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
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
|
(***********************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
(* \VV/ *************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
(* $Id$ *)
open Pp
open Util
open Names
open Nameops
open Sign
open Term
open Termops
open Declarations
open Inductive
open Inductiveops
open Reductionops
open Environ
open Libnames
open Declare
open Evd
open Pfedit
open Tacred
open Rawterm
open Tacmach
open Proof_trees
open Proof_type
open Logic
open Evar_refiner
open Clenv
open Refiner
open Tacticals
open Hipattern
open Coqlib
open Nametab
open Tacexpr
open Decl_kinds
exception Bound
let rec nb_prod x =
let rec count n c =
match kind_of_term c with
Prod(_,_,t) -> count (n+1) t
| LetIn(_,a,_,t) -> count n (subst1 a t)
| Cast(c,_) -> count n c
| _ -> n
in count 0 x
(*********************************************)
(* Tactics *)
(*********************************************)
(****************************************)
(* General functions *)
(****************************************)
(*
let get_pairs_from_bindings =
let pair_from_binding = function
| [(Bindings binds)] -> binds
| _ -> error "not a binding list!"
in
List.map pair_from_binding
*)
let string_of_inductive c =
try match kind_of_term c with
| Ind ind_sp ->
let (mib,mip) = Global.lookup_inductive ind_sp in
string_of_id mip.mind_typename
| _ -> raise Bound
with Bound -> error "Bound head variable"
let rec head_constr_bound t l =
let t = strip_outer_cast(collapse_appl t) in
match kind_of_term t with
| Prod (_,_,c2) -> head_constr_bound c2 l
| LetIn (_,_,_,c2) -> head_constr_bound c2 l
| App (f,args) ->
head_constr_bound f (Array.fold_right (fun a l -> a::l) args l)
| Const _ -> t::l
| Ind _ -> t::l
| Construct _ -> t::l
| Var _ -> t::l
| _ -> raise Bound
let head_constr c =
try head_constr_bound c [] with Bound -> error "Bound head variable"
(*
let bad_tactic_args s l =
raise (RefinerError (BadTacticArgs (s,l)))
*)
(******************************************)
(* Primitive tactics *)
(******************************************)
let introduction = Tacmach.introduction
let intro_replacing = Tacmach.intro_replacing
let internal_cut = Tacmach.internal_cut
let internal_cut_rev = Tacmach.internal_cut_rev
let refine = Tacmach.refine
let convert_concl = Tacmach.convert_concl
let convert_hyp = Tacmach.convert_hyp
let thin = Tacmach.thin
let thin_body = Tacmach.thin_body
(* Moving hypotheses *)
let move_hyp = Tacmach.move_hyp
(* Renaming hypotheses *)
let rename_hyp = Tacmach.rename_hyp
(* Refine as a fixpoint *)
let mutual_fix = Tacmach.mutual_fix
let fix ido n = match ido with
| None -> mutual_fix (Pfedit.get_current_proof_name ()) n []
| Some id -> mutual_fix id n []
(* Refine as a cofixpoint *)
let mutual_cofix = Tacmach.mutual_cofix
let cofix = function
| None -> mutual_cofix (Pfedit.get_current_proof_name ()) []
| Some id -> mutual_cofix id []
(**************************************************************)
(* Reduction and conversion tactics *)
(**************************************************************)
type tactic_reduction = env -> evar_map -> constr -> constr
(* The following two tactics apply an arbitrary
reduction function either to the conclusion or to a
certain hypothesis *)
let reduct_in_concl redfun gl =
convert_concl_no_check (pf_reduce redfun gl (pf_concl gl)) gl
let reduct_in_hyp redfun idref gl =
let inhyp,id = match idref with
| InHyp id -> true, id
| InHypType id -> false, id in
let (_,c, ty) = pf_get_hyp gl id in
let redfun' = under_casts (pf_reduce redfun gl) in
match c with
| None -> convert_hyp_no_check (id,None,redfun' ty) gl
| Some b ->
if inhyp then (* Default for defs: reduce in body *)
convert_hyp_no_check (id,Some (redfun' b),ty) gl
else
convert_hyp_no_check (id,Some b,redfun' ty) gl
let reduct_option redfun = function
| Some id -> reduct_in_hyp redfun id
| None -> reduct_in_concl redfun
(* The following tactic determines whether the reduction
function has to be applied to the conclusion or
to the hypotheses. *)
let redin_combinator redfun = function
| [] -> reduct_in_concl redfun
| x -> (tclMAP (reduct_in_hyp redfun) x)
(* Now we introduce different instances of the previous tacticals *)
let change_and_check cv_pb t env sigma c =
if is_fconv cv_pb env sigma t c then
t
else
errorlabstrm "convert-check-hyp" (str "Not convertible")
(* Use cumulutavity only if changing the conclusion not a subterm *)
let change_on_subterm cv_pb t = function
| None -> change_and_check cv_pb t
| Some occl -> contextually occl (change_and_check CONV t)
let change_in_concl occl t = reduct_in_concl (change_on_subterm CUMUL t occl)
let change_in_hyp occl t = reduct_in_hyp (change_on_subterm CONV t occl)
let change occl c = function
| [] -> change_in_concl occl c
| l ->
if List.tl l <> [] & occl <> None then
error "No occurrences expected when changing several hypotheses";
tclMAP (change_in_hyp occl c) l
(* Pour usage interne (le niveau User est pris en compte par reduce) *)
let red_in_concl = reduct_in_concl red_product
let red_in_hyp = reduct_in_hyp red_product
let red_option = reduct_option red_product
let hnf_in_concl = reduct_in_concl hnf_constr
let hnf_in_hyp = reduct_in_hyp hnf_constr
let hnf_option = reduct_option hnf_constr
let simpl_in_concl = reduct_in_concl nf
let simpl_in_hyp = reduct_in_hyp nf
let simpl_option = reduct_option nf
let normalise_in_concl = reduct_in_concl compute
let normalise_in_hyp = reduct_in_hyp compute
let normalise_option = reduct_option compute
let unfold_in_concl loccname = reduct_in_concl (unfoldn loccname)
let unfold_in_hyp loccname = reduct_in_hyp (unfoldn loccname)
let unfold_option loccname = reduct_option (unfoldn loccname)
let pattern_option l = reduct_option (pattern_occs l)
(* A function which reduces accordingly to a reduction expression,
as the command Eval does. *)
let reduce redexp cl goal =
redin_combinator (reduction_of_redexp redexp) cl goal
(* Unfolding occurrences of a constant *)
let unfold_constr = function
| ConstRef sp -> unfold_in_concl [[],EvalConstRef sp]
| VarRef id -> unfold_in_concl [[],EvalVarRef id]
| _ -> errorlabstrm "unfold_constr" (str "Cannot unfold a non-constant.")
