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
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: Mapcard.v 8733 2006-04-25 22:52:18Z letouzey $ i*)
Require Import Bool.
Require Import Sumbool.
Require Import Arith.
Require Import NArith.
Require Import Ndigits.
Require Import Ndec.
Require Import Map.
Require Import Mapaxioms.
Require Import Mapiter.
Require Import Fset.
Require Import Mapsubset.
Require Import List.
Require Import Lsort.
Require Import Peano_dec.
Section MapCard.
Variables A B : Set.
Lemma MapCard_M0 : MapCard A (M0 A) = 0.
Proof.
trivial.
Qed.
Lemma MapCard_M1 : forall (a:ad) (y:A), MapCard A (M1 A a y) = 1.
Proof.
trivial.
Qed.
Lemma MapCard_is_O :
forall m:Map A, MapCard A m = 0 -> forall a:ad, MapGet A m a = None.
Proof.
simple induction m. trivial.
intros a y H. discriminate H.
intros. simpl in H1. elim (plus_is_O _ _ H1). intros. rewrite (MapGet_M2_bit_0_if A m0 m1 a).
case (Nbit0 a). apply H0. assumption.
apply H. assumption.
Qed.
Lemma MapCard_is_not_O :
forall (m:Map A) (a:ad) (y:A),
MapGet A m a = Some y -> {n : nat | MapCard A m = S n}.
Proof.
simple induction m. intros. discriminate H.
intros a y a0 y0 H. simpl in H. elim (sumbool_of_bool (Neqb a a0)). intro H0. split with 0.
reflexivity.
intro H0. rewrite H0 in H. discriminate H.
intros. elim (sumbool_of_bool (Nbit0 a)). intro H2.
rewrite (MapGet_M2_bit_0_1 A a H2 m0 m1) in H1. elim (H0 (Ndiv2 a) y H1). intros n H3.
simpl in |- *. rewrite H3. split with (MapCard A m0 + n).
rewrite <- (plus_Snm_nSm (MapCard A m0) n). reflexivity.
intro H2. rewrite (MapGet_M2_bit_0_0 A a H2 m0 m1) in H1. elim (H (Ndiv2 a) y H1).
intros n H3. simpl in |- *. rewrite H3. split with (n + MapCard A m1). reflexivity.
Qed.
Lemma MapCard_is_one :
forall m:Map A,
MapCard A m = 1 -> {a : ad & {y : A | MapGet A m a = Some y}}.
Proof.
simple induction m. intro. discriminate H.
intros a y H. split with a. split with y. apply M1_semantics_1.
intros. simpl in H1. elim (plus_is_one (MapCard A m0) (MapCard A m1) H1).
intro H2. elim H2. intros. elim (H0 H4). intros a H5. split with (Ndouble_plus_one a).
rewrite (MapGet_M2_bit_0_1 A _ (Ndouble_plus_one_bit0 a) m0 m1).
rewrite Ndouble_plus_one_div2. exact H5.
intro H2. elim H2. intros. elim (H H3). intros a H5. split with (Ndouble a).
rewrite (MapGet_M2_bit_0_0 A _ (Ndouble_bit0 a) m0 m1).
rewrite Ndouble_div2. exact H5.
Qed.
Lemma MapCard_is_one_unique :
forall m:Map A,
MapCard A m = 1 ->
forall (a a':ad) (y y':A),
MapGet A m a = Some y ->
MapGet A m a' = Some y' -> a = a' /\ y = y'.
Proof.
simple induction m. intro. discriminate H.
intros. elim (sumbool_of_bool (Neqb a a1)). intro H2. rewrite (Neqb_complete _ _ H2) in H0.
rewrite (M1_semantics_1 A a1 a0) in H0. inversion H0. elim (sumbool_of_bool (Neqb a a')).
intro H5. rewrite (Neqb_complete _ _ H5) in H1. rewrite (M1_semantics_1 A a' a0) in H1.
inversion H1. rewrite <- (Neqb_complete _ _ H2). rewrite <- (Neqb_complete _ _ H5).
rewrite <- H4. rewrite <- H6. split; reflexivity.
intro H5. rewrite (M1_semantics_2 A a a' a0 H5) in H1. discriminate H1.
intro H2. rewrite (M1_semantics_2 A a a1 a0 H2) in H0. discriminate H0.
intros. simpl in H1. elim (plus_is_one _ _ H1). intro H4. elim H4. intros.
rewrite (MapGet_M2_bit_0_if A m0 m1 a) in H2. elim (sumbool_of_bool (Nbit0 a)).
intro H7. rewrite H7 in H2. rewrite (MapGet_M2_bit_0_if A m0 m1 a') in H3.
elim (sumbool_of_bool (Nbit0 a')). intro H8. rewrite H8 in H3. elim (H0 H6 _ _ _ _ H2 H3).
intros. split. rewrite <- (Ndiv2_double_plus_one a H7).
rewrite <- (Ndiv2_double_plus_one a' H8). rewrite H9. reflexivity.
assumption.
intro H8. rewrite H8 in H3. rewrite (MapCard_is_O m0 H5 (Ndiv2 a')) in H3.
discriminate H3.
intro H7. rewrite H7 in H2. rewrite (MapCard_is_O m0 H5 (Ndiv2 a)) in H2.
discriminate H2.
