# HG changeset patch
# User haftmann
# Date 1266916309 -3600
# Node ID f80aee1ed4750061425b5702179b6dd511aadc8c
# Parent  4140f31b2ed207024c86f9af7b9a1d64e0cc62f2
dropped axclass; dropped Id; session theory Hoare.thy

diff -r 4140f31b2ed2 -r f80aee1ed475 src/HOL/Hoare/Hoare_Logic_Abort.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Hoare/Hoare_Logic_Abort.thy	Tue Feb 23 10:11:49 2010 +0100
@@ -0,0 +1,269 @@
+(*  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"
+
+consts iter :: "nat => 'a bexp => 'a sem => 'a sem"
+primrec
+"iter 0 b S = (\<lambda>s s'. s \<notin> Some ` b \<and> s=s')"
+"iter (Suc n) b S =
+  (\<lambda>s s'. s \<in> Some ` b \<and> (\<exists>s''. S s s'' \<and> iter n b S s'' s'))"
+
+consts Sem :: "'a com => 'a sem"
+primrec
+"Sem(Basic f) s s' = (case s of None \<Rightarrow> s' = None | Some t \<Rightarrow> s' = Some(f t))"
+"Sem Abort s s' = (s' = None)"
+"Sem(c1;c2) s s' = (\<exists>s''. Sem c1 s s'' \<and> Sem c2 s'' s')"
+"Sem(IF b THEN c1 ELSE c2 FI) s s' =
+ (case s of None \<Rightarrow> s' = None
+  | Some t \<Rightarrow> ((t \<in> b \<longrightarrow> Sem c1 s s') \<and> (t \<notin> b \<longrightarrow> Sem c2 s s')))"
+"Sem(While b x c) s s' =
+ (if s = None then s' = None else \<exists>n. iter n b (Sem c) s s')"
+
+constdefs Valid :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool"
+  "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 split} $ 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 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 (@{const_syntax split},_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t
+  | abs2list (Abs(x,T,t)) = [Free (x, T)]
+  | abs2list _ = [];
+
+fun mk_ts (Const (@{const_syntax split},_) $ (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 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 (@{const_syntax split},_) $ (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 iter_aux:
+ "! s s'. Sem c s s' \<longrightarrow> s \<in> Some ` (I \<inter> b) \<longrightarrow> s' \<in> Some ` I \<Longrightarrow>
+  (\<And>s s'. s \<in> Some ` I \<Longrightarrow> iter n b (Sem c) s s' \<Longrightarrow> s' \<in> Some ` (I \<inter> -b))";
+apply(unfold image_def)
+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(simp add:Valid_def)
+apply(simp (no_asm) add:image_def)
+apply clarify
+apply(drule iter_aux)
+  prefer 2 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