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.
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14 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
15 * GNU General Public License for more details.
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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 (******************** OTHER USEFUL TACTICS **********************)
29 let rewrite_tac ~term:equality ~status:(proof,goal) =
31 let module U = UriManager in
32 let curi,metasenv,pbo,pty = proof in
33 let metano,context,gty = List.find (function (m,_,_) -> m=goal) metasenv in
34 let eq_ind_r,ty,t1,t2 =
35 match CicTypeChecker.type_of_aux' metasenv context equality with
36 C.Appl [C.MutInd (uri,0,[]) ; ty ; t1 ; t2]
37 when U.eq uri (U.uri_of_string "cic:/Coq/Init/Logic/eq.ind") ->
40 (U.uri_of_string "cic:/Coq/Init/Logic/eq_ind_r.con",[])
43 | C.Appl [C.MutInd (uri,0,[]) ; ty ; t1 ; t2]
44 when U.eq uri (U.uri_of_string "cic:/Coq/Init/Logic_Type/eqT.ind") ->
47 (U.uri_of_string "cic:/Coq/Init/Logic_Type/eqT_ind_r.con",[])
52 (ProofEngineTypes.Fail
53 "Rewrite: the argument is not a proof of an equality")
56 let gty' = CicSubstitution.lift 1 gty in
57 let t1' = CicSubstitution.lift 1 t1 in
59 ProofEngineReduction.replace_lifting
60 ~equality:ProofEngineReduction.alpha_equivalence
61 ~what:t1' ~with_what:(C.Rel 1) ~where:gty'
63 C.Lambda (C.Name "dummy_for_rewrite", ty, gty'')
65 prerr_endline ("#### Sintetizzato: " ^ CicPp.ppterm pred);
66 let fresh_meta = ProofEngineHelpers.new_meta proof in
68 ProofEngineHelpers.identity_relocation_list_for_metavariable context in
69 let metasenv' = (fresh_meta,context,C.Appl [pred ; t2])::metasenv in
72 PrimitiveTactics.exact_tac
74 [eq_ind_r ; ty ; t2 ; pred ; C.Meta (fresh_meta,irl) ; t1 ;equality])
75 ~status:((curi,metasenv',pbo,pty),goal)
77 assert (List.length goals = 0) ;
82 let rewrite_simpl_tac ~term ~status =
83 Tacticals.then_ ~start:(rewrite_tac ~term)
85 (ReductionTactics.simpl_tac ~also_in_hypotheses:false ~term:None)
89 (******************** THE FOURIER TACTIC ***********************)
91 (* La tactique Fourier ne fonctionne de manière sûre que si les coefficients
92 des inéquations et équations sont entiers. En attendant la tactique Field.
98 let debug x = print_string ("____ "^x) ; flush stdout;;
100 let debug_pcontext x =
102 List.iter (fun y -> match y with Some(Cic.Name(a),_) -> str := !str ^
103 a ^ " " | _ ->()) x ;
104 debug ("contesto : "^ (!str) ^ "\n")
107 (******************************************************************************
108 Operations on linear combinations.
110 Opérations sur les combinaisons linéaires affines.
111 La partie homogène d'une combinaison linéaire est en fait une table de hash
112 qui donne le coefficient d'un terme du calcul des constructions,
113 qui est zéro si le terme n'y est pas.
119 The type for linear combinations
121 type flin = {fhom:(Cic.term , rational)Hashtbl.t;fcste:rational}
125 @return an empty flin
127 let flin_zero () = {fhom = Hashtbl.create 50;fcste=r0}
133 @return the rational associated with x (coefficient)
137 (Hashtbl.find f.fhom x)
143 Adds c to the coefficient of x
150 let cx = flin_coef f x in
151 Hashtbl.remove f.fhom x;
152 Hashtbl.add f.fhom x (rplus cx c);
161 let flin_add_cste f c =
163 fcste=rplus f.fcste c}
167 @return a empty flin with r1 in fcste
169 let flin_one () = flin_add_cste (flin_zero()) r1;;
174 let flin_plus f1 f2 =
175 let f3 = flin_zero() in
176 Hashtbl.iter (fun x c -> let _=flin_add f3 x c in ()) f1.fhom;
177 Hashtbl.iter (fun x c -> let _=flin_add f3 x c in ()) f2.fhom;
