2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department, University of Bologna, Italy.
6 ||T|| HELM is free software; you can redistribute it and/or
7 ||A|| modify it under the terms of the GNU General Public License
8 \ / version 2 or (at your option) any later version.
9 \ / This software is distributed as is, NO WARRANTY.
10 V_______________________________________________________________ *)
12 (* $Id: index.mli 9822 2009-06-03 15:37:06Z tassi $ *)
14 module Superposition (B : Terms.Blob) =
16 module IDX = Index.Index(B)
17 module Unif = FoUnif.Founif(B)
18 module Subst = FoSubst (*.Subst(B)*)
19 module Order = Orderings.Orderings(B)
20 module Utils = FoUtils.Utils(B)
23 exception Success of B.t Terms.bag * int * B.t Terms.unit_clause
25 (* let debug s = prerr_endline s;;*)
28 let rec list_first f = function
30 | x::tl -> match f x with Some _ as x -> x | _ -> list_first f tl
33 let first_position pos ctx t f =
34 let rec aux pos ctx = function
35 | Terms.Leaf _ as t -> f t pos ctx
38 match f t pos ctx with
41 let rec first pre post = function
44 let newctx = fun x -> ctx (Terms.Node (pre@[x]@post)) in
45 match aux (List.length pre :: pos) newctx t with
48 if post = [] then None (* tl is also empty *)
49 else first (pre @ [t]) (List.tl post) tl
51 first [] (List.tl l) l
56 let all_positions pos ctx t f =
57 let rec aux pos ctx = function
58 | Terms.Leaf _ as t -> f t pos ctx
63 (fun (acc,pre,post) t -> (* Invariant: pre @ [t] @ post = l *)
64 let newctx = fun x -> ctx (Terms.Node (pre@[x]@post)) in
65 let acc = aux (List.length pre :: pos) newctx t @ acc in
66 if post = [] then acc, l, []
67 else acc, pre @ [t], List.tl post)
68 (f t pos ctx, [], List.tl l) l
76 let rec aux acc = function
78 | Terms.Var i -> if (List.mem i acc) then acc else i::acc
79 | Terms.Node l -> List.fold_left aux acc l
83 let build_clause bag filter rule t subst vl id id2 pos dir =
84 let proof = Terms.Step(rule,id,id2,dir,pos,subst) in
85 let t = Subst.apply_subst subst t in
89 | Terms.Node [ Terms.Leaf eq ; ty; l; r ] when B.eq B.eqP eq ->
90 let o = Order.compare_terms l r in
91 Terms.Equation (l, r, ty, o)
92 | t -> Terms.Predicate t
95 Utils.add_to_bag bag (0, literal, vars_of_term t, proof)
99 ((*prerr_endline ("Filtering: " ^ Pp.pp_foterm t);*)None)
103 (* ============ simplification ================= *)
105 let demod table varlist subterm pos context =
106 let cands = IDX.DT.retrieve_generalizations table subterm in
108 (fun (dir, (id,lit,vl,_)) ->
110 | Terms.Predicate _ -> assert false
111 | Terms.Equation (l,r,_,o) ->
112 let side, newside = if dir=Terms.Left2Right then l,r else r,l in
115 Unif.unification (varlist@vl) varlist subterm side
117 if o = Terms.Incomparable then
118 let side = Subst.apply_subst subst side in
119 let newside = Subst.apply_subst subst newside in
120 let o = Order.compare_terms newside side in
121 (* Riazanov, pp. 45 (ii) *)
123 Some (context newside, subst, varlist, id, pos, dir)
125 ((*prerr_endline ("Filtering: " ^
126 Pp.pp_foterm side ^ " =(< || =)" ^
127 Pp.pp_foterm newside ^ " coming from " ^
128 Pp.pp_unit_clause uc );*)None)
130 Some (context newside, subst, varlist, id, pos, dir)
131 with FoUnif.UnificationFailure _ -> None)
132 (IDX.ClauseSet.elements cands)
135 let demodulate_once ~jump_to_right bag (id, literal, vl, pr) table =
136 (* debug ("Demodulating : " ^ (Pp.pp_unit_clause (id, literal, vl, pr)));*)
138 | Terms.Predicate t -> assert false
139 | Terms.Equation (l,r,ty,_) ->
140 let left_position = if jump_to_right then None else
