summaryrefslogtreecommitdiff
path: root/theories/MSets/MSetProperties.v
blob: c0038a4f52551c02a3b7011fc9a2dfded19237fb (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
(***********************************************************************)
(*  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       *)
(***********************************************************************)

(* $Id$ *)

(** * Finite sets library *)

(** This functor derives additional properties from [MSetInterface.S].
    Contrary to the functor in [MSetEqProperties] it uses
    predicates over sets instead of sets operations, i.e.
    [In x s] instead of [mem x s=true],
    [Equal s s'] instead of [equal s s'=true], etc. *)

Require Export MSetInterface.
Require Import DecidableTypeEx OrdersLists MSetFacts MSetDecide.
Set Implicit Arguments.
Unset Strict Implicit.

Hint Unfold transpose.

(** First, a functor for Weak Sets in functorial version. *)

Module WPropertiesOn (Import E : DecidableType)(M : WSetsOn E).
  Module Import Dec := WDecideOn E M.
  Module Import FM := Dec.F (* MSetFacts.WFactsOn E M *).
  Import M.

  Lemma In_dec : forall x s, {In x s} + {~ In x s}.
  Proof.
  intros; generalize (mem_iff s x); case (mem x s); intuition.
  Qed.

  Definition Add x s s' := forall y, In y s' <-> E.eq x y \/ In y s.

  Lemma Add_Equal : forall x s s', Add x s s' <-> s' [=] add x s.
  Proof.
  unfold Add.
  split; intros.
  red; intros.
  rewrite H; clear H.
  fsetdec.
  fsetdec.
  Qed.

  Ltac expAdd := repeat rewrite Add_Equal.

  Section BasicProperties.

  Variable s s' s'' s1 s2 s3 : t.
  Variable x x' : elt.

  Lemma equal_refl : s[=]s.
  Proof. fsetdec. Qed.

  Lemma equal_sym : s[=]s' -> s'[=]s.
  Proof. fsetdec. Qed.

  Lemma equal_trans : s1[=]s2 -> s2[=]s3 -> s1[=]s3.
  Proof. fsetdec. Qed.

  Lemma subset_refl : s[<=]s.
  Proof. fsetdec. Qed.

  Lemma subset_trans : s1[<=]s2 -> s2[<=]s3 -> s1[<=]s3.
  Proof. fsetdec. Qed.

  Lemma subset_antisym : s[<=]s' -> s'[<=]s -> s[=]s'.
  Proof. fsetdec. Qed.

  Lemma subset_equal : s[=]s' -> s[<=]s'.
  Proof. fsetdec. Qed.

  Lemma subset_empty : empty[<=]s.
  Proof. fsetdec. Qed.

  Lemma subset_remove_3 : s1[<=]s2 -> remove x s1 [<=] s2.
  Proof. fsetdec. Qed.

  Lemma subset_diff : s1[<=]s3 -> diff s1 s2 [<=] s3.
  Proof. fsetdec. Qed.

  Lemma subset_add_3 : In x s2 -> s1[<=]s2 -> add x s1 [<=] s2.
  Proof. fsetdec. Qed.

  Lemma subset_add_2 : s1[<=]s2 -> s1[<=] add x s2.
  Proof. fsetdec. Qed.

  Lemma in_subset : In x s1 -> s1[<=]s2 -> In x s2.
  Proof. fsetdec. Qed.

  Lemma double_inclusion : s1[=]s2 <-> s1[<=]s2 /\ s2[<=]s1.
  Proof. intuition fsetdec. Qed.

  Lemma empty_is_empty_1 : Empty s -> s[=]empty.
  Proof. fsetdec. Qed.

  Lemma empty_is_empty_2 : s[=]empty -> Empty s.
  Proof. fsetdec. Qed.

  Lemma add_equal : In x s -> add x s [=] s.
  Proof. fsetdec. Qed.

  Lemma add_add : add x (add x' s) [=] add x' (add x s).
  Proof. fsetdec. Qed.

  Lemma remove_equal : ~ In x s -> remove x s [=] s.
  Proof. fsetdec. Qed.

  Lemma Equal_remove : s[=]s' -> remove x s [=] remove x s'.
  Proof. fsetdec. Qed.

  Lemma add_remove : In x s -> add x (remove x s) [=] s.
  Proof. fsetdec. Qed.

  Lemma remove_add : ~In x s -> remove x (add x s) [=] s.
  Proof. fsetdec. Qed.

  Lemma singleton_equal_add : singleton x [=] add x empty.
  Proof. fsetdec. Qed.

  Lemma remove_singleton_empty :
   In x s -> remove x s [=] empty -> singleton x [=] s.
  Proof. fsetdec. Qed.

  Lemma union_sym : union s s' [=] union s' s.
  Proof. fsetdec. Qed.

  Lemma union_subset_equal : s[<=]s' -> union s s' [=] s'.
  Proof. fsetdec. Qed.

  Lemma union_equal_1 : s[=]s' -> union s s'' [=] union s' s''.
  Proof. fsetdec. Qed.

  Lemma union_equal_2 : s'[=]s'' -> union s s' [=] union s s''.
  Proof. fsetdec. Qed.

  Lemma union_assoc : union (union s s') s'' [=] union s (union s' s'').
  Proof. fsetdec. Qed.

  Lemma add_union_singleton : add x s [=] union (singleton x) s.
  Proof. fsetdec. Qed.

  Lemma union_add : union (add x s) s' [=] add x (union s s').
  Proof. fsetdec. Qed.

  Lemma union_remove_add_1 :
   union (remove x s) (add x s') [=] union (add x s) (remove x s').
  Proof. fsetdec. Qed.

  Lemma union_remove_add_2 : In x s ->
   union (remove x s) (add x s') [=] union s s'.
  Proof. fsetdec. Qed.

