(* Title: HOL/ex/meson
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
The MESON resolution proof procedure for HOL
When making clauses, avoids using the rewriter -- instead uses RS recursively
*)
writeln"File HOL/ex/meson.";
(*Prove theorems using fast_tac*)
fun prove_fun s =
prove_goal HOL.thy s
(fn prems => [ cut_facts_tac prems 1, fast_tac HOL_cs 1 ]);
(**** Negation Normal Form ****)
(*** de Morgan laws ***)
val not_conjD = prove_fun "~(P&Q) ==> ~P | ~Q";
val not_disjD = prove_fun "~(P|Q) ==> ~P & ~Q";
val not_notD = prove_fun "~~P ==> P";
val not_allD = prove_fun "~(! x.P(x)) ==> ? x. ~P(x)";
val not_exD = prove_fun "~(? x.P(x)) ==> ! x. ~P(x)";
(*** Removal of --> and <-> (positive and negative occurrences) ***)
val imp_to_disjD = prove_fun "P-->Q ==> ~P | Q";
val not_impD = prove_fun "~(P-->Q) ==> P & ~Q";
val iff_to_disjD = prove_fun "P=Q ==> (~P | Q) & (~Q | P)";
(*Much more efficient than (P & ~Q) | (Q & ~P) for computing CNF*)
val not_iffD = prove_fun "~(P=Q) ==> (P | Q) & (~P | ~Q)";
(**** Pulling out the existential quantifiers ****)
(*** Conjunction ***)
val conj_exD1 = prove_fun "(? x.P(x)) & Q ==> ? x. P(x) & Q";
val conj_exD2 = prove_fun "P & (? x.Q(x)) ==> ? x. P & Q(x)";
(*** Disjunction ***)
(*DO NOT USE with forall-Skolemization: makes fewer schematic variables!!
With ex-Skolemization, makes fewer Skolem constants*)
val disj_exD = prove_fun "(? x.P(x)) | (? x.Q(x)) ==> ? x. P(x) | Q(x)";
val disj_exD1 = prove_fun "(? x.P(x)) | Q ==> ? x. P(x) | Q";
val disj_exD2 = prove_fun "P | (? x.Q(x)) ==> ? x. P | Q(x)";
(**** Skolemization -- pulling "?" over "!" ****)
(*"Axiom" of Choice, proved using the description operator*)
val [major] = goal HOL.thy
"! x. ? y. Q x y ==> ? f. ! x. Q x (f x)";
by (cut_facts_tac [major] 1);
by (fast_tac (HOL_cs addEs [selectI]) 1);
qed "choice";
(***** Generating clauses for the Meson Proof Procedure *****)
(*** Disjunctions ***)
val disj_assoc = prove_fun "(P|Q)|R ==> P|(Q|R)";
val disj_comm = prove_fun "P|Q ==> Q|P";
val disj_FalseD1 = prove_fun "False|P ==> P";
val disj_FalseD2 = prove_fun "P|False ==> P";
(*** Generation of contrapositives ***)
(*Inserts negated disjunct after removing the negation; P is a literal*)
val [major,minor] = goal HOL.thy "~P|Q ==> ((~P==>P) ==> Q)";
by (rtac (major RS disjE) 1);
by (rtac notE 1);
by (etac minor 2);
by (ALLGOALS assume_tac);
qed "make_neg_rule";
(*For Plaisted's "Postive refinement" of the MESON procedure*)
val [major,minor] = goal HOL.thy "~P|Q ==> (P ==> Q)";
by (rtac (major RS disjE) 1);
by (rtac notE 1);
by (rtac minor 2);
by (ALLGOALS assume_tac);
qed "make_refined_neg_rule";
(*P should be a literal*)
val [major,minor] = goal HOL.thy "P|Q ==> ((P==>~P) ==> Q)";
by (rtac (major RS disjE) 1);
by (rtac notE 1);
by (etac minor 1);
by (ALLGOALS assume_tac);
qed "make_pos_rule";
(*** Generation of a goal clause -- put away the final literal ***)
val [major,minor] = goal HOL.thy "~P ==> ((~P==>P) ==> False)";
by (rtac notE 1);
by (rtac minor 2);
by (ALLGOALS (rtac major));
qed "make_neg_goal";
val [major,minor] = goal HOL.thy "P ==> ((P==>~P) ==> False)";
by (rtac notE 1);
by (rtac minor 1);
by (ALLGOALS (rtac major));
qed "make_pos_goal";
(**** Lemmas for forward proof (like congruence rules) ****)
(*NOTE: could handle conjunctions (faster?) by
nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
val major::prems = goal HOL.thy
