1 (* Copyright (C) 2002, HELM Team.
3 * This file is part of HELM, an Hypertextual, Electronic
4 * Library of Mathematics, developed at the Computer Science
5 * Department, University of Bologna, Italy.
7 * HELM is free software; you can redistribute it and/or
8 * modify it under the terms of the GNU General Public License
9 * as published by the Free Software Foundation; either version 2
10 * of the License, or (at your option) any later version.
12 * HELM is distributed in the hope that it will be useful,
13 * but WITHOUT ANY WARRANTY; without even the implied warranty of
14 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
15 * GNU General Public License for more details.
17 * You should have received a copy of the GNU General Public License
18 * along with HELM; if not, write to the Free Software
19 * Foundation, Inc., 59 Temple Place - Suite 330, Boston,
22 * For details, see the HELM World-Wide-Web page,
23 * http://cs.unibo.it/helm/.
27 (******************** THE FOURIER TACTIC ***********************)
29 (* La tactique Fourier ne fonctionne de manière sûre que si les coefficients
30 des inéquations et équations sont entiers. En attendant la tactique Field.
37 let debug x = print_string ("____ "^x) ; flush stdout;;
39 let debug_pcontext x =
41 List.iter (fun y -> match y with Some(Cic.Name(a),_) -> str := !str ^
43 debug ("contesto : "^ (!str) ^ "\n")
46 (******************************************************************************
47 Operations on linear combinations.
49 Opérations sur les combinaisons linéaires affines.
50 La partie homogène d'une combinaison linéaire est en fait une table de hash
51 qui donne le coefficient d'un terme du calcul des constructions,
52 qui est zéro si le terme n'y est pas.
58 The type for linear combinations
60 type flin = {fhom:(Cic.term , rational)Hashtbl.t;fcste:rational}
66 let flin_zero () = {fhom = Hashtbl.create 50;fcste=r0}
72 @return the rational associated with x (coefficient)
76 (Hashtbl.find f.fhom x)
82 Adds c to the coefficient of x
91 let cx = flin_coef f x in
92 Hashtbl.remove f.fhom x;
93 Hashtbl.add f.fhom x (rplus cx c);
95 |_->debug ("Internal error in Fourier! this is not a Rel "^CicPp.ppterm x^"\n");
96 let cx = flin_coef f x in
97 Hashtbl.remove f.fhom x;
98 Hashtbl.add f.fhom x (rplus cx c);
107 let flin_add_cste f c =
109 fcste=rplus f.fcste c}
113 @return a empty flin with r1 in fcste
115 let flin_one () = flin_add_cste (flin_zero()) r1;;
120 let flin_plus f1 f2 =
121 let f3 = flin_zero() in
122 Hashtbl.iter (fun x c -> let _=flin_add f3 x c in ()) f1.fhom;
123 Hashtbl.iter (fun x c -> let _=flin_add f3 x c in ()) f2.fhom;
124 flin_add_cste (flin_add_cste f3 f1.fcste) f2.fcste;
130 let flin_minus f1 f2 =
131 let f3 = flin_zero() in
132 Hashtbl.iter (fun x c -> let _=flin_add f3 x c in ()) f1.fhom;
133 Hashtbl.iter (fun x c -> let _=flin_add f3 x (rop c) in ()) f2.fhom;
134 flin_add_cste (flin_add_cste f3 f1.fcste) (rop f2.fcste);
141 let f2 = flin_zero() in
142 Hashtbl.iter (fun x c -> let _=flin_add f2 x (rmult a c) in ()) f.fhom;
143 flin_add_cste f2 (rmult a f.fcste);
147 (*****************************************************************************)
152 @raise Failure if conversion is impossible
153 @return rational proiection of t
155 let rec rational_of_term t =
156 (* fun to apply f to the first and second rational-term of l *)
157 let rat_of_binop f l =
158 let a = List.hd l and
159 b = List.hd(List.tl l) in
160 f (rational_of_term a) (rational_of_term b)
162 (* as before, but f is unary *)
163 let rat_of_unop f l =
164 f (rational_of_term (List.hd l))
167 | Cic.Cast (t1,t2) -> (rational_of_term t1)
168 | Cic.Appl (t1::next) ->
171 if UriManager.eq u HelmLibraryObjects.Reals.ropp_URI then
173 else if UriManager.eq u HelmLibraryObjects.Reals.rinv_URI then
174 rat_of_unop rinv next
175 else if UriManager.eq u HelmLibraryObjects.Reals.rmult_URI then
176 rat_of_binop rmult next
177 else if UriManager.eq u HelmLibraryObjects.Reals.rdiv_URI then
178 rat_of_binop rdiv next
179 else if UriManager.eq u HelmLibraryObjects.Reals.rplus_URI then
180 rat_of_binop rplus next
181 else if UriManager.eq u HelmLibraryObjects.Reals.rminus_URI then
182 rat_of_binop rminus next
183 else failwith "not a rational"
184 | _ -> failwith "not a rational")
185 | Cic.Const (u,boh) ->
186 if UriManager.eq u HelmLibraryObjects.Reals.r1_URI then r1
187 else if UriManager.eq u HelmLibraryObjects.Reals.r0_URI then r0
188 else failwith "not a rational"
189 | _ -> failwith "not a rational"
193 let rational_of_const = rational_of_term;;
203 let rec flin_of_term t =
204 let fl_of_binop f l =
205 let a = List.hd l and
206 b = List.hd(List.tl l) in
207 f (flin_of_term a) (flin_of_term b)
211 | Cic.Cast (t1,t2) -> (flin_of_term t1)
212 | Cic.Appl (t1::next) ->
217 if UriManager.eq u HelmLibraryObjects.Reals.ropp_URI then
218 flin_emult (rop r1) (flin_of_term (List.hd next))
219 else if UriManager.eq u HelmLibraryObjects.Reals.rplus_URI then