142 (fun x -> Terms.Node [ Terms.Leaf B.eqP; ty; x; r ]) l
145 match left_position with
146 | Some (newt, subst, varlist, id2, pos, dir) ->
148 match build_clause bag (fun _ -> true) Terms.Demodulation
149 newt subst varlist id id2 pos dir
151 | None -> assert false
152 | Some x -> Some (x,false)
156 [3] (fun x -> Terms.Node [ Terms.Leaf B.eqP; ty; l; x ]) r
160 | Some (newt, subst, varlist, id2, pos, dir) ->
161 match build_clause bag (fun _ -> true)
162 Terms.Demodulation newt subst varlist id id2 pos dir
164 | None -> assert false
165 | Some x -> Some (x,true)
168 let rec demodulate ~jump_to_right bag clause table =
169 match demodulate_once ~jump_to_right bag clause table with
170 | None -> bag, clause
171 | Some ((bag, clause),r) -> demodulate ~jump_to_right:r
175 let demodulate bag clause table = demodulate ~jump_to_right:false
180 let is_identity_clause ~unify = function
181 | _, Terms.Equation (_,_,_,Terms.Eq), _, _ -> true
182 | _, Terms.Equation (l,r,_,_), vl, proof when unify ->
183 (try ignore(Unif.unification vl [] l r); true
184 with FoUnif.UnificationFailure _ -> false)
185 | _, Terms.Equation (_,_,_,_), _, _ -> false
186 | _, Terms.Predicate _, _, _ -> assert false
189 let build_new_clause bag maxvar filter rule t subst vl id id2 pos dir =
190 let maxvar, vl, relocsubst = Utils.relocate maxvar vl in
191 let subst = Subst.concat relocsubst subst in
192 match build_clause bag filter rule t subst vl id id2 pos dir with
193 | Some (bag, c) -> Some ((bag, maxvar), c)
197 let fold_build_new_clause bag maxvar id rule filter res =
198 let (bag, maxvar), res =
199 HExtlib.filter_map_acc
200 (fun (bag, maxvar) (t,subst,vl,id2,pos,dir) ->
201 build_new_clause bag maxvar filter rule t subst vl id id2 pos dir)
208 let rewrite_eq ~unify l r ty vl table =
209 let retrieve = if unify then IDX.DT.retrieve_unifiables
210 else IDX.DT.retrieve_generalizations in
211 let lcands = retrieve table l in
212 let rcands = retrieve table r in
214 let id, dir, l, r, vl =
216 | (d, (id,Terms.Equation (l,r,ty,_),vl,_))-> id, d, l, r, vl
219 let reverse = (dir = Terms.Left2Right) = b in
220 let l, r, proof_rewrite_dir = if reverse then l,r,Terms.Left2Right
221 else r,l, Terms.Right2Left in
222 (id,proof_rewrite_dir,Terms.Node [ Terms.Leaf B.eqP; ty; l; r ], vl)
224 let cands1 = List.map (f true) (IDX.ClauseSet.elements lcands) in
225 let cands2 = List.map (f false) (IDX.ClauseSet.elements rcands) in
226 let t = Terms.Node [ Terms.Leaf B.eqP; ty; l; r ] in
227 let locked_vars = if unify then [] else vl in
228 let rec aux = function
230 | (id2,dir,c,vl1)::tl ->
232 let subst,vl1 = Unif.unification (vl@vl1) locked_vars c t in
233 Some (id2, dir, subst)
234 with FoUnif.UnificationFailure _ -> aux tl
236 aux (cands1 @ cands2)
239 let is_subsumed ~unify bag maxvar (id, lit, vl, _) table =
241 | Terms.Predicate _ -> assert false
242 | Terms.Equation (l,r,ty,_) ->
243 match rewrite_eq ~unify l r ty vl table with
245 | Some (id2, dir, subst) ->
246 let id_t = Terms.Node [ Terms.Leaf B.eqP; ty; r; r ] in
247 build_new_clause bag maxvar (fun _ -> true)
248 Terms.Superposition id_t subst [] id id2 [2] dir
250 (* id refers to a clause proving contextl l = contextr r *)
252 let rec deep_eq ~unify l r ty pos contextl contextr table acc =
255 | Some(bag,maxvar,(id,lit,vl,p),subst) ->
256 let l = Subst.apply_subst subst l in
257 let r = Subst.apply_subst subst r in
259 let subst1,vl1 = Unif.unification vl [] l r in
261 match lit with Terms.Predicate _ -> assert false
262 | Terms.Equation (l,r,ty,o) ->
263 Terms.Equation (FoSubst.apply_subst subst1 l,
