added parentheses to cope with a possible reduction of the precedence of unary
minus
(* Title: HOL/Hoare/Hoare.ML
ID: $Id$
Author: Leonor Prensa Nieto & Tobias Nipkow
Copyright 1998 TUM
Derivation of the proof rules and, most importantly, the VCG tactic.
*)
(*** The proof rules ***)
Goalw [Valid_def] "p <= q ==> Valid p SKIP q";
by (Auto_tac);
qed "SkipRule";
Goalw [Valid_def] "p <= {s. (f s):q} ==> Valid p (Basic f) q";
by (Auto_tac);
qed "BasicRule";
Goalw [Valid_def] "[| Valid P c1 Q; Valid Q c2 R |] ==> Valid P (c1;c2) R";
by (Asm_simp_tac 1);
by (Blast_tac 1);
qed "SeqRule";
Goalw [Valid_def]
"[| p <= {s. (s:b --> s:w) & (s~:b --> s:w')}; \
\ Valid w c1 q; Valid w' c2 q |] \
\ ==> Valid p (IF b THEN c1 ELSE c2 FI) q";
by (Asm_simp_tac 1);
by (Blast_tac 1);
qed "CondRule";
Goal "! s s'. Sem c s s' --> s : I Int b --> s' : I ==> \
\ ! s s'. s : I --> iter n b (Sem c) s s' --> s' : I & s' ~: b";
by (induct_tac "n" 1);
by (Asm_simp_tac 1);
by (Simp_tac 1);
by (Blast_tac 1);
val lemma = result() RS spec RS spec RS mp RS mp;
Goalw [Valid_def]
"[| p <= i; Valid (i Int b) c i; i Int (-b) <= q |] \
\ ==> Valid p (WHILE b INV {i} DO c OD) q";
by (Asm_simp_tac 1);
by (Clarify_tac 1);
by (dtac lemma 1);
by (assume_tac 2);
by (Blast_tac 1);
by (Blast_tac 1);
qed "WhileRule";
(*** The tactics ***)
(*****************************************************************************)
(** The function Mset makes the theorem **)
(** "?Mset <= {(x1,...,xn). ?P (x1,...,xn)} ==> ?Mset <= {s. ?P s}", **)
(** where (x1,...,xn) are the variables of the particular program we are **)
(** working on at the moment of the call. For instance, (found,x,y) are **)
(** the variables of the Zero Search program. **)
(*****************************************************************************)
local open HOLogic in
(** maps (%x1 ... xn. t) to [x1,...,xn] **)
fun abs2list (Const ("split",_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t
| abs2list (Abs(x,T,t)) = [Free (x, T)]
| abs2list _ = [];
(** maps {(x1,...,xn). t} to [x1,...,xn] **)
fun mk_vars (Const ("Collect",_) $ T) = abs2list T
| mk_vars _ = [];
(** abstraction of body over a tuple formed from a list of free variables.
Types are also built **)
fun mk_abstupleC [] body = absfree ("x", unitT, body)
| mk_abstupleC (v::w) body = let val (n,T) = dest_Free v
in if w=[] then absfree (n, T, body)
else let val z = mk_abstupleC w body;
val T2 = case z of Abs(_,T,_) => T
| Const (_, Type (_,[_, Type (_,[T,_])])) $ _ => T;
in Const ("split", (T --> T2 --> boolT) --> mk_prodT (T,T2) --> boolT)
$ absfree (n, T, z) end end;
(** maps [x1,...,xn] to (x1,...,xn) and types**)
fun mk_bodyC [] = Const ("()", unitT)
| mk_bodyC (x::xs) = if xs=[] then x
else let val (n, T) = dest_Free x ;
val z = mk_bodyC xs;
val T2 = case z of Free(_, T) => T
| Const ("Pair", Type ("fun", [_, Type
("fun", [_, T])])) $ _ $ _ => T;
in Const ("Pair", [T, T2] ---> mk_prodT (T, T2)) $ x $ z end;
fun dest_Goal (Const ("Goal", _) $ P) = P;
(** maps a goal of the form:
1. [| P |] ==> |- VARS x1 ... xn. {._.} _ {._.} or to [x1,...,xn]**)
fun get_vars thm = let val c = dest_Goal (concl_of (thm));
val d = Logic.strip_assums_concl c;
val Const _ $ pre $ _ $ _ = dest_Trueprop d;
in mk_vars pre end;
(** Makes Collect with type **)
fun mk_CollectC trm = let val T as Type ("fun",[t,_]) = fastype_of trm
in Collect_const t $ trm end;
fun inclt ty = Const ("op <=", [ty,ty] ---> boolT);
(** Makes "Mset <= t" **)
fun Mset_incl t = let val MsetT = fastype_of t
in mk_Trueprop ((inclt MsetT) $ Free ("Mset", MsetT) $ t) end;
fun Mset thm = let val vars = get_vars(thm);
val varsT = fastype_of (mk_bodyC vars);
val big_Collect = mk_CollectC (mk_abstupleC vars