220 fl_of_binop flin_plus next
221 else if UriManager.eq u HelmLibraryObjects.Reals.rminus_URI then
222 fl_of_binop flin_minus next
223 else if UriManager.eq u HelmLibraryObjects.Reals.rmult_URI then
225 let arg1 = (List.hd next) and
226 arg2 = (List.hd(List.tl next))
228 if fails rational_of_term arg1
230 if fails rational_of_term arg2
232 ( (* prodotto tra 2 incognite ????? impossibile*)
233 failwith "Sistemi lineari!!!!\n"
238 Cic.Rel(n) -> (*trasformo al volo*)
239 (flin_add (flin_zero()) arg1 (rational_of_term arg2))
241 let tmp = flin_of_term arg1 in
242 flin_emult (rational_of_term arg2) (tmp)
245 if fails rational_of_term arg2
249 Cic.Rel(n) -> (*trasformo al volo*)
250 (flin_add (flin_zero()) arg2 (rational_of_term arg1))
252 let tmp = flin_of_term arg2 in
253 flin_emult (rational_of_term arg1) (tmp)
257 ( (*prodotto tra razionali*)
258 (flin_add_cste (flin_zero()) (rmult (rational_of_term arg1) (rational_of_term arg2)))
262 (*let a = rational_of_term arg1 in
263 debug("ho fatto rational of term di "^CicPp.ppterm arg1^
264 " e ho ottenuto "^string_of_int a.num^"/"^string_of_int a.den^"\n");*)
265 let a = flin_of_term arg1
268 let b = (rational_of_term arg2) in
269 debug("ho fatto rational of term di "^CicPp.ppterm arg2^
270 " e ho ottenuto "^string_of_int b.num^"/"^string_of_int b.den^"\n");
271 (flin_add_cste (flin_zero()) (rmult a b))
274 _ -> debug ("ho fallito2 su "^CicPp.ppterm arg2^"\n");
275 (flin_add (flin_zero()) arg2 a)
278 _-> debug ("ho fallito1 su "^CicPp.ppterm arg1^"\n");
279 (flin_add(flin_zero()) arg1 (rational_of_term arg2))
282 else if UriManager.eq u HelmLibraryObjects.Reals.rinv_URI then
283 let a=(rational_of_term (List.hd next)) in
284 flin_add_cste (flin_zero()) (rinv a)
285 else if UriManager.eq u HelmLibraryObjects.Reals.rdiv_URI then
287 let b=(rational_of_term (List.hd(List.tl next))) in
290 let a = (rational_of_term (List.hd next)) in
291 (flin_add_cste (flin_zero()) (rdiv a b))
294 _-> (flin_add (flin_zero()) (List.hd next) (rinv b))
300 | Cic.Const (u,boh) ->
302 if UriManager.eq u HelmLibraryObjects.Reals.r1_URI then flin_one ()
303 else if UriManager.eq u HelmLibraryObjects.Reals.r0_URI then flin_zero ()
307 with _ -> debug("eccezione = "^CicPp.ppterm t^"\n");flin_add (flin_zero()) t r1
311 let flin_of_constr = flin_of_term;;
315 Translates a flin to (c,x) list
317 @return something like (c1,x1)::(c2,x2)::...::(cn,xn)
319 let flin_to_alist f =
321 Hashtbl.iter (fun x c -> res:=(c,x)::(!res)) f;
325 (* Représentation des hypothèses qui sont des inéquations ou des équations.
329 The structure for ineq
331 type hineq={hname:Cic.term; (* le nom de l'hypothèse *)
332 htype:string; (* Rlt, Rgt, Rle, Rge, eqTLR ou eqTRL *)
339 (* Transforme une hypothese h:t en inéquation flin<0 ou flin<=0
342 let ineq1_of_term (h,t) =
343 match t with (* match t *)
344 Cic.Appl (t1::next) ->
345 let arg1= List.hd next in
346 let arg2= List.hd(List.tl next) in
347 (match t1 with (* match t1 *)
349 if UriManager.eq u HelmLibraryObjects.Reals.rlt_URI then
354 hflin= flin_minus (flin_of_term arg1)
357 else if UriManager.eq u HelmLibraryObjects.Reals.rgt_URI then
362 hflin= flin_minus (flin_of_term arg2)
365 else if UriManager.eq u HelmLibraryObjects.Reals.rle_URI then
370 hflin= flin_minus (flin_of_term arg1)
373 else if UriManager.eq u HelmLibraryObjects.Reals.rge_URI then
378 hflin= flin_minus (flin_of_term arg2)
382 | Cic.MutInd (u,i,o) ->
383 if UriManager.eq u HelmLibraryObjects.Logic.eq_URI then
386 let arg2= List.hd(List.tl (List.tl next)) in
389 if UriManager.eq u HelmLibraryObjects.Reals.r_URI then
394 hflin= flin_minus (flin_of_term arg1)
401 hflin= flin_minus (flin_of_term arg2)
407 |_-> assert false)(* match t1 *)
408 |_-> assert false (* match t *)
411 let ineq1_of_constr = ineq1_of_term;;
414 (* Applique la méthode de Fourier à une liste d'hypothèses (type hineq)
420 | a::next -> Fourier.print_rational a ; print_string " " ; print_rl next
423 let rec print_sys l =
426 | (a,b)::next -> (print_rl a;
427 print_string (if b=true then "strict\n"else"\n");
432 Hashtbl.iter (fun x y -> print_string ("("^"-"^","^"-"^")")) h
435 let fourier_lineq lineq1 =
437 let hvar=Hashtbl.create 50 in (* la table des variables des inéquations *)
439 Hashtbl.iter (fun x c ->
440 try (Hashtbl.find hvar x;())
441 with _-> nvar:=(!nvar)+1;
442 Hashtbl.add hvar x (!nvar);
443 debug("aggiungo una var "^
444 string_of_int !nvar^" per "^
445 CicPp.ppterm x^"\n"))
449 debug("Il numero di incognite e' "^string_of_int (!nvar+1)^"\n");
450 let sys= List.map (fun h->
451 let v=Array.create ((!nvar)+1) r0 in
452 Hashtbl.iter (fun x c -> v.(Hashtbl.find hvar x) <- c)
454 ((Array.to_list v)@[rop h.hflin.fcste],h.hstrict))
456 debug ("chiamo unsolvable sul sistema di "^
457 string_of_int (List.length sys) ^"\n");
462 (*****************************************************************************
463 Construction de la preuve en cas de succès de la méthode de Fourier,
464 i.e. on obtient une contradiction.