264 FoSubst.apply_subst subst1 r, ty, o)
266 Some(bag,maxvar,(id,lit,vl1,p),Subst.concat subst1 subst)
267 with FoUnif.UnificationFailure _ ->
268 match rewrite_eq ~unify l r ty vl table with
269 | Some (id2, dir, subst1) ->
270 let newsubst = Subst.concat subst1 subst in
272 FoSubst.apply_subst newsubst
273 (Terms.Node[Terms.Leaf B.eqP;ty;contextl r;contextr r])
276 build_new_clause bag maxvar (fun _ -> true)
277 Terms.Superposition id_t
278 subst1 [] id id2 (pos@[2]) dir
280 | Some ((bag, maxvar), c) ->
281 Some(bag,maxvar,c,newsubst)
282 | None -> assert false)
285 | Terms.Node (a::la), Terms.Node (b::lb) when
286 a = b && List.length la = List.length lb ->
289 (fun (acc,pre,postl,postr) a b ->
291 fun x -> contextl(Terms.Node (pre@(x::postl))) in
293 fun x -> contextr(Terms.Node (pre@(x::postr))) in
294 let newpos = List.length pre::pos in
296 if l = [] then [] else List.tl l in
297 (deep_eq ~unify a b ty
298 newpos newcl newcr table acc,pre@[b],
299 footail postl, footail postr))
300 (acc,[a],List.tl la,List.tl lb) la lb
305 let rec orphan_murder bag acc i =
306 match Terms.M.find i bag with
307 | (_,_,_,Terms.Exact _),discarded -> (discarded,acc)
308 | (_,_,_,Terms.Step (_,i1,i2,_,_,_)),true -> (true,acc)
309 | (_,_,_,Terms.Step (_,i1,i2,_,_,_)),false ->
310 if (List.mem i acc) then (false,acc)
311 else match orphan_murder bag acc i1 with
312 | (true,acc) -> (true,acc)
314 let (res,acc) = orphan_murder bag acc i2 in
315 if res then res,acc else res,i::acc
318 let orphan_murder bag cl =
319 let (id,_,_,_) = cl in
320 let (res,_) = orphan_murder bag [] id in
321 if res then debug "Orphan murdered"; res
324 (* demodulate and check for subsumption *)
325 let simplify table maxvar bag clause =
326 if is_identity_clause ~unify:false clause then bag,None
327 (* else if orphan_murder bag clause then bag,None *)
328 else let bag, clause = demodulate bag clause table in
329 if is_identity_clause ~unify:false clause then bag,None
331 match is_subsumed ~unify:false bag maxvar clause table with
332 | None -> bag, Some clause
333 | Some _ -> bag, None
336 let simplify table maxvar bag clause =
337 match simplify table maxvar bag clause with
338 | bag, None -> let (id,_,_,_) = clause in
339 Terms.M.add id (clause,true) bag, None
340 | bag, Some clause -> bag, Some clause
343 let one_pass_simplification new_clause (alist,atable) bag maxvar =
344 match simplify atable maxvar bag new_clause with
345 | bag,None -> bag,None (* new_clause has been discarded *)
346 | bag,(Some clause) ->
347 let ctable = IDX.index_unit_clause IDX.DT.empty clause in
348 let bag, alist, atable =
350 (fun (bag, alist, atable) c ->
351 match simplify ctable maxvar bag c with
352 |bag,None -> (bag,alist,atable)
353 (* an active clause as been discarded *)
355 bag, c :: alist, IDX.index_unit_clause atable c)
356 (bag,[],IDX.DT.empty) alist
358 bag, Some (clause, (alist,atable))
361 let simplification_step ~new_cl cl (alist,atable) bag maxvar new_clause =
363 if new_cl then atable else
364 IDX.index_unit_clause atable cl
366 (* Simplification of new_clause with : *
367 * - actives and cl if new_clause is not cl *
368 * - only actives otherwise *)
369 match simplify atable1 maxvar bag new_clause with
370 | bag,None -> bag,(Some cl, None) (* new_clause has been discarded *)
372 (* Simplification of each active clause with clause *
373 * which is the simplified form of new_clause *)
374 let ctable = IDX.index_unit_clause IDX.DT.empty clause in
375 let bag, newa, alist, atable =
377 (fun (bag, newa, alist, atable) c ->
