aboutsummaryrefslogtreecommitdiffhomepage
path: root/ltac/extratactics.ml4
blob: d6ba670d83017c45e7f0fe697ba7619f8d47191c (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
(************************************************************************)
(*  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        *)
(************************************************************************)

(*i camlp4deps: "grammar/grammar.cma" i*)

open Pp
open Genarg
open Stdarg
open Constrarg
open Extraargs
open Pcoq.Prim
open Pcoq.Tactic
open Mod_subst
open Names
open Tacexpr
open Glob_ops
open CErrors
open Util
open Evd
open Termops
open Equality
open Misctypes
open Sigma.Notations
open Proofview.Notations
open Constrarg

DECLARE PLUGIN "extratactics"

(**********************************************************************)
(* replace, discriminate, injection, simplify_eq                      *)
(* cutrewrite, dependent rewrite                                      *)

let with_delayed_uconstr ist c tac =
  let flags = {
    Pretyping.use_typeclasses = false;
    use_unif_heuristics = true;
    use_hook = Some Pfedit.solve_by_implicit_tactic;
    fail_evar = false;
    expand_evars = true
  } in
  let c = Pretyping.type_uconstr ~flags ist c in
  Tacticals.New.tclDELAYEDWITHHOLES false c tac

let replace_in_clause_maybe_by ist c1 c2 cl tac =
  with_delayed_uconstr ist c1
  (fun c1 -> replace_in_clause_maybe_by c1 c2 cl (Option.map (Tacinterp.tactic_of_value ist) tac))

let replace_term ist dir_opt c cl =
  with_delayed_uconstr ist c (fun c -> replace_term dir_opt c cl)

let clause = Pcoq.Tactic.clause_dft_concl

TACTIC EXTEND replace
   ["replace" uconstr(c1) "with" constr(c2) clause(cl) by_arg_tac(tac) ]
-> [ replace_in_clause_maybe_by ist c1 c2 cl tac ]
END

TACTIC EXTEND replace_term_left
  [ "replace"  "->" uconstr(c) clause(cl) ]
  -> [ replace_term ist (Some true) c cl ]
END

TACTIC EXTEND replace_term_right
  [ "replace"  "<-" uconstr(c) clause(cl) ]
  -> [ replace_term ist (Some false) c cl ]
END

TACTIC EXTEND replace_term
  [ "replace" uconstr(c) clause(cl) ]
  -> [ replace_term ist None c cl ]
END

let induction_arg_of_quantified_hyp = function
  | AnonHyp n -> None,ElimOnAnonHyp n
  | NamedHyp id -> None,ElimOnIdent (Loc.ghost,id)

(* Versions *_main must come first!! so that "1" is interpreted as a
   ElimOnAnonHyp and not as a "constr", and "id" is interpreted as a
   ElimOnIdent and not as "constr" *)

let mytclWithHoles tac with_evars c =
  Proofview.Goal.enter { enter = begin fun gl ->
    let env = Tacmach.New.pf_env gl in
    let sigma = Tacmach.New.project gl in
    let sigma',c = Tactics.force_destruction_arg with_evars env sigma c in
    Tacticals.New.tclWITHHOLES with_evars (tac with_evars (Some c)) sigma'
  end }

let elimOnConstrWithHoles tac with_evars c =
  Tacticals.New.tclDELAYEDWITHHOLES with_evars c
    (fun c -> tac with_evars (Some (None,ElimOnConstr c)))

TACTIC EXTEND simplify_eq
  [ "simplify_eq" ] -> [ dEq false None ]
| [ "simplify_eq" destruction_arg(c) ] -> [ mytclWithHoles dEq false c ]
END
TACTIC EXTEND esimplify_eq
| [ "esimplify_eq" ] -> [ dEq true None ]
| [ "esimplify_eq" destruction_arg(c) ] -> [ mytclWithHoles dEq true c ]
END

let discr_main c = elimOnConstrWithHoles discr_tac false c

TACTIC EXTEND discriminate
| [ "discriminate" ] -> [ discr_tac false None ]
| [ "discriminate" destruction_arg(c) ] ->
    [ mytclWithHoles discr_tac false c ]
END
TACTIC EXTEND ediscriminate
| [ "ediscriminate" ] -> [ discr_tac true None ]
| [ "ediscriminate" destruction_arg(c) ] ->
    [ mytclWithHoles discr_tac true c ]
END

let discrHyp id =
  Proofview.tclEVARMAP >>= fun sigma ->
  discr_main { delayed = fun env sigma -> Sigma.here (Term.mkVar id, NoBindings) sigma }

let injection_main with_evars c =
 elimOnConstrWithHoles (injClause None) with_evars c

TACTIC EXTEND injection
| [ "injection" ] -> [ injClause None false None ]
| [ "injection" destruction_arg(c) ] -> [ mytclWithHoles (injClause None) false c ]
END
TACTIC EXTEND einjection
| [ "einjection" ] -> [ injClause None true None ]
| [ "einjection" destruction_arg(c) ] -> [ mytclWithHoles (injClause None) true c ]
END
TACTIC EXTEND injection_as
| [ "injection" "as" intropattern_list(ipat)] ->
    [ injClause (Some ipat) false None ]
| [ "injection" destruction_arg(c) "as" intropattern_list(ipat)] ->
    [ mytclWithHoles (injClause (Some ipat)) false c ]
END
TACTIC EXTEND einjection_as
| [ "einjection" "as" intropattern_list(ipat)] ->
    [ injClause (Some ipat) true None ]
| [ "einjection" destruction_arg(c) "as" intropattern_list(ipat)] ->
    [ mytclWithHoles (injClause (Some ipat)) true c ]
END
TACTIC EXTEND simple_injection
| [ "simple" "injection" ] -> [ simpleInjClause false None ]
| [ "simple" "injection" destruction_arg(c) ] -> [ mytclWithHoles simpleInjClause false c ]
END

let injHyp id =
  Proofview.tclEVARMAP >>= fun sigma ->
  injection_main false { delayed = fun env sigma -> Sigma.here (Term.mkVar id, NoBindings) sigma }

