src/HOL/Tools/meson.ML
author paulson
Fri, 28 May 2004 11:20:04 +0200
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new skolemize_tac and skolemize method
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(*  Title:      HOL/Tools/meson.ML
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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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local
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 val not_conjD = thm "meson_not_conjD";
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 val not_disjD = thm "meson_not_disjD";
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 val not_notD = thm "meson_not_notD";
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 val not_allD = thm "meson_not_allD";
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 val not_exD = thm "meson_not_exD";
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 val imp_to_disjD = thm "meson_imp_to_disjD";
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 val not_impD = thm "meson_not_impD";
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 val iff_to_disjD = thm "meson_iff_to_disjD";
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 val not_iffD = thm "meson_not_iffD";
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 val conj_exD1 = thm "meson_conj_exD1";
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 val conj_exD2 = thm "meson_conj_exD2";
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 val disj_exD = thm "meson_disj_exD";
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 val disj_exD1 = thm "meson_disj_exD1";
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 val disj_exD2 = thm "meson_disj_exD2";
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 val disj_assoc = thm "meson_disj_assoc";
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 val disj_comm = thm "meson_disj_comm";
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 val disj_FalseD1 = thm "meson_disj_FalseD1";
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 val disj_FalseD2 = thm "meson_disj_FalseD2";
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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 Seq.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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 (*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 (Const ("Hilbert_Choice.Eps",_) $ _) = false
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                      (*ignore constants within @-terms*)
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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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 (**** 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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 (*Include False as a literal: an occurrence of ~False is a tautology*)
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 fun is_taut th = taut_lits ((true, HOLogic.false_const) ::
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                             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)  
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   then ths     (*tautology ignored*)
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   else if not (has_consts ["All","op &"] (prop_of th))  
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   then th::ths (*no work to do, terminate*)
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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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         (fn st' => st' |>
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                 METAHYPS (resop (cnf_nil (literals (concl_of st') @ seen))) 1)
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     in  Seq.list_of (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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 (*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 Seq.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 ordinary resolution. *)
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 val resolution_clause_rules = [disj_assoc, make_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 (make_ord 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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 (*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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 (*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 = Seq.single st
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   | TRYALL_eq_assume_tac i st =
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        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  Seq.empty  else  Seq.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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in
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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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(*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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(*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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(*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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(*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 Drule.eq_thm_prop (foldr (add_contras clause_rules) (ths,[])));
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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 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 Thm.no_prems 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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    end;
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fun iter_deepen_prolog_tac horns =
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    ITER_DEEPEN (has_fewer_prems 1) (prolog_step_tac' horns);
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val iter_deepen_meson_tac =
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  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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fun meson_claset_tac cs =
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  SELECT_GOAL (TRY (safe_tac cs) THEN TRYALL iter_deepen_meson_tac);
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val meson_tac = CLASET' meson_claset_tac;
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(**** Code to support ordinary resolution, rather than Model Elimination ****)
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(*Convert a list of clauses to meta-level clauses, with no contrapositives,
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  for ordinary resolution.*)
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(*Rules to convert the head literal into a negated assumption. If the head
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  literal is already negated, then using notEfalse instead of notEfalse'
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  prevents a double negation.*)
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val notEfalse = read_instantiate [("R","False")] notE;
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val notEfalse' = rotate_prems 1 notEfalse;
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fun negate_nead th = 
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    th RS notEfalse handle THM _ => th RS notEfalse';
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(*Converting one clause*)
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fun make_meta_clause th = negate_nead (make_horn resolution_clause_rules th);
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fun make_meta_clauses ths =
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    name_thms "MClause#"
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      (gen_distinct Drule.eq_thm_prop (map make_meta_clause ths));
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(*Permute a rule's premises to move the i-th premise to the last position.*)
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fun make_last i th =
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  let val n = nprems_of th 
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  in  if 1 <= i andalso i <= n 
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      then Thm.permute_prems (i-1) 1 th
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      else raise THM("make_last", i, [th])
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  end;
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(*Maps a rule that ends "... ==> P ==> False" to "... ==> ~P" while suppressing
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  double-negations.*)
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val negate_head = rewrite_rule [atomize_not, not_not RS eq_reflection];
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(*Maps the clause  [P1,...Pn]==>False to [P1,...,P(i-1),P(i+1),...Pn] ==> ~P*)
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fun select_literal i cl = negate_head (make_last i cl);
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(*Top-level Skolemization. Allows part of the conversion to clauses to be
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  expressed as a tactic (or Isar method).  Each assumption of the selected 
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  goal is converted to NNF and then its existential quantifiers are pulled
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  to the front. Finally, all existential quantifiers are eliminated, 
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  leaving !!-quantified variables. Perhaps Safe_tac should follow, but it
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  might generate many subgoals.*)
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val skolemize_tac = 
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  SUBGOAL
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    (fn (prop,_) =>
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     let val ts = Logic.strip_assums_hyp prop
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     in EVERY1 
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	 [METAHYPS
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	    (fn hyps => (cut_facts_tac (map (skolemize o make_nnf) hyps) 1
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                         THEN REPEAT (etac exE 1))),
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	  REPEAT_DETERM_N (length ts) o (etac thin_rl)]
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     end);
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(** proof method setup **)
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local
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fun meson_meth ctxt =
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dcb75538f542 CHANGED_PROP;
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  Method.SIMPLE_METHOD' HEADGOAL
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    (CHANGED_PROP o meson_claset_tac (Classical.get_local_claset ctxt));
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in
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val meson_setup =
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 [Method.add_methods
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  [("meson", Method.ctxt_args meson_meth, 
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    "The MESON resolution proof procedure"),
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   ("skolemize", Method.no_args (Method.METHOD (fn _ => skolemize_tac 1)), 
826edbc317e6 new skolemize_tac and skolemize method
paulson
parents: 14763
diff changeset
   418
    "Skolemization into existential quantifiers")]];
9840
9dfcb0224f8c meson.ML moved from HOL/ex to HOL/Tools: meson_tac installed by default
paulson
parents:
diff changeset
   419
9dfcb0224f8c meson.ML moved from HOL/ex to HOL/Tools: meson_tac installed by default
paulson
parents:
diff changeset
   420
end;
9869
95dca9f991f2 improved meson setup;
wenzelm
parents: 9840
diff changeset
   421
95dca9f991f2 improved meson setup;
wenzelm
parents: 9840
diff changeset
   422
end;