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
26 () (* prerr_endline s *)
29 let rec list_first f = function
31 | x::tl -> match f x with Some _ as x -> x | _ -> list_first f tl
34 let first_position pos ctx t f =
35 let rec aux pos ctx = function
36 | Terms.Leaf _ as t -> f t pos ctx
39 match f t pos ctx with
42 let rec first pre post = function
45 let newctx = fun x -> ctx (Terms.Node (pre@[x]@post)) in
46 match aux (List.length pre :: pos) newctx t with
49 if post = [] then None (* tl is also empty *)
50 else first (pre @ [t]) (List.tl post) tl
52 first [] (List.tl l) l
57 let all_positions pos ctx t f =
58 let rec aux pos ctx = function
59 | Terms.Leaf _ as t -> f t pos ctx
64 (fun (acc,pre,post) t -> (* Invariant: pre @ [t] @ post = l *)
65 let newctx = fun x -> ctx (Terms.Node (pre@[x]@post)) in
66 let acc = aux (List.length pre :: pos) newctx t @ acc in
67 if post = [] then acc, l, []
68 else acc, pre @ [t], List.tl post)
69 (f t pos ctx, [], List.tl l) l
77 let rec aux acc = function
79 | Terms.Var i -> if (List.mem i acc) then acc else i::acc
80 | Terms.Node l -> List.fold_left aux acc l
84 let build_clause bag filter rule t subst vl id id2 pos dir =
85 let proof = Terms.Step(rule,id,id2,dir,pos,subst) in
86 let t = Subst.apply_subst subst t in
90 | Terms.Node [ Terms.Leaf eq ; ty; l; r ] when B.eq B.eqP eq ->
91 let o = Order.compare_terms l r in
92 Terms.Equation (l, r, ty, o)
93 | t -> Terms.Predicate t
96 Utils.add_to_bag bag (0, literal, vars_of_term t, proof)
100 ((*prerr_endline ("Filtering: " ^ Pp.pp_foterm t);*)None)
104 (* ============ simplification ================= *)
106 let demod table varlist subterm pos context =
107 let cands = IDX.DT.retrieve_generalizations table subterm in
109 (fun (dir, (id,lit,vl,_)) ->
111 | Terms.Predicate _ -> assert false
112 | Terms.Equation (l,r,_,o) ->
113 let side, newside = if dir=Terms.Left2Right then l,r else r,l in
116 Unif.unification (varlist@vl) varlist subterm side
118 if o = Terms.Incomparable then
119 let side = Subst.apply_subst subst side in
120 let newside = Subst.apply_subst subst newside in
121 let o = Order.compare_terms newside side in
122 (* Riazanov, pp. 45 (ii) *)
124 Some (context newside, subst, varlist, id, pos, dir)
126 ((*prerr_endline ("Filtering: " ^
127 Pp.pp_foterm side ^ " =(< || =)" ^
128 Pp.pp_foterm newside ^ " coming from " ^
129 Pp.pp_unit_clause uc );*)None)
131 Some (context newside, subst, varlist, id, pos, dir)
132 with FoUnif.UnificationFailure _ -> None)
133 (IDX.ClauseSet.elements cands)
136 let demodulate_once ~jump_to_right bag (id, literal, vl, pr) table =
137 (* debug ("Demodulating : " ^ (Pp.pp_unit_clause (id, literal, vl, pr)));*)
139 | Terms.Predicate t -> assert false
140 | Terms.Equation (l,r,ty,_) ->
141 let left_position = if jump_to_right then None else
143 (fun x -> Terms.Node [ Terms.Leaf B.eqP; ty; x; r ]) l
146 match left_position with
147 | Some (newt, subst, varlist, id2, pos, dir) ->
149 match build_clause bag (fun _ -> true) Terms.Demodulation
150 newt subst varlist id id2 pos dir
152 | None -> assert false
153 | Some x -> Some (x,false)
157 [3] (fun x -> Terms.Node [ Terms.Leaf B.eqP; ty; l; x ]) r
