summaryrefslogtreecommitdiff
path: root/plugins/funind/invfun.ml
blob: d10924f8860862ba3c90043a6ec6f6ed47f4a78a (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)
open Tacexpr
open Declarations
open Errors
open Util
open Names
open Term
open Vars
open Pp
open Globnames
open Tacticals
open Tactics
open Indfun_common
open Tacmach
open Misctypes

(* Some pretty printing function for debugging purpose *)

let pr_binding prc  =
  function
    | loc, NamedHyp id, c -> hov 1 (Ppconstr.pr_id id ++ str " := " ++ Pp.cut () ++ prc c)
    | loc, AnonHyp n, c -> hov 1 (int n ++ str " := " ++ Pp.cut () ++ prc c)

let pr_bindings prc prlc = function
  | ImplicitBindings l ->
      brk (1,1) ++ str "with" ++ brk (1,1) ++
      pr_sequence prc l
  | ExplicitBindings l ->
      brk (1,1) ++ str "with" ++ brk (1,1) ++
        pr_sequence (fun b -> str"(" ++ pr_binding prlc b ++ str")") l
  | NoBindings -> mt ()


let pr_with_bindings prc prlc (c,bl) =
  prc c ++ hv 0 (pr_bindings prc prlc bl)



let pr_constr_with_binding prc (c,bl) :  Pp.std_ppcmds =
  pr_with_bindings prc prc  (c,bl)

(* The local debugging mechanism *)
(* let msgnl = Pp.msgnl *)

let observe strm =
  if do_observe ()
  then Pp.msg_debug strm
  else ()

(*let observennl strm =
  if do_observe ()
  then begin Pp.msg strm;Pp.pp_flush () end
  else ()*)


let do_observe_tac s tac g =
  let goal =
    try Printer.pr_goal g
    with e when Errors.noncritical e -> assert false
  in
  try
    let v = tac g in
    msgnl (goal ++ fnl () ++ s ++(str " ")++(str "finished")); v
  with reraise ->
    let reraise = Errors.push reraise in
    let e = Cerrors.process_vernac_interp_error reraise in
    observe (str "observation "++ s++str " raised exception " ++
	     Errors.iprint e ++ str " on goal " ++ goal );
    iraise reraise;;


let observe_tac_strm s tac g =
  if do_observe ()
  then do_observe_tac s tac g
  else tac g

let observe_tac s tac g =
  if do_observe ()
  then do_observe_tac (str s) tac g
  else tac g

(* [nf_zeta] $\zeta$-normalization of a term *)
let nf_zeta =
  Reductionops.clos_norm_flags  (Closure.RedFlags.mkflags [Closure.RedFlags.fZETA])
    Environ.empty_env
    Evd.empty


(* (\* [id_to_constr id] finds the term associated to [id] in the global environment *\) *)
(* let id_to_constr id = *)
(*   try *)
(*     Constrintern.global_reference id *)
(*   with Not_found -> *)
(*     raise (UserError ("",str "Cannot find " ++ Ppconstr.pr_id id)) *)


let make_eq () =
  try
    Universes.constr_of_global (Coqlib.build_coq_eq ())
  with _ -> assert false 
let make_eq_refl () =
  try
    Universes.constr_of_global (Coqlib.build_coq_eq_refl ())
  with _ -> assert false

	  
(* [generate_type g_to_f f graph i] build the completeness (resp. correctness) lemma type if [g_to_f = true]
   (resp. g_to_f = false) where [graph]  is the graph of [f] and is the [i]th function in the block.

   [generate_type true f i] returns
   \[\forall (x_1:t_1)\ldots(x_n:t_n), let fv := f x_1\ldots x_n in, forall res,
   graph\ x_1\ldots x_n\ res \rightarrow res = fv \] decomposed as the context and the conclusion

   [generate_type false f i] returns
   \[\forall (x_1:t_1)\ldots(x_n:t_n), let fv := f x_1\ldots x_n in, forall res,
   res = fv \rightarrow graph\ x_1\ldots x_n\ res\] decomposed as the context and the conclusion
 *)

let generate_type evd g_to_f f graph i =
  (*i we deduce the number of arguments of the function and its returned type from the graph i*)
  let evd',graph =
    Evd.fresh_global  (Global.env ()) !evd  (Globnames.IndRef (fst (destInd graph)))
  in
  let evd',graph_arity = Typing.e_type_of (Global.env ()) evd' graph in
  evd:=evd';
  let ctxt,_ = decompose_prod_assum graph_arity in
  let fun_ctxt,res_type =
    match ctxt with
      | [] | [_] -> anomaly (Pp.str "Not a valid context")
      | (_,_,res_type)::fun_ctxt -> fun_ctxt,res_type
  in
  let rec args_from_decl i accu = function
  | [] -> accu
  | (_, Some _, _) :: l ->
    args_from_decl (succ i) accu l
  | _ :: l ->
    let t = mkRel i in
    args_from_decl (succ i) (t :: accu) l
  in
  (*i We need to name the vars [res] and [fv] i*)
  let filter = function (Name id,_,_) -> Some id | (Anonymous,_,_) -> None in
  let named_ctxt = List.map_filter filter fun_ctxt in
  let res_id = Namegen.next_ident_away_in_goal (Id.of_string "_res") named_ctxt in
  let fv_id = Namegen.next_ident_away_in_goal (Id.of_string "fv") (res_id :: named_ctxt) in
  (*i we can then type the argument to be applied to the function [f] i*)
  let args_as_rels = Array.of_list (args_from_decl 1 [] fun_ctxt) in
  (*i
    the hypothesis [res = fv] can then be computed
    We will need to lift it by one in order to use it as a conclusion
    i*)
  let make_eq = make_eq ()
  in
  let res_eq_f_of_args =
    mkApp(make_eq ,[|lift 2 res_type;mkRel 1;mkRel 2|])
  in
  (*i
    The hypothesis [graph\ x_1\ldots x_n\ res] can then be computed
    We will need to lift it by one in order to use it as a conclusion
    i*)
  let args_and_res_as_rels = Array.of_list (args_from_decl 3 [] fun_ctxt) in
  let args_and_res_as_rels = Array.append args_and_res_as_rels [|mkRel 1|] in
  let graph_applied = mkApp(graph, args_and_res_as_rels) in
  (*i The [pre_context]  is the defined to be the context corresponding to
    \[\forall (x_1:t_1)\ldots(x_n:t_n), let fv := f x_1\ldots x_n in, forall res,  \]
    i*)
  let pre_ctxt =
    (Name res_id,None,lift 1 res_type)::(Name fv_id,Some (mkApp(f,args_as_rels)),res_type)::fun_ctxt
  in
  (*i and we can return the solution depending on which lemma type we are defining i*)
  if g_to_f
  then (Anonymous,None,graph_applied)::pre_ctxt,(lift 1 res_eq_f_of_args),graph
  else (Anonymous,None,res_eq_f_of_args)::pre_ctxt,(lift 1 graph_applied),graph


