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
|
(************************************************************************)
(* 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 $Id$ i*)
Require Import Bool.
Require Import Sumbool.
Require Import Arith.
Require Import ZArith.
Require Import Addr.
Require Import Adist.
Require Import Addec.
Require Import Map.
Require Import Fset.
Require Import Mapaxioms.
Require Import Mapiter.
Section MapSubsetDef.
Variables A B : Set.
Definition MapSubset (m:Map A) (m':Map B) :=
forall a:ad, in_dom A a m = true -> in_dom B a m' = true.
Definition MapSubset_1 (m:Map A) (m':Map B) :=
match MapSweep A (fun (a:ad) (_:A) => negb (in_dom B a m')) m with
| NONE => true
| _ => false
end.
Definition MapSubset_2 (m:Map A) (m':Map B) :=
eqmap A (MapDomRestrBy A B m m') (M0 A).
Lemma MapSubset_imp_1 :
forall (m:Map A) (m':Map B), MapSubset m m' -> MapSubset_1 m m' = true.
Proof.
unfold MapSubset, MapSubset_1 in |- *. intros.
elim
(option_sum _ (MapSweep A (fun (a:ad) (_:A) => negb (in_dom B a m')) m)).
intro H0. elim H0. intro r. elim r. intros a y H1. cut (negb (in_dom B a m') = true).
intro. cut (in_dom A a m = false). intro. unfold in_dom in H3.
rewrite (MapSweep_semantics_2 _ _ m a y H1) in H3. discriminate H3.
elim (sumbool_of_bool (in_dom A a m)). intro H3. rewrite (H a H3) in H2. discriminate H2.
trivial.
exact (MapSweep_semantics_1 _ _ m a y H1).
intro H0. rewrite H0. reflexivity.
Qed.
Lemma MapSubset_1_imp :
forall (m:Map A) (m':Map B), MapSubset_1 m m' = true -> MapSubset m m'.
Proof.
unfold MapSubset, MapSubset_1 in |- *. unfold in_dom at 2 in |- *. intros. elim (option_sum _ (MapGet A m a)).
intro H1. elim H1. intros y H2.
elim
(option_sum _ (MapSweep A (fun (a:ad) (_:A) => negb (in_dom B a m')) m)). intro H3.
elim H3. intro r. elim r. intros a' y' H4. rewrite H4 in H. discriminate H.
intro H3. cut (negb (in_dom B a m') = false). intro. rewrite (negb_intro (in_dom B a m')).
rewrite H4. reflexivity.
exact (MapSweep_semantics_3 _ _ m H3 a y H2).
intro H1. rewrite H1 in H0. discriminate H0.
Qed.
Lemma map_dom_empty_1 :
forall m:Map A, eqmap A m (M0 A) -> forall a:ad, in_dom _ a m = false.
Proof.
unfold eqmap, eqm, in_dom in |- *. intros. rewrite (H a). reflexivity.
Qed.
Lemma map_dom_empty_2 :
forall m:Map A, (forall a:ad, in_dom _ a m = false) -> eqmap A m (M0 A).
Proof.
unfold eqmap, eqm, in_dom in |- *. intros.
cut
(match MapGet A m a with
| NONE => false
| SOME _ => true
end = false).
case (MapGet A m a). trivial.
intros. discriminate H0.
exact (H a).
Qed.
Lemma MapSubset_imp_2 :
forall (m:Map A) (m':Map B), MapSubset m m' -> MapSubset_2 m m'.
Proof.
unfold MapSubset, MapSubset_2 in |- *. intros. apply map_dom_empty_2. intro. rewrite in_dom_restrby.
elim (sumbool_of_bool (in_dom A a m)). intro H0. rewrite H0. rewrite (H a H0). reflexivity.
intro H0. rewrite H0. reflexivity.
Qed.
Lemma MapSubset_2_imp :
forall (m:Map A) (m':Map B), MapSubset_2 m m' -> MapSubset m m'.