161 | Some (newt, subst, varlist, id2, pos, dir) ->
162 match build_clause bag (fun _ -> true)
163 Terms.Demodulation newt subst varlist id id2 pos dir
165 | None -> assert false
166 | Some x -> Some (x,true)
169 let rec demodulate ~jump_to_right bag clause table =
170 match demodulate_once ~jump_to_right bag clause table with
171 | None -> bag, clause
172 | Some ((bag, clause),r) -> demodulate ~jump_to_right:r
176 let demodulate bag clause table = demodulate ~jump_to_right:false
181 let is_identity_clause ~unify = function
182 | _, Terms.Equation (_,_,_,Terms.Eq), _, _ -> true
183 | _, Terms.Equation (l,r,_,_), vl, proof when unify ->
184 (try ignore(Unif.unification vl [] l r); true
185 with FoUnif.UnificationFailure _ -> false)
186 | _, Terms.Equation (_,_,_,_), _, _ -> false
187 | _, Terms.Predicate _, _, _ -> assert false
190 let build_new_clause bag maxvar filter rule t subst vl id id2 pos dir =
191 let maxvar, vl, relocsubst = Utils.relocate maxvar vl in
192 let subst = Subst.concat relocsubst subst in
193 match build_clause bag filter rule t subst vl id id2 pos dir with
194 | Some (bag, c) -> Some ((bag, maxvar), c)
198 let fold_build_new_clause bag maxvar id rule filter res =
199 let (bag, maxvar), res =
200 HExtlib.filter_map_acc
201 (fun (bag, maxvar) (t,subst,vl,id2,pos,dir) ->
202 build_new_clause bag maxvar filter rule t subst vl id id2 pos dir)
209 let rewrite_eq ~unify l r ty vl table =
210 let retrieve = if unify then IDX.DT.retrieve_unifiables
211 else IDX.DT.retrieve_generalizations in
212 let lcands = retrieve table l in
213 let rcands = retrieve table r in
215 let id, dir, l, r, vl =
217 | (d, (id,Terms.Equation (l,r,ty,_),vl,_))-> id, d, l, r, vl
220 let reverse = (dir = Terms.Left2Right) = b in
221 let l, r, proof_rewrite_dir = if reverse then l,r,Terms.Left2Right
222 else r,l, Terms.Right2Left in
223 (id,proof_rewrite_dir,Terms.Node [ Terms.Leaf B.eqP; ty; l; r ], vl)
225 let cands1 = List.map (f true) (IDX.ClauseSet.elements lcands) in
226 let cands2 = List.map (f false) (IDX.ClauseSet.elements rcands) in
227 let t = Terms.Node [ Terms.Leaf B.eqP; ty; l; r ] in
228 let locked_vars = if unify then [] else vl in
229 let rec aux = function
231 | (id2,dir,c,vl1)::tl ->
233 let subst,vl1 = Unif.unification (vl@vl1) locked_vars c t in
234 Some (id2, dir, subst)
235 with FoUnif.UnificationFailure _ -> aux tl
237 aux (cands1 @ cands2)
240 let is_subsumed ~unify bag maxvar (id, lit, vl, _) table =
242 | Terms.Predicate _ -> assert false
243 | Terms.Equation (l,r,ty,_) ->
244 match rewrite_eq ~unify l r ty vl table with
246 | Some (id2, dir, subst) ->
247 let id_t = Terms.Node [ Terms.Leaf B.eqP; ty; r; r ] in
248 build_new_clause bag maxvar (fun _ -> true)
249 Terms.Superposition id_t subst [] id id2 [2] dir
251 (* id refers to a clause proving contextl l = contextr r *)
253 let rec deep_eq ~unify l r ty pos contextl contextr table acc =
256 | Some(bag,maxvar,[],subst) -> assert false
257 | Some(bag,maxvar,(id,_,vl,_)::clauses,subst) ->
258 let l = Subst.apply_subst subst l in
259 let r = Subst.apply_subst subst r in
261 let subst1,vl1 = Unif.unification vl [] l r in
262 Some(bag,maxvar,clauses,Subst.concat subst1 subst)