TACTIC EXTEND dependent_rewrite
| [ "dependent" "rewrite" orient(b) constr(c) ] -> [ rewriteInConcl b c ]
| [ "dependent" "rewrite" orient(b) constr(c) "in" hyp(id) ]
    -> [ rewriteInHyp b c id ]
END

(** To be deprecated?, "cutrewrite (t=u) as <-" is equivalent to
    "replace u with t" or "enough (t=u) as <-" and 
    "cutrewrite (t=u) as ->" is equivalent to "enough (t=u) as ->". *)

TACTIC EXTEND cut_rewrite
| [ "cutrewrite" orient(b) constr(eqn) ] -> [ cutRewriteInConcl b eqn ]
| [ "cutrewrite" orient(b) constr(eqn) "in" hyp(id) ]
    -> [ cutRewriteInHyp b eqn id ]
END

(**********************************************************************)
(* Decompose                                                          *)

TACTIC EXTEND decompose_sum
| [ "decompose" "sum" constr(c) ] -> [ Elim.h_decompose_or c ]
END

TACTIC EXTEND decompose_record
| [ "decompose" "record" constr(c) ] -> [ Elim.h_decompose_and c ]
END

(**********************************************************************)
(* Contradiction                                                      *)

open Contradiction

TACTIC EXTEND absurd
 [ "absurd" constr(c) ] -> [ absurd c ]
END

let onSomeWithHoles tac = function
  | None -> tac None
  | Some c -> Tacticals.New.tclDELAYEDWITHHOLES false c (fun c -> tac (Some c))

TACTIC EXTEND contradiction
 [ "contradiction" constr_with_bindings_opt(c) ] ->
    [ onSomeWithHoles contradiction c ]
END

(**********************************************************************)
(* AutoRewrite                                                        *)

open Autorewrite

let pr_orient _prc _prlc _prt = function
  | true -> Pp.mt ()
  | false -> Pp.str " <-"

let pr_orient_string _prc _prlc _prt (orient, s) =
  pr_orient _prc _prlc _prt orient ++ Pp.spc () ++ Pp.str s

ARGUMENT EXTEND orient_string TYPED AS (bool * string) PRINTED BY pr_orient_string
| [ orient(r) preident(i) ] -> [ r, i ]
END

TACTIC EXTEND autorewrite
| [ "autorewrite" "with" ne_preident_list(l) clause(cl) ] ->
    [ auto_multi_rewrite  l ( cl) ]
| [ "autorewrite" "with" ne_preident_list(l) clause(cl) "using" tactic(t) ] ->
    [
      auto_multi_rewrite_with (Tacinterp.tactic_of_value ist t) l cl
    ]
END

TACTIC EXTEND autorewrite_star
| [ "autorewrite" "*" "with" ne_preident_list(l) clause(cl) ] ->
    [ auto_multi_rewrite ~conds:AllMatches l cl ]
| [ "autorewrite" "*" "with" ne_preident_list(l) clause(cl) "using" tactic(t) ] ->
  [ auto_multi_rewrite_with ~conds:AllMatches (Tacinterp.tactic_of_value ist t) l cl ]
END

(**********************************************************************)
(* Rewrite star                                                       *)

let rewrite_star ist clause orient occs c (tac : Geninterp.Val.t option) =
  let tac' = Option.map (fun t -> Tacinterp.tactic_of_value ist t, FirstSolved) tac in
  with_delayed_uconstr ist c
    (fun c -> general_rewrite_ebindings_clause clause orient occs ?tac:tac' true true (c,NoBindings) true)

TACTIC EXTEND rewrite_star
| [ "rewrite" "*" orient(o) uconstr(c) "in" hyp(id) "at" occurrences(occ) by_arg_tac(tac) ] ->
    [ rewrite_star ist (Some id) o (occurrences_of occ) c tac ]
| [ "rewrite" "*" orient(o) uconstr(c) "at" occurrences(occ) "in" hyp(id) by_arg_tac(tac) ] ->
    [ rewrite_star ist (Some id) o (occurrences_of occ) c tac ]
| [ "rewrite" "*" orient(o) uconstr(c) "in" hyp(id) by_arg_tac(tac) ] ->
    [ rewrite_star ist (Some id) o Locus.AllOccurrences c tac ]
| [ "rewrite" "*" orient(o) uconstr(c) "at" occurrences(occ) by_arg_tac(tac) ] ->
    [ rewrite_star ist None o (occurrences_of occ) c tac ]
| [ "rewrite" "*" orient(o) uconstr(c) by_arg_tac(tac) ] ->
    [ rewrite_star ist None o Locus.AllOccurrences c tac ]
    END

