aboutsummaryrefslogtreecommitdiffhomepage
path: root/vernac/auto_ind_decl.ml
blob: ec6b62ee274a3391bee642490e79a87ff4825191 (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
(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(* This file is about the automatic generation of schemes about
   decidable equality, created by Vincent Siles, Oct 2007 *)

open CErrors
open Util
open Pp
open Term
open Constr
open Vars
open Termops
open Declarations
open Names
open Globnames
open Inductiveops
open Tactics
open Ind_tables
open Misctypes
open Proofview.Notations

module RelDecl = Context.Rel.Declaration

(**********************************************************************)
(* Generic synthesis of boolean equality *)

let quick_chop n l =
  let rec kick_last = function
    | t::[] -> []
    | t::q -> t::(kick_last q)
    | [] -> failwith "kick_last"
and aux = function
    | (0,l') -> l'
    | (n,h::t) -> aux (n-1,t)
    | _ -> failwith "quick_chop"
  in
  if n > (List.length l) then failwith "quick_chop args"
  else kick_last (aux (n,l) )

let deconstruct_type t =
  let l,r = decompose_prod t in
  (List.rev_map snd l)@[r]

exception EqNotFound of inductive * inductive
exception EqUnknown of string
exception UndefinedCst of string
exception InductiveWithProduct
exception InductiveWithSort
exception ParameterWithoutEquality of global_reference
exception NonSingletonProp of inductive
exception DecidabilityMutualNotSupported
exception NoDecidabilityCoInductive

let constr_of_global g = lazy (Universes.constr_of_global g)

(* Some pre declaration of constant we are going to use *)
let bb = constr_of_global Coqlib.glob_bool

let andb_prop = fun _ -> Universes.constr_of_global (Coqlib.build_bool_type()).Coqlib.andb_prop

let andb_true_intro = fun _ ->
  Universes.constr_of_global
    (Coqlib.build_bool_type()).Coqlib.andb_true_intro

let tt = constr_of_global Coqlib.glob_true

let ff = constr_of_global Coqlib.glob_false

let eq = constr_of_global Coqlib.glob_eq

let sumbool () = Universes.constr_of_global (Coqlib.build_coq_sumbool ())

let andb = fun _ -> Universes.constr_of_global (Coqlib.build_bool_type()).Coqlib.andb

let induct_on c = induction false None c None None

let destruct_on c = destruct false None c None None

let destruct_on_using c id =
  destruct false None c
    (Some (Loc.tag @@ IntroOrPattern [[Loc.tag @@ IntroNaming IntroAnonymous];
               [Loc.tag @@ IntroNaming (IntroIdentifier id)]]))
    None

let destruct_on_as c l =
  destruct false None c (Some (Loc.tag l)) None

let inj_flags = Some {
    Equality.keep_proof_equalities = true; (* necessary *)
    injection_in_context = true; (* does not matter here *)
    Equality.injection_pattern_l2r_order = true; (* does not matter here *)
  }

let my_discr_tac = Equality.discr_tac false None
let my_inj_tac x = Equality.inj inj_flags None false None (EConstr.mkVar x,NoBindings)

(* reconstruct the inductive with the correct de Bruijn indexes *)
let mkFullInd (ind,u) n =
  let mib = Global.lookup_mind (fst ind) in
  let nparams = mib.mind_nparams in
  let nparrec = mib.mind_nparams_rec in
  (* params context divided *)
  let lnonparrec,lnamesparrec =
    context_chop (nparams-nparrec) mib.mind_params_ctxt in
  if nparrec > 0
    then mkApp (mkIndU (ind,u),
      Array.of_list(Context.Rel.to_extended_list mkRel (nparrec+n) lnamesparrec))
    else mkIndU (ind,u)

let check_bool_is_defined () =
  try let _ = Global.type_of_global_in_context (Global.env ()) Coqlib.glob_bool in ()
  with e when CErrors.noncritical e -> raise (UndefinedCst "bool")

let beq_scheme_kind_aux = ref (fun _ -> failwith "Undefined")

let build_beq_scheme mode kn =
  check_bool_is_defined ();
  (* fetching global env *)
  let env = Global.env() in
  (* fetching the mutual inductive body *)
  let mib = Global.lookup_mind kn in
  (* number of inductives in the mutual *)
  let nb_ind = Array.length mib.mind_packets in
  (* number of params in the type *)
  let nparams = mib.mind_nparams in
  let nparrec = mib.mind_nparams_rec in
  (* params context divided *)
  let lnonparrec,lnamesparrec =
    context_chop (nparams-nparrec) mib.mind_params_ctxt in
  (* predef coq's boolean type *)
  (* rec name *)
  let rec_name i =(Id.to_string (Array.get mib.mind_packets i).mind_typename)^
                    "_eqrec"
  in
  (* construct the "fun A B ... N, eqA eqB eqC ... N => fixpoint" part *)
  let create_input c =
    let myArrow u v = mkArrow u (lift 1 v)
    and eqName = function
        | Name s -> Id.of_string ("eq_"^(Id.to_string s))
        | Anonymous -> Id.of_string "eq_A"
    in
    let ext_rel_list = Context.Rel.to_extended_list mkRel 0 lnamesparrec in
      let lift_cnt = ref 0 in
      let eqs_typ = List.map (fun aa ->
                                let a = lift !lift_cnt aa in
                                  incr lift_cnt;
                                  myArrow a (myArrow a (Lazy.force bb))
                             ) ext_rel_list in