(*******************************************)
(* Introduction tactics *)
(*******************************************)
let is_section_variable id =
try let _ = Declare.find_section_variable id in true with Not_found -> false
let next_global_ident_from id avoid =
let rec next_rec id =
let id = next_ident_away_from id avoid in
if is_section_variable id || not (Declare.is_global id) then
id
else
next_rec (lift_ident id)
in
next_rec id
let next_global_ident_away id avoid =
let id = next_ident_away id avoid in
if is_section_variable id || not (Declare.is_global id) then
id
else
next_global_ident_from (lift_ident id) avoid
let fresh_id avoid id gl =
next_global_ident_away id (avoid@(pf_ids_of_hyps gl))
let id_of_name_with_default s = function
| Anonymous -> id_of_string s
| Name id -> id
let default_id gl = function
| (name,None,t) ->
(match kind_of_term (pf_whd_betadeltaiota gl (pf_type_of gl t)) with
| Sort (Prop _) -> (id_of_name_with_default "H" name)
| Sort (Type _) -> (id_of_name_with_default "X" name)
| _ -> anomaly "Wrong sort")
| (name,Some b,_) -> id_of_name_using_hdchar (pf_env gl) b name
(* Non primitive introduction tactics are treated by central_intro
There is possibly renaming, with possibly names to avoid and
possibly a move to do after the introduction *)
type intro_name_flag =
| IntroAvoid of identifier list
| IntroBasedOn of identifier * identifier list
| IntroMustBe of identifier
let find_name decl gl = function
| IntroAvoid idl -> fresh_id idl (default_id gl decl) gl
| IntroBasedOn (id,idl) -> fresh_id idl id gl
| IntroMustBe id ->
let id' = fresh_id [] id gl in
if id' <> id then error ((string_of_id id)^" is already used");
id'
let build_intro_tac id = function
| None -> introduction id
| Some dest -> tclTHEN (introduction id) (move_hyp true id dest)
let rec intro_gen name_flag move_flag force_flag gl =
match kind_of_term (pf_concl gl) with
| Prod (name,t,_) ->
build_intro_tac (find_name (name,None,t) gl name_flag) move_flag gl
| LetIn (name,b,t,_) ->
build_intro_tac (find_name (name,Some b,t) gl name_flag) move_flag gl
| _ ->
if not force_flag then raise (RefinerError IntroNeedsProduct);
try
tclTHEN
(reduce (Red true) [])
(intro_gen name_flag move_flag force_flag) gl
with Redelimination ->
errorlabstrm "Intro" (str "No product even after head-reduction")
let intro_mustbe_force id = intro_gen (IntroMustBe id) None true
let intro_using id = intro_gen (IntroBasedOn (id,[])) None false
let intro_force force_flag = intro_gen (IntroAvoid []) None force_flag
let intro = intro_force false
let introf = intro_force true
let introf_move_name destopt = intro_gen (IntroAvoid []) destopt true
(* For backwards compatibility *)
let central_intro = intro_gen
(**** Multiple introduction tactics ****)
let rec intros_using = function
[] -> tclIDTAC
| str::l -> tclTHEN (intro_using str) (intros_using l)
let intros = tclREPEAT (intro_force false)
let intro_erasing id = tclTHEN (thin [id]) (intro_using id)
let intros_replacing ids gls =
let rec introrec = function
| [] -> tclIDTAC
| id::tl ->
(tclTHEN (tclORELSE (intro_replacing id)
(tclORELSE (intro_erasing id) (* ?? *)
(intro_using id)))
(introrec tl))
in
introrec ids gls
(* User-level introduction tactics *)
let intro_move idopt idopt' = match idopt with
| None -> intro_gen (IntroAvoid []) idopt' true
| Some id -> intro_gen (IntroMustBe id) idopt' true
let pf_lookup_hypothesis_as_renamed env ccl = function
| AnonHyp n -> pf_lookup_index_as_renamed env ccl n
| NamedHyp id -> pf_lookup_name_as_renamed env ccl id
let pf_lookup_hypothesis_as_renamed_gen red h gl =
let env = pf_env gl in
let rec aux ccl =
match pf_lookup_hypothesis_as_renamed env ccl h with
| None when red -> aux (reduction_of_redexp (Red true) env Evd.empty ccl)
| x -> x
in
try aux (pf_concl gl)
with Redelimination -> None
let is_quantified_hypothesis id g =
match pf_lookup_hypothesis_as_renamed_gen true (NamedHyp id) g with
| Some _ -> true
| None -> false
let msg_quantified_hypothesis = function
| NamedHyp id ->
str "hypothesis " ++ pr_id id
| AnonHyp n ->
int n ++ str (match n with 1 -> "st" | 2 -> "nd" | _ -> "th") ++
str " non dependent hypothesis"
let depth_of_quantified_hypothesis red h gl =
match pf_lookup_hypothesis_as_renamed_gen red h gl with
| Some depth -> depth
| None ->
errorlabstrm "lookup_quantified_hypothesis"
(str "No " ++ msg_quantified_hypothesis h ++
str " in current goal" ++
if red then str " even after head-reduction" else mt ())
let intros_until_gen red h g =
tclDO (depth_of_quantified_hypothesis red h g) intro g
let intros_until_id id = intros_until_gen true (NamedHyp id)
let intros_until_n_gen red n = intros_until_gen red (AnonHyp n)
let intros_until = intros_until_gen true
let intros_until_n = intros_until_n_gen true
let intros_until_n_wored = intros_until_n_gen false
let try_intros_until tac = function
| NamedHyp id -> tclTHEN (tclTRY (intros_until_id id)) (tac id)
| AnonHyp n -> tclTHEN (intros_until_n n) (onLastHyp tac)
let rec intros_move = function
| [] -> tclIDTAC
| (hyp,destopt) :: rest ->
tclTHEN (intro_gen (IntroMustBe hyp) destopt false)
(intros_move rest)
let dependent_in_decl a (_,c,t) =
match c with
| None -> dependent a t
| Some body -> dependent a body || dependent a t
let move_to_rhyp rhyp gl =
let rec get_lhyp lastfixed depdecls = function
| [] ->
(match rhyp with
| None -> lastfixed
| Some h -> anomaly ("Hypothesis should occur: "^ (string_of_id h)))
| (hyp,c,typ) as ht :: rest ->
if Some hyp = rhyp then
lastfixed
else if List.exists (occur_var_in_decl (pf_env gl) hyp) depdecls then
get_lhyp lastfixed (ht::depdecls) rest
else
get_lhyp (Some hyp) depdecls rest
in
let sign = pf_hyps gl in
let (hyp,c,typ as decl) = List.hd sign in
match get_lhyp None [decl] (List.tl sign) with
| None -> tclIDTAC gl
| Some hypto -> move_hyp true hyp hypto gl
let rec intros_rmove = function
| [] -> tclIDTAC
| (hyp,destopt) :: rest ->
tclTHENLIST [ introduction hyp;
move_to_rhyp destopt;
intros_rmove rest ]
(****************************************************)
(* Resolution tactics *)
(****************************************************)
(* Refinement tactic: unification with the head of the head normal form
* of the type of a term. *)
let apply_type hdcty argl gl =
refine (applist (mkCast (mkMeta (new_meta()),hdcty),argl)) gl
let apply_term hdc argl gl =
refine (applist (hdc,argl)) gl
let bring_hyps hyps =
if hyps = [] then Refiner.tclIDTAC
else
(fun gl ->