263 with FoUnif.UnificationFailure _ ->
264 match rewrite_eq ~unify l r ty vl table with
265 | Some (id2, dir, subst1) ->
267 Terms.Node[Terms.Leaf B.eqP;ty;contextl r;contextr r] in
269 build_new_clause bag maxvar (fun _ -> true)
270 Terms.Superposition id_t subst1 [] id id2 (2::pos) dir
272 | Some ((bag, maxvar), c) ->
273 Some(bag,maxvar,c::clauses,Subst.concat subst1 subst)
274 | None -> assert false)
277 | Terms.Node (a::la), Terms.Node (b::lb) when
278 a = b && List.length la = List.length lb ->
281 (fun (acc,pre,postl,postr) a b ->
283 fun x -> contextl(Terms.Node (pre@(x::postl))) in
285 fun x -> contextr(Terms.Node (pre@(x::postr))) in
286 let newpos = List.length pre::pos in
288 if l = [] then [] else List.tl l in
289 (deep_eq ~unify a b ty
290 newpos newcl newcr table acc,pre@[b],
291 footail postl, footail postr))
292 (acc,[a],List.tl la,List.tl lb) la lb
295 | _, Terms.Var _ -> assert false
299 (* demodulate and check for subsumption *)
300 let simplify table maxvar bag clause =
301 let bag, clause = demodulate bag clause table in
302 if is_identity_clause ~unify:false clause then bag,None
304 match is_subsumed ~unify:false bag maxvar clause table with
305 | None -> bag, Some clause
306 | Some _ -> bag, None
309 let one_pass_simplification new_clause (alist,atable) bag maxvar =
310 match simplify atable maxvar bag new_clause with
311 | bag,None -> None (* new_clause has been discarded *)
312 | bag,(Some clause) ->
313 let ctable = IDX.index_unit_clause IDX.DT.empty clause in
314 let bag, alist, atable =
316 (fun (bag, alist, atable as acc) c ->
317 match simplify ctable maxvar bag c with
318 |bag,None -> acc (* an active clause as been discarded *)
320 bag, c :: alist, IDX.index_unit_clause atable c)
321 (bag,[],IDX.DT.empty) alist
323 Some (clause, bag, (alist,atable))
326 let simplification_step ~new_cl cl (alist,atable) bag maxvar new_clause =
328 if new_cl then atable else
329 IDX.index_unit_clause atable cl
331 (* Simplification of new_clause with : *
332 * - actives and cl if new_clause is not cl *
333 * - only actives otherwise *)
334 match simplify atable1 maxvar bag new_clause with
335 | bag,None -> (Some cl, None) (* new_clause has been discarded *)
337 (* Simplification of each active clause with clause *
338 * which is the simplified form of new_clause *)
339 let ctable = IDX.index_unit_clause IDX.DT.empty clause in
340 let bag, newa, alist, atable =
342 (fun (bag, newa, alist, atable as acc) c ->
343 match simplify ctable maxvar bag c with
344 |bag,None -> acc (* an active clause as been discarded *)
347 bag, newa, c :: alist,
348 IDX.index_unit_clause atable c
350 bag, c1 :: newa, alist, atable)
351 (bag,[],[],IDX.DT.empty) alist
354 (Some cl, Some (clause, (alist,atable), newa, bag))
356 (* if new_clause is not cl, we simplify cl with clause *)
357 match simplify ctable maxvar bag cl with
359 (* cl has been discarded *)
360 (None, Some (clause, (alist,atable), newa, bag))
362 (Some cl1, Some (clause, (alist,atable), newa, bag))
365 let keep_simplified cl (alist,atable) bag maxvar =
366 let rec keep_simplified_aux ~new_cl cl (alist,atable) bag newc =
368 match simplification_step ~new_cl cl (alist,atable) bag maxvar cl with