        let eq_input = List.fold_left2
          ( fun a b decl -> (* mkLambda(n,b,a) ) *)
                (* here I leave the Naming thingy so that the type of
                  the function is more readable for the user *)
                mkNamedLambda (eqName (RelDecl.get_name decl)) b a )
                c (List.rev eqs_typ) lnamesparrec
       in
        List.fold_left (fun a decl ->(* mkLambda(n,t,a)) eq_input rel_list *)
          (* Same here , hoping the auto renaming will do something good ;)  *)
          mkNamedLambda
                (match RelDecl.get_name decl with Name s -> s | Anonymous ->  Id.of_string "A")
                (RelDecl.get_type decl)  a) eq_input lnamesparrec
 in
 let make_one_eq cur =
  let u = Univ.Instance.empty in
  let ind = (kn,cur),u (* FIXME *) in
  (* current inductive we are working on *)
  let cur_packet = mib.mind_packets.(snd (fst ind)) in
  (* Inductive toto : [rettyp] := *)
  let rettyp = Inductive.type_of_inductive env ((mib,cur_packet),u) in
  (* split rettyp in a list without the non rec params and the last ->
  e.g. Inductive vec (A:Set) : nat -> Set := ... will do [nat] *)
  let rettyp_l = quick_chop nparrec (deconstruct_type rettyp) in
    (* give a type A, this function tries to find the equality on A declared
       previously *)
    (*  nlist = the number of args (A , B , ... )
        eqA   = the de Bruijn index of the first eq param
        ndx   = how much to translate due to the 2nd Case
    *)
    let compute_A_equality rel_list nlist eqA ndx t =
      let lifti = ndx in
      let sigma = Evd.empty (** FIXME *) in
      let rec aux c =
	let (c,a) = Reductionops.whd_betaiota_stack Evd.empty c in
	match EConstr.kind sigma c with
        | Rel x -> mkRel (x-nlist+ndx), Safe_typing.empty_private_constants
        | Var x ->
          let eid = Id.of_string ("eq_"^(Id.to_string x)) in
          let () =
            try ignore (Environ.lookup_named eid env)
            with Not_found -> raise (ParameterWithoutEquality (VarRef x))
          in
          mkVar eid, Safe_typing.empty_private_constants
        | Cast (x,_,_) -> aux (EConstr.applist (x,a))
        | App _ -> assert false
        | Ind ((kn',i as ind'),u) (*FIXME: universes *) -> 
            if MutInd.equal kn kn' then mkRel(eqA-nlist-i+nb_ind-1), Safe_typing.empty_private_constants
            else begin
              try
                let eq, eff =
                  let c, eff = find_scheme ~mode (!beq_scheme_kind_aux()) (kn',i) in
                  mkConst c, eff in
                let eqa, eff =
                  let eqa, effs = List.split (List.map aux a) in
                  Array.of_list eqa,
                  List.fold_left Safe_typing.concat_private eff (List.rev effs)
                  in
                let args =
                  Array.append 
                    (Array.of_list (List.map (fun x -> lift lifti (EConstr.Unsafe.to_constr x)) a)) eqa in
                if Int.equal (Array.length args) 0 then eq, eff
                else mkApp (eq, args), eff
              with Not_found -> raise(EqNotFound (ind', fst ind))
            end
        | Sort _  -> raise InductiveWithSort
        | Prod _ -> raise InductiveWithProduct
        | Lambda _-> raise (EqUnknown "abstraction")
        | LetIn _ -> raise (EqUnknown "let-in")
        | Const (kn, u) ->
            let u = EConstr.EInstance.kind sigma u in
	    (match Environ.constant_opt_value_in env (kn, u) with
	      | None -> raise (ParameterWithoutEquality (ConstRef kn))
	      | Some c -> aux (EConstr.applist (EConstr.of_constr c,a)))
        | Proj _ -> raise (EqUnknown "projection")
        | Construct _ -> raise (EqUnknown "constructor")
        | Case _ -> raise (EqUnknown "match")
        | CoFix _ -> raise (EqUnknown "cofix")
        | Fix _   -> raise (EqUnknown "fix")
        | Meta _  -> raise (EqUnknown "meta-variable")
        | Evar _  -> raise (EqUnknown "existential variable")
    in
      aux t
  in
  (* construct the predicate for the Case part*)
  let do_predicate rel_list n =
     List.fold_left (fun a b -> mkLambda(Anonymous,b,a))
      (mkLambda (Anonymous,
                 mkFullInd ind (n+3+(List.length rettyp_l)+nb_ind-1),
                 (Lazy.force bb)))
      (List.rev rettyp_l) in
  (* make_one_eq *)
  (* do the [| C1 ... =>  match Y with ... end
               ...
               Cn => match Y with ... end |]  part *)
    let ci = make_case_info env (fst ind) MatchStyle in
    let constrs n = get_constructors env (make_ind_family (ind,
      Context.Rel.to_extended_list mkRel (n+nb_ind-1) mib.mind_params_ctxt)) in
    let constrsi = constrs (3+nparrec) in
    let n = Array.length constrsi in
    let ar = Array.make n (Lazy.force ff) in
    let eff = ref Safe_typing.empty_private_constants in
	for i=0 to n-1 do
	  let nb_cstr_args = List.length constrsi.(i).cs_args in
	  let ar2 = Array.make n (Lazy.force ff) in
          let constrsj = constrs (3+nparrec+nb_cstr_args) in
	    for j=0 to n-1 do
	      if Int.equal i j then
		ar2.(j) <- let cc = (match nb_cstr_args with
                    | 0 -> Lazy.force tt
                    | _ -> let eqs = Array.make nb_cstr_args (Lazy.force tt) in
                      for ndx = 0 to nb_cstr_args-1 do
                        let cc = RelDecl.get_type (List.nth constrsi.(i).cs_args ndx) in
                          let eqA, eff' = compute_A_equality rel_list
                                          nparrec
                                          (nparrec+3+2*nb_cstr_args)
                                          (nb_cstr_args+ndx+1)
                                          (EConstr.of_constr cc)
                          in
                          eff := Safe_typing.concat_private eff' !eff;
                          Array.set eqs ndx
                              (mkApp (eqA,
                                [|mkRel (ndx+1+nb_cstr_args);mkRel (ndx+1)|]
                              ))
                      done;
                      Array.fold_left
                          (fun a b -> mkApp (andb(),[|b;a|]))
                          (eqs.(0))
                          (Array.sub eqs 1 (nb_cstr_args - 1))
                  )
   		  in
		    (List.fold_left (fun a decl -> mkLambda (RelDecl.get_name decl, RelDecl.get_type decl, a)) cc
                    (constrsj.(j).cs_args)
		)
	      else ar2.(j) <- (List.fold_left (fun a decl ->
			mkLambda (RelDecl.get_name decl, RelDecl.get_type decl, a)) (Lazy.force ff) (constrsj.(j).cs_args) )
	    done;