Proof.
unfold MapSubset, MapSubset_2 in |- *. intros. cut (in_dom _ a (MapDomRestrBy A B m m') = false).
rewrite in_dom_restrby. intro. elim (andb_false_elim _ _ H1). rewrite H0.
intro H2. discriminate H2.
intro H2. rewrite (negb_intro (in_dom B a m')). rewrite H2. reflexivity.
exact (map_dom_empty_1 _ H a).
Qed.
End MapSubsetDef.
Section MapSubsetOrder.
Variables A B C : Set.
Lemma MapSubset_refl : forall m:Map A, MapSubset A A m m.
Proof.
unfold MapSubset in |- *. trivial.
Qed.
Lemma MapSubset_antisym :
forall (m:Map A) (m':Map B),
MapSubset A B m m' ->
MapSubset B A m' m -> eqmap unit (MapDom A m) (MapDom B m').
Proof.
unfold MapSubset, eqmap, eqm in |- *. intros. elim (option_sum _ (MapGet _ (MapDom A m) a)).
intro H1. elim H1. intro t. elim t. intro H2. elim (option_sum _ (MapGet _ (MapDom B m') a)).
intro H3. elim H3. intro t'. elim t'. intro H4. rewrite H4. exact H2.
intro H3. cut (in_dom B a m' = true). intro. rewrite (MapDom_Dom B m' a) in H4.
unfold in_FSet, in_dom in H4. rewrite H3 in H4. discriminate H4.
apply H. rewrite (MapDom_Dom A m a). unfold in_FSet, in_dom in |- *. rewrite H2. reflexivity.
intro H1. elim (option_sum _ (MapGet _ (MapDom B m') a)). intro H2. elim H2. intros t H3.
cut (in_dom A a m = true). intro. rewrite (MapDom_Dom A m a) in H4. unfold in_FSet, in_dom in H4.
rewrite H1 in H4. discriminate H4.
apply H0. rewrite (MapDom_Dom B m' a). unfold in_FSet, in_dom in |- *. rewrite H3. reflexivity.
intro H2. rewrite H2. exact H1.
Qed.
Lemma MapSubset_trans :
forall (m:Map A) (m':Map B) (m'':Map C),
MapSubset A B m m' -> MapSubset B C m' m'' -> MapSubset A C m m''.
Proof.
unfold MapSubset in |- *. intros. apply H0. apply H. assumption.
Qed.
End MapSubsetOrder.
Section FSubsetOrder.
Lemma FSubset_refl : forall s:FSet, MapSubset _ _ s s.
Proof.
exact (MapSubset_refl unit).
Qed.
Lemma FSubset_antisym :
forall s s':FSet,
MapSubset _ _ s s' -> MapSubset _ _ s' s -> eqmap unit s s'.
Proof.
intros. rewrite <- (FSet_Dom s). rewrite <- (FSet_Dom s').
exact (MapSubset_antisym _ _ s s' H H0).
Qed.
Lemma FSubset_trans :
forall s s' s'':FSet,
MapSubset _ _ s s' -> MapSubset _ _ s' s'' -> MapSubset _ _ s s''.
Proof.
exact (MapSubset_trans unit unit unit).
Qed.
End FSubsetOrder.
Section MapSubsetExtra.
Variables A B : Set.
Lemma MapSubset_Dom_1 :
forall (m:Map A) (m':Map B),
MapSubset A B m m' -> MapSubset unit unit (MapDom A m) (MapDom B m').
Proof.
unfold MapSubset in |- *. intros. elim (MapDom_semantics_2 _ m a H0). intros y H1.
cut (in_dom A a m = true -> in_dom B a m' = true). intro. unfold in_dom in H2.
rewrite H1 in H2. elim (option_sum _ (MapGet B m' a)). intro H3. elim H3.
intros y' H4. exact (MapDom_semantics_1 _ m' a y' H4).
intro H3. rewrite H3 in H2. cut (false = true). intro. discriminate H4.
apply H2. reflexivity.
exact (H a).
Qed.
Lemma MapSubset_Dom_2 :
forall (m:Map A) (m':Map B),
MapSubset unit unit (MapDom A m) (MapDom B m') -> MapSubset A B m m'.