let newcl = List.fold_right mkNamedProd_or_LetIn hyps (pf_concl gl) in
let f = mkCast (mkMeta (new_meta()),newcl) in
refine_no_check (mkApp (f, instance_from_named_context hyps)) gl)
(* Resolution with missing arguments *)
let apply_with_bindings (c,lbind) gl =
let apply =
match kind_of_term c with
| Lambda _ -> res_pf_cast
| _ -> res_pf
in
let (wc,kONT) = startWalk gl in
(* The actual type of the theorem. It will be matched against the
goal. If this fails, then the head constant will be unfolded step by
step. *)
let thm_ty0 = nf_betaiota (w_type_of wc c) in
let rec try_apply thm_ty =
try
let n = nb_prod thm_ty - nb_prod (pf_concl gl) in
if n<0 then error "Apply: theorem has not enough premisses.";
let clause = make_clenv_binding_apply wc n (c,thm_ty) lbind in
apply kONT clause gl
with (RefinerError _|UserError _|Failure _) as exn ->
let red_thm =
try red_product (w_env wc) (w_Underlying wc) thm_ty
with (Redelimination | UserError _) -> raise exn in
try_apply red_thm in
try try_apply thm_ty0
with (RefinerError _|UserError _|Failure _) ->
(* Last chance: if the head is a variable, apply may try
second order unification *)
let clause = make_clenv_binding_apply wc (-1) (c,thm_ty0) lbind in
apply kONT clause gl
let apply c = apply_with_bindings (c,NoBindings)
let apply_list = function
| c::l -> apply_with_bindings (c,ImplicitBindings l)
| _ -> assert false
(* Resolution with no reduction on the type *)
let apply_without_reduce c gl =
let (wc,kONT) = startWalk gl in
let clause = mk_clenv_type_of wc c in
res_pf kONT clause gl
(* Dead code
let refinew_scheme kONT clause gl = res_pf kONT clause gl
*)
(* A useful resolution tactic which, if c:A->B, transforms |- C into
|- B -> C and |- A (which is realized by Cut B;[Idtac|Apply c]
-------------------
Gamma |- c : A -> B Gamma |- ?2 : A
----------------------------------------
Gamma |- B Gamma |- ?1 : B -> C
-----------------------------------------------------
Gamma |- ? : C
*)
let cut_and_apply c gl =
let goal_constr = pf_concl gl in
match kind_of_term (pf_hnf_constr gl (pf_type_of gl c)) with
| Prod (_,c1,c2) when not (dependent (mkRel 1) c2) ->
tclTHENLAST
(apply_type (mkProd (Anonymous,c2,goal_constr)) [mkMeta(new_meta())])
(apply_term c [mkMeta (new_meta())]) gl
| _ -> error "Imp_elim needs a non-dependent product"
(**************************)
(* Cut tactics *)
(**************************)
let true_cut idopt c gl =
match kind_of_term (hnf_type_of gl c) with
| Sort s ->
let id =
match idopt with
| None ->
let d = match s with Prop _ -> "H" | Type _ -> "X" in
fresh_id [] (id_of_string d) gl
| Some id -> id
in
internal_cut id c gl
| _ -> error "Not a proposition or a type"
let cut c gl =
match kind_of_term (hnf_type_of gl c) with
| Sort _ ->
let id=next_name_away_with_default "H" Anonymous (pf_ids_of_hyps gl) in
let t = mkProd (Anonymous, c, pf_concl gl) in
tclTHENFIRST
(internal_cut_rev id c)
(tclTHEN (apply_type t [mkVar id]) (thin [id]))
gl
| _ -> error "Not a proposition or a type"
let cut_intro t = tclTHENFIRST (cut t) intro
let cut_replacing id t =
tclTHENFIRST
(cut t)
(tclORELSE
(intro_replacing id)
(tclORELSE (intro_erasing id)
(intro_using id)))
let cut_in_parallel l =
let rec prec = function
| [] -> tclIDTAC
| h::t -> tclTHENFIRST (cut h) (prec t)
in
prec (List.rev l)
(**************************)
(* Generalize tactics *)
(**************************)
let generalize_goal gl c cl =
let t = pf_type_of gl c in
match kind_of_term c with
| Var id ->
(* The choice of remembering or not a non dependent name has an impact
on the future Intro naming strategy! *)
(* if dependent c cl then mkNamedProd id t cl
else mkProd (Anonymous,t,cl) *)
mkNamedProd id t cl
| _ ->
let cl' = subst_term c cl in
if noccurn 1 cl' then
mkProd (Anonymous,t,cl)
(* On ne se casse pas la tete : on prend pour nom de variable
la premiere lettre du type, meme si "ci" est une
constante et qu'on pourrait prendre directement son nom *)
else
prod_name (Global.env()) (Anonymous, t, cl')
let generalize_dep c gl =
let env = pf_env gl in
let sign = pf_hyps gl in
let init_ids = ids_of_named_context (Global.named_context()) in
let rec seek toquant d =
if List.exists (fun (id,_,_) -> occur_var_in_decl env id d) toquant
or dependent_in_decl c d then
d::toquant
else
toquant in
let toq_rev = Sign.fold_named_context_reverse seek ~init:[] sign in
let qhyps = List.map (fun (id,_,_) -> id) toq_rev in
let to_quantify =
List.fold_left
(fun sign d -> add_named_decl d sign)
empty_named_context
toq_rev in
let tothin = List.filter (fun id -> not (List.mem id init_ids)) qhyps in
let tothin' =
match kind_of_term c with
| Var id when mem_named_context id sign & not (List.mem id init_ids)
-> id::tothin
| _ -> tothin
in
let cl' = it_mkNamedProd_or_LetIn (pf_concl gl) to_quantify in
let cl'' = generalize_goal gl c cl' in
let args = List.map mkVar qhyps in
tclTHEN
(apply_type cl'' (c::args))
(thin (List.rev tothin'))
gl
let generalize lconstr gl =
let newcl = List.fold_right (generalize_goal gl) lconstr (pf_concl gl) in
apply_type newcl lconstr gl
(* Faudra-t-il une version avec plusieurs args de generalize_dep ?
Cela peut-être troublant de faire "Generalize Dependent H n" dans
"n:nat; H:n=n |- P(n)" et d'échouer parce que H a disparu après la
généralisation dépendante par n.
let quantify lconstr =
List.fold_right
(fun com tac -> tclTHEN tac (tactic_com generalize_dep c))
lconstr
tclIDTAC
*)
(* A dependent cut rule à la sequent calculus
------------------------------------------
Sera simplifiable le jour où il y aura un let in primitif dans constr
[letin_tac b na c (occ_hyp,occ_ccl) gl] transforms
[...x1:T1(c),...,x2:T2(c),... |- G(c)] into
[...x:T;x1:T1(x),...,x2:T2(x),... |- G(x)] if [b] is false or
[...x:=c:T;x1:T1(x),...,x2:T2(x),... |- G(x)] if [b] is true
[occ_hyp,occ_ccl] tells which occurrences of [c] have to be substituted;
if [occ_hyp = []] and [occ_ccl = None] then [c] is substituted
wherever it occurs, otherwise [c] is substituted only in hyps
present in [occ_hyps] at the specified occurrences (everywhere if
the list of occurrences is empty), and in the goal at the specified
occurrences if [occ_goal] is not [None];
if name = Anonymous, the name is build from the first letter of the type;
The tactic first quantify the goal over x1, x2,... then substitute then
re-intro x1, x2,... at their initial place ([marks] is internally
used to remember the place of x1, x2, ...: it is the list of hypotheses on
the left of each x1, ...).
