diff -r 000000000000 -r 7949f97df77a ex/meson.ML --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/ex/meson.ML Thu Sep 16 12:21:07 1993 +0200 @@ -0,0 +1,417 @@ +(* 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); +val choice = result(); + + +(***** 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); +val make_neg_rule = result(); + +(*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); +val make_refined_neg_rule = result(); + +(*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); +val make_pos_rule = result(); + +(*** 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)); +val make_neg_goal = result(); + +val [major,minor] = goal HOL.thy "P ==> ((P==>~P) ==> False)"; +by (rtac notE 1); +by (rtac minor 1); +by (ALLGOALS (rtac major)); +val make_pos_goal = result(); + + +(**** 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)); +val conj_forward = result(); + +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])); +val disj_forward = result(); + +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); +val all_forward = result(); + +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); +val ex_forward = result(); + + +(**** 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); +val disj_forward2 = result(); + +(*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.";