468 let _eqT = Cic.MutInd(HelmLibraryObjects.Logic.eq_URI, 0, []) ;;
469 let _False = Cic.MutInd (HelmLibraryObjects.Logic.false_URI, 0, []) ;;
470 let _not = Cic.Const (HelmLibraryObjects.Logic.not_URI,[]);;
471 let _R0 = Cic.Const (HelmLibraryObjects.Reals.r0_URI,[]);;
472 let _R1 = Cic.Const (HelmLibraryObjects.Reals.r1_URI,[]);;
473 let _R = Cic.Const (HelmLibraryObjects.Reals.r_URI,[]);;
474 let _Rfourier_eqLR_to_le=Cic.Const ((UriManager.uri_of_string
475 "cic:/Coq/fourier/Fourier_util/Rfourier_eqLR_to_le.con"), []) ;;
476 let _Rfourier_eqRL_to_le=Cic.Const ((UriManager.uri_of_string
477 "cic:/Coq/fourier/Fourier_util/Rfourier_eqRL_to_le.con"), []) ;;
478 let _Rfourier_ge_to_le =Cic.Const ((UriManager.uri_of_string
479 "cic:/Coq/fourier/Fourier_util/Rfourier_ge_to_le.con"), []) ;;
480 let _Rfourier_gt_to_lt =Cic.Const ((UriManager.uri_of_string
481 "cic:/Coq/fourier/Fourier_util/Rfourier_gt_to_lt.con"), []) ;;
482 let _Rfourier_le=Cic.Const ((UriManager.uri_of_string
483 "cic:/Coq/fourier/Fourier_util/Rfourier_le.con"), []) ;;
484 let _Rfourier_le_le =Cic.Const ((UriManager.uri_of_string
485 "cic:/Coq/fourier/Fourier_util/Rfourier_le_le.con"), []) ;;
486 let _Rfourier_le_lt =Cic.Const ((UriManager.uri_of_string
487 "cic:/Coq/fourier/Fourier_util/Rfourier_le_lt.con"), []) ;;
488 let _Rfourier_lt=Cic.Const ((UriManager.uri_of_string
489 "cic:/Coq/fourier/Fourier_util/Rfourier_lt.con"), []) ;;
490 let _Rfourier_lt_le =Cic.Const ((UriManager.uri_of_string
491 "cic:/Coq/fourier/Fourier_util/Rfourier_lt_le.con"), []) ;;
492 let _Rfourier_lt_lt =Cic.Const ((UriManager.uri_of_string
493 "cic:/Coq/fourier/Fourier_util/Rfourier_lt_lt.con"), []) ;;
494 let _Rfourier_not_ge_lt = Cic.Const ((UriManager.uri_of_string
495 "cic:/Coq/fourier/Fourier_util/Rfourier_not_ge_lt.con"), []) ;;
496 let _Rfourier_not_gt_le = Cic.Const ((UriManager.uri_of_string
497 "cic:/Coq/fourier/Fourier_util/Rfourier_not_gt_le.con"), []) ;;
498 let _Rfourier_not_le_gt = Cic.Const ((UriManager.uri_of_string
499 "cic:/Coq/fourier/Fourier_util/Rfourier_not_le_gt.con"), []) ;;
500 let _Rfourier_not_lt_ge = Cic.Const ((UriManager.uri_of_string
501 "cic:/Coq/fourier/Fourier_util/Rfourier_not_lt_ge.con"), []) ;;
502 let _Rinv = Cic.Const (HelmLibraryObjects.Reals.rinv_URI, []);;
503 let _Rinv_R1 = Cic.Const(HelmLibraryObjects.Reals.rinv_r1_URI, []);;
504 let _Rle = Cic.Const (HelmLibraryObjects.Reals.rle_URI, []);;
505 let _Rle_mult_inv_pos = Cic.Const ((UriManager.uri_of_string
506 "cic:/Coq/fourier/Fourier_util/Rle_mult_inv_pos.con"), []) ;;
507 let _Rle_not_lt = Cic.Const ((UriManager.uri_of_string
508 "cic:/Coq/fourier/Fourier_util/Rle_not_lt.con"), []) ;;
509 let _Rle_zero_1 = Cic.Const ((UriManager.uri_of_string
510 "cic:/Coq/fourier/Fourier_util/Rle_zero_1.con"), []) ;;
511 let _Rle_zero_pos_plus1 = Cic.Const ((UriManager.uri_of_string
512 "cic:/Coq/fourier/Fourier_util/Rle_zero_pos_plus1.con"), []) ;;
513 let _Rlt = Cic.Const (HelmLibraryObjects.Reals.rlt_URI, []);;
514 let _Rlt_mult_inv_pos = Cic.Const ((UriManager.uri_of_string
515 "cic:/Coq/fourier/Fourier_util/Rlt_mult_inv_pos.con"), []) ;;
516 let _Rlt_not_le = Cic.Const ((UriManager.uri_of_string
517 "cic:/Coq/fourier/Fourier_util/Rlt_not_le.con"), []) ;;
518 let _Rlt_zero_1 = Cic.Const ((UriManager.uri_of_string
519 "cic:/Coq/fourier/Fourier_util/Rlt_zero_1.con"), []) ;;
520 let _Rlt_zero_pos_plus1 = Cic.Const ((UriManager.uri_of_string
521 "cic:/Coq/fourier/Fourier_util/Rlt_zero_pos_plus1.con"), []) ;;
522 let _Rminus = Cic.Const (HelmLibraryObjects.Reals.rminus_URI, []);;
523 let _Rmult = Cic.Const (HelmLibraryObjects.Reals.rmult_URI, []);;
524 let _Rnot_le_le =Cic.Const ((UriManager.uri_of_string