178 flin_add_cste (flin_add_cste f3 f1.fcste) f2.fcste;
184 let flin_minus f1 f2 =
185 let f3 = flin_zero() in
186 Hashtbl.iter (fun x c -> let _=flin_add f3 x c in ()) f1.fhom;
187 Hashtbl.iter (fun x c -> let _=flin_add f3 x (rop c) in ()) f2.fhom;
188 flin_add_cste (flin_add_cste f3 f1.fcste) (rop f2.fcste);
195 let f2 = flin_zero() in
196 Hashtbl.iter (fun x c -> let _=flin_add f2 x (rmult a c) in ()) f.fhom;
197 flin_add_cste f2 (rmult a f.fcste);
201 (*****************************************************************************)
206 @raise Failure if conversion is impossible
207 @return rational proiection of t
209 let rec rational_of_term t =
210 (* fun to apply f to the first and second rational-term of l *)
211 let rat_of_binop f l =
212 let a = List.hd l and
213 b = List.hd(List.tl l) in
214 f (rational_of_term a) (rational_of_term b)
216 (* as before, but f is unary *)
217 let rat_of_unop f l =
218 f (rational_of_term (List.hd l))
221 | Cic.Cast (t1,t2) -> (rational_of_term t1)
222 | Cic.Appl (t1::next) ->
225 (match (UriManager.string_of_uri u) with
226 "cic:/Coq/Reals/Rdefinitions/Ropp.con" ->
228 |"cic:/Coq/Reals/Rdefinitions/Rinv.con" ->
229 rat_of_unop rinv next
230 |"cic:/Coq/Reals/Rdefinitions/Rmult.con" ->
231 rat_of_binop rmult next
232 |"cic:/Coq/Reals/Rdefinitions/Rdiv.con" ->
233 rat_of_binop rdiv next
234 |"cic:/Coq/Reals/Rdefinitions/Rplus.con" ->
235 rat_of_binop rplus next
236 |"cic:/Coq/Reals/Rdefinitions/Rminus.con" ->
237 rat_of_binop rminus next
238 | _ -> failwith "not a rational")
239 | _ -> failwith "not a rational")
240 | Cic.Const (u,boh) ->
241 (match (UriManager.string_of_uri u) with
242 "cic:/Coq/Reals/Rdefinitions/R1.con" -> r1
243 |"cic:/Coq/Reals/Rdefinitions/R0.con" -> r0
244 | _ -> failwith "not a rational")
245 | _ -> failwith "not a rational"
249 let rational_of_const = rational_of_term;;
253 let rec flin_of_term t =
254 let fl_of_binop f l =
255 let a = List.hd l and
256 b = List.hd(List.tl l) in
257 f (flin_of_term a) (flin_of_term b)
261 | Cic.Cast (t1,t2) -> (flin_of_term t1)
262 | Cic.Appl (t1::next) ->
267 match (UriManager.string_of_uri u) with
268 "cic:/Coq/Reals/Rdefinitions/Ropp.con" ->
269 flin_emult (rop r1) (flin_of_term (List.hd next))
270 |"cic:/Coq/Reals/Rdefinitions/Rplus.con"->
271 fl_of_binop flin_plus next
272 |"cic:/Coq/Reals/Rdefinitions/Rminus.con"->
273 fl_of_binop flin_minus next
274 |"cic:/Coq/Reals/Rdefinitions/Rmult.con"->
276 let arg1 = (List.hd next) and
277 arg2 = (List.hd(List.tl next))
281 let a = rational_of_term arg1 in
284 let b = (rational_of_term arg2) in
285 (flin_add_cste (flin_zero()) (rmult a b))
288 _ -> (flin_add (flin_zero()) arg2 a)
291 _-> (flin_add(flin_zero()) arg1 (rational_of_term arg2))
293 |"cic:/Coq/Reals/Rdefinitions/Rinv.con"->
294 let a=(rational_of_term (List.hd next)) in
295 flin_add_cste (flin_zero()) (rinv a)
296 |"cic:/Coq/Reals/Rdefinitions/Rdiv.con"->
298 let b=(rational_of_term (List.hd(List.tl next))) in
301 let a = (rational_of_term (List.hd next)) in
302 (flin_add_cste (flin_zero()) (rdiv a b))
305 _-> (flin_add (flin_zero()) (List.hd next) (rinv b))
311 | Cic.Const (u,boh) ->
313 match (UriManager.string_of_uri u) with
314 "cic:/Coq/Reals/Rdefinitions/R1.con" -> flin_one ()
315 |"cic:/Coq/Reals/Rdefinitions/R0.con" -> flin_zero ()
319 with _ -> flin_add (flin_zero()) t r1
323 let flin_of_constr = flin_of_term;;
327 Translates a flin to (c,x) list
329 @return something like (c1,x1)::(c2,x2)::...::(cn,xn)
331 let flin_to_alist f =
333 Hashtbl.iter (fun x c -> res:=(c,x)::(!res)) f;
337 (* Représentation des hypothèses qui sont des inéquations ou des équations.