intro H4. elim H4. intros. rewrite (MapGet_M2_bit_0_if A m0 m1 a) in H2.
elim (sumbool_of_bool (Nbit0 a)). intro H7. rewrite H7 in H2.
rewrite (MapCard_is_O m1 H6 (Ndiv2 a)) in H2. discriminate H2.
intro H7. rewrite H7 in H2. rewrite (MapGet_M2_bit_0_if A m0 m1 a') in H3.
elim (sumbool_of_bool (Nbit0 a')). intro H8. rewrite H8 in H3.
rewrite (MapCard_is_O m1 H6 (Ndiv2 a')) in H3. discriminate H3.
intro H8. rewrite H8 in H3. elim (H H5 _ _ _ _ H2 H3). intros. split.
rewrite <- (Ndiv2_double a H7). rewrite <- (Ndiv2_double a' H8).
rewrite H9. reflexivity.
assumption.
Qed.
Lemma length_as_fold :
forall (C:Set) (l:list C),
length l = fold_right (fun (_:C) (n:nat) => S n) 0 l.
Proof.
simple induction l. reflexivity.
intros. simpl in |- *. rewrite H. reflexivity.
Qed.
Lemma length_as_fold_2 :
forall l:alist A,
length l =
fold_right (fun (r:ad * A) (n:nat) => let (a, y) := r in 1 + n) 0 l.
Proof.
simple induction l. reflexivity.
intros. simpl in |- *. rewrite H. elim a; reflexivity.
Qed.
Lemma MapCard_as_Fold_1 :
forall (m:Map A) (pf:ad -> ad),
MapCard A m = MapFold1 A nat 0 plus (fun (_:ad) (_:A) => 1) pf m.
Proof.
simple induction m. trivial.
trivial.
intros. simpl in |- *. rewrite <- (H (fun a0:ad => pf (Ndouble a0))).
rewrite <- (H0 (fun a0:ad => pf (Ndouble_plus_one a0))). reflexivity.
Qed.
Lemma MapCard_as_Fold :
forall m:Map A,
MapCard A m = MapFold A nat 0 plus (fun (_:ad) (_:A) => 1) m.
Proof.
intro. exact (MapCard_as_Fold_1 m (fun a0:ad => a0)).
Qed.
Lemma MapCard_as_length :
forall m:Map A, MapCard A m = length (alist_of_Map A m).
Proof.
intro. rewrite MapCard_as_Fold. rewrite length_as_fold_2.
apply MapFold_as_fold with
(op := plus) (neutral := 0) (f := fun (_:ad) (_:A) => 1). exact plus_assoc_reverse.
trivial.
intro. rewrite <- plus_n_O. reflexivity.
Qed.
Lemma MapCard_Put1_equals_2 :
forall (p:positive) (a a':ad) (y y':A),
MapCard A (MapPut1 A a y a' y' p) = 2.
Proof.
simple induction p. intros. simpl in |- *. case (Nbit0 a); reflexivity.
intros. simpl in |- *. case (Nbit0 a). exact (H (Ndiv2 a) (Ndiv2 a') y y').
simpl in |- *. rewrite <- plus_n_O. exact (H (Ndiv2 a) (Ndiv2 a') y y').
intros. simpl in |- *. case (Nbit0 a); reflexivity.
Qed.
Lemma MapCard_Put_sum :
forall (m m':Map A) (a:ad) (y:A) (n n':nat),
m' = MapPut A m a y ->
n = MapCard A m -> n' = MapCard A m' -> {n' = n} + {n' = S n}.
Proof.
simple induction m. simpl in |- *. intros. rewrite H in H1. simpl in H1. right.
rewrite H0. rewrite H1. reflexivity.
intros a y m' a0 y0 n n' H H0 H1. simpl in H. elim (Ndiscr (Nxor a a0)). intro H2.
elim H2. intros p H3. rewrite H3 in H. rewrite H in H1.
rewrite (MapCard_Put1_equals_2 p a a0 y y0) in H1. simpl in H0. right.
rewrite H0. rewrite H1. reflexivity.
intro H2. rewrite H2 in H. rewrite H in H1. simpl in H1. simpl in H0. left.
rewrite H0. rewrite H1. reflexivity.
intros. simpl in H2. rewrite (MapPut_semantics_3_1 A m0 m1 a y) in H1.
elim (sumbool_of_bool (Nbit0 a)). intro H4. rewrite H4 in H1.
elim
(H0 (MapPut A m1 (Ndiv2 a) y) (Ndiv2 a) y (
MapCard A m1) (MapCard A (MapPut A m1 (Ndiv2 a) y)) (
refl_equal _) (refl_equal _) (refl_equal _)).
intro H5. rewrite H1 in H3. simpl in H3. rewrite H5 in H3. rewrite <- H2 in H3. left.
assumption.
intro H5. rewrite H1 in H3. simpl in H3. rewrite H5 in H3.
rewrite <- (plus_Snm_nSm (MapCard A m0) (MapCard A m1)) in H3.
simpl in H3. rewrite <- H2 in H3. right. assumption.
intro H4. rewrite H4 in H1.
elim
(H (MapPut A m0 (Ndiv2 a) y) (Ndiv2 a) y (
MapCard A m0) (MapCard A (MapPut A m0 (Ndiv2 a) y)) (
refl_equal _) (refl_equal _) (refl_equal _)).
intro H5. rewrite H1 in H3. simpl in H3. rewrite H5 in H3. rewrite <- H2 in H3.
left. assumption.
intro H5. rewrite H1 in H3. simpl in H3. rewrite H5 in H3. simpl in H3. rewrite <- H2 in H3.
right. assumption.
