(*  Title:      HOL/Hoare/HoareAbort.thy

     Author:     Leonor Prensa Nieto & Tobias Nipkow

     Copyright   2003 TUM

 Like Hoare.thy, but with an Abort statement for modelling run time errors.

 *)

 theory Hoare_Logic_Abort

 imports Main

 uses ("hoare_tac.ML")

 begin

 types

     'a bexp = "'a set"

     'a assn = "'a set"

 datatype

  'a com = Basic "'a \<Rightarrow> 'a"

    | Abort

    | 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)

 abbreviation annskip ("SKIP") where "SKIP == Basic id"

 types 'a sem = "'a option => 'a option => bool"

 inductive Sem :: "'a com \<Rightarrow> 'a sem"

 where

   "Sem (Basic f) None None"

 | "Sem (Basic f) (Some s) (Some (f s))"

 | "Sem Abort s None"

 | "Sem c1 s s'' \<Longrightarrow> Sem c2 s'' s' \<Longrightarrow> Sem (c1;c2) s s'"

 | "Sem (IF b THEN c1 ELSE c2 FI) None None"

 | "s \<in> b \<Longrightarrow> Sem c1 (Some s) s' \<Longrightarrow> Sem (IF b THEN c1 ELSE c2 FI) (Some s) s'"

 | "s \<notin> b \<Longrightarrow> Sem c2 (Some s) s' \<Longrightarrow> Sem (IF b THEN c1 ELSE c2 FI) (Some s) s'"

 | "Sem (While b x c) None None"

 | "s \<notin> b \<Longrightarrow> Sem (While b x c) (Some s) (Some s)"

 | "s \<in> b \<Longrightarrow> Sem c (Some s) s'' \<Longrightarrow> Sem (While b x c) s'' s' \<Longrightarrow>

    Sem (While b x c) (Some s) s'"

 inductive_cases [elim!]:

   "Sem (Basic f) s s'" "Sem (c1;c2) s s'"

   "Sem (IF b THEN c1 ELSE c2 FI) s s'"

 definition Valid :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool" where

   "Valid p c q == \<forall>s s'. Sem c s s' \<longrightarrow> s : Some ` p \<longrightarrow> s' : Some ` q"

 (** parse translations **)

 syntax

   "_assign"  :: "id => 'b => 'a com"        ("(2_ :=/ _)" [70,65] 61)

 syntax

   "_hoare_abort_vars" :: "[idts, 'a assn,'a com,'a assn] => bool"

                  ("VARS _// {_} // _ // {_}" [0,0,55,0] 50)

 syntax ("" output)

   "_hoare_abort"      :: "['a assn,'a com,'a assn] => bool"

                  ("{_} // _ // {_}" [0,55,0] 50)

 ML {*

 local

 fun free a = Free(a,dummyT)

 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 @{const_syntax prod_case} $ abs (x, mk_abstuple xs body);

 fun mk_fbody a e [x as (b,_)] = if a=b then e else free b

   | mk_fbody a e ((b,_)::xs) =

       Syntax.const @{const_syntax Pair} $ (if a=b then e else 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"   (* FIXME !? *)

   | bexp_tr b xs = Syntax.const @{const_syntax Collect} $ mk_abstuple xs b;

 fun assn_tr r xs = Syntax.const @{const_syntax Collect} $ mk_abstuple xs r;

 *}

 (* com_tr *)

 ML{*

 fun com_tr (Const (@{syntax_const "_assign"},_) $ Free (a,_) $ e) xs =

       Syntax.const @{const_syntax Basic} $ mk_fexp a e xs

   | com_tr (Const (@{const_syntax Basic},_) $ f) xs = Syntax.const @{const_syntax Basic} $ f

   | com_tr (Const (@{const_syntax Seq},_) $ c1 $ c2) xs =

       Syntax.const @{const_syntax Seq} $ com_tr c1 xs $ com_tr c2 xs

   | com_tr (Const (@{const_syntax Cond},_) $ b $ c1 $ c2) xs =

       Syntax.const @{const_syntax Cond} $ bexp_tr b xs $ com_tr c1 xs $ com_tr c2 xs

   | com_tr (Const (@{const_syntax While},_) $ b $ I $ c) xs =

       Syntax.const @{const_syntax 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 (@{syntax_const "_constrain"}, _) $ Free (a, _) $ T) = (a, T);

 fun vars_tr (Const (@{syntax_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 @{const_syntax 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 {* [(@{syntax_const "_hoare_abort_vars"}, hoare_vars_tr)] *}

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

 (*** print translations ***)