Proof.
unfold MapSubset in |- *. intros. unfold in_dom in H0. elim (option_sum _ (MapGet A m a)).
intro H1. elim H1. intros y H2.
elim (MapDom_semantics_2 _ _ _ (H a (MapDom_semantics_1 _ _ _ _ H2))). intros y' H3.
unfold in_dom in |- *. rewrite H3. reflexivity.
intro H1. rewrite H1 in H0. discriminate H0.
Qed.
Lemma MapSubset_1_Dom :
forall (m:Map A) (m':Map B),
MapSubset_1 A B m m' = MapSubset_1 unit unit (MapDom A m) (MapDom B m').
Proof.
intros. elim (sumbool_of_bool (MapSubset_1 A B m m')). intro H. rewrite H.
apply sym_eq. apply MapSubset_imp_1. apply MapSubset_Dom_1. exact (MapSubset_1_imp _ _ _ _ H).
intro H. rewrite H. elim (sumbool_of_bool (MapSubset_1 unit unit (MapDom A m) (MapDom B m'))).
intro H0.
rewrite
(MapSubset_imp_1 _ _ _ _
(MapSubset_Dom_2 _ _ (MapSubset_1_imp _ _ _ _ H0)))
in H.
discriminate H.
intro. apply sym_eq. assumption.
Qed.
Lemma MapSubset_Put :
forall (m:Map A) (a:ad) (y:A), MapSubset A A m (MapPut A m a y).
Proof.
unfold MapSubset in |- *. intros. rewrite in_dom_put. rewrite H. apply orb_b_true.
Qed.
Lemma MapSubset_Put_mono :
forall (m:Map A) (m':Map B) (a:ad) (y:A) (y':B),
MapSubset A B m m' -> MapSubset A B (MapPut A m a y) (MapPut B m' a y').
Proof.
unfold MapSubset in |- *. intros. rewrite in_dom_put. rewrite (in_dom_put A m a y a0) in H0.
elim (orb_true_elim _ _ H0). intro H1. rewrite H1. reflexivity.
intro H1. rewrite (H _ H1). apply orb_b_true.
Qed.
Lemma MapSubset_Put_behind :
forall (m:Map A) (a:ad) (y:A), MapSubset A A m (MapPut_behind A m a y).
Proof.
unfold MapSubset in |- *. intros. rewrite in_dom_put_behind. rewrite H. apply orb_b_true.
Qed.
Lemma MapSubset_Put_behind_mono :
forall (m:Map A) (m':Map B) (a:ad) (y:A) (y':B),
MapSubset A B m m' ->
MapSubset A B (MapPut_behind A m a y) (MapPut_behind B m' a y').
Proof.
unfold MapSubset in |- *. intros. rewrite in_dom_put_behind.
rewrite (in_dom_put_behind A m a y a0) in H0.
elim (orb_true_elim _ _ H0). intro H1. rewrite H1. reflexivity.
intro H1. rewrite (H _ H1). apply orb_b_true.
Qed.
Lemma MapSubset_Remove :
forall (m:Map A) (a:ad), MapSubset A A (MapRemove A m a) m.
Proof.
unfold MapSubset in |- *. intros. unfold MapSubset in |- *. intros. rewrite (in_dom_remove _ m a a0) in H.
elim (andb_prop _ _ H). trivial.
Qed.
Lemma MapSubset_Remove_mono :
forall (m:Map A) (m':Map B) (a:ad),
MapSubset A B m m' -> MapSubset A B (MapRemove A m a) (MapRemove B m' a).
Proof.
unfold MapSubset in |- *. intros. rewrite in_dom_remove. rewrite (in_dom_remove A m a a0) in H0.
elim (andb_prop _ _ H0). intros. rewrite H1. rewrite (H _ H2). reflexivity.
Qed.
Lemma MapSubset_Merge_l :
forall m m':Map A, MapSubset A A m (MapMerge A m m').
