src/HOL/Hoare/Hoare.thy
author wenzelm
Wed Aug 29 11:10:28 2007 +0200 (2007-08-29)
changeset 24470 41c81e23c08d
parent 21588 cd0dc678a205
child 24472 943ef707396c
permissions -rw-r--r--
removed Hoare/hoare.ML, Hoare/hoareAbort.ML, ex/svc_oracle.ML (which can be mistaken as attached ML script on case-insensitive file-system);
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(*  Title:      HOL/Hoare/Hoare.thy
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    ID:         $Id$
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    Author:     Leonor Prensa Nieto & Tobias Nipkow
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    Copyright   1998 TUM
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Sugared semantic embedding of Hoare logic.
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Strictly speaking a shallow embedding (as implemented by Norbert Galm
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following Mike Gordon) would suffice. Maybe the datatype com comes in useful
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later.
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*)
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theory Hoare  imports Main
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begin
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types
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    'a bexp = "'a set"
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    'a assn = "'a set"
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datatype
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 'a com = Basic "'a \<Rightarrow> 'a"         
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   | Seq "'a com" "'a com"               ("(_;/ _)"      [61,60] 60)
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   | Cond "'a bexp" "'a com" "'a com"    ("(1IF _/ THEN _ / ELSE _/ FI)"  [0,0,0] 61)
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   | While "'a bexp" "'a assn" "'a com"  ("(1WHILE _/ INV {_} //DO _ /OD)"  [0,0,0] 61)
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syntax
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  "@assign"  :: "id => 'b => 'a com"        ("(2_ :=/ _)" [70,65] 61)
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  "@annskip" :: "'a com"                    ("SKIP")
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translations
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            "SKIP" == "Basic id"
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types 'a sem = "'a => 'a => bool"
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consts iter :: "nat => 'a bexp => 'a sem => 'a sem"
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primrec
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"iter 0 b S = (%s s'. s ~: b & (s=s'))"
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"iter (Suc n) b S = (%s s'. s : b & (? s''. S s s'' & iter n b S s'' s'))"
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consts Sem :: "'a com => 'a sem"
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primrec
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"Sem(Basic f) s s' = (s' = f s)"
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"Sem(c1;c2) s s' = (? s''. Sem c1 s s'' & Sem c2 s'' s')"
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"Sem(IF b THEN c1 ELSE c2 FI) s s' = ((s  : b --> Sem c1 s s') &
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                                      (s ~: b --> Sem c2 s s'))"
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"Sem(While b x c) s s' = (? n. iter n b (Sem c) s s')"
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constdefs Valid :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool"
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  "Valid p c q == !s s'. Sem c s s' --> s : p --> s' : q"
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syntax
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 "@hoare_vars" :: "[idts, 'a assn,'a com,'a assn] => bool"
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                 ("VARS _// {_} // _ // {_}" [0,0,55,0] 50)
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syntax ("" output)
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 "@hoare"      :: "['a assn,'a com,'a assn] => bool"
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                 ("{_} // _ // {_}" [0,55,0] 50)
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(** parse translations **)
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ML{*
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local
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fun abs((a,T),body) =
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  let val a = absfree(a, dummyT, body)
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  in if T = Bound 0 then a else Const(Syntax.constrainAbsC,dummyT) $ a $ T end
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in
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fun mk_abstuple [x] body = abs (x, body)
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  | mk_abstuple (x::xs) body =
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      Syntax.const "split" $ abs (x, mk_abstuple xs body);
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fun mk_fbody a e [x as (b,_)] = if a=b then e else Syntax.free b
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  | mk_fbody a e ((b,_)::xs) =
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      Syntax.const "Pair" $ (if a=b then e else Syntax.free b) $ mk_fbody a e xs;
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fun mk_fexp a e xs = mk_abstuple xs (mk_fbody a e xs)
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end
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*}
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(* bexp_tr & assn_tr *)
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(*all meta-variables for bexp except for TRUE are translated as if they
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  were boolean expressions*)
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ML{*
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fun bexp_tr (Const ("TRUE", _)) xs = Syntax.const "TRUE"
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  | bexp_tr b xs = Syntax.const "Collect" $ mk_abstuple xs b;
