diff -r f04b33ce250f -r a4dc62a46ee4 ex/meson.ML --- 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.";