Proof.
unfold MapSubset in |- *. intros. rewrite in_dom_merge. rewrite H. reflexivity.
Qed.
Lemma MapSubset_Merge_r :
forall m m':Map A, MapSubset A A m' (MapMerge A m m').
Proof.
unfold MapSubset in |- *. intros. rewrite in_dom_merge. rewrite H. apply orb_b_true.
Qed.
Lemma MapSubset_Merge_mono :
forall (m m':Map A) (m'' m''':Map B),
MapSubset A B m m'' ->
MapSubset A B m' m''' ->
MapSubset A B (MapMerge A m m') (MapMerge B m'' m''').
Proof.
unfold MapSubset in |- *. intros. rewrite in_dom_merge. rewrite (in_dom_merge A m m' a) in H1.
elim (orb_true_elim _ _ H1). intro H2. rewrite (H _ H2). reflexivity.
intro H2. rewrite (H0 _ H2). apply orb_b_true.
Qed.
Lemma MapSubset_DomRestrTo_l :
forall (m:Map A) (m':Map B), MapSubset A A (MapDomRestrTo A B m m') m.
Proof.
unfold MapSubset in |- *. intros. rewrite (in_dom_restrto _ _ m m' a) in H. elim (andb_prop _ _ H).
trivial.
Qed.
Lemma MapSubset_DomRestrTo_r :
forall (m:Map A) (m':Map B), MapSubset A B (MapDomRestrTo A B m m') m'.
Proof.
unfold MapSubset in |- *. intros. rewrite (in_dom_restrto _ _ m m' a) in H. elim (andb_prop _ _ H).
trivial.
Qed.
Lemma MapSubset_ext :
forall (m0 m1:Map A) (m2 m3:Map B),
eqmap A m0 m1 ->
eqmap B m2 m3 -> MapSubset A B m0 m2 -> MapSubset A B m1 m3.
Proof.
intros. apply MapSubset_2_imp. unfold MapSubset_2 in |- *.
apply eqmap_trans with (m' := MapDomRestrBy A B m0 m2). apply MapDomRestrBy_ext. apply eqmap_sym.
assumption.
apply eqmap_sym. assumption.
exact (MapSubset_imp_2 _ _ _ _ H1).
Qed.
Variables C D : Set.
Lemma MapSubset_DomRestrTo_mono :
forall (m:Map A) (m':Map B) (m'':Map C) (m''':Map D),
MapSubset _ _ m m'' ->
MapSubset _ _ m' m''' ->
MapSubset _ _ (MapDomRestrTo _ _ m m') (MapDomRestrTo _ _ m'' m''').
Proof.
unfold MapSubset in |- *. intros. rewrite in_dom_restrto. rewrite (in_dom_restrto A B m m' a) in H1.
elim (andb_prop _ _ H1). intros. rewrite (H _ H2). rewrite (H0 _ H3). reflexivity.
Qed.
Lemma MapSubset_DomRestrBy_l :
forall (m:Map A) (m':Map B), MapSubset A A (MapDomRestrBy A B m m') m.
Proof.
unfold MapSubset in |- *. intros. rewrite (in_dom_restrby _ _ m m' a) in H. elim (andb_prop _ _ H).
trivial.
Qed.
Lemma MapSubset_DomRestrBy_mono :
forall (m:Map A) (m':Map B) (m'':Map C) (m''':Map D),
MapSubset _ _ m m'' ->
MapSubset _ _ m''' m' ->
MapSubset _ _ (MapDomRestrBy _ _ m m') (MapDomRestrBy _ _ m'' m''').
Proof.
unfold MapSubset in |- *. intros. rewrite in_dom_restrby. rewrite (in_dom_restrby A B m m' a) in H1.
elim (andb_prop _ _ H1). intros. rewrite (H _ H2). elim (sumbool_of_bool (in_dom D a m''')).
intro H4. rewrite (H0 _ H4) in H3. discriminate H3.
intro H4. rewrite H4. reflexivity.
Qed.
End MapSubsetExtra.
Section MapDisjointDef.
Variables A B : Set.