341 The structure for ineq
343 type hineq={hname:Cic.term; (* le nom de l'hypothèse *)
344 htype:string; (* Rlt, Rgt, Rle, Rge, eqTLR ou eqTRL *)
351 (* Transforme une hypothese h:t en inéquation flin<0 ou flin<=0
354 let ineq1_of_term (h,t) =
355 match t with (* match t *)
356 Cic.Appl (t1::next) ->
357 let arg1= List.hd next in
358 let arg2= List.hd(List.tl next) in
359 (match t1 with (* match t1 *)
361 (match UriManager.string_of_uri u with (* match u *)
362 "cic:/Coq/Reals/Rdefinitions/Rlt.con" ->
367 hflin= flin_minus (flin_of_term arg1)
370 |"cic:/Coq/Reals/Rdefinitions/Rgt.con" ->
375 hflin= flin_minus (flin_of_term arg2)
378 |"cic:/Coq/Reals/Rdefinitions/Rle.con" ->
383 hflin= flin_minus (flin_of_term arg1)
386 |"cic:/Coq/Reals/Rdefinitions/Rge.con" ->
391 hflin= flin_minus (flin_of_term arg2)
394 |_->assert false)(* match u *)
395 | Cic.MutInd (u,i,o) ->
396 (match UriManager.string_of_uri u with
397 "cic:/Coq/Init/Logic_Type/eqT.ind" ->
400 let arg2= List.hd(List.tl (List.tl next)) in
403 (match UriManager.string_of_uri u with
404 "cic:/Coq/Reals/Rdefinitions/R.con"->
409 hflin= flin_minus (flin_of_term arg1)
416 hflin= flin_minus (flin_of_term arg2)
422 |_-> assert false)(* match t1 *)
423 |_-> assert false (* match t *)
426 let ineq1_of_constr = ineq1_of_term;;
429 (* Applique la méthode de Fourier à une liste d'hypothèses (type hineq)
435 | a::next -> Fourier.print_rational a ; print_string " " ; print_rl next
438 let rec print_sys l =
441 | (a,b)::next -> (print_rl a;
442 print_string (if b=true then "strict\n"else"\n");
447 Hashtbl.iter (fun x y -> print_string ("("^"-"^","^"-"^")")) h
450 let fourier_lineq lineq1 =
452 let hvar=Hashtbl.create 50 in (* la table des variables des inéquations *)
454 Hashtbl.iter (fun x c ->
455 try (Hashtbl.find hvar x;())
456 with _-> nvar:=(!nvar)+1;
457 Hashtbl.add hvar x (!nvar))
461 debug("Il numero di incognite e' "^string_of_int (!nvar+1)^"\n");
462 let sys= List.map (fun h->
463 let v=Array.create ((!nvar)+1) r0 in
464 Hashtbl.iter (fun x c -> v.(Hashtbl.find hvar x)<-c)
466 ((Array.to_list v)@[rop h.hflin.fcste],h.hstrict))
468 debug ("chiamo unsolvable sul sistema di "^
469 string_of_int (List.length sys) ^"\n");
474 (*****************************************************************************
475 Construction de la preuve en cas de succès de la méthode de Fourier,
476 i.e. on obtient une contradiction.
480 let _eqT = Cic.MutInd(UriManager.uri_of_string
481 "cic:/Coq/Init/Logic_Type/eqT.ind") 0 [] ;;
482 let _False = Cic.MutInd (UriManager.uri_of_string
483 "cic:/Coq/Init/Logic/False.ind") 0 [] ;;
484 let _not = Cic.Const (UriManager.uri_of_string
485 "cic:/Coq/Init/Logic/not.con") [];;
486 let _R0 = Cic.Const (UriManager.uri_of_string
487 "cic:/Coq/Reals/Rdefinitions/R0.con") [] ;;
488 let _R1 = Cic.Const (UriManager.uri_of_string
489 "cic:/Coq/Reals/Rdefinitions/R1.con") [] ;;
490 let _R = Cic.Const (UriManager.uri_of_string
491 "cic:/Coq/Reals/Rdefinitions/R.con") [] ;;
492 let _Rfourier_eqLR_to_le=Cic.Const (UriManager.uri_of_string
493 "cic:/Coq/fourier/Fourier_util/Rfourier_eqLR_to_le.con") [] ;;
494 let _Rfourier_eqRL_to_le=Cic.Const (UriManager.uri_of_string
495 "cic:/Coq/fourier/Fourier_util/Rfourier_eqRL_to_le.con") [] ;;
496 let _Rfourier_ge_to_le =Cic.Const (UriManager.uri_of_string
497 "cic:/Coq/fourier/Fourier_util/Rfourier_ge_to_le.con") [] ;;
498 let _Rfourier_gt_to_lt =Cic.Const (UriManager.uri_of_string
499 "cic:/Coq/fourier/Fourier_util/Rfourier_gt_to_lt.con") [] ;;
500 let _Rfourier_le=Cic.Const (UriManager.uri_of_string
501 "cic:/Coq/fourier/Fourier_util/Rfourier_le.con") [] ;;
502 let _Rfourier_le_le =Cic.Const (UriManager.uri_of_string
503 "cic:/Coq/fourier/Fourier_util/Rfourier_le_le.con") [] ;;
504 let _Rfourier_le_lt =Cic.Const (UriManager.uri_of_string
505 "cic:/Coq/fourier/Fourier_util/Rfourier_le_lt.con") [] ;;
506 let _Rfourier_lt=Cic.Const (UriManager.uri_of_string
507 "cic:/Coq/fourier/Fourier_util/Rfourier_lt.con") [] ;;
508 let _Rfourier_lt_le =Cic.Const (UriManager.uri_of_string