525 "cic:/Coq/fourier/Fourier_util/Rnot_le_le.con"), []) ;;
526 let _Rnot_lt0 = Cic.Const ((UriManager.uri_of_string
527 "cic:/Coq/fourier/Fourier_util/Rnot_lt0.con"), []) ;;
528 let _Rnot_lt_lt =Cic.Const ((UriManager.uri_of_string
529 "cic:/Coq/fourier/Fourier_util/Rnot_lt_lt.con"), []) ;;
530 let _Ropp = Cic.Const (HelmLibraryObjects.Reals.ropp_URI, []);;
531 let _Rplus = Cic.Const (HelmLibraryObjects.Reals.rplus_URI, []);;
533 (******************************************************************************)
535 let is_int x = (x.den)=1
538 (* fraction = couple (num,den) *)
539 let rec rational_to_fraction x= (x.num,x.den)
542 (* traduction -3 -> (Ropp (Rplus R1 (Rplus R1 R1)))
545 let rec int_to_real_aux n =
547 0 -> _R0 (* o forse R0 + R0 ????? *)
549 | _ -> Cic.Appl [ _Rplus ; _R1 ; int_to_real_aux (n-1) ]
554 let x = int_to_real_aux (abs n) in
556 Cic.Appl [ _Ropp ; x ]
562 (* -1/2 -> (Rmult (Ropp R1) (Rinv (Rplus R1 R1)))
565 let rational_to_real x =
566 let (n,d)=rational_to_fraction x in
567 Cic.Appl [ _Rmult ; int_to_real n ; Cic.Appl [ _Rinv ; int_to_real d ] ]
570 (* preuve que 0<n*1/d
573 let tac_zero_inf_pos (n,d) =
574 let tac_zero_inf_pos (n,d) status =
575 (*let cste = pf_parse_constr gl in*)
576 let pall str (proof,goal) t =
577 debug ("tac "^str^" :\n" );
578 let curi,metasenv,pbo,pty = proof in
579 let metano,context,ty = CicUtil.lookup_meta goal metasenv in
580 debug ("th = "^ CicPp.ppterm t ^"\n");
581 debug ("ty = "^ CicPp.ppterm ty^"\n");
583 let tacn=ref (mk_tactic (fun status ->
584 pall "n0" status _Rlt_zero_1 ;
585 apply_tactic (PrimitiveTactics.apply_tac ~term:_Rlt_zero_1) status )) in
586 let tacd=ref (mk_tactic (fun status ->
587 pall "d0" status _Rlt_zero_1 ;
588 apply_tactic (PrimitiveTactics.apply_tac ~term:_Rlt_zero_1) status )) in
592 tacn:=(Tacticals.then_
593 ~start:(mk_tactic (fun status ->
594 pall ("n"^string_of_int i) status _Rlt_zero_pos_plus1;
596 (PrimitiveTactics.apply_tac ~term:_Rlt_zero_pos_plus1)
598 ~continuation:!tacn);
601 tacd:=(Tacticals.then_
602 ~start:(mk_tactic (fun status ->
603 pall "d" status _Rlt_zero_pos_plus1 ;
605 (PrimitiveTactics.apply_tac ~term:_Rlt_zero_pos_plus1) status))
606 ~continuation:!tacd);
609 debug("TAC ZERO INF POS\n");
612 ~start:(PrimitiveTactics.apply_tac ~term:_Rlt_mult_inv_pos)
613 ~continuations:[!tacn ;!tacd ] )
616 mk_tactic (tac_zero_inf_pos (n,d))
621 (* preuve que 0<=n*1/d
624 let tac_zero_infeq_pos gl (n,d) =
625 let tac_zero_infeq_pos gl (n,d) status =
626 (*let cste = pf_parse_constr gl in*)
627 debug("inizio tac_zero_infeq_pos\n");
630 (PrimitiveTactics.apply_tac ~term:_Rle_zero_zero )
632 (PrimitiveTactics.apply_tac ~term:_Rle_zero_1 )
635 let tacd=ref (PrimitiveTactics.apply_tac ~term:_Rlt_zero_1 ) in
637 tacn:=(Tacticals.then_ ~start:(PrimitiveTactics.apply_tac
638 ~term:_Rle_zero_pos_plus1) ~continuation:!tacn);
641 tacd:=(Tacticals.then_ ~start:(PrimitiveTactics.apply_tac
642 ~term:_Rlt_zero_pos_plus1) ~continuation:!tacd);
646 ~start:(PrimitiveTactics.apply_tac ~term:_Rle_mult_inv_pos)
647 ~continuations:[!tacn;!tacd]) status
649 mk_tactic (tac_zero_infeq_pos gl (n,d))
654 (* preuve que 0<(-n)*(1/d) => False
657 let tac_zero_inf_false gl (n,d) =
658 let tac_zero_inf_false gl (n,d) status =
660 apply_tactic (PrimitiveTactics.apply_tac ~term:_Rnot_lt0) status
662 apply_tactic (Tacticals.then_
663 ~start:( mk_tactic (fun status ->
664 let (proof, goal) = status in
665 let curi,metasenv,pbo,pty = proof in
666 let metano,context,ty = CicUtil.lookup_meta goal metasenv in
667 apply_tactic (PrimitiveTactics.apply_tac ~term:_Rle_not_lt) status))
668 ~continuation:(tac_zero_infeq_pos gl (-n,d)))
671 mk_tactic (tac_zero_inf_false gl (n,d))
674 (* preuve que 0<=n*(1/d) => False ; n est negatif