*)
let occurrences_of_hyp id = function
| None, [] -> (* Everywhere *) Some []
| _, occ_hyps -> try Some (List.assoc id occ_hyps) with Not_found -> None
let occurrences_of_goal = function
| None, [] -> (* Everywhere *) Some []
| Some gocc as x, _ -> x
| None, _ -> None
let everywhere (occ_ccl,occ_hyps) = (occ_ccl = None) & (occ_hyps = [])
let letin_abstract id c occs gl =
let env = pf_env gl in
let compute_dependency _ (hyp,_,_ as d) ctxt =
let d' =
try
match occurrences_of_hyp hyp occs with
| None -> raise Not_found
| Some occ ->
let newdecl = subst_term_occ_decl occ c d in
if d = newdecl then
if not (everywhere occs)
then raise (RefinerError (DoesNotOccurIn (c,hyp)))
else raise Not_found
else
(subst1_decl (mkVar id) newdecl, true)
with Not_found ->
(d,List.exists
(fun ((id,_,_),dep) -> dep && occur_var_in_decl env id d) ctxt)
in d'::ctxt
in
let ctxt' = fold_named_context compute_dependency env ~init:[] in
let compute_marks ((depdecls,marks as accu),lhyp) ((hyp,_,_) as d,b) =
if b then ((d::depdecls,(hyp,lhyp)::marks), lhyp)
else (accu, Some hyp) in
let (depdecls,marks),_ = List.fold_left compute_marks (([],[]),None) ctxt' in
let ccl = match occurrences_of_goal occs with
| None -> pf_concl gl
| Some occ -> subst1 (mkVar id) (subst_term_occ occ c (pf_concl gl))
in
(depdecls,marks,ccl)
let letin_tac with_eq name c occs gl =
let x = id_of_name_using_hdchar (Global.env()) (pf_type_of gl c) name in
let id =
if name = Anonymous then fresh_id [] x gl else
if not (mem_named_context x (pf_hyps gl)) then x else
error ("The variable "^(string_of_id x)^" is already declared") in
let (depdecls,marks,ccl)= letin_abstract id c occs gl in
let ctxt =
List.fold_left
(fun sign d -> add_named_decl d sign)
empty_named_context
depdecls in
let t = pf_type_of gl c in
let tmpcl = List.fold_right mkNamedProd_or_LetIn depdecls ccl in
let args = Array.to_list (instance_from_named_context depdecls) in
let newcl = mkNamedLetIn id c t tmpcl in
let lastlhyp = if marks=[] then None else snd (List.hd marks) in
tclTHENLIST
[ apply_type newcl args;
thin (List.map (fun (id,_,_) -> id) depdecls);
intro_gen (IntroMustBe id) lastlhyp false;
if with_eq then tclIDTAC else thin_body [id];
intros_move marks ] gl
let check_hypotheses_occurrences_list env (_,occl) =
let rec check acc = function
| (hyp,_) :: rest ->
if List.mem hyp acc then
error ("Hypothesis "^(string_of_id hyp)^" occurs twice");
if not (mem_named_context hyp (named_context env)) then
error ("No such hypothesis: " ^ (string_of_id hyp));
check (hyp::acc) rest
| [] -> ()
in check [] occl
let nowhere = (Some [],[])
let forward b na c = letin_tac b na c nowhere
(********************************************************************)
(* Exact tactics *)
(********************************************************************)
let exact_check c gl =
let concl = (pf_concl gl) in
let ct = pf_type_of gl c in
if pf_conv_x_leq gl ct concl then
refine_no_check c gl
else
error "Not an exact proof"
let exact_no_check = refine_no_check
let exact_proof c gl =
(* on experimente la synthese d'ise dans exact *)
let c = Constrintern.interp_casted_constr (project gl) (pf_env gl) c (pf_concl gl)
in refine_no_check c gl
let (assumption : tactic) = fun gl ->
let concl = pf_concl gl in
let rec arec = function
| [] -> error "No such assumption"
| (id,c,t)::rest ->
if pf_conv_x_leq gl t concl then refine_no_check (mkVar id) gl
else arec rest
in
arec (pf_hyps gl)
(*****************************************************************)
(* Modification of a local context *)
(*****************************************************************)
(* This tactic enables the user to remove hypotheses from the signature.
* Some care is taken to prevent him from removing variables that are
* subsequently used in other hypotheses or in the conclusion of the
* goal. *)
let clear ids gl = (* avant seul dyn_clear n'echouait pas en [] *)
if ids=[] then tclIDTAC gl else with_check (thin ids) gl
let clear_body = thin_body
(* Takes a list of booleans, and introduces all the variables
* quantified in the goal which are associated with a value
* true in the boolean list. *)
let rec intros_clearing = function
| [] -> tclIDTAC
| (false::tl) -> tclTHEN intro (intros_clearing tl)
| (true::tl) ->
tclTHENLIST
[ intro; onLastHyp (fun id -> clear [id]); intros_clearing tl]
(* Adding new hypotheses *)
let new_hyp mopt (c,lbind) g =
let (wc,kONT) = startWalk g in
let clause = make_clenv_binding wc (c,w_type_of wc c) lbind in
let (thd,tstack) = whd_stack (clenv_instance_template clause) in
let nargs = List.length tstack in
let cut_pf =
applist(thd,
match mopt with
| Some m -> if m < nargs then list_firstn m tstack else tstack
| None -> tstack)
in
(tclTHENLAST (tclTHEN (kONT clause.hook)
(cut (pf_type_of g cut_pf)))
((tclORELSE (apply cut_pf) (exact_no_check cut_pf)))) g
(************************)
(* Introduction tactics *)
(************************)
let constructor_tac boundopt i lbind gl =
let cl = pf_concl gl in
let (mind,redcl) = pf_reduce_to_quantified_ind gl cl in
let nconstr =
Array.length (snd (Global.lookup_inductive mind)).mind_consnames
and sigma = project gl in
if i=0 then error "The constructors are numbered starting from 1";
if i > nconstr then error "Not enough constructors";
begin match boundopt with
| Some expctdnum ->
if expctdnum <> nconstr then
error "Not the expected number of constructors"
| None -> ()
end;
let cons = mkConstruct (ith_constructor_of_inductive mind i) in
let apply_tac = apply_with_bindings (cons,lbind) in
(tclTHENLIST [convert_concl_no_check redcl; intros; apply_tac]) gl
let one_constructor i = constructor_tac None i
(* Try to apply the constructor of the inductive definition followed by
a tactic t given as an argument.
Should be generalize in Constructor (Fun c : I -> tactic)
*)
let any_constructor tacopt gl =
let t = match tacopt with None -> tclIDTAC | Some t -> t in
let mind = fst (pf_reduce_to_quantified_ind gl (pf_concl gl)) in
let nconstr =
Array.length (snd (Global.lookup_inductive mind)).mind_consnames in
if nconstr = 0 then error "The type has no constructors";
tclFIRST (List.map (fun i -> tclTHEN (one_constructor i NoBindings) t)
(interval 1 nconstr)) gl
let left = constructor_tac (Some 2) 1
let simplest_left = left NoBindings
let right = constructor_tac (Some 2) 2
let simplest_right = right NoBindings
let split = constructor_tac (Some 1) 1
let simplest_split = split NoBindings
(********************************************)
(* Elimination tactics *)
(********************************************)
(* kONT : ??
* wc : ??
* elimclause : ??
* inclause : ??
* gl : the current goal
*)
let last_arg c = match kind_of_term c with
| App (f,cl) -> array_last cl
| _ -> anomaly "last_arg"
let elimination_clause_scheme kONT elimclause indclause allow_K gl =
let indmv =
(match kind_of_term (last_arg (clenv_template elimclause).rebus) with
| Meta mv -> mv
| _ -> errorlabstrm "elimination_clause"
(str "The type of elimination clause is not well-formed"))
in
let elimclause' = clenv_fchain indmv elimclause indclause in
elim_res_pf kONT elimclause' allow_K gl
(* cast added otherwise tactics Case (n1,n2) generates (?f x y) and
* refine fails *)
let type_clenv_binding wc (c,t) lbind =
clenv_instance_template_type (make_clenv_binding wc (c,t) lbind)
(*
* Elimination tactic with bindings and using an arbitrary
* elimination constant called elimc. This constant should end
* with a clause (x:I)(P .. ), where P is a bound variable.