"[| P'&Q'; P' ==> P; Q' ==> Q |] ==> P&Q";
by (rtac (major RS conjE) 1);
by (rtac conjI 1);
by (ALLGOALS (eresolve_tac prems));
qed "conj_forward";
val major::prems = goal HOL.thy
"[| P'|Q'; P' ==> P; Q' ==> Q |] ==> P|Q";
by (rtac (major RS disjE) 1);
by (ALLGOALS (dresolve_tac prems));
by (ALLGOALS (eresolve_tac [disjI1,disjI2]));
qed "disj_forward";
val major::prems = goal HOL.thy
"[| ! x. P'(x); !!x. P'(x) ==> P(x) |] ==> ! x. P(x)";
by (rtac allI 1);
by (resolve_tac prems 1);
by (rtac (major RS spec) 1);
qed "all_forward";
val major::prems = goal HOL.thy
"[| ? x. P'(x); !!x. P'(x) ==> P(x) |] ==> ? x. P(x)";
by (rtac (major RS exE) 1);
by (rtac exI 1);
by (eresolve_tac prems 1);
qed "ex_forward";
(**** Operators for forward proof ****)
(*raises exception if no rules apply -- unlike RL*)
fun tryres (th, rl::rls) = (th RS rl handle THM _ => tryres(th,rls))
| tryres (th, []) = raise THM("tryres", 0, [th]);
val prop_of = #prop o rep_thm;
(*Permits forward proof from rules that discharge assumptions*)
fun forward_res nf st =
case Sequence.pull (ALLGOALS (METAHYPS (fn [prem] => rtac (nf prem) 1)) st)
of Some(th,_) => th
| None => raise THM("forward_res", 0, [st]);
(*Negation Normal Form*)
val nnf_rls = [imp_to_disjD, iff_to_disjD, not_conjD, not_disjD,
not_impD, not_iffD, not_allD, not_exD, not_notD];
fun make_nnf th = make_nnf (tryres(th, nnf_rls))
handle THM _ =>
forward_res make_nnf
(tryres(th, [conj_forward,disj_forward,all_forward,ex_forward]))
handle THM _ => th;
(*Are any of the constants in "bs" present in the term?*)
fun has_consts bs =
let fun has (Const(a,_)) = a mem bs
| has (f$u) = has f orelse has u
| has (Abs(_,_,t)) = has t
| has _ = false
in has end;
(*Pull existential quantifiers (Skolemization)*)
fun skolemize th =
if not (has_consts ["Ex"] (prop_of th)) then th
else skolemize (tryres(th, [choice, conj_exD1, conj_exD2,
disj_exD, disj_exD1, disj_exD2]))
handle THM _ =>
skolemize (forward_res skolemize
(tryres (th, [conj_forward, disj_forward, all_forward])))
handle THM _ => forward_res skolemize (th RS ex_forward);
(**** Clause handling ****)
fun literals (Const("Trueprop",_) $ P) = literals P
| literals (Const("op |",_) $ P $ Q) = literals P @ literals Q
| literals (Const("not",_) $ P) = [(false,P)]
| literals P = [(true,P)];
(*number of literals in a term*)
val nliterals = length o literals;
(*to detect, and remove, tautologous clauses*)
fun taut_lits [] = false
| taut_lits ((flg,t)::ts) = (not flg,t) mem ts orelse taut_lits ts;
val term_False = term_of (read_cterm (sign_of HOL.thy)
("False", Type("bool",[])));
(*Include False as a literal: an occurrence of ~False is a tautology*)
fun is_taut th = taut_lits ((true,term_False) :: literals (prop_of th));
(*Generation of unique names -- maxidx cannot be relied upon to increase!
Cannot rely on "variant", since variables might coincide when literals
are joined to make a clause...
19 chooses "U" as the first variable name*)
val name_ref = ref 19;
(*Replaces universally quantified variables by FREE variables -- because
assumptions may not contain scheme variables. Later, call "generalize". *)
fun freeze_spec th =
let val sth = th RS spec
val newname = (name_ref := !name_ref + 1;
radixstring(26, "A", !name_ref))
in read_instantiate [("x", newname)] sth end;
fun resop nf [prem] = resolve_tac (nf prem) 1;
(*Conjunctive normal form, detecting tautologies early.