  Lemma union_subset_1 : s [<=] union s s'.
  Proof. fsetdec. Qed.

  Lemma union_subset_2 : s' [<=] union s s'.
  Proof. fsetdec. Qed.

  Lemma union_subset_3 : s[<=]s'' -> s'[<=]s'' -> union s s' [<=] s''.
  Proof. fsetdec. Qed.

  Lemma union_subset_4 : s[<=]s' -> union s s'' [<=] union s' s''.
  Proof. fsetdec. Qed.

  Lemma union_subset_5 : s[<=]s' -> union s'' s [<=] union s'' s'.
  Proof. fsetdec. Qed.

  Lemma empty_union_1 : Empty s -> union s s' [=] s'.
  Proof. fsetdec. Qed.

  Lemma empty_union_2 : Empty s -> union s' s [=] s'.
  Proof. fsetdec. Qed.

  Lemma not_in_union : ~In x s -> ~In x s' -> ~In x (union s s').
  Proof. fsetdec. Qed.

  Lemma inter_sym : inter s s' [=] inter s' s.
  Proof. fsetdec. Qed.

  Lemma inter_subset_equal : s[<=]s' -> inter s s' [=] s.
  Proof. fsetdec. Qed.

  Lemma inter_equal_1 : s[=]s' -> inter s s'' [=] inter s' s''.
  Proof. fsetdec. Qed.

  Lemma inter_equal_2 : s'[=]s'' -> inter s s' [=] inter s s''.
  Proof. fsetdec. Qed.

  Lemma inter_assoc : inter (inter s s') s'' [=] inter s (inter s' s'').
  Proof. fsetdec. Qed.

  Lemma union_inter_1 : inter (union s s') s'' [=] union (inter s s'') (inter s' s'').
  Proof. fsetdec. Qed.

  Lemma union_inter_2 : union (inter s s') s'' [=] inter (union s s'') (union s' s'').
  Proof. fsetdec. Qed.

  Lemma inter_add_1 : In x s' -> inter (add x s) s' [=] add x (inter s s').
  Proof. fsetdec. Qed.

  Lemma inter_add_2 : ~ In x s' -> inter (add x s) s' [=] inter s s'.
  Proof. fsetdec. Qed.

  Lemma empty_inter_1 : Empty s -> Empty (inter s s').
  Proof. fsetdec. Qed.

  Lemma empty_inter_2 : Empty s' -> Empty (inter s s').
  Proof. fsetdec. Qed.

  Lemma inter_subset_1 : inter s s' [<=] s.
  Proof. fsetdec. Qed.

  Lemma inter_subset_2 : inter s s' [<=] s'.
  Proof. fsetdec. Qed.

  Lemma inter_subset_3 :
   s''[<=]s -> s''[<=]s' -> s''[<=] inter s s'.
  Proof. fsetdec. Qed.

  Lemma empty_diff_1 : Empty s -> Empty (diff s s').
  Proof. fsetdec. Qed.

  Lemma empty_diff_2 : Empty s -> diff s' s [=] s'.
  Proof. fsetdec. Qed.

  Lemma diff_subset : diff s s' [<=] s.
  Proof. fsetdec. Qed.

  Lemma diff_subset_equal : s[<=]s' -> diff s s' [=] empty.
  Proof. fsetdec. Qed.

  Lemma remove_diff_singleton :
   remove x s [=] diff s (singleton x).
  Proof. fsetdec. Qed.

  Lemma diff_inter_empty : inter (diff s s') (inter s s') [=] empty.
  Proof. fsetdec. Qed.

  Lemma diff_inter_all : union (diff s s') (inter s s') [=] s.
  Proof. fsetdec. Qed.

  Lemma Add_add : Add x s (add x s).
  Proof. expAdd; fsetdec. Qed.

  Lemma Add_remove : In x s -> Add x (remove x s) s.
  Proof. expAdd; fsetdec. Qed.

  Lemma union_Add : Add x s s' -> Add x (union s s'') (union s' s'').
  Proof. expAdd; fsetdec. Qed.

  Lemma inter_Add :
   In x s'' -> Add x s s' -> Add x (inter s s'') (inter s' s'').
  Proof. expAdd; fsetdec. Qed.

  Lemma union_Equal :
   In x s'' -> Add x s s' -> union s s'' [=] union s' s''.
  Proof. expAdd; fsetdec. Qed.

  Lemma inter_Add_2 :
   ~In x s'' -> Add x s s' -> inter s s'' [=] inter s' s''.
  Proof. expAdd; fsetdec. Qed.

  End BasicProperties.

  Hint Immediate equal_sym add_remove remove_add union_sym inter_sym: set.
  Hint Resolve equal_refl equal_trans subset_refl subset_equal subset_antisym
    subset_trans subset_empty subset_remove_3 subset_diff subset_add_3
    subset_add_2 in_subset empty_is_empty_1 empty_is_empty_2 add_equal
    remove_equal singleton_equal_add union_subset_equal union_equal_1
    union_equal_2 union_assoc add_union_singleton union_add union_subset_1
    union_subset_2 union_subset_3 inter_subset_equal inter_equal_1 inter_equal_2
    inter_assoc union_inter_1 union_inter_2 inter_add_1 inter_add_2
    empty_inter_1 empty_inter_2 empty_union_1 empty_union_2 empty_diff_1
    empty_diff_2 union_Add inter_Add union_Equal inter_Add_2 not_in_union
    inter_subset_1 inter_subset_2 inter_subset_3 diff_subset diff_subset_equal
    remove_diff_singleton diff_inter_empty diff_inter_all Add_add Add_remove
    Equal_remove add_add : set.