(*
   [find_induction_principle f] searches and returns the [body] and the [type] of [f_rect]

   WARNING: while convertible, [type_of body] and [type] can be non equal
*)
let find_induction_principle evd f =
  let f_as_constant,u =  match kind_of_term f with
    | Const c' -> c'
    | _ -> error "Must be used with a function"
  in
  let infos = find_Function_infos f_as_constant in
  match infos.rect_lemma with
    | None -> raise Not_found
    | Some rect_lemma ->
       let evd',rect_lemma = Evd.fresh_global  (Global.env ()) !evd  (Globnames.ConstRef rect_lemma) in
       let evd',typ = Typing.e_type_of ~refresh:true (Global.env ()) evd' rect_lemma in
       evd:=evd';
       rect_lemma,typ


let rec generate_fresh_id x avoid i =
  if i == 0
  then []
  else
    let id = Namegen.next_ident_away_in_goal x avoid in
    id::(generate_fresh_id x (id::avoid) (pred i))


(* [prove_fun_correct functional_induction funs_constr graphs_constr schemes lemmas_types_infos i ]
   is the tactic used to prove correctness lemma.

   [functional_induction] is the tactic defined in [indfun] (dependency problem)
   [funs_constr], [graphs_constr] [schemes] [lemmas_types_infos] are the mutually recursive functions
   (resp. graphs of the functions and principles and correctness lemma types) to prove correct.

   [i] is the indice of the function to prove correct

   The lemma to prove if suppose to have been generated by [generate_type] (in $\zeta$ normal form that is
   it looks like~:
   [\forall (x_1:t_1)\ldots(x_n:t_n), forall res,
   res = f x_1\ldots x_n in, \rightarrow graph\ x_1\ldots x_n\ res]


   The sketch of the proof is the following one~:
   \begin{enumerate}
   \item intros until $x_n$
   \item $functional\ induction\ (f.(i)\ x_1\ldots x_n)$ using schemes.(i)
   \item for each generated branch intro [res] and [hres :res = f x_1\ldots x_n], rewrite [hres] and the
   apply the corresponding constructor of the corresponding graph inductive.
   \end{enumerate}