* The term c is of type t, which is a product ending with a type
* matching I, lbindc are the expected terms for c arguments
*)
let general_elim (c,lbindc) (elimc,lbindelimc) ?(allow_K=true) gl =
let (wc,kONT) = startWalk gl in
let ct = pf_type_of gl c in
let t = try snd (pf_reduce_to_quantified_ind gl ct) with UserError _ -> ct in
let indclause = make_clenv_binding wc (c,t) lbindc in
let elimt = w_type_of wc elimc in
let elimclause = make_clenv_binding wc (elimc,elimt) lbindelimc in
elimination_clause_scheme kONT elimclause indclause allow_K gl
(* Elimination tactic with bindings but using the default elimination
* constant associated with the type. *)
let find_eliminator c gl =
let env = pf_env gl in
let (ind,t) = reduce_to_quantified_ind env (project gl) (pf_type_of gl c) in
let s = elimination_sort_of_goal gl in
Indrec.lookup_eliminator ind s
(* with Not_found ->
let dir, base = repr_path (path_of_inductive env ind) in
let id = Indrec.make_elimination_ident base s in
errorlabstrm "default_elim"
(str "Cannot find the elimination combinator :" ++
pr_id id ++ spc () ++
str "The elimination of the inductive definition :" ++
pr_id base ++ spc () ++ str "on sort " ++
spc () ++ print_sort (new_sort_in_family s) ++
str " is probably not allowed")
(* lookup_eliminator prints the message *) *)
let default_elim (c,lbindc) gl =
general_elim (c,lbindc) (find_eliminator c gl,NoBindings) gl
let elim (c,lbindc) elim gl =
match elim with
| Some (elimc,lbindelimc) -> general_elim (c,lbindc) (elimc,lbindelimc) gl
| None -> general_elim (c,lbindc) (find_eliminator c gl,NoBindings) gl
(* The simplest elimination tactic, with no substitutions at all. *)
let simplest_elim c = default_elim (c,NoBindings)
(* Elimination in hypothesis *)
let elimination_in_clause_scheme kONT id elimclause indclause =
let (hypmv,indmv) =
match clenv_independent elimclause with
[k1;k2] -> (k1,k2)
| _ -> errorlabstrm "elimination_clause"
(str "The type of elimination clause is not well-formed") in
let elimclause' = clenv_fchain indmv elimclause indclause in
let hyp = mkVar id in
let hyp_typ = clenv_type_of elimclause' hyp in
let hypclause =
mk_clenv_from_n elimclause'.hook (Some 0) (hyp, hyp_typ) in
let elimclause'' = clenv_fchain hypmv elimclause' hypclause in
let new_hyp_prf = clenv_instance_template elimclause'' in
let new_hyp_typ = clenv_instance_template_type elimclause'' in
if eq_constr hyp_typ new_hyp_typ then
errorlabstrm "general_rewrite_in"
(str "Nothing to rewrite in " ++ pr_id id);
tclTHEN
(kONT elimclause''.hook)
(tclTHENS
(cut new_hyp_typ)
[ (* Try to insert the new hyp at the same place *)
tclORELSE (intro_replacing id)
(tclTHEN (clear [id]) (introduction id));
refine_no_check new_hyp_prf])
let general_elim_in id (c,lbindc) (elimc,lbindelimc) gl =
let (wc,kONT) = startWalk gl in
let ct = pf_type_of gl c in
let t = try snd (pf_reduce_to_quantified_ind gl ct) with UserError _ -> ct in
let indclause = make_clenv_binding wc (c,t) lbindc in
let elimt = w_type_of wc elimc in
let elimclause = make_clenv_binding wc (elimc,elimt) lbindelimc in
elimination_in_clause_scheme kONT id elimclause indclause gl
(* Case analysis tactics *)
let general_case_analysis (c,lbindc) gl =
let env = pf_env gl in
let (mind,_) = pf_reduce_to_quantified_ind gl (pf_type_of gl c) in
let sigma = project gl in
let sort = elimination_sort_of_goal gl in
let case = if occur_term c (pf_concl gl) then Indrec.make_case_dep
else Indrec.make_case_gen in
let elim = case env sigma mind sort in
general_elim (c,lbindc) (elim,NoBindings) gl
let simplest_case c = general_case_analysis (c,NoBindings)
(*****************************)
(* Decomposing introductions *)
(*****************************)
let clear_last = tclLAST_HYP (fun c -> (clear [destVar c]))
let case_last = tclLAST_HYP simplest_case
let rec intro_pattern destopt = function
| IntroWildcard ->
tclTHEN intro clear_last
| IntroIdentifier id ->
intro_gen (IntroMustBe id) destopt true
| IntroOrAndPattern l ->
tclTHEN introf
(tclTHENS
(tclTHEN case_last clear_last)
(List.map (intros_pattern destopt) l))
and intros_pattern destopt l = tclMAP (intro_pattern destopt) l
let intro_patterns = function
| [] -> tclREPEAT intro
| l -> intros_pattern None l
let h_intro_patterns l = abstract_tactic (TacIntroPattern l) (intro_patterns l)
(*
* A "natural" induction tactic
*
- [H0:T0, ..., Hi:Ti, hyp0:P->I(args), Hi+1:Ti+1, ..., Hn:Tn |-G] is the goal
- [hyp0] is the induction hypothesis
- we extract from [args] the variables which are not rigid parameters
of the inductive type, this is [indvars] (other terms are forgotten);
[indhyps] are the ones which actually are declared in context
(done in [find_atomic_param_of_ind])
- we look for all hyps depending of [hyp0] or one of [indvars]:
this is [dephyps] of types [deptyps] respectively
- [statuslist] tells for each hyps in [dephyps] after which other hyp
fixed in the context they must be moved (when induction is done)
- [hyp0succ] is the name of the hyp fixed in the context after which to
move the subterms of [hyp0succ] in the i-th branch where it is supposed
to be the i-th constructor of the inductive type.
Strategy: (cf in [induction_from_context])
- requantify and clear all [dephyps]
- apply induction on [hyp0]
- clear [indhyps] and [hyp0]
- in the i-th subgoal, intro the arguments of the i-th constructor
of the inductive type after [hyp0succ] (done in
[induct_discharge]) let the induction hypotheses on top of the
hyps because they may depend on variables between [hyp0] and the
top. A counterpart is that the dep hyps programmed to be intro-ed
on top must now be intro-ed after the induction hypotheses
- move each of [dephyps] at the right place following the
[statuslist]
*)
let rec str_intro_pattern = function
| IntroOrAndPattern pll ->
"["^(String.concat "|"
(List.map
(fun pl -> String.concat " " (List.map str_intro_pattern pl)) pll))
^"]"
| IntroWildcard -> "_"
| IntroIdentifier id -> string_of_id id
let check_unused_names names =
if names <> [] & Options.is_verbose () then
let s = if List.tl names = [] then " " else "s " in
let names = String.concat " " (List.map str_intro_pattern names) in
warning ("Unused introduction pattern"^s^": "^names)
let rec first_name_buggy = function
| IntroOrAndPattern [] -> None
| IntroOrAndPattern ([]::l) -> first_name_buggy (IntroOrAndPattern l)
| IntroOrAndPattern ((p::_)::_) -> first_name_buggy p
| IntroWildcard -> None
| IntroIdentifier id -> Some id
(* We recompute recargs because we are not sure the elimination lemma
comes from a canonically generated one *)
(* dead code ?