369 | (None, _) -> assert false
370 | (Some _, None) -> None
371 | (Some _, Some (clause, (alist,atable), newa, bag)) ->
372 keep_simplified_aux ~new_cl:(cl!=clause) clause (alist,atable)
376 | [] -> Some (cl, bag, (alist,atable))
378 match simplification_step ~new_cl cl
379 (alist,atable) bag maxvar hd with
380 | (None,None) -> assert false
382 keep_simplified_aux ~new_cl cl (alist,atable) bag tl
383 | (None, Some _) -> None
384 | (Some cl1, Some (clause, (alist,atable), newa, bag)) ->
386 (clause::alist, IDX.index_unit_clause atable clause)
388 keep_simplified_aux ~new_cl:(cl!=cl1) cl1 (alist,atable)
391 keep_simplified_aux ~new_cl:true cl (alist,atable) bag []
394 let are_alpha_eq cl1 cl2 =
395 let get_term (_,lit,_,_) =
397 | Terms.Predicate _ -> assert false
398 | Terms.Equation (l,r,ty,_) ->
399 Terms.Node [Terms.Leaf B.eqP; ty; l ; r]
401 try ignore(Unif.alpha_eq (get_term cl1) (get_term cl2)) ; true
402 with FoUnif.UnificationFailure _ -> false
405 (* this is like simplify but raises Success *)
406 let simplify_goal maxvar table bag g_actives clause =
407 let bag, clause = demodulate bag clause table in
408 if (is_identity_clause ~unify:true clause)
409 then raise (Success (bag, maxvar, clause))
412 let (id,lit,vl,_) = clause in
415 | Terms.Equation(l,r,ty,_) -> l,r,ty
418 match deep_eq ~unify:true l r ty [] (fun x -> x) (fun x -> x)
419 table (Some(bag,maxvar,[clause],Subst.id_subst)) with
421 if List.exists (are_alpha_eq clause) g_actives then None
422 else Some (bag, clause)
423 | Some (bag,maxvar,cl,subst) ->
424 debug "Goal subsumed";
425 raise (Success (bag,maxvar,List.hd cl))
427 else match is_subsumed ~unify:true bag maxvar clause table with
429 if List.exists (are_alpha_eq clause) g_actives then None
430 else Some (bag, clause)
431 | Some ((bag,maxvar),c) ->
432 debug "Goal subsumed";
433 raise (Success (bag,maxvar,c))
436 (* =================== inference ===================== *)
438 (* this is OK for both the sup_left and sup_right inference steps *)
439 let superposition table varlist subterm pos context =
440 let cands = IDX.DT.retrieve_unifiables table subterm in
442 (fun (dir, (id,lit,vl,_ (*as uc*))) ->
444 | Terms.Predicate _ -> assert false
445 | Terms.Equation (l,r,_,o) ->
446 let side, newside = if dir=Terms.Left2Right then l,r else r,l in
449 Unif.unification (varlist@vl) [] subterm side
451 if o = Terms.Incomparable then
452 let side = Subst.apply_subst subst side in
453 let newside = Subst.apply_subst subst newside in
454 let o = Order.compare_terms side newside in
455 (* XXX: check Riazanov p. 33 (iii) *)
456 if o <> Terms.Lt && o <> Terms.Eq then
457 Some (context newside, subst, varlist, id, pos, dir)
459 ((*prerr_endline ("Filtering: " ^
460 Pp.pp_foterm side ^ " =(< || =)" ^
461 Pp.pp_foterm newside ^ " coming from " ^
462 Pp.pp_unit_clause uc );*)None)
464 Some (context newside, subst, varlist, id, pos, dir)
465 with FoUnif.UnificationFailure _ -> None)
466 (IDX.ClauseSet.elements cands)
469 (* Superposes selected equation with equalities in table *)
470 let superposition_with_table bag maxvar (id,selected,vl,_) table =
472 | Terms.Predicate _ -> assert false