*)
let prove_fun_correct evd functional_induction funs_constr graphs_constr schemes lemmas_types_infos i : tactic =
  fun g ->
    (* first of all we recreate the lemmas types to be used as predicates of the induction principle
       that is~:
       \[fun (x_1:t_1)\ldots(x_n:t_n)=> fun  fv => fun res => res = fv \rightarrow graph\ x_1\ldots x_n\ res\]
    *)
    (* we the get the definition of the graphs block *)
    let graph_ind,u = destInd graphs_constr.(i) in
    let kn = fst graph_ind in
    let mib,_ = Global.lookup_inductive graph_ind in
    (* and the principle to use in this lemma in $\zeta$ normal form *)
    let f_principle,princ_type = schemes.(i) in
    let princ_type =  nf_zeta princ_type in
    let princ_infos = Tactics.compute_elim_sig princ_type in
    (* The number of args of the function is then easily computable *)
    let nb_fun_args = nb_prod (pf_concl g)  - 2  in
    let args_names = generate_fresh_id (Id.of_string "x") [] nb_fun_args in
    let ids = args_names@(pf_ids_of_hyps g) in
    (* Since we cannot ensure that the functional principle is defined in the
       environment and due to the bug #1174, we will need to pose the principle
       using a name
    *)
    let principle_id = Namegen.next_ident_away_in_goal (Id.of_string "princ") ids in
    let ids = principle_id :: ids in
    (* We get the branches of the principle *)
    let branches = List.rev princ_infos.branches in
    (* and built the intro pattern for each of them *)
    let intro_pats =
      List.map
	(fun (_,_,br_type) ->
	   List.map
	     (fun id -> Loc.ghost, IntroNaming (IntroIdentifier id))
	     (generate_fresh_id (Id.of_string "y") ids (List.length (fst (decompose_prod_assum br_type))))
	)
	branches
    in
    (* before building the full intro pattern for the principle *)
    let eq_ind = make_eq () in
    let eq_construct = mkConstructUi (destInd eq_ind, 1) in
    (* The next to referencies will be used to find out which constructor to apply in each branch *)
    let ind_number = ref 0
    and min_constr_number = ref 0 in
    (* The tactic to prove the ith branch of the principle *)
    let prove_branche i g =
      (* We get the identifiers of this branch *)
      let pre_args =
      	List.fold_right
      	  (fun (_,pat) acc ->
      	     match pat with
	       | IntroNaming (IntroIdentifier id) -> id::acc
      	       | _ -> anomaly (Pp.str "Not an identifier")
      	  )
      	  (List.nth intro_pats (pred i))
      	  []
      in
      (* and get the real args of the branch by unfolding the defined constant *)
      (*
	 We can then recompute the arguments of the constructor.
	 For each [hid] introduced by this branch, if [hid] has type
	 $forall res, res=fv -> graph.(j)\ x_1\ x_n res$ the corresponding arguments of the constructor are
	 [ fv (hid fv (refl_equal fv)) ].
	 If [hid] has another type the corresponding argument of the constructor is [hid]
      *)
      let constructor_args g =
	List.fold_right 
	  (fun hid acc ->
	     let type_of_hid = pf_type_of g (mkVar hid) in
	     match kind_of_term type_of_hid with
	       | Prod(_,_,t') ->
		   begin
		     match kind_of_term t' with
		       | Prod(_,t'',t''') ->
			   begin
			     match kind_of_term t'',kind_of_term t''' with
			       | App(eq,args), App(graph',_)
				   when
				     (eq_constr eq eq_ind) &&
				       Array.exists  (Constr.eq_constr_nounivs graph') graphs_constr ->
				   (args.(2)::(mkApp(mkVar hid,[|args.(2);(mkApp(eq_construct,[|args.(0);args.(2)|]))|]))
				    ::acc)
			       | _ -> mkVar hid ::  acc
			   end
		       | _ -> mkVar hid :: acc
		   end
	       | _ -> mkVar hid :: acc
	  ) pre_args []
      in
      (* in fact we must also add the parameters to the constructor args *)
      let constructor_args g =
	let params_id = fst (List.chop princ_infos.nparams args_names) in
	(List.map mkVar params_id)@((constructor_args g))
      in
      (* We then get the constructor corresponding to this branch and
	 modifies the references has needed i.e.
	 if the constructor is the last one of the current inductive then
	 add one the number of the inductive to take and add the number of constructor of the previous
	 graph to the minimal constructor number
      *)
      let constructor =
	let constructor_num = i - !min_constr_number in
	let length = Array.length (mib.Declarations.mind_packets.(!ind_number).Declarations.mind_consnames) in
	if constructor_num <= length
	then
	  begin
	    (kn,!ind_number),constructor_num
	  end
	else
	  begin
	    incr ind_number;
	    min_constr_number := !min_constr_number + length ;
	    (kn,!ind_number),1
	  end
      in
      (* we can then build the final proof term *)
      let app_constructor g = applist((mkConstructU(constructor,u)),constructor_args g) in
      (* an apply the tactic *)
      let res,hres =
	match generate_fresh_id (Id.of_string "z") (ids(* @this_branche_ids *)) 2 with
	  | [res;hres] -> res,hres
	  | _ -> assert false
      in
      (* observe (str "constructor := " ++ Printer.pr_lconstr_env (pf_env g) app_constructor); *)
      (
	tclTHENSEQ
	  [
	    observe_tac("h_intro_patterns ")  (let l = (List.nth intro_pats (pred i)) in 
					       match l with 
						 | [] -> tclIDTAC 
						 | _ -> Proofview.V82.of_tactic (intro_patterns l));
	    (* unfolding of all the defined variables introduced by this branch *)
	    (* observe_tac "unfolding" pre_tac; *)
	    (* $zeta$ normalizing of the conclusion *)
	    reduce
	      (Genredexpr.Cbv
		 { Redops.all_flags with
		     Genredexpr.rDelta = false ;
		     Genredexpr.rConst = []
		 }
	      )
	      Locusops.onConcl;
	    observe_tac ("toto ") tclIDTAC;
    