	  ar.(i) <- (List.fold_left (fun a decl -> mkLambda (RelDecl.get_name decl, RelDecl.get_type decl, a))
			(mkCase (ci,do_predicate rel_list nb_cstr_args,
				  mkVar (Id.of_string "Y") ,ar2))
			 (constrsi.(i).cs_args))
	done;
        mkNamedLambda (Id.of_string "X") (mkFullInd ind (nb_ind-1+1))  (
          mkNamedLambda (Id.of_string "Y") (mkFullInd ind (nb_ind-1+2))  (
 	    mkCase (ci, do_predicate rel_list 0,mkVar (Id.of_string "X"),ar))),
        !eff
    in (* build_beq_scheme *)
    let names = Array.make nb_ind Anonymous and
        types = Array.make nb_ind mkSet and
        cores = Array.make nb_ind mkSet in
    let eff = ref Safe_typing.empty_private_constants in
    let u = Univ.Instance.empty in
    for i=0 to (nb_ind-1) do
        names.(i) <- Name (Id.of_string (rec_name i));
	types.(i) <- mkArrow (mkFullInd ((kn,i),u) 0)
                     (mkArrow (mkFullInd ((kn,i),u) 1) (Lazy.force bb));
        let c, eff' = make_one_eq i in
        cores.(i) <- c;
        eff := Safe_typing.concat_private eff' !eff
    done;
      (Array.init nb_ind (fun i ->
      let kelim = Inductive.elim_sorts (mib,mib.mind_packets.(i)) in
	if not (Sorts.List.mem InSet kelim) then
	  raise (NonSingletonProp (kn,i));
        if mib.mind_finite = CoFinite then
	  raise NoDecidabilityCoInductive;
        let fix = mkFix (((Array.make nb_ind 0),i),(names,types,cores)) in
        create_input fix),
       Evd.make_evar_universe_context (Global.env ()) None),
      !eff

let beq_scheme_kind = declare_mutual_scheme_object "_beq" build_beq_scheme

let _ = beq_scheme_kind_aux := fun () -> beq_scheme_kind

(* This function tryies to get the [inductive] between a constr
  the constr should be Ind i or App(Ind i,[|args|])
*)
let destruct_ind sigma c =
  let open EConstr in
    try let u,v =  destApp sigma c in
          let indc = destInd sigma u in
            indc,v
    with DestKO -> let indc = destInd sigma c in
            indc,[||]

(*
  In the following, avoid is the list of names to avoid.
  If the args of the Inductive type are A1 ... An
  then avoid should be
 [| lb_An ... lb _A1  (resp. bl_An ... bl_A1)
    eq_An .... eq_A1 An ... A1 |]
so from Ai we can find the the correct eq_Ai bl_ai or lb_ai
*)
(* used in the leib -> bool side*)
let do_replace_lb mode lb_scheme_key aavoid narg p q =
  let open EConstr in
  let avoid = Array.of_list aavoid in
  let do_arg sigma v offset =
  try
    let x = narg*offset in
    let s = destVar sigma v in
    let n = Array.length avoid in
    let rec find i =
      if Id.equal avoid.(n-i) s then avoid.(n-i-x)
      else (if i<n then find (i+1)
            else user_err ~hdr:"AutoIndDecl.do_replace_lb"
                   (str "Var " ++ Id.print s ++ str " seems unknown.")
      )
    in mkVar (find 1)
  with e when CErrors.noncritical e ->
      (* if this happen then the args have to be already declared as a
              Parameter*)
      (
        let mp,dir,lbl = Constant.repr3 (fst (destConst sigma v)) in
          mkConst (Constant.make3 mp dir (Label.make (
          if Int.equal offset 1 then ("eq_"^(Label.to_string lbl))
                       else ((Label.to_string lbl)^"_lb")
        )))
      )
  in
  Proofview.Goal.enter begin fun gl ->
    let type_of_pq = Tacmach.New.pf_unsafe_type_of gl p in
    let sigma = Tacmach.New.project gl in
    let env = Tacmach.New.pf_env gl in
    let u,v = destruct_ind sigma type_of_pq
    in let lb_type_of_p =
        try
          let c, eff = find_scheme ~mode lb_scheme_key (fst u) (*FIXME*) in
          Proofview.tclUNIT (mkConst c, eff)
        with Not_found ->
          (* spiwack: the format of this error message should probably
	              be improved. *)
          let err_msg =
	    (str "Leibniz->boolean:" ++
             str "You have to declare the" ++
	     str "decidability over " ++
             Printer.pr_econstr_env env sigma type_of_pq ++
	     str " first.")
          in
          Tacticals.New.tclZEROMSG err_msg
       in
       lb_type_of_p >>= fun (lb_type_of_p,eff) ->
       Proofview.tclEVARMAP >>= fun sigma ->
       let lb_args = Array.append (Array.append
                          (Array.map (fun x -> x) v)
                          (Array.map (fun x -> do_arg sigma x 1) v))
                          (Array.map (fun x -> do_arg sigma x 2) v)
        in let app =  if Array.is_empty lb_args
                       then lb_type_of_p else mkApp (lb_type_of_p,lb_args)
           in
           Tacticals.New.tclTHENLIST [
             Proofview.tclEFFECTS eff;
             Equality.replace p q ; apply app ; Auto.default_auto]
  end

