src/HOL/ex/meson.ML
author paulson
Fri, 24 May 1996 11:42:04 +0200
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Augmented comment about conversion to clauses
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(*  Title:      HOL/ex/meson
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1992  University of Cambridge
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The MESON resolution proof procedure for HOL
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When making clauses, avoids using the rewriter -- instead uses RS recursively
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NEED TO SORT LITERALS BY # OF VARS, USING ==>I/E.  ELIMINATES NEED FOR
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FUNCTION nodups -- if done to goal clauses too!
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*)
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writeln"File HOL/ex/meson.";
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(*Prove theorems using fast_tac*)
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fun prove_fun s = 
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    prove_goal HOL.thy s
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         (fn prems => [ cut_facts_tac prems 1, fast_tac HOL_cs 1 ]);
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(**** Negation Normal Form ****)
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(*** de Morgan laws ***)
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val not_conjD = prove_fun "~(P&Q) ==> ~P | ~Q";
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val not_disjD = prove_fun "~(P|Q) ==> ~P & ~Q";
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val not_notD = prove_fun "~~P ==> P";
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val not_allD = prove_fun  "~(! x.P(x)) ==> ? x. ~P(x)";
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val not_exD = prove_fun   "~(? x.P(x)) ==> ! x. ~P(x)";
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(*** Removal of --> and <-> (positive and negative occurrences) ***)
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val imp_to_disjD = prove_fun "P-->Q ==> ~P | Q";
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val not_impD = prove_fun   "~(P-->Q) ==> P & ~Q";
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val iff_to_disjD = prove_fun "P=Q ==> (~P | Q) & (~Q | P)";
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(*Much more efficient than (P & ~Q) | (Q & ~P) for computing CNF*)
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val not_iffD = prove_fun "~(P=Q) ==> (P | Q) & (~P | ~Q)";
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(**** Pulling out the existential quantifiers ****)
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(*** Conjunction ***)
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val conj_exD1 = prove_fun "(? x.P(x)) & Q ==> ? x. P(x) & Q";
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val conj_exD2 = prove_fun "P & (? x.Q(x)) ==> ? x. P & Q(x)";
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(*** Disjunction ***)
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(*DO NOT USE with forall-Skolemization: makes fewer schematic variables!!
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  With ex-Skolemization, makes fewer Skolem constants*)
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val disj_exD = prove_fun "(? x.P(x)) | (? x.Q(x)) ==> ? x. P(x) | Q(x)";
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val disj_exD1 = prove_fun "(? x.P(x)) | Q ==> ? x. P(x) | Q";
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val disj_exD2 = prove_fun "P | (? x.Q(x)) ==> ? x. P | Q(x)";
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(**** Skolemization -- pulling "?" over "!" ****)
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(*"Axiom" of Choice, proved using the description operator*)
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val [major] = goal HOL.thy
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    "! x. ? y. Q x y ==> ? f. ! x. Q x (f x)";
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by (cut_facts_tac [major] 1);
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by (fast_tac (HOL_cs addEs [selectI]) 1);
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qed "choice";
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(***** Generating clauses for the Meson Proof Procedure *****)
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(*** Disjunctions ***)
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val disj_assoc = prove_fun "(P|Q)|R ==> P|(Q|R)";
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val disj_comm = prove_fun "P|Q ==> Q|P";
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val disj_FalseD1 = prove_fun "False|P ==> P";
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val disj_FalseD2 = prove_fun "P|False ==> P";
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(*** Generation of contrapositives ***)
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(*Inserts negated disjunct after removing the negation; P is a literal*)
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val [major,minor] = goal HOL.thy "~P|Q ==> ((~P==>P) ==> Q)";
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by (rtac (major RS disjE) 1);
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by (rtac notE 1);
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by (etac minor 2);
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by (ALLGOALS assume_tac);
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qed "make_neg_rule";
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(*For Plaisted's "Postive refinement" of the MESON procedure*)
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val [major,minor] = goal HOL.thy "~P|Q ==> (P ==> Q)";
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by (rtac (major RS disjE) 1);
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by (rtac notE 1);
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by (rtac minor 2);
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by (ALLGOALS assume_tac);
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qed "make_refined_neg_rule";
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(*P should be a literal*)
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val [major,minor] = goal HOL.thy "P|Q ==> ((P==>~P) ==> Q)";
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by (rtac (major RS disjE) 1);
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by (rtac notE 1);
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by (etac minor 1);
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by (ALLGOALS assume_tac);
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qed "make_pos_rule";
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(*** Generation of a goal clause -- put away the final literal ***)
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val [major,minor] = goal HOL.thy "~P ==> ((~P==>P) ==> False)";
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by (rtac notE 1);
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by (rtac minor 2);
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by (ALLGOALS (rtac major));
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qed "make_neg_goal";
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val [major,minor] = goal HOL.thy "P ==> ((P==>~P) ==> False)";