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fun assn_tr r xs = Syntax.const "Collect" $ mk_abstuple xs r;
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*}
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(* com_tr *)
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ML{*
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fun com_tr (Const("@assign",_) $ Free (a,_) $ e) xs =
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      Syntax.const "Basic" $ mk_fexp a e xs
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  | com_tr (Const ("Basic",_) $ f) xs = Syntax.const "Basic" $ f
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  | com_tr (Const ("Seq",_) $ c1 $ c2) xs =
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      Syntax.const "Seq" $ com_tr c1 xs $ com_tr c2 xs
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  | com_tr (Const ("Cond",_) $ b $ c1 $ c2) xs =
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      Syntax.const "Cond" $ bexp_tr b xs $ com_tr c1 xs $ com_tr c2 xs
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  | com_tr (Const ("While",_) $ b $ I $ c) xs =
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      Syntax.const "While" $ bexp_tr b xs $ assn_tr I xs $ com_tr c xs
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  | com_tr t _ = t (* if t is just a Free/Var *)
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*}
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(* triple_tr *)    (* FIXME does not handle "_idtdummy" *)
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ML{*
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local
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fun var_tr(Free(a,_)) = (a,Bound 0) (* Bound 0 = dummy term *)
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  | var_tr(Const ("_constrain", _) $ (Free (a,_)) $ T) = (a,T);
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fun vars_tr (Const ("_idts", _) $ idt $ vars) = var_tr idt :: vars_tr vars
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  | vars_tr t = [var_tr t]
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in
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fun hoare_vars_tr [vars, pre, prg, post] =
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      let val xs = vars_tr vars
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      in Syntax.const "Valid" $
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         assn_tr pre xs $ com_tr prg xs $ assn_tr post xs
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      end
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  | hoare_vars_tr ts = raise TERM ("hoare_vars_tr", ts);
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end
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*}
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parse_translation {* [("@hoare_vars", hoare_vars_tr)] *}
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(*****************************************************************************)
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(*** print translations ***)
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ML{*
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fun dest_abstuple (Const ("split",_) $ (Abs(v,_, body))) =
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                            subst_bound (Syntax.free v, dest_abstuple body)
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  | dest_abstuple (Abs(v,_, body)) = subst_bound (Syntax.free v, body)
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  | dest_abstuple trm = trm;
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fun abs2list (Const ("split",_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t
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  | abs2list (Abs(x,T,t)) = [Free (x, T)]
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  | abs2list _ = [];
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fun mk_ts (Const ("split",_) $ (Abs(x,_,t))) = mk_ts t
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  | mk_ts (Abs(x,_,t)) = mk_ts t
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  | mk_ts (Const ("Pair",_) $ a $ b) = a::(mk_ts b)
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  | mk_ts t = [t];
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fun mk_vts (Const ("split",_) $ (Abs(x,_,t))) = 
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           ((Syntax.free x)::(abs2list t), mk_ts t)
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  | mk_vts (Abs(x,_,t)) = ([Syntax.free x], [t])
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  | mk_vts t = raise Match;
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fun find_ch [] i xs = (false, (Syntax.free "not_ch",Syntax.free "not_ch" ))
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  | find_ch ((v,t)::vts) i xs = if t=(Bound i) then find_ch vts (i-1) xs
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              else (true, (v, subst_bounds (xs,t)));
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fun is_f (Const ("split",_) $ (Abs(x,_,t))) = true
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  | is_f (Abs(x,_,t)) = true
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  | is_f t = false;
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*}
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(* assn_tr' & bexp_tr'*)
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ML{*  
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fun assn_tr' (Const ("Collect",_) $ T) = dest_abstuple T
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  | assn_tr' (Const ("op Int",_) $ (Const ("Collect",_) $ T1) $ 
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                                   (Const ("Collect",_) $ T2)) =  
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            Syntax.const "op Int" $ dest_abstuple T1 $ dest_abstuple T2
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  | assn_tr' t = t;
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fun bexp_tr' (Const ("Collect",_) $ T) = dest_abstuple T 
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  | bexp_tr' t = t;
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*}
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(*com_tr' *)