  (** * Properties of elements *)

  Lemma elements_Empty : forall s, Empty s <-> elements s = nil.
  Proof.
  intros.
  unfold Empty.
  split; intros.
  assert (forall a, ~ List.In a (elements s)).
   red; intros.
   apply (H a).
   rewrite elements_iff.
   rewrite InA_alt; exists a; auto with relations.
  destruct (elements s); auto.
  elim (H0 e); simpl; auto.
  red; intros.
  rewrite elements_iff in H0.
  rewrite InA_alt in H0; destruct H0.
  rewrite H in H0; destruct H0 as (_,H0); inversion H0.
  Qed.

  Lemma elements_empty : elements empty = nil.
  Proof.
  rewrite <-elements_Empty; auto with set.
  Qed.

  (** * Conversions between lists and sets *)

  Definition of_list (l : list elt) := List.fold_right add empty l.

  Definition to_list := elements.

  Lemma of_list_1 : forall l x, In x (of_list l) <-> InA E.eq x l.
  Proof.
  induction l; simpl; intro x.
  rewrite empty_iff, InA_nil. intuition.
  rewrite add_iff, InA_cons, IHl. intuition.
  Qed.

  Lemma of_list_2 : forall l, equivlistA E.eq (to_list (of_list l)) l.
  Proof.
  unfold to_list; red; intros.
  rewrite <- elements_iff; apply of_list_1.
  Qed.

  Lemma of_list_3 : forall s, of_list (to_list s) [=] s.
  Proof.
  unfold to_list; red; intros.
  rewrite of_list_1; symmetry; apply elements_iff.
  Qed.

  (** * Fold *)

  Section Fold.

  Notation NoDup := (NoDupA E.eq).
  Notation InA := (InA E.eq).

  (** ** Induction principles for fold (contributed by S. Lescuyer) *)

  (** In the following lemma, the step hypothesis is deliberately restricted
      to the precise set s we are considering. *)

  Theorem fold_rec :
    forall (A:Type)(P : t -> A -> Type)(f : elt -> A -> A)(i:A)(s:t),
     (forall s', Empty s' -> P s' i) ->
     (forall x a s' s'', In x s -> ~In x s' -> Add x s' s'' ->
       P s' a -> P s'' (f x a)) ->
     P s (fold f s i).
  Proof.
  intros A P f i s Pempty Pstep.
  rewrite fold_1; unfold flip; rewrite <- fold_left_rev_right.
  set (l:=rev (elements s)).
  assert (Pstep' : forall x a s' s'', InA x l -> ~In x s' -> Add x s' s'' ->
           P s' a -> P s'' (f x a)).
   intros; eapply Pstep; eauto.
   rewrite elements_iff, <- InA_rev; auto with *.
  assert (Hdup : NoDup l) by
    (unfold l; eauto using elements_3w, NoDupA_rev with *).
  assert (Hsame : forall x, In x s <-> InA x l) by
    (unfold l; intros; rewrite elements_iff, InA_rev; intuition).
  clear Pstep; clearbody l; revert s Hsame; induction l.
  (* empty *)
  intros s Hsame; simpl.
  apply Pempty. intro x. rewrite Hsame, InA_nil; intuition.
  (* step *)
  intros s Hsame; simpl.
  apply Pstep' with (of_list l); auto with relations.
   inversion_clear Hdup; rewrite of_list_1; auto.
   red. intros. rewrite Hsame, of_list_1, InA_cons; intuition.
  apply IHl.
   intros; eapply Pstep'; eauto.
   inversion_clear Hdup; auto.
   exact (of_list_1 l).
  Qed.

  (** Same, with [empty] and [add] instead of [Empty] and [Add]. In this
      case, [P] must be compatible with equality of sets *)

  Theorem fold_rec_bis :
    forall (A:Type)(P : t -> A -> Type)(f : elt -> A -> A)(i:A)(s:t),
     (forall s s' a, s[=]s' -> P s a -> P s' a) ->
     (P empty i) ->
     (forall x a s', In x s -> ~In x s' -> P s' a -> P (add x s') (f x a)) ->
     P s (fold f s i).
  Proof.
  intros A P f i s Pmorphism Pempty Pstep.
  apply fold_rec; intros.
  apply Pmorphism with empty; auto with set.
  rewrite Add_Equal in H1; auto with set.
  apply Pmorphism with (add x s'); auto with set.
  Qed.

  Lemma fold_rec_nodep :
    forall (A:Type)(P : A -> Type)(f : elt -> A -> A)(i:A)(s:t),
     P i -> (forall x a, In x s -> P a -> P (f x a)) ->
     P (fold f s i).
  Proof.
  intros; apply fold_rec_bis with (P:=fun _ => P); auto.
  Qed.

  (** [fold_rec_weak] is a weaker principle than [fold_rec_bis] :
      the step hypothesis must here be applicable to any [x].
      At the same time, it looks more like an induction principle,
      and hence can be easier to use. *)

  Lemma fold_rec_weak :
    forall (A:Type)(P : t -> A -> Type)(f : elt -> A -> A)(i:A),
    (forall s s' a, s[=]s' -> P s a -> P s' a) ->
    P empty i ->
    (forall x a s, ~In x s -> P s a -> P (add x s) (f x a)) ->
    forall s, P s (fold f s i).
  Proof.
  intros; apply fold_rec_bis; auto.
  Qed.