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by (rtac notE 1);
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by (rtac minor 1);
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by (ALLGOALS (rtac major));
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qed "make_pos_goal";
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(**** Lemmas for forward proof (like congruence rules) ****)
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(*NOTE: could handle conjunctions (faster?) by
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    nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
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val major::prems = goal HOL.thy
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    "[| P'&Q';  P' ==> P;  Q' ==> Q |] ==> P&Q";
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by (rtac (major RS conjE) 1);
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by (rtac conjI 1);
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by (ALLGOALS (eresolve_tac prems));
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qed "conj_forward";
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val major::prems = goal HOL.thy
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    "[| P'|Q';  P' ==> P;  Q' ==> Q |] ==> P|Q";
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by (rtac (major RS disjE) 1);
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by (ALLGOALS (dresolve_tac prems));
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by (ALLGOALS (eresolve_tac [disjI1,disjI2]));
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qed "disj_forward";
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val major::prems = goal HOL.thy
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    "[| ! x. P'(x);  !!x. P'(x) ==> P(x) |] ==> ! x. P(x)";
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by (rtac allI 1);
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by (resolve_tac prems 1);
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by (rtac (major RS spec) 1);
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qed "all_forward";
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val major::prems = goal HOL.thy
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    "[| ? x. P'(x);  !!x. P'(x) ==> P(x) |] ==> ? x. P(x)";
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by (rtac (major RS exE) 1);
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by (rtac exI 1);
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by (eresolve_tac prems 1);
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qed "ex_forward";
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(**** Operators for forward proof ****)
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(*raises exception if no rules apply -- unlike RL*)
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fun tryres (th, rl::rls) = (th RS rl handle THM _ => tryres(th,rls))
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  | tryres (th, []) = raise THM("tryres", 0, [th]);
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val prop_of = #prop o rep_thm;
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(*Permits forward proof from rules that discharge assumptions*)
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fun forward_res nf st =
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  case Sequence.pull (ALLGOALS (METAHYPS (fn [prem] => rtac (nf prem) 1)) st)
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  of Some(th,_) => th
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   | None => raise THM("forward_res", 0, [st]);
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(*Negation Normal Form*)
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val nnf_rls = [imp_to_disjD, iff_to_disjD, not_conjD, not_disjD,
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               not_impD, not_iffD, not_allD, not_exD, not_notD];
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fun make_nnf th = make_nnf (tryres(th, nnf_rls))
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    handle THM _ => 
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        forward_res make_nnf
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           (tryres(th, [conj_forward,disj_forward,all_forward,ex_forward]))
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    handle THM _ => th;
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(*Are any of the constants in "bs" present in the term?*)
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fun has_consts bs = 
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  let fun has (Const(a,_)) = a mem bs
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        | has (f$u) = has f orelse has u
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        | has (Abs(_,_,t)) = has t
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        | has _ = false
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  in  has  end;
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(*Pull existential quantifiers (Skolemization)*)
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fun skolemize th = 
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  if not (has_consts ["Ex"] (prop_of th)) then th
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  else skolemize (tryres(th, [choice, conj_exD1, conj_exD2,
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			      disj_exD, disj_exD1, disj_exD2]))
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    handle THM _ => 
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        skolemize (forward_res skolemize
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		   (tryres (th, [conj_forward, disj_forward, all_forward])))
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    handle THM _ => forward_res skolemize (th RS ex_forward);
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(**** Clause handling ****)
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fun literals (Const("Trueprop",_) $ P) = literals P
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  | literals (Const("op |",_) $ P $ Q) = literals P @ literals Q
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  | literals (Const("not",_) $ P) = [(false,P)]
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  | literals P = [(true,P)];
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(*number of literals in a term*)
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val nliterals = length o literals;
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(*to detect, and remove, tautologous clauses*)
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fun taut_lits [] = false
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  | taut_lits ((flg,t)::ts) = (not flg,t) mem ts orelse taut_lits ts;
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val term_False = term_of (read_cterm (sign_of HOL.thy) 
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			  ("False", Type("bool",[])));
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(*Include False as a literal: an occurrence of ~False is a tautology*)
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fun is_taut th = taut_lits ((true,term_False) :: literals (prop_of th));
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(*Generation of unique names -- maxidx cannot be relied upon to increase!
