src/HOL/Tools/meson.ML
author paulson
Wed May 18 10:24:11 2005 +0200 (2005-05-18)
changeset 15998 bc916036cf84
parent 15965 f422f8283491
child 16012 4ae42d8f2fea
permissions -rw-r--r--
new cnf function taking Skolemization theorems as an extra argument
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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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signature BASIC_MESON =
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sig
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  val size_of_subgoals	: thm -> int
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  val make_cnf		: thm list -> thm -> thm list
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  val make_nnf		: thm -> thm
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  val skolemize		: thm -> thm
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  val make_clauses	: thm list -> thm list
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  val make_horns	: thm list -> thm list
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  val best_prolog_tac	: (thm -> int) -> thm list -> tactic
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  val depth_prolog_tac	: thm list -> tactic
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  val gocls		: thm list -> thm list
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  val skolemize_prems_tac	: thm list -> int -> tactic
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  val MESON		: (thm list -> tactic) -> int -> tactic
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  val best_meson_tac	: (thm -> int) -> int -> tactic
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  val safe_best_meson_tac	: int -> tactic
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  val depth_meson_tac	: int -> tactic
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  val prolog_step_tac'	: thm list -> int -> tactic
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  val iter_deepen_prolog_tac	: thm list -> tactic
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  val iter_deepen_meson_tac	: int -> tactic
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  val meson_tac		: int -> tactic
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  val negate_head	: thm -> thm
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  val select_literal	: int -> thm -> thm
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  val skolemize_tac	: int -> tactic
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  val make_clauses_tac	: int -> tactic
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  val meson_setup	: (theory -> theory) list
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end
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structure Meson =
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struct
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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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(*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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(* for tracing: encloses each string element in brackets. *)
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fun concat_with_and [] = ""
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  | concat_with_and [x] = "(" ^ x ^ ")"
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  | concat_with_and (x::xs) = "(" ^ x ^ ")" ^ " & " ^ concat_with_and xs;
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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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(*Used with METAHYPS below. There is one assumption, which gets bound to prem
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  and then normalized via function nf. The normal form is given to resolve_tac,
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  presumably to instantiate a Boolean variable.*)
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fun resop nf [prem] = resolve_tac (nf prem) 1;
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(*Conjunctive normal form, adding clauses from th in front of ths (for foldr).
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  Detects tautologies early, with "seen" holding the other literals of a clause.
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  Strips universal quantifiers and breaks up conjunctions.
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  Eliminates existential quantifiers using skoths: Skolemization theorems.*)
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fun cnf skoths (th,ths) =
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  let 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","Ex","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 quantifier?*)
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	      cnf_aux seen (freeze_spec th,  ths)
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	handle THM _ => (*existential quantifier? Insert Skolem functions.*)
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	      cnf_aux seen (tryres (th,skoths), 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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  in 
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    name_ref := 19;  (*It's safe to reset this in a top-level call to cnf*)
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    (cnf skoths (th RS conjunct1, cnf skoths (th RS conjunct2, ths))
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     handle THM _ => (*not a conjunction*) cnf_aux [] (th, ths))
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  end;
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fun make_cnf skoths th = cnf skoths (th, []);
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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) (List.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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(*True if the given type contains bool anywhere*)
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fun has_bool (Type("bool",_)) = true
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  | has_bool (Type(_, Ts)) = exists has_bool Ts
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  | has_bool _ = false;
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(*Is the string the name of a connective? It doesn't matter if this list is
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  incomplete, since when actually called, the only connectives likely to
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  remain are & | Not.*)  
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fun is_conn c = c mem_string
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    ["Trueprop", "op &", "op |", "op -->", "op =", "Not", 
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     "All", "Ex", "Ball", "Bex"];
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(*True if the term contains a function where the type of any argument contains
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  bool.*)
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val has_bool_arg_const = 
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    exists_Const
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      (fn (c,T) => not(is_conn c) andalso exists (has_bool) (binder_types T));
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val has_meta_conn = 
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    exists_Const (fn (c,_) => c mem_string ["==", "==>", "all", "Goal"]);
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(*Raises an exception if any Vars in the theorem mention type bool. That would mean
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  they are higher-order, and in addition, they could cause make_horn to loop!