Qed.
Lemma MapCard_Put_lb :
forall (m:Map A) (a:ad) (y:A), MapCard A (MapPut A m a y) >= MapCard A m.
Proof.
unfold ge in |- *. intros.
elim
(MapCard_Put_sum m (MapPut A m a y) a y (MapCard A m)
(MapCard A (MapPut A m a y)) (refl_equal _) (
refl_equal _) (refl_equal _)).
intro H. rewrite H. apply le_n.
intro H. rewrite H. apply le_n_Sn.
Qed.
Lemma MapCard_Put_ub :
forall (m:Map A) (a:ad) (y:A),
MapCard A (MapPut A m a y) <= S (MapCard A m).
Proof.
intros.
elim
(MapCard_Put_sum m (MapPut A m a y) a y (MapCard A m)
(MapCard A (MapPut A m a y)) (refl_equal _) (
refl_equal _) (refl_equal _)).
intro H. rewrite H. apply le_n_Sn.
intro H. rewrite H. apply le_n.
Qed.
Lemma MapCard_Put_1 :
forall (m:Map A) (a:ad) (y:A),
MapCard A (MapPut A m a y) = MapCard A m ->
{y : A | MapGet A m a = Some y}.
Proof.
simple induction m. intros. discriminate H.
intros a y a0 y0 H. simpl in H. elim (Ndiscr (Nxor a a0)). intro H0. elim H0.
intros p H1. rewrite H1 in H. rewrite (MapCard_Put1_equals_2 p a a0 y y0) in H.
discriminate H.
intro H0. rewrite H0 in H. rewrite (Nxor_eq _ _ H0). split with y. apply M1_semantics_1.
intros. rewrite (MapPut_semantics_3_1 A m0 m1 a y) in H1. elim (sumbool_of_bool (Nbit0 a)).
intro H2. rewrite H2 in H1. simpl in H1. elim (H0 (Ndiv2 a) y ((fun n m p:nat => plus_reg_l m p n) _ _ _ H1)).
intros y0 H3. split with y0. rewrite <- H3. exact (MapGet_M2_bit_0_1 A a H2 m0 m1).
intro H2. rewrite H2 in H1. simpl in H1.
rewrite
(plus_comm (MapCard A (MapPut A m0 (Ndiv2 a) y)) (MapCard A m1))
in H1.
rewrite (plus_comm (MapCard A m0) (MapCard A m1)) in H1.
elim (H (Ndiv2 a) y ((fun n m p:nat => plus_reg_l m p n) _ _ _ H1)). intros y0 H3. split with y0.
rewrite <- H3. exact (MapGet_M2_bit_0_0 A a H2 m0 m1).
Qed.
Lemma MapCard_Put_2 :
forall (m:Map A) (a:ad) (y:A),
MapCard A (MapPut A m a y) = S (MapCard A m) -> MapGet A m a = None.
Proof.
simple induction m. trivial.
intros. simpl in H. elim (sumbool_of_bool (Neqb a a1)). intro H0.
rewrite (Neqb_complete _ _ H0) in H. rewrite (Nxor_nilpotent a1) in H. discriminate H.
intro H0. exact (M1_semantics_2 A a a1 a0 H0).
intros. elim (sumbool_of_bool (Nbit0 a)). intro H2.
rewrite (MapGet_M2_bit_0_1 A a H2 m0 m1). apply (H0 (Ndiv2 a) y).
apply (fun n m p:nat => plus_reg_l m p n) with (n := MapCard A m0).
rewrite <- (plus_Snm_nSm (MapCard A m0) (MapCard A m1)). simpl in H1. simpl in |- *. rewrite <- H1.
clear H1.
induction a. discriminate H2.
induction p. reflexivity.
discriminate H2.
reflexivity.
intro H2. rewrite (MapGet_M2_bit_0_0 A a H2 m0 m1). apply (H (Ndiv2 a) y).
cut
(MapCard A (MapPut A m0 (Ndiv2 a) y) + MapCard A m1 =
S (MapCard A m0) + MapCard A m1).
intro. rewrite (plus_comm (MapCard A (MapPut A m0 (Ndiv2 a) y)) (MapCard A m1))
in H3.
rewrite (plus_comm (S (MapCard A m0)) (MapCard A m1)) in H3. exact ((fun n m p:nat => plus_reg_l m p n) _ _ _ H3).
simpl in |- *. simpl in H1. rewrite <- H1. induction a. trivial.
induction p. discriminate H2.
reflexivity.
discriminate H2.
Qed.
Lemma MapCard_Put_1_conv :
forall (m:Map A) (a:ad) (y y':A),
MapGet A m a = Some y -> MapCard A (MapPut A m a y') = MapCard A m.
Proof.
intros.
elim
(MapCard_Put_sum m (MapPut A m a y') a y' (MapCard A m)
(MapCard A (MapPut A m a y')) (refl_equal _) (
refl_equal _) (refl_equal _)).
trivial.
intro H0. rewrite (MapCard_Put_2 m a y' H0) in H. discriminate H.
Qed.