let rec is_rec_arg env sigma indpath t =
try
let (ind_sp,_) = find_mrectype env sigma t in
path_of_inductive env ind_sp = indpath
with Not_found -> false
let rec recargs indpath env sigma t =
match kind_of_term (whd_betadeltaiota env sigma t) with
| Prod (na,t,c2) ->
(is_rec_arg env sigma indpath t)
::(recargs indpath (push_rel_assum (na,t) env) sigma c2)
| _ -> []
*)
let induct_discharge old_style mind statuslists cname destopt avoid ra names gl
=
let (lstatus,rstatus) = statuslists in
let tophyp = ref None in
let n = List.fold_left (fun n b -> if b then n+1 else n) 0 ra in
let recvarname, hyprecname, avoid =
if old_style (* = V6.3 version of Induction on hypotheses *)
then
let recvarname =
if n=1 then
cname
else (* To force renumbering if there is only one *)
make_ident (string_of_id cname) (Some 1) in
recvarname, add_prefix "Hrec" recvarname, avoid
else
let hyprecname =
add_prefix "IH"
(if atompart_of_id cname <> "H"
then cname
else (snd (Global.lookup_inductive mind)).mind_typename) in
let avoid =
if n=1 (* Only one recursive argument *)
or
(* Rem: no recursive argument (especially if Destruct) *)
n=0 (* & atompart_of_id cname <> "H" (* for 7.1 compatibility *)*)
then avoid
else
(* Forbid to use cname, cname0, hyprecname and hyprecname0 *)
(* in order to get names such as f1, f2, ... *)
let avoid =
(make_ident (string_of_id cname) (Some 0)) ::(*here for 7.1 cmpat*)
(make_ident (string_of_id hyprecname) None) ::
(make_ident (string_of_id hyprecname) (Some 0)) :: avoid in
if atompart_of_id cname <> "H" then
(make_ident (string_of_id cname) None) :: avoid
else avoid in
cname, hyprecname, avoid
in
let rec peel_tac ra names gl = match ra with
| true :: ra' ->
let recpat,hyprec,names = match names with
| [] ->
(IntroIdentifier (fresh_id avoid recvarname gl),
IntroIdentifier (fresh_id avoid hyprecname gl), [])
| [IntroIdentifier id as pat] ->
(pat,
IntroIdentifier (next_ident_away (add_prefix "IH" id) avoid),
[])
| [pat] ->
(pat, IntroIdentifier (fresh_id avoid hyprecname gl), [])
| pat1::pat2::names -> (pat1,pat2,names) in
(* This is buggy for intro-or-patterns with different first hypnames *)
if !tophyp=None then tophyp := first_name_buggy hyprec;
tclTHENLIST
[ intros_pattern destopt [recpat];
intros_pattern None [hyprec];
peel_tac ra' names ] gl
| false :: ra' ->
let introtac,names = match names with
| [] -> intro_gen (IntroAvoid avoid) destopt false, []
| pat::names -> intros_pattern destopt [pat],names in
tclTHEN introtac (peel_tac ra' names) gl
| [] ->
check_unused_names names;
tclIDTAC gl
in
let intros_move lstatus =
let newlstatus = (* if some IH has taken place at the top of hyps *)
List.map (function (hyp,None) -> (hyp,!tophyp) | x -> x) lstatus in
intros_move newlstatus
in
tclTHENLIST [ peel_tac ra names;
intros_rmove rstatus;
intros_move lstatus ] gl
(* - le recalcul de indtyp à chaque itération de atomize_one est pour ne pas
s'embêter à regarder si un letin_tac ne fait pas des
substitutions aussi sur l'argument voisin *)
(* Marche pas... faut prendre en compte l'occurrence précise... *)
let atomize_param_of_ind hyp0 gl =
let tmptyp0 = pf_get_hyp_typ gl hyp0 in
let (mind,typ0) = pf_reduce_to_quantified_ind gl tmptyp0 in
let (mib,mip) = Global.lookup_inductive mind in
let nparams = mip.mind_nparams in
let prods, indtyp = decompose_prod typ0 in
let argl = snd (decompose_app indtyp) in
let params = list_firstn nparams argl in
(* le gl est important pour ne pas préévaluer *)
let rec atomize_one i avoid gl =
if i<>nparams then
let tmphyp0 = pf_get_hyp_typ gl hyp0 in
(* If argl <> [], we expect typ0 not to be quantified, in order to
avoid bound parameters... then we call pf_reduce_to_atomic_ind *)
let (_,indtyp) = pf_reduce_to_atomic_ind gl tmptyp0 in
let argl = snd (decompose_app indtyp) in
let c = List.nth argl (i-1) in
match kind_of_term c with
| Var id when not (List.exists (occur_var (pf_env gl) id) avoid) ->
atomize_one (i-1) ((mkVar id)::avoid) gl
| Var id ->
let x = fresh_id [] id gl in
tclTHEN
(letin_tac true (Name x) (mkVar id) (None,[]))
(atomize_one (i-1) ((mkVar x)::avoid)) gl
| _ ->
let id = id_of_name_using_hdchar (Global.env()) (pf_type_of gl c)
Anonymous in
let x = fresh_id [] id gl in
tclTHEN
(letin_tac true (Name x) c (None,[]))
(atomize_one (i-1) ((mkVar x)::avoid)) gl
else
tclIDTAC gl
in
atomize_one (List.length argl) params gl
let find_atomic_param_of_ind mind indtyp =
let (mib,mip) = Global.lookup_inductive mind in
let nparams = mip.mind_nparams in
let argl = snd (decompose_app indtyp) in
let argv = Array.of_list argl in
let params = list_firstn nparams argl in
let indvars = ref Idset.empty in
for i = nparams to (Array.length argv)-1 do
match kind_of_term argv.(i) with
| Var id
when not (List.exists (occur_var (Global.env()) id) params) ->
indvars := Idset.add id !indvars
| _ -> ()
done;
Idset.elements !indvars
(* [cook_sign] builds the lists [indhyps] of hyps that must be
erased, the lists of hyps to be generalize [(hdeps,tdeps)] on the
goal together with the places [(lstatus,rstatus)] where to re-intro
them after induction. To know where to re-intro the dep hyp, we
remember the name of the hypothesis [lhyp] after which (if the dep
hyp is more recent than [hyp0]) or [rhyp] before which (if older
than [hyp0]) its equivalent must be moved when the induction has
been applied. Since computation of dependencies and [rhyp] is from
more ancient (on the right) to more recent hyp (on the left) but
the computation of [lhyp] progresses from the other way, [cook_hyp]
is in two passes (an alternative would have been to write an
higher-order algorithm). We strongly use references to reduce
the accumulation of arguments.
To summarize, the situation looks like this
Goal(n,x) -| H6:(Q n); x:A; H5:True; H4:(le O n); H3:(P n); H2:True; n:nat
Left Right
Induction hypothesis is H4 ([hyp0])
Variable parameters of (le O n) is the singleton list with "n" ([indvars])
Part of [indvars] really in context is the same ([indhyps])
The dependent hyps are H3 and H6 ([dephyps])
For H3 the memorized places are H5 ([lhyp]) and H2 ([rhyp])
because these names are among the hyp which are fixed through the induction
For H6 the neighbours are None ([lhyp]) and H5 ([rhyp])
For H3, because on the right of H4, we remember rhyp (here H2)
For H6, because on the left of H4, we remember lhyp (here None)
For H4, we remember lhyp (here H5)
The right neighbour is then translated into the left neighbour
because move_hyp tactic needs the name of the hyp _after_ which we
move the hyp to move.
But, say in the 2nd subgoal of the hypotheses, the goal will be
(m:nat)((P m)->(Q m)->(Goal m)) -> (P Sm)-> (Q Sm)-> (Goal Sm)
^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^
both go where H4 was goes where goes where
H3 was H6 was
We have to intro and move m and the recursive hyp first, but then
where to move H3 ??? Only the hyp on its right is relevant, but we
have to translate it into the name of the hyp on the left
Note: this case where some hyp(s) in [dephyps] has(have) the same
left neighbour as [hyp0] is the only problematic case with right
neighbours. For the other cases (e.g. an hyp H1:(R n) between n and H2
would have posed no problem. But for uniformity, we decided to use
the right hyp for all hyps on the right of H4.