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  Functions taking Boolean arguments are also rejected.*)
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fun check_no_bool th =
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  let val {prop,...} = rep_thm th
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  in if exists (has_bool o fastype_of) (term_vars prop)  orelse
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        has_bool_arg_const prop  orelse  
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        has_meta_conn prop
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  then raise THM ("check_no_bool", 0, [th]) else th
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  end;
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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 (length ths, []) 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 = List.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 = (Library.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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(*Sums the sizes of the subgoals, ignoring hypotheses (ancestors)*)
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fun addconcl(prem,sz) = size_of_term(Logic.strip_assums_concl prem) + sz
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fun size_of_subgoals st = foldr addconcl 0 (prems_of 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_nnf1 th = make_nnf1 (tryres(th, nnf_rls))
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    handle THM _ =>
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        forward_res make_nnf1
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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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(*The simplification removes defined quantifiers and occurrences of True and False.*)
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val nnf_ss = HOL_basic_ss addsimps Ex1_def::Ball_def::Bex_def::simp_thms;
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fun make_nnf th = th |> simplify nnf_ss
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                     |> check_no_bool |> make_nnf1
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(*Pull existential quantifiers to front. This accomplishes Skolemization for
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  clauses that arise from a subgoal.*)
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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_prems_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_prems_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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   391
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   392
(** Depth-first search version **)
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   393
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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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   397
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   398
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   399
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   400
(** Iterative deepening version **)
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   401
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   402
(*This version does only one inference per call;
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   403
  having only one eq_assume_tac speeds it up!*)
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   404
fun prolog_step_tac' horns =
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    let val (horn0s, hornps) = (*0 subgoals vs 1 or more*)
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   406
            take_prefix Thm.no_prems horns
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        val nrtac = net_resolve_tac horns
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   408
    in  fn i => eq_assume_tac i ORELSE
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   409
                match_tac horn0s i ORELSE  (*no backtracking if unit MATCHES*)
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   410
                ((assume_tac i APPEND nrtac i) THEN check_tac)
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   411
    end;
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   412
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   413
fun iter_deepen_prolog_tac horns =
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    ITER_DEEPEN (has_fewer_prems 1) (prolog_step_tac' horns);
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   415
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   416
val iter_deepen_meson_tac =
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  MESON (fn cls =>
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   418
         (THEN_ITER_DEEPEN (resolve_tac (gocls cls) 1)
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   419
                           (has_fewer_prems 1)
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   420
                           (prolog_step_tac' (make_horns cls))));
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   421
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   422
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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   424
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   425
val meson_tac = CLASET' meson_claset_tac;
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   426
wenzelm@9869
   427
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   428
(**** Code to support ordinary resolution, rather than Model Elimination ****)
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   429
paulson@15008
   430
(*Convert a list of clauses (disjunctions) to meta-level clauses (==>), 