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  Cannot rely on "variant", since variables might coincide when literals
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  are joined to make a clause... 
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  19 chooses "U" as the first variable name*)
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val name_ref = ref 19;
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(*Replaces universally quantified variables by FREE variables -- because
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  assumptions may not contain scheme variables.  Later, call "generalize". *)
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fun freeze_spec th =
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  let val sth = th RS spec
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      val newname = (name_ref := !name_ref + 1;
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                     radixstring(26, "A", !name_ref))
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  in  read_instantiate [("x", newname)] sth  end;
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fun resop nf [prem] = resolve_tac (nf prem) 1;
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(*Conjunctive normal form, detecting tautologies early.
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  Strips universal quantifiers and breaks up conjunctions. *)
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fun cnf_aux seen (th,ths) = 
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  if taut_lits (literals(prop_of th) @ seen)  then ths
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  else if not (has_consts ["All","op &"] (prop_of th))  then th::ths
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  else (*conjunction?*)
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        cnf_aux seen (th RS conjunct1, 
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                      cnf_aux seen (th RS conjunct2, ths))
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  handle THM _ => (*universal quant?*)
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        cnf_aux  seen (freeze_spec th,  ths)
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  handle THM _ => (*disjunction?*)
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    let val tac = 
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        (METAHYPS (resop (cnf_nil seen)) 1) THEN
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        (STATE (fn st' => 
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                METAHYPS (resop (cnf_nil (literals (concl_of st') @ seen))) 1))
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    in  Sequence.list_of_s (tac (th RS disj_forward)) @ ths  end
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and cnf_nil seen th = cnf_aux seen (th,[]);
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(*Top-level call to cnf -- it's safe to reset name_ref*)
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fun cnf (th,ths) = 
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   (name_ref := 19;  cnf (th RS conjunct1, cnf (th RS conjunct2, ths))
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    handle THM _ => (*not a conjunction*) cnf_aux [] (th, ths));
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(**** Removal of duplicate literals ****)
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(*Version for removal of duplicate literals*)
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val major::prems = goal HOL.thy
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    "[| P'|Q';  P' ==> P;  [| Q'; P==>False |] ==> Q |] ==> P|Q";
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by (rtac (major RS disjE) 1);
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by (rtac disjI1 1);
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by (rtac (disjCI RS disj_comm) 2);
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by (ALLGOALS (eresolve_tac prems));
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by (etac notE 1);
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by (assume_tac 1);
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qed "disj_forward2";
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(*Forward proof, passing extra assumptions as theorems to the tactic*)
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fun forward_res2 nf hyps st =
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  case Sequence.pull
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        (REPEAT 
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	 (METAHYPS (fn major::minors => rtac (nf (minors@hyps) major) 1) 1) 
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	 st)
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  of Some(th,_) => th
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   | None => raise THM("forward_res2", 0, [st]);
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(*Remove duplicates in P|Q by assuming ~P in Q
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  rls (initially []) accumulates assumptions of the form P==>False*)
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fun nodups_aux rls th = nodups_aux rls (th RS disj_assoc)
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    handle THM _ => tryres(th,rls)
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    handle THM _ => tryres(forward_res2 nodups_aux rls (th RS disj_forward2),
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                           [disj_FalseD1, disj_FalseD2, asm_rl])
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    handle THM _ => th;
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(*Remove duplicate literals, if there are any*)
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fun nodups th =
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    if null(findrep(literals(prop_of th))) then th
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    else nodups_aux [] th;
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(**** Generation of contrapositives ****)
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(*Associate disjuctions to right -- make leftmost disjunct a LITERAL*)
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fun assoc_right th = assoc_right (th RS disj_assoc)
