aboutsummaryrefslogtreecommitdiffhomepage
path: root/plugins/funind/recdef.ml
blob: f43251bc50d7187b09ceb04b7bbe52419970ef71 (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
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

open Term
open Vars
open Namegen
open Environ
open Entries
open Pp
open Names
open Libnames
open Globnames
open Nameops
open CErrors
open Util
open Tacticals
open Tacmach
open Tactics
open Nametab
open Declare
open Decl_kinds
open Tacred
open Proof_type
open Pfedit
open Glob_term
open Pretyping
open Termops
open Constrintern
open Misctypes
open Genredexpr

open Equality
open Auto
open Eauto

open Indfun_common
open Sigma.Notations
open Context.Rel.Declaration


(* Ugly things which should not be here *)

let coq_constant m s =
  Coqlib.coq_constant "RecursiveDefinition" m s

let arith_Nat = ["Arith";"PeanoNat";"Nat"]
let arith_Lt = ["Arith";"Lt"]

let coq_init_constant s =
  Coqlib.gen_constant_in_modules "RecursiveDefinition" Coqlib.init_modules s

let find_reference sl s =
  let dp = Names.DirPath.make (List.rev_map Id.of_string sl) in
  locate (make_qualid dp (Id.of_string s))

let declare_fun f_id kind ?(ctx=Univ.UContext.empty) value =
  let ce = definition_entry ~univs:ctx value (*FIXME *) in
    ConstRef(declare_constant f_id (DefinitionEntry ce, kind));;

let defined () = Lemmas.save_proof (Vernacexpr.(Proved (Transparent,None)))

let def_of_const t =
   match (kind_of_term t) with
    Const sp ->
      (try (match constant_opt_value_in (Global.env ()) sp with
             | Some c -> c
	     | _ -> raise Not_found)
       with Not_found ->
	 anomaly (str "Cannot find definition of constant " ++
		    (Id.print (Label.to_id (con_label (fst sp)))))
      )
     |_ -> assert false

let type_of_const t =
   match (kind_of_term t) with
    Const sp -> Typeops.type_of_constant (Global.env()) sp
    |_ -> assert false

let constr_of_global x = 
  fst (Universes.unsafe_constr_of_global x)

let constant sl s = constr_of_global (find_reference sl s)

let const_of_ref = function
    ConstRef kn -> kn
  | _ -> anomaly (Pp.str "ConstRef expected")


let nf_zeta env =
  Reductionops.clos_norm_flags  (CClosure.RedFlags.mkflags [CClosure.RedFlags.fZETA])
    env
    Evd.empty


let nf_betaiotazeta = (* Reductionops.local_strong Reductionops.whd_betaiotazeta  *)
  let clos_norm_flags flgs env sigma t =
    CClosure.norm_val (CClosure.create_clos_infos flgs env) (CClosure.inject (Reductionops.nf_evar sigma t)) in
  clos_norm_flags CClosure.betaiotazeta  Environ.empty_env Evd.empty






(* Generic values *)
let pf_get_new_ids idl g =
  let ids = pf_ids_of_hyps g in
  List.fold_right
    (fun id acc -> next_global_ident_away id (acc@ids)::acc)
    idl
    []

let compute_renamed_type gls c =
  rename_bound_vars_as_displayed (*no avoid*) [] (*no rels*) []
    (pf_unsafe_type_of gls c)
let h'_id = Id.of_string "h'"
let teq_id = Id.of_string "teq"
let ano_id = Id.of_string "anonymous"
let x_id = Id.of_string "x"
let k_id = Id.of_string "k"
let v_id = Id.of_string "v"
let def_id = Id.of_string "def"
let p_id = Id.of_string "p"
let rec_res_id = Id.of_string "rec_res";;
let lt = function () -> (coq_init_constant "lt")
let le = function () -> (coq_init_constant "le")
let ex = function () -> (coq_init_constant "ex")
let nat = function () -> (coq_init_constant "nat")
let iter_ref () =  
  try find_reference ["Recdef"] "iter" 
  with Not_found -> error "module Recdef not loaded"
let iter = function () -> (constr_of_global (delayed_force iter_ref))
let eq = function () -> (coq_init_constant "eq")
let le_lt_SS = function () -> (constant ["Recdef"] "le_lt_SS")
let le_lt_n_Sm = function () -> (coq_constant arith_Lt "le_lt_n_Sm")
let le_trans = function () -> (coq_constant arith_Nat "le_trans")
let le_lt_trans = function () -> (coq_constant arith_Nat "le_lt_trans")
let lt_S_n = function () -> (coq_constant arith_Lt "lt_S_n")
let le_n = function () -> (coq_init_constant "le_n")
let coq_sig_ref = function () -> (find_reference ["Coq";"Init";"Specif"] "sig")
let coq_O = function () -> (coq_init_constant "O")
let coq_S = function () -> (coq_init_constant "S")
let lt_n_O = function () -> (coq_constant arith_Nat "nlt_0_r")
let max_ref = function () -> (find_reference ["Recdef"] "max")
let max_constr = function () -> (constr_of_global (delayed_force max_ref))
let coq_conj = function () -> find_reference Coqlib.logic_module_name "conj"

let f_S t = mkApp(delayed_force coq_S, [|t|]);;

let rec n_x_id ids n =
  if Int.equal n 0 then []
  else let x = next_ident_away_in_goal x_id ids in
          x::n_x_id (x::ids) (n-1);;


let simpl_iter clause =
  reduce
    (Lazy
       {rBeta=true;rMatch=true;rFix=true;rCofix=true;rZeta=true;rDelta=false;
        rConst = [ EvalConstRef (const_of_ref (delayed_force iter_ref))]})
    clause

(* Others ugly things ... *)
let (value_f:constr list -> global_reference -> constr) =
  fun al fterm ->
    let d0 = Loc.ghost in
    let rev_x_id_l =
      (
	List.fold_left
	  (fun x_id_l _ ->
	     let x_id = next_ident_away_in_goal x_id x_id_l in
	     x_id::x_id_l
	  )
	  []
	  al
      )
    in
    let context = List.map
      (fun (x, c) -> LocalAssum (Name x, c)) (List.combine rev_x_id_l (List.rev al))
    in
    let env = Environ.push_rel_context context (Global.env ()) in
    let glob_body =
      GCases
	(d0,RegularStyle,None,
	 [GApp(d0, GRef(d0,fterm,None), List.rev_map (fun x_id -> GVar(d0, x_id)) rev_x_id_l),
	  (Anonymous,None)],
	 [d0, [v_id], [PatCstr(d0,(destIndRef
				     (delayed_force coq_sig_ref),1),
			       [PatVar(d0, Name v_id);
				PatVar(d0, Anonymous)],
			       Anonymous)],
	  GVar(d0,v_id)])
    in
    let body = fst (understand env (Evd.from_env env) glob_body)(*FIXME*) in
    it_mkLambda_or_LetIn body context

let (declare_f : Id.t -> logical_kind -> constr list -> global_reference -> global_reference) =
  fun f_id kind input_type fterm_ref ->
    declare_fun f_id kind (value_f input_type fterm_ref);;



(* Debugging mechanism *)
let debug_queue = Stack.create ()

let print_debug_queue b e = 
  if  not (Stack.is_empty debug_queue) 
  then
    begin
      let lmsg,goal = Stack.pop debug_queue in 
      if b then 
	Feedback.msg_debug (hov 1 (lmsg ++ (str " raised exception " ++ CErrors.print e) ++ str " on goal" ++ fnl() ++ goal))
      else
	begin
	  Feedback.msg_debug (hov 1 (str " from " ++ lmsg ++ str " on goal"++fnl() ++ goal));
	end;
      (* print_debug_queue false e; *)
    end