(**********************************************************************)
(* Hint Rewrite                                                       *)

let add_rewrite_hint bases ort t lcsr =
  let env = Global.env() in
  let sigma = Evd.from_env env in
  let poly = Flags.use_polymorphic_flag () in
  let f ce = 
    let c, ctx = Constrintern.interp_constr env sigma ce in
    let ctx =
      let ctx = UState.context_set ctx in
        if poly then ctx
	else (Declare.declare_universe_context false ctx;
              Univ.ContextSet.empty)
    in
      Constrexpr_ops.constr_loc ce, (c, ctx), ort, Option.map (in_gen (rawwit wit_ltac)) t in
  let eqs = List.map f lcsr in
  let add_hints base = add_rew_rules base eqs in
  List.iter add_hints bases

let classify_hint _ = Vernacexpr.VtSideff [], Vernacexpr.VtLater

VERNAC COMMAND EXTEND HintRewrite CLASSIFIED BY classify_hint
  [ "Hint" "Rewrite" orient(o) ne_constr_list(l) ":" preident_list(bl) ] ->
  [ add_rewrite_hint bl o None l ]
| [ "Hint" "Rewrite" orient(o) ne_constr_list(l) "using" tactic(t)
    ":" preident_list(bl) ] ->
  [ add_rewrite_hint bl o (Some t) l ]
| [ "Hint" "Rewrite" orient(o) ne_constr_list(l) ] ->
  [ add_rewrite_hint ["core"] o None l ]
| [ "Hint" "Rewrite" orient(o) ne_constr_list(l) "using" tactic(t) ] ->
  [ add_rewrite_hint ["core"] o (Some t) l ]
END

(**********************************************************************)
(* Hint Resolve                                                       *)

open Term
open Vars
open Coqlib

let project_hint pri l2r r =
  let gr = Smartlocate.global_with_alias r in
  let env = Global.env() in
  let sigma = Evd.from_env env in
  let sigma, c = Evd.fresh_global env sigma gr in
  let t = Retyping.get_type_of env sigma c in
  let t =
    Tacred.reduce_to_quantified_ref env sigma (Lazy.force coq_iff_ref) t in
  let sign,ccl = decompose_prod_assum t in
  let (a,b) = match snd (decompose_app ccl) with
    | [a;b] -> (a,b)
    | _ -> assert false in
  let p =
    if l2r then build_coq_iff_left_proj () else build_coq_iff_right_proj () in
  let c = Reductionops.whd_beta Evd.empty (mkApp (c, Context.Rel.to_extended_vect 0 sign)) in
  let c = it_mkLambda_or_LetIn
    (mkApp (p,[|mkArrow a (lift 1 b);mkArrow b (lift 1 a);c|])) sign in
  let id =
    Nameops.add_suffix (Nametab.basename_of_global gr) ("_proj_" ^ (if l2r then "l2r" else "r2l"))
  in
  let ctx = Evd.universe_context_set sigma in
  let c = Declare.declare_definition ~internal:Declare.InternalTacticRequest id (c,ctx) in
    (pri,false,true,Hints.PathAny, Hints.IsGlobRef (Globnames.ConstRef c))

let add_hints_iff l2r lc n bl =
  Hints.add_hints true bl
    (Hints.HintsResolveEntry (List.map (project_hint n l2r) lc))

VERNAC COMMAND EXTEND HintResolveIffLR CLASSIFIED AS SIDEFF
  [ "Hint" "Resolve" "->" ne_global_list(lc) natural_opt(n)
    ":" preident_list(bl) ] ->
  [ add_hints_iff true lc n bl ]
| [ "Hint" "Resolve" "->" ne_global_list(lc) natural_opt(n) ] ->
  [ add_hints_iff true lc n ["core"] ]
END
VERNAC COMMAND EXTEND HintResolveIffRL CLASSIFIED AS SIDEFF
  [ "Hint" "Resolve" "<-" ne_global_list(lc) natural_opt(n)
    ":" preident_list(bl) ] ->
  [ add_hints_iff false lc n bl ]
| [ "Hint" "Resolve" "<-" ne_global_list(lc) natural_opt(n) ] ->
  [ add_hints_iff false lc n ["core"] ]
END

(**********************************************************************)
(* Refine                                                             *)

let constr_flags = {
  Pretyping.use_typeclasses = true;
  Pretyping.use_unif_heuristics = true;
  Pretyping.use_hook = Some Pfedit.solve_by_implicit_tactic;
  Pretyping.fail_evar = false;
  Pretyping.expand_evars = true }

let refine_tac ist simple c =
  Proofview.Goal.nf_enter { enter = begin fun gl ->
    let concl = Proofview.Goal.concl gl in
    let env = Proofview.Goal.env gl in
    let flags = constr_flags in
    let expected_type = Pretyping.OfType concl in
    let c = Pretyping.type_uconstr ~flags ~expected_type ist c in
    let update = { run = fun sigma -> c.delayed env sigma } in
    let refine = Refine.refine ~unsafe:false update in
    if simple then refine
    else refine <*>
           Tactics.New.reduce_after_refine <*>
           Proofview.shelve_unifiable
  end }

TACTIC EXTEND refine
| [ "refine" uconstr(c) ] -> [ refine_tac ist false c ]
END

TACTIC EXTEND simple_refine
| [ "simple" "refine" uconstr(c) ] -> [ refine_tac ist true c ]
END

(**********************************************************************)
(* Inversion lemmas (Leminv)                                          *)

open Inv
open Leminv

let seff id = Vernacexpr.VtSideff [id], Vernacexpr.VtLater

VERNAC ARGUMENT EXTEND sort
| [ "Set" ] -> [ GSet ]
| [ "Prop" ] -> [ GProp ]
| [ "Type" ] -> [ GType [] ]
END