Lemma MapCard_Put_2_conv :
forall (m:Map A) (a:ad) (y:A),
MapGet A m a = None -> MapCard A (MapPut A m a y) = S (MapCard A m).
Proof.
intros.
elim
(MapCard_Put_sum m (MapPut A m a y) a y (MapCard A m)
(MapCard A (MapPut A m a y)) (refl_equal _) (
refl_equal _) (refl_equal _)).
intro H0. elim (MapCard_Put_1 m a y H0). intros y' H1. rewrite H1 in H. discriminate H.
trivial.
Qed.
Lemma MapCard_ext :
forall m m':Map A,
eqm A (MapGet A m) (MapGet A m') -> MapCard A m = MapCard A m'.
Proof.
unfold eqm in |- *. intros. rewrite (MapCard_as_length m). rewrite (MapCard_as_length m').
rewrite (alist_canonical A (alist_of_Map A m) (alist_of_Map A m')). reflexivity.
unfold eqm in |- *. intro. rewrite (Map_of_alist_semantics A (alist_of_Map A m) a).
rewrite (Map_of_alist_semantics A (alist_of_Map A m') a). rewrite (Map_of_alist_of_Map A m' a).
rewrite (Map_of_alist_of_Map A m a). exact (H a).
apply alist_of_Map_sorts2.
apply alist_of_Map_sorts2.
Qed.
Lemma MapCard_Dom : forall m:Map A, MapCard A m = MapCard unit (MapDom A m).
Proof.
simple induction m; trivial. intros. simpl in |- *. rewrite H. rewrite H0. reflexivity.
Qed.
Lemma MapCard_Dom_Put_behind :
forall (m:Map A) (a:ad) (y:A),
MapDom A (MapPut_behind A m a y) = MapDom A (MapPut A m a y).
Proof.
simple induction m. trivial.
intros a y a0 y0. simpl in |- *. elim (Ndiscr (Nxor a a0)). intro H. elim H.
intros p H0. rewrite H0. reflexivity.
intro H. rewrite H. rewrite (Nxor_eq _ _ H). reflexivity.
intros. simpl in |- *. elim (Ndiscr a). intro H1. elim H1. intros p H2. rewrite H2. case p.
intro p0. simpl in |- *. rewrite H0. reflexivity.
intro p0. simpl in |- *. rewrite H. reflexivity.
simpl in |- *. rewrite H0. reflexivity.
intro H1. rewrite H1. simpl in |- *. rewrite H. reflexivity.
Qed.
Lemma MapCard_Put_behind_Put :
forall (m:Map A) (a:ad) (y:A),
MapCard A (MapPut_behind A m a y) = MapCard A (MapPut A m a y).
Proof.
intros. rewrite MapCard_Dom. rewrite MapCard_Dom. rewrite MapCard_Dom_Put_behind.
reflexivity.
Qed.
Lemma MapCard_Put_behind_sum :
forall (m m':Map A) (a:ad) (y:A) (n n':nat),
m' = MapPut_behind A m a y ->
n = MapCard A m -> n' = MapCard A m' -> {n' = n} + {n' = S n}.
Proof.
intros. apply (MapCard_Put_sum m (MapPut A m a y) a y n n'); trivial.
rewrite <- MapCard_Put_behind_Put. rewrite <- H. assumption.
Qed.
Lemma MapCard_makeM2 :
forall m m':Map A, MapCard A (makeM2 A m m') = MapCard A m + MapCard A m'.
Proof.
intros. rewrite (MapCard_ext _ _ (makeM2_M2 A m m')). reflexivity.
Qed.
Lemma MapCard_Remove_sum :
forall (m m':Map A) (a:ad) (n n':nat),
m' = MapRemove A m a ->
n = MapCard A m -> n' = MapCard A m' -> {n = n'} + {n = S n'}.
Proof.
simple induction m. simpl in |- *. intros. rewrite H in H1. simpl in H1. left. rewrite H1. assumption.
simpl in |- *. intros. elim (sumbool_of_bool (Neqb a a1)). intro H2. rewrite H2 in H.
rewrite H in H1. simpl in H1. right. rewrite H1. assumption.
intro H2. rewrite H2 in H. rewrite H in H1. simpl in H1. left. rewrite H1. assumption.
intros. simpl in H1. simpl in H2. elim (sumbool_of_bool (Nbit0 a)). intro H4.
rewrite H4 in H1. rewrite H1 in H3.
rewrite (MapCard_makeM2 m0 (MapRemove A m1 (Ndiv2 a))) in H3.
elim
(H0 (MapRemove A m1 (Ndiv2 a)) (Ndiv2 a) (
MapCard A m1) (MapCard A (MapRemove A m1 (Ndiv2 a)))
(refl_equal _) (refl_equal _) (refl_equal _)).
intro H5. rewrite H5 in H2. left. rewrite H3. exact H2.
intro H5. rewrite H5 in H2.
rewrite <-
(plus_Snm_nSm (MapCard A m0) (MapCard A (MapRemove A m1 (Ndiv2 a))))
in H2.
right. rewrite H3. exact H2.
intro H4. rewrite H4 in H1. rewrite H1 in H3.
rewrite (MapCard_makeM2 (MapRemove A m0 (Ndiv2 a)) m1) in H3.
elim
(H (MapRemove A m0 (Ndiv2 a)) (Ndiv2 a) (
MapCard A m0) (MapCard A (MapRemove A m0 (Ndiv2 a)))
(refl_equal _) (refl_equal _) (refl_equal _)).
intro H5. rewrite H5 in H2. left. rewrite H3. exact H2.
intro H5. rewrite H5 in H2. right. rewrite H3. exact H2.