Others solutions are welcome *)
exception Shunt of identifier option
let cook_sign hyp0 indvars env =
(* First phase from L to R: get [indhyps], [decldep] and [statuslist]
for the hypotheses before (= more ancient than) hyp0 (see above) *)
let allindhyps = hyp0::indvars in
let indhyps = ref [] in
let decldeps = ref [] in
let ldeps = ref [] in
let rstatus = ref [] in
let lstatus = ref [] in
let before = ref true in
let seek_deps env (hyp,_,_ as decl) rhyp =
if hyp = hyp0 then begin
before:=false;
None (* fake value *)
end else if List.mem hyp indvars then begin
indhyps := hyp::!indhyps;
rhyp
end else
if (List.exists (fun id -> occur_var_in_decl env id decl) allindhyps
or List.exists (fun (id,_,_) -> occur_var_in_decl env id decl)
!decldeps)
then begin
decldeps := decl::!decldeps;
if !before then
rstatus := (hyp,rhyp)::!rstatus
else
ldeps := hyp::!ldeps; (* status computed in 2nd phase *)
Some hyp end
else
Some hyp
in
let _ = fold_named_context seek_deps env ~init:None in
(* 2nd phase from R to L: get left hyp of [hyp0] and [lhyps] *)
let compute_lstatus lhyp (hyp,_,_ as d) =
if hyp = hyp0 then raise (Shunt lhyp);
if List.mem hyp !ldeps then begin
lstatus := (hyp,lhyp)::!lstatus;
lhyp
end else
if List.mem hyp !indhyps then lhyp else (Some hyp)
in
try
let _ = fold_named_context_reverse compute_lstatus ~init:None env in
anomaly "hyp0 not found"
with Shunt lhyp0 ->
let statuslists = (!lstatus,List.rev !rstatus) in
(statuslists, lhyp0, !indhyps, !decldeps)
let induction_tac varname typ (elimc,elimt,lbindelimc) gl =
let c = mkVar varname in
let (wc,kONT) = startWalk gl in
let indclause = make_clenv_binding wc (c,typ) NoBindings in
let elimclause =
make_clenv_binding wc (mkCast (elimc,elimt),elimt) lbindelimc in
elimination_clause_scheme kONT elimclause indclause true gl
let compute_induction_names n names =
let names = if names = [] then Array.make n [] else Array.of_list names in
if Array.length names <> n then
errorlabstrm "induction_from_context"
(str "Expect " ++ int n ++ str " lists of names");
names
let is_indhyp p n t =
let c,_ = decompose_app t in
match kind_of_term c with
| Rel k when p < k & k <= p + n -> true
| _ -> false
(* We check that the eliminator has been build by Coq (usual *)
(* eliminator _ind, _rec or _rect, or eliminator built by Scheme) *)
let compute_elim_signature_and_roughly_check elimt mind =
let (mib,mip) = Global.lookup_inductive mind in
let lra = dest_subterms mip.mind_recargs in
let nconstr = Array.length mip.mind_consnames in
let _,elimt2 = decompose_prod_n mip.mind_nparams elimt in
let n = nb_prod elimt2 in
let npred = n - nconstr - mip.mind_nrealargs - 1 in
let rec check_branch p c ra = match kind_of_term c, ra with
| Prod (_,_,c), r :: ra' ->
(match dest_recarg r, kind_of_term c with
| Mrec i, Prod (_,t,c) when is_indhyp (p+1) npred t ->
true::(check_branch (p+2) c ra')
| _ -> false::(check_branch (p+1) c ra'))
| LetIn (_,_,_,c), ra' -> false::(check_branch (p+1) c ra)
| _, [] -> []
| _ ->
error"Not a recursive eliminator: some constructor argument is lacking"
in
let rec check_elim c n =
if n = nconstr then []
else match kind_of_term c with
| Prod (_,t,c) -> (check_branch n t lra.(n)) :: (check_elim c (n+1))
| _ -> error "Not an eliminator: some constructor case is lacking" in
let _,elimt3 = decompose_prod_n npred elimt2 in
Array.of_list (check_elim elimt3 0)
let induction_from_context isrec style elim hyp0 names gl =
(*test suivant sans doute inutile car refait par le letin_tac*)
if List.mem hyp0 (ids_of_named_context (Global.named_context())) then
errorlabstrm "induction"
(str "Cannot generalize a global variable");
let tmptyp0 = pf_get_hyp_typ gl hyp0 in
let env = pf_env gl in
let (mind,typ0) = pf_reduce_to_quantified_ind gl tmptyp0 in
let elimc,lbindelimc = match elim with
| None ->
let s = elimination_sort_of_goal gl in
(if isrec then Indrec.lookup_eliminator mind s
else Indrec.make_case_gen env (project gl) mind s),
NoBindings
| Some elim ->
(* Not really robust: no control on the form of the combinator *)
elim in
let elimt = pf_type_of gl elimc in
let indvars = find_atomic_param_of_ind mind (snd (decompose_prod typ0)) in
let (statlists,lhyp0,indhyps,deps) = cook_sign hyp0 indvars env in
let tmpcl = it_mkNamedProd_or_LetIn (pf_concl gl) deps in
let lr = compute_elim_signature_and_roughly_check elimt mind in
let names = compute_induction_names (Array.length lr) names in
let dephyps = List.map (fun (id,_,_) -> id) deps in
let args =
List.fold_left
(fun a (id,b,_) -> if b = None then (mkVar id)::a else a) [] deps in
(* Magistral effet de bord: si hyp0 a des arguments, ceux d'entre
eux qui ouvrent de nouveaux buts arrivent en premier dans la
liste des sous-buts du fait qu'ils sont le plus à gauche dans le
combinateur engendré par make_case_gen (un "Cases (hyp0 ?) of
...") et il faut alors appliquer tclTHENLASTn; en revanche,
comme lookup_eliminator renvoie un combinateur de la forme
"ind_rec ... (hyp0 ?)", les buts correspondant à des arguments de
hyp0 sont maintenant à la fin et c'est tclTHENFIRSTn qui marche !!! *)
tclTHENLIST
[ if deps = [] then tclIDTAC else apply_type tmpcl args;
thin dephyps;
(if isrec then tclTHENFIRSTn else tclTHENLASTn)
(tclTHEN
(induction_tac hyp0 typ0 (elimc,elimt,lbindelimc))
(thin (hyp0::indhyps)))
(array_map2
(induct_discharge style mind statlists hyp0 lhyp0
(List.rev dephyps)) lr names)
]
gl
let induction_with_atomization_of_ind_arg isrec elim names hyp0 =
tclTHEN
(atomize_param_of_ind hyp0)
(induction_from_context isrec false elim hyp0 names)
(* This is Induction since V7 ("natural" induction both in quantified
premisses and introduced ones) *)
let new_induct_gen isrec elim names c gl =
match kind_of_term c with
| Var id when not (mem_named_context id (Global.named_context())) ->
induction_with_atomization_of_ind_arg isrec elim names id gl
| _ ->
let x = id_of_name_using_hdchar (Global.env()) (pf_type_of gl c)
Anonymous in
let id = fresh_id [] x gl in
tclTHEN
(letin_tac true (Name id) c (None,[]))
(induction_with_atomization_of_ind_arg isrec elim names id) gl
let new_induct_destruct isrec c elim names = match c with
| ElimOnConstr c -> new_induct_gen isrec elim names c
| ElimOnAnonHyp n ->
tclTHEN (intros_until_n n)
(tclLAST_HYP (new_induct_gen isrec elim names))
(* Identifier apart because id can be quantified in goal and not typable *)
| ElimOnIdent (_,id) ->
tclTHEN (tclTRY (intros_until_id id))
(new_induct_gen isrec elim names (mkVar id))
let new_induct = new_induct_destruct true
let new_destruct = new_induct_destruct false
(* The registered tactic, which calls the default elimination
* if no elimination constant is provided. *)
(* Induction tactics *)
(* This was Induction before 6.3 (induction only in quantified premisses) *)
let raw_induct s = tclTHEN (intros_until_id s) (tclLAST_HYP simplest_elim)
let raw_induct_nodep n = tclTHEN (intros_until_n n) (tclLAST_HYP simplest_elim)
(* This was Induction in 6.3 (hybrid form) *)
let old_induct_id s =
tclORELSE (raw_induct s) (induction_from_context true true None s [])
let old_induct_nodep = raw_induct_nodep
let old_induct = function
| NamedHyp id -> old_induct_id id
| AnonHyp n -> old_induct_nodep n
(* Destruction tactics *)
let old_destruct_id s =
(tclTHEN (intros_until_id s) (tclLAST_HYP simplest_case))
let old_destruct_nodep n =
(tclTHEN (intros_until_n n) (tclLAST_HYP simplest_case))
let old_destruct = function
| NamedHyp id -> old_destruct_id id
| AnonHyp n -> old_destruct_nodep n
(*
* Eliminations giving the type instead of the proof.
* These tactics use the default elimination constant and
* no substitutions at all.
* May be they should be integrated into Elim ...