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ML{*
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fun mk_assign f =
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  let val (vs, ts) = mk_vts f;
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      val (ch, which) = find_ch (vs~~ts) ((length vs)-1) (rev vs)
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  in if ch then Syntax.const "@assign" $ fst(which) $ snd(which)
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     else Syntax.const "@skip" end;
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fun com_tr' (Const ("Basic",_) $ f) = if is_f f then mk_assign f
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                                           else Syntax.const "Basic" $ f
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  | com_tr' (Const ("Seq",_) $ c1 $ c2) = Syntax.const "Seq" $
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                                                 com_tr' c1 $ com_tr' c2
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  | com_tr' (Const ("Cond",_) $ b $ c1 $ c2) = Syntax.const "Cond" $
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                                           bexp_tr' b $ com_tr' c1 $ com_tr' c2
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  | com_tr' (Const ("While",_) $ b $ I $ c) = Syntax.const "While" $
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                                               bexp_tr' b $ assn_tr' I $ com_tr' c
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  | com_tr' t = t;
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fun spec_tr' [p, c, q] =
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  Syntax.const "@hoare" $ assn_tr' p $ com_tr' c $ assn_tr' q
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*}
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print_translation {* [("Valid", spec_tr')] *}
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lemma SkipRule: "p \<subseteq> q \<Longrightarrow> Valid p (Basic id) q"
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by (auto simp:Valid_def)
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lemma BasicRule: "p \<subseteq> {s. f s \<in> q} \<Longrightarrow> Valid p (Basic f) q"
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by (auto simp:Valid_def)
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lemma SeqRule: "Valid P c1 Q \<Longrightarrow> Valid Q c2 R \<Longrightarrow> Valid P (c1;c2) R"
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by (auto simp:Valid_def)
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lemma CondRule:
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 "p \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> w) \<and> (s \<notin> b \<longrightarrow> s \<in> w')}
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  \<Longrightarrow> Valid w c1 q \<Longrightarrow> Valid w' c2 q \<Longrightarrow> Valid p (Cond b c1 c2) q"
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by (auto simp:Valid_def)
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lemma iter_aux: "! s s'. Sem c s s' --> s : I & s : b --> s' : I ==>
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       (\<And>s s'. s : I \<Longrightarrow> iter n b (Sem c) s s' \<Longrightarrow> s' : I & s' ~: b)";
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apply(induct n)
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 apply clarsimp
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apply(simp (no_asm_use))
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apply blast
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done
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lemma WhileRule:
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 "p \<subseteq> i \<Longrightarrow> Valid (i \<inter> b) c i \<Longrightarrow> i \<inter> (-b) \<subseteq> q \<Longrightarrow> Valid p (While b i c) q"
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apply (clarsimp simp:Valid_def)
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apply(drule iter_aux)
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  prefer 2 apply assumption
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 apply blast
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apply blast
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done
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subsection {* Derivation of the proof rules and, most importantly, the VCG tactic *}
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ML {*
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(*** The tactics ***)
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(*****************************************************************************)
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(** The function Mset makes the theorem                                     **)
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(** "?Mset <= {(x1,...,xn). ?P (x1,...,xn)} ==> ?Mset <= {s. ?P s}",        **)
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(** where (x1,...,xn) are the variables of the particular program we are    **)
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(** working on at the moment of the call                                    **)
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(*****************************************************************************)
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local open HOLogic in
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(** maps (%x1 ... xn. t) to [x1,...,xn] **)
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fun abs2list (Const ("split",_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t
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  | abs2list (Abs(x,T,t)) = [Free (x, T)]
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  | abs2list _ = [];
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(** maps {(x1,...,xn). t} to [x1,...,xn] **)
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fun mk_vars (Const ("Collect",_) $ T) = abs2list T
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  | mk_vars _ = [];
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(** abstraction of body over a tuple formed from a list of free variables. 