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        handle THM _ => th;
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(*Must check for negative literal first!*)
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val clause_rules = [disj_assoc, make_neg_rule, make_pos_rule];
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(*For Plaisted's postive refinement.  [currently unused] *)
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val refined_clause_rules = [disj_assoc, make_refined_neg_rule, make_pos_rule];
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(*Create a goal or support clause, conclusing False*)
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fun make_goal th =   (*Must check for negative literal first!*)
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    make_goal (tryres(th, clause_rules)) 
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  handle THM _ => tryres(th, [make_neg_goal, make_pos_goal]);
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(*Sort clauses by number of literals*)
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fun fewerlits(th1,th2) = nliterals(prop_of th1) < nliterals(prop_of th2);
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(*TAUTOLOGY CHECK SHOULD NOT BE NECESSARY!*)
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fun sort_clauses ths = sort fewerlits (filter (not o is_taut) ths);
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(*Convert all suitable free variables to schematic variables*)
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fun generalize th = forall_elim_vars 0 (forall_intr_frees th);
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(*Make clauses from a list of theorems, previously Skolemized and put into nnf.
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  The resulting clauses are HOL disjunctions.*)
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fun make_clauses ths = 
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    sort_clauses (map (generalize o nodups) (foldr cnf (ths,[])));
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(*Create a meta-level Horn clause*)
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fun make_horn crules th = make_horn crules (tryres(th,crules)) 
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                          handle THM _ => th;
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(*Generate Horn clauses for all contrapositives of a clause*)
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fun add_contras crules (th,hcs) = 
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  let fun rots (0,th) = hcs
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        | rots (k,th) = zero_var_indexes (make_horn crules th) ::
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                        rots(k-1, assoc_right (th RS disj_comm))
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  in case nliterals(prop_of th) of
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        1 => th::hcs
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      | n => rots(n, assoc_right th)
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  end;
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(*Use "theorem naming" to label the clauses*)
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fun name_thms label = 
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    let fun name1 (th, (k,ths)) =
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          (k-1, Thm.name_thm (label ^ string_of_int k, th) :: ths)
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    in  fn ths => #2 (foldr name1 (ths, (length ths, [])))  end;
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(*Convert a list of clauses to (contrapositive) Horn clauses*)
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fun make_horns ths = 
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    name_thms "Horn#"
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      (gen_distinct eq_thm (foldr (add_contras clause_rules) (ths,[])));
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(*Find an all-negative support clause*)
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fun is_negative th = forall (not o #1) (literals (prop_of th));
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val neg_clauses = filter is_negative;
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(***** MESON PROOF PROCEDURE *****)
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fun rhyps (Const("==>",_) $ (Const("Trueprop",_) $ A) $ phi,
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           As) = rhyps(phi, A::As)
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  | rhyps (_, As) = As;
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(** Detecting repeated assumptions in a subgoal **)
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(*The stringtree detects repeated assumptions.*)
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fun ins_term (net,t) = Net.insert_term((t,t), net, op aconv);
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(*detects repetitions in a list of terms*)
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fun has_reps [] = false
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  | has_reps [_] = false
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  | has_reps [t,u] = (t aconv u)
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  | has_reps ts = (foldl ins_term (Net.empty, ts);  false)
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                  handle INSERT => true; 
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(*Like TRYALL eq_assume_tac, but avoids expensive THEN calls*)
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fun TRYALL_eq_assume_tac 0 st = Sequence.single st
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  | TRYALL_eq_assume_tac i st = TRYALL_eq_assume_tac (i-1) (eq_assumption i st)
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                                handle THM _ => TRYALL_eq_assume_tac (i-1) st;
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(*Loop checking: FAIL if trying to prove the same thing twice
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  -- if *ANY* subgoal has repeated literals*)
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fun check_tac st = 
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  if exists (fn prem => has_reps (rhyps(prem,[]))) (prems_of st)