378 match simplify ctable maxvar bag c with
379 |bag,None -> (bag, newa, alist, atable)
380 (* an active clause as been discarded *)
383 bag, newa, c :: alist,
384 IDX.index_unit_clause atable c
386 bag, c1 :: newa, alist, atable)
387 (bag,[],[],IDX.DT.empty) alist
390 bag, (Some cl, Some (clause, (alist,atable), newa))
392 (* if new_clause is not cl, we simplify cl with clause *)
393 match simplify ctable maxvar bag cl with
395 (* cl has been discarded *)
396 bag,(None, Some (clause, (alist,atable), newa))
398 bag,(Some cl1, Some (clause, (alist,atable), newa))
401 let keep_simplified cl (alist,atable) bag maxvar =
402 let rec keep_simplified_aux ~new_cl cl (alist,atable) bag newc =
404 match simplification_step ~new_cl cl (alist,atable) bag maxvar cl with
405 | _,(None, _) -> assert false
406 | bag,(Some _, None) -> bag,None
407 | bag,(Some _, Some (clause, (alist,atable), newa)) ->
408 keep_simplified_aux ~new_cl:(cl!=clause) clause (alist,atable)
412 | [] -> bag, Some (cl, (alist,atable))
414 match simplification_step ~new_cl cl
415 (alist,atable) bag maxvar hd with
416 | _,(None,None) -> assert false
417 | bag,(Some _,None) ->
418 keep_simplified_aux ~new_cl cl (alist,atable) bag tl
419 | bag,(None, Some _) -> bag,None
420 | bag,(Some cl1, Some (clause, (alist,atable), newa)) ->
422 (clause::alist, IDX.index_unit_clause atable clause)
424 keep_simplified_aux ~new_cl:(cl!=cl1) cl1 (alist,atable)
427 keep_simplified_aux ~new_cl:true cl (alist,atable) bag []
430 let are_alpha_eq cl1 cl2 =
431 let get_term (_,lit,_,_) =
433 | Terms.Predicate _ -> assert false
434 | Terms.Equation (l,r,ty,_) ->
435 Terms.Node [Terms.Leaf B.eqP; ty; l ; r]
437 try ignore(Unif.alpha_eq (get_term cl1) (get_term cl2)) ; true
438 with FoUnif.UnificationFailure _ -> false
441 (* this is like simplify but raises Success *)
442 let simplify_goal ~no_demod maxvar table bag g_actives clause =
444 if no_demod then bag, clause else demodulate bag clause table
446 if List.exists (are_alpha_eq clause) g_actives then None else
447 if (is_identity_clause ~unify:true clause)
448 then raise (Success (bag, maxvar, clause))
450 let (id,lit,vl,_) = clause in
453 | Terms.Equation(l,r,ty,_) -> l,r,ty
456 match deep_eq ~unify:true l r ty [] (fun x -> x) (fun x -> x)
457 table (Some(bag,maxvar,clause,Subst.id_subst)) with
458 | None -> Some (bag,clause)
459 | Some (bag,maxvar,cl,subst) ->
460 prerr_endline "Goal subsumed";
461 raise (Success (bag,maxvar,cl))
463 else match is_subsumed ~unify:true bag maxvar clause table with
464 | None -> Some (bag, clause)
465 | Some ((bag,maxvar),c) ->
466 prerr_endline "Goal subsumed";
467 raise (Success (bag,maxvar,c))
471 (* =================== inference ===================== *)
473 (* this is OK for both the sup_left and sup_right inference steps *)
474 let superposition table varlist subterm pos context =
475 let cands = IDX.DT.retrieve_unifiables table subterm in
477 (fun (dir, (id,lit,vl,_ (*as uc*))) ->
479 | Terms.Predicate _ -> assert false
480 | Terms.Equation (l,r,_,o) ->
481 let side, newside = if dir=Terms.Left2Right then l,r else r,l in
484 Unif.unification (varlist@vl) [] subterm side
486 if o = Terms.Incomparable then
487 let side = Subst.apply_subst subst side in
488 let newside = Subst.apply_subst subst newside in
489 let o = Order.compare_terms side newside in
490 (* XXX: check Riazanov p. 33 (iii) *)
491 if o <> Terms.Lt && o <> Terms.Eq then
492 Some (context newside, subst, varlist, id, pos, dir)
494 ((*prerr_endline ("Filtering: " ^
495 Pp.pp_foterm side ^ " =(< || =)" ^
496 Pp.pp_foterm newside ^ " coming from " ^
