(* Title: HOL/Hoare/HoareAbort.thy
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 ("hoare_tac.ML")
begin
types
'a bexp = "'a set"
'a assn = "'a set"
datatype
'a com = Basic "'a \<Rightarrow> '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 = (\<lambda>s s'. s \<notin> Some ` b \<and> s=s')"
"iter (Suc n) b S =
(\<lambda>s s'. s \<in> Some ` b \<and> (\<exists>s''. S s s'' \<and> iter n b S s'' s'))"
consts Sem :: "'a com => 'a sem"
primrec
"Sem(Basic f) s s' = (case s of None \<Rightarrow> s' = None | Some t \<Rightarrow> s' = Some(f t))"
"Sem Abort s s' = (s' = None)"
"Sem(c1;c2) s s' = (\<exists>s''. Sem c1 s s'' \<and> Sem c2 s'' s')"
"Sem(IF b THEN c1 ELSE c2 FI) s s' =
(case s of None \<Rightarrow> s' = None
| Some t \<Rightarrow> ((t \<in> b \<longrightarrow> Sem c1 s s') \<and> (t \<notin> b \<longrightarrow> Sem c2 s s')))"
"Sem(While b x c) s s' =
(if s = None then s' = None else \<exists>n. iter n b (Sem c) s s')"
constdefs Valid :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool"
"Valid p c q == \<forall>s s'. Sem c s s' \<longrightarrow> s : Some ` p \<longrightarrow> 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 *) (* FIXME does not handle "_idtdummy" *)
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 (@{const_name inter},_) $ (Const ("Collect",_) $ T1) $
(Const ("Collect",_) $ T2)) =
Syntax.const "Set.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 \<subseteq> q \<Longrightarrow> Valid p (Basic id) q"
by (auto simp:Valid_def)
lemma BasicRule: "p \<subseteq> {s. f s \<in> q} \<Longrightarrow> Valid p (Basic f) q"
by (auto simp:Valid_def)
lemma SeqRule: "Valid P c1 Q \<Longrightarrow> Valid Q c2 R \<Longrightarrow> Valid P (c1;c2) R"
by (auto simp:Valid_def)
lemma CondRule:
"p \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> w) \<and> (s \<notin> b \<longrightarrow> s \<in> w')}
\<Longrightarrow> Valid w c1 q \<Longrightarrow> Valid w' c2 q \<Longrightarrow> Valid p (Cond b c1 c2) q"
by (fastsimp simp:Valid_def image_def)
lemma iter_aux:
"! s s'. Sem c s s' \<longrightarrow> s \<in> Some ` (I \<inter> b) \<longrightarrow> s' \<in> Some ` I \<Longrightarrow>
(\<And>s s'. s \<in> Some ` I \<Longrightarrow> iter n b (Sem c) s s' \<Longrightarrow> s' \<in> Some ` (I \<inter> -b))";
apply(unfold image_def)
apply(induct n)
apply clarsimp
apply(simp (no_asm_use))
apply blast
done
lemma WhileRule:
"p \<subseteq> i \<Longrightarrow> Valid (i \<inter> b) c i \<Longrightarrow> i \<inter> (-b) \<subseteq> q \<Longrightarrow> 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 \<subseteq> {s. False} \<Longrightarrow> Valid p Abort q"
by(auto simp:Valid_def)
subsection {* Derivation of the proof rules and, most importantly, the VCG tactic *}
lemma Compl_Collect: "-(Collect b) = {x. ~(b x)}"
by blast
use "hoare_tac.ML"
method_setup vcg = {*
Scan.succeed (fn ctxt => SIMPLE_METHOD' (hoare_tac ctxt (K all_tac))) *}
"verification condition generator"
method_setup vcg_simp = {*
Scan.succeed (fn ctxt =>
SIMPLE_METHOD' (hoare_tac ctxt (asm_full_simp_tac (simpset_of ctxt)))) *}
"verification condition generator plus simplification"
(* Special syntax for guarded statements and guarded array updates: *)
syntax
guarded_com :: "bool \<Rightarrow> 'a com \<Rightarrow> 'a com" ("(2_ \<rightarrow>/ _)" 71)
array_update :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a com" ("(2_[_] :=/ _)" [70,65] 61)
translations
"P \<rightarrow> c" == "IF P THEN c ELSE Abort FI"
"a[i] := v" => "(i < CONST length a) \<rightarrow> (a := CONST 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 \<rightarrow> a[i] := a!j"}. *}
end