  Lemma fold_rel :
    forall (A B:Type)(R : A -> B -> Type)
     (f : elt -> A -> A)(g : elt -> B -> B)(i : A)(j : B)(s : t),
     R i j ->
     (forall x a b, In x s -> R a b -> R (f x a) (g x b)) ->
     R (fold f s i) (fold g s j).
  Proof.
  intros A B R f g i j s Rempty Rstep.
  do 2 (rewrite fold_1; unfold flip; rewrite <- fold_left_rev_right).
  set (l:=rev (elements s)).
  assert (Rstep' : forall x a b, InA x l -> R a b -> R (f x a) (g x b)) by
    (intros; apply Rstep; auto; rewrite elements_iff, <- InA_rev; auto with *).
  clearbody l; clear Rstep s.
  induction l; simpl; auto with relations.
  Qed.

  (** From the induction principle on [fold], we can deduce some general
      induction principles on sets. *)

  Lemma set_induction :
   forall P : t -> Type,
   (forall s, Empty s -> P s) ->
   (forall s s', P s -> forall x, ~In x s -> Add x s s' -> P s') ->
   forall s, P s.
  Proof.
  intros. apply (@fold_rec _ (fun s _ => P s) (fun _ _ => tt) tt s); eauto.
  Qed.

  Lemma set_induction_bis :
   forall P : t -> Type,
   (forall s s', s [=] s' -> P s -> P s') ->
   P empty ->
   (forall x s, ~In x s -> P s -> P (add x s)) ->
   forall s, P s.
  Proof.
  intros.
  apply (@fold_rec_bis _ (fun s _ => P s) (fun _ _ => tt) tt s); eauto.
  Qed.

  (** [fold] can be used to reconstruct the same initial set. *)

  Lemma fold_identity : forall s, fold add s empty [=] s.
  Proof.
  intros.
  apply fold_rec with (P:=fun s acc => acc[=]s); auto with set.
  intros. rewrite H2; rewrite Add_Equal in H1; auto with set.
  Qed.

  (** ** Alternative (weaker) specifications for [fold] *)

  (** When [MSets] was first designed, the order in which Ocaml's [Set.fold]
      takes the set elements was unspecified. This specification reflects
      this fact:
  *)

  Lemma fold_0 :
      forall s (A : Type) (i : A) (f : elt -> A -> A),
      exists l : list elt,
        NoDup l /\
        (forall x : elt, In x s <-> InA x l) /\
        fold f s i = fold_right f i l.
  Proof.
  intros; exists (rev (elements s)); split.
  apply NoDupA_rev; auto with *.
  split; intros.
  rewrite elements_iff; do 2 rewrite InA_alt.
  split; destruct 1; generalize (In_rev (elements s) x0); exists x0; intuition.
  rewrite fold_left_rev_right.
  apply fold_1.
  Qed.

  (** An alternate (and previous) specification for [fold] was based on
      the recursive structure of a set. It is now lemmas [fold_1] and
      [fold_2]. *)

  Lemma fold_1 :
   forall s (A : Type) (eqA : A -> A -> Prop)
     (st : Equivalence eqA) (i : A) (f : elt -> A -> A),
   Empty s -> eqA (fold f s i) i.
  Proof.
  unfold Empty; intros; destruct (fold_0 s i f) as (l,(H1, (H2, H3))).
  rewrite H3; clear H3.
  generalize H H2; clear H H2; case l; simpl; intros.
  reflexivity.
  elim (H e).
  elim (H2 e); intuition.
  Qed.

  Lemma fold_2 :
   forall s s' x (A : Type) (eqA : A -> A -> Prop)
     (st : Equivalence eqA) (i : A) (f : elt -> A -> A),
   Proper (E.eq==>eqA==>eqA) f ->
   transpose eqA f ->
   ~ In x s -> Add x s s' -> eqA (fold f s' i) (f x (fold f s i)).
  Proof.
  intros; destruct (fold_0 s i f) as (l,(Hl, (Hl1, Hl2)));
    destruct (fold_0 s' i f) as (l',(Hl', (Hl'1, Hl'2))).
  rewrite Hl2; rewrite Hl'2; clear Hl2 Hl'2.
  apply fold_right_add with (eqA:=E.eq)(eqB:=eqA); auto.
  eauto with *.
  rewrite <- Hl1; auto.
  intros a; rewrite InA_cons; rewrite <- Hl1; rewrite <- Hl'1;
   rewrite (H2 a); intuition.
  Qed.

  (** In fact, [fold] on empty sets is more than equivalent to
      the initial element, it is Leibniz-equal to it. *)

  Lemma fold_1b :
   forall s (A : Type)(i : A) (f : elt -> A -> A),
   Empty s -> (fold f s i) = i.
  Proof.
  intros.
  rewrite FM.fold_1.
  rewrite elements_Empty in H; rewrite H; simpl; auto.
  Qed.

  Section Fold_More.

  Variables (A:Type)(eqA:A->A->Prop)(st:Equivalence eqA).
  Variables (f:elt->A->A)(Comp:Proper (E.eq==>eqA==>eqA) f)(Ass:transpose eqA f).

  Lemma fold_commutes : forall i s x,
   eqA (fold f s (f x i)) (f x (fold f s i)).
  Proof.
  intros.
  apply fold_rel with (R:=fun u v => eqA u (f x v)); intros.
  reflexivity.
  transitivity (f x0 (f x b)); auto.
  apply Comp; auto with relations.
  Qed.

  (** ** Fold is a morphism *)

  Lemma fold_init : forall i i' s, eqA i i' ->
   eqA (fold f s i) (fold f s i').
  Proof.
  intros. apply fold_rel with (R:=eqA); auto.
  intros; apply Comp; auto with relations.
  Qed.

  Lemma fold_equal :
   forall i s s', s[=]s' -> eqA (fold f s i) (fold f s' i).
  Proof.
  intros i s; pattern s; apply set_induction; clear s; intros.
  transitivity i.
  apply fold_1; auto.
  symmetry; apply fold_1; auto.
  rewrite <- H0; auto.
  transitivity (f x (fold f s i)).
  apply fold_2 with (eqA := eqA); auto.
  symmetry; apply fold_2 with (eqA := eqA); auto.
  unfold Add in *; intros.
  rewrite <- H2; auto.
  Qed.