Qed.
Lemma MapCard_Remove_ub :
forall (m:Map A) (a:ad), MapCard A (MapRemove A m a) <= MapCard A m.
Proof.
intros.
elim
(MapCard_Remove_sum m (MapRemove A m a) a (MapCard A m)
(MapCard A (MapRemove A m a)) (refl_equal _) (
refl_equal _) (refl_equal _)).
intro H. rewrite H. apply le_n.
intro H. rewrite H. apply le_n_Sn.
Qed.
Lemma MapCard_Remove_lb :
forall (m:Map A) (a:ad), S (MapCard A (MapRemove A m a)) >= MapCard A m.
Proof.
unfold ge in |- *. intros.
elim
(MapCard_Remove_sum m (MapRemove A m a) a (MapCard A m)
(MapCard A (MapRemove A m a)) (refl_equal _) (
refl_equal _) (refl_equal _)).
intro H. rewrite H. apply le_n_Sn.
intro H. rewrite H. apply le_n.
Qed.
Lemma MapCard_Remove_1 :
forall (m:Map A) (a:ad),
MapCard A (MapRemove A m a) = MapCard A m -> MapGet A m a = None.
Proof.
simple induction m. trivial.
simpl in |- *. intros a y a0 H. elim (sumbool_of_bool (Neqb a a0)). intro H0.
rewrite H0 in H. discriminate H.
intro H0. rewrite H0. reflexivity.
intros. simpl in H1. elim (sumbool_of_bool (Nbit0 a)). intro H2. rewrite H2 in H1.
rewrite (MapCard_makeM2 m0 (MapRemove A m1 (Ndiv2 a))) in H1.
rewrite (MapGet_M2_bit_0_1 A a H2 m0 m1). apply H0. exact ((fun n m p:nat => plus_reg_l m p n) _ _ _ H1).
intro H2. rewrite H2 in H1.
rewrite (MapCard_makeM2 (MapRemove A m0 (Ndiv2 a)) m1) in H1.
rewrite (MapGet_M2_bit_0_0 A a H2 m0 m1). apply H.
rewrite
(plus_comm (MapCard A (MapRemove A m0 (Ndiv2 a))) (MapCard A m1))
in H1.
rewrite (plus_comm (MapCard A m0) (MapCard A m1)) in H1. exact ((fun n m p:nat => plus_reg_l m p n) _ _ _ H1).
Qed.
Lemma MapCard_Remove_2 :
forall (m:Map A) (a:ad),
S (MapCard A (MapRemove A m a)) = MapCard A m ->
{y : A | MapGet A m a = Some y}.
Proof.
simple induction m. intros. discriminate H.
intros a y a0 H. simpl in H. elim (sumbool_of_bool (Neqb a a0)). intro H0.
rewrite (Neqb_complete _ _ H0). split with y. exact (M1_semantics_1 A a0 y).
intro H0. rewrite H0 in H. discriminate H.
intros. simpl in H1. elim (sumbool_of_bool (Nbit0 a)). intro H2. rewrite H2 in H1.
rewrite (MapCard_makeM2 m0 (MapRemove A m1 (Ndiv2 a))) in H1.
rewrite (MapGet_M2_bit_0_1 A a H2 m0 m1). apply H0.
change
(S (MapCard A m0) + MapCard A (MapRemove A m1 (Ndiv2 a)) =
MapCard A m0 + MapCard A m1) in H1.
rewrite
(plus_Snm_nSm (MapCard A m0) (MapCard A (MapRemove A m1 (Ndiv2 a))))
in H1.
exact ((fun n m p:nat => plus_reg_l m p n) _ _ _ H1).
intro H2. rewrite H2 in H1. rewrite (MapGet_M2_bit_0_0 A a H2 m0 m1). apply H.
rewrite (MapCard_makeM2 (MapRemove A m0 (Ndiv2 a)) m1) in H1.
change
(S (MapCard A (MapRemove A m0 (Ndiv2 a))) + MapCard A m1 =
MapCard A m0 + MapCard A m1) in H1.
rewrite
(plus_comm (S (MapCard A (MapRemove A m0 (Ndiv2 a)))) (MapCard A m1))
in H1.
rewrite (plus_comm (MapCard A m0) (MapCard A m1)) in H1. exact ((fun n m p:nat => plus_reg_l m p n) _ _ _ H1).
Qed.
Lemma MapCard_Remove_1_conv :
forall (m:Map A) (a:ad),
MapGet A m a = None -> MapCard A (MapRemove A m a) = MapCard A m.
Proof.
intros.
elim
(MapCard_Remove_sum m (MapRemove A m a) a (MapCard A m)
(MapCard A (MapRemove A m a)) (refl_equal _) (
refl_equal _) (refl_equal _)).
intro H0. rewrite H0. reflexivity.
intro H0. elim (MapCard_Remove_2 m a (sym_eq H0)). intros y H1. rewrite H1 in H.
discriminate H.
Qed.