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Types are also built **)
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fun mk_abstupleC []     body = absfree ("x", unitT, body)
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  | mk_abstupleC (v::w) body = let val (n,T) = dest_Free v
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                               in if w=[] then absfree (n, T, body)
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        else let val z  = mk_abstupleC w body;
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                 val T2 = case z of Abs(_,T,_) => T
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                        | Const (_, Type (_,[_, Type (_,[T,_])])) $ _ => T;
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       in Const ("split", (T --> T2 --> boolT) --> mk_prodT (T,T2) --> boolT) 
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          $ absfree (n, T, z) end end;
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(** maps [x1,...,xn] to (x1,...,xn) and types**)
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fun mk_bodyC []      = HOLogic.unit
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  | mk_bodyC (x::xs) = if xs=[] then x 
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               else let val (n, T) = dest_Free x ;
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                        val z = mk_bodyC xs;
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                        val T2 = case z of Free(_, T) => T
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                                         | Const ("Pair", Type ("fun", [_, Type
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                                            ("fun", [_, T])])) $ _ $ _ => T;
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                 in Const ("Pair", [T, T2] ---> mk_prodT (T, T2)) $ x $ z end;
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(** maps a goal of the form:
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        1. [| P |] ==> VARS x1 ... xn {._.} _ {._.} or to [x1,...,xn]**) 
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fun get_vars thm = let  val c = Logic.unprotect (concl_of (thm));
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                        val d = Logic.strip_assums_concl c;
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                        val Const _ $ pre $ _ $ _ = dest_Trueprop d;
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      in mk_vars pre end;
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(** Makes Collect with type **)
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fun mk_CollectC trm = let val T as Type ("fun",[t,_]) = fastype_of trm 
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                      in Collect_const t $ trm end;
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fun inclt ty = Const (@{const_name HOL.less_eq}, [ty,ty] ---> boolT);
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(** Makes "Mset <= t" **)
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fun Mset_incl t = let val MsetT = fastype_of t 
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                 in mk_Trueprop ((inclt MsetT) $ Free ("Mset", MsetT) $ t) end;
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fun Mset thm = let val vars = get_vars(thm);
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                   val varsT = fastype_of (mk_bodyC vars);
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                   val big_Collect = mk_CollectC (mk_abstupleC vars 
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                         (Free ("P",varsT --> boolT) $ mk_bodyC vars));
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                   val small_Collect = mk_CollectC (Abs("x",varsT,
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                           Free ("P",varsT --> boolT) $ Bound 0));
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                   val impl = implies $ (Mset_incl big_Collect) $ 
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                                          (Mset_incl small_Collect);
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   in Goal.prove (ProofContext.init (Thm.theory_of_thm thm)) ["Mset", "P"] [] impl (K (CLASET' blast_tac 1)) end;
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end;
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*}
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(*****************************************************************************)
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(** Simplifying:                                                            **)
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(** Some useful lemmata, lists and simplification tactics to control which  **)
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(** theorems are used to simplify at each moment, so that the original      **)
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(** input does not suffer any unexpected transformation                     **)
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(*****************************************************************************)
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lemma Compl_Collect: "-(Collect b) = {x. ~(b x)}"
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  by blast
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ML {*
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(**Simp_tacs**)
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val before_set2pred_simp_tac =
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  (simp_tac (HOL_basic_ss addsimps [@{thm Collect_conj_eq} RS sym, @{thm Compl_Collect}]));
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val split_simp_tac = (simp_tac (HOL_basic_ss addsimps [split_conv]));
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(*****************************************************************************)
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(** set2pred transforms sets inclusion into predicates implication,         **)
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(** maintaining the original variable names.                                **)