Definition MapDisjoint (m:Map A) (m':Map B) :=
forall a:ad, in_dom A a m = true -> in_dom B a m' = true -> False.
Definition MapDisjoint_1 (m:Map A) (m':Map B) :=
match MapSweep A (fun (a:ad) (_:A) => in_dom B a m') m with
| NONE => true
| _ => false
end.
Definition MapDisjoint_2 (m:Map A) (m':Map B) :=
eqmap A (MapDomRestrTo A B m m') (M0 A).
Lemma MapDisjoint_imp_1 :
forall (m:Map A) (m':Map B), MapDisjoint m m' -> MapDisjoint_1 m m' = true.
Proof.
unfold MapDisjoint, MapDisjoint_1 in |- *. intros.
elim (option_sum _ (MapSweep A (fun (a:ad) (_:A) => in_dom B a m') m)). intro H0. elim H0.
intro r. elim r. intros a y H1. cut (in_dom A a m = true -> in_dom B a m' = true -> False).
intro. unfold in_dom at 1 in H2. rewrite (MapSweep_semantics_2 _ _ _ _ _ H1) in H2.
rewrite (MapSweep_semantics_1 _ _ _ _ _ H1) in H2. elim (H2 (refl_equal _) (refl_equal _)).
exact (H a).
intro H0. rewrite H0. reflexivity.
Qed.
Lemma MapDisjoint_1_imp :
forall (m:Map A) (m':Map B), MapDisjoint_1 m m' = true -> MapDisjoint m m'.
Proof.
unfold MapDisjoint, MapDisjoint_1 in |- *. intros.
elim (option_sum _ (MapSweep A (fun (a:ad) (_:A) => in_dom B a m') m)). intro H2. elim H2.
intro r. elim r. intros a' y' H3. rewrite H3 in H. discriminate H.
intro H2. unfold in_dom in H0. elim (option_sum _ (MapGet A m a)). intro H3. elim H3.
intros y H4. rewrite (MapSweep_semantics_3 _ _ _ H2 a y H4) in H1. discriminate H1.
intro H3. rewrite H3 in H0. discriminate H0.
Qed.
Lemma MapDisjoint_imp_2 :
forall (m:Map A) (m':Map B), MapDisjoint m m' -> MapDisjoint_2 m m'.
Proof.
unfold MapDisjoint, MapDisjoint_2 in |- *. unfold eqmap, eqm in |- *. intros.
rewrite (MapDomRestrTo_semantics A B m m' a).
cut (in_dom A a m = true -> in_dom B a m' = true -> False). intro.
elim (option_sum _ (MapGet A m a)). intro H1. elim H1. intros y H2. unfold in_dom at 1 in H0.
elim (option_sum _ (MapGet B m' a)). intro H3. elim H3. intros y' H4. unfold in_dom at 1 in H0.
rewrite H4 in H0. rewrite H2 in H0. elim (H0 (refl_equal _) (refl_equal _)).
intro H3. rewrite H3. reflexivity.
intro H1. rewrite H1. case (MapGet B m' a); reflexivity.
exact (H a).
Qed.
Lemma MapDisjoint_2_imp :
forall (m:Map A) (m':Map B), MapDisjoint_2 m m' -> MapDisjoint m m'.
Proof.
unfold MapDisjoint, MapDisjoint_2 in |- *. unfold eqmap, eqm in |- *. intros. elim (in_dom_some _ _ _ H0).
intros y H2. elim (in_dom_some _ _ _ H1). intros y' H3.
cut (MapGet A (MapDomRestrTo A B m m') a = NONE A). intro.
rewrite (MapDomRestrTo_semantics _ _ m m' a) in H4. rewrite H3 in H4. rewrite H2 in H4.
discriminate H4.
exact (H a).
Qed.
Lemma Map_M0_disjoint : forall m:Map B, MapDisjoint (M0 A) m.
Proof.
unfold MapDisjoint, in_dom in |- *. intros. discriminate H.
Qed.
Lemma Map_disjoint_M0 : forall m:Map A, MapDisjoint m (M0 B).