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


let do_observe_tac s tac g = 
  let goal = Printer.pr_goal g in
  let lmsg = (str "recdef : ") ++ s in 
  observe (s++fnl());
  Stack.push (lmsg,goal) debug_queue;
  try 
    let v = tac g in 
    ignore(Stack.pop debug_queue);
    v
  with reraise ->
    let reraise = CErrors.push reraise in
    if not (Stack.is_empty debug_queue)
    then print_debug_queue true (fst (ExplainErr.process_vernac_interp_error reraise));
    iraise reraise

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


let observe_tclTHENLIST s tacl =
  if do_observe ()
  then
    let rec aux n = function
      | [] -> tclIDTAC
      | [tac] -> observe_tac (s ++ spc () ++ int n) tac
      | tac::tacl -> observe_tac (s ++ spc () ++ int n) (tclTHEN tac (aux (succ n) tacl))
    in
    aux 0 tacl
  else tclTHENLIST tacl
	   
(* Conclusion tactics *)

(* The boolean value is_mes expresses that the termination is expressed
  using a measure function instead of a well-founded relation. *)
let tclUSER tac is_mes l g =
  let clear_tac =
    match l with
      | None -> tclIDTAC
      | Some l -> tclMAP (fun id -> tclTRY (Proofview.V82.of_tactic (clear [id]))) (List.rev l)
  in
  observe_tclTHENLIST (str "tclUSER1")
    [
      clear_tac;
      if is_mes
      then observe_tclTHENLIST (str "tclUSER2")
        [
	  Proofview.V82.of_tactic (unfold_in_concl [(Locus.AllOccurrences, evaluable_of_global_reference
            (delayed_force Indfun_common.ltof_ref))]);
         tac
	 ]
      else tac
    ]
    g

let tclUSER_if_not_mes concl_tac is_mes names_to_suppress =
  if is_mes
  then tclCOMPLETE (fun gl -> Proofview.V82.of_tactic (Simple.apply (delayed_force well_founded_ltof)) gl)
  else (* tclTHEN (Simple.apply (delayed_force acc_intro_generator_function) ) *) (tclUSER concl_tac is_mes names_to_suppress)



  

(* Traveling term.
   Both definitions of [f_terminate] and [f_equation] use the same generic 
   traveling mechanism.
*)

(* [check_not_nested forbidden e] checks that [e] does not contains any variable 
   of [forbidden]
*)
let check_not_nested forbidden e =
  let rec check_not_nested e =  
    match kind_of_term e with 
      | Rel _ -> ()
      | Var x ->
        if Id.List.mem x forbidden
        then user_err ~hdr:"Recdef.check_not_nested"
               (str "check_not_nested: failure " ++ pr_id x)
      | Meta _ | Evar _ | Sort _ -> ()
      | Cast(e,_,t) -> check_not_nested e;check_not_nested t
      | Prod(_,t,b) -> check_not_nested t;check_not_nested b
      | Lambda(_,t,b) -> check_not_nested t;check_not_nested b
      | LetIn(_,v,t,b) -> check_not_nested t;check_not_nested b;check_not_nested v
      | App(f,l) -> check_not_nested f;Array.iter check_not_nested l
      | Proj (p,c) -> check_not_nested c
      | Const _ -> ()
      | Ind _ -> ()
      | Construct _ -> ()
      | Case(_,t,e,a) -> 
	check_not_nested t;check_not_nested e;Array.iter check_not_nested a
      | Fix _ -> error "check_not_nested : Fix"
      | CoFix _ -> error "check_not_nested : Fix"
  in
  try 
    check_not_nested e 
  with UserError(_,p) -> 
    user_err ~hdr:"_" (str "on expr : " ++ Printer.pr_lconstr e ++ str " " ++ p)

(* ['a info] contains the local information for traveling *)
type 'a infos = 
    { nb_arg : int; (* function number of arguments *)
      concl_tac : tactic; (* final tactic to finish proofs *)
      rec_arg_id : Id.t; (*name of the declared recursive argument *)
      is_mes : bool; (* type of recursion *)
      ih : Id.t; (* induction hypothesis name *)
      f_id : Id.t;  (* function name *)
      f_constr : constr;  (* function term *)
      f_terminate : constr; (* termination proof term *)
      func : global_reference; (* functional reference *)
      info : 'a;
      is_main_branch : bool; (* on the main branch or on a matched expression *)
      is_final : bool; (* final first order term or not *)
      values_and_bounds : (Id.t*Id.t) list; 
      eqs : Id.t list; 
      forbidden_ids : Id.t list;
      acc_inv : constr lazy_t;
      acc_id : Id.t;
      args_assoc : ((constr list)*constr) list;
    }


type ('a,'b) journey_info_tac = 
    'a -> (* the arguments of the constructor *)
    'b infos -> (* infos of the caller *)
    ('b infos -> tactic) -> (* the continuation tactic of the caller *)
    'b infos -> (* argument of the tactic *)
    tactic
       
(* journey_info : specifies the actions to do on the different term constructors during the traveling of the term
*)
type journey_info = 
    { letiN : ((Name.t*constr*types*constr),constr) journey_info_tac;
      lambdA : ((Name.t*types*constr),constr) journey_info_tac;
      casE : ((constr infos -> tactic) -> constr infos -> tactic) -> 
	((case_info * constr * constr * constr array),constr) journey_info_tac;
      otherS : (unit,constr) journey_info_tac;
      apP : (constr*(constr list),constr) journey_info_tac;
      app_reC : (constr*(constr list),constr) journey_info_tac;
      message : string
    }

	       

let rec add_vars forbidden e = 
  match kind_of_term e with 
    | Var x -> x::forbidden 
    | _ -> Term.fold_constr add_vars forbidden e


let treat_case forbid_new_ids to_intros finalize_tac nb_lam e infos : tactic = 
  fun g -> 
    let rev_context,b = decompose_lam_n nb_lam e in 
    let ids = List.fold_left (fun acc (na,_) -> 	
      let pre_id = 
	match na with 
	  | Name x -> x 
	  | Anonymous -> ano_id 
      in
      pre_id::acc
    ) [] rev_context in 
    let rev_ids = pf_get_new_ids (List.rev ids) g in 
    let new_b = substl (List.map mkVar rev_ids) b in 
    observe_tclTHENLIST (str "treat_case1") 
      [
	h_intros (List.rev rev_ids);
	Proofview.V82.of_tactic (intro_using teq_id);
	onLastHypId (fun heq -> 
	  observe_tclTHENLIST (str "treat_case2")[
	    Proofview.V82.of_tactic (clear to_intros);
	    h_intros to_intros;
	    (fun g' -> 
	      let ty_teq = pf_unsafe_type_of g' (mkVar heq) in
	      let teq_lhs,teq_rhs =
		let _,args = try destApp ty_teq with DestKO -> assert false in
		args.(1),args.(2)
	      in
	      let new_b' = Termops.replace_term teq_lhs teq_rhs new_b in 
	      let new_infos = {
		infos with 
		  info = new_b';
		  eqs = heq::infos.eqs;
		  forbidden_ids = 
		  if forbid_new_ids 
		  then add_vars infos.forbidden_ids new_b'
		  else infos.forbidden_ids
	      } in 
	      finalize_tac new_infos g' 
	    )
	  ]
	)
      ] g