509 "cic:/Coq/fourier/Fourier_util/Rfourier_lt_le.con") [] ;;
510 let _Rfourier_lt_lt =Cic.Const (UriManager.uri_of_string
511 "cic:/Coq/fourier/Fourier_util/Rfourier_lt_lt.con") [] ;;
512 let _Rfourier_not_ge_lt = Cic.Const (UriManager.uri_of_string
513 "cic:/Coq/fourier/Fourier_util/Rfourier_not_ge_lt.con") [] ;;
514 let _Rfourier_not_gt_le = Cic.Const (UriManager.uri_of_string
515 "cic:/Coq/fourier/Fourier_util/Rfourier_not_gt_le.con") [] ;;
516 let _Rfourier_not_le_gt = Cic.Const (UriManager.uri_of_string
517 "cic:/Coq/fourier/Fourier_util/Rfourier_not_le_gt.con") [] ;;
518 let _Rfourier_not_lt_ge = Cic.Const (UriManager.uri_of_string
519 "cic:/Coq/fourier/Fourier_util/Rfourier_not_lt_ge.con") [] ;;
520 let _Rinv = Cic.Const (UriManager.uri_of_string
521 "cic:/Coq/Reals/Rdefinitions/Rinv.con") [] ;;
522 let _Rinv_R1 = Cic.Const(UriManager.uri_of_string
523 "cic:/Coq/Reals/Rbase/Rinv_R1.con" ) [] ;;
524 let _Rle = Cic.Const (UriManager.uri_of_string
525 "cic:/Coq/Reals/Rdefinitions/Rle.con") [] ;;
526 let _Rle_mult_inv_pos = Cic.Const (UriManager.uri_of_string
527 "cic:/Coq/fourier/Fourier_util/Rle_mult_inv_pos.con") [] ;;
528 let _Rle_not_lt = Cic.Const (UriManager.uri_of_string
529 "cic:/Coq/fourier/Fourier_util/Rle_not_lt.con") [] ;;
530 let _Rle_zero_1 = Cic.Const (UriManager.uri_of_string
531 "cic:/Coq/fourier/Fourier_util/Rle_zero_1.con") [] ;;
532 let _Rle_zero_pos_plus1 = Cic.Const (UriManager.uri_of_string
533 "cic:/Coq/fourier/Fourier_util/Rle_zero_pos_plus1.con") [] ;;
534 (*let _Rle_zero_zero = Cic.Const (UriManager.uri_of_string
535 "cic:/Coq/fourier/Fourier_util/Rle_zero_zero.con") [] ;;*)
536 let _Rlt = Cic.Const (UriManager.uri_of_string
537 "cic:/Coq/Reals/Rdefinitions/Rlt.con") [] ;;
538 let _Rlt_mult_inv_pos = Cic.Const (UriManager.uri_of_string
539 "cic:/Coq/fourier/Fourier_util/Rlt_mult_inv_pos.con") [] ;;
540 let _Rlt_not_le = Cic.Const (UriManager.uri_of_string
541 "cic:/Coq/fourier/Fourier_util/Rlt_not_le.con") [] ;;
542 let _Rlt_zero_1 = Cic.Const (UriManager.uri_of_string
543 "cic:/Coq/fourier/Fourier_util/Rlt_zero_1.con") [] ;;
544 let _Rlt_zero_pos_plus1 = Cic.Const (UriManager.uri_of_string
545 "cic:/Coq/fourier/Fourier_util/Rlt_zero_pos_plus1.con") [] ;;
546 let _Rminus = Cic.Const (UriManager.uri_of_string
547 "cic:/Coq/Reals/Rdefinitions/Rminus.con") [] ;;
548 let _Rmult = Cic.Const (UriManager.uri_of_string
549 "cic:/Coq/Reals/Rdefinitions/Rmult.con") [] ;;
550 let _Rnot_le_le =Cic.Const (UriManager.uri_of_string
551 "cic:/Coq/fourier/Fourier_util/Rnot_le_le.con") [] ;;
552 let _Rnot_lt0 = Cic.Const (UriManager.uri_of_string
553 "cic:/Coq/fourier/Fourier_util/Rnot_lt0.con") [] ;;
554 let _Rnot_lt_lt =Cic.Const (UriManager.uri_of_string
555 "cic:/Coq/fourier/Fourier_util/Rnot_lt_lt.con") [] ;;
556 let _Ropp = Cic.Const (UriManager.uri_of_string
557 "cic:/Coq/Reals/Rdefinitions/Ropp.con") [] ;;
558 let _Rplus = Cic.Const (UriManager.uri_of_string
559 "cic:/Coq/Reals/Rdefinitions/Rplus.con") [] ;;
561 (******************************************************************************)
563 let is_int x = (x.den)=1
566 (* fraction = couple (num,den) *)
567 let rec rational_to_fraction x= (x.num,x.den)
570 (* traduction -3 -> (Ropp (Rplus R1 (Rplus R1 R1)))
573 let rec int_to_real_aux n =
575 0 -> _R0 (* o forse R0 + R0 ????? *)
577 | _ -> Cic.Appl [ _Rplus ; _R1 ; int_to_real_aux (n-1) ]
582 let x = int_to_real_aux (abs n) in
584 Cic.Appl [ _Ropp ; x ]
590 (* -1/2 -> (Rmult (Ropp R1) (Rinv (Rplus R1 R1)))
593 let rational_to_real x =
594 let (n,d)=rational_to_fraction x in
595 Cic.Appl [ _Rmult ; int_to_real n ; Cic.Appl [ _Rinv ; int_to_real d ] ]
598 (* preuve que 0<n*1/d
601 let tac_zero_inf_pos (n,d) ~status =
602 (*let cste = pf_parse_constr gl in*)
603 let pall str ~status:(proof,goal) t =
604 debug ("tac "^str^" :\n" );
605 let curi,metasenv,pbo,pty = proof in
606 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
607 debug ("th = "^ CicPp.ppterm t ^"\n");
608 debug ("ty = "^ CicPp.ppterm ty^"\n");
611 (fun ~status -> pall "n0" ~status _Rlt_zero_1 ;
