src/HOL/Hoare/Hoare.thy
author wenzelm
Thu Feb 11 22:19:58 2010 +0100 (2010-02-11)
changeset 35113 1a0c129bb2e0
parent 35054 a5db9779b026
child 35321 c298a4fc324b
permissions -rw-r--r--
modernized translations;
formal markup of @{syntax_const} and @{const_syntax};
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(*  Title:      HOL/Hoare/Hoare.thy
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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
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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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   | 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 => '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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(** 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 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 @{const_syntax 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 @{const_syntax 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"   (* 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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*}
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(* com_tr *)
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ML{*
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fun com_tr (Const(@{syntax_const "_assign"},_) $ Free (a,_) $ e) xs =
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      Syntax.const @{const_syntax Basic} $ mk_fexp a 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 (* 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 (@{syntax_const "_constrain"}, _) $ (Free (a,_)) $ T) = (a,T);
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fun vars_tr (Const (@{syntax_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 @{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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end
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*}
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parse_translation {* [(@{syntax_const "_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 (@{const_syntax 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 (@{const_syntax 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 (@{const_syntax 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 (@{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 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 =
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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 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 (@{const_syntax Collect},_) $ T) = dest_abstuple T
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  | assn_tr' (Const (@{const_syntax inter}, _) $
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        (Const (@{const_syntax Collect},_) $ T1) $ (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
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      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"} $ 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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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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lemma Compl_Collect: "-(Collect b) = {x. ~(b x)}"
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  by blast
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lemmas AbortRule = SkipRule  -- "dummy version"
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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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end