  (** ** Fold and other set operators *)

  Lemma fold_empty : forall i, fold f empty i = i.
  Proof.
  intros i; apply fold_1b; auto with set.
  Qed.

  Lemma fold_add : forall i s x, ~In x s ->
   eqA (fold f (add x s) i) (f x (fold f s i)).
  Proof.
  intros; apply fold_2 with (eqA := eqA); auto with set.
  Qed.

  Lemma add_fold : forall i s x, In x s ->
   eqA (fold f (add x s) i) (fold f s i).
  Proof.
  intros; apply fold_equal; auto with set.
  Qed.

  Lemma remove_fold_1: forall i s x, In x s ->
   eqA (f x (fold f (remove x s) i)) (fold f s i).
  Proof.
  intros.
  symmetry.
  apply fold_2 with (eqA:=eqA); auto with set relations.
  Qed.

  Lemma remove_fold_2: forall i s x, ~In x s ->
   eqA (fold f (remove x s) i) (fold f s i).
  Proof.
  intros.
  apply fold_equal; auto with set.
  Qed.

  Lemma fold_union_inter : forall i s s',
   eqA (fold f (union s s') (fold f (inter s s') i))
       (fold f s (fold f s' i)).
  Proof.
  intros; pattern s; apply set_induction; clear s; intros.
  transitivity (fold f s' (fold f (inter s s') i)).
  apply fold_equal; auto with set.
  transitivity (fold f s' i).
  apply fold_init; auto.
  apply fold_1; auto with set.
  symmetry; apply fold_1; auto.
  rename s'0 into s''.
  destruct (In_dec x s').
  (* In x s' *)
  transitivity (fold f (union s'' s') (f x (fold f (inter s s') i))); auto with set.
  apply fold_init; auto.
  apply fold_2 with (eqA:=eqA); auto with set.
  rewrite inter_iff; intuition.
  transitivity (f x (fold f s (fold f s' i))).
  transitivity (fold f (union s s') (f x (fold f (inter s s') i))).
  apply fold_equal; auto.
  apply equal_sym; apply union_Equal with x; auto with set.
  transitivity (f x (fold f (union s s') (fold f (inter s s') i))).
  apply fold_commutes; auto.
  apply Comp; auto with relations.
  symmetry; apply fold_2 with (eqA:=eqA); auto.
  (* ~(In x s') *)
  transitivity (f x (fold f (union s s') (fold f (inter s'' s') i))).
  apply fold_2 with (eqA:=eqA); auto with set.
  transitivity (f x (fold f (union s s') (fold f (inter s s') i))).
  apply Comp;auto with relations.
  apply fold_init;auto.
  apply fold_equal;auto.
  apply equal_sym; apply inter_Add_2 with x; auto with set.
  transitivity (f x (fold f s (fold f s' i))).
  apply Comp; auto with relations.
  symmetry; apply fold_2 with (eqA:=eqA); auto.
  Qed.

  Lemma fold_diff_inter : forall i s s',
   eqA (fold f (diff s s') (fold f (inter s s') i)) (fold f s i).
  Proof.
  intros.
  transitivity (fold f (union (diff s s') (inter s s'))
               (fold f (inter (diff s s') (inter s s')) i)).
  symmetry; apply fold_union_inter; auto.
  transitivity (fold f s (fold f (inter (diff s s') (inter s s')) i)).
  apply fold_equal; auto with set.
  apply fold_init; auto.
  apply fold_1; auto with set.
  Qed.

  Lemma fold_union: forall i s s',
   (forall x, ~(In x s/\In x s')) ->
   eqA (fold f (union s s') i) (fold f s (fold f s' i)).
  Proof.
  intros.
  transitivity (fold f (union s s') (fold f (inter s s') i)).
  apply fold_init; auto.
  symmetry; apply fold_1; auto with set.
  unfold Empty; intro a; generalize (H a); set_iff; tauto.
  apply fold_union_inter; auto.
  Qed.

  End Fold_More.

  Lemma fold_plus :
   forall s p, fold (fun _ => S) s p = fold (fun _ => S) s 0 + p.
  Proof.
  intros. apply fold_rel with (R:=fun u v => u = v + p); simpl; auto.
  Qed.

  End Fold.

  (** * Cardinal *)

  (** ** Characterization of cardinal in terms of fold *)

  Lemma cardinal_fold : forall s, cardinal s = fold (fun _ => S) s 0.
  Proof.
  intros; rewrite cardinal_1; rewrite FM.fold_1.
  symmetry; apply fold_left_length; auto.
  Qed.

  (** ** Old specifications for [cardinal]. *)

  Lemma cardinal_0 :
     forall s, exists l : list elt,
        NoDupA E.eq l /\
        (forall x : elt, In x s <-> InA E.eq x l) /\
        cardinal s = length l.
  Proof.
  intros; exists (elements s); intuition; apply cardinal_1.
  Qed.

  Lemma cardinal_1 : forall s, Empty s -> cardinal s = 0.
  Proof.
  intros; rewrite cardinal_fold; apply fold_1; auto with *.
  Qed.

  Lemma cardinal_2 :
    forall s s' x, ~ In x s -> Add x s s' -> cardinal s' = S (cardinal s).
  Proof.
  intros; do 2 rewrite cardinal_fold.
  change S with ((fun _ => S) x).
  apply fold_2; auto.
  split; congruence.
  congruence.
  Qed.