(* used in the bool -> leib side *)
let do_replace_bl mode bl_scheme_key (ind,u as indu) aavoid narg lft rgt =
  let open EConstr in
  let avoid = Array.of_list aavoid in
  let do_arg sigma v offset =
  try
    let x = narg*offset in
    let s = destVar sigma v in
    let n = Array.length avoid in
    let rec find i =
      if Id.equal avoid.(n-i) s then avoid.(n-i-x)
      else (if i<n then find (i+1)
            else user_err ~hdr:"AutoIndDecl.do_replace_bl"
                   (str "Var " ++ Id.print s ++ str " seems unknown.")
      )
    in mkVar (find 1)
  with e when CErrors.noncritical e ->
      (* if this happen then the args have to be already declared as a
         Parameter*)
      (
        let mp,dir,lbl = Constant.repr3 (fst (destConst sigma v)) in
          mkConst (Constant.make3 mp dir (Label.make (
          if Int.equal offset 1 then ("eq_"^(Label.to_string lbl))
                       else ((Label.to_string lbl)^"_bl")
        )))
      )
  in

  let rec aux l1 l2 =
    match (l1,l2) with
    | (t1::q1,t2::q2) ->
        Proofview.Goal.enter begin fun gl ->
        let tt1 = Tacmach.New.pf_unsafe_type_of gl t1 in
        let sigma = Tacmach.New.project gl in
        let env = Tacmach.New.pf_env gl in
        if EConstr.eq_constr sigma t1 t2 then aux q1 q2
        else (
          let u,v = try  destruct_ind sigma tt1
          (* trick so that the good sequence is returned*)
                with e when CErrors.noncritical e -> indu,[||]
          in if eq_ind (fst u) ind
             then Tacticals.New.tclTHENLIST [Equality.replace t1 t2; Auto.default_auto ; aux q1 q2 ]
             else (
               let bl_t1, eff =
               try 
                 let c, eff = find_scheme bl_scheme_key (fst u) (*FIXME*) in
                 mkConst c, eff
               with Not_found ->
		 (* spiwack: the format of this error message should probably
	                     be improved. *)
		 let err_msg =
	                                (str "boolean->Leibniz:" ++
                                         str "You have to declare the" ++
			   	         str "decidability over " ++
                                         Printer.pr_econstr_env env sigma tt1 ++
				         str " first.")
		 in
		 user_err err_msg
               in let bl_args =
                        Array.append (Array.append
                          (Array.map (fun x -> x) v)
                          (Array.map (fun x -> do_arg sigma x 1) v))
                          (Array.map (fun x -> do_arg sigma x 2) v )
                in
                let app =  if Array.is_empty bl_args
                           then bl_t1 else mkApp (bl_t1,bl_args)
                in
                Tacticals.New.tclTHENLIST [
                  Proofview.tclEFFECTS eff;
                  Equality.replace_by t1 t2
                    (Tacticals.New.tclTHEN (apply app) (Auto.default_auto)) ;
                  aux q1 q2 ]
              )
        )
        end
    | ([],[]) -> Proofview.tclUNIT ()
    | _ -> Tacticals.New.tclZEROMSG (str "Both side of the equality must have the same arity.")
  in
  Proofview.tclEVARMAP >>= fun sigma ->
  begin try Proofview.tclUNIT (destApp sigma lft)
    with DestKO -> Tacticals.New.tclZEROMSG (str "replace failed.")
  end >>= fun (ind1,ca1) ->
  begin try Proofview.tclUNIT (destApp sigma rgt)
    with DestKO -> Tacticals.New.tclZEROMSG (str "replace failed.")
  end >>= fun (ind2,ca2) ->
  begin try Proofview.tclUNIT (fst (destInd sigma ind1))
    with DestKO ->
      begin try Proofview.tclUNIT (fst (fst (destConstruct sigma ind1)))
        with DestKO -> Tacticals.New.tclZEROMSG (str "The expected type is an inductive one.")
      end
  end >>= fun (sp1,i1) ->
  begin try Proofview.tclUNIT (fst (destInd sigma ind2))
    with DestKO ->
      begin try Proofview.tclUNIT (fst (fst (destConstruct sigma ind2)))
        with DestKO -> Tacticals.New.tclZEROMSG (str "The expected type is an inductive one.")
      end
  end >>= fun (sp2,i2) ->
  if not (MutInd.equal sp1 sp2) || not (Int.equal i1 i2)
  then Tacticals.New.tclZEROMSG (str "Eq should be on the same type")
  else aux (Array.to_list ca1) (Array.to_list ca2)

(*
  create, from a list of ids [i1,i2,...,in] the list
  [(in,eq_in,in_bl,in_al),,...,(i1,eq_i1,i1_bl_i1_al  )]
*)
let list_id l = List.fold_left ( fun a decl -> let s' =
      match RelDecl.get_name decl with
        Name s -> Id.to_string s
      | Anonymous -> "A" in
          (Id.of_string s',Id.of_string ("eq_"^s'),
              Id.of_string (s'^"_bl"),
              Id.of_string (s'^"_lb"))
            ::a
    ) [] l
(*
  build the right eq_I A B.. N eq_A .. eq_N
*)
let eqI ind l =
  let list_id = list_id l in
  let eA = Array.of_list((List.map (fun (s,_,_,_) -> mkVar s) list_id)@
                           (List.map (fun (_,seq,_,_)-> mkVar seq) list_id ))
  and e, eff = 
    try let c, eff = find_scheme beq_scheme_kind ind in mkConst c, eff 
    with Not_found -> user_err ~hdr:"AutoIndDecl.eqI"
      (str "The boolean equality on " ++ MutInd.print (fst ind) ++ str " is needed.");
  in (if Array.equal Constr.equal eA [||] then e else mkApp(e,eA)), eff