let rec travel_aux jinfo continuation_tac (expr_info:constr infos) =
  match kind_of_term expr_info.info with 
    | CoFix _ | Fix _ -> error "Function cannot treat local fixpoint or cofixpoint"
    | Proj _ -> error "Function cannot treat projections"
    | LetIn(na,b,t,e) -> 
      begin
	let new_continuation_tac = 
	  jinfo.letiN (na,b,t,e) expr_info continuation_tac
	in
	travel jinfo new_continuation_tac 
	  {expr_info with info = b; is_final=false}
      end
    | Rel _ -> anomaly (Pp.str "Free var in goal conclusion !")
    | Prod _ -> 
      begin
	try
	  check_not_nested (expr_info.f_id::expr_info.forbidden_ids) expr_info.info;
	  jinfo.otherS () expr_info continuation_tac expr_info
	with e when CErrors.noncritical e ->
	  user_err ~hdr:"Recdef.travel" (str "the term " ++ Printer.pr_lconstr expr_info.info ++ str " can not contain a recursive call to " ++ pr_id expr_info.f_id)
      end
    | Lambda(n,t,b) -> 
      begin
	try
	  check_not_nested (expr_info.f_id::expr_info.forbidden_ids) expr_info.info;
	  jinfo.otherS () expr_info continuation_tac expr_info
	with e when CErrors.noncritical e ->
	  user_err ~hdr:"Recdef.travel" (str "the term " ++ Printer.pr_lconstr expr_info.info ++ str " can not contain a recursive call to " ++ pr_id expr_info.f_id)
      end
    | Case(ci,t,a,l) -> 
      begin
	let continuation_tac_a = 
	  jinfo.casE 
	    (travel jinfo) (ci,t,a,l) 
	    expr_info continuation_tac in 
	travel 
	  jinfo continuation_tac_a 
	  {expr_info with info = a; is_main_branch = false; 
	    is_final = false}
      end
    | App _ -> 
      let f,args = decompose_app expr_info.info in 
      if eq_constr f (expr_info.f_constr) 
      then jinfo.app_reC (f,args) expr_info continuation_tac expr_info 
      else
      begin
	match kind_of_term f with 
	  | App _ -> assert false (* f is coming from a decompose_app *)
	  | Const _ | Construct _ | Rel _ | Evar _ | Meta _  | Ind _ 
	  | Sort _ | Prod _ | Var _ -> 
	    let new_infos = {expr_info with info=(f,args)} in 
	    let new_continuation_tac = 
	      jinfo.apP (f,args) expr_info continuation_tac in 
	    travel_args jinfo
	      expr_info.is_main_branch new_continuation_tac new_infos
	  | Case _ ->  user_err ~hdr:"Recdef.travel" (str "the term " ++ Printer.pr_lconstr expr_info.info ++ str " can not contain an applied match (See Limitation in Section 2.3 of refman)")
	  | _ -> anomaly (Pp.str "travel_aux : unexpected "++ Printer.pr_lconstr expr_info.info)
      end
    | Cast(t,_,_) -> travel jinfo continuation_tac {expr_info with info=t}
    | Const _ | Var _ | Meta _ | Evar _ | Sort _ | Construct _ | Ind _ ->
      let new_continuation_tac = 
	jinfo.otherS () expr_info continuation_tac in
      new_continuation_tac expr_info
and travel_args jinfo is_final continuation_tac infos = 
  let (f_args',args) = infos.info in 
  match args with 
    | [] -> 
      continuation_tac {infos with info = f_args'; is_final = is_final}
    | arg::args' -> 
      let new_continuation_tac new_infos = 
	let new_arg = new_infos.info in 
	travel_args jinfo is_final
	  continuation_tac
	  {new_infos with info = (mkApp(f_args',[|new_arg|]),args')}
      in
      travel jinfo new_continuation_tac 
	{infos with info=arg;is_final=false} 
and travel jinfo continuation_tac expr_info = 
  observe_tac 
    (str jinfo.message ++ Printer.pr_lconstr expr_info.info) 
    (travel_aux jinfo continuation_tac expr_info)

(* Termination proof *) 

let rec prove_lt hyple g = 
  begin
    try
      let (varx,varz) = match decompose_app (pf_concl g) with
        | _, x::z::_ when isVar x && isVar z -> x, z
        | _ -> assert false
      in
      let h =
	List.find (fun id ->
          match decompose_app (pf_unsafe_type_of g (mkVar id)) with
            | _, t::_ -> eq_constr t varx
            | _ -> false
	) hyple
      in
      let y =
	List.hd (List.tl (snd (decompose_app (pf_unsafe_type_of g (mkVar h))))) in
      observe_tclTHENLIST (str "prove_lt1")[
	Proofview.V82.of_tactic (apply (mkApp(le_lt_trans (),[|varx;y;varz;mkVar h|])));
	observe_tac (str "prove_lt") (prove_lt hyple)
      ]
    with Not_found -> 
      (
	(
	  observe_tclTHENLIST (str "prove_lt2")[
	    Proofview.V82.of_tactic (apply (delayed_force lt_S_n));
	    (observe_tac (str "assumption: " ++ Printer.pr_goal g) (Proofview.V82.of_tactic assumption))
	  ])
      )
  end
    g

let rec destruct_bounds_aux infos (bound,hyple,rechyps) lbounds g = 
  match lbounds with 
    | [] -> 
      let ids = pf_ids_of_hyps g in 
      let s_max = mkApp(delayed_force coq_S, [|bound|]) in 
      let k = next_ident_away_in_goal k_id ids in
      let ids = k::ids in
      let h' = next_ident_away_in_goal (h'_id) ids in
      let ids = h'::ids in
      let def = next_ident_away_in_goal def_id ids in
      observe_tclTHENLIST (str "destruct_bounds_aux1")[
	Proofview.V82.of_tactic (split (ImplicitBindings [s_max]));
	Proofview.V82.of_tactic (intro_then
	  (fun id -> 
            Proofview.V82.tactic begin
	    observe_tac (str "destruct_bounds_aux") 
	      (tclTHENS (Proofview.V82.of_tactic (simplest_case (mkVar id)))
		 [
		   observe_tclTHENLIST (str "")[Proofview.V82.of_tactic (intro_using h_id);
			       Proofview.V82.of_tactic (simplest_elim(mkApp(delayed_force lt_n_O,[|s_max|])));
			       Proofview.V82.of_tactic default_full_auto];
		   observe_tclTHENLIST (str "destruct_bounds_aux2")[
		     observe_tac (str "clearing k ") (Proofview.V82.of_tactic (clear [id]));
		     h_intros [k;h';def];
		     observe_tac (str "simple_iter") (Proofview.V82.of_tactic (simpl_iter Locusops.onConcl));
		     observe_tac (str "unfold functional")
		       (Proofview.V82.of_tactic (unfold_in_concl[(Locus.OnlyOccurrences [1],
					 evaluable_of_global_reference infos.func)]));
		     (
		       observe_tclTHENLIST (str "test")[
			 list_rewrite true
			   (List.fold_right 
			      (fun e acc -> (mkVar e,true)::acc)
			      infos.eqs
			      (List.map (fun e -> (e,true)) rechyps)
			   );
		      (* list_rewrite true *)
		      (*   (List.map (fun e -> (mkVar e,true)) infos.eqs) *)
		      (*   ; *)
			 
			 (observe_tac (str "finishing") 
			    (tclORELSE
			       (Proofview.V82.of_tactic intros_reflexivity)
			       (observe_tac (str "calling prove_lt") (prove_lt hyple))))])
		   ]
	      ]
	      )end))
	  ] g
      | (_,v_bound)::l -> 
      observe_tclTHENLIST (str "destruct_bounds_aux3")[
	Proofview.V82.of_tactic (simplest_elim (mkVar v_bound));
	Proofview.V82.of_tactic (clear [v_bound]);
	tclDO 2 (Proofview.V82.of_tactic intro);
	onNthHypId 1 
	  (fun p_hyp -> 
	    (onNthHypId 2 
	       (fun p ->  
		 observe_tclTHENLIST (str "destruct_bounds_aux4")[
		   Proofview.V82.of_tactic (simplest_elim 
		     (mkApp(delayed_force max_constr, [| bound; mkVar p|])));
		   tclDO 3 (Proofview.V82.of_tactic intro);
		   onNLastHypsId 3 (fun lids -> 
		     match lids with
		       [hle2;hle1;pmax] -> 
			 destruct_bounds_aux infos 
			   ((mkVar pmax),
			    hle1::hle2::hyple,(mkVar p_hyp)::rechyps) 
			   l 
		       | _ -> assert false) ;
		 ]
	       )
	    )
	  )
      ] g

let destruct_bounds infos = 
  destruct_bounds_aux infos (delayed_force coq_O,[],[]) infos.values_and_bounds