612 PrimitiveTactics.apply_tac ~term:_Rlt_zero_1 ~status ) in
614 (fun ~status -> pall "d0" ~status _Rlt_zero_1 ;
615 PrimitiveTactics.apply_tac ~term:_Rlt_zero_1 ~status ) in
619 tacn:=(Tacticals.then_ ~start:(fun ~status -> pall ("n"^string_of_int i)
620 ~status _Rlt_zero_pos_plus1;
621 PrimitiveTactics.apply_tac ~term:_Rlt_zero_pos_plus1 ~status)
622 ~continuation:!tacn);
625 tacd:=(Tacticals.then_ ~start:(fun ~status -> pall "d"
626 ~status _Rlt_zero_pos_plus1 ;PrimitiveTactics.apply_tac
627 ~term:_Rlt_zero_pos_plus1 ~status) ~continuation:!tacd);
632 debug("TAC ZERO INF POS\n");
634 (Tacticals.thens ~start:(PrimitiveTactics.apply_tac ~term:_Rlt_mult_inv_pos)
643 (* preuve que 0<=n*1/d
646 let tac_zero_infeq_pos gl (n,d) ~status =
647 (*let cste = pf_parse_constr gl in*)
648 debug("inizio tac_zero_infeq_pos\n");
651 (PrimitiveTactics.apply_tac ~term:_Rle_zero_zero )
653 (PrimitiveTactics.apply_tac ~term:_Rle_zero_1 )
656 let tacd=ref (PrimitiveTactics.apply_tac ~term:_Rlt_zero_1 ) in
658 tacn:=(Tacticals.then_ ~start:(PrimitiveTactics.apply_tac
659 ~term:_Rle_zero_pos_plus1) ~continuation:!tacn);
662 tacd:=(Tacticals.then_ ~start:(PrimitiveTactics.apply_tac
663 ~term:_Rlt_zero_pos_plus1) ~continuation:!tacd);
666 (Tacticals.thens ~start:(PrimitiveTactics.apply_tac
667 ~term:_Rle_mult_inv_pos) ~continuations:[!tacn;!tacd]) ~status in
668 debug("fine tac_zero_infeq_pos\n");
674 (* preuve que 0<(-n)*(1/d) => False
677 let tac_zero_inf_false gl (n,d) ~status=
678 debug("inizio tac_zero_inf_false\n");
680 (debug "1\n";let r =(PrimitiveTactics.apply_tac ~term:_Rnot_lt0 ~status) in
684 (debug "2\n";let r = (Tacticals.then_ ~start:(
685 fun ~status:(proof,goal as status) ->
686 let curi,metasenv,pbo,pty = proof in
687 let metano,context,ty =List.find (function (m,_,_) -> m=goal) metasenv in
688 debug("!!!!!!!!!1: unify "^CicPp.ppterm _Rle_not_lt^" with "
689 ^ CicPp.ppterm ty ^"\n");
690 let r = PrimitiveTactics.apply_tac ~term:_Rle_not_lt ~status in
691 debug("!!!!!!!!!2\n");
694 ~continuation:(tac_zero_infeq_pos gl (-n,d))) ~status in
700 (* preuve que 0<=n*(1/d) => False ; n est negatif
703 let tac_zero_infeq_false gl (n,d) ~status:(proof,goal as status)=
704 debug("stat tac_zero_infeq_false\n");
706 let curi,metasenv,pbo,pty = proof in
707 let metano,context,ty =List.find (function (m,_,_) -> m=goal) metasenv in
709 debug("faccio fold di " ^ CicPp.ppterm
713 [_Rmult ; int_to_real n ; Cic.Appl [_Rinv ; int_to_real d]]
716 debug("apply di _Rlt_not_le a "^ CicPp.ppterm ty ^"\n");
717 (*CSC: Patch to undo the over-simplification of RewriteSimpl *)
720 (ReductionTactics.fold_tac ~also_in_hypotheses:false
725 [_Rmult ; int_to_real n ; Cic.Appl [_Rinv ; int_to_real d]]
730 (Tacticals.then_ ~start:(PrimitiveTactics.apply_tac ~term:_Rlt_not_le)
731 ~continuation:(tac_zero_inf_pos (-n,d))) ~status in
732 debug("end tac_zero_infeq_false\n");
735 Tacticals.id_tac ~status
740 (* *********** ********** ******** ??????????????? *********** **************)
742 let apply_type_tac ~cast:t ~applist:al ~status:(proof,goal) =
743 let curi,metasenv,pbo,pty = proof in
744 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
745 let fresh_meta = ProofEngineHelpers.new_meta proof in
747 ProofEngineHelpers.identity_relocation_list_for_metavariable context in
748 let metasenv' = (fresh_meta,context,t)::metasenv in
749 let proof' = curi,metasenv',pbo,pty in
751 PrimitiveTactics.apply_tac
752 (*~term:(Cic.Appl ((Cic.Cast (Cic.Meta (fresh_meta,irl),t))::al)) (* ??? *)*)
753 ~term:(Cic.Appl ((Cic.Meta (fresh_meta,irl))::al)) (* ??? *)
754 ~status:(proof',goal)
756 proof'',fresh_meta::goals
763 let my_cut ~term:c ~status:(proof,goal)=
764 let curi,metasenv,pbo,pty = proof in
765 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
767 debug("my_cut di "^CicPp.ppterm c^"\n");
770 let fresh_meta = ProofEngineHelpers.new_meta proof in
772 ProofEngineHelpers.identity_relocation_list_for_metavariable context in
773 let metasenv' = (fresh_meta,context,c)::metasenv in
774 let proof' = curi,metasenv',pbo,pty in