(**********************************************************************)
(* Boolean->Leibniz *)

open Namegen

let compute_bl_goal ind lnamesparrec nparrec =
  let eqI, eff = eqI ind lnamesparrec in
  let list_id = list_id lnamesparrec in
  let avoid = List.fold_right (Nameops.Name.fold_right (fun id l -> Id.Set.add id l)) (List.map RelDecl.get_name lnamesparrec) Id.Set.empty in
  let create_input c =
    let x = next_ident_away (Id.of_string "x") avoid and
        y = next_ident_away (Id.of_string "y") avoid in
      let bl_typ = List.map (fun (s,seq,_,_) ->
        mkNamedProd x (mkVar s) (
            mkNamedProd y (mkVar s) (
              mkArrow
               ( mkApp(Lazy.force eq,[|(Lazy.force bb);mkApp(mkVar seq,[|mkVar x;mkVar y|]);(Lazy.force tt)|]))
               ( mkApp(Lazy.force eq,[|mkVar s;mkVar x;mkVar y|]))
          ))
        ) list_id in
      let bl_input = List.fold_left2 ( fun a (s,_,sbl,_) b ->
        mkNamedProd sbl b a
      ) c (List.rev list_id) (List.rev bl_typ) in
      let eqs_typ = List.map (fun (s,_,_,_) ->
          mkProd(Anonymous,mkVar s,mkProd(Anonymous,mkVar s,(Lazy.force bb)))
          ) list_id in
      let eq_input = List.fold_left2 ( fun a (s,seq,_,_) b ->
        mkNamedProd seq b a
      ) bl_input (List.rev list_id) (List.rev eqs_typ) in
      List.fold_left (fun a decl -> mkNamedProd
                (match RelDecl.get_name decl with Name s -> s | Anonymous -> next_ident_away (Id.of_string "A") avoid)
                (RelDecl.get_type decl) a) eq_input lnamesparrec
    in
      let n = next_ident_away (Id.of_string "x") avoid and
          m = next_ident_away (Id.of_string "y") avoid in
      let u = Univ.Instance.empty in
     create_input (
        mkNamedProd n (mkFullInd (ind,u) nparrec) (
          mkNamedProd m (mkFullInd (ind,u) (nparrec+1)) (
            mkArrow
              (mkApp(Lazy.force eq,[|(Lazy.force bb);mkApp(eqI,[|mkVar n;mkVar m|]);(Lazy.force tt)|]))
              (mkApp(Lazy.force eq,[|mkFullInd (ind,u) (nparrec+3);mkVar n;mkVar m|]))
        ))), eff

let compute_bl_tact mode bl_scheme_key ind lnamesparrec nparrec =
  let list_id = list_id lnamesparrec in
  let avoid = ref [] in
      let first_intros =
        ( List.map (fun (s,_,_,_) -> s ) list_id ) @
        ( List.map (fun (_,seq,_,_ ) -> seq) list_id ) @
        ( List.map (fun (_,_,sbl,_ ) -> sbl) list_id )
      in
      let fresh_id s gl =
          let fresh = fresh_id_in_env (Id.Set.of_list !avoid) s (Proofview.Goal.env gl) in
          avoid := fresh::(!avoid); fresh
      in
      Proofview.Goal.enter begin fun gl ->
      let fresh_first_intros = List.map (fun id -> fresh_id id gl) first_intros in
      let freshn = fresh_id (Id.of_string "x") gl in
      let freshm = fresh_id (Id.of_string "y") gl in
      let freshz = fresh_id (Id.of_string "Z") gl in
  (* try with *)
      Tacticals.New.tclTHENLIST [ intros_using fresh_first_intros;
                     intro_using freshn ;
                     induct_on (EConstr.mkVar freshn);
                     intro_using freshm;
                     destruct_on (EConstr.mkVar freshm);
                     intro_using freshz;
                     intros;
                     Tacticals.New.tclTRY (
                      Tacticals.New.tclORELSE reflexivity my_discr_tac
                     );
                     simpl_in_hyp (freshz,Locus.InHyp);
(*
repeat ( apply andb_prop in z;let z1:= fresh "Z" in destruct z as [z1 z]).
*)
                    Tacticals.New.tclREPEAT (
                      Tacticals.New.tclTHENLIST [
                         Simple.apply_in freshz (EConstr.of_constr (andb_prop()));
                         Proofview.Goal.enter begin fun gl ->
                           let fresht = fresh_id (Id.of_string "Z") gl in
                            destruct_on_as (EConstr.mkVar freshz)
                                  (IntroOrPattern [[Loc.tag @@ IntroNaming (IntroIdentifier fresht);
                                    Loc.tag @@ IntroNaming (IntroIdentifier freshz)]])
                         end
                        ]);
(*
  Ci a1 ... an = Ci b1 ... bn
 replace bi with ai; auto || replace bi with ai by  apply typeofbi_prod ; auto
*)
                      Proofview.Goal.enter begin fun gl ->
                        let concl = Proofview.Goal.concl gl in
                        let sigma = Tacmach.New.project gl in
                        match EConstr.kind sigma concl with
                        | App (c,ca) -> (
                          match EConstr.kind sigma c with
                          | Ind (indeq, u) ->
                              if eq_gr (IndRef indeq) Coqlib.glob_eq
                              then
                                Tacticals.New.tclTHEN
                                  (do_replace_bl mode bl_scheme_key ind
				     (!avoid)
                                     nparrec (ca.(2))
                                     (ca.(1)))
                                  Auto.default_auto
                              else
                                Tacticals.New.tclZEROMSG (str "Failure while solving Boolean->Leibniz.")
                          | _ -> Tacticals.New.tclZEROMSG (str" Failure while solving Boolean->Leibniz.")
                        )
                        | _ -> Tacticals.New.tclZEROMSG (str "Failure while solving Boolean->Leibniz.")
                      end