677 let tac_zero_infeq_false gl (n,d) =
678 let tac_zero_infeq_false gl (n,d) status =
679 let (proof, goal) = status in
680 let curi,metasenv,pbo,pty = proof in
681 let metano,context,ty = CicUtil.lookup_meta goal metasenv in
683 debug("faccio fold di " ^ CicPp.ppterm
687 [_Rmult ; int_to_real n ; Cic.Appl [_Rinv ; int_to_real d]]
690 debug("apply di _Rlt_not_le a "^ CicPp.ppterm ty ^"\n");
691 (*CSC: Patch to undo the over-simplification of RewriteSimpl *)
695 (ReductionTactics.fold_tac ~reduction:CicReduction.whd
697 (ProofEngineTypes.conclusion_pattern
702 [_Rmult ; int_to_real n ; Cic.Appl [_Rinv ; int_to_real d]]
708 ~start:(PrimitiveTactics.apply_tac ~term:_Rlt_not_le)
709 ~continuation:(tac_zero_inf_pos (-n,d))))
712 mk_tactic (tac_zero_infeq_false gl (n,d))
716 (* *********** ********** ******** ??????????????? *********** **************)
718 let apply_type_tac ~cast:t ~applist:al =
719 let apply_type_tac ~cast:t ~applist:al (proof,goal) =
720 let curi,metasenv,pbo,pty = proof in
721 let metano,context,ty = CicUtil.lookup_meta goal metasenv in
722 let fresh_meta = ProofEngineHelpers.new_meta_of_proof proof in
724 CicMkImplicit.identity_relocation_list_for_metavariable context in
725 let metasenv' = (fresh_meta,context,t)::metasenv in
726 let proof' = curi,metasenv',pbo,pty in
729 (PrimitiveTactics.apply_tac
730 (*~term:(Cic.Appl ((Cic.Cast (Cic.Meta (fresh_meta,irl),t))::al)) *)
731 ~term:(Cic.Appl ((Cic.Meta (fresh_meta,irl))::al))) (* ??? *)
734 proof'',fresh_meta::goals
736 mk_tactic (apply_type_tac ~cast:t ~applist:al)
740 let my_cut ~term:c (proof,goal) =
741 let curi,metasenv,pbo,pty = proof in
742 let metano,context,ty = CicUtil.lookup_meta goal metasenv in
743 let fresh_meta = ProofEngineHelpers.new_meta_of_proof proof in
745 CicMkImplicit.identity_relocation_list_for_metavariable context in
746 let metasenv' = (fresh_meta,context,c)::metasenv in
747 let proof' = curi,metasenv',pbo,pty in
751 ~cast:(Cic.Prod(Cic.Name "Anonymous",c,CicSubstitution.lift 1 ty))
752 ~applist:[Cic.Meta(fresh_meta,irl)])
755 (* We permute the generated goals to be consistent with Coq *)
758 | he::tl -> proof'',he::fresh_meta::tl
760 mk_tactic (my_cut ~term:c)
763 let exact = PrimitiveTactics.exact_tac;;
766 let tac_use h status =
767 let (proof, goal) = status in
768 debug("Inizio TC_USE\n");
769 let curi,metasenv,pbo,pty = proof in
770 let metano,context,ty = CicUtil.lookup_meta goal metasenv in
771 debug ("hname = "^ CicPp.ppterm h.hname ^"\n");
772 debug ("ty = "^ CicPp.ppterm ty^"\n");
775 "Rlt" -> exact ~term:h.hname
776 | "Rle" -> exact ~term:h.hname
777 | "Rgt" -> (Tacticals.then_
778 ~start:(PrimitiveTactics.apply_tac ~term:_Rfourier_gt_to_lt)
779 ~continuation:(exact ~term:h.hname))
780 | "Rge" -> (Tacticals.then_
781 ~start:(PrimitiveTactics.apply_tac ~term:_Rfourier_ge_to_le)
782 ~continuation:(exact ~term:h.hname))
783 | "eqTLR" -> (Tacticals.then_
784 ~start:(PrimitiveTactics.apply_tac ~term:_Rfourier_eqLR_to_le)
785 ~continuation:(exact ~term:h.hname))
786 | "eqTRL" -> (Tacticals.then_
787 ~start:(PrimitiveTactics.apply_tac ~term:_Rfourier_eqRL_to_le)
788 ~continuation:(exact ~term:h.hname))
792 mk_tactic (tac_use h)
797 Cic.Appl ( Cic.Const(u,boh)::next) ->
798 (if UriManager.eq u HelmLibraryObjects.Reals.rlt_URI or
799 UriManager.eq u HelmLibraryObjects.Reals.rgt_URI or
800 UriManager.eq u HelmLibraryObjects.Reals.rle_URI or
801 UriManager.eq u HelmLibraryObjects.Reals.rge_URI then true
802 else if UriManager.eq u HelmLibraryObjects.Logic.eq_URI then
803 (match (List.hd next) with
804 Cic.Const (uri,_) when
805 UriManager.eq uri HelmLibraryObjects.Reals.r_URI
812 let list_of_sign s = List.map (fun (x,_,z)->(x,z)) s;;
