b) | mk_ts t = [t]; fun mk_vts (Const ("split",_) $ (Abs(x,_,t))) = ((Syntax.free x)::(abs2list t), mk_ts t) | mk_vts (Abs(x,_,t)) = ([Syntax.free x], [t]) | mk_vts t = raise Match; fun find_ch [] i xs = (false, (Syntax.free "not_ch",Syntax.free "not_ch" )) | find_ch ((v,t)::vts) i xs = if t=(Bound i) then find_ch vts (i-1) xs else (true, (v, subst_bounds (xs,t))); fun is_f (Const ("split",_) $ (Abs(x,_,t))) = true | is_f (Abs(x,_,t)) = true | is_f t = false; *} (* assn_tr' & bexp_tr'*) ML{* fun assn_tr' (Const ("Collect",_) $ T) = dest_abstuple T | assn_tr' (Const ("op Int",_) $ (Const ("Collect",_) $ T1) $ (Const ("Collect",_) $ T2)) = Syntax.const "op Int" $ dest_abstuple T1 $ dest_abstuple T2 | assn_tr' t = t; fun bexp_tr' (Const ("Collect",_) $ T) = dest_abstuple T | bexp_tr' t = t; *} (*com_tr' *) ML{* fun mk_assign f = let val (vs, ts) = mk_vts f; val (ch, which) = find_ch (vs~~ts) ((length vs)-1) (rev vs) in if ch then Syntax.const "@assign" $ fst(which) $ snd(which) else Syntax.const "@skip" end; fun com_tr' (Const ("Basic",_) $ f) = if is_f f then mk_assign f else Syntax.const "Basic" $ f | com_tr' (Const ("Seq",_) $ c1 $ c2) = Syntax.const "Seq" $ com_tr' c1 $ com_tr' c2 | com_tr' (Const ("Cond",_) $ b $ c1 $ c2) = Syntax.const "Cond" $ bexp_tr' b $ com_tr' c1 $ com_tr' c2 | com_tr' (Const ("While",_) $ b $ I $ c) = Syntax.const "While" $ bexp_tr' b $ assn_tr' I $ com_tr' c | com_tr' t = t; fun spec_tr' [p, c, q] = Syntax.const "@hoare" $ assn_tr' p $ com_tr' c $ assn_tr' q *} print_translation {* [("Valid", spec_tr')] *} (*** The proof rules ***) lemma SkipRule: "p ⊆ q ==> Valid p (Basic id) q" by (auto simp:Valid_def) lemma BasicRule: "p ⊆ {s. f s ∈ q} ==> Valid p (Basic f) q" by (auto simp:Valid_def) lemma SeqRule: "Valid P c1 Q ==> Valid Q c2 R ==> Valid P (c1;c2) R" by (auto simp:Valid_def) lemma CondRule: "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 (fastsimp simp:Valid_def image_def) lemma iter_aux: "! s s'. Sem c s s' --> s ∈ Some ` (I ∩ b) --> s' ∈ Some ` I ==> (!!s s'. s ∈ Some ` I ==> iter n b (Sem c) s s' ==> s' ∈ Some ` (I ∩ -b))"; apply(unfold image_def) apply(induct n) apply clarsimp apply(simp (no_asm_use)) apply blast done lemma WhileRule: "p ⊆ i ==> Valid (i ∩ b) c i ==> i ∩ (-b) ⊆ q ==> Valid p (While b i c) q" apply(simp add:Valid_def) apply(simp (no_asm) add:image_def) apply clarify apply(drule iter_aux) prefer 2 apply assumption apply blast apply blast done lemma AbortRule: "p ⊆ {s. False} ==> Valid p Abort q" by(auto simp:Valid_def) use "hoareAbort.ML" method_setup vcg = {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (hoare_tac (K all_tac))) *} "verification condition generator" method_setup vcg_simp = {* Method.ctxt_args (fn ctxt => Method.METHOD (fn facts => hoare_tac (asm_full_simp_tac (local_simpset_of ctxt))1)) *} "verification condition generator plus simplification" (* Special syntax for guarded statements and guarded array updates: *) syntax guarded_com :: "bool => 'a com => 'a com" ("(2_ ->/ _)" 71) array_update :: "'a list => nat => 'a => 'a com" ("(2_[_] :=/ _)" [70,65] 61) translations "P -> c" == "IF P THEN c ELSE Abort FI" "a[i] := v" => "(i < length a) -> (a := list_update a i v)" (* reverse translation not possible because of duplicate "a" *) text{* Note: there is no special syntax for guarded array access. Thus you must write @{text"j < length a -> a[i] := a!j"}. *} end
lemma SkipRule:
p ⊆ q ==> {p} SKIP {q}
lemma BasicRule:
p ⊆ {s. f s ∈ q} ==> {p} Basic f {q}
lemma SeqRule:
[| {P} c1.0 {Q}; {Q} c2.0 {R} |] ==> {P} c1.0; c2.0 {R}
lemma CondRule:
[| p ⊆ {s. (s ∈ b --> s ∈ w) ∧ (s ∉ b --> s ∈ w')}; {w} c1.0 {q}; {w'} c2.0 {q} |] ==> {p} IF b THEN c1.0 ELSE c2.0 FI {q}
lemma iter_aux:
[| ∀s s'. Sem c s s' --> s ∈ Some ` (I ∩ b) --> s' ∈ Some ` I; s ∈ Some ` I; iter n b (Sem c) s s' |] ==> s' ∈ Some ` (I ∩ - b)
lemma WhileRule:
[| p ⊆ i; {i ∩ b} c {i}; i ∩ - b ⊆ q |] ==> {p} WHILE b INV {i} DO c OD {q}
lemma AbortRule:
p ⊆ {s. False} ==> {p} Abort {q}
theorem Compl_Collect:
- Collect b = {x. ¬ b x}