776 apply_type_tac ~cast:(Cic.Prod(Cic.Name "Anonymous",c,
777 CicSubstitution.lift 1 ty)) ~applist:[Cic.Meta(fresh_meta,irl)]
778 ~status:(proof',goal)
780 (* We permute the generated goals to be consistent with Coq *)
783 | he::tl -> proof'',he::fresh_meta::tl
787 let exact = PrimitiveTactics.exact_tac;;
789 let tac_use h ~status:(proof,goal as status) =
790 debug("Inizio TC_USE\n");
791 let curi,metasenv,pbo,pty = proof in
792 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
793 debug ("hname = "^ CicPp.ppterm h.hname ^"\n");
794 debug ("ty = "^ CicPp.ppterm ty^"\n");
798 "Rlt" -> exact ~term:h.hname ~status
799 |"Rle" -> exact ~term:h.hname ~status
800 |"Rgt" -> (Tacticals.then_ ~start:(PrimitiveTactics.apply_tac
801 ~term:_Rfourier_gt_to_lt)
802 ~continuation:(exact ~term:h.hname)) ~status
803 |"Rge" -> (Tacticals.then_ ~start:(PrimitiveTactics.apply_tac
804 ~term:_Rfourier_ge_to_le)
805 ~continuation:(exact ~term:h.hname)) ~status
806 |"eqTLR" -> (Tacticals.then_ ~start:(PrimitiveTactics.apply_tac
807 ~term:_Rfourier_eqLR_to_le)
808 ~continuation:(exact ~term:h.hname)) ~status
809 |"eqTRL" -> (Tacticals.then_ ~start:(PrimitiveTactics.apply_tac
810 ~term:_Rfourier_eqRL_to_le)
811 ~continuation:(exact ~term:h.hname)) ~status
814 debug("Fine TAC_USE\n");
822 Cic.Appl ( Cic.Const(u,boh)::next) ->
823 (match (UriManager.string_of_uri u) with
824 "cic:/Coq/Reals/Rdefinitions/Rlt.con" -> true
825 |"cic:/Coq/Reals/Rdefinitions/Rgt.con" -> true
826 |"cic:/Coq/Reals/Rdefinitions/Rle.con" -> true
827 |"cic:/Coq/Reals/Rdefinitions/Rge.con" -> true
828 |"cic:/Coq/Init/Logic_Type/eqT.con" ->
829 (match (List.hd next) with
830 Cic.Const (uri,_) when
831 UriManager.string_of_uri uri =
832 "cic:/Coq/Reals/Rdefinitions/R.con" -> true
838 let list_of_sign s = List.map (fun (x,_,z)->(x,z)) s;;
841 Cic.Appl(Array.to_list a)
844 (* Résolution d'inéquations linéaires dans R *)
845 let rec strip_outer_cast c = match c with
846 | Cic.Cast (c,_) -> strip_outer_cast c
850 let find_in_context id context =
851 let rec find_in_context_aux c n =
853 [] -> failwith (id^" not found in context")
854 | a::next -> (match a with
855 Some (Cic.Name(name),_) when name = id -> n
856 (*? magari al posto di _ qualcosaltro?*)
857 | _ -> find_in_context_aux next (n+1))
859 find_in_context_aux context 1
862 (* mi sembra quadratico *)
863 let rec filter_real_hyp context cont =
866 | Some(Cic.Name(h),Cic.Decl(t))::next -> (
867 let n = find_in_context h cont in
868 [(Cic.Rel(n),t)] @ filter_real_hyp next cont)
869 | a::next -> debug(" no\n"); filter_real_hyp next cont
872 (* lifts everithing at the conclusion level *)
873 let rec superlift c n=
876 | Some(name,Cic.Decl(a))::next -> [Some(name,Cic.Decl(
877 CicSubstitution.lift n a))] @ superlift next (n+1)
878 | Some(name,Cic.Def(a))::next -> [Some(name,Cic.Def(
879 CicSubstitution.lift n a))] @ superlift next (n+1)
880 | _::next -> superlift next (n+1) (*?? ??*)
884 let equality_replace a b ~status =
885 debug("inizio EQ\n");
886 let module C = Cic in
887 let proof,goal = status in
888 let curi,metasenv,pbo,pty = proof in
889 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
890 let a_eq_b = C.Appl [ _eqT ; _R ; a ; b ] in
891 let fresh_meta = ProofEngineHelpers.new_meta proof in
893 ProofEngineHelpers.identity_relocation_list_for_metavariable context in
894 let metasenv' = (fresh_meta,context,a_eq_b)::metasenv in
895 debug("chamo rewrite tac su"^CicPp.ppterm (C.Meta (fresh_meta,irl)));
897 rewrite_simpl_tac ~term:(C.Meta (fresh_meta,irl))
898 ~status:((curi,metasenv',pbo,pty),goal)
900 let new_goals = fresh_meta::goals in
901 debug("fine EQ -> goals : "^string_of_int( List.length new_goals) ^" = "
902 ^string_of_int( List.length goals)^"+ meta\n");
906 let tcl_fail a ~status:(proof,goal) =
908 1 -> raise (ProofEngineTypes.Fail "fail-tactical")
913 let assumption_tac ~status:(proof,goal)=
914 let curi,metasenv,pbo,pty = proof in
915 let metano,context,ty = List.find (function (m,_,_) -> m=goal) metasenv in
917 let tac_list = List.map
918 ( fun x -> num := !num + 1;