Proof.
unfold MapDisjoint, in_dom in |- *. intros. discriminate H0.
Qed.
End MapDisjointDef.
Section MapDisjointExtra.
Variables A B : Set.
Lemma MapDisjoint_ext :
forall (m0 m1:Map A) (m2 m3:Map B),
eqmap A m0 m1 ->
eqmap B m2 m3 -> MapDisjoint A B m0 m2 -> MapDisjoint A B m1 m3.
Proof.
intros. apply MapDisjoint_2_imp. unfold MapDisjoint_2 in |- *.
apply eqmap_trans with (m' := MapDomRestrTo A B m0 m2). apply eqmap_sym. apply MapDomRestrTo_ext.
assumption.
assumption.
exact (MapDisjoint_imp_2 _ _ _ _ H1).
Qed.
Lemma MapMerge_disjoint :
forall m m':Map A,
MapDisjoint A A m m' ->
forall a:ad,
in_dom A a (MapMerge A m m') =
orb (andb (in_dom A a m) (negb (in_dom A a m')))
(andb (in_dom A a m') (negb (in_dom A a m))).
Proof.
unfold MapDisjoint in |- *. intros. rewrite in_dom_merge. elim (sumbool_of_bool (in_dom A a m)).
intro H0. rewrite H0. elim (sumbool_of_bool (in_dom A a m')). intro H1. elim (H a H0 H1).
intro H1. rewrite H1. reflexivity.
intro H0. rewrite H0. simpl in |- *. rewrite andb_b_true. reflexivity.
Qed.
Lemma MapDisjoint_M2_l :
forall (m0 m1:Map A) (m2 m3:Map B),
MapDisjoint A B (M2 A m0 m1) (M2 B m2 m3) -> MapDisjoint A B m0 m2.
Proof.
unfold MapDisjoint, in_dom in |- *. intros. elim (option_sum _ (MapGet A m0 a)). intro H2.
elim H2. intros y H3. elim (option_sum _ (MapGet B m2 a)). intro H4. elim H4.
intros y' H5. apply (H (ad_double a)).
rewrite (MapGet_M2_bit_0_0 _ (ad_double a) (ad_double_bit_0 a) m0 m1).
rewrite (ad_double_div_2 a). rewrite H3. reflexivity.
rewrite (MapGet_M2_bit_0_0 _ (ad_double a) (ad_double_bit_0 a) m2 m3).
rewrite (ad_double_div_2 a). rewrite H5. reflexivity.
intro H4. rewrite H4 in H1. discriminate H1.
intro H2. rewrite H2 in H0. discriminate H0.
Qed.
Lemma MapDisjoint_M2_r :
forall (m0 m1:Map A) (m2 m3:Map B),
MapDisjoint A B (M2 A m0 m1) (M2 B m2 m3) -> MapDisjoint A B m1 m3.
Proof.
unfold MapDisjoint, in_dom in |- *. intros. elim (option_sum _ (MapGet A m1 a)). intro H2.
elim H2. intros y H3. elim (option_sum _ (MapGet B m3 a)). intro H4. elim H4.
intros y' H5. apply (H (ad_double_plus_un a)).
rewrite
(MapGet_M2_bit_0_1 _ (ad_double_plus_un a) (ad_double_plus_un_bit_0 a)
m0 m1).
rewrite (ad_double_plus_un_div_2 a). rewrite H3. reflexivity.
rewrite
(MapGet_M2_bit_0_1 _ (ad_double_plus_un a) (ad_double_plus_un_bit_0 a)
m2 m3).
rewrite (ad_double_plus_un_div_2 a). rewrite H5. reflexivity.
intro H4. rewrite H4 in H1. discriminate H1.
intro H2. rewrite H2 in H0. discriminate H0.
Qed.
Lemma MapDisjoint_M2 :
forall (m0 m1:Map A) (m2 m3:Map B),
MapDisjoint A B m0 m2 ->
MapDisjoint A B m1 m3 -> MapDisjoint A B (M2 A m0 m1) (M2 B m2 m3).