815 Cic.Appl(Array.to_list a)
818 (* Résolution d'inéquations linéaires dans R *)
819 let rec strip_outer_cast c = match c with
820 | Cic.Cast (c,_) -> strip_outer_cast c
824 (*let find_in_context id context =
825 let rec find_in_context_aux c n =
827 [] -> failwith (id^" not found in context")
828 | a::next -> (match a with
829 Some (Cic.Name(name),_) when name = id -> n
830 (*? magari al posto di _ qualcosaltro?*)
831 | _ -> find_in_context_aux next (n+1))
833 find_in_context_aux context 1
836 (* mi sembra quadratico *)
837 let rec filter_real_hyp context cont =
840 | Some(Cic.Name(h),Cic.Decl(t))::next -> (
841 let n = find_in_context h cont in
842 debug("assegno "^string_of_int n^" a "^CicPp.ppterm t^"\n");
843 [(Cic.Rel(n),t)] @ filter_real_hyp next cont)
844 | a::next -> debug(" no\n"); filter_real_hyp next cont
847 let filter_real_hyp context _ =
848 let rec filter_aux context num =
851 | Some(Cic.Name(h),Cic.Decl(t))::next ->
852 [(Cic.Rel(num),t)] @ filter_aux next (num+1)
853 | a::next -> filter_aux next (num+1)
859 (* lifts everithing at the conclusion level *)
860 let rec superlift c n=
863 | Some(name,Cic.Decl(a))::next ->
864 [Some(name,Cic.Decl(CicSubstitution.lift n a))]@ superlift next (n+1)
865 | Some(name,Cic.Def(a,None))::next ->
866 [Some(name,Cic.Def((CicSubstitution.lift n a),None))]@ superlift next (n+1)
867 | Some(name,Cic.Def(a,Some ty))::next ->
869 Cic.Def((CicSubstitution.lift n a),Some (CicSubstitution.lift n ty)))
870 ] @ superlift next (n+1)
871 | _::next -> superlift next (n+1) (*?? ??*)
875 let equality_replace a b =
876 let equality_replace a b status =
877 debug("inizio EQ\n");
878 let module C = Cic in
879 let proof,goal = status in
880 let curi,metasenv,pbo,pty = proof in
881 let metano,context,ty = CicUtil.lookup_meta goal metasenv in
882 let a_eq_b = C.Appl [ _eqT ; _R ; a ; b ] in
883 let fresh_meta = ProofEngineHelpers.new_meta_of_proof proof in
885 CicMkImplicit.identity_relocation_list_for_metavariable context in
886 let metasenv' = (fresh_meta,context,a_eq_b)::metasenv in
887 debug("chamo rewrite tac su"^CicPp.ppterm (C.Meta (fresh_meta,irl)));
888 let (proof,goals) = apply_tactic
889 (EqualityTactics.rewrite_simpl_tac
890 ~direction:`LeftToRight
891 ~pattern:(ProofEngineTypes.conclusion_pattern None)
892 (C.Meta (fresh_meta,irl)))
893 ((curi,metasenv',pbo,pty),goal)
895 let new_goals = fresh_meta::goals in
896 debug("fine EQ -> goals : "^string_of_int( List.length new_goals) ^" = "
897 ^string_of_int( List.length goals)^"+ meta\n");
900 mk_tactic (equality_replace a b)
903 let tcl_fail a (proof,goal) =
905 1 -> raise (ProofEngineTypes.Fail "fail-tactical")
906 | _ -> (proof,[goal])
909 (* Galla: moved in variousTactics.ml
910 let assumption_tac (proof,goal)=
911 let curi,metasenv,pbo,pty = proof in
912 let metano,context,ty = CicUtil.lookup_meta goal metasenv in
914 let tac_list = List.map
915 ( fun x -> num := !num + 1;
917 Some(Cic.Name(nm),t) -> (nm,exact ~term:(Cic.Rel(!num)))
918 | _ -> ("fake",tcl_fail 1)
922 Tacticals.try_tactics ~tactics:tac_list (proof,goal)
925 (* Galla: moved in negationTactics.ml
926 (* !!!!! fix !!!!!!!!!! *)
927 let contradiction_tac (proof,goal)=
929 (*inutile sia questo che quello prima della chiamata*)
930 ~start:PrimitiveTactics.intros_tac
931 ~continuation:(Tacticals.then_
932 ~start:(VariousTactics.elim_type_tac ~term:_False)
933 ~continuation:(assumption_tac))
938 (* ********************* TATTICA ******************************** *)
940 let rec fourier (s_proof,s_goal)=
941 let s_curi,s_metasenv,s_pbo,s_pty = s_proof in
942 let s_metano,s_context,s_ty = CicUtil.lookup_meta s_goal s_metasenv in
943 debug ("invoco fourier_tac sul goal "^string_of_int(s_goal)^" e contesto:\n");