let terminate_app f_and_args expr_info continuation_tac infos = 
    if expr_info.is_final && expr_info.is_main_branch 
    then 
      observe_tclTHENLIST (str "terminate_app1")[
	continuation_tac infos;
	observe_tac (str "first split") 
	  (Proofview.V82.of_tactic (split (ImplicitBindings [infos.info])));
	observe_tac (str "destruct_bounds (1)") (destruct_bounds infos)
      ]
    else continuation_tac infos

let terminate_others _ expr_info continuation_tac infos = 
  if expr_info.is_final && expr_info.is_main_branch 
  then 
    observe_tclTHENLIST (str "terminate_others")[
	    continuation_tac infos;
      observe_tac (str "first split") 
	(Proofview.V82.of_tactic (split (ImplicitBindings [infos.info])));
      observe_tac (str "destruct_bounds") (destruct_bounds infos)
    ]
  else continuation_tac infos

let terminate_letin (na,b,t,e) expr_info continuation_tac info = 
  let new_e = subst1 info.info e in 
  let new_forbidden = 
    let forbid = 
      try 
	check_not_nested (expr_info.f_id::expr_info.forbidden_ids) b;
	true
      with e when CErrors.noncritical e -> false
    in
    if forbid 
    then 
      match na with
	| Anonymous -> info.forbidden_ids
	| Name id -> id::info.forbidden_ids
    else info.forbidden_ids 
  in
  continuation_tac {info with info = new_e; forbidden_ids = new_forbidden} 

let pf_type c tac gl = 
  let evars, ty = Typing.type_of (pf_env gl) (project gl) c in
    tclTHEN (Refiner.tclEVARS evars) (tac ty) gl

let pf_typel l tac =
  let rec aux tys l =
    match l with
    | [] -> tac (List.rev tys)
    | hd :: tl -> pf_type hd (fun ty -> aux (ty::tys) tl)
  in aux [] l

(* This is like the previous one except that it also rewrite on all
  hypotheses except the ones given in the first argument.  All the
  modified hypotheses are generalized in the process and should be
  introduced back later; the result is the pair of the tactic and the
  list of hypotheses that have been generalized and cleared. *)
let mkDestructEq :
  Id.t list -> constr -> goal sigma -> tactic * Id.t list =
  fun not_on_hyp expr g ->
  let hyps = pf_hyps g in
  let to_revert =
    Util.List.map_filter
      (fun decl ->
        let open Context.Named.Declaration in
        let id = get_id decl in
        if Id.List.mem id not_on_hyp || not (Termops.occur_term expr (get_type decl))
        then None else Some id) hyps in
  let to_revert_constr = List.rev_map mkVar to_revert in
  let type_of_expr = pf_unsafe_type_of g expr in
  let new_hyps = mkApp(Lazy.force refl_equal, [|type_of_expr; expr|])::
           to_revert_constr in
    pf_typel new_hyps (fun _ ->
    observe_tclTHENLIST (str "mkDestructEq")
     [Proofview.V82.of_tactic (generalize new_hyps);
      (fun g2 ->
        let changefun patvars = { run = fun sigma ->
          let redfun = pattern_occs [Locus.AllOccurrencesBut [1], expr] in
          redfun.Reductionops.e_redfun (pf_env g2) sigma (pf_concl g2)
        } in
	Proofview.V82.of_tactic (change_in_concl None changefun) g2);
      Proofview.V82.of_tactic (simplest_case expr)]), to_revert


let terminate_case next_step (ci,a,t,l) expr_info continuation_tac infos g =
  let f_is_present =
    try
      check_not_nested (expr_info.f_id::expr_info.forbidden_ids) a;
      false
    with e when CErrors.noncritical e ->
      true
  in
  let a' = infos.info in
  let new_info =
    {infos with
      info = mkCase(ci,t,a',l);
      is_main_branch = expr_info.is_main_branch;
      is_final = expr_info.is_final} in
  let destruct_tac,rev_to_thin_intro = 
    mkDestructEq [expr_info.rec_arg_id] a' g in 
  let to_thin_intro = List.rev rev_to_thin_intro in 
  observe_tac (str "treating cases (" ++ int (Array.length l) ++ str")" ++ spc () ++ Printer.pr_lconstr a') 
    (try
      (tclTHENS
	 destruct_tac
	 (List.map_i (fun i e -> observe_tac (str "do treat case") (treat_case f_is_present to_thin_intro (next_step continuation_tac) ci.ci_cstr_ndecls.(i) e new_info)) 0 (Array.to_list l)
	 ))
    with 
      | UserError(Some "Refiner.thensn_tac3",_) 
      | UserError(Some "Refiner.tclFAIL_s",_) ->
	(observe_tac (str "is computable " ++ Printer.pr_lconstr new_info.info) (next_step continuation_tac {new_info with info = nf_betaiotazeta new_info.info} )
	))
    g
    
let terminate_app_rec (f,args) expr_info continuation_tac _ = 
  List.iter (check_not_nested (expr_info.f_id::expr_info.forbidden_ids))
    args;
  begin
    try 
      let v = List.assoc_f (List.equal Constr.equal) args expr_info.args_assoc in
      let new_infos = {expr_info with info = v} in 
      observe_tclTHENLIST (str "terminate_app_rec")[
	continuation_tac new_infos;
	if expr_info.is_final && expr_info.is_main_branch 
	then
	  observe_tclTHENLIST (str "terminate_app_rec1")[ 
	    observe_tac (str "first split")
	      (Proofview.V82.of_tactic (split (ImplicitBindings [new_infos.info])));
	    observe_tac (str "destruct_bounds (3)")
	      (destruct_bounds new_infos)
	  ]
	else
	  tclIDTAC			       
      ]    
    with Not_found -> 
      observe_tac (str "terminate_app_rec not found") (tclTHENS
	(Proofview.V82.of_tactic (simplest_elim (mkApp(mkVar expr_info.ih,Array.of_list args))))
	[		
	  observe_tclTHENLIST (str "terminate_app_rec2")[
	    Proofview.V82.of_tactic (intro_using rec_res_id);
	    Proofview.V82.of_tactic intro;
	    onNthHypId 1 
	      (fun v_bound -> 
		(onNthHypId 2 
		   (fun v ->  
		     let new_infos = { expr_info with 
		       info = (mkVar v); 
		       values_and_bounds = 
			 (v,v_bound)::expr_info.values_and_bounds; 
		       args_assoc=(args,mkVar v)::expr_info.args_assoc
		     } in
		     observe_tclTHENLIST (str "terminate_app_rec3")[
		       continuation_tac new_infos;
		       if expr_info.is_final && expr_info.is_main_branch 
		       then
			 observe_tclTHENLIST (str "terminate_app_rec4")[ 
			   observe_tac (str "first split") 
			     (Proofview.V82.of_tactic (split (ImplicitBindings [new_infos.info])));
			   observe_tac (str "destruct_bounds (2)") 
			     (destruct_bounds new_infos)
			 ]
		       else
			 tclIDTAC			       
		     ]    
		   )
		)
	      )
	  ];
	  observe_tac (str "proving decreasing") (
	    tclTHENS (* proof of args < formal args *)
	      (Proofview.V82.of_tactic (apply (Lazy.force expr_info.acc_inv)))
	      [ 
		observe_tac (str "assumption") (Proofview.V82.of_tactic assumption);
		observe_tclTHENLIST (str "terminate_app_rec5")
		  [
		    tclTRY(list_rewrite true 
			     (List.map 
				(fun e -> mkVar e,true) 
				expr_info.eqs
			     )
		    );
		    tclUSER expr_info.concl_tac true 
		      (Some (
		      expr_info.ih::expr_info.acc_id::
			(fun (x,y) -> y) 
			(List.split expr_info.values_and_bounds)
		       )
		      );
		  ]
	      ])
	])
  end

let terminate_info = 
  { message = "prove_terminate with term ";
    letiN = terminate_letin;
    lambdA = (fun _ _ _ _ -> assert false);
    casE = terminate_case;
    otherS = terminate_others;
    apP = terminate_app;
    app_reC = terminate_app_rec;
  }