  (** ** Cardinal and (non-)emptiness *)

  Lemma cardinal_Empty : forall s, Empty s <-> cardinal s = 0.
  Proof.
  intros.
  rewrite elements_Empty, FM.cardinal_1.
  destruct (elements s); intuition; discriminate.
  Qed.

  Lemma cardinal_inv_1 : forall s, cardinal s = 0 -> Empty s.
  Proof.
  intros; rewrite cardinal_Empty; auto.
  Qed.
  Hint Resolve cardinal_inv_1.

  Lemma cardinal_inv_2 :
   forall s n, cardinal s = S n -> { x : elt | In x s }.
  Proof.
  intros; rewrite FM.cardinal_1 in H.
  generalize (elements_2 (s:=s)).
  destruct (elements s); try discriminate.
  exists e; auto with relations.
  Qed.

  Lemma cardinal_inv_2b :
   forall s, cardinal s <> 0 -> { x : elt | In x s }.
  Proof.
  intro; generalize (@cardinal_inv_2 s); destruct cardinal;
   [intuition|eauto].
  Qed.

  (** ** Cardinal is a morphism *)

  Lemma Equal_cardinal : forall s s', s[=]s' -> cardinal s = cardinal s'.
  Proof.
  symmetry.
  remember (cardinal s) as n; symmetry in Heqn; revert s s' Heqn H.
  induction n; intros.
  apply cardinal_1; rewrite <- H; auto.
  destruct (cardinal_inv_2 Heqn) as (x,H2).
  revert Heqn.
  rewrite (cardinal_2 (s:=remove x s) (s':=s) (x:=x));
   auto with set relations.
  rewrite (cardinal_2 (s:=remove x s') (s':=s') (x:=x));
   eauto with set relations.
  Qed.

  Instance cardinal_m : Proper (Equal==>Logic.eq) cardinal.
  Proof.
  exact Equal_cardinal.
  Qed.

  Hint Resolve Add_add Add_remove Equal_remove cardinal_inv_1 Equal_cardinal.

  (** ** Cardinal and set operators *)

  Lemma empty_cardinal : cardinal empty = 0.
  Proof.
  rewrite cardinal_fold; apply fold_1; auto with *.
  Qed.

  Hint Immediate empty_cardinal cardinal_1 : set.

  Lemma singleton_cardinal : forall x, cardinal (singleton x) = 1.
  Proof.
  intros.
  rewrite (singleton_equal_add x).
  replace 0 with (cardinal empty); auto with set.
  apply cardinal_2 with x; auto with set.
  Qed.

  Hint Resolve singleton_cardinal: set.

  Lemma diff_inter_cardinal :
   forall s s', cardinal (diff s s') + cardinal (inter s s') = cardinal s .
  Proof.
  intros; do 3 rewrite cardinal_fold.
  rewrite <- fold_plus.
  apply fold_diff_inter with (eqA:=@Logic.eq nat); auto with *.
  congruence.
  Qed.

  Lemma union_cardinal:
   forall s s', (forall x, ~(In x s/\In x s')) ->
   cardinal (union s s')=cardinal s+cardinal s'.
  Proof.
  intros; do 3 rewrite cardinal_fold.
  rewrite <- fold_plus.
  apply fold_union; auto.
  split; congruence.
  congruence.
  Qed.

  Lemma subset_cardinal :
   forall s s', s[<=]s' -> cardinal s <= cardinal s' .
  Proof.
  intros.
  rewrite <- (diff_inter_cardinal s' s).
  rewrite (inter_sym s' s).
  rewrite (inter_subset_equal H); auto with arith.
  Qed.

  Lemma subset_cardinal_lt :
   forall s s' x, s[<=]s' -> In x s' -> ~In x s -> cardinal s < cardinal s'.
  Proof.
  intros.
  rewrite <- (diff_inter_cardinal s' s).
  rewrite (inter_sym s' s).
  rewrite (inter_subset_equal H).
  generalize (@cardinal_inv_1 (diff s' s)).
  destruct (cardinal (diff s' s)).
  intro H2; destruct (H2 (refl_equal _) x).
  set_iff; auto.
  intros _.
  change (0 + cardinal s < S n + cardinal s).
  apply Plus.plus_lt_le_compat; auto with arith.
  Qed.

  Theorem union_inter_cardinal :
   forall s s', cardinal (union s s') + cardinal (inter s s')  = cardinal s  + cardinal s' .
  Proof.
  intros.
  do 4 rewrite cardinal_fold.
  do 2 rewrite <- fold_plus.
  apply fold_union_inter with (eqA:=@Logic.eq nat); auto with *.
  congruence.
  Qed.

  Lemma union_cardinal_inter :
   forall s s', cardinal (union s s') = cardinal s + cardinal s' - cardinal (inter s s').
  Proof.
  intros.
  rewrite <- union_inter_cardinal.
  rewrite Plus.plus_comm.
  auto with arith.
  Qed.

  Lemma union_cardinal_le :
   forall s s', cardinal (union s s') <= cardinal s  + cardinal s'.
  Proof.
   intros; generalize (union_inter_cardinal s s').
   intros; rewrite <- H; auto with arith.
  Qed.

  Lemma add_cardinal_1 :
   forall s x, In x s -> cardinal (add x s) = cardinal s.
  Proof.
  auto with set.
  Qed.

  Lemma add_cardinal_2 :
   forall s x, ~In x s -> cardinal (add x s) = S (cardinal s).
  Proof.
  intros.
  do 2 rewrite cardinal_fold.
  change S with ((fun _ => S) x);
   apply fold_add with (eqA:=@Logic.eq nat); auto with *.
  congruence.
  Qed.

  Lemma remove_cardinal_1 :
   forall s x, In x s -> S (cardinal (remove x s)) = cardinal s.
  Proof.
  intros.
  do 2 rewrite cardinal_fold.
  change S with ((fun _ =>S) x).
  apply remove_fold_1 with (eqA:=@Logic.eq nat); auto with *.
  congruence.
  Qed.

