--- a/src/HOL/Hoare/Hoare.thy Tue Feb 23 12:14:46 2010 +0100
+++ b/src/HOL/Hoare/Hoare.thy Tue Feb 23 14:11:32 2010 +0100
@@ -1,245 +1,9 @@
-(* Title: HOL/Hoare/Hoare.thy
- Author: Leonor Prensa Nieto & Tobias Nipkow
- Copyright 1998 TUM
-
-Sugared semantic embedding of Hoare logic.
-Strictly speaking a shallow embedding (as implemented by Norbert Galm
-following Mike Gordon) would suffice. Maybe the datatype com comes in useful
-later.
+(* Author: Tobias Nipkow
+ Copyright 1998-2003 TUM
*)
theory Hoare
-imports Main
-uses ("hoare_tac.ML")
+imports Examples ExamplesAbort Pointers0 Pointer_Examples Pointer_ExamplesAbort SchorrWaite Separation
begin
-types
- 'a bexp = "'a set"
- 'a assn = "'a set"
-
-datatype
- 'a com = Basic "'a \<Rightarrow> 'a"
- | 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)
-
-abbreviation annskip ("SKIP") where "SKIP == Basic id"
-
-types 'a sem = "'a => 'a => bool"
-
-consts iter :: "nat => 'a bexp => 'a sem => 'a sem"
-primrec
-"iter 0 b S = (%s s'. s ~: b & (s=s'))"
-"iter (Suc n) b S = (%s s'. s : b & (? s''. S s s'' & iter n b S s'' s'))"
-
-consts Sem :: "'a com => 'a sem"
-primrec
-"Sem(Basic f) s s' = (s' = f s)"
-"Sem(c1;c2) s s' = (? s''. Sem c1 s s'' & Sem c2 s'' s')"
-"Sem(IF b THEN c1 ELSE c2 FI) s s' = ((s : b --> Sem c1 s s') &
- (s ~: b --> Sem c2 s s'))"
-"Sem(While b x c) s s' = (? n. iter n b (Sem c) s s')"
-
-constdefs Valid :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool"
- "Valid p c q == !s s'. Sem c s s' --> s : p --> s' : q"
-
-
-
-(** parse translations **)
-
-syntax
- "_assign" :: "id => 'b => 'a com" ("(2_ :=/ _)" [70,65] 61)
-
-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)
-ML {*
-
-local
-
-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 @{const_syntax split} $ abs (x, mk_abstuple xs body);
-
-fun mk_fbody a e [x as (b,_)] = if a=b then e else Syntax.free b
- | mk_fbody a e ((b,_)::xs) =
- Syntax.const @{const_syntax Pair} $ (if a=b then e else Syntax.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" (* FIXME !? *)
- | bexp_tr b xs = Syntax.const @{const_syntax Collect} $ mk_abstuple xs b;
-
-fun assn_tr r xs = Syntax.const @{const_syntax Collect} $ mk_abstuple xs r;
-*}
-(* com_tr *)
-ML{*
-fun com_tr (Const(@{syntax_const "_assign"},_) $ Free (a,_) $ e) xs =
- Syntax.const @{const_syntax Basic} $ mk_fexp a e xs
- | com_tr (Const (@{const_syntax Basic},_) $ f) xs = Syntax.const @{const_syntax Basic} $ f
- | com_tr (Const (@{const_syntax Seq},_) $ c1 $ c2) xs =
- Syntax.const @{const_syntax Seq} $ com_tr c1 xs $ com_tr c2 xs
- | com_tr (Const (@{const_syntax Cond},_) $ b $ c1 $ c2) xs =
- Syntax.const @{const_syntax Cond} $ bexp_tr b xs $ com_tr c1 xs $ com_tr c2 xs
- | com_tr (Const (@{const_syntax While},_) $ b $ I $ c) xs =
- Syntax.const @{const_syntax 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 (@{syntax_const "_constrain"}, _) $ (Free (a,_)) $ T) = (a,T);
-
-fun vars_tr (Const (@{syntax_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 @{const_syntax 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 {* [(@{syntax_const "_hoare_vars"}, hoare_vars_tr)] *}
-
-
-(*****************************************************************************)
-
-(*** print translations ***)
-ML{*
-fun dest_abstuple (Const (@{const_syntax 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 (@{const_syntax split},_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t
- | abs2list (Abs(x,T,t)) = [Free (x, T)]
- | abs2list _ = [];
-
-fun mk_ts (Const (@{const_syntax split},_) $ (Abs(x,_,t))) = mk_ts t
- | mk_ts (Abs(x,_,t)) = mk_ts t
- | mk_ts (Const (@{const_syntax Pair},_) $ a $ b) = a::(mk_ts b)
- | mk_ts t = [t];
-
-fun mk_vts (Const (@{const_syntax 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 (@{const_syntax split},_) $ (Abs(x,_,t))) = true
- | is_f (Abs(x,_,t)) = true
- | is_f t = false;
-*}
-
-(* assn_tr' & bexp_tr'*)
-ML{*
-fun assn_tr' (Const (@{const_syntax Collect},_) $ T) = dest_abstuple T
- | assn_tr' (Const (@{const_syntax inter}, _) $
- (Const (@{const_syntax Collect},_) $ T1) $ (Const (@{const_syntax Collect},_) $ T2)) =
- Syntax.const @{const_syntax inter} $ dest_abstuple T1 $ dest_abstuple T2
- | assn_tr' t = t;
-
-fun bexp_tr' (Const (@{const_syntax 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 @{syntax_const "_assign"} $ fst which $ snd which
- else Syntax.const @{const_syntax annskip}
- end;
-
-fun com_tr' (Const (@{const_syntax Basic},_) $ f) =
- if is_f f then mk_assign f
- else Syntax.const @{const_syntax Basic} $ f
- | com_tr' (Const (@{const_syntax Seq},_) $ c1 $ c2) =
- Syntax.const @{const_syntax Seq} $ com_tr' c1 $ com_tr' c2
- | com_tr' (Const (@{const_syntax Cond},_) $ b $ c1 $ c2) =
- Syntax.const @{const_syntax Cond} $ bexp_tr' b $ com_tr' c1 $ com_tr' c2
- | com_tr' (Const (@{const_syntax While},_) $ b $ I $ c) =
- Syntax.const @{const_syntax While} $ bexp_tr' b $ assn_tr' I $ com_tr' c
- | com_tr' t = t;
-
-fun spec_tr' [p, c, q] =
- Syntax.const @{syntax_const "_hoare"} $ assn_tr' p $ com_tr' c $ assn_tr' q
-*}
-
-print_translation {* [(@{const_syntax Valid}, spec_tr')] *}
-
-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 (auto simp:Valid_def)
-
-lemma iter_aux: "! s s'. Sem c s s' --> s : I & s : b --> s' : I ==>
- (\<And>s s'. s : I \<Longrightarrow> iter n b (Sem c) s s' \<Longrightarrow> s' : I & s' ~: b)";
-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 (clarsimp simp:Valid_def)
-apply(drule iter_aux)
- prefer 2 apply assumption
- apply blast
-apply blast
-done
-
-
-lemma Compl_Collect: "-(Collect b) = {x. ~(b x)}"
- by blast
-
-lemmas AbortRule = SkipRule -- "dummy version"
-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"
-
-end