473 | Terms.Equation (l,r,ty,Terms.Lt) ->
474 fold_build_new_clause bag maxvar id Terms.Superposition
477 (fun x -> Terms.Node [ Terms.Leaf B.eqP; ty; l; x ])
478 r (superposition table vl))
479 | Terms.Equation (l,r,ty,Terms.Gt) ->
480 fold_build_new_clause bag maxvar id Terms.Superposition
483 (fun x -> Terms.Node [ Terms.Leaf B.eqP; ty; x; r ])
484 l (superposition table vl))
485 | Terms.Equation (l,r,ty,Terms.Incomparable) ->
486 fold_build_new_clause bag maxvar id Terms.Superposition
487 (function (* Riazanov: p.33 condition (iv) *)
488 | Terms.Node [Terms.Leaf eq; ty; l; r ] when B.eq B.eqP eq ->
489 Order.compare_terms l r <> Terms.Eq
492 (fun x -> Terms.Node [ Terms.Leaf B.eqP; ty; l; x ])
493 r (superposition table vl)) @
495 (fun x -> Terms.Node [ Terms.Leaf B.eqP; ty; x; r ])
496 l (superposition table vl)))
500 (* the current equation is normal w.r.t. demodulation with atable
501 * (and is not the identity) *)
502 let infer_right bag maxvar current (alist,atable) =
503 (* We demodulate actives clause with current until all *
504 * active clauses are reduced w.r.t each other *)
505 (* let bag, (alist,atable) = keep_simplified (alist,atable) bag [current] in *)
506 let ctable = IDX.index_unit_clause IDX.DT.empty current in
507 (* let bag, (alist, atable) =
509 HExtlib.filter_map_acc (simplify ctable) bag alist
511 bag, (alist, List.fold_left IDX.index_unit_clause IDX.DT.empty alist)
513 debug "Simplified active clauses with fact";
514 (* We superpose active clauses with current *)
515 let bag, maxvar, new_clauses =
517 (fun (bag, maxvar, acc) active ->
518 let bag, maxvar, newc =
519 superposition_with_table bag maxvar active ctable
521 bag, maxvar, newc @ acc)
522 (bag, maxvar, []) alist
524 debug "First superpositions";
525 (* We add current to active clauses so that it can be *
526 * superposed with itself *)
528 current :: alist, IDX.index_unit_clause atable current
531 let fresh_current, maxvar = Utils.fresh_unit_clause maxvar current in
532 (* We need to put fresh_current into the bag so that all *
533 * variables clauses refer to are known. *)
534 let bag, fresh_current = Utils.add_to_bag bag fresh_current in
535 (* We superpose current with active clauses *)
536 let bag, maxvar, additional_new_clauses =
537 superposition_with_table bag maxvar fresh_current atable
539 debug "Another superposition";
540 let new_clauses = new_clauses @ additional_new_clauses in
541 debug (Printf.sprintf "Demodulating %d clauses"
542 (List.length new_clauses));
543 let bag, new_clauses =
544 HExtlib.filter_map_monad (simplify atable maxvar) bag new_clauses
546 debug "Demodulated new clauses";
547 bag, maxvar, (alist, atable), new_clauses
550 let infer_left bag maxvar goal (_alist, atable) =
551 (* We superpose the goal with active clauses *)
552 let bag, maxvar, new_goals =
553 superposition_with_table bag maxvar goal atable
555 debug "Superposed goal with active clauses";
556 (* We simplify the new goals with active clauses *)
560 match simplify_goal maxvar atable bag [] g with
561 | None -> assert false
562 | Some (bag,g) -> bag,g::acc)
565 debug "Simplified new goals with active clauses";
566 bag, maxvar, List.rev new_goals