(*  Title:      HOL/Tools/meson.ML

     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

 NEED TO SORT LITERALS BY # OF VARS, USING ==>I/E.  ELIMINATES NEED FOR

 FUNCTION nodups -- if done to goal clauses too!

 *)

 local

  val not_conjD = thm "meson_not_conjD";

  val not_disjD = thm "meson_not_disjD";

  val not_notD = thm "meson_not_notD";

  val not_allD = thm "meson_not_allD";

  val not_exD = thm "meson_not_exD";

  val imp_to_disjD = thm "meson_imp_to_disjD";

  val not_impD = thm "meson_not_impD";

  val iff_to_disjD = thm "meson_iff_to_disjD";

  val not_iffD = thm "meson_not_iffD";

  val conj_exD1 = thm "meson_conj_exD1";

  val conj_exD2 = thm "meson_conj_exD2";

  val disj_exD = thm "meson_disj_exD";

  val disj_exD1 = thm "meson_disj_exD1";

  val disj_exD2 = thm "meson_disj_exD2";

  val disj_assoc = thm "meson_disj_assoc";

  val disj_comm = thm "meson_disj_comm";

  val disj_FalseD1 = thm "meson_disj_FalseD1";

  val disj_FalseD2 = thm "meson_disj_FalseD2";

  (**** 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 Seq.pull (ALLGOALS (METAHYPS (fn [prem] => rtac (nf prem) 1)) st)

    of Some(th,_) => th

     | None => raise THM("forward_res", 0, [st]);

  (*Are any of the constants in "bs" present in the term?*)

  fun has_consts bs =

    let fun has (Const(a,_)) = a mem bs

 	 | has (Const ("Hilbert_Choice.Eps",_) $ _) = false

                       (*ignore constants within @-terms*)

          | has (f$u) = has f orelse has u

          | has (Abs(_,_,t)) = has t

          | has _ = false

    in  has  end;

  (**** 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;

  (*Include False as a literal: an occurrence of ~False is a tautology*)

  fun is_taut th = taut_lits ((true, HOLogic.false_const) ::

                              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     (*tautology ignored*)

    else if not (has_consts ["All","op &"] (prop_of th))  

    then th::ths (*no work to do, terminate*)

    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

          (fn st' => st' |>

                  METAHYPS (resop (cnf_nil (literals (concl_of st') @ seen))) 1)

      in  Seq.list_of (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 ****)

  (*Forward proof, passing extra assumptions as theorems to the tactic*)

  fun forward_res2 nf hyps st =

    case Seq.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 ordinary resolution. *)

  val resolution_clause_rules = [disj_assoc, make_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 (make_ord 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);

  (*Create a meta-level 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;

  (*Use "theorem naming" to label the clauses*)

  fun name_thms label =

      let fun name1 (th, (k,ths)) =

            (k-1, Thm.name_thm (label ^ string_of_int k, th) :: ths)

      in  fn ths => #2 (foldr name1 (ths, (length ths, [])))  end;

  (*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 = Seq.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  Seq.empty  else  Seq.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;

 in

 (*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;

 (*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;

 (*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);

 (*Make clauses from a list of theorems, previously Skolemized and put into nnf.

   The resulting clauses are HOL disjunctions.*)

 fun make_clauses ths =

     sort_clauses (map (generalize o nodups) (foldr cnf (ths,[])));

 (*Convert a list of clauses to (contrapositive) Horn clauses*)

 fun make_horns ths =

     name_thms "Horn#"

       (gen_distinct Drule.eq_thm_prop (foldr (add_contras clause_rules) (ths,[])));

 (*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 = name_thms "Goal#" (map make_goal (neg_clauses cls));

 fun skolemize_tac prems =

     cut_facts_tac (map (skolemize o make_nnf) prems)  THEN'

     REPEAT o (etac exE);

 (*Shell of all meson-tactics.  Supplies cltac with clauses: HOL disjunctions*)

 fun MESON cltac = SELECT_GOAL

  (EVERY1 [rtac ccontr,

           METAHYPS (fn negs =>

                     EVERY1 [skolemize_tac negs,

                             METAHYPS (cltac 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 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 Thm.no_prems 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))));

 fun meson_claset_tac cs =

   SELECT_GOAL (TRY (safe_tac cs) THEN TRYALL iter_deepen_meson_tac);

 val meson_tac = CLASET' meson_claset_tac;

 (** Code to support ordinary resolution, rather than Model Elimination **)

 (*Convert a list of clauses to meta-level clauses, with no contrapositives,

   for ordinary resolution.*)

 (*Rules to convert the head literal into a negated assumption. If the head

   literal is already negated, then using notEfalse instead of notEfalse'

   prevents a double negation.*)

 val notEfalse = read_instantiate [("R","False")] notE;

 val notEfalse' = rotate_prems 1 notEfalse;

 fun negate_nead th = 

     th RS notEfalse handle THM _ => th RS notEfalse';

 (*Converting one clause*)

 fun make_meta_clause th = negate_nead (make_horn resolution_clause_rules th);

 fun make_meta_clauses ths =

     name_thms "MClause#"

       (gen_distinct Drule.eq_thm_prop (map make_meta_clause ths));

 (*Permute a rule's premises to move the i-th premise to the last position.*)

 fun make_last i th =

   let val n = nprems_of th 

   in  if 1 <= i andalso i <= n 

       then Thm.permute_prems (i-1) 1 th

       else raise THM("make_last", i, [th])

   end;

 (*Maps a rule that ends "... ==> P ==> False" to "... ==> ~P" while suppressing

   double-negations.*)

 val negate_head = rewrite_rule [atomize_not, not_not RS eq_reflection];

 (*Maps the clause  [P1,...Pn]==>False to [P1,...,P(i-1),P(i+1),...Pn] ==> ~P*)

 fun select_literal i cl = negate_head (make_last i cl);

 (** proof method setup **)

 local

 fun meson_meth ctxt =

   Method.SIMPLE_METHOD' HEADGOAL

     (CHANGED_PROP o meson_claset_tac (Classical.get_local_claset ctxt));

 in

 val meson_setup =

  [Method.add_methods

   [("meson", Method.ctxt_args meson_meth, "The MESON resolution proof procedure")]];

 end;

 end;