                    ]
      end

let bl_scheme_kind_aux = ref (fun _ -> failwith "Undefined")

let side_effect_of_mode = function
  | Declare.UserAutomaticRequest -> false
  | Declare.InternalTacticRequest -> true
  | Declare.UserIndividualRequest -> false

let make_bl_scheme mode mind =
  let mib = Global.lookup_mind mind in
  if not (Int.equal (Array.length mib.mind_packets) 1) then
    user_err 
      (str "Automatic building of boolean->Leibniz lemmas not supported");
  let ind = (mind,0) in
  let nparams = mib.mind_nparams in
  let nparrec = mib.mind_nparams_rec in
  let lnonparrec,lnamesparrec = (* TODO subst *)
    context_chop (nparams-nparrec) mib.mind_params_ctxt in
  let bl_goal, eff = compute_bl_goal ind lnamesparrec nparrec in
  let ctx = Evd.make_evar_universe_context (Global.env ()) None in
  let side_eff = side_effect_of_mode mode in
  let bl_goal = EConstr.of_constr bl_goal in
  let (ans, _, ctx) = Pfedit.build_by_tactic ~side_eff (Global.env()) ctx bl_goal
    (compute_bl_tact mode (!bl_scheme_kind_aux()) (ind, EConstr.EInstance.empty) lnamesparrec nparrec)
  in
  ([|ans|], ctx), eff

let bl_scheme_kind = declare_mutual_scheme_object "_dec_bl" make_bl_scheme

let _ = bl_scheme_kind_aux := fun () -> bl_scheme_kind

(**********************************************************************)
(* Leibniz->Boolean *)

let compute_lb_goal ind lnamesparrec nparrec =
  let list_id = list_id lnamesparrec in
  let eq = Lazy.force eq and tt = Lazy.force tt and bb = Lazy.force bb in
  let avoid = List.fold_right (Nameops.Name.fold_right (fun id l -> Id.Set.add id l)) (List.map RelDecl.get_name lnamesparrec) Id.Set.empty in
  let eqI, eff = eqI ind lnamesparrec in
    let create_input c =
      let x = next_ident_away (Id.of_string "x") avoid and
          y = next_ident_away (Id.of_string "y") avoid in
      let lb_typ = List.map (fun (s,seq,_,_) ->
        mkNamedProd x (mkVar s) (
            mkNamedProd y (mkVar s) (
              mkArrow
               ( mkApp(eq,[|mkVar s;mkVar x;mkVar y|]))
               ( mkApp(eq,[|bb;mkApp(mkVar seq,[|mkVar x;mkVar y|]);tt|]))
          ))
        ) list_id in
      let lb_input = List.fold_left2 ( fun a (s,_,_,slb) b ->
        mkNamedProd slb b a
      ) c (List.rev list_id) (List.rev lb_typ) in
      let eqs_typ = List.map (fun (s,_,_,_) ->
          mkProd(Anonymous,mkVar s,mkProd(Anonymous,mkVar s,bb))
          ) list_id in
      let eq_input = List.fold_left2 ( fun a (s,seq,_,_) b ->
        mkNamedProd seq b a
      ) lb_input (List.rev list_id) (List.rev eqs_typ) in
      List.fold_left (fun a decl -> mkNamedProd
                (match (RelDecl.get_name decl) with Name s -> s | Anonymous ->  Id.of_string "A")
                (RelDecl.get_type decl)  a) eq_input lnamesparrec
    in
      let n = next_ident_away (Id.of_string "x") avoid and
          m = next_ident_away (Id.of_string "y") avoid in
      let u = Univ.Instance.empty in
      create_input (
        mkNamedProd n (mkFullInd (ind,u) nparrec) (
          mkNamedProd m (mkFullInd (ind,u) (nparrec+1)) (
            mkArrow
              (mkApp(eq,[|mkFullInd (ind,u) (nparrec+2);mkVar n;mkVar m|]))
              (mkApp(eq,[|bb;mkApp(eqI,[|mkVar n;mkVar m|]);tt|]))
        ))), eff