VERNAC COMMAND EXTEND DeriveInversionClear
| [ "Derive" "Inversion_clear" ident(na) "with" constr(c) "Sort" sort(s) ]
  => [ seff na ]
  -> [ add_inversion_lemma_exn na c s false inv_clear_tac ]

| [ "Derive" "Inversion_clear" ident(na) "with" constr(c) ] => [ seff na ]
  -> [ add_inversion_lemma_exn na c GProp false inv_clear_tac ]
END

open Term

VERNAC COMMAND EXTEND DeriveInversion
| [ "Derive" "Inversion" ident(na) "with" constr(c) "Sort" sort(s) ]
  => [ seff na ]
  -> [ add_inversion_lemma_exn na c s false inv_tac ]

| [ "Derive" "Inversion" ident(na) "with" constr(c) ] => [ seff na ]
  -> [ add_inversion_lemma_exn na c GProp false inv_tac ]
END

VERNAC COMMAND EXTEND DeriveDependentInversion
| [ "Derive" "Dependent" "Inversion" ident(na) "with" constr(c) "Sort" sort(s) ]
  => [ seff na ]
  -> [ add_inversion_lemma_exn na c s true dinv_tac ]
END

VERNAC COMMAND EXTEND DeriveDependentInversionClear
| [ "Derive" "Dependent" "Inversion_clear" ident(na) "with" constr(c) "Sort" sort(s) ]
  => [ seff na ]
  -> [ add_inversion_lemma_exn na c s true dinv_clear_tac ]
END

(**********************************************************************)
(* Subst                                                              *)

TACTIC EXTEND subst
| [ "subst" ne_var_list(l) ] -> [ subst l ]
| [ "subst" ] -> [ subst_all () ]
END

let simple_subst_tactic_flags =
  { only_leibniz = true; rewrite_dependent_proof = false }

TACTIC EXTEND simple_subst
| [ "simple" "subst" ] -> [ subst_all ~flags:simple_subst_tactic_flags () ]
END

open Evar_tactics

(**********************************************************************)
(* Evar creation                                                      *)

(* TODO: add support for some test similar to g_constr.name_colon so that
   expressions like "evar (list A)" do not raise a syntax error *)
TACTIC EXTEND evar
  [ "evar" "(" ident(id) ":" lconstr(typ) ")" ] -> [ let_evar (Name id) typ ]
| [ "evar" constr(typ) ] -> [ let_evar Anonymous typ ]
END

TACTIC EXTEND instantiate
  [ "instantiate" "(" ident(id) ":=" lglob(c) ")" ] ->
    [ Tacticals.New.tclTHEN (instantiate_tac_by_name id c) Proofview.V82.nf_evar_goals ]
| [ "instantiate" "(" integer(i) ":=" lglob(c) ")" hloc(hl) ] ->
    [ Tacticals.New.tclTHEN (instantiate_tac i c hl) Proofview.V82.nf_evar_goals ]
| [ "instantiate" ] -> [ Proofview.V82.nf_evar_goals ]
END

(**********************************************************************)
(** Nijmegen "step" tactic for setoid rewriting                       *)

open Tactics
open Glob_term
open Libobject
open Lib

(* Registered lemmas are expected to be of the form
     x R y -> y == z -> x R z    (in the right table)
     x R y -> x == z -> z R y    (in the left table)
*)

let transitivity_right_table = Summary.ref [] ~name:"transitivity-steps-r"
let transitivity_left_table = Summary.ref [] ~name:"transitivity-steps-l"

(* [step] tries to apply a rewriting lemma; then apply [tac] intended to
   complete to proof of the last hypothesis (assumed to state an equality) *)

let step left x tac =
  let l =
    List.map (fun lem ->
      Tacticals.New.tclTHENLAST
        (apply_with_bindings (lem, ImplicitBindings [x]))
        tac)
      !(if left then transitivity_left_table else transitivity_right_table)
  in
  Tacticals.New.tclFIRST l

(* Main function to push lemmas in persistent environment *)

let cache_transitivity_lemma (_,(left,lem)) =
  if left then
    transitivity_left_table  := lem :: !transitivity_left_table
  else
    transitivity_right_table := lem :: !transitivity_right_table

let subst_transitivity_lemma (subst,(b,ref)) = (b,subst_mps subst ref)

let inTransitivity : bool * constr -> obj =
  declare_object {(default_object "TRANSITIVITY-STEPS") with
    cache_function = cache_transitivity_lemma;
    open_function = (fun i o -> if Int.equal i 1 then cache_transitivity_lemma o);
    subst_function = subst_transitivity_lemma;
    classify_function = (fun o -> Substitute o) }

(* Main entry points *)

let add_transitivity_lemma left lem =
  let env = Global.env () in
  let sigma = Evd.from_env env in
  let lem',ctx (*FIXME*) = Constrintern.interp_constr env sigma lem in
  add_anonymous_leaf (inTransitivity (left,lem'))

(* Vernacular syntax *)

TACTIC EXTEND stepl
| ["stepl" constr(c) "by" tactic(tac) ] -> [ step true c (Tacinterp.tactic_of_value ist tac) ]
| ["stepl" constr(c) ] -> [ step true c (Proofview.tclUNIT ()) ]
END

TACTIC EXTEND stepr
| ["stepr" constr(c) "by" tactic(tac) ] -> [ step false c (Tacinterp.tactic_of_value ist tac) ]
| ["stepr" constr(c) ] -> [ step false c (Proofview.tclUNIT ()) ]
END

VERNAC COMMAND EXTEND AddStepl CLASSIFIED AS SIDEFF
| [ "Declare" "Left" "Step" constr(t) ] ->
    [ add_transitivity_lemma true t ]
END