944 debug_pcontext s_context;
946 let fhyp = String.copy "new_hyp_for_fourier" in
948 (* here we need to negate the thesis, but to do this we need to apply the
949 right theoreme,so let's parse our thesis *)
951 let th_to_appl = ref _Rfourier_not_le_gt in
953 Cic.Appl ( Cic.Const(u,boh)::args) ->
955 (if UriManager.eq u HelmLibraryObjects.Reals.rlt_URI then
957 else if UriManager.eq u HelmLibraryObjects.Reals.rle_URI then
959 else if UriManager.eq u HelmLibraryObjects.Reals.rgt_URI then
961 else if UriManager.eq u HelmLibraryObjects.Reals.rge_URI then
963 else failwith "fourier can't be applyed")
964 |_-> failwith "fourier can't be applyed");
965 (* fix maybe strip_outer_cast goes here?? *)
967 (* now let's change our thesis applying the th and put it with hp *)
969 let proof,gl = apply_tactic
971 ~start:(PrimitiveTactics.apply_tac ~term:!th_to_appl)
972 ~continuation:(PrimitiveTactics.intros_tac ()))
975 let goal = if List.length gl = 1 then List.hd gl
976 else failwith "a new goal" in
978 debug ("port la tesi sopra e la nego. contesto :\n");
979 debug_pcontext s_context;
981 (* now we have all the right environment *)
983 let curi,metasenv,pbo,pty = proof in
984 let metano,context,ty = CicUtil.lookup_meta goal metasenv in
986 (* now we want to convert hp to inequations, but first we must lift
987 everyting to thesis level, so that a variable has the save Rel(n)
988 in each hp ( needed by ineq1_of_term ) *)
990 (* ? fix if None ?????*)
991 (* fix change superlift with a real name *)
993 let l_context = superlift context 1 in
994 let hyps = filter_real_hyp l_context l_context in
996 debug ("trasformo in diseq. "^ string_of_int (List.length hyps)^" ipotesi\n");
1000 (* transform hyps into inequations *)
1002 List.iter (fun h -> try (lineq:=(ineq1_of_term h)@(!lineq))
1006 debug ("applico fourier a "^ string_of_int (List.length !lineq)^
1009 let res=fourier_lineq (!lineq) in
1010 let tac=ref Tacticals.id_tac in
1012 (print_string "Tactic Fourier fails.\n";flush stdout;
1013 failwith "fourier_tac fails")
1016 match res with (*match res*)
1019 (* in lc we have the coefficient to "reduce" the system *)
1021 print_string "Fourier's method can prove the goal...\n";flush stdout;
1023 debug "I coeff di moltiplicazione rit sono: ";
1027 (fun (h,c) -> if c<>r0 then (lutil:=(h,c)::(!lutil);
1028 (* DBG *)Fourier.print_rational(c);print_string " "(* DBG *))
1030 (List.combine (!lineq) lc);
1032 print_string (" quindi lutil e' lunga "^
1033 string_of_int (List.length (!lutil))^"\n");
1035 (* on construit la combinaison linéaire des inéquation *)
1037 (match (!lutil) with (*match (!lutil) *)
1039 debug ("elem di lutil ");Fourier.print_rational c1;print_string "\n";
1041 let s=ref (h1.hstrict) in
1044 let t1 = ref (Cic.Appl [_Rmult;rational_to_real c1;h1.hleft] ) in
1045 let t2 = ref (Cic.Appl [_Rmult;rational_to_real c1;h1.hright]) in
1047 List.iter (fun (h,c) ->
1048 s:=(!s)||(h.hstrict);
1049 t1:=(Cic.Appl [_Rplus;!t1;Cic.Appl
1050 [_Rmult;rational_to_real c;h.hleft ] ]);
1051 t2:=(Cic.Appl [_Rplus;!t2;Cic.Appl
1052 [_Rmult;rational_to_real c;h.hright] ]))
1055 let ineq=Cic.Appl [(if (!s) then _Rlt else _Rle);!t1;!t2 ] in
1056 let tc=rational_to_real cres in
1059 (* ora ho i termini che descrivono i passi di fourier per risolvere il sistema *)
1061 debug "inizio a costruire tac1\n";
1062 Fourier.print_rational(c1);
1064 let tac1=ref ( mk_tactic (fun status ->
1068 ~start:(mk_tactic (fun status ->
1069 debug ("inizio t1 strict\n");
1070 let curi,metasenv,pbo,pty = proof in
1071 let metano,context,ty = CicUtil.lookup_meta goal metasenv in
1072 debug ("th = "^ CicPp.ppterm _Rfourier_lt ^"\n");
1073 debug ("ty = "^ CicPp.ppterm ty^"\n");