	    (* introducing the the result of the graph and the equality hypothesis *)
	    observe_tac "introducing" (tclMAP (fun x -> Proofview.V82.of_tactic (Simple.intro x)) [res;hres]);
	    (* replacing [res] with its value *)
	    observe_tac "rewriting res value" (Proofview.V82.of_tactic (Equality.rewriteLR (mkVar hres)));
	    (* Conclusion *)
	    observe_tac "exact" (fun g ->
				 Proofview.V82.of_tactic (exact_check (app_constructor g)) g)  
	  ]
      )
	g
    in
    (* end of branche proof *)
    let lemmas =
      Array.map
	(fun ((_,(ctxt,concl))) ->
	   match ctxt with
	     | [] | [_] | [_;_] -> anomaly (Pp.str "bad context")
	     | hres::res::(x,_,t)::ctxt ->
		let res = Termops.it_mkLambda_or_LetIn
			    (Termops.it_mkProd_or_LetIn concl [hres;res])
			    ((x,None,t)::ctxt)
		in
		res
	)
	lemmas_types_infos
    in
    let param_names = fst (List.chop princ_infos.nparams args_names) in
    let params = List.map mkVar param_names in
    let lemmas = Array.to_list (Array.map (fun c -> applist(c,params)) lemmas) in
    (* The bindings of the principle
       that is the params of the principle and the different lemma types
    *)
    let bindings =
      let params_bindings,avoid =
	List.fold_left2
	  (fun (bindings,avoid) (x,_,_) p ->
	     let id = Namegen.next_ident_away (Nameops.out_name x) avoid in
	     p::bindings,id::avoid
	  )
	  ([],pf_ids_of_hyps g)
	  princ_infos.params
	  (List.rev params)
      in
      let lemmas_bindings =
	List.rev (fst  (List.fold_left2
	  (fun (bindings,avoid) (x,_,_) p ->
	     let id = Namegen.next_ident_away (Nameops.out_name x) avoid in
	     (nf_zeta p)::bindings,id::avoid)
	  ([],avoid)
	  princ_infos.predicates
	  (lemmas)))
      in
      (params_bindings@lemmas_bindings)
    in
    tclTHENSEQ
      [ 
	observe_tac "principle" (Proofview.V82.of_tactic (assert_by
	  (Name principle_id)
	  princ_type
	  (exact_check f_principle)));
	observe_tac "intro args_names" (tclMAP (fun id -> Proofview.V82.of_tactic (Simple.intro id)) args_names);
	(* observe_tac "titi" (pose_proof (Name (Id.of_string "__")) (Reductionops.nf_beta Evd.empty  ((mkApp (mkVar principle_id,Array.of_list bindings))))); *)
	observe_tac "idtac" tclIDTAC;
	tclTHEN_i
	  (observe_tac
	     "functional_induction" (
	       (fun gl ->
		let term = mkApp (mkVar principle_id,Array.of_list bindings) in
		let gl', _ty = pf_eapply (Typing.e_type_of ~refresh:true)  gl term in
		Proofview.V82.of_tactic (apply term) gl')
	   ))
	  (fun i g -> observe_tac ("proving branche "^string_of_int i) (prove_branche i) g )
      ]
      g




(* [generalize_dependent_of x hyp g]
   generalize every hypothesis which depends of [x] but [hyp]
*)
let generalize_dependent_of x hyp g =
  tclMAP
    (function
       | (id,None,t) when not (Id.equal id hyp) &&
	   (Termops.occur_var (pf_env g) x t) -> tclTHEN (Tactics.Simple.generalize [mkVar id]) (thin [id])
       | _ -> tclIDTAC
    )
    (pf_hyps g)
    g


(* [intros_with_rewrite] do the intros in each branch and treat each new hypothesis
       (unfolding, substituting, destructing cases \ldots)
 *)
let  rec intros_with_rewrite g =
  observe_tac "intros_with_rewrite" intros_with_rewrite_aux g
and intros_with_rewrite_aux : tactic =
  fun g ->
    let eq_ind = make_eq () in
    match kind_of_term (pf_concl g) with
	  | Prod(_,t,t') ->
	      begin
		match kind_of_term t with
		  | App(eq,args) when (eq_constr eq eq_ind)  ->
 		      if Reductionops.is_conv (pf_env g) (project g) args.(1) args.(2)
		      then
			let id = pf_get_new_id (Id.of_string "y") g  in
			tclTHENSEQ [ Proofview.V82.of_tactic (Simple.intro id); thin [id]; intros_with_rewrite ] g
		      else if isVar args.(1) && (Environ.evaluable_named (destVar args.(1)) (pf_env g)) 
		      then tclTHENSEQ[
			unfold_in_concl [(Locus.AllOccurrences, Names.EvalVarRef (destVar args.(1)))];
			tclMAP (fun id -> tclTRY(unfold_in_hyp [(Locus.AllOccurrences, Names.EvalVarRef (destVar args.(1)))] ((destVar args.(1)),Locus.InHyp) ))
			  (pf_ids_of_hyps g);
			intros_with_rewrite
		      ] g
		      else if isVar args.(2) && (Environ.evaluable_named (destVar args.(2)) (pf_env g)) 
		      then tclTHENSEQ[
			unfold_in_concl [(Locus.AllOccurrences, Names.EvalVarRef (destVar args.(2)))];
			tclMAP (fun id -> tclTRY(unfold_in_hyp [(Locus.AllOccurrences, Names.EvalVarRef (destVar args.(2)))] ((destVar args.(2)),Locus.InHyp) ))
			  (pf_ids_of_hyps g);
			intros_with_rewrite
		      ] g
		      else if isVar args.(1)
		      then
			let id = pf_get_new_id (Id.of_string "y") g  in
			tclTHENSEQ [ Proofview.V82.of_tactic (Simple.intro id);
				     generalize_dependent_of (destVar args.(1)) id;
				     tclTRY (Proofview.V82.of_tactic (Equality.rewriteLR (mkVar id)));
				     intros_with_rewrite
				   ]
			  g
		      else if isVar args.(2) 
		      then 
			let id = pf_get_new_id (Id.of_string "y") g  in
			tclTHENSEQ [ Proofview.V82.of_tactic (Simple.intro id);
				     generalize_dependent_of (destVar args.(2)) id;
				     tclTRY (Proofview.V82.of_tactic (Equality.rewriteRL (mkVar id)));
				     intros_with_rewrite
				   ]
			  g
		      else
			begin
			  let id = pf_get_new_id (Id.of_string "y") g  in
			  tclTHENSEQ[
			    Proofview.V82.of_tactic (Simple.intro id);
			    tclTRY (Proofview.V82.of_tactic (Equality.rewriteLR (mkVar id)));
			    intros_with_rewrite
			  ] g
			end
		  | Ind _ when eq_constr t (Coqlib.build_coq_False ()) ->
		      Proofview.V82.of_tactic Tauto.tauto g
		  | Case(_,_,v,_) ->
		      tclTHENSEQ[
			Proofview.V82.of_tactic (simplest_case v);
			intros_with_rewrite
		      ] g
		  | LetIn _ ->
		      tclTHENSEQ[
			reduce
			  (Genredexpr.Cbv
			     {Redops.all_flags
			      with Genredexpr.rDelta = false;
			     })
			  Locusops.onConcl
			;
			intros_with_rewrite
		      ] g
		  | _ ->
		      let id = pf_get_new_id (Id.of_string "y") g  in
		      tclTHENSEQ [ Proofview.V82.of_tactic (Simple.intro id);intros_with_rewrite] g
	      end
	  | LetIn _ ->
	      tclTHENSEQ[
		reduce
		  (Genredexpr.Cbv
		     {Redops.all_flags
		      with Genredexpr.rDelta = false;
		     })
		  Locusops.onConcl
		;
		intros_with_rewrite
	      ] g
	  | _ -> tclIDTAC g