VERNAC COMMAND EXTEND AddStepr CLASSIFIED AS SIDEFF
| [ "Declare" "Right" "Step" constr(t) ] ->
    [ add_transitivity_lemma false t ]
END

let cache_implicit_tactic (_,tac) = match tac with
  | Some tac -> Pfedit.declare_implicit_tactic (Tacinterp.eval_tactic tac)
  | None -> Pfedit.clear_implicit_tactic ()

let subst_implicit_tactic (subst,tac) =
  Option.map (Tacsubst.subst_tactic subst) tac

let inImplicitTactic : glob_tactic_expr option -> obj =
  declare_object {(default_object "IMPLICIT-TACTIC") with
       open_function = (fun i o -> if Int.equal i 1 then cache_implicit_tactic o);
       cache_function = cache_implicit_tactic;
       subst_function = subst_implicit_tactic;
       classify_function = (fun o -> Dispose)}

let declare_implicit_tactic tac =
  Lib.add_anonymous_leaf (inImplicitTactic (Some (Tacintern.glob_tactic tac)))

let clear_implicit_tactic () =
  Lib.add_anonymous_leaf (inImplicitTactic None)

VERNAC COMMAND EXTEND ImplicitTactic CLASSIFIED AS SIDEFF
| [ "Declare" "Implicit" "Tactic" tactic(tac) ] -> [ declare_implicit_tactic tac ]
| [ "Clear" "Implicit" "Tactic" ] -> [ clear_implicit_tactic () ]
END




(**********************************************************************)
(*spiwack : Vernac commands for retroknowledge                        *)

VERNAC COMMAND EXTEND RetroknowledgeRegister CLASSIFIED AS SIDEFF
 | [ "Register" constr(c) "as" retroknowledge_field(f) "by" constr(b)] ->
           [ let tc,ctx = Constrintern.interp_constr (Global.env ()) Evd.empty c in
             let tb,ctx(*FIXME*) = Constrintern.interp_constr (Global.env ()) Evd.empty b in
             Global.register f tc tb ]
END



(**********************************************************************)
(* sozeau: abs/gen for induction on instantiated dependent inductives, using "Ford" induction as
  defined by Conor McBride *)
TACTIC EXTEND generalize_eqs
| ["generalize_eqs" hyp(id) ] -> [ abstract_generalize ~generalize_vars:false id ]
END
TACTIC EXTEND dep_generalize_eqs
| ["dependent" "generalize_eqs" hyp(id) ] -> [ abstract_generalize ~generalize_vars:false ~force_dep:true id ]
END
TACTIC EXTEND generalize_eqs_vars
| ["generalize_eqs_vars" hyp(id) ] -> [ abstract_generalize ~generalize_vars:true id ]
END
TACTIC EXTEND dep_generalize_eqs_vars
| ["dependent" "generalize_eqs_vars" hyp(id) ] -> [ abstract_generalize ~force_dep:true ~generalize_vars:true id ]
END

(** Tactic to automatically simplify hypotheses of the form [Π Δ, x_i = t_i -> T] 
    where [t_i] is closed w.r.t. Δ. Such hypotheses are automatically generated
    during dependent induction. For internal use. *)

TACTIC EXTEND specialize_eqs
[ "specialize_eqs" hyp(id) ] -> [ specialize_eqs id ]
END

(**********************************************************************)
(* A tactic that considers a given occurrence of [c] in [t] and       *)
(* abstract the minimal set of all the occurrences of [c] so that the *)
(* abstraction [fun x -> t[x/c]] is well-typed                        *)
(*                                                                    *)
(* Contributed by Chung-Kil Hur (Winter 2009)                         *)
(**********************************************************************)

let subst_var_with_hole occ tid t = 
  let occref = if occ > 0 then ref occ else Find_subterm.error_invalid_occurrence [occ] in
  let locref = ref 0 in
  let rec substrec = function
    | GVar (_,id) as x -> 
        if Id.equal id tid 
        then
	  (decr occref;
	   if Int.equal !occref 0 then x
           else
	     (incr locref;
	      GHole (Loc.make_loc (!locref,0),
		     Evar_kinds.QuestionMark(Evar_kinds.Define true),
                     Misctypes.IntroAnonymous, None)))
        else x
    | c -> map_glob_constr_left_to_right substrec c in
  let t' = substrec t
  in
  if !occref > 0 then Find_subterm.error_invalid_occurrence [occ] else t'

let subst_hole_with_term occ tc t =
  let locref = ref 0 in
  let occref = ref occ in
  let rec substrec = function
    | GHole (_,Evar_kinds.QuestionMark(Evar_kinds.Define true),Misctypes.IntroAnonymous,s) ->
        decr occref;
        if Int.equal !occref 0 then tc
        else
	  (incr locref;
	   GHole (Loc.make_loc (!locref,0),
		  Evar_kinds.QuestionMark(Evar_kinds.Define true),Misctypes.IntroAnonymous,s))
    | c -> map_glob_constr_left_to_right substrec c
  in
  substrec t

open Tacmach

let hResolve id c occ t =
  Proofview.Goal.nf_s_enter { s_enter = begin fun gl ->
  let sigma = Proofview.Goal.sigma gl in
  let sigma = Sigma.to_evar_map sigma in
  let env = Termops.clear_named_body id (Proofview.Goal.env gl) in
  let concl = Proofview.Goal.concl gl in
  let env_ids = Termops.ids_of_context env in
  let c_raw = Detyping.detype true env_ids env sigma c in
  let t_raw = Detyping.detype true env_ids env sigma t in
  let rec resolve_hole t_hole =
    try 
      Pretyping.understand env sigma t_hole
    with
      | Pretype_errors.PretypeError (_,_,Pretype_errors.UnsolvableImplicit _) as e ->
          let (e, info) = CErrors.push e in
          let loc = match Loc.get_loc info with None -> Loc.ghost | Some loc -> loc in
          resolve_hole (subst_hole_with_term (fst (Loc.unloc loc)) c_raw t_hole)
  in
  let t_constr,ctx = resolve_hole (subst_var_with_hole occ id t_raw) in
  let sigma = Evd.merge_universe_context sigma ctx in
  let t_constr_type = Retyping.get_type_of env sigma t_constr in
  let tac =
    (change_concl (mkLetIn (Anonymous,t_constr,t_constr_type,concl)))
  in
  Sigma.Unsafe.of_pair (tac, sigma)
  end }

let hResolve_auto id c t =
  let rec resolve_auto n = 
    try
      hResolve id c n t
    with
    | UserError _ as e -> raise e
    | e when CErrors.noncritical e -> resolve_auto (n+1)
  in
  resolve_auto 1