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  then  Sequence.null  else  Sequence.single st;
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(* net_resolve_tac actually made it slower... *)
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fun prolog_step_tac horns i = 
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    (assume_tac i APPEND resolve_tac horns i) THEN check_tac THEN
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    TRYALL eq_assume_tac;
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(*Sums the sizes of the subgoals, ignoring hypotheses (ancestors)*)
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local fun addconcl(prem,sz) = size_of_term (Logic.strip_assums_concl prem) + sz
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in
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fun size_of_subgoals st = foldr addconcl (prems_of st, 0)
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end;
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(*Could simply use nprems_of, which would count remaining subgoals -- no
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  discrimination as to their size!  With BEST_FIRST, fails for problem 41.*)
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fun best_prolog_tac sizef horns = 
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    BEST_FIRST (has_fewer_prems 1, sizef) (prolog_step_tac horns 1);
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fun depth_prolog_tac horns = 
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    DEPTH_FIRST (has_fewer_prems 1) (prolog_step_tac horns 1);
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(*Return all negative clauses, as possible goal clauses*)
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fun gocls cls = name_thms "Goal#" (map make_goal (neg_clauses cls));
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fun skolemize_tac prems = 
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    cut_facts_tac (map (skolemize o make_nnf) prems)  THEN'
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    REPEAT o (etac exE);
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(*Shell of all meson-tactics.  Supplies cltac with clauses: HOL disjunctions*)
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fun MESON cltac = SELECT_GOAL
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 (EVERY1 [rtac ccontr,
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          METAHYPS (fn negs =>
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                    EVERY1 [skolemize_tac negs,
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                            METAHYPS (cltac o make_clauses)])]);
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(** Best-first search versions **)
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fun best_meson_tac sizef = 
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  MESON (fn cls => 
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         THEN_BEST_FIRST (resolve_tac (gocls cls) 1)
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                         (has_fewer_prems 1, sizef)
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			 (prolog_step_tac (make_horns cls) 1));
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(*First, breaks the goal into independent units*)
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val safe_best_meson_tac =
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     SELECT_GOAL (TRY (safe_tac HOL_cs) THEN 
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                  TRYALL (best_meson_tac size_of_subgoals));
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(** Depth-first search version **)
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val depth_meson_tac =
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     MESON (fn cls => EVERY [resolve_tac (gocls cls) 1, 
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                             depth_prolog_tac (make_horns cls)]);
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(** Iterative deepening version **)
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(*This version does only one inference per call;
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  having only one eq_assume_tac speeds it up!*)
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fun prolog_step_tac' horns = 
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    let val (horn0s, hornps) = (*0 subgoals vs 1 or more*)
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            take_prefix (fn rl => nprems_of rl=0) horns
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        val nrtac = net_resolve_tac horns
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    in  fn i => eq_assume_tac i ORELSE
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                match_tac horn0s i ORELSE  (*no backtracking if unit MATCHES*)
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	        ((assume_tac i APPEND nrtac i) THEN check_tac)
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   455
    end;
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   456
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   457
fun iter_deepen_prolog_tac horns = 
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   458
    ITER_DEEPEN (has_fewer_prems 1) (prolog_step_tac' horns);
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   459
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   460
val iter_deepen_meson_tac = 
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   461
  MESON (fn cls => 
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         (THEN_ITER_DEEPEN (resolve_tac (gocls cls) 1)
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	                   (has_fewer_prems 1)
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	                   (prolog_step_tac' (make_horns cls))));
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   465
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   466
val safe_meson_tac =
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     SELECT_GOAL (TRY (safe_tac HOL_cs) THEN 
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                  TRYALL (iter_deepen_meson_tac));
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writeln"Reached end of file.";