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(** Ex. "{x. x=0} <= {x. x <= 1}" -set2pred-> "x=0 --> x <= 1"              **)
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(** Subgoals containing intersections (A Int B) or complement sets (-A)     **)
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(** are first simplified by "before_set2pred_simp_tac", that returns only   **)
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(** subgoals of the form "{x. P x} <= {x. Q x}", which are easily           **)
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(** transformed.                                                            **)
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(** This transformation may solve very easy subgoals due to a ligth         **)
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(** simplification done by (split_all_tac)                                  **)
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(*****************************************************************************)
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fun set2pred i thm =
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  let val var_names = map (fst o dest_Free) (get_vars thm) in
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    ((before_set2pred_simp_tac i) THEN_MAYBE
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     (EVERY [rtac subsetI i, 
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             rtac CollectI i,
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             dtac CollectD i,
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             (TRY(split_all_tac i)) THEN_MAYBE
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             ((rename_params_tac var_names i) THEN
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              (full_simp_tac (HOL_basic_ss addsimps [split_conv]) i)) ])) thm
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  end;
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(*****************************************************************************)
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(** BasicSimpTac is called to simplify all verification conditions. It does **)
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(** a light simplification by applying "mem_Collect_eq", then it calls      **)
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(** MaxSimpTac, which solves subgoals of the form "A <= A",                 **)
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(** and transforms any other into predicates, applying then                 **)
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(** the tactic chosen by the user, which may solve the subgoal completely.  **)
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(*****************************************************************************)
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fun MaxSimpTac tac = FIRST'[rtac subset_refl, set2pred THEN_MAYBE' tac];
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fun BasicSimpTac tac =
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  simp_tac
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    (HOL_basic_ss addsimps [mem_Collect_eq,split_conv] addsimprocs [record_simproc])
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  THEN_MAYBE' MaxSimpTac tac;
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(** HoareRuleTac **)
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fun WlpTac Mlem tac i =
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  rtac @{thm SeqRule} i THEN  HoareRuleTac Mlem tac false (i+1)
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and HoareRuleTac Mlem tac pre_cond i st = st |>
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        (*abstraction over st prevents looping*)
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    ( (WlpTac Mlem tac i THEN HoareRuleTac Mlem tac pre_cond i)
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      ORELSE
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      (FIRST[rtac @{thm SkipRule} i,
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             EVERY[rtac @{thm BasicRule} i,
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                   rtac Mlem i,
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                   split_simp_tac i],
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             EVERY[rtac @{thm CondRule} i,
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                   HoareRuleTac Mlem tac false (i+2),
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                   HoareRuleTac Mlem tac false (i+1)],
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             EVERY[rtac @{thm WhileRule} i,
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                   BasicSimpTac tac (i+2),
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                   HoareRuleTac Mlem tac true (i+1)] ] 
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       THEN (if pre_cond then (BasicSimpTac tac i) else (rtac subset_refl i)) ));
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(** tac:(int -> tactic) is the tactic the user chooses to solve or simplify **)
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(** the final verification conditions                                       **)
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fun hoare_tac tac i thm =
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  let val Mlem = Mset(thm)
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  in SELECT_GOAL(EVERY[HoareRuleTac Mlem tac true 1]) i thm end;
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*}
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method_setup vcg = {*
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  Method.no_args (Method.SIMPLE_METHOD' (hoare_tac (K all_tac))) *}
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  "verification condition generator"
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method_setup vcg_simp = {*
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  Method.ctxt_args (fn ctxt =>
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    Method.SIMPLE_METHOD' (hoare_tac (asm_full_simp_tac (local_simpset_of ctxt)))) *}
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  "verification condition generator plus simplification"
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end