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   431
  with no contrapositives, for ordinary resolution.*)
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   432
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   433
(*Rules to convert the head literal into a negated assumption. If the head
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   434
  literal is already negated, then using notEfalse instead of notEfalse'
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   435
  prevents a double negation.*)
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   436
val notEfalse = read_instantiate [("R","False")] notE;
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   437
val notEfalse' = rotate_prems 1 notEfalse;
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   438
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   439
fun negated_asm_of_head th = 
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   440
    th RS notEfalse handle THM _ => th RS notEfalse';
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   441
paulson@14744
   442
(*Converting one clause*)
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   443
fun make_meta_clause th = 
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   444
	negated_asm_of_head (make_horn resolution_clause_rules (check_no_bool th));
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   445
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   446
fun make_meta_clauses ths =
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   447
    name_thms "MClause#"
paulson@14744
   448
      (gen_distinct Drule.eq_thm_prop (map make_meta_clause ths));
paulson@14744
   449
paulson@14744
   450
(*Permute a rule's premises to move the i-th premise to the last position.*)
paulson@14744
   451
fun make_last i th =
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   452
  let val n = nprems_of th 
paulson@14744
   453
  in  if 1 <= i andalso i <= n 
paulson@14744
   454
      then Thm.permute_prems (i-1) 1 th
paulson@15118
   455
      else raise THM("select_literal", i, [th])
paulson@14744
   456
  end;
paulson@14744
   457
paulson@14744
   458
(*Maps a rule that ends "... ==> P ==> False" to "... ==> ~P" while suppressing
paulson@14744
   459
  double-negations.*)
paulson@14744
   460
val negate_head = rewrite_rule [atomize_not, not_not RS eq_reflection];
paulson@14744
   461
paulson@14744
   462
(*Maps the clause  [P1,...Pn]==>False to [P1,...,P(i-1),P(i+1),...Pn] ==> ~P*)
paulson@14744
   463
fun select_literal i cl = negate_head (make_last i cl);
paulson@14744
   464
paulson@14813
   465
(*Top-level Skolemization. Allows part of the conversion to clauses to be
paulson@14813
   466
  expressed as a tactic (or Isar method).  Each assumption of the selected 
paulson@14813
   467
  goal is converted to NNF and then its existential quantifiers are pulled
paulson@14813
   468
  to the front. Finally, all existential quantifiers are eliminated, 
paulson@14813
   469
  leaving !!-quantified variables. Perhaps Safe_tac should follow, but it
paulson@14813
   470
  might generate many subgoals.*)
paulson@14813
   471
val skolemize_tac = 
paulson@14813
   472
  SUBGOAL
paulson@14813
   473
    (fn (prop,_) =>
paulson@14813
   474
     let val ts = Logic.strip_assums_hyp prop
paulson@14813
   475
     in EVERY1 
paulson@14813
   476
	 [METAHYPS
quigley@15773
   477
	    (fn hyps => (cut_facts_tac (map (skolemize o make_nnf) hyps) 1
paulson@14813
   478
                         THEN REPEAT (etac exE 1))),
paulson@14813
   479
	  REPEAT_DETERM_N (length ts) o (etac thin_rl)]
paulson@14813
   480
     end);
paulson@14813
   481
paulson@15118
   482
(*Top-level conversion to meta-level clauses. Each clause has  
paulson@15118
   483
  leading !!-bound universal variables, to express generality. To get 
paulson@15118
   484
  disjunctions instead of meta-clauses, remove "make_meta_clauses" below.*)
paulson@15008
   485
val make_clauses_tac = 
paulson@15008
   486
  SUBGOAL
paulson@15008
   487
    (fn (prop,_) =>
paulson@15008
   488
     let val ts = Logic.strip_assums_hyp prop
paulson@15008
   489
     in EVERY1 
paulson@15008
   490
	 [METAHYPS
paulson@15008
   491
	    (fn hyps => 
paulson@15151
   492
              (Method.insert_tac
paulson@15118
   493
                (map forall_intr_vars 
paulson@15118
   494
                  (make_meta_clauses (make_clauses hyps))) 1)),
paulson@15008
   495
	  REPEAT_DETERM_N (length ts) o (etac thin_rl)]
paulson@15008
   496
     end);
paulson@15008
   497
paulson@14744
   498
paulson@15579
   499
(*** proof method setup ***)
wenzelm@9869
   500
wenzelm@9869
   501
fun meson_meth ctxt =
wenzelm@10821
   502
  Method.SIMPLE_METHOD' HEADGOAL
wenzelm@15032
   503
    (CHANGED_PROP o meson_claset_tac (local_claset_of ctxt));
wenzelm@9869
   504
paulson@14890
   505
val skolemize_meth =
paulson@14890
   506
  Method.SIMPLE_METHOD' HEADGOAL
paulson@14890
   507
    (CHANGED_PROP o skolemize_tac);
paulson@14890
   508
paulson@15008
   509
val make_clauses_meth =
paulson@15008
   510
  Method.SIMPLE_METHOD' HEADGOAL
paulson@15008
   511
    (CHANGED_PROP o make_clauses_tac);
paulson@15008
   512
wenzelm@9869
   513
wenzelm@9869
   514
val meson_setup =
wenzelm@9869
   515
 [Method.add_methods
paulson@14813
   516
  [("meson", Method.ctxt_args meson_meth, 
paulson@14813
   517
    "The MESON resolution proof procedure"),
paulson@14890
   518
   ("skolemize", Method.no_args skolemize_meth, 
paulson@15008
   519
    "Skolemization into existential quantifiers"),
paulson@15008
   520
   ("make_clauses", Method.no_args make_clauses_meth, 
paulson@15118
   521
    "Conversion to !!-quantified meta-level clauses")]];
paulson@9840
   522
paulson@9840
   523
end;
wenzelm@9869
   524
paulson@15579
   525
structure BasicMeson: BASIC_MESON = Meson;
paulson@15579
   526
open BasicMeson;