Strips universal quantifiers and breaks up conjunctions. *)
fun cnf_aux seen (th,ths) =
if taut_lits (literals(prop_of th) @ seen) then ths
else if not (has_consts ["All","op &"] (prop_of th)) then th::ths
else (*conjunction?*)
cnf_aux seen (th RS conjunct1,
cnf_aux seen (th RS conjunct2, ths))
handle THM _ => (*universal quant?*)
cnf_aux seen (freeze_spec th, ths)
handle THM _ => (*disjunction?*)
let val tac =
(METAHYPS (resop (cnf_nil seen)) 1) THEN
(STATE (fn st' =>
METAHYPS (resop (cnf_nil (literals (concl_of st') @ seen))) 1))
in Sequence.list_of_s (tac (th RS disj_forward)) @ ths end
and cnf_nil seen th = cnf_aux seen (th,[]);
(*Top-level call to cnf -- it's safe to reset name_ref*)
fun cnf (th,ths) =
(name_ref := 19; cnf (th RS conjunct1, cnf (th RS conjunct2, ths))
handle THM _ => (*not a conjunction*) cnf_aux [] (th, ths));
(**** Removal of duplicate literals ****)
(*Version for removal of duplicate literals*)
val major::prems = goal HOL.thy
"[| P'|Q'; P' ==> P; [| Q'; P==>False |] ==> Q |] ==> P|Q";
by (rtac (major RS disjE) 1);
by (rtac disjI1 1);
by (rtac (disjCI RS disj_comm) 2);
by (ALLGOALS (eresolve_tac prems));
by (etac notE 1);
by (assume_tac 1);
qed "disj_forward2";
(*Forward proof, passing extra assumptions as theorems to the tactic*)
fun forward_res2 nf hyps st =
case Sequence.pull
(REPEAT
(METAHYPS (fn major::minors => rtac (nf (minors@hyps) major) 1) 1)
st)
of Some(th,_) => th
| None => raise THM("forward_res2", 0, [st]);
(*Remove duplicates in P|Q by assuming ~P in Q
rls (initially []) accumulates assumptions of the form P==>False*)
fun nodups_aux rls th = nodups_aux rls (th RS disj_assoc)
handle THM _ => tryres(th,rls)
handle THM _ => tryres(forward_res2 nodups_aux rls (th RS disj_forward2),
[disj_FalseD1, disj_FalseD2, asm_rl])
handle THM _ => th;
(*Remove duplicate literals, if there are any*)
fun nodups th =
if null(findrep(literals(prop_of th))) then th
else nodups_aux [] th;
(**** Generation of contrapositives ****)
(*Associate disjuctions to right -- make leftmost disjunct a LITERAL*)
fun assoc_right th = assoc_right (th RS disj_assoc)
handle THM _ => th;
(*Must check for negative literal first!*)
val clause_rules = [disj_assoc, make_neg_rule, make_pos_rule];
(*For Plaisted's postive refinement. [currently unused] *)
val refined_clause_rules = [disj_assoc, make_refined_neg_rule, make_pos_rule];
(*Create a goal or support clause, conclusing False*)
fun make_goal th = (*Must check for negative literal first!*)
make_goal (tryres(th, clause_rules))
handle THM _ => tryres(th, [make_neg_goal, make_pos_goal]);
(*Sort clauses by number of literals*)
fun fewerlits(th1,th2) = nliterals(prop_of th1) < nliterals(prop_of th2);
(*TAUTOLOGY CHECK SHOULD NOT BE NECESSARY!*)
fun sort_clauses ths = sort fewerlits (filter (not o is_taut) ths);
(*Convert all suitable free variables to schematic variables*)
fun generalize th = forall_elim_vars 0 (forall_intr_frees th);
(*make clauses from a list of theorems*)
fun make_clauses ths =
sort_clauses (map (generalize o nodups) (foldr cnf (ths,[])));
(*Create a Horn clause*)
fun make_horn crules th = make_horn crules (tryres(th,crules))
handle THM _ => th;
(*Generate Horn clauses for all contrapositives of a clause*)
fun add_contras crules (th,hcs) =
let fun rots (0,th) = hcs
| rots (k,th) = zero_var_indexes (make_horn crules th) ::
rots(k-1, assoc_right (th RS disj_comm))
in case nliterals(prop_of th) of
1 => th::hcs
| n => rots(n, assoc_right th)
end;
(*Convert a list of clauses to (contrapositive) Horn clauses*)
fun make_horns ths =