  Lemma remove_cardinal_2 :
   forall s x, ~In x s -> cardinal (remove x s) = cardinal s.
  Proof.
  auto with set.
  Qed.

  Hint Resolve subset_cardinal union_cardinal add_cardinal_1 add_cardinal_2.

End WPropertiesOn.

(** Now comes variants for self-contained weak sets and for full sets.
    For these variants, only one argument is necessary. Thanks to
    the subtyping [WS<=S], the [Properties] functor which is meant to be
    used on modules [(M:S)] can simply be an alias of [WProperties]. *)

Module WProperties (M:WSets) := WPropertiesOn M.E M.
Module Properties := WProperties.


(** Now comes some properties specific to the element ordering,
    invalid for Weak Sets. *)

Module OrdProperties (M:Sets).
  Module Import ME:=OrderedTypeFacts(M.E).
  Module Import ML:=OrderedTypeLists(M.E).
  Module Import P := Properties M.
  Import FM.
  Import M.E.
  Import M.

  Hint Resolve elements_spec2.
  Hint Immediate
    min_elt_spec1 min_elt_spec2 min_elt_spec3
    max_elt_spec1 max_elt_spec2 max_elt_spec3 : set.

  (** First, a specialized version of SortA_equivlistA_eqlistA: *)
  Lemma sort_equivlistA_eqlistA : forall l l' : list elt,
   sort E.lt l -> sort E.lt l' -> equivlistA E.eq l l' -> eqlistA E.eq l l'.
  Proof.
  apply SortA_equivlistA_eqlistA; eauto with *.
  Qed.

  Definition gtb x y := match E.compare x y with Gt => true | _ => false end.
  Definition leb x := fun y => negb (gtb x y).

  Definition elements_lt x s := List.filter (gtb x) (elements s).
  Definition elements_ge x s := List.filter (leb x) (elements s).

  Lemma gtb_1 : forall x y, gtb x y = true <-> E.lt y x.
  Proof.
   intros; rewrite <- compare_gt_iff. unfold gtb.
   destruct E.compare; intuition; try discriminate.
  Qed.

  Lemma leb_1 : forall x y, leb x y = true <-> ~E.lt y x.
  Proof.
   intros; rewrite <- compare_gt_iff. unfold leb, gtb.
   destruct E.compare; intuition; try discriminate.
  Qed.

  Instance gtb_compat x : Proper (E.eq==>Logic.eq) (gtb x).
  Proof.
   intros a b H. unfold gtb. rewrite H; auto.
  Qed.

  Instance leb_compat x : Proper (E.eq==>Logic.eq) (leb x).
  Proof.
   intros a b H; unfold leb. rewrite H; auto.
  Qed.
  Hint Resolve gtb_compat leb_compat.

  Lemma elements_split : forall x s,
   elements s = elements_lt x s ++ elements_ge x s.
  Proof.
  unfold elements_lt, elements_ge, leb; intros.
  eapply (@filter_split _ E.eq); eauto with *.
  intros.
  rewrite gtb_1 in H.
  assert (~E.lt y x).
   unfold gtb in *; elim_compare x y; intuition;
   try discriminate; order.
  order.
  Qed.

  Lemma elements_Add : forall s s' x, ~In x s -> Add x s s' ->
    eqlistA E.eq (elements s') (elements_lt x s ++ x :: elements_ge x s).
  Proof.
  intros; unfold elements_ge, elements_lt.
  apply sort_equivlistA_eqlistA; auto with set.
  apply (@SortA_app _ E.eq); auto with *.
  apply (@filter_sort _ E.eq); auto with *; eauto with *.
  constructor; auto.
  apply (@filter_sort _ E.eq); auto with *; eauto with *.
  rewrite Inf_alt by (apply (@filter_sort _ E.eq); eauto with *).
  intros.
  rewrite filter_InA in H1; auto with *; destruct H1.
  rewrite leb_1 in H2.
  rewrite <- elements_iff in H1.
  assert (~E.eq x y).
   contradict H; rewrite H; auto.
  order.
  intros.
  rewrite filter_InA in H1; auto with *; destruct H1.
  rewrite gtb_1 in H3.
  inversion_clear H2.
  order.
  rewrite filter_InA in H4; auto with *; destruct H4.
  rewrite leb_1 in H4.
  order.
  red; intros a.
  rewrite InA_app_iff, InA_cons, !filter_InA, <-!elements_iff,
   leb_1, gtb_1, (H0 a) by (auto with *).
  intuition.
  elim_compare a x; intuition.
  right; right; split; auto.
  order.
  Qed.

  Definition Above x s := forall y, In y s -> E.lt y x.
  Definition Below x s := forall y, In y s -> E.lt x y.

  Lemma elements_Add_Above : forall s s' x,
   Above x s -> Add x s s' ->
     eqlistA E.eq (elements s') (elements s ++ x::nil).
  Proof.
  intros.
  apply sort_equivlistA_eqlistA; auto with set.
  apply (@SortA_app _ E.eq); auto with *.
  intros.
  invlist InA.
  rewrite <- elements_iff in H1.
  setoid_replace y with x; auto.
  red; intros a.
  rewrite InA_app_iff, InA_cons, InA_nil, <-!elements_iff, (H0 a)
   by (auto with *).
  intuition.
  Qed.