let compute_lb_tact mode lb_scheme_key ind lnamesparrec nparrec =
  let list_id = list_id lnamesparrec in
    let avoid = ref [] in
      let first_intros =
        ( List.map (fun (s,_,_,_) -> s ) list_id ) @
        ( List.map (fun (_,seq,_,_) -> seq) list_id ) @
        ( List.map (fun (_,_,_,slb) -> slb) list_id )
      in
      let fresh_id s gl =
          let fresh = fresh_id_in_env (Id.Set.of_list !avoid) s (Proofview.Goal.env gl) in
          avoid := fresh::(!avoid); fresh
      in
      Proofview.Goal.enter begin fun gl ->
      let fresh_first_intros = List.map (fun id -> fresh_id id gl) first_intros in
      let freshn = fresh_id (Id.of_string "x") gl in
      let freshm = fresh_id (Id.of_string "y") gl in
      let freshz = fresh_id (Id.of_string "Z") gl in
  (* try with *)
      Tacticals.New.tclTHENLIST [ intros_using fresh_first_intros;
                     intro_using freshn ;
                     induct_on (EConstr.mkVar freshn);
                     intro_using freshm;
                     destruct_on (EConstr.mkVar freshm);
                     intro_using freshz;
                     intros;
                     Tacticals.New.tclTRY (
                      Tacticals.New.tclORELSE reflexivity my_discr_tac
                     );
                     my_inj_tac freshz;
		     intros; simpl_in_concl;
                     Auto.default_auto;
                     Tacticals.New.tclREPEAT (
                      Tacticals.New.tclTHENLIST [apply (EConstr.of_constr (andb_true_intro()));
                                  simplest_split ;Auto.default_auto ]
                      );
                      Proofview.Goal.enter begin fun gls ->
                        let concl = Proofview.Goal.concl gls in
                        let sigma = Tacmach.New.project gl in
                        (* assume the goal to be eq (eq_type ...) = true *)
                        match EConstr.kind sigma concl with
                        | App(c,ca) -> (match (EConstr.kind sigma ca.(1)) with
                          | App(c',ca') ->
                              let n = Array.length ca' in
                              do_replace_lb mode lb_scheme_key
				(!avoid)
                                nparrec
                                ca'.(n-2) ca'.(n-1)
                          | _ ->
                              Tacticals.New.tclZEROMSG (str "Failure while solving Leibniz->Boolean.")
                        )
                        | _ ->
                            Tacticals.New.tclZEROMSG (str "Failure while solving Leibniz->Boolean.")
                      end
                    ]
      end

let lb_scheme_kind_aux = ref (fun () -> failwith "Undefined")

let make_lb_scheme mode mind =
  let mib = Global.lookup_mind mind in
  if not (Int.equal (Array.length mib.mind_packets) 1) then
    user_err 
      (str "Automatic building of Leibniz->boolean lemmas not supported");
  let ind = (mind,0) in
  let nparams = mib.mind_nparams in
  let nparrec = mib.mind_nparams_rec in
  let lnonparrec,lnamesparrec =
    context_chop (nparams-nparrec) mib.mind_params_ctxt in
  let lb_goal, eff = compute_lb_goal ind lnamesparrec nparrec in
  let ctx = Evd.make_evar_universe_context (Global.env ()) None in
  let side_eff = side_effect_of_mode mode in
  let lb_goal = EConstr.of_constr lb_goal in
  let (ans, _, ctx) = Pfedit.build_by_tactic ~side_eff (Global.env()) ctx lb_goal
    (compute_lb_tact mode (!lb_scheme_kind_aux()) ind lnamesparrec nparrec)
  in
  ([|ans|], ctx), eff

let lb_scheme_kind = declare_mutual_scheme_object "_dec_lb" make_lb_scheme

let _ = lb_scheme_kind_aux := fun () -> lb_scheme_kind

(**********************************************************************)
(* Decidable equality *)

let check_not_is_defined () =
  try ignore (Coqlib.build_coq_not ())
  with e when CErrors.noncritical e -> raise (UndefinedCst "not")

(* {n=m}+{n<>m}  part  *)
let compute_dec_goal ind lnamesparrec nparrec =
  check_not_is_defined ();
  let eq = Lazy.force eq and tt = Lazy.force tt and bb = Lazy.force bb in
  let list_id = list_id lnamesparrec in
  let avoid = List.fold_right (Nameops.Name.fold_right (fun id l -> Id.Set.add id l)) (List.map RelDecl.get_name lnamesparrec) Id.Set.empty in
    let create_input c =
      let x = next_ident_away (Id.of_string "x") avoid and
          y = next_ident_away (Id.of_string "y") avoid in
      let lb_typ = List.map (fun (s,seq,_,_) ->
        mkNamedProd x (mkVar s) (
            mkNamedProd y (mkVar s) (
              mkArrow
               ( mkApp(eq,[|mkVar s;mkVar x;mkVar y|]))
               ( mkApp(eq,[|bb;mkApp(mkVar seq,[|mkVar x;mkVar y|]);tt|]))
          ))
        ) list_id in
      let bl_typ = List.map (fun (s,seq,_,_) ->
        mkNamedProd x (mkVar s) (
            mkNamedProd y (mkVar s) (
              mkArrow
               ( mkApp(eq,[|bb;mkApp(mkVar seq,[|mkVar x;mkVar y|]);tt|]))
               ( mkApp(eq,[|mkVar s;mkVar x;mkVar y|]))
          ))
        ) list_id in

      let lb_input = List.fold_left2 ( fun a (s,_,_,slb) b ->
        mkNamedProd slb b a
      ) c (List.rev list_id) (List.rev lb_typ) in
      let bl_input = List.fold_left2 ( fun a (s,_,sbl,_) b ->
        mkNamedProd sbl b a
      ) lb_input (List.rev list_id) (List.rev bl_typ) in

      let eqs_typ = List.map (fun (s,_,_,_) ->
          mkProd(Anonymous,mkVar s,mkProd(Anonymous,mkVar s,bb))
          ) list_id in
      let eq_input = List.fold_left2 ( fun a (s,seq,_,_) b ->
        mkNamedProd seq b a
      ) bl_input (List.rev list_id) (List.rev eqs_typ) in
      List.fold_left (fun a decl -> mkNamedProd
                (match RelDecl.get_name decl with Name s -> s | Anonymous ->  Id.of_string "A")
                (RelDecl.get_type decl) a) eq_input lnamesparrec
    in
      let n = next_ident_away (Id.of_string "x") avoid and
          m = next_ident_away (Id.of_string "y") avoid in
        let eqnm = mkApp(eq,[|mkFullInd ind (2*nparrec+2);mkVar n;mkVar m|]) in
        create_input (
          mkNamedProd n (mkFullInd ind (2*nparrec)) (
            mkNamedProd m (mkFullInd ind (2*nparrec+1)) (
              mkApp(sumbool(),[|eqnm;mkApp (Universes.constr_of_global @@ Coqlib.build_coq_not(),[|eqnm|])|])
          )
        )
      )