let prove_terminate = travel terminate_info


(* Equation proof *)

let equation_case next_step (ci,a,t,l) expr_info continuation_tac infos = 
  observe_tac (str "equation case") (terminate_case next_step (ci,a,t,l) expr_info continuation_tac infos)

let rec prove_le g = 
  let x,z = 
    let _,args = decompose_app (pf_concl g) in 
    (List.hd args,List.hd (List.tl args))
  in 
  tclFIRST[
    Proofview.V82.of_tactic assumption;
    Proofview.V82.of_tactic (apply (delayed_force le_n));
    begin
      try
	let matching_fun = 
	  pf_is_matching g
	    (Pattern.PApp(Pattern.PRef (reference_of_constr (le ())),[|Pattern.PVar (destVar x);Pattern.PMeta None|])) in 
	let (h,t) = List.find (fun (_,t) -> matching_fun t) (pf_hyps_types g)
	in 
	let y = 
	  let _,args = decompose_app t in 
	  List.hd (List.tl args)
	in 
	observe_tclTHENLIST (str "prove_le")[
	  Proofview.V82.of_tactic (apply(mkApp(le_trans (),[|x;y;z;mkVar h|])));
	  observe_tac (str "prove_le (rec)") (prove_le)
	] 
      with Not_found -> tclFAIL 0 (mt())
    end;
  ]
    g

let rec make_rewrite_list expr_info max = function 
  | [] -> tclIDTAC
  | (_,p,hp)::l -> 
    observe_tac (str "make_rewrite_list") (tclTHENS
      (observe_tac (str "rewrite heq on " ++ pr_id p ) (
	(fun g -> 
	  let t_eq = compute_renamed_type g (mkVar hp) in
	  let k,def =
	    let k_na,_,t = destProd t_eq in
	    let _,_,t  = destProd t in
	    let def_na,_,_ = destProd t in
	    Nameops.out_name k_na,Nameops.out_name def_na
	  in
	  Proofview.V82.of_tactic (general_rewrite_bindings false Locus.AllOccurrences
	    true (* dep proofs also: *) true 
	    (mkVar hp,
	     ExplicitBindings[Loc.ghost,NamedHyp def,
			      expr_info.f_constr;Loc.ghost,NamedHyp k,
			      (f_S max)]) false) g) )
      )
      [make_rewrite_list expr_info max l;
       observe_tclTHENLIST (str "make_rewrite_list")[ (* x < S max proof *)
	 Proofview.V82.of_tactic (apply (delayed_force le_lt_n_Sm));
	 observe_tac (str "prove_le(2)") prove_le
       ]
      ] )

let make_rewrite expr_info l hp max = 
  tclTHENFIRST
    (observe_tac (str "make_rewrite") (make_rewrite_list expr_info max l))
    (observe_tac (str "make_rewrite") (tclTHENS
       (fun g -> 
	  let t_eq = compute_renamed_type g (mkVar hp) in
	  let k,def =
	    let k_na,_,t = destProd t_eq in
	    let _,_,t  = destProd t in
	    let def_na,_,_ = destProd t in
	    Nameops.out_name k_na,Nameops.out_name def_na
	  in
	 observe_tac (str "general_rewrite_bindings")
	   (Proofview.V82.of_tactic (general_rewrite_bindings false Locus.AllOccurrences
	    true (* dep proofs also: *) true 
	    (mkVar hp,
	     ExplicitBindings[Loc.ghost,NamedHyp def,
			      expr_info.f_constr;Loc.ghost,NamedHyp k,
			      (f_S (f_S max))]) false)) g)
       [observe_tac(str "make_rewrite finalize") (
	 (* tclORELSE( h_reflexivity) *)
	 (observe_tclTHENLIST (str "make_rewrite")[
	   Proofview.V82.of_tactic (simpl_iter Locusops.onConcl);
	   observe_tac (str "unfold functional")
	     (Proofview.V82.of_tactic (unfold_in_concl[(Locus.OnlyOccurrences [1],
			       evaluable_of_global_reference expr_info.func)]));
	   
	   (list_rewrite true 
	      (List.map (fun e -> mkVar e,true) expr_info.eqs));
	   (observe_tac (str "h_reflexivity")
			(Proofview.V82.of_tactic intros_reflexivity)
	   )
	    ]))
       ;
	 observe_tclTHENLIST (str "make_rewrite1")[ (* x < S (S max) proof *)
	 Proofview.V82.of_tactic (apply (delayed_force le_lt_SS));
	 observe_tac (str "prove_le (3)") prove_le
	 ]
       ])  
    )

let rec compute_max rew_tac max l = 
  match l with 
    | [] -> rew_tac max
    | (_,p,_)::l -> 
      observe_tclTHENLIST (str "compute_max")[
	Proofview.V82.of_tactic (simplest_elim 
	  (mkApp(delayed_force max_constr, [| max; mkVar p|])));
	tclDO 3 (Proofview.V82.of_tactic intro);
	onNLastHypsId 3 (fun lids -> 
	  match lids with
	    | [hle2;hle1;pmax] -> compute_max rew_tac (mkVar pmax) l 
	    | _ -> assert false
	)]

let rec destruct_hex expr_info acc l = 
  match l with 
    | [] -> 
      begin
	match List.rev acc with 
	  | [] -> tclIDTAC 
	  | (_,p,hp)::tl  -> 
	    observe_tac (str "compute max ") (compute_max (make_rewrite expr_info tl hp) (mkVar p) tl)
      end
    | (v,hex)::l -> 
      observe_tclTHENLIST (str "destruct_hex")[
	Proofview.V82.of_tactic (simplest_case (mkVar hex));
	Proofview.V82.of_tactic (clear [hex]);
	tclDO 2 (Proofview.V82.of_tactic intro);
	onNthHypId 1 (fun hp -> 
	  onNthHypId 2 (fun p -> 
	    observe_tac 
	      (str "destruct_hex after " ++ pr_id hp ++ spc () ++ pr_id p)
	      (destruct_hex expr_info ((v,p,hp)::acc) l)
	  )
	)
      ]
	
let rec intros_values_eq expr_info acc = 
  tclORELSE(
    observe_tclTHENLIST (str "intros_values_eq")[
      tclDO 2 (Proofview.V82.of_tactic intro);
      onNthHypId 1 (fun hex -> 
	(onNthHypId 2 (fun v -> intros_values_eq expr_info ((v,hex)::acc)))
      )
    ])
    (tclCOMPLETE (
      destruct_hex expr_info [] acc
    ))

let equation_others _ expr_info continuation_tac infos = 
  if expr_info.is_final && expr_info.is_main_branch 
  then 
    observe_tac (str "equation_others (cont_tac +intros) " ++ Printer.pr_lconstr expr_info.info)
		(tclTHEN 
      (continuation_tac infos) 
      (observe_tac (str "intros_values_eq equation_others "  ++ Printer.pr_lconstr expr_info.info) (intros_values_eq expr_info [])))
  else observe_tac (str "equation_others (cont_tac) " ++ Printer.pr_lconstr expr_info.info) (continuation_tac infos)

let equation_app f_and_args expr_info continuation_tac infos = 
    if expr_info.is_final && expr_info.is_main_branch 
    then ((observe_tac (str "intros_values_eq equation_app") (intros_values_eq expr_info [])))
    else continuation_tac infos
	    
let equation_app_rec (f,args) expr_info continuation_tac info = 
  begin
    try
      let v = List.assoc_f (List.equal Constr.equal) args expr_info.args_assoc in
      let new_infos = {expr_info with info = v} in
      observe_tac (str "app_rec found") (continuation_tac new_infos)
    with Not_found ->
      if expr_info.is_final && expr_info.is_main_branch 
      then 
	observe_tclTHENLIST (str "equation_app_rec")
	  [ Proofview.V82.of_tactic (simplest_case (mkApp (expr_info.f_terminate,Array.of_list args)));
	    continuation_tac {expr_info with args_assoc = (args,delayed_force coq_O)::expr_info.args_assoc};
	    observe_tac (str "app_rec intros_values_eq") (intros_values_eq expr_info [])
	  ]
      else 
	observe_tclTHENLIST (str "equation_app_rec1")[
  	  Proofview.V82.of_tactic (simplest_case (mkApp (expr_info.f_terminate,Array.of_list args)));
	  observe_tac (str "app_rec not_found") (continuation_tac {expr_info with args_assoc = (args,delayed_force coq_O)::expr_info.args_assoc})
	]
  end

