src/HOL/Hoare/HoareAbort.thy
author wenzelm
Mon Feb 08 21:28:27 2010 +0100 (2010-02-08)
changeset 35054 a5db9779b026
parent 34940 3e80eab831a1
child 35101 6ce9177d6b38
permissions -rw-r--r--
modernized some syntax translations;
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(*  Title:      HOL/Hoare/HoareAbort.thy
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    Author:     Leonor Prensa Nieto & Tobias Nipkow
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    Copyright   2003 TUM
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Like Hoare.thy, but with an Abort statement for modelling run time errors.
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*)
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theory HoareAbort
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imports Main
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uses ("hoare_tac.ML")
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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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   | Abort
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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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abbreviation annskip ("SKIP") where "SKIP == Basic id"
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types 'a sem = "'a option => 'a option => 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 = (\<lambda>s s'. s \<notin> Some ` b \<and> s=s')"
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"iter (Suc n) b S =
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  (\<lambda>s s'. s \<in> Some ` b \<and> (\<exists>s''. S s s'' \<and> 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' = (case s of None \<Rightarrow> s' = None | Some t \<Rightarrow> s' = Some(f t))"
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"Sem Abort s s' = (s' = None)"
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"Sem(c1;c2) s s' = (\<exists>s''. Sem c1 s s'' \<and> Sem c2 s'' s')"
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"Sem(IF b THEN c1 ELSE c2 FI) s s' =
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 (case s of None \<Rightarrow> s' = None
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  | Some t \<Rightarrow> ((t \<in> b \<longrightarrow> Sem c1 s s') \<and> (t \<notin> b \<longrightarrow> Sem c2 s s')))"
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"Sem(While b x c) s s' =
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 (if s = None then s' = None else \<exists>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 == \<forall>s s'. Sem c s s' \<longrightarrow> s : Some ` p \<longrightarrow> s' : Some ` q"
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(** parse translations **)
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syntax
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  "_assign"  :: "id => 'b => 'a com"        ("(2_ :=/ _)" [70,65] 61)
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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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ML {*
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local
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fun free a = Free(a,dummyT)
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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 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 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 (@{const_name inter},_) $ (Const ("Collect",_) $ T1) $ 
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                                   (Const ("Collect",_) $ T2)) =  
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            Syntax.const "Set.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 @{const_syntax annskip} 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 {* [(@{const_syntax Valid}, spec_tr')] *}
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(*** The proof rules ***)
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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 (fastsimp simp:Valid_def image_def)
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lemma iter_aux:
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 "! s s'. Sem c s s' \<longrightarrow> s \<in> Some ` (I \<inter> b) \<longrightarrow> s' \<in> Some ` I \<Longrightarrow>
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  (\<And>s s'. s \<in> Some ` I \<Longrightarrow> iter n b (Sem c) s s' \<Longrightarrow> s' \<in> Some ` (I \<inter> -b))";
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apply(unfold image_def)
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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(simp add:Valid_def)
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apply(simp (no_asm) add:image_def)
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apply clarify
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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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lemma AbortRule: "p \<subseteq> {s. False} \<Longrightarrow> Valid p Abort q"
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by(auto simp:Valid_def)
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subsection {* Derivation of the proof rules and, most importantly, the VCG tactic *}
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lemma Compl_Collect: "-(Collect b) = {x. ~(b x)}"
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  by blast
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use "hoare_tac.ML"
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method_setup vcg = {*
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  Scan.succeed (fn ctxt => SIMPLE_METHOD' (hoare_tac ctxt (K all_tac))) *}
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  "verification condition generator"
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method_setup vcg_simp = {*
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  Scan.succeed (fn ctxt =>
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    SIMPLE_METHOD' (hoare_tac ctxt (asm_full_simp_tac (simpset_of ctxt)))) *}
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  "verification condition generator plus simplification"
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(* Special syntax for guarded statements and guarded array updates: *)
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syntax
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  guarded_com :: "bool \<Rightarrow> 'a com \<Rightarrow> 'a com"  ("(2_ \<rightarrow>/ _)" 71)
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  array_update :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a com"  ("(2_[_] :=/ _)" [70, 65] 61)
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translations
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  "P \<rightarrow> c" == "IF P THEN c ELSE Abort FI"
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  "a[i] := v" => "(i < CONST length a) \<rightarrow> (a := CONST list_update a i v)"
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  (* reverse translation not possible because of duplicate "a" *)
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text{* Note: there is no special syntax for guarded array access. Thus
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you must write @{text"j < length a \<rightarrow> a[i] := a!j"}. *}
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end