920 Some(Cic.Name(nm),t) -> (nm,exact ~term:(Cic.Rel(!num)))
921 | _ -> ("fake",tcl_fail 1)
925 Tacticals.try_tactics ~tactics:tac_list ~status:(proof,goal)
928 (* !!!!! fix !!!!!!!!!! *)
929 let contradiction_tac ~status:(proof,goal)=
931 ~start:(PrimitiveTactics.intros_tac ~name:"bo?" ) (*inutile sia questo che quello prima della chiamata*)
932 ~continuation:(Tacticals.then_
933 ~start:(Ring.elim_type_tac ~term:_False)
934 ~continuation:(assumption_tac))
938 (* ********************* TATTICA ******************************** *)
940 let rec fourier ~status:(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 = List.find (function (m,_,_) -> m=s_goal)
945 debug ("invoco fourier_tac sul goal "^string_of_int(s_goal)^" e contesto :\n");
946 debug_pcontext s_context;
948 let fhyp = String.copy "new_hyp_for_fourier" in
950 (* here we need to negate the thesis, but to do this we need to apply the right
951 theoreme,so let's parse our thesis *)
953 let th_to_appl = ref _Rfourier_not_le_gt in
955 Cic.Appl ( Cic.Const(u,boh)::args) ->
956 (match UriManager.string_of_uri u with
957 "cic:/Coq/Reals/Rdefinitions/Rlt.con" -> th_to_appl :=
959 |"cic:/Coq/Reals/Rdefinitions/Rle.con" -> th_to_appl :=
961 |"cic:/Coq/Reals/Rdefinitions/Rgt.con" -> th_to_appl :=
963 |"cic:/Coq/Reals/Rdefinitions/Rge.con" -> th_to_appl :=
965 |_-> failwith "fourier can't be applyed")
966 |_-> failwith "fourier can't be applyed");
967 (* fix maybe strip_outer_cast goes here?? *)
969 (* now let's change our thesis applying the th and put it with hp *)
971 let proof,gl = Tacticals.then_
972 ~start:(PrimitiveTactics.apply_tac ~term:!th_to_appl)
973 ~continuation:(PrimitiveTactics.intros_tac ~name:fhyp)
974 ~status:(s_proof,s_goal) in
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 = List.find (function (m,_,_) -> m=goal) metasenv in
987 (* now we want to convert hp to inequations, but first we must lift
988 everyting to thesis level, so that a variable has the save Rel(n)
989 in each hp ( needed by ineq1_of_term ) *)
991 (* ? fix if None ?????*)
992 (* fix change superlift with a real name *)
994 let l_context = superlift context 1 in
995 let hyps = filter_real_hyp l_context l_context in
997 debug ("trasformo in diseq. "^ string_of_int (List.length hyps)^" ipotesi\n");
1001 (* transform hyps into inequations *)
1003 List.iter (fun h -> try (lineq:=(ineq1_of_term h)@(!lineq))
1008 debug ("applico fourier a "^ string_of_int (List.length !lineq)^
1011 let res=fourier_lineq (!lineq) in
1012 let tac=ref Tacticals.id_tac in
1014 (print_string "Tactic Fourier fails.\n";flush stdout;
1015 failwith "fourier_tac fails")
1018 match res with (*match res*)
1021 (* in lc we have the coefficient to "reduce" the system *)
1023 print_string "Fourier's method can prove the goal...\n";flush stdout;
1025 debug "I coeff di moltiplicazione rit sono: ";
1029 (fun (h,c) -> if c<>r0 then (lutil:=(h,c)::(!lutil);
1030 (* DBG *)Fourier.print_rational(c);print_string " "(* DBG *))
1032 (List.combine (!lineq) lc);
1034 print_string (" quindi lutil e' lunga "^
1035 string_of_int (List.length (!lutil))^"\n");
1037 (* on construit la combinaison linéaire des inéquation *)
1039 (match (!lutil) with (*match (!lutil) *)
1041 debug ("elem di lutil ");Fourier.print_rational c1;print_string "\n";
1043 let s=ref (h1.hstrict) in
1046 let t1 = ref (Cic.Appl [_Rmult;rational_to_real c1;h1.hleft] ) in
1047 let t2 = ref (Cic.Appl [_Rmult;rational_to_real c1;h1.hright]) in
1049 List.iter (fun (h,c) ->
1050 s:=(!s)||(h.hstrict);
1051 t1:=(Cic.Appl [_Rplus;!t1;Cic.Appl
1052 [_Rmult;rational_to_real c;h.hleft ] ]);
1053 t2:=(Cic.Appl [_Rplus;!t2;Cic.Appl
1054 [_Rmult;rational_to_real c;h.hright] ]))
1057 let ineq=Cic.Appl [(if (!s) then _Rlt else _Rle);!t1;!t2 ] in
1058 let tc=rational_to_real cres in
1061 (* ora ho i termini che descrivono i passi di fourier per risolvere il sistema *)
1063 debug "inizio a costruire tac1\n";
1064 Fourier.print_rational(c1);
1066 let tac1=ref ( fun ~status ->
1071 debug ("inizio t1 strict\n");
1072 let curi,metasenv,pbo,pty = proof in