let compute_dec_tact ind lnamesparrec nparrec =
  let eq = Lazy.force eq and tt = Lazy.force tt 
  and ff = Lazy.force ff and bb = Lazy.force bb in
  let list_id = list_id lnamesparrec in
  let eqI, eff = eqI ind lnamesparrec in
  let avoid = ref [] in
  let eqtrue x = mkApp(eq,[|bb;x;tt|]) in
  let eqfalse x = mkApp(eq,[|bb;x;ff|]) in
  let first_intros =
      ( List.map (fun (s,_,_,_) -> s ) list_id ) @
      ( List.map (fun (_,seq,_,_) -> seq) list_id ) @
      ( List.map (fun (_,_,sbl,_) -> sbl) list_id ) @
      ( List.map (fun (_,_,_,slb) -> slb) list_id )
  in
  let fresh_id s gl =
      let fresh = fresh_id_in_env (Id.Set.of_list !avoid) s (Proofview.Goal.env gl) in
      avoid := fresh::(!avoid); fresh
  in
  Proofview.Goal.enter begin fun gl ->
  let fresh_first_intros = List.map (fun id -> fresh_id id gl) first_intros in
  let freshn = fresh_id (Id.of_string "x") gl in
  let freshm = fresh_id (Id.of_string "y") gl in
  let freshH = fresh_id (Id.of_string "H") gl in
  let eqbnm = mkApp(eqI,[|mkVar freshn;mkVar freshm|]) in
  let arfresh = Array.of_list fresh_first_intros in
  let xargs = Array.sub arfresh 0 (2*nparrec) in
  begin try
          let c, eff = find_scheme bl_scheme_kind ind in
          Proofview.tclUNIT (mkConst c,eff) with
    Not_found ->
      Tacticals.New.tclZEROMSG (str "Error during the decidability part, boolean to leibniz equality is required.")
  end >>= fun (blI,eff') ->
  begin try
          let c, eff = find_scheme lb_scheme_kind ind in
          Proofview.tclUNIT (mkConst c,eff) with
    Not_found ->
      Tacticals.New.tclZEROMSG (str "Error during the decidability part, leibniz to boolean equality is required.")
  end >>= fun (lbI,eff'') ->
  let eff = (Safe_typing.concat_private eff'' (Safe_typing.concat_private eff' eff)) in
  Tacticals.New.tclTHENLIST [
        Proofview.tclEFFECTS eff;
        intros_using fresh_first_intros;
        intros_using [freshn;freshm];
	(*we do this so we don't have to prove the same goal twice *)
        assert_by (Name freshH) (EConstr.of_constr (
          mkApp(sumbool(),[|eqtrue eqbnm; eqfalse eqbnm|])
	))
	  (Tacticals.New.tclTHEN (destruct_on (EConstr.of_constr eqbnm)) Auto.default_auto);

        Proofview.Goal.enter begin fun gl ->
          let freshH2 = fresh_id (Id.of_string "H") gl in
	  Tacticals.New.tclTHENS (destruct_on_using (EConstr.mkVar freshH) freshH2) [
	    (* left *)
	    Tacticals.New.tclTHENLIST [
	      simplest_left;
              apply (EConstr.of_constr (mkApp(blI,Array.map(fun x->mkVar x) xargs)));
              Auto.default_auto
            ]
            ;

	    (*right *)
            Proofview.Goal.enter begin fun gl ->
            let freshH3 = fresh_id (Id.of_string "H") gl in
            Tacticals.New.tclTHENLIST [
	      simplest_right ;
              unfold_constr (Lazy.force Coqlib.coq_not_ref);
              intro;
              Equality.subst_all ();
              assert_by (Name freshH3)
		(EConstr.of_constr (mkApp(eq,[|bb;mkApp(eqI,[|mkVar freshm;mkVar freshm|]);tt|])))
		(Tacticals.New.tclTHENLIST [
		  apply (EConstr.of_constr (mkApp(lbI,Array.map (fun x->mkVar x) xargs)));
                  Auto.default_auto
		]);
	      Equality.general_rewrite_bindings_in true
	                      Locus.AllOccurrences true false
                              (List.hd !avoid)
                              ((EConstr.mkVar (List.hd (List.tl !avoid))),
                                NoBindings
                              )
                              true;
              my_discr_tac
	    ]
            end
	  ]
        end
  ]
  end

let make_eq_decidability mode mind =
  let mib = Global.lookup_mind mind in
  if not (Int.equal (Array.length mib.mind_packets) 1) then
    raise DecidabilityMutualNotSupported;
  let ind = (mind,0) in
  let nparams = mib.mind_nparams in
  let nparrec = mib.mind_nparams_rec in
  let u = Univ.Instance.empty in
  let ctx = Evd.make_evar_universe_context (Global.env ()) None in
  let lnonparrec,lnamesparrec =
    context_chop (nparams-nparrec) mib.mind_params_ctxt in
  let side_eff = side_effect_of_mode mode in
  let (ans, _, ctx) = Pfedit.build_by_tactic ~side_eff (Global.env()) ctx
    (EConstr.of_constr (compute_dec_goal (ind,u) lnamesparrec nparrec))
    (compute_dec_tact ind lnamesparrec nparrec)
  in
  ([|ans|], ctx), Safe_typing.empty_private_constants

let eq_dec_scheme_kind =
  declare_mutual_scheme_object "_eq_dec" make_eq_decidability

(* The eq_dec_scheme proofs depend on the equality and discr tactics
   but the inj tactics, that comes with discr, depends on the
   eq_dec_scheme... *)

let _ = Equality.set_eq_dec_scheme_kind eq_dec_scheme_kind