src/HOL/Hoare/Hoare_Logic_Abort.thy
author wenzelm
Tue Mar 29 21:11:02 2011 +0200 (2011-03-29)
changeset 42152 6c17259724b2
parent 42054 8cd4783904d8
child 42153 fa108629d132
permissions -rw-r--r--
Hoare syntax: standard abstraction syntax admits source positions;
re-unified some clones (!);
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(*  Title:      HOL/Hoare/Hoare_Logic_Abort.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 Hoare_Logic_Abort
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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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inductive Sem :: "'a com \<Rightarrow> 'a sem"
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where
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  "Sem (Basic f) None None"
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| "Sem (Basic f) (Some s) (Some (f s))"
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| "Sem Abort s None"
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| "Sem c1 s s'' \<Longrightarrow> Sem c2 s'' s' \<Longrightarrow> Sem (c1;c2) s s'"
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| "Sem (IF b THEN c1 ELSE c2 FI) None None"
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| "s \<in> b \<Longrightarrow> Sem c1 (Some s) s' \<Longrightarrow> Sem (IF b THEN c1 ELSE c2 FI) (Some s) s'"
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| "s \<notin> b \<Longrightarrow> Sem c2 (Some s) s' \<Longrightarrow> Sem (IF b THEN c1 ELSE c2 FI) (Some s) s'"
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| "Sem (While b x c) None None"
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| "s \<notin> b \<Longrightarrow> Sem (While b x c) (Some s) (Some s)"
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| "s \<in> b \<Longrightarrow> Sem c (Some s) s'' \<Longrightarrow> Sem (While b x c) s'' s' \<Longrightarrow>
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   Sem (While b x c) (Some s) s'"
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inductive_cases [elim!]:
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  "Sem (Basic f) s s'" "Sem (c1;c2) s s'"
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  "Sem (IF b THEN c1 ELSE c2 FI) s s'"
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definition Valid :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool" where
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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" :: "idt => 'b => 'a com"  ("(2_ :=/ _)" [70, 65] 61)
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syntax
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  "_hoare_abort_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_abort"      :: "['a assn,'a com,'a assn] => bool"
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                 ("{_} // _ // {_}" [0,55,0] 50)
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parse_translation {*
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  let
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    fun mk_abstuple [x] body = Syntax.abs_tr [x, body]
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      | mk_abstuple (x :: xs) body =
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          Syntax.const @{const_syntax prod_case} $ Syntax.abs_tr [x, mk_abstuple xs body];
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    fun mk_fbody x e [y] = if Syntax.eq_idt (x, y) then e else y
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      | mk_fbody x e (y :: xs) =
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          Syntax.const @{const_syntax Pair} $
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            (if Syntax.eq_idt (x, y) then e else y) $ mk_fbody x e xs;
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    fun mk_fexp x e xs = mk_abstuple xs (mk_fbody x e xs);
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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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    fun bexp_tr (Const ("TRUE", _)) xs = Syntax.const "TRUE"   (* FIXME !? *)
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      | bexp_tr b xs = Syntax.const @{const_syntax Collect} $ mk_abstuple xs b;
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    fun assn_tr r xs = Syntax.const @{const_syntax Collect} $ mk_abstuple xs r;
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    (* com_tr *)
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    fun com_tr (Const (@{syntax_const "_assign"}, _) $ x $ e) xs =
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          Syntax.const @{const_syntax Basic} $ mk_fexp x e xs
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      | com_tr (Const (@{const_syntax Basic},_) $ f) xs = Syntax.const @{const_syntax Basic} $ f
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      | com_tr (Const (@{const_syntax Seq},_) $ c1 $ c2) xs =
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          Syntax.const @{const_syntax Seq} $ com_tr c1 xs $ com_tr c2 xs
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      | com_tr (Const (@{const_syntax Cond},_) $ b $ c1 $ c2) xs =
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          Syntax.const @{const_syntax Cond} $ bexp_tr b xs $ com_tr c1 xs $ com_tr c2 xs
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      | com_tr (Const (@{const_syntax While},_) $ b $ I $ c) xs =
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          Syntax.const @{const_syntax While} $ bexp_tr b xs $ assn_tr I xs $ com_tr c xs
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      | com_tr t _ = t;
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    fun vars_tr (Const (@{syntax_const "_idts"}, _) $ idt $ vars) = idt :: vars_tr vars
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      | vars_tr t = [t];
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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 @{const_syntax 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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  in [(@{syntax_const "_hoare_abort_vars"}, hoare_vars_tr)] end
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*}
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(*****************************************************************************)
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(*** print translations ***)
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ML{*
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fun dest_abstuple (Const (@{const_syntax prod_case},_) $ (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 (@{const_syntax prod_case},_) $ (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 (@{const_syntax prod_case},_) $ (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 (@{const_syntax Pair},_) $ a $ b) = a::(mk_ts b)
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  | mk_ts t = [t];
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fun mk_vts (Const (@{const_syntax prod_case},_) $ (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 =
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      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 (@{const_syntax prod_case},_) $ (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 (@{const_syntax Collect},_) $ T) = dest_abstuple T
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  | assn_tr' (Const (@{const_syntax inter},_) $ (Const (@{const_syntax Collect},_) $ T1) $
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        (Const (@{const_syntax Collect},_) $ T2)) =
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      Syntax.const @{const_syntax inter} $ dest_abstuple T1 $ dest_abstuple T2
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  | assn_tr' t = t;
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fun bexp_tr' (Const (@{const_syntax 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
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    if ch then Syntax.const @{syntax_const "_assign"} $ fst which $ snd which
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    else Syntax.const @{const_syntax annskip}
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  end;
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fun com_tr' (Const (@{const_syntax Basic},_) $ f) =
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      if is_f f then mk_assign f else Syntax.const @{const_syntax Basic} $ f
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  | com_tr' (Const (@{const_syntax Seq},_) $ c1 $ c2) =
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      Syntax.const @{const_syntax Seq} $ com_tr' c1 $ com_tr' c2
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  | com_tr' (Const (@{const_syntax Cond},_) $ b $ c1 $ c2) =
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      Syntax.const @{const_syntax Cond} $ bexp_tr' b $ com_tr' c1 $ com_tr' c2
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  | com_tr' (Const (@{const_syntax While},_) $ b $ I $ c) =
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      Syntax.const @{const_syntax While} $ 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 @{syntax_const "_hoare_abort"} $ 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 While_aux:
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  assumes "Sem (WHILE b INV {i} DO c OD) s s'"
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  shows "\<forall>s s'. Sem c s s' \<longrightarrow> s \<in> Some ` (I \<inter> b) \<longrightarrow> s' \<in> Some ` I \<Longrightarrow>
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    s \<in> Some ` I \<Longrightarrow> s' \<in> Some ` (I \<inter> -b)"
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  using assms
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  by (induct "WHILE b INV {i} DO c OD" s s') auto
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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 While_aux)
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  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 CONST 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