Lemma MapCard_Remove_2_conv :
forall (m:Map A) (a:ad) (y:A),
MapGet A m a = Some y -> S (MapCard A (MapRemove A m a)) = MapCard A m.
Proof.
intros.
elim
(MapCard_Remove_sum m (MapRemove A m a) a (MapCard A m)
(MapCard A (MapRemove A m a)) (refl_equal _) (
refl_equal _) (refl_equal _)).
intro H0. rewrite (MapCard_Remove_1 m a (sym_eq H0)) in H. discriminate H.
intro H0. rewrite H0. reflexivity.
Qed.
Lemma MapMerge_Restr_Card :
forall m m':Map A,
MapCard A m + MapCard A m' =
MapCard A (MapMerge A m m') + MapCard A (MapDomRestrTo A A m m').
Proof.
simple induction m. simpl in |- *. intro. apply plus_n_O.
simpl in |- *. intros a y m'. elim (option_sum A (MapGet A m' a)). intro H. elim H. intros y0 H0.
rewrite H0. rewrite MapCard_Put_behind_Put. rewrite (MapCard_Put_1_conv m' a y0 y H0).
simpl in |- *. rewrite <- plus_Snm_nSm. apply plus_n_O.
intro H. rewrite H. rewrite MapCard_Put_behind_Put. rewrite (MapCard_Put_2_conv m' a y H).
apply plus_n_O.
intros.
change
(MapCard A m0 + MapCard A m1 + MapCard A m' =
MapCard A (MapMerge A (M2 A m0 m1) m') +
MapCard A (MapDomRestrTo A A (M2 A m0 m1) m'))
in |- *.
elim m'. reflexivity.
intros a y. unfold MapMerge in |- *. unfold MapDomRestrTo in |- *.
elim (option_sum A (MapGet A (M2 A m0 m1) a)). intro H1. elim H1. intros y0 H2. rewrite H2.
rewrite (MapCard_Put_1_conv (M2 A m0 m1) a y0 y H2). reflexivity.
intro H1. rewrite H1. rewrite (MapCard_Put_2_conv (M2 A m0 m1) a y H1). simpl in |- *.
rewrite <- (plus_Snm_nSm (MapCard A m0 + MapCard A m1) 0). reflexivity.
intros. simpl in |- *.
rewrite
(plus_permute_2_in_4 (MapCard A m0) (MapCard A m1) (
MapCard A m2) (MapCard A m3)).
rewrite (H m2). rewrite (H0 m3).
rewrite
(MapCard_makeM2 (MapDomRestrTo A A m0 m2) (MapDomRestrTo A A m1 m3))
.
apply plus_permute_2_in_4.
Qed.
Lemma MapMerge_disjoint_Card :
forall m m':Map A,
MapDisjoint A A m m' ->
MapCard A (MapMerge A m m') = MapCard A m + MapCard A m'.
Proof.
intros. rewrite (MapMerge_Restr_Card m m').
rewrite (MapCard_ext _ _ (MapDisjoint_imp_2 _ _ _ _ H)). apply plus_n_O.
Qed.
Lemma MapSplit_Card :
forall (m:Map A) (m':Map B),
MapCard A m =
MapCard A (MapDomRestrTo A B m m') + MapCard A (MapDomRestrBy A B m m').
Proof.
intros. rewrite (MapCard_ext _ _ (MapDom_Split_1 A B m m')). apply MapMerge_disjoint_Card.
apply MapDisjoint_2_imp. unfold MapDisjoint_2 in |- *. apply MapDom_Split_3.
Qed.
Lemma MapMerge_Card_ub :
forall m m':Map A,
MapCard A (MapMerge A m m') <= MapCard A m + MapCard A m'.
Proof.
intros. rewrite MapMerge_Restr_Card. apply le_plus_l.
Qed.
Lemma MapDomRestrTo_Card_ub_l :
forall (m:Map A) (m':Map B),
MapCard A (MapDomRestrTo A B m m') <= MapCard A m.
Proof.
intros. rewrite (MapSplit_Card m m'). apply le_plus_l.
Qed.
Lemma MapDomRestrBy_Card_ub_l :
forall (m:Map A) (m':Map B),
MapCard A (MapDomRestrBy A B m m') <= MapCard A m.
Proof.
intros. rewrite (MapSplit_Card m m'). apply le_plus_r.
Qed.
Lemma MapMerge_Card_disjoint :
forall m m':Map A,
MapCard A (MapMerge A m m') = MapCard A m + MapCard A m' ->
MapDisjoint A A m m'.
Proof.
simple induction m. intros. apply Map_M0_disjoint.
simpl in |- *. intros. rewrite (MapCard_Put_behind_Put m' a a0) in H. unfold MapDisjoint, in_dom in |- *.
simpl in |- *. intros. elim (sumbool_of_bool (Neqb a a1)). intro H2.
rewrite (Neqb_complete _ _ H2) in H. rewrite (MapCard_Put_2 m' a1 a0 H) in H1.
discriminate H1.
intro H2. rewrite H2 in H0. discriminate H0.
simple induction m'. intros. apply Map_disjoint_M0.
intros a y H1. rewrite <- (MapCard_ext _ _ (MapPut_as_Merge A (M2 A m0 m1) a y)) in H1.
unfold MapCard at 3 in H1. rewrite <- (plus_Snm_nSm (MapCard A (M2 A m0 m1)) 0) in H1.
