Hoare syntax: standard abstraction syntax admits source positions;
re-unified some clones (!);
(* Title: HOL/Hoare/Hoare_Logic.thy
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_Logic
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)
abbreviation annskip ("SKIP") where "SKIP == Basic id"
types 'a sem = "'a => 'a => bool"
inductive Sem :: "'a com \<Rightarrow> 'a sem"
where
"Sem (Basic f) s (f s)"
| "Sem c1 s s'' \<Longrightarrow> Sem c2 s'' s' \<Longrightarrow> Sem (c1;c2) s s'"
| "s \<in> b \<Longrightarrow> Sem c1 s s' \<Longrightarrow> Sem (IF b THEN c1 ELSE c2 FI) s s'"
| "s \<notin> b \<Longrightarrow> Sem c2 s s' \<Longrightarrow> Sem (IF b THEN c1 ELSE c2 FI) s s'"
| "s \<notin> b \<Longrightarrow> Sem (While b x c) s s"
| "s \<in> b \<Longrightarrow> Sem c s s'' \<Longrightarrow> Sem (While b x c) s'' s' \<Longrightarrow>
Sem (While b x c) 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 \<longleftrightarrow> (!s s'. Sem c s s' --> s : p --> s' : q)"
(** parse translations **)
syntax
"_assign" :: "idt => 'b => 'a com" ("(2_ :=/ _)" [70, 65] 61)
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_translation {*
let
fun mk_abstuple [x] body = Syntax.abs_tr [x, body]
| mk_abstuple (x :: xs) body =
Syntax.const @{const_syntax prod_case} $ Syntax.abs_tr [x, mk_abstuple xs body];
fun mk_fbody x e [y] = if Syntax.eq_idt (x, y) then e else y
| mk_fbody x e (y :: xs) =
Syntax.const @{const_syntax Pair} $
(if Syntax.eq_idt (x, y) then e else y) $ mk_fbody x e xs;
fun mk_fexp x e xs = mk_abstuple xs (mk_fbody x e xs);
(* bexp_tr & assn_tr *)
(*all meta-variables for bexp except for TRUE are translated as if they
were boolean expressions*)
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 *)
fun com_tr (Const (@{syntax_const "_assign"}, _) $ x $ e) xs =
Syntax.const @{const_syntax Basic} $ mk_fexp x 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;
fun vars_tr (Const (@{syntax_const "_idts"}, _) $ idt $ vars) = idt :: vars_tr vars
| vars_tr t = [t];
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);
in [(@{syntax_const "_hoare_vars"}, hoare_vars_tr)] end
*}
(*****************************************************************************)
(*** 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"} $ assn_tr' p $ com_tr' c $ assn_tr' q
*}
print_translation {* [(@{const_syntax 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 While_aux:
assumes "Sem (WHILE b INV {i} DO c OD) s s'"
shows "\<forall>s s'. Sem c s s' \<longrightarrow> s \<in> I \<and> s \<in> b \<longrightarrow> s' \<in> I \<Longrightarrow>
s \<in> I \<Longrightarrow> s' \<in> I \<and> s' \<notin> 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 (clarsimp simp:Valid_def)
apply(drule While_aux)
apply assumption
apply blast
apply blast
done
lemma Compl_Collect: "-(Collect b) = {x. ~(b x)}"
by blast
lemmas AbortRule = SkipRule -- "dummy version"
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"
end