  Lemma elements_Add_Below : forall s s' x,
   Below x s -> Add x s s' ->
     eqlistA E.eq (elements s') (x::elements s).
  Proof.
  intros.
  apply sort_equivlistA_eqlistA; auto with set.
  change (sort E.lt ((x::nil) ++ elements s)).
  apply (@SortA_app _ E.eq); auto with *.
  intros.
  invlist InA.
  rewrite <- elements_iff in H2.
  setoid_replace x0 with x; auto.
  red; intros a.
  rewrite InA_cons, <- !elements_iff, (H0 a); intuition.
  Qed.

  (** Two other induction principles on sets: we can be more restrictive
      on the element we add at each step.  *)

  Lemma set_induction_max :
   forall P : t -> Type,
   (forall s : t, Empty s -> P s) ->
   (forall s s', P s -> forall x, Above x s -> Add x s s' -> P s') ->
   forall s : t, P s.
  Proof.
  intros; remember (cardinal s) as n; revert s Heqn; induction n; intros; auto.
  case_eq (max_elt s); intros.
  apply X0 with (remove e s) e; auto with set.
  apply IHn.
  assert (S n = S (cardinal (remove e s))).
   rewrite Heqn; apply cardinal_2 with e; auto with set relations.
  inversion H0; auto.
  red; intros.
  rewrite remove_iff in H0; destruct H0.
  generalize (@max_elt_spec2 s e y H H0); order.

  assert (H0:=max_elt_spec3 H).
  rewrite cardinal_Empty in H0; rewrite H0 in Heqn; inversion Heqn.
  Qed.

  Lemma set_induction_min :
   forall P : t -> Type,
   (forall s : t, Empty s -> P s) ->
   (forall s s', P s -> forall x, Below x s -> Add x s s' -> P s') ->
   forall s : t, P s.
  Proof.
  intros; remember (cardinal s) as n; revert s Heqn; induction n; intros; auto.
  case_eq (min_elt s); intros.
  apply X0 with (remove e s) e; auto with set.
  apply IHn.
  assert (S n = S (cardinal (remove e s))).
   rewrite Heqn; apply cardinal_2 with e; auto with set relations.
  inversion H0; auto.
  red; intros.
  rewrite remove_iff in H0; destruct H0.
  generalize (@min_elt_spec2 s e y H H0); order.

  assert (H0:=min_elt_spec3 H).
  rewrite cardinal_Empty in H0; auto; rewrite H0 in Heqn; inversion Heqn.
  Qed.

  (** More properties of [fold] : behavior with respect to Above/Below *)

  Lemma fold_3 :
   forall s s' x (A : Type) (eqA : A -> A -> Prop)
     (st : Equivalence eqA) (i : A) (f : elt -> A -> A),
   Proper (E.eq==>eqA==>eqA) f ->
   Above x s -> Add x s s' -> eqA (fold f s' i) (f x (fold f s i)).
  Proof.
  intros.
  rewrite !FM.fold_1.
  unfold flip; rewrite <-!fold_left_rev_right.
  change (f x (fold_right f i (rev (elements s)))) with
    (fold_right f i (rev (x::nil)++rev (elements s))).
  apply (@fold_right_eqlistA E.t E.eq A eqA st); auto with *.
  rewrite <- distr_rev.
  apply eqlistA_rev.
  apply elements_Add_Above; auto.
  Qed.

  Lemma fold_4 :
   forall s s' x (A : Type) (eqA : A -> A -> Prop)
     (st : Equivalence eqA) (i : A) (f : elt -> A -> A),
   Proper (E.eq==>eqA==>eqA) f ->
   Below x s -> Add x s s' -> eqA (fold f s' i) (fold f s (f x i)).
  Proof.
  intros.
  rewrite !FM.fold_1.
  change (eqA (fold_left (flip f) (elements s') i)
              (fold_left (flip f) (x::elements s) i)).
  unfold flip; rewrite <-!fold_left_rev_right.
  apply (@fold_right_eqlistA E.t E.eq A eqA st); auto.
  apply eqlistA_rev.
  apply elements_Add_Below; auto.
  Qed.

  (** The following results have already been proved earlier,
    but we can now prove them with one hypothesis less:
    no need for [(transpose eqA f)]. *)

  Section FoldOpt.
  Variables (A:Type)(eqA:A->A->Prop)(st:Equivalence eqA).
  Variables (f:elt->A->A)(Comp:Proper (E.eq==>eqA==>eqA) f).

  Lemma fold_equal :
   forall i s s', s[=]s' -> eqA (fold f s i) (fold f s' i).
  Proof.
  intros.
  rewrite !FM.fold_1.
  unfold flip; rewrite <- !fold_left_rev_right.
  apply (@fold_right_eqlistA E.t E.eq A eqA st); auto.
  apply eqlistA_rev.
  apply sort_equivlistA_eqlistA; auto with set.
  red; intro a; do 2 rewrite <- elements_iff; auto.
  Qed.

  Lemma add_fold : forall i s x, In x s ->
   eqA (fold f (add x s) i) (fold f s i).
  Proof.
  intros; apply fold_equal; auto with set.
  Qed.

  Lemma remove_fold_2: forall i s x, ~In x s ->
   eqA (fold f (remove x s) i) (fold f s i).
  Proof.
  intros.
  apply fold_equal; auto with set.
  Qed.

  End FoldOpt.

  (** An alternative version of [choose_3] *)

  Lemma choose_equal : forall s s', Equal s s' ->
    match choose s, choose s' with
      | Some x, Some x' => E.eq x x'
      | None, None => True
      | _, _ => False
     end.
  Proof.
  intros s s' H;
  generalize (@choose_spec1 s)(@choose_spec2 s)
             (@choose_spec1 s')(@choose_spec2 s')(@choose_spec3 s s');
  destruct (choose s); destruct (choose s'); simpl; intuition.
  apply H5 with e; rewrite <-H; auto.
  apply H5 with e; rewrite H; auto.
  Qed.

End OrdProperties.