rewrite <- (plus_n_O (S (MapCard A (M2 A m0 m1)))) in H1. unfold MapDisjoint, in_dom in |- *.
unfold MapGet at 2 in |- *. intros. elim (sumbool_of_bool (Neqb a a0)). intro H4.
rewrite <- (Neqb_complete _ _ H4) in H2. rewrite (MapCard_Put_2 _ _ _ H1) in H2.
discriminate H2.
intro H4. rewrite H4 in H3. discriminate H3.
intros. unfold MapDisjoint in |- *. intros. elim (sumbool_of_bool (Nbit0 a)). intro H6.
unfold MapDisjoint in H0. apply H0 with (m' := m3) (a := Ndiv2 a). apply le_antisym.
apply MapMerge_Card_ub.
apply (fun p n m:nat => plus_le_reg_l n m p) with
(p := MapCard A m0 + MapCard A m2).
rewrite
(plus_permute_2_in_4 (MapCard A m0) (MapCard A m2) (
MapCard A m1) (MapCard A m3)).
change
(MapCard A (M2 A (MapMerge A m0 m2) (MapMerge A m1 m3)) =
MapCard A m0 + MapCard A m1 + (MapCard A m2 + MapCard A m3))
in H3.
rewrite <- H3. simpl in |- *. apply plus_le_compat_r. apply MapMerge_Card_ub.
elim (in_dom_some _ _ _ H4). intros y H7. rewrite (MapGet_M2_bit_0_1 _ a H6 m0 m1) in H7.
unfold in_dom in |- *. rewrite H7. reflexivity.
elim (in_dom_some _ _ _ H5). intros y H7. rewrite (MapGet_M2_bit_0_1 _ a H6 m2 m3) in H7.
unfold in_dom in |- *. rewrite H7. reflexivity.
intro H6. unfold MapDisjoint in H. apply H with (m' := m2) (a := Ndiv2 a). apply le_antisym.
apply MapMerge_Card_ub.
apply (fun p n m:nat => plus_le_reg_l n m p) with
(p := MapCard A m1 + MapCard A m3).
rewrite
(plus_comm (MapCard A m1 + MapCard A m3) (MapCard A m0 + MapCard A m2))
.
rewrite
(plus_permute_2_in_4 (MapCard A m0) (MapCard A m2) (
MapCard A m1) (MapCard A m3)).
rewrite
(plus_comm (MapCard A m1 + MapCard A m3) (MapCard A (MapMerge A m0 m2)))
.
change
(MapCard A (MapMerge A m0 m2) + MapCard A (MapMerge A m1 m3) =
MapCard A m0 + MapCard A m1 + (MapCard A m2 + MapCard A m3))
in H3.
rewrite <- H3. apply plus_le_compat_l. apply MapMerge_Card_ub.
elim (in_dom_some _ _ _ H4). intros y H7. rewrite (MapGet_M2_bit_0_0 _ a H6 m0 m1) in H7.
unfold in_dom in |- *. rewrite H7. reflexivity.
elim (in_dom_some _ _ _ H5). intros y H7. rewrite (MapGet_M2_bit_0_0 _ a H6 m2 m3) in H7.
unfold in_dom in |- *. rewrite H7. reflexivity.
Qed.
Lemma MapCard_is_Sn :
forall (m:Map A) (n:nat),
MapCard _ m = S n -> {a : ad | in_dom _ a m = true}.
Proof.
simple induction m. intros. discriminate H.
intros a y n H. split with a. unfold in_dom in |- *. rewrite (M1_semantics_1 _ a y). reflexivity.
intros. simpl in H1. elim (O_or_S (MapCard _ m0)). intro H2. elim H2. intros m2 H3.
elim (H _ (sym_eq H3)). intros a H4. split with (Ndouble a). unfold in_dom in |- *.
rewrite (MapGet_M2_bit_0_0 A (Ndouble a) (Ndouble_bit0 a) m0 m1).
rewrite (Ndouble_div2 a). elim (in_dom_some _ _ _ H4). intros y H5. rewrite H5. reflexivity.
intro H2. rewrite <- H2 in H1. simpl in H1. elim (H0 _ H1). intros a H3.
split with (Ndouble_plus_one a). unfold in_dom in |- *.
rewrite
(MapGet_M2_bit_0_1 A (Ndouble_plus_one a) (Ndouble_plus_one_bit0 a)
m0 m1).
rewrite (Ndouble_plus_one_div2 a). elim (in_dom_some _ _ _ H3). intros y H4. rewrite H4.
reflexivity.
Qed.
End MapCard.
Section MapCard2.
Variables A B : Set.
Lemma MapSubset_card_eq_1 :
forall (n:nat) (m:Map A) (m':Map B),
MapSubset _ _ m m' ->
MapCard _ m = n -> MapCard _ m' = n -> MapSubset _ _ m' m.