497 Pp.pp_unit_clause uc );*)None)
499 Some (context newside, subst, varlist, id, pos, dir)
500 with FoUnif.UnificationFailure _ -> None)
501 (IDX.ClauseSet.elements cands)
504 (* Superposes selected equation with equalities in table *)
505 let superposition_with_table bag maxvar (id,selected,vl,_) table =
507 | Terms.Predicate _ -> assert false
508 | Terms.Equation (l,r,ty,Terms.Lt) ->
509 fold_build_new_clause bag maxvar id Terms.Superposition
512 (fun x -> Terms.Node [ Terms.Leaf B.eqP; ty; l; x ])
513 r (superposition table vl))
514 | Terms.Equation (l,r,ty,Terms.Gt) ->
515 fold_build_new_clause bag maxvar id Terms.Superposition
518 (fun x -> Terms.Node [ Terms.Leaf B.eqP; ty; x; r ])
519 l (superposition table vl))
520 | Terms.Equation (l,r,ty,Terms.Incomparable) ->
521 fold_build_new_clause bag maxvar id Terms.Superposition
522 (function (* Riazanov: p.33 condition (iv) *)
523 | Terms.Node [Terms.Leaf eq; ty; l; r ] when B.eq B.eqP eq ->
524 Order.compare_terms l r <> Terms.Eq
527 (fun x -> Terms.Node [ Terms.Leaf B.eqP; ty; l; x ])
528 r (superposition table vl)) @
530 (fun x -> Terms.Node [ Terms.Leaf B.eqP; ty; x; r ])
531 l (superposition table vl)))
535 (* the current equation is normal w.r.t. demodulation with atable
536 * (and is not the identity) *)
537 let infer_right bag maxvar current (alist,atable) =
538 (* We demodulate actives clause with current until all *
539 * active clauses are reduced w.r.t each other *)
540 (* let bag, (alist,atable) = keep_simplified (alist,atable) bag [current] in *)
541 let ctable = IDX.index_unit_clause IDX.DT.empty current in
542 (* let bag, (alist, atable) =
544 HExtlib.filter_map_acc (simplify ctable) bag alist
546 bag, (alist, List.fold_left IDX.index_unit_clause IDX.DT.empty alist)
548 debug "Simplified active clauses with fact";
549 (* We superpose active clauses with current *)
550 let bag, maxvar, new_clauses =
552 (fun (bag, maxvar, acc) active ->
553 let bag, maxvar, newc =
554 superposition_with_table bag maxvar active ctable
556 bag, maxvar, newc @ acc)
557 (bag, maxvar, []) alist
559 debug "First superpositions";
560 (* We add current to active clauses so that it can be *
561 * superposed with itself *)
563 current :: alist, IDX.index_unit_clause atable current
566 let fresh_current, maxvar = Utils.fresh_unit_clause maxvar current in
567 (* We need to put fresh_current into the bag so that all *
568 * variables clauses refer to are known. *)
569 let bag, fresh_current = Utils.add_to_bag bag fresh_current in
570 (* We superpose current with active clauses *)
571 let bag, maxvar, additional_new_clauses =
572 superposition_with_table bag maxvar fresh_current atable
574 debug "Another superposition";
575 let new_clauses = new_clauses @ additional_new_clauses in
576 debug (Printf.sprintf "Demodulating %d clauses"
577 (List.length new_clauses));
578 let bag, new_clauses =
579 HExtlib.filter_map_monad (simplify atable maxvar) bag new_clauses
581 debug "Demodulated new clauses";
582 bag, maxvar, (alist, atable), new_clauses
585 let infer_left bag maxvar goal (_alist, atable) =
586 (* We superpose the goal with active clauses *)
587 let bag, maxvar, new_goals =
588 superposition_with_table bag maxvar goal atable
590 debug "Superposed goal with active clauses";
591 (* We simplify the new goals with active clauses *)
595 match simplify_goal ~no_demod:false maxvar atable bag [] g with
596 | None -> assert false
597 | Some (bag,g) -> bag,g::acc)
600 debug "Simplified new goals with active clauses";
601 bag, maxvar, List.rev new_goals