let rec reflexivity_with_destruct_cases g =
  let destruct_case () =
    try
      match kind_of_term (snd (destApp (pf_concl g))).(2) with
	| Case(_,_,v,_) ->
	    tclTHENSEQ[
	      Proofview.V82.of_tactic (simplest_case v);
	      Proofview.V82.of_tactic intros;
	      observe_tac "reflexivity_with_destruct_cases" reflexivity_with_destruct_cases
	    ]
        | _ -> Proofview.V82.of_tactic reflexivity
    with e when Errors.noncritical e -> Proofview.V82.of_tactic reflexivity
  in
  let eq_ind = make_eq () in
  let discr_inject =
    Tacticals.onAllHypsAndConcl (
       fun sc g ->
	 match sc with
	     None -> tclIDTAC g
	   | Some id ->
	       match kind_of_term  (pf_type_of g (mkVar id)) with
		 | App(eq,[|_;t1;t2|]) when eq_constr eq eq_ind ->
		     if Equality.discriminable (pf_env g) (project g) t1 t2
		     then Proofview.V82.of_tactic (Equality.discrHyp id) g
		     else if Equality.injectable (pf_env g) (project g) t1 t2
		     then tclTHENSEQ [Proofview.V82.of_tactic (Equality.injHyp None id);thin [id];intros_with_rewrite]  g
		     else tclIDTAC g
		 | _ -> tclIDTAC g
    )
  in
  (tclFIRST
    [ observe_tac "reflexivity_with_destruct_cases : reflexivity" (Proofview.V82.of_tactic reflexivity);
      observe_tac "reflexivity_with_destruct_cases : destruct_case" ((destruct_case ()));
      (*  We reach this point ONLY if
	  the same value is matched (at least) two times
	  along binding path.
	  In this case, either we have a discriminable hypothesis and we are done,
	  either at least an injectable one and we do the injection before continuing
      *)
      observe_tac "reflexivity_with_destruct_cases : others" (tclTHEN (tclPROGRESS discr_inject ) reflexivity_with_destruct_cases)
    ])
    g


(* [prove_fun_complete funs graphs schemes lemmas_types_infos i]
   is the tactic used to prove completness lemma.

   [funcs], [graphs] [schemes] [lemmas_types_infos] are the mutually recursive functions
   (resp. definitions of the graphs of the functions, principles and correctness lemma types) to prove correct.

   [i] is the indice of the function to prove complete

   The lemma to prove if suppose to have been generated by [generate_type] (in $\zeta$ normal form that is
   it looks like~:
   [\forall (x_1:t_1)\ldots(x_n:t_n), forall res,
   graph\ x_1\ldots x_n\ res, \rightarrow  res = f x_1\ldots x_n in]


   The sketch of the proof is the following one~:
   \begin{enumerate}
   \item intros until $H:graph\ x_1\ldots x_n\ res$
   \item $elim\ H$ using schemes.(i)
   \item for each generated branch, intro  the news hyptohesis, for each such hyptohesis [h], if [h] has
   type [x=?] with [x] a variable, then subst [x],
   if [h] has type [t=?] with [t] not a variable then rewrite [t] in the subterms, else
   if [h] is a match then destruct it, else do just introduce it,
   after all intros, the conclusion should be a reflexive equality.
   \end{enumerate}

*)