1075 (PrimitiveTactics.apply_tac ~term:_Rfourier_lt) status))
1076 ~continuations:[tac_use h1;
1077 tac_zero_inf_pos (rational_to_fraction c1)])
1080 ~start:(PrimitiveTactics.apply_tac ~term:_Rfourier_le)
1081 ~continuations:[tac_use h1;tac_zero_inf_pos
1082 (rational_to_fraction c1)]))
1087 List.iter (fun (h,c) ->
1091 tac1:=(Tacticals.thens
1092 ~start:(PrimitiveTactics.apply_tac ~term:_Rfourier_lt_lt)
1093 ~continuations:[!tac1;tac_use h;tac_zero_inf_pos
1094 (rational_to_fraction c)]))
1097 Fourier.print_rational(c1);
1098 tac1:=(Tacticals.thens
1099 ~start:(mk_tactic (fun status ->
1100 debug("INIZIO TAC 1 2\n");
1101 let curi,metasenv,pbo,pty = proof in
1102 let metano,context,ty = CicUtil.lookup_meta goal metasenv in
1103 debug ("th = "^ CicPp.ppterm _Rfourier_lt_le ^"\n");
1104 debug ("ty = "^ CicPp.ppterm ty^"\n");
1106 (PrimitiveTactics.apply_tac ~term:_Rfourier_lt_le)
1108 ~continuations:[!tac1;tac_use h;tac_zero_inf_pos
1109 (rational_to_fraction c)])))
1113 tac1:=(Tacticals.thens
1114 ~start:(PrimitiveTactics.apply_tac ~term:_Rfourier_le_lt)
1115 ~continuations:[!tac1;tac_use h;tac_zero_inf_pos
1116 (rational_to_fraction c)]))
1119 tac1:=(Tacticals.thens
1120 ~start:(PrimitiveTactics.apply_tac ~term:_Rfourier_le_le)
1121 ~continuations:[!tac1;tac_use h;tac_zero_inf_pos
1122 (rational_to_fraction c)]))));
1123 s:=(!s)||(h.hstrict)) (* end fun -> *)
1124 lutil;(*end List.iter*)
1128 tac_zero_inf_false goal (rational_to_fraction cres)
1130 tac_zero_infeq_false goal (rational_to_fraction cres)
1132 tac:=(Tacticals.thens
1133 ~start:(my_cut ~term:ineq)
1134 ~continuations:[Tacticals.then_
1135 ~start:( mk_tactic (fun status ->
1136 let (proof, goal) = status in
1137 let curi,metasenv,pbo,pty = proof in
1138 let metano,context,ty = CicUtil.lookup_meta goal metasenv in
1140 (PrimitiveTactics.change_tac
1141 ~pattern:(ProofEngineTypes.conclusion_pattern (Some ty))
1142 (Cic.Appl [ _not; ineq]))
1144 ~continuation:(Tacticals.then_
1145 ~start:(PrimitiveTactics.apply_tac ~term:
1146 (if sres then _Rnot_lt_lt else _Rnot_le_le))
1147 ~continuation:(Tacticals.thens
1148 ~start:(mk_tactic (fun status ->
1149 debug("t1 ="^CicPp.ppterm !t1 ^"t2 ="^
1150 CicPp.ppterm !t2 ^"tc="^ CicPp.ppterm tc^"\n");
1151 let r = apply_tactic
1152 (equality_replace (Cic.Appl [_Rminus;!t2;!t1] ) tc)
1155 (match r with (p,gl) ->
1156 debug("eq1 ritorna "^string_of_int(List.length gl)^"\n" ));
1158 ~continuations:[(Tacticals.thens
1159 ~start:(mk_tactic (fun status ->
1160 let r = apply_tactic
1161 (equality_replace (Cic.Appl[_Rinv;_R1]) _R1)
1164 (match r with (p,gl) ->
1165 debug("eq2 ritorna "^string_of_int(List.length gl)^"\n" ));
1168 [PrimitiveTactics.apply_tac ~term:_Rinv_R1;
1169 Tacticals.try_tactics
1170 ~tactics:[ "ring",Ring.ring_tac; "id", Tacticals.id_tac]
1172 ;(*Tacticals.id_tac*)
1174 ~start:(mk_tactic (fun status ->
1175 let (proof, goal) = status in
1176 let curi,metasenv,pbo,pty = proof in
1177 let metano,context,ty = CicUtil.lookup_meta goal metasenv in
1178 (* check if ty is of type *)
1180 debug("qui c'e' gia' l'or "^CicPp.ppterm ty^"\n");
1182 Cic.Prod (Cic.Anonymous,a,b) -> (Cic.Appl [_not;a])
1185 let r = apply_tactic
1186 (PrimitiveTactics.change_tac
1187 ~pattern:(ProofEngineTypes.conclusion_pattern (Some ty))
1190 debug("fine MY_CHNGE\n");
1192 ~continuation:(*PORTINGTacticals.id_tac*)tac2]))
1193 ;(*Tacticals.id_tac*)!tac1]);(*end tac:=*)
1195 |_-> assert false)(*match (!lutil) *)
1196 |_-> assert false); (*match res*)
1197 debug ("finalmente applico tac\n");
1199 let r = apply_tactic !tac (proof,goal) in
1200 debug("\n\n]]]]]]]]]]]]]]]]]) That's all folks ([[[[[[[[[[[[[[[[[[[\n\n");r
1205 let fourier_tac = mk_tactic fourier