--- a/ex/meson.ML Tue Oct 24 14:59:17 1995 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,417 +0,0 @@
-(* 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 state =
- case Sequence.pull
- (tapply(ALLGOALS (METAHYPS (fn [prem] => rtac (nf prem) 1)),
- state))
- of Some(th,_) => th
- | None => raise THM("forward_res", 0, [state]);
-
-
-(*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 delete tautologous clauses*)
-fun taut_lits [] = false
- | taut_lits ((flg,t)::ts) = (not flg,t) mem ts orelse taut_lits ts;
-
-val is_taut = taut_lits o literals o prop_of;
-
-
-(*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 (tapply(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 state =
- case Sequence.pull
- (tapply(REPEAT
- (METAHYPS (fn major::minors => rtac (nf (minors@hyps) major) 1) 1),
- state))
- of Some(th,_) => th
- | None => raise THM("forward_res2", 0, [state]);
-
-(*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];
-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 = 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;
-
-(*Loop checking: FAIL if trying to prove the same thing twice
- -- repeated literals*)
-val check_tac = SUBGOAL (fn (prem,_) =>
- if has_reps (rhyps(prem,[])) then no_tac else all_tac);
-
-(* net_resolve_tac actually made it slower... *)
-fun prolog_step_tac horns i =
- (assume_tac i APPEND resolve_tac horns i) THEN
- (ALLGOALS 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)])]);
-
-fun best_meson_tac sizef =
- MESON (fn cls =>
- resolve_tac (gocls cls) 1
- THEN_BEST_FIRST
- (has_fewer_prems 1, sizef,
- prolog_step_tac (make_horns cls) 1));
-
-(*First, breaks the goal into independent units*)
-val safe_meson_tac =
- SELECT_GOAL (TRY (safe_tac HOL_cs) THEN
- TRYALL (best_meson_tac size_of_subgoals));
-
-val depth_meson_tac =
- MESON (fn cls => EVERY [resolve_tac (gocls cls) 1,
- depth_prolog_tac (make_horns cls)]);
-
-writeln"Reached end of file.";