let prove_fun_complete funcs graphs schemes lemmas_types_infos i : tactic =
  fun g ->
    (* We compute the types of the different mutually recursive lemmas
       in $\zeta$ normal form
    *)
    let lemmas =
      Array.map
	(fun (_,(ctxt,concl)) -> nf_zeta (Termops.it_mkLambda_or_LetIn concl ctxt))
	lemmas_types_infos
    in
    (* We get the constant and the principle corresponding to this lemma *)
    let f = funcs.(i) in
    let graph_principle = nf_zeta schemes.(i)  in
    let princ_type = pf_type_of g graph_principle in
    let princ_infos = Tactics.compute_elim_sig princ_type in
    (* Then we get the number of argument of the function
       and compute a fresh name for each of them
    *)
    let nb_fun_args = nb_prod (pf_concl g)  - 2  in
    let args_names = generate_fresh_id (Id.of_string "x") [] nb_fun_args in
    let ids = args_names@(pf_ids_of_hyps g) in
    (* and fresh names for res H and the principle (cf bug bug #1174) *)
    let res,hres,graph_principle_id =
      match generate_fresh_id (Id.of_string "z") ids 3 with
	| [res;hres;graph_principle_id] -> res,hres,graph_principle_id
	| _ -> assert false
    in
    let ids = res::hres::graph_principle_id::ids in
    (* we also compute fresh names for each hyptohesis of each branch
       of the principle *)
    let branches = List.rev princ_infos.branches in
    let intro_pats =
      List.map
	(fun (_,_,br_type) ->
	   List.map
	     (fun id -> id)
	     (generate_fresh_id (Id.of_string "y") ids (nb_prod br_type))
	)
	branches
    in
    (* We will need to change the function by its body
       using [f_equation] if it is recursive (that is the graph is infinite
       or unfold if the graph is finite
    *)
    let rewrite_tac j ids : tactic =
      let graph_def = graphs.(j) in
      let infos =
        try find_Function_infos (fst (destConst funcs.(j)))
        with Not_found ->  error "No graph found"
      in
      if infos.is_general
        || Rtree.is_infinite Declareops.eq_recarg graph_def.mind_recargs
      then
	let eq_lemma =
	  try Option.get (infos).equation_lemma
	  with Option.IsNone -> anomaly (Pp.str "Cannot find equation lemma")
	in
	tclTHENSEQ[
	  tclMAP (fun id -> Proofview.V82.of_tactic (Simple.intro id)) ids;
	  Proofview.V82.of_tactic (Equality.rewriteLR (mkConst eq_lemma));
	  (* Don't forget to $\zeta$ normlize the term since the principles
             have been $\zeta$-normalized *)
	  reduce
	    (Genredexpr.Cbv
	       {Redops.all_flags
		with Genredexpr.rDelta = false;
	       })
	    Locusops.onConcl
	  ;
	  Simple.generalize (List.map mkVar ids);
	  thin ids
	]
      else
        unfold_in_concl [(Locus.AllOccurrences, Names.EvalConstRef (fst (destConst f)))]
    in
    (* The proof of each branche itself *)
    let ind_number = ref 0 in
    let min_constr_number = ref 0 in
    let prove_branche i g =
      (* we fist compute the inductive corresponding to the branch *)
      let this_ind_number =
	let constructor_num = i - !min_constr_number in
	let length = Array.length (graphs.(!ind_number).Declarations.mind_consnames) in
	if constructor_num <= length
	then !ind_number
	else
	  begin
	    incr ind_number;
	    min_constr_number := !min_constr_number + length;
	    !ind_number
	  end
      in
      let this_branche_ids = List.nth intro_pats (pred i) in
      tclTHENSEQ[
	(* we expand the definition of the function *)
        observe_tac "rewrite_tac" (rewrite_tac this_ind_number this_branche_ids);
	(* introduce hypothesis with some rewrite *)
        observe_tac "intros_with_rewrite (all)" intros_with_rewrite;
	(* The proof is (almost) complete *)
        observe_tac "reflexivity" (reflexivity_with_destruct_cases)
      ]
	g
    in
    let params_names = fst (List.chop princ_infos.nparams args_names) in
    let params = List.map mkVar params_names in
    tclTHENSEQ
      [ tclMAP (fun id -> Proofview.V82.of_tactic (Simple.intro id)) (args_names@[res;hres]);
	observe_tac "h_generalize"
	(Simple.generalize [mkApp(applist(graph_principle,params),Array.map (fun c -> applist(c,params)) lemmas)]);
	Proofview.V82.of_tactic (Simple.intro graph_principle_id);
	observe_tac "" (tclTHEN_i
	  (observe_tac "elim" (Proofview.V82.of_tactic (elim false None (mkVar hres,NoBindings) (Some (mkVar graph_principle_id,NoBindings)))))
	  (fun i g -> observe_tac "prove_branche" (prove_branche i) g ))
      ]
      g


(* [derive_correctness make_scheme functional_induction funs graphs] create correctness and completeness
   lemmas for each function in [funs] w.r.t. [graphs]

   [make_scheme] is Functional_principle_types.make_scheme (dependency pb) and
   [functional_induction] is Indfun.functional_induction (same pb)
*)

let derive_correctness make_scheme functional_induction (funs: pconstant list) (graphs:inductive list) =
  assert (funs <> []);
  assert (graphs <> []);
  let funs = Array.of_list funs and graphs = Array.of_list graphs in
  let funs_constr = Array.map mkConstU funs  in
  States.with_state_protection_on_exception
    (fun () ->
     let evd = ref Evd.empty in 
     let graphs_constr = Array.map mkInd graphs in
     let lemmas_types_infos =
       Util.Array.map2_i
	 (fun i f_constr graph ->
	 (* let const_of_f,u = destConst f_constr in *)
	 let (type_of_lemma_ctxt,type_of_lemma_concl,graph) =
	   generate_type evd false f_constr graph i
	 in
	 let type_info = (type_of_lemma_ctxt,type_of_lemma_concl) in
	 graphs_constr.(i) <- graph;
	 let type_of_lemma = Termops.it_mkProd_or_LetIn type_of_lemma_concl type_of_lemma_ctxt in
	 let _ = evd := fst (Typing.e_type_of (Global.env ()) !evd type_of_lemma) in 
	   let type_of_lemma = nf_zeta type_of_lemma in
	   observe (str "type_of_lemma := " ++ Printer.pr_lconstr_env (Global.env ()) !evd type_of_lemma);
	   type_of_lemma,type_info
	)
	funs_constr
	graphs_constr
    in
    let schemes =
      (* The functional induction schemes are computed and not saved if there is more that one function
	 if the block contains only one function we can safely reuse [f_rect]
       *)
      try
	if not (Int.equal (Array.length funs_constr) 1) then raise Not_found;
	[| find_induction_principle evd funs_constr.(0) |]
      with Not_found ->
	(
	