 ML{*

 fun dest_abstuple (Const (@{const_syntax prod_case},_) $ (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 (@{const_syntax prod_case},_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t

   | abs2list (Abs(x,T,t)) = [Free (x, T)]

   | abs2list _ = [];

 fun mk_ts (Const (@{const_syntax prod_case},_) $ (Abs(x,_,t))) = mk_ts t

   | mk_ts (Abs(x,_,t)) = mk_ts t

   | mk_ts (Const (@{const_syntax Pair},_) $ a $ b) = a::(mk_ts b)

   | mk_ts t = [t];

 fun mk_vts (Const (@{const_syntax prod_case},_) $ (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 (@{const_syntax prod_case},_) $ (Abs(x,_,t))) = true

   | is_f (Abs(x,_,t)) = true

   | is_f t = false;

 *}

 (* assn_tr' & bexp_tr'*)

 ML{*

 fun assn_tr' (Const (@{const_syntax Collect},_) $ T) = dest_abstuple T

   | assn_tr' (Const (@{const_syntax inter},_) $ (Const (@{const_syntax Collect},_) $ T1) $

         (Const (@{const_syntax Collect},_) $ T2)) =

       Syntax.const @{const_syntax inter} $ dest_abstuple T1 $ dest_abstuple T2

   | assn_tr' t = t;

 fun bexp_tr' (Const (@{const_syntax 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 @{syntax_const "_assign"} $ fst which $ snd which

     else Syntax.const @{const_syntax annskip}

   end;

 fun com_tr' (Const (@{const_syntax Basic},_) $ f) =

       if is_f f then mk_assign f else Syntax.const @{const_syntax Basic} $ f

   | com_tr' (Const (@{const_syntax Seq},_) $ c1 $ c2) =

       Syntax.const @{const_syntax Seq} $ com_tr' c1 $ com_tr' c2

   | com_tr' (Const (@{const_syntax Cond},_) $ b $ c1 $ c2) =

       Syntax.const @{const_syntax Cond} $ bexp_tr' b $ com_tr' c1 $ com_tr' c2

   | com_tr' (Const (@{const_syntax While},_) $ b $ I $ c) =

       Syntax.const @{const_syntax While} $ bexp_tr' b $ assn_tr' I $ com_tr' c

   | com_tr' t = t;

 fun spec_tr' [p, c, q] =

   Syntax.const @{syntax_const "_hoare_abort"} $ assn_tr' p $ com_tr' c $ assn_tr' q

 *}

 print_translation {* [(@{const_syntax Valid}, spec_tr')] *}

 (*** The proof rules ***)

 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 (fastsimp simp:Valid_def image_def)

 lemma While_aux:

   assumes "Sem (WHILE b INV {i} DO c OD) s s'"

   shows "\<forall>s s'. Sem c s s' \<longrightarrow> s \<in> Some ` (I \<inter> b) \<longrightarrow> s' \<in> Some ` I \<Longrightarrow>

     s \<in> Some ` I \<Longrightarrow> s' \<in> Some ` (I \<inter> -b)"

   using assms

   by (induct "WHILE b INV {i} DO c OD" s s') auto

 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(simp add:Valid_def)

 apply(simp (no_asm) add:image_def)

 apply clarify

 apply(drule While_aux)

   apply assumption

  apply blast

 apply blast

 done

 lemma AbortRule: "p \<subseteq> {s. False} \<Longrightarrow> Valid p Abort q"

 by(auto simp:Valid_def)

 subsection {* Derivation of the proof rules and, most importantly, the VCG tactic *}

 lemma Compl_Collect: "-(Collect b) = {x. ~(b x)}"

   by blast

 use "hoare_tac.ML"

 method_setup vcg = {*

   Scan.succeed (fn ctxt => SIMPLE_METHOD' (hoare_tac ctxt (K all_tac))) *}

   "verification condition generator"

 method_setup vcg_simp = {*

   Scan.succeed (fn ctxt =>

     SIMPLE_METHOD' (hoare_tac ctxt (asm_full_simp_tac (simpset_of ctxt)))) *}

   "verification condition generator plus simplification"

 (* Special syntax for guarded statements and guarded array updates: *)

 syntax

   "_guarded_com" :: "bool \<Rightarrow> 'a com \<Rightarrow> 'a com"  ("(2_ \<rightarrow>/ _)" 71)

   "_array_update" :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a com"  ("(2_[_] :=/ _)" [70, 65] 61)

 translations

   "P \<rightarrow> c" == "IF P THEN c ELSE CONST Abort FI"

   "a[i] := v" => "(i < CONST length a) \<rightarrow> (a := CONST list_update a i v)"

   (* reverse translation not possible because of duplicate "a" *)

 text{* Note: there is no special syntax for guarded array access. Thus

 you must write @{text"j < length a \<rightarrow> a[i] := a!j"}. *}

 end