(Free ("P",varsT --> boolT) $ mk_bodyC vars));
val small_Collect = mk_CollectC (Abs("x",varsT,
Free ("P",varsT --> boolT) $ Bound 0));
val impl = implies $ (Mset_incl big_Collect) $
(Mset_incl small_Collect);
val cimpl = cterm_of (#sign (rep_thm thm)) impl
in prove_goalw_cterm [] cimpl (fn prems =>
[cut_facts_tac prems 1,Blast_tac 1]) end;
end;
(*****************************************************************************)
(** Simplifying: **)
(** Some useful lemmata, lists and simplification tactics to control which **)
(** theorems are used to simplify at each moment, so that the original **)
(** input does not suffer any unexpected transformation **)
(*****************************************************************************)
val Compl_Collect = prove_goal thy "-(Collect b) = {x. ~(b x)}"
(fn _ => [Fast_tac 1]);
(**Simp_tacs**)
val before_set2pred_simp_tac =
(simp_tac (HOL_basic_ss addsimps [Collect_conj_eq RS sym,Compl_Collect]));
val split_simp_tac = (simp_tac (HOL_basic_ss addsimps [split]));
(*****************************************************************************)
(** set2pred transforms sets inclusion into predicates implication, **)
(** maintaining the original variable names. **)
(** Ex. "{x. x=0} <= {x. x <= 1}" -set2pred-> "x=0 --> x <= 1" **)
(** Subgoals containing intersections (A Int B) or complement sets (-A) **)
(** are first simplified by "before_set2pred_simp_tac", that returns only **)
(** subgoals of the form "{x. P x} <= {x. Q x}", which are easily **)
(** transformed. **)
(** This transformation may solve very easy subgoals due to a ligth **)
(** simplification done by (split_all_tac) **)
(*****************************************************************************)
fun set2pred i thm = let fun mk_string [] = ""
| mk_string (x::xs) = x^" "^mk_string xs;
val vars=get_vars(thm);
val var_string = mk_string (map (fst o dest_Free) vars);
in ((before_set2pred_simp_tac i) THEN_MAYBE
(EVERY [rtac subsetI i,
rtac CollectI i,
dtac CollectD i,
(TRY(split_all_tac i)) THEN_MAYBE
((rename_tac var_string i) THEN
(full_simp_tac (HOL_basic_ss addsimps [split]) i)) ])) thm
end;
(*****************************************************************************)
(** BasicSimpTac is called to simplify all verification conditions. It does **)
(** a light simplification by applying "mem_Collect_eq", then it calls **)
(** MaxSimpTac, which solves subgoals of the form "A <= A", **)
(** and transforms any other into predicates, applying then **)
(** the tactic chosen by the user, which may solve the subgoal completely. **)
(*****************************************************************************)
fun MaxSimpTac tac = FIRST'[rtac subset_refl, set2pred THEN_MAYBE' tac];
fun BasicSimpTac tac =
simp_tac (HOL_basic_ss addsimps [mem_Collect_eq,split])
THEN_MAYBE' MaxSimpTac tac;
(** HoareRuleTac **)
fun WlpTac Mlem tac i = rtac SeqRule i THEN HoareRuleTac Mlem tac false (i+1)
and HoareRuleTac Mlem tac pre_cond i st = st |>
(*abstraction over st prevents looping*)
( (WlpTac Mlem tac i THEN HoareRuleTac Mlem tac pre_cond i)
ORELSE
(FIRST[rtac SkipRule i,
EVERY[rtac BasicRule i,
rtac Mlem i,
split_simp_tac i],
EVERY[rtac CondRule i,
HoareRuleTac Mlem tac false (i+2),
HoareRuleTac Mlem tac false (i+1)],
EVERY[rtac WhileRule i,
BasicSimpTac tac (i+2),
HoareRuleTac Mlem tac true (i+1)] ]
THEN (if pre_cond then (BasicSimpTac tac i) else (rtac subset_refl i)) ));
(** tac:(int -> tactic) is the tactic the user chooses to solve or simplify **)
(** the final verification conditions **)
fun hoare_tac tac i thm =
let val Mlem = Mset(thm)
in SELECT_GOAL(EVERY[HoareRuleTac Mlem tac true 1]) i thm end;