	  Array.of_list
	    (List.map
	       (fun entry ->
		  (fst (fst(Future.force entry.Entries.const_entry_body)), Option.get entry.Entries.const_entry_type )
	       )
	       (make_scheme evd (Array.map_to_list (fun const -> const,GType []) funs))
	    )
	)
    in
    let proving_tac =
      prove_fun_correct !evd functional_induction funs_constr graphs_constr schemes lemmas_types_infos
    in
    Array.iteri
      (fun i f_as_constant ->
	 let f_id = Label.to_id (con_label (fst f_as_constant)) in
	 (*i The next call to mk_correct_id is valid since we are constructing the lemma
	     Ensures by: obvious
	 i*)
	 let lem_id = mk_correct_id f_id in
	 let (typ,_) = lemmas_types_infos.(i) in 
	 Lemmas.start_proof
	   lem_id
	   (Decl_kinds.Global,Flags.is_universe_polymorphism (),((Decl_kinds.Proof Decl_kinds.Theorem)))
           !evd
	   typ
           (Lemmas.mk_hook (fun _ _ -> ()));
	 ignore (Pfedit.by
		   (Proofview.V82.tactic (observe_tac ("prove correctness ("^(Id.to_string f_id)^")")
						      (proving_tac i))));
	 (Lemmas.save_proof (Vernacexpr.(Proved(Transparent,None))));
	 let finfo = find_Function_infos (fst f_as_constant) in
	 (* let lem_cst = fst (destConst (Constrintern.global_reference lem_id)) in *)
	 let _,lem_cst_constr = Evd.fresh_global
				  (Global.env ()) !evd (Constrintern.locate_reference (Libnames.qualid_of_ident lem_id)) in
	 let (lem_cst,_) = destConst lem_cst_constr in
	 update_Function {finfo with correctness_lemma = Some lem_cst};

      )
      funs;
    (* let evd = ref Evd.empty in  *)
    let lemmas_types_infos =
      Util.Array.map2_i
	(fun i f_constr graph ->
	 let (type_of_lemma_ctxt,type_of_lemma_concl,graph)   =
	   generate_type evd true f_constr graph i
	 in
	 let type_info = (type_of_lemma_ctxt,type_of_lemma_concl) in
	 graphs_constr.(i) <- graph;
	 let type_of_lemma =
	   Termops.it_mkProd_or_LetIn type_of_lemma_concl type_of_lemma_ctxt
	 in
	 let type_of_lemma = nf_zeta type_of_lemma in
	 observe (str "type_of_lemma := " ++ Printer.pr_lconstr type_of_lemma);
	 type_of_lemma,type_info
	)
	funs_constr
	graphs_constr
    in

    let (kn,_) as graph_ind,u  = (destInd graphs_constr.(0)) in
    let mib,mip = Global.lookup_inductive graph_ind in
    let sigma, scheme = 
	(Indrec.build_mutual_induction_scheme (Global.env ()) !evd
	   (Array.to_list
	      (Array.mapi
		 (fun i _ -> ((kn,i),u(* Univ.Instance.empty *)),true,InType)
		 mib.Declarations.mind_packets
	      )
	   )
	)
    in
    let schemes =
      Array.of_list scheme
    in
    let proving_tac =
      prove_fun_complete funs_constr mib.Declarations.mind_packets schemes lemmas_types_infos
    in
    Array.iteri
      (fun i f_as_constant ->
	 let f_id = Label.to_id (con_label (fst f_as_constant)) in
	 (*i The next call to mk_complete_id is valid since we are constructing the lemma
	     Ensures by: obvious
	   i*)
	 let lem_id = mk_complete_id f_id in
	 Lemmas.start_proof lem_id
	   (Decl_kinds.Global,Flags.is_universe_polymorphism (),(Decl_kinds.Proof Decl_kinds.Theorem))  !evd
	 (fst lemmas_types_infos.(i))
           (Lemmas.mk_hook (fun _ _ -> ()));
	 ignore (Pfedit.by
	   (Proofview.V82.tactic (observe_tac ("prove completeness ("^(Id.to_string f_id)^")")
	      (proving_tac i)))) ;
	 (Lemmas.save_proof (Vernacexpr.(Proved(Transparent,None))));
	 let finfo = find_Function_infos (fst f_as_constant) in
	 let _,lem_cst_constr = Evd.fresh_global
				  (Global.env ()) !evd (Constrintern.locate_reference (Libnames.qualid_of_ident lem_id)) in
	 let (lem_cst,_) = destConst lem_cst_constr in
	 update_Function {finfo with completeness_lemma = Some lem_cst}
      )
      funs)
    ()

(***********************************************)

(* [revert_graph kn post_tac hid] transforme an hypothesis [hid] having type Ind(kn,num) t1 ... tn res
   when [kn] denotes a graph block  into
   f_num t1... tn = res (by applying [f_complete] to the first type) before apply post_tac on the result

   if the type of hypothesis has not this form or if we cannot find the completeness lemma then we do nothing
*)
let revert_graph kn post_tac hid g =
    let typ = pf_type_of g (mkVar hid) in
    match kind_of_term typ with
      | App(i,args) when isInd i ->
	  let ((kn',num) as ind'),u = destInd i in
	  if MutInd.equal kn kn'
	  then (* We have generated a graph hypothesis so that we must change it if we can *)
	    let info =
	      try find_Function_of_graph ind'
	      with Not_found -> (* The graphs are mutually recursive but we cannot find one of them !*)
		anomaly (Pp.str "Cannot retrieve infos about a mutual block")
	    in
	    (* if we can find a completeness lemma for this function
	       then we can come back to the functional form. If not, we do nothing
	    *)
	    match info.completeness_lemma with
	      | None -> tclIDTAC g
	      | Some f_complete ->
		  let f_args,res = Array.chop (Array.length args - 1) args in
		  tclTHENSEQ
		    [
		      Simple.generalize [applist(mkConst f_complete,(Array.to_list f_args)@[res.(0);mkVar hid])];
		      thin [hid];
		      Proofview.V82.of_tactic (Simple.intro hid);
		      post_tac hid
		    ]
		    g

	  else tclIDTAC g
      | _ -> tclIDTAC g


(*
   [functional_inversion hid fconst f_correct ] is the functional version of [inversion]

   [hid] is the hypothesis to invert, [fconst] is the function to invert and  [f_correct]
   is the correctness lemma for [fconst].