*)
let elim_scheme_type elim t gl =
let (wc,kONT) = startWalk gl in
let clause = mk_clenv_type_of wc elim in
match kind_of_term (last_arg (clenv_template clause).rebus) with
| Meta mv ->
let clause' =
(* t is inductive, then CUMUL or CONV is irrelevant *)
clenv_unify true CUMUL t (clenv_instance_type clause mv) clause in
elim_res_pf kONT clause' true gl
| _ -> anomaly "elim_scheme_type"
let elim_type t gl =
let (ind,t) = pf_reduce_to_atomic_ind gl t in
let elimc = Indrec.lookup_eliminator ind (elimination_sort_of_goal gl) in
elim_scheme_type elimc t gl
let case_type t gl =
let (ind,t) = pf_reduce_to_atomic_ind gl t in
let env = pf_env gl in
let elimc = Indrec.make_case_gen env (project gl) ind (elimination_sort_of_goal gl) in
elim_scheme_type elimc t gl
(* Some eliminations frequently used *)
(* These elimination tactics are particularly adapted for sequent
calculus. They take a clause as argument, and yield the
elimination rule if the clause is of the form (Some id) and a
suitable introduction rule otherwise. They do not depend on
the name of the eliminated constant, so they can be also
used on ad-hoc disjunctions and conjunctions introduced by
the user.
-- Eduardo Gimenez (11/8/97)
HH (29/5/99) replaces failures by specific error messages
*)
let andE id gl =
let t = pf_get_hyp_typ gl id in
if is_conjunction (pf_hnf_constr gl t) then
(tclTHEN (simplest_elim (mkVar id)) (tclDO 2 intro)) gl
else
errorlabstrm "andE"
(str("Tactic andE expects "^(string_of_id id)^" is a conjunction."))
let dAnd cls gl =
match cls with
| None -> simplest_split gl
| Some id -> andE id gl
let orE id gl =
let t = pf_get_hyp_typ gl id in
if is_disjunction (pf_hnf_constr gl t) then
(tclTHEN (simplest_elim (mkVar id)) intro) gl
else
errorlabstrm "orE"
(str("Tactic orE expects "^(string_of_id id)^" is a disjunction."))
let dorE b cls gl =
match cls with
| (Some id) -> orE id gl
| None -> (if b then right else left) NoBindings gl
let impE id gl =
let t = pf_get_hyp_typ gl id in
if is_imp_term (pf_hnf_constr gl t) then
let (dom, _, rng) = destProd (pf_hnf_constr gl t) in
tclTHENLAST
(cut_intro rng)
(apply_term (mkVar id) [mkMeta (new_meta())]) gl
else
errorlabstrm "impE"
(str("Tactic impE expects "^(string_of_id id)^
" is a an implication."))
let dImp cls gl =
match cls with
| None -> intro gl
| Some id -> impE id gl
(************************************************)
(* Tactics related with logic connectives *)
(************************************************)
(* Reflexivity tactics *)
let reflexivity gl =
match match_with_equation (pf_concl gl) with
| None -> error "The conclusion is not a substitutive equation"
| Some (hdcncl,args) -> one_constructor 1 NoBindings gl
let intros_reflexivity = (tclTHEN intros reflexivity)
(* Symmetry tactics *)
(* This tactic first tries to apply a constant named sym_eq, where eq
is the name of the equality predicate. If this constant is not
defined and the conclusion is a=b, it solves the goal doing (Cut
b=a;Intro H;Case H;Constructor 1) *)
let symmetry gl =
match match_with_equation (pf_concl gl) with
| None -> error "The conclusion is not a substitutive equation"
| Some (hdcncl,args) ->
let hdcncls = string_of_inductive hdcncl in
begin
try
(apply (pf_parse_const gl ("sym_"^hdcncls)) gl)
with _ ->
let symc = match args with
| [t1; c1; t2; c2] -> mkApp (hdcncl, [| t2; c2; t1; c1 |])
| [typ;c1;c2] -> mkApp (hdcncl, [| typ; c2; c1 |])
| [c1;c2] -> mkApp (hdcncl, [| c2; c1 |])
| _ -> assert false
in
tclTHENLAST (cut symc)
(tclTHENLIST
[ intro;
tclLAST_HYP simplest_case;
one_constructor 1 NoBindings ])
gl
end
let symmetry_in id gl =
let ctype = pf_type_of gl (mkVar id) in
let sign,t = decompose_prod_assum ctype in
match match_with_equation t with
| None -> (* Do not deal with setoids yet *)
error "The term provided does not end with an equation"
| Some (hdcncl,args) ->
let symccl = match args with
| [t1; c1; t2; c2] -> mkApp (hdcncl, [| t2; c2; t1; c1 |])
| [typ;c1;c2] -> mkApp (hdcncl, [| typ; c2; c1 |])
| [c1;c2] -> mkApp (hdcncl, [| c2; c1 |])
| _ -> assert false in
tclTHENS (cut (it_mkProd_or_LetIn symccl sign))
[ intro_replacing id;
tclTHENLIST [ intros; symmetry; apply (mkVar id); assumption ] ]
gl
let intros_symmetry = function
| None -> tclTHEN intros symmetry
| Some id -> symmetry_in id
(* Transitivity tactics *)
(* This tactic first tries to apply a constant named trans_eq, where eq
is the name of the equality predicate. If this constant is not
defined and the conclusion is a=b, it solves the goal doing
Cut x1=x2;
[Cut x2=x3; [Intros e1 e2; Case e2;Assumption
| Idtac]
| Idtac]
--Eduardo (19/8/97)
*)
let transitivity t gl =
match match_with_equation (pf_concl gl) with
| None -> error "The conclusion is not a substitutive equation"
| Some (hdcncl,args) ->
let hdcncls = string_of_inductive hdcncl in
begin
try
apply_list [(pf_parse_const gl ("trans_"^hdcncls));t] gl
with _ ->
let eq1, eq2 = match args with
| [typ1;c1;typ2;c2] -> let typt = pf_type_of gl t in
( mkApp(hdcncl, [| typ1; c1; typt ;t |]),
mkApp(hdcncl, [| typt; t; typ2; c2 |]) )
| [typ;c1;c2] ->
( mkApp (hdcncl, [| typ; c1; t |]),
mkApp (hdcncl, [| typ; t; c2 |]) )
| [c1;c2] ->
( mkApp (hdcncl, [| c1; t|]),
mkApp (hdcncl, [| t; c2 |]) )
| _ -> assert false
in
tclTHENFIRST (cut eq2)
(tclTHENFIRST (cut eq1)
(tclTHENLIST
[ tclDO 2 intro;
tclLAST_HYP simplest_case;
assumption ])) gl
end
let intros_transitivity n = tclTHEN intros (transitivity n)
(* tactical to save as name a subproof such that the generalisation of
the current goal, abstracted with respect to the local signature,
is solved by tac *)
let abstract_subproof name tac gls =
let env = Global.env() in
let current_sign = Global.named_context()
and global_sign = pf_hyps gls in
let sign =
List.fold_right
(fun (id,_,_ as d) s ->
if mem_named_context id current_sign then s else add_named_decl d s)
global_sign empty_named_context
in
let na = next_global_ident_away name (ids_of_named_context global_sign) in
let concl =
List.fold_left (fun t d -> mkNamedProd_or_LetIn d t) (pf_concl gls) sign
in
if occur_existential concl then error "Abstract cannot handle existentials";
let lemme =
start_proof na (IsGlobal (Proof Lemma)) current_sign concl (fun _ _ -> ());
let _,(const,kind,_) =
try
by (tclCOMPLETE (tclTHEN (tclDO (List.length sign) intro) tac));
let r = cook_proof () in
delete_current_proof (); r
with e when catchable_exception e ->
(delete_current_proof(); raise e)
in (* Faudrait un peu fonctionnaliser cela *)
let cd = Entries.DefinitionEntry const in
let sp = Declare.declare_constant na (cd,IsProof Lemma) in
let newenv = Global.env() in
constr_of_reference (ConstRef (snd sp))
in
exact_no_check
(applist (lemme,
List.map mkVar (List.rev (ids_of_named_context sign))))
gls
let tclABSTRACT name_op tac gls =
let s = match name_op with
| Some s -> s
| None -> add_suffix (get_current_proof_name ()) "_subproof"
in
abstract_subproof s tac gls
|