Proof.
simple induction n. intros. unfold MapSubset, in_dom in |- *. intro. rewrite (MapCard_is_O _ m H0 a).
rewrite (MapCard_is_O _ m' H1 a). intro H2. discriminate H2.
intros. elim (MapCard_is_Sn A m n0 H1). intros a H3. elim (in_dom_some _ _ _ H3).
intros y H4. elim (in_dom_some _ _ _ (H0 _ H3)). intros y' H6.
cut (eqmap _ (MapPut _ (MapRemove _ m a) a y) m). intro.
cut (eqmap _ (MapPut _ (MapRemove _ m' a) a y') m'). intro.
apply MapSubset_ext with
(m0 := MapPut _ (MapRemove _ m' a) a y')
(m2 := MapPut _ (MapRemove _ m a) a y).
assumption.
assumption.
apply MapSubset_Put_mono. apply H. apply MapSubset_Remove_mono. assumption.
rewrite <- (MapCard_Remove_2_conv _ m a y H4) in H1. inversion_clear H1. reflexivity.
rewrite <- (MapCard_Remove_2_conv _ m' a y' H6) in H2. inversion_clear H2. reflexivity.
unfold eqmap, eqm in |- *. intro. rewrite (MapPut_semantics _ (MapRemove B m' a) a y' a0).
elim (sumbool_of_bool (Neqb a a0)). intro H7. rewrite H7. rewrite <- (Neqb_complete _ _ H7).
apply sym_eq. assumption.
intro H7. rewrite H7. rewrite (MapRemove_semantics _ m' a a0). rewrite H7. reflexivity.
unfold eqmap, eqm in |- *. intro. rewrite (MapPut_semantics _ (MapRemove A m a) a y a0).
elim (sumbool_of_bool (Neqb a a0)). intro H7. rewrite H7. rewrite <- (Neqb_complete _ _ H7).
apply sym_eq. assumption.
intro H7. rewrite H7. rewrite (MapRemove_semantics A m a a0). rewrite H7. reflexivity.
Qed.
Lemma MapDomRestrTo_Card_ub_r :
forall (m:Map A) (m':Map B),
MapCard A (MapDomRestrTo A B m m') <= MapCard B m'.
Proof.
simple induction m. intro. simpl in |- *. apply le_O_n.
intros a y m'. simpl in |- *. elim (option_sum B (MapGet B m' a)). intro H. elim H. intros y0 H0.
rewrite H0. elim (MapCard_is_not_O B m' a y0 H0). intros n H1. rewrite H1. simpl in |- *.
apply le_n_S. apply le_O_n.
intro H. rewrite H. simpl in |- *. apply le_O_n.
simple induction m'. simpl in |- *. apply le_O_n.
intros a y. unfold MapDomRestrTo in |- *. case (MapGet A (M2 A m0 m1) a). simpl in |- *.
intro. simpl in |- *. apply le_n.
apply le_O_n.
intros. simpl in |- *. rewrite
(MapCard_makeM2 A (MapDomRestrTo A B m0 m2) (MapDomRestrTo A B m1 m3))
.
apply plus_le_compat. apply H.
apply H0.
Qed.
End MapCard2.
Section MapCard3.
Variables A B : Set.
Lemma MapMerge_Card_lb_l :
forall m m':Map A, MapCard A (MapMerge A m m') >= MapCard A m.
Proof.
unfold ge in |- *. intros. apply ((fun p n m:nat => plus_le_reg_l n m p) (MapCard A m')).
rewrite (plus_comm (MapCard A m') (MapCard A m)).
rewrite (plus_comm (MapCard A m') (MapCard A (MapMerge A m m'))).
rewrite (MapMerge_Restr_Card A m m'). apply plus_le_compat_l. apply MapDomRestrTo_Card_ub_r.
Qed.
Lemma MapMerge_Card_lb_r :
forall m m':Map A, MapCard A (MapMerge A m m') >= MapCard A m'.
Proof.
unfold ge in |- *. intros. apply ((fun p n m:nat => plus_le_reg_l n m p) (MapCard A m)). rewrite (MapMerge_Restr_Card A m m').
rewrite
(plus_comm (MapCard A (MapMerge A m m'))
(MapCard A (MapDomRestrTo A A m m'))).
apply plus_le_compat_r. apply MapDomRestrTo_Card_ub_l.
Qed.
Lemma MapDomRestrBy_Card_lb :
forall (m:Map A) (m':Map B),
MapCard B m' + MapCard A (MapDomRestrBy A B m m') >= MapCard A m.
Proof.
unfold ge in |- *. intros. rewrite (MapSplit_Card A B m m'). apply plus_le_compat_r.
apply MapDomRestrTo_Card_ub_r.
Qed.
Lemma MapSubset_Card_le :
forall (m:Map A) (m':Map B),
MapSubset A B m m' -> MapCard A m <= MapCard B m'.
Proof.
intros. apply le_trans with (m := MapCard B m' + MapCard A (MapDomRestrBy A B m m')).
exact (MapDomRestrBy_Card_lb m m').
rewrite (MapCard_ext _ _ _ (MapSubset_imp_2 _ _ _ _ H)). simpl in |- *. rewrite <- plus_n_O.
apply le_n.
Qed.
Lemma MapSubset_card_eq :
forall (m:Map A) (m':Map B),
MapSubset _ _ m m' ->
MapCard _ m' <= MapCard _ m -> eqmap _ (MapDom _ m) (MapDom _ m').
Proof.
intros. apply MapSubset_antisym. assumption.
cut (MapCard B m' = MapCard A m). intro. apply (MapSubset_card_eq_1 A B (MapCard A m)).
assumption.
reflexivity.
assumption.
apply le_antisym. assumption.
apply MapSubset_Card_le. assumption.
Qed.
End MapCard3.
|