gen_distinct eq_thm (foldr (add_contras clause_rules) (ths,[]));
(*Find an all-negative support clause*)
fun is_negative th = forall (not o #1) (literals (prop_of th));
val neg_clauses = filter is_negative;
(***** MESON PROOF PROCEDURE *****)
fun rhyps (Const("==>",_) $ (Const("Trueprop",_) $ A) $ phi,
As) = rhyps(phi, A::As)
| rhyps (_, As) = As;
(** Detecting repeated assumptions in a subgoal **)
(*The stringtree detects repeated assumptions.*)
fun ins_term (net,t) = Net.insert_term((t,t), net, op aconv);
(*detects repetitions in a list of terms*)
fun has_reps [] = false
| has_reps [_] = false
| has_reps [t,u] = (t aconv u)
| has_reps ts = (foldl ins_term (Net.empty, ts); false)
handle INSERT => true;
(*Like TRYALL eq_assume_tac, but avoids expensive THEN calls*)
fun TRYALL_eq_assume_tac 0 st = Sequence.single st
| TRYALL_eq_assume_tac i st = TRYALL_eq_assume_tac (i-1) (eq_assumption i st)
handle THM _ => TRYALL_eq_assume_tac (i-1) st;
(*Loop checking: FAIL if trying to prove the same thing twice
-- if *ANY* subgoal has repeated literals*)
fun check_tac st =
if exists (fn prem => has_reps (rhyps(prem,[]))) (prems_of st)
then Sequence.null else Sequence.single st;
(* net_resolve_tac actually made it slower... *)
fun prolog_step_tac horns i =
(assume_tac i APPEND resolve_tac horns i) THEN check_tac THEN
TRYALL eq_assume_tac;
(*Sums the sizes of the subgoals, ignoring hypotheses (ancestors)*)
local fun addconcl(prem,sz) = size_of_term (Logic.strip_assums_concl prem) + sz
in
fun size_of_subgoals st = foldr addconcl (prems_of st, 0)
end;
(*Could simply use nprems_of, which would count remaining subgoals -- no
discrimination as to their size! With BEST_FIRST, fails for problem 41.*)
fun best_prolog_tac sizef horns =
BEST_FIRST (has_fewer_prems 1, sizef) (prolog_step_tac horns 1);
fun depth_prolog_tac horns =
DEPTH_FIRST (has_fewer_prems 1) (prolog_step_tac horns 1);
(*Return all negative clauses, as possible goal clauses*)
fun gocls cls = map make_goal (neg_clauses cls);
fun skolemize_tac prems =
cut_facts_tac (map (skolemize o make_nnf) prems) THEN'
REPEAT o (etac exE);
fun MESON sko_tac = SELECT_GOAL
(EVERY1 [rtac ccontr,
METAHYPS (fn negs =>
EVERY1 [skolemize_tac negs,
METAHYPS (sko_tac o make_clauses)])]);
(** Best-first search versions **)
fun best_meson_tac sizef =
MESON (fn cls =>
THEN_BEST_FIRST (resolve_tac (gocls cls) 1)
(has_fewer_prems 1, sizef)
(prolog_step_tac (make_horns cls) 1));
(*First, breaks the goal into independent units*)
val safe_best_meson_tac =
SELECT_GOAL (TRY (safe_tac HOL_cs) THEN
TRYALL (best_meson_tac size_of_subgoals));
(** Depth-first search version **)
val depth_meson_tac =
MESON (fn cls => EVERY [resolve_tac (gocls cls) 1,
depth_prolog_tac (make_horns cls)]);
(** Iterative deepening version **)
(*This version does only one inference per call;
having only one eq_assume_tac speeds it up!*)
fun prolog_step_tac' horns =
let val (horn0s, hornps) = (*0 subgoals vs 1 or more*)
take_prefix (fn rl => nprems_of rl=0) horns
val nrtac = net_resolve_tac horns
in fn i => eq_assume_tac i ORELSE
match_tac horn0s i ORELSE (*no backtracking if unit MATCHES*)
((assume_tac i APPEND nrtac i) THEN check_tac)
end;
fun iter_deepen_prolog_tac horns =
ITER_DEEPEN (has_fewer_prems 1) (prolog_step_tac' horns);
val iter_deepen_meson_tac =
MESON (fn cls =>
(THEN_ITER_DEEPEN (resolve_tac (gocls cls) 1)
(has_fewer_prems 1)
(prolog_step_tac' (make_horns cls))));
val safe_meson_tac =
SELECT_GOAL (TRY (safe_tac HOL_cs) THEN
TRYALL (iter_deepen_meson_tac));
writeln"Reached end of file.";