   The sketch is the follwing~:
   \begin{enumerate}
   \item Transforms the hypothesis [hid] such that its type is now $res\ =\ f\ t_1 \ldots t_n$
   (fails if it is not possible)
   \item replace [hid] with $R\_f t_1 \ldots t_n res$ using [f_correct]
   \item apply [inversion] on [hid]
   \item finally in each branch, replace each  hypothesis [R\_f ..]  by [f ...] using [f_complete] (whenever
   such a lemma exists)
   \end{enumerate}
*)

let functional_inversion kn hid fconst f_correct : tactic =
  fun g ->
    let old_ids = List.fold_right Id.Set.add  (pf_ids_of_hyps g) Id.Set.empty in
    let type_of_h = pf_type_of g (mkVar hid) in
    match kind_of_term type_of_h with
      | App(eq,args) when eq_constr eq (make_eq ())  ->
	  let pre_tac,f_args,res =
	    match kind_of_term args.(1),kind_of_term args.(2) with
	      | App(f,f_args),_ when eq_constr f fconst ->
		  ((fun hid -> Proofview.V82.of_tactic (intros_symmetry (Locusops.onHyp hid))),f_args,args.(2))
	      |_,App(f,f_args) when eq_constr f fconst ->
		 ((fun hid -> tclIDTAC),f_args,args.(1))
	      | _ -> (fun hid -> tclFAIL 1 (mt ())),[||],args.(2)
	  in
	  tclTHENSEQ[
	    pre_tac hid;
	    Simple.generalize [applist(f_correct,(Array.to_list f_args)@[res;mkVar hid])];
	    thin [hid];
	    Proofview.V82.of_tactic (Simple.intro hid);
	    Proofview.V82.of_tactic (Inv.inv FullInversion None (NamedHyp hid));
	    (fun g ->
	       let new_ids = List.filter (fun id -> not (Id.Set.mem id old_ids)) (pf_ids_of_hyps g) in
	       tclMAP (revert_graph kn pre_tac)  (hid::new_ids)  g
	    );
	  ] g
      | _ -> tclFAIL 1 (mt ()) g



let invfun qhyp f  =
  let f =
    match f with
      | ConstRef f -> f
      | _ -> raise (Errors.UserError("",str "Not a function"))
  in
  try
    let finfos = find_Function_infos f in
    let f_correct = mkConst(Option.get finfos.correctness_lemma)
    and kn = fst finfos.graph_ind
    in
    Proofview.V82.of_tactic (
      Tactics.try_intros_until (fun hid -> Proofview.V82.tactic (functional_inversion kn hid (mkConst f)  f_correct)) qhyp
    )
  with
    | Not_found ->  error "No graph found"
    | Option.IsNone  -> error "Cannot use equivalence with graph!"


let invfun qhyp f g =
  match f with
    | Some f -> invfun qhyp f g
    | None ->
       Proofview.V82.of_tactic begin
	Tactics.try_intros_until
	  (fun hid -> Proofview.V82.tactic begin fun g ->
	     let hyp_typ = pf_type_of g (mkVar hid)  in
	     match kind_of_term hyp_typ with
	       | App(eq,args) when eq_constr eq (make_eq ()) ->
		   begin
		     let f1,_ = decompose_app args.(1) in
		     try
		       if not (isConst f1) then failwith "";
		       let finfos = find_Function_infos (fst (destConst f1)) in
		       let f_correct = mkConst(Option.get finfos.correctness_lemma)
		       and kn = fst finfos.graph_ind
		       in
		       functional_inversion kn hid f1 f_correct g
		     with | Failure "" | Option.IsNone | Not_found ->
		       try
			 let f2,_ = decompose_app args.(2) in
			 if not (isConst f2) then failwith "";
			 let finfos = find_Function_infos (fst (destConst f2)) in
			 let f_correct = mkConst(Option.get finfos.correctness_lemma)
			 and kn = fst finfos.graph_ind
			 in
			 functional_inversion kn hid  f2 f_correct g
		       with
			 | Failure "" ->
			     errorlabstrm "" (str "Hypothesis " ++ Ppconstr.pr_id hid ++ str " must contain at least one Function")
			 | Option.IsNone  ->
			     if do_observe ()
			     then
			       error "Cannot use equivalence with graph for any side of the equality"
			     else errorlabstrm "" (str "Cannot find inversion information for hypothesis " ++ Ppconstr.pr_id hid)
			 | Not_found ->
			     if do_observe ()
			     then
			       error "No graph found for any side of equality"
			     else errorlabstrm "" (str "Cannot find inversion information for hypothesis " ++ Ppconstr.pr_id hid)
		   end
	       | _ -> errorlabstrm "" (Ppconstr.pr_id hid ++ str " must be an equality ")
	  end)
	  qhyp
          end
	  g