let equation_info = 
  {message = "prove_equation with term ";
   letiN = (fun _ -> assert false);
   lambdA = (fun _ _ _ _ -> assert false);
   casE = equation_case;
   otherS = equation_others;
   apP = equation_app;
   app_reC = equation_app_rec
}
    
let prove_eq = travel equation_info

(* wrappers *)
(* [compute_terminate_type] computes the type of the Definition f_terminate from the type of f_F
*)
let compute_terminate_type nb_args func =
  let _,a_arrow_b,_ = destLambda(def_of_const (constr_of_global func)) in
  let rev_args,b = decompose_prod_n nb_args a_arrow_b in
  let left =
    mkApp(delayed_force iter,
	  Array.of_list
	    (lift 5 a_arrow_b:: mkRel 3::
	       constr_of_global func::mkRel 1::
	       List.rev (List.map_i (fun i _ -> mkRel (6+i)) 0 rev_args)
	    )
	 )
  in
  let right = mkRel 5 in
  let equality = mkApp(delayed_force eq, [|lift 5 b; left; right|]) in
  let result = (mkProd ((Name def_id) , lift 4 a_arrow_b, equality)) in
  let cond = mkApp(delayed_force lt, [|(mkRel 2); (mkRel 1)|]) in
  let nb_iter =
    mkApp(delayed_force ex,
	  [|delayed_force nat;
	    (mkLambda
	       (Name
		  p_id,
		  delayed_force nat,
		  (mkProd (Name k_id, delayed_force nat,
			   mkArrow cond result))))|])in
  let value = mkApp(constr_of_global (delayed_force coq_sig_ref),
		    [|b;
		      (mkLambda (Name v_id, b, nb_iter))|]) in
  compose_prod rev_args value


let termination_proof_header is_mes input_type ids args_id relation
    rec_arg_num rec_arg_id tac wf_tac : tactic =
  begin
    fun g ->
      let nargs = List.length args_id in
      let pre_rec_args =
	List.rev_map
	  mkVar (fst (List.chop (rec_arg_num - 1) args_id))
      in
      let relation = substl pre_rec_args relation in
      let input_type = substl pre_rec_args input_type in
      let wf_thm = next_ident_away_in_goal (Id.of_string ("wf_R")) ids in
      let wf_rec_arg =
	next_ident_away_in_goal
	  (Id.of_string ("Acc_"^(Id.to_string rec_arg_id)))
	  (wf_thm::ids)
      in
      let hrec = next_ident_away_in_goal hrec_id
	(wf_rec_arg::wf_thm::ids) in
      let acc_inv =
	  lazy (
	    mkApp (
	      delayed_force acc_inv_id,
	      [|input_type;relation;mkVar rec_arg_id|]
	    )
	  )
      in
      tclTHEN
	(h_intros args_id)
	(tclTHENS
	   (observe_tac
	      (str "first assert")
	      (Proofview.V82.of_tactic (assert_before
		 (Name wf_rec_arg)
		 (mkApp (delayed_force acc_rel,
			 [|input_type;relation;mkVar rec_arg_id|])
		 )
	      ))
	   )
	   [
	     (* accesibility proof *)
	     tclTHENS
	       (observe_tac
		  (str "second assert")
		  (Proofview.V82.of_tactic (assert_before
		     (Name wf_thm)
		     (mkApp (delayed_force well_founded,[|input_type;relation|]))
		  ))
	       )
	       [
		 (* interactive proof that the relation is well_founded *)
		 observe_tac (str "wf_tac") (wf_tac is_mes (Some args_id));
		 (* this gives the accessibility argument *)
		 observe_tac
		   (str "apply wf_thm")
		   (Proofview.V82.of_tactic (Simple.apply (mkApp(mkVar wf_thm,[|mkVar rec_arg_id|])))
		   )
	       ]
	     ;
	     (* rest of the proof *)
	     observe_tclTHENLIST (str "rest of proof")
	       [observe_tac (str "generalize")
		  (onNLastHypsId (nargs+1)
		     (tclMAP (fun id ->
			tclTHEN (Proofview.V82.of_tactic (Tactics.generalize [mkVar id])) (Proofview.V82.of_tactic (clear [id])))
		     ))
	       ;
		observe_tac (str "fix") (Proofview.V82.of_tactic (fix (Some hrec) (nargs+1)));
		h_intros args_id;
		Proofview.V82.of_tactic (Simple.intro wf_rec_arg);
		observe_tac (str "tac") (tac wf_rec_arg hrec wf_rec_arg acc_inv)
	       ]
	   ]
	) g
  end



let rec instantiate_lambda t l =
  match l with
  | [] -> t
  | a::l ->
      let (_, _, body) = destLambda t in
      instantiate_lambda (subst1 a body) l

let whole_start (concl_tac:tactic) nb_args is_mes func input_type relation rec_arg_num  : tactic =
  begin
    fun g ->
      let ids = Termops.ids_of_named_context (pf_hyps g) in
      let func_body = (def_of_const (constr_of_global func)) in
      let (f_name, _, body1) = destLambda func_body in
      let f_id =
	match f_name with
	  | Name f_id -> next_ident_away_in_goal f_id ids
	  | Anonymous -> anomaly (Pp.str "Anonymous function")
      in
      let n_names_types,_ = decompose_lam_n nb_args body1 in
      let n_ids,ids =
	List.fold_left
	  (fun (n_ids,ids) (n_name,_) ->
	     match n_name with
	       | Name id ->
		   let n_id = next_ident_away_in_goal id ids in
		   n_id::n_ids,n_id::ids
	       | _ -> anomaly (Pp.str "anonymous argument")
	  )
	  ([],(f_id::ids))
	  n_names_types
      in
      let rec_arg_id = List.nth n_ids (rec_arg_num - 1) in
      let expr = instantiate_lambda func_body (mkVar f_id::(List.map mkVar n_ids)) in
      termination_proof_header
	is_mes
	input_type
	ids
	n_ids
	relation
	rec_arg_num
	rec_arg_id
	(fun rec_arg_id hrec acc_id acc_inv g -> 	      
	  (prove_terminate (fun infos -> tclIDTAC) 
	     { is_main_branch = true; (* we are on the main branche (i.e. still on a match ... with .... end *)
	       is_final = true;      (* and on leaf (more or less) *)
	       f_terminate = delayed_force coq_O;
	       nb_arg = nb_args;
	       concl_tac = concl_tac;
	       rec_arg_id = rec_arg_id;
	       is_mes = is_mes;
	       ih = hrec;
	       f_id = f_id;
	       f_constr = mkVar f_id;
	       func = func;
	       info = expr;
	       acc_inv = acc_inv;
	       acc_id = acc_id;
	       values_and_bounds = [];
	       eqs = [];
	       forbidden_ids = [];
	       args_assoc = []
	     }
	  ) 
	     g
	)
	(tclUSER_if_not_mes concl_tac)
	g
  end

let get_current_subgoals_types () =
  let p = Proof_global.give_me_the_proof () in
  let { Evd.it=sgs ; sigma=sigma } = Proof.V82.subgoals p in
    sigma, List.map (Goal.V82.abstract_type sigma) sgs