TACTIC EXTEND hresolve_core
| [ "hresolve_core" "(" ident(id) ":=" constr(c) ")" "at" int_or_var(occ) "in" constr(t) ] -> [ hResolve id c occ t ]
| [ "hresolve_core" "(" ident(id) ":=" constr(c) ")" "in" constr(t) ] -> [ hResolve_auto id c t ]
END

(**
   hget_evar
*)

let hget_evar n =
  Proofview.Goal.nf_enter { enter = begin fun gl ->
  let sigma = Tacmach.New.project gl in
  let concl = Proofview.Goal.concl gl in
  let evl = evar_list concl in
  if List.length evl < n then
    error "Not enough uninstantiated existential variables.";
  if n <= 0 then error "Incorrect existential variable index.";
  let ev = List.nth evl (n-1) in
  let ev_type = existential_type sigma ev in
  change_concl (mkLetIn (Anonymous,mkEvar ev,ev_type,concl))
  end }

TACTIC EXTEND hget_evar
| [ "hget_evar" int_or_var(n) ] -> [ hget_evar n ]
END

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

(**********************************************************************)
(* A tactic that reduces one match t with ... by doing destruct t.    *)
(* if t is not a variable, the tactic does                            *)
(* case_eq t;intros ... heq;rewrite heq in *|-. (but heq itself is    *)
(* preserved).                                                        *)
(* Contributed by Julien Forest and Pierre Courtieu (july 2010)       *)
(**********************************************************************)

exception Found of unit Proofview.tactic

let rewrite_except h =
  Proofview.Goal.nf_enter { enter = begin fun gl ->
  let hyps = Tacmach.New.pf_ids_of_hyps gl in
  Tacticals.New.tclMAP (fun id -> if Id.equal id h then Proofview.tclUNIT () else 
      Tacticals.New.tclTRY (Equality.general_rewrite_in true Locus.AllOccurrences true true id (mkVar h) false))
    hyps
  end }


let refl_equal = 
  let coq_base_constant s =
    Coqlib.gen_constant_in_modules "RecursiveDefinition"
      (Coqlib.init_modules @ [["Coq";"Arith";"Le"];["Coq";"Arith";"Lt"]]) s in
  function () -> (coq_base_constant "eq_refl")


(* This is simply an implementation of the case_eq tactic.  this code
  should be replaced by a call to the tactic but I don't know how to
  call it before it is defined. *)
let  mkCaseEq a  : unit Proofview.tactic =
  Proofview.Goal.nf_enter { enter = begin fun gl ->
    let type_of_a = Tacmach.New.of_old (fun g -> Tacmach.pf_unsafe_type_of g a) gl in
       Tacticals.New.tclTHENLIST
         [Tactics.generalize [mkApp(delayed_force refl_equal, [| type_of_a; a|])];
          Proofview.Goal.nf_enter { enter = begin fun gl ->
            let concl = Proofview.Goal.concl gl in
            let env = Proofview.Goal.env gl in
            (** FIXME: this looks really wrong. Does anybody really use this tactic? *)
            let Sigma (c, _, _) = (Tacred.pattern_occs [Locus.OnlyOccurrences [1], a]).Reductionops.e_redfun env (Sigma.Unsafe.of_evar_map Evd.empty) concl in
	    change_concl c
          end };
	  simplest_case a]
  end }


let case_eq_intros_rewrite x =
  Proofview.Goal.nf_enter { enter = begin fun gl ->
  let n = nb_prod (Proofview.Goal.concl gl) in
  (* Pp.msgnl (Printer.pr_lconstr x); *)
  Tacticals.New.tclTHENLIST [
      mkCaseEq x;
    Proofview.Goal.nf_enter { enter = begin fun gl ->
      let concl = Proofview.Goal.concl gl in
      let hyps = Tacmach.New.pf_ids_of_hyps gl in
      let n' = nb_prod concl in
      let h = Tacmach.New.of_old (fun g -> fresh_id hyps (Id.of_string "heq") g) gl in
      Tacticals.New.tclTHENLIST [
                    Tacticals.New.tclDO (n'-n-1) intro;
		    introduction h;
		    rewrite_except h]
    end }
  ]
  end }

let rec find_a_destructable_match t =
  let cl = induction_arg_of_quantified_hyp (NamedHyp (Id.of_string "x")) in
  let cl = [cl, (None, None), None], None in
  let dest = TacAtom (Loc.ghost, TacInductionDestruct(false, false, cl)) in
  match kind_of_term t with
    | Case (_,_,x,_) when closed0 x ->
	if isVar x then
	  (* TODO check there is no rel n. *)
	  raise (Found (Tacinterp.eval_tactic dest))
	else
	  (* let _ = Pp.msgnl (Printer.pr_lconstr x)  in *)
	  raise (Found (case_eq_intros_rewrite x))
    | _ -> iter_constr find_a_destructable_match t
	

let destauto t =
  try find_a_destructable_match t;
    Tacticals.New.tclZEROMSG (str "No destructable match found")
  with Found tac -> tac

let destauto_in id = 
  Proofview.Goal.nf_enter { enter = begin fun gl ->
  let ctype = Tacmach.New.of_old (fun g -> Tacmach.pf_unsafe_type_of g (mkVar id)) gl in
(*  Pp.msgnl (Printer.pr_lconstr (mkVar id)); *)
(*  Pp.msgnl (Printer.pr_lconstr (ctype)); *)
  destauto ctype
  end }

