src/HOL/Hoare/Hoare.thy

add lemma contI2

(*  Title:      HOL/Hoare/Hoare.thy
    ID:         $Id$
    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.
*)

theory Hoare  imports Main
uses ("hoare_tac.ML")
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)
  
syntax
  "@assign"  :: "id => 'b => 'a com"        ("(2_ :=/ _)" [70,65] 61)
  "@annskip" :: "'a com"                    ("SKIP")

translations
            "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"


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)

(** parse translations **)

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 "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 "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"
  | bexp_tr b xs = Syntax.const "Collect" $ mk_abstuple xs b;
  
fun assn_tr r xs = Syntax.const "Collect" $ mk_abstuple xs r;
*}
(* com_tr *)
ML{*
fun com_tr (Const("@assign",_) $ Free (a,_) $ e) xs =
      Syntax.const "Basic" $ mk_fexp a e xs
  | com_tr (Const ("Basic",_) $ f) xs = Syntax.const "Basic" $ f
  | com_tr (Const ("Seq",_) $ c1 $ c2) xs =
      Syntax.const "Seq" $ com_tr c1 xs $ com_tr c2 xs
  | com_tr (Const ("Cond",_) $ b $ c1 $ c2) xs =
      Syntax.const "Cond" $ bexp_tr b xs $ com_tr c1 xs $ com_tr c2 xs
  | com_tr (Const ("While",_) $ b $ I $ c) xs =
      Syntax.const "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 ("_constrain", _) $ (Free (a,_)) $ T) = (a,T);

fun vars_tr (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 "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 {* [("@hoare_vars", hoare_vars_tr)] *}


(*****************************************************************************)

(*** print translations ***)
ML{*
fun dest_abstuple (Const ("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 ("split",_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t
  | abs2list (Abs(x,T,t)) = [Free (x, T)]
  | abs2list _ = [];

fun mk_ts (Const ("split",_) $ (Abs(x,_,t))) = mk_ts t
  | mk_ts (Abs(x,_,t)) = mk_ts t
  | mk_ts (Const ("Pair",_) $ a $ b) = a::(mk_ts b)
  | mk_ts t = [t];

fun mk_vts (Const ("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 ("split",_) $ (Abs(x,_,t))) = true
  | is_f (Abs(x,_,t)) = true
  | is_f t = false;
*}

(* assn_tr' & bexp_tr'*)
ML{*  
fun assn_tr' (Const ("Collect",_) $ T) = dest_abstuple T
  | assn_tr' (Const ("op Int",_) $ (Const ("Collect",_) $ T1) $ 
                                   (Const ("Collect",_) $ T2)) =  
            Syntax.const "op Int" $ dest_abstuple T1 $ dest_abstuple T2
  | assn_tr' t = t;

fun bexp_tr' (Const ("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 "@assign" $ fst(which) $ snd(which)
     else Syntax.const "@skip" end;

fun com_tr' (Const ("Basic",_) $ f) = if is_f f then mk_assign f
                                           else Syntax.const "Basic" $ f
  | com_tr' (Const ("Seq",_) $ c1 $ c2) = Syntax.const "Seq" $
                                                 com_tr' c1 $ com_tr' c2
  | com_tr' (Const ("Cond",_) $ b $ c1 $ c2) = Syntax.const "Cond" $
                                           bexp_tr' b $ com_tr' c1 $ com_tr' c2
  | com_tr' (Const ("While",_) $ b $ I $ c) = Syntax.const "While" $
                                               bexp_tr' b $ assn_tr' I $ com_tr' c
  | com_tr' t = t;


fun spec_tr' [p, c, q] =
  Syntax.const "@hoare" $ assn_tr' p $ com_tr' c $ assn_tr' q
*}

print_translation {* [("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

use "hoare_tac.ML"

method_setup vcg = {*
  Method.no_args (Method.SIMPLE_METHOD' (hoare_tac (K all_tac))) *}
  "verification condition generator"

method_setup vcg_simp = {*
  Method.ctxt_args (fn ctxt =>
    Method.SIMPLE_METHOD' (hoare_tac (asm_full_simp_tac (local_simpset_of ctxt)))) *}
  "verification condition generator plus simplification"

end