let build_and_l l =
  let and_constr =  Coqlib.build_coq_and () in
  let conj_constr = coq_conj () in
  let mk_and p1 p2 =
    Term.mkApp(and_constr,[|p1;p2|]) in
  let rec is_well_founded t = 
    match kind_of_term t with 
      | Prod(_,_,t') -> is_well_founded t'
      | App(_,_) -> 
	let (f,_) = decompose_app t in 
	eq_constr f (well_founded ())
      | _ -> 
	false
  in
  let compare t1 t2 = 
    let b1,b2= is_well_founded t1,is_well_founded t2 in 
    if (b1&&b2) || not (b1 || b2) then 0
    else if b1 && not b2 then 1 else -1
  in
  let l = List.sort compare l in 
  let rec f  = function
    | [] -> failwith "empty list of subgoals!"
    | [p] -> p,tclIDTAC,1
    | p1::pl ->
	let c,tac,nb = f pl in
	mk_and p1 c,
	tclTHENS
	  (Proofview.V82.of_tactic (apply (constr_of_global conj_constr)))
	  [tclIDTAC;
	   tac
	  ],nb+1
  in f l


let is_rec_res id =
  let rec_res_name = Id.to_string rec_res_id   in
  let id_name = Id.to_string id in
  try
    String.equal (String.sub id_name 0 (String.length rec_res_name)) rec_res_name
  with Invalid_argument _ -> false
 
let clear_goals =
  let rec clear_goal t =
    match kind_of_term t with
      | Prod(Name id as na,t',b) ->
	  let b' = clear_goal b in
	  if noccurn 1 b' && (is_rec_res id)
	  then Termops.pop b'
	  else if b' == b then t
	  else mkProd(na,t',b')
      | _ -> Term.map_constr clear_goal t
  in
  List.map clear_goal


let build_new_goal_type () =
  let sigma, sub_gls_types = get_current_subgoals_types () in
  (* Pp.msgnl (str "sub_gls_types1 := " ++ Util.prlist_with_sep (fun () -> Pp.fnl () ++ Pp.fnl ()) Printer.pr_lconstr sub_gls_types); *)
  let sub_gls_types = clear_goals sub_gls_types in
  (* Pp.msgnl (str "sub_gls_types2 := " ++ Pp.prlist_with_sep (fun () -> Pp.fnl () ++ Pp.fnl ()) Printer.pr_lconstr sub_gls_types); *)
  let res = build_and_l sub_gls_types in
    sigma, res

let is_opaque_constant c =
  let cb = Global.lookup_constant c in
  match cb.Declarations.const_body with
    | Declarations.OpaqueDef _ -> Vernacexpr.Opaque None
    | Declarations.Undef _ -> Vernacexpr.Opaque None
    | Declarations.Def _ -> Vernacexpr.Transparent

let open_new_goal build_proof sigma using_lemmas ref_ goal_name (gls_type,decompose_and_tac,nb_goal)   =
  (* Pp.msgnl (str "gls_type := " ++ Printer.pr_lconstr gls_type); *)
  let current_proof_name = get_current_proof_name () in
  let name = match goal_name with
    | Some s -> s
    | None   ->
	try add_suffix current_proof_name "_subproof"
	with e when CErrors.noncritical e ->
          anomaly (Pp.str "open_new_goal with an unamed theorem")
  in
  let na = next_global_ident_away name [] in
  if Termops.occur_existential gls_type then
    CErrors.error "\"abstract\" cannot handle existentials";
  let hook _ _ =
    let opacity =
      let na_ref = Libnames.Ident (Loc.ghost,na) in
      let na_global = Smartlocate.global_with_alias na_ref in
      match na_global with
	  ConstRef c -> is_opaque_constant c
	| _ -> anomaly ~label:"equation_lemma" (Pp.str "not a constant")
    in
    let lemma = mkConst (Names.Constant.make1 (Lib.make_kn na)) in
    ref_ := Some lemma ;
    let lid = ref [] in
    let h_num = ref (-1) in
    let env = Global.env () in
    Proof_global.discard_all ();
    build_proof (Evd.from_env env)
      (  fun gls ->
	   let hid = next_ident_away_in_goal h_id (pf_ids_of_hyps gls) in
	   observe_tclTHENLIST (str "")
	     [
	       Proofview.V82.of_tactic (generalize [lemma]);
	       Proofview.V82.of_tactic (Simple.intro hid);
	       (fun g ->
		  let ids = pf_ids_of_hyps g in
		  tclTHEN
		    (Proofview.V82.of_tactic (Elim.h_decompose_and (mkVar hid)))
		    (fun g ->
		       let ids' = pf_ids_of_hyps g in
		       lid := List.rev (List.subtract Id.equal ids' ids);
		       if List.is_empty !lid then lid := [hid];
		       tclIDTAC g
		    )
		    g
	       );
	     ] gls)
      (fun g ->
	 match kind_of_term (pf_concl g) with
	   | App(f,_) when eq_constr f (well_founded ()) ->
	       Proofview.V82.of_tactic (Auto.h_auto None [] (Some []))  g
	   | _ ->
	       incr h_num;
	       (observe_tac (str "finishing using")
		  (
		    tclCOMPLETE(
		      tclFIRST[
			tclTHEN
			  (Proofview.V82.of_tactic (eapply_with_bindings (mkVar (List.nth !lid !h_num), NoBindings)))
			  (Proofview.V82.of_tactic e_assumption);
		      Eauto.eauto_with_bases
			(true,5)
			[{ Tacexpr.delayed = fun _ sigma -> Sigma.here (Lazy.force refl_equal) sigma}]
			[Hints.Hint_db.empty empty_transparent_state false]
		      ]
		    )
		  )
	       )
      		 g)
;
    Lemmas.save_proof (Vernacexpr.Proved(opacity,None));
  in
  Lemmas.start_proof
    na
    (Decl_kinds.Global, false (* FIXME *), Decl_kinds.Proof Decl_kinds.Lemma)
    sigma gls_type
    (Lemmas.mk_hook hook);
  if Indfun_common.is_strict_tcc  ()
  then
    ignore (by (Proofview.V82.tactic (tclIDTAC)))
  else 
    begin
      ignore (by (Proofview.V82.tactic begin
	fun g ->
	  tclTHEN
	    (decompose_and_tac)
	    (tclORELSE
	       (tclFIRST
	 	  (List.map
	 	     (fun c ->
	 		Proofview.V82.of_tactic (Tacticals.New.tclTHENLIST 
	 		  [intros;
	 		   Simple.apply (fst (interp_constr (Global.env()) Evd.empty c)) (*FIXME*);
	 		   Tacticals.New.tclCOMPLETE Auto.default_auto
	 		  ])
	 	     )
	 	     using_lemmas)
	       ) tclIDTAC)
	    g end))
    end;
  try
    ignore (by (Proofview.V82.tactic tclIDTAC)); (* raises UserError _ if the proof is complete *)
  with UserError _ ->
    defined ()



let com_terminate
    tcc_lemma_name
    tcc_lemma_ref
    is_mes
    fonctional_ref
    input_type
    relation
    rec_arg_num
    thm_name using_lemmas
    nb_args ctx
    hook =
  let start_proof ctx (tac_start:tactic) (tac_end:tactic) =
    let (evmap, env) = Lemmas.get_current_context() in
    Lemmas.start_proof thm_name
      (Global, false (* FIXME *), Proof Lemma) ~sign:(Environ.named_context_val env)
      ctx (compute_terminate_type nb_args fonctional_ref) hook;

    ignore (by (Proofview.V82.tactic (observe_tac (str "starting_tac") tac_start)));
    ignore (by (Proofview.V82.tactic (observe_tac (str "whole_start") (whole_start tac_end nb_args is_mes fonctional_ref
    				   input_type relation rec_arg_num ))))
  in
  start_proof ctx tclIDTAC tclIDTAC;
  try
    let sigma, new_goal_type = build_new_goal_type () in
    let sigma = Evd.from_ctx (Evd.evar_universe_context sigma) in
    open_new_goal start_proof sigma
      using_lemmas tcc_lemma_ref
      (Some tcc_lemma_name)
      (new_goal_type);
  with Failure "empty list of subgoals!" ->
    (* a non recursive function declared with measure ! *)
    defined ()