TACTIC EXTEND destauto
| [ "destauto" ] -> [ Proofview.Goal.nf_enter { enter = begin fun gl -> destauto (Proofview.Goal.concl gl) end } ]
| [ "destauto" "in" hyp(id) ] -> [ destauto_in id ]
END


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

let eq_constr x y = 
  Proofview.Goal.enter { enter = begin fun gl ->
    let evd = Tacmach.New.project gl in
      if Evarutil.eq_constr_univs_test evd evd x y then Proofview.tclUNIT () 
      else Tacticals.New.tclFAIL 0 (str "Not equal")
  end }

TACTIC EXTEND constr_eq
| [ "constr_eq" constr(x) constr(y) ] -> [ eq_constr x y ]
END

TACTIC EXTEND constr_eq_nounivs
| [ "constr_eq_nounivs" constr(x) constr(y) ] -> [
    if eq_constr_nounivs x y then Proofview.tclUNIT () else Tacticals.New.tclFAIL 0 (str "Not equal") ]
END

TACTIC EXTEND is_evar
| [ "is_evar" constr(x) ] ->
    [ Proofview.tclBIND Proofview.tclEVARMAP begin fun sigma ->
      match Evarutil.kind_of_term_upto sigma x with
        | Evar _ -> Proofview.tclUNIT ()
        | _ -> Tacticals.New.tclFAIL 0 (str "Not an evar")
      end
    ]
END

let rec has_evar x =
  match kind_of_term x with
    | Evar _ -> true
    | Rel _ | Var _ | Meta _ | Sort _ | Const _ | Ind _ | Construct _ ->
      false
    | Cast (t1, _, t2) | Prod (_, t1, t2) | Lambda (_, t1, t2) ->
      has_evar t1 || has_evar t2
    | LetIn (_, t1, t2, t3) ->
      has_evar t1 || has_evar t2 || has_evar t3
    | App (t1, ts) ->
      has_evar t1 || has_evar_array ts
    | Case (_, t1, t2, ts) ->
      has_evar t1 || has_evar t2 || has_evar_array ts
    | Fix ((_, tr)) | CoFix ((_, tr)) ->
      has_evar_prec tr
    | Proj (p, c) -> has_evar c
and has_evar_array x =
  Array.exists has_evar x
and has_evar_prec (_, ts1, ts2) =
  Array.exists has_evar ts1 || Array.exists has_evar ts2

TACTIC EXTEND has_evar
| [ "has_evar" constr(x) ] ->
    [ if has_evar x then Proofview.tclUNIT () else Tacticals.New.tclFAIL 0 (str "No evars") ]
END

TACTIC EXTEND is_hyp
| [ "is_var" constr(x) ] ->
  [ match kind_of_term x with
    | Var _ ->  Proofview.tclUNIT ()
    | _ -> Tacticals.New.tclFAIL 0 (str "Not a variable or hypothesis") ]
END

TACTIC EXTEND is_fix
| [ "is_fix" constr(x) ] ->
  [ match kind_of_term x with
    | Fix _ -> Proofview.tclUNIT ()
    | _ -> Tacticals.New.tclFAIL 0 (Pp.str "not a fix definition") ]
END;;

TACTIC EXTEND is_cofix
| [ "is_cofix" constr(x) ] ->
  [ match kind_of_term x with
    | CoFix _ -> Proofview.tclUNIT ()
    | _ -> Tacticals.New.tclFAIL 0 (Pp.str "not a cofix definition") ]
END;;

TACTIC EXTEND is_ind
| [ "is_ind" constr(x) ] ->
  [ match kind_of_term x with
    | Ind _ -> Proofview.tclUNIT ()
    | _ -> Tacticals.New.tclFAIL 0 (Pp.str "not an (co)inductive datatype") ]
END;;

TACTIC EXTEND is_constructor
| [ "is_constructor" constr(x) ] ->
  [ match kind_of_term x with
    | Construct _ -> Proofview.tclUNIT ()
    | _ -> Tacticals.New.tclFAIL 0 (Pp.str "not a constructor") ]
END;;

TACTIC EXTEND is_proj
| [ "is_proj" constr(x) ] ->
  [ match kind_of_term x with
    | Proj _ -> Proofview.tclUNIT ()
    | _ -> Tacticals.New.tclFAIL 0 (Pp.str "not a primitive projection") ]
END;;

TACTIC EXTEND is_const
| [ "is_const" constr(x) ] ->
  [ match kind_of_term x with
    | Const _ -> Proofview.tclUNIT ()
    | _ -> Tacticals.New.tclFAIL 0 (Pp.str "not a constant") ]
END;;

(* Command to grab the evars left unresolved at the end of a proof. *)
(* spiwack: I put it in extratactics because it is somewhat tied with
   the semantics of the LCF-style tactics, hence with the classic tactic
   mode. *)
VERNAC COMMAND EXTEND GrabEvars
[ "Grab" "Existential" "Variables" ]
  => [ Vernac_classifier.classify_as_proofstep ]
  -> [ Proof_global.simple_with_current_proof (fun _ p  -> Proof.V82.grab_evars p) ]
END

(* Shelves all the goals under focus. *)
TACTIC EXTEND shelve
| [ "shelve" ] ->
    [ Proofview.shelve ]
END

