(* 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 = Main
files ("hoare.ML"):
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"
nonterminals
vars
syntax
"" :: "id => vars" ("_")
"_vars" :: "[id, vars] => vars" ("_ _")
syntax
"@hoare_vars" :: "[vars, '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{*
fun mk_abstuple [] body = absfree ("x", dummyT, body)
| mk_abstuple [v] body = absfree ((fst o dest_Free) v, dummyT, body)
| mk_abstuple (v::w) body = Syntax.const "split" $
absfree ((fst o dest_Free) v, dummyT, mk_abstuple w body);
fun mk_fbody v e [] = Syntax.const "Unity"
| mk_fbody v e [x] = if v=x then e else x
| mk_fbody v e (x::xs) = Syntax.const "Pair" $ (if v=x then e else x) $
mk_fbody v e xs;
fun mk_fexp v e xs = mk_abstuple xs (mk_fbody v e xs);
*}
(* 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 assign_tr [Free (V,_),E] xs = Syntax.const "Basic" $
mk_fexp (Free(V,dummyT)) E xs
| assign_tr ts _ = raise TERM ("assign_tr", ts);
fun com_tr (Const("@assign",_) $ Free (V,_) $ E) xs =
assign_tr [Free (V,dummyT),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 trm _ = trm;
*}
(* triple_tr *)
ML{*
fun vars_tr (x as Free _) = [x]
| vars_tr (Const ("_vars", _) $ (x as Free _) $ vars) = x :: vars_tr vars
| vars_tr t = raise TERM ("vars_tr", [t]);
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);
*}
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')] *}
use "hoare.ML"
method_setup vcg = {*
Method.no_args
(Method.SIMPLE_METHOD' HEADGOAL (hoare_tac (K all_tac))) *}
"verification condition generator"
method_setup vcg_simp = {*
Method.ctxt_args (fn ctxt =>
Method.METHOD (fn facts =>
hoare_tac (asm_full_simp_tac (Simplifier.get_local_simpset ctxt))1)) *}
"verification condition generator plus simplification"
end