let start_equation (f:global_reference) (term_f:global_reference)
  (cont_tactic:Id.t list -> tactic) g =
  let ids = pf_ids_of_hyps g in
  let terminate_constr = constr_of_global term_f in
  let nargs = nb_prod (fst (type_of_const terminate_constr)) (*FIXME*) in
  let x = n_x_id ids nargs in
  observe_tac (str "start_equation") (observe_tclTHENLIST (str "start_equation") [
    h_intros x;
    Proofview.V82.of_tactic (unfold_in_concl [(Locus.AllOccurrences, evaluable_of_global_reference f)]);
    observe_tac (str "simplest_case")
      (Proofview.V82.of_tactic (simplest_case (mkApp (terminate_constr,
                             Array.of_list (List.map mkVar x)))));
    observe_tac (str "prove_eq") (cont_tactic x)]) g;;

let (com_eqn : int -> Id.t ->
       global_reference -> global_reference -> global_reference
	 -> constr -> unit) =
  fun nb_arg eq_name functional_ref f_ref terminate_ref equation_lemma_type ->
    let opacity =
      match terminate_ref with
	| ConstRef c -> is_opaque_constant c
	| _ -> anomaly ~label:"terminate_lemma" (Pp.str "not a constant")
    in
    let (evmap, env) = Lemmas.get_current_context() in
    let evmap = Evd.from_ctx (Evd.evar_universe_context evmap) in
    let f_constr = constr_of_global f_ref in
    let equation_lemma_type = subst1 f_constr equation_lemma_type in
    (Lemmas.start_proof eq_name (Global, false, Proof Lemma)
       ~sign:(Environ.named_context_val env)
       evmap
       equation_lemma_type
       (Lemmas.mk_hook (fun _ _ -> ()));
     ignore (by
       (Proofview.V82.tactic (start_equation f_ref terminate_ref
	  (fun  x ->
	     prove_eq (fun _ -> tclIDTAC)
	       {nb_arg=nb_arg;
		f_terminate = constr_of_global terminate_ref; 
	        f_constr = f_constr; 
		concl_tac = tclIDTAC;
		func=functional_ref;
		info=(instantiate_lambda
	       		(def_of_const (constr_of_global functional_ref))
	       		(f_constr::List.map mkVar x)
		);
		is_main_branch = true;
		is_final = true;
		values_and_bounds = [];
		eqs = [];
		forbidden_ids = [];
		acc_inv = lazy (assert false);
		acc_id = Id.of_string "____";
		args_assoc = [];
		f_id = Id.of_string "______";
		rec_arg_id = Id.of_string "______";
		is_mes = false;
		ih = Id.of_string "______";
	       }
	  )
       ))); 
     (* (try Vernacentries.interp (Vernacexpr.VernacShow Vernacexpr.ShowProof) with _ -> ()); *)
(*      Vernacentries.interp (Vernacexpr.VernacShow Vernacexpr.ShowScript); *)
     Flags.silently (fun () -> Lemmas.save_proof (Vernacexpr.Proved(opacity,None))) () ; 
(*      Pp.msgnl (str "eqn finished"); *)
    );;


let recursive_definition is_mes function_name rec_impls type_of_f r rec_arg_num eq
    generate_induction_principle using_lemmas : unit =
  let env = Global.env() in
  let evd = ref (Evd.from_env env) in
  let function_type = interp_type_evars env evd type_of_f in
  let env = push_named (Context.Named.Declaration.LocalAssum (function_name,function_type)) env in
  (* Pp.msgnl (str "function type := " ++ Printer.pr_lconstr function_type);  *)
  let ty = interp_type_evars env evd ~impls:rec_impls eq in
  let evm, nf = Evarutil.nf_evars_and_universes !evd in
  let equation_lemma_type = nf_betaiotazeta (nf ty) in
  let function_type = nf function_type in
 (* Pp.msgnl (str "lemma type := " ++ Printer.pr_lconstr equation_lemma_type ++ fnl ()); *)
  let res_vars,eq' = decompose_prod equation_lemma_type in
  let env_eq' = Environ.push_rel_context (List.map (fun (x,y) -> LocalAssum (x,y)) res_vars) env in
  let eq' = nf_zeta env_eq' eq'  in
  let res =
(*     Pp.msgnl (str "res_var :=" ++ Printer.pr_lconstr_env (push_rel_context (List.map (function (x,t) -> (x,None,t)) res_vars) env) eq'); *)
(*     Pp.msgnl (str "rec_arg_num := " ++ str (string_of_int rec_arg_num)); *)
(*     Pp.msgnl (str "eq' := " ++ str (string_of_int rec_arg_num)); *)
    match kind_of_term eq' with
      | App(e,[|_;_;eq_fix|]) ->
	  mkLambda (Name function_name,function_type,subst_var function_name (compose_lam res_vars  eq_fix))
      | _ -> failwith "Recursive Definition (res not eq)"
  in
  let pre_rec_args,function_type_before_rec_arg = decompose_prod_n (rec_arg_num - 1) function_type in
  let (_, rec_arg_type, _) = destProd function_type_before_rec_arg in
  let arg_types = List.rev_map snd (fst (decompose_prod_n (List.length res_vars) function_type)) in
  let equation_id = add_suffix function_name "_equation" in
  let functional_id =  add_suffix function_name "_F" in
  let term_id = add_suffix function_name "_terminate" in
  let functional_ref = declare_fun functional_id (IsDefinition Decl_kinds.Definition) ~ctx:(snd (Evd.universe_context evm)) res in
  (* Refresh the global universes, now including those of _F *)
  let evm = Evd.from_env (Global.env ()) in
  let env_with_pre_rec_args = push_rel_context(List.map (function (x,t) -> LocalAssum (x,t)) pre_rec_args) env in
  let relation, evuctx =
    interp_constr env_with_pre_rec_args evm r
  in
  let evm = Evd.from_ctx evuctx in
  let tcc_lemma_name = add_suffix function_name "_tcc" in
  let tcc_lemma_constr = ref None in
  (* let _ = Pp.msgnl (str "relation := " ++ Printer.pr_lconstr_env env_with_pre_rec_args relation) in *)
  let hook _ _ = 
    let term_ref = Nametab.locate (qualid_of_ident term_id) in
    let f_ref = declare_f function_name (IsProof Lemma) arg_types term_ref in
    let _ = Extraction_plugin.Table.extraction_inline true [Ident (Loc.ghost,term_id)] in
    (*     message "start second proof"; *)
    let stop = 
      try com_eqn (List.length res_vars) equation_id functional_ref f_ref term_ref (subst_var function_name equation_lemma_type);
	  false
      with e when CErrors.noncritical e ->
	begin
	  if do_observe ()
	  then Feedback.msg_debug (str "Cannot create equation Lemma " ++ CErrors.print e)
	  else anomaly (Pp.str "Cannot create equation Lemma")
	  ;
	  true
	end
    in
    if not stop
    then
      let eq_ref = Nametab.locate (qualid_of_ident equation_id ) in
      let f_ref = destConst (constr_of_global f_ref)
      and functional_ref = destConst (constr_of_global functional_ref)
      and eq_ref = destConst (constr_of_global eq_ref) in
      generate_induction_principle f_ref tcc_lemma_constr
	functional_ref eq_ref rec_arg_num rec_arg_type (nb_prod res) relation;
      if Flags.is_verbose ()
      then msgnl (h 1 (Ppconstr.pr_id function_name ++
			 spc () ++ str"is defined" )++ fnl () ++
		    h 1 (Ppconstr.pr_id equation_id ++
			   spc () ++ str"is defined" )
      )
  in
  States.with_state_protection_on_exception (fun () ->
    com_terminate
      tcc_lemma_name
      tcc_lemma_constr
      is_mes functional_ref
      rec_arg_type
      relation rec_arg_num
      term_id
      using_lemmas
      (List.length res_vars)
      evm (Lemmas.mk_hook hook))
    ()