(* Shelves the unifiable goals under focus, i.e. the goals which
   appear in other goals under focus (the unfocused goals are not
   considered). *)
TACTIC EXTEND shelve_unifiable
| [ "shelve_unifiable" ] ->
    [ Proofview.shelve_unifiable ]
END

(* Unshelves the goal shelved by the tactic. *)
TACTIC EXTEND unshelve
| [ "unshelve" tactic1(t) ] ->
    [
      Proofview.with_shelf (Tacinterp.tactic_of_value ist t) >>= fun (gls, ()) ->
      Proofview.Unsafe.tclGETGOALS >>= fun ogls ->
      Proofview.Unsafe.tclSETGOALS (gls @ ogls)
    ]
END

(* Command to add every unshelved variables to the focus *)
VERNAC COMMAND EXTEND Unshelve
[ "Unshelve" ]
  => [ Vernac_classifier.classify_as_proofstep ]
  -> [ Proof_global.simple_with_current_proof (fun _ p  -> Proof.unshelve p) ]
END

(* Gives up on the goals under focus: the goals are considered solved,
   but the proof cannot be closed until the user goes back and solve
   these goals. *)
TACTIC EXTEND give_up
| [ "give_up" ] ->
    [ Proofview.give_up ]
END

(* cycles [n] goals *)
TACTIC EXTEND cycle
| [ "cycle" int_or_var(n) ] -> [ Proofview.cycle n ]
END

(* swaps goals number [i] and [j] *)
TACTIC EXTEND swap
| [ "swap" int_or_var(i) int_or_var(j) ] -> [ Proofview.swap i j ]
END

(* reverses the list of focused goals *)
TACTIC EXTEND revgoals
| [ "revgoals" ] -> [ Proofview.revgoals ]
END

type cmp =
  | Eq
  | Lt | Le
  | Gt | Ge

type 'i test =
  | Test of cmp * 'i * 'i

let pr_cmp = function
  | Eq -> Pp.str"="
  | Lt -> Pp.str"<"
  | Le -> Pp.str"<="
  | Gt -> Pp.str">"
  | Ge -> Pp.str">="

let pr_cmp' _prc _prlc _prt = pr_cmp

let pr_test_gen f (Test(c,x,y)) =
  Pp.(f x ++ pr_cmp c ++ f y)

let pr_test = pr_test_gen (Pptactic.pr_or_var Pp.int)

let pr_test' _prc _prlc _prt = pr_test

let pr_itest = pr_test_gen Pp.int

let pr_itest' _prc _prlc _prt = pr_itest



ARGUMENT EXTEND comparison PRINTED BY pr_cmp'
| [ "="  ] -> [ Eq ]
| [ "<"  ] -> [ Lt ]
| [ "<=" ] -> [ Le ]
| [ ">"  ] -> [ Gt ]
| [ ">=" ] -> [ Ge ]
    END

let interp_test ist gls = function
  | Test (c,x,y) ->
      project gls ,
      Test(c,Tacinterp.interp_int_or_var ist x,Tacinterp.interp_int_or_var ist y)

ARGUMENT EXTEND test
  PRINTED BY pr_itest'
  INTERPRETED BY interp_test
  RAW_PRINTED BY pr_test'
  GLOB_PRINTED BY pr_test'
| [ int_or_var(x) comparison(c) int_or_var(y) ] -> [ Test(c,x,y) ]
END

let interp_cmp = function
  | Eq -> Int.equal
  | Lt -> ((<):int->int->bool)
  | Le -> ((<=):int->int->bool)
  | Gt -> ((>):int->int->bool)
  | Ge -> ((>=):int->int->bool)

let run_test = function
  | Test(c,x,y) -> interp_cmp c x y

let guard tst =
  if run_test tst then
    Proofview.tclUNIT ()
  else
    let msg = Pp.(str"Condition not satisfied:"++ws 1++(pr_itest tst)) in
    Tacticals.New.tclZEROMSG msg


TACTIC EXTEND guard
| [ "guard" test(tst) ] -> [ guard tst ]
END

let decompose l c =
  Proofview.Goal.enter { enter = begin fun gl ->
    let to_ind c =
      if isInd c then Univ.out_punivs (destInd c)
      else error "not an inductive type"
    in
    let l = List.map to_ind l in
    Elim.h_decompose l c
  end }

TACTIC EXTEND decompose
| [ "decompose" "[" ne_constr_list(l) "]" constr(c) ] -> [ decompose l c ]
END

(** library/keys *)

VERNAC COMMAND EXTEND Declare_keys CLASSIFIED AS SIDEFF
| [ "Declare" "Equivalent" "Keys" constr(c) constr(c') ] -> [ 
  let it c = snd (Constrintern.interp_open_constr (Global.env ()) Evd.empty c) in 
  let k1 = Keys.constr_key (it c) in
  let k2 = Keys.constr_key (it c') in
    match k1, k2 with
    | Some k1, Some k2 -> Keys.declare_equiv_keys k1 k2
    | _ -> () ]
END

VERNAC COMMAND EXTEND Print_keys CLASSIFIED AS QUERY
| [ "Print" "Equivalent" "Keys" ] -> [ Feedback.msg_info (Keys.pr_keys Printer.pr_global) ]
END


VERNAC COMMAND EXTEND OptimizeProof
| [ "Optimize" "Proof" ] => [ Vernac_classifier.classify_as_proofstep ] ->
  [ Proof_global.compact_the_proof () ]
| [ "Optimize" "Heap" ] => [ Vernac_classifier.classify_as_proofstep ] ->
  [ Gc.compact () ]
END