1073 let metano,context,ty = List.find
1074 (function (m,_,_) -> m=goal) metasenv in
1075 debug ("th = "^ CicPp.ppterm _Rfourier_lt ^"\n");
1076 debug ("ty = "^ CicPp.ppterm ty^"\n");
1077 PrimitiveTactics.apply_tac ~term:_Rfourier_lt ~status)
1078 ~continuations:[tac_use h1;tac_zero_inf_pos
1079 (rational_to_fraction c1)]
1084 ~start:(PrimitiveTactics.apply_tac ~term:_Rfourier_le)
1085 ~continuations:[tac_use h1;tac_zero_inf_pos
1086 (rational_to_fraction c1)] ~status
1092 List.iter (fun (h,c) ->
1096 tac1:=(Tacticals.thens
1097 ~start:(PrimitiveTactics.apply_tac
1098 ~term:_Rfourier_lt_lt)
1099 ~continuations:[!tac1;tac_use h;tac_zero_inf_pos
1100 (rational_to_fraction c)])
1104 Fourier.print_rational(c1);
1105 tac1:=(Tacticals.thens
1108 debug("INIZIO TAC 1 2\n");
1109 let curi,metasenv,pbo,pty = proof in
1110 let metano,context,ty = List.find (function (m,_,_) -> m=goal)
1112 debug ("th = "^ CicPp.ppterm _Rfourier_lt_le ^"\n");
1113 debug ("ty = "^ CicPp.ppterm ty^"\n");
1114 PrimitiveTactics.apply_tac ~term:_Rfourier_lt_le ~status)
1115 ~continuations:[!tac1;tac_use h;tac_zero_inf_pos
1116 (rational_to_fraction c)])
1122 tac1:=(Tacticals.thens
1123 ~start:(PrimitiveTactics.apply_tac ~term:_Rfourier_le_lt)
1124 ~continuations:[!tac1;tac_use h;tac_zero_inf_pos
1125 (rational_to_fraction c)])
1129 tac1:=(Tacticals.thens
1130 ~start:(PrimitiveTactics.apply_tac ~term:_Rfourier_le_le)
1131 ~continuations:[!tac1;tac_use h;tac_zero_inf_pos
1132 (rational_to_fraction c)])
1136 s:=(!s)||(h.hstrict)) lutil;(*end List.iter*)
1140 tac_zero_inf_false goal (rational_to_fraction cres)
1142 tac_zero_infeq_false goal (rational_to_fraction cres)
1144 tac:=(Tacticals.thens
1145 ~start:(my_cut ~term:ineq)
1146 ~continuations:[(*Tacticals.id_tac;Tacticals.id_tac*)(**)Tacticals.then_
1147 ~start:(fun ~status:(proof,goal as status) ->
1148 let curi,metasenv,pbo,pty = proof in
1149 let metano,context,ty = List.find (function (m,_,_) -> m=goal)
1151 PrimitiveTactics.change_tac ~what:ty
1152 ~with_what:(Cic.Appl [ _not; ineq]) ~status)
1153 ~continuation:(Tacticals.then_
1154 ~start:(PrimitiveTactics.apply_tac ~term:
1155 (if sres then _Rnot_lt_lt else _Rnot_le_le))
1156 ~continuation:(Tacticals.thens
1159 debug("t1 ="^CicPp.ppterm !t1 ^"t2 ="^CicPp.ppterm !t2 ^"tc="^ CicPp.ppterm tc^"\n");
1160 let r = equality_replace (Cic.Appl [_Rminus;!t2;!t1] ) tc
1163 (match r with (p,gl) ->
1164 debug("eq1 ritorna "^string_of_int(List.length gl)^"\n" ));
1166 ~continuations:[(Tacticals.thens
1169 let r = equality_replace (Cic.Appl[_Rinv;_R1]) _R1 ~status in
1170 (match r with (p,gl) ->
1171 debug("eq2 ritorna "^string_of_int(List.length gl)^"\n" ));
1174 [PrimitiveTactics.apply_tac ~term:_Rinv_R1
1175 ;Tacticals.try_tactics
1176 ~tactics:[ "ring", (fun ~status ->
1177 debug("begin RING\n");
1178 let r = Ring.ring_tac ~status in
1179 debug ("end RING\n");
1181 ; "id", Tacticals.id_tac]
1183 ;(*Tacticals.id_tac*)
1187 fun ~status:(proof,goal as status) ->
1188 let curi,metasenv,pbo,pty = proof in
1189 let metano,context,ty = List.find (function (m,_,_) -> m=
1191 (* check if ty is of type *)
1193 debug("qui c'e' gia' l'or "^CicPp.ppterm ty^"\n");
1195 (* Fix: aspetta mail di Claudio per capire cosa comporta anonimous*)
1196 Cic.Prod (Cic.Anonymous,a,b) -> (Cic.Appl [_not;a])
1199 let r = PrimitiveTactics.change_tac ~what:ty ~with_what:w1 ~status in
1200 debug("fine MY_CHNGE\n");
1204 ~continuation:(*PORTINGTacticals.id_tac*)tac2]))
1205 ;(*Tacticals.id_tac*)!tac1]);(*end tac:=*)
1206 (*tac:=(Tacticals.thens
1207 ~start:(PrimitiveTactics.cut_tac ~term:_False)
1208 ~continuations:[Tacticals.then_
1209 ~start:(PrimitiveTactics.intros_tac ~name:"??")
1210 ~continuation:contradiction_tac
1211 ;!tac]) FIXED - this was useless*)
1215 |_-> assert false)(*match (!lutil) *)
1216 |_-> assert false); (*match res*)
1217 debug ("finalmente applico tac\n");
1219 let r = !tac ~status:(proof,goal) in
1220 debug("\n\n]]]]]]]]]]]]]]]]]) That's all folks ([[[[[[[[[[[[[[[[[[[\n\n");r
1225 let fourier_tac ~status:(proof,goal) = fourier ~status:(proof,goal);;