Proof.
unfold MapDisjoint, in_dom in |- *. intros. elim (sumbool_of_bool (ad_bit_0 a)). intro H3.
rewrite (MapGet_M2_bit_0_1 A a H3 m0 m1) in H1.
rewrite (MapGet_M2_bit_0_1 B a H3 m2 m3) in H2. exact (H0 (ad_div_2 a) H1 H2).
intro H3. rewrite (MapGet_M2_bit_0_0 A a H3 m0 m1) in H1.
rewrite (MapGet_M2_bit_0_0 B a H3 m2 m3) in H2. exact (H (ad_div_2 a) H1 H2).
Qed.
Lemma MapDisjoint_M1_l :
forall (m:Map A) (a:ad) (y:B),
MapDisjoint B A (M1 B a y) m -> in_dom A a m = false.
Proof.
unfold MapDisjoint in |- *. intros. elim (sumbool_of_bool (in_dom A a m)). intro H0.
elim (H a (in_dom_M1_1 B a y) H0).
trivial.
Qed.
Lemma MapDisjoint_M1_r :
forall (m:Map A) (a:ad) (y:B),
MapDisjoint A B m (M1 B a y) -> in_dom A a m = false.
Proof.
unfold MapDisjoint in |- *. intros. elim (sumbool_of_bool (in_dom A a m)). intro H0.
elim (H a H0 (in_dom_M1_1 B a y)).
trivial.
Qed.
Lemma MapDisjoint_M1_conv_l :
forall (m:Map A) (a:ad) (y:B),
in_dom A a m = false -> MapDisjoint B A (M1 B a y) m.
Proof.
unfold MapDisjoint in |- *. intros. rewrite (in_dom_M1_2 B a a0 y H0) in H. rewrite H1 in H.
discriminate H.
Qed.
Lemma MapDisjoint_M1_conv_r :
forall (m:Map A) (a:ad) (y:B),
in_dom A a m = false -> MapDisjoint A B m (M1 B a y).
Proof.
unfold MapDisjoint in |- *. intros. rewrite (in_dom_M1_2 B a a0 y H1) in H. rewrite H0 in H.
discriminate H.
Qed.
Lemma MapDisjoint_sym :
forall (m:Map A) (m':Map B), MapDisjoint A B m m' -> MapDisjoint B A m' m.
Proof.
unfold MapDisjoint in |- *. intros. exact (H _ H1 H0).
Qed.
Lemma MapDisjoint_empty :
forall m:Map A, MapDisjoint A A m m -> eqmap A m (M0 A).
Proof.
unfold eqmap, eqm in |- *. intros. rewrite <- (MapDomRestrTo_idempotent A m a).
exact (MapDisjoint_imp_2 A A m m H a).
Qed.
Lemma MapDelta_disjoint :
forall m m':Map A,
MapDisjoint A A m m' -> eqmap A (MapDelta A m m') (MapMerge A m m').
Proof.
intros.
apply eqmap_trans with
(m' := MapDomRestrBy A A (MapMerge A m m') (MapDomRestrTo A A m m')).
apply MapDelta_as_DomRestrBy.
apply eqmap_trans with (m' := MapDomRestrBy A A (MapMerge A m m') (M0 A)).
apply MapDomRestrBy_ext. apply eqmap_refl.
exact (MapDisjoint_imp_2 A A m m' H).
apply MapDomRestrBy_m_empty.
Qed.
Variable C : Set.
Lemma MapDomRestr_disjoint :
forall (m:Map A) (m':Map B) (m'':Map C),
MapDisjoint A B (MapDomRestrTo A C m m'') (MapDomRestrBy B C m' m'').
Proof.
unfold MapDisjoint in |- *. intros m m' m'' a. rewrite in_dom_restrto. rewrite in_dom_restrby.
intros. elim (andb_prop _ _ H). elim (andb_prop _ _ H0). intros. rewrite H4 in H2.
discriminate H2.
Qed.
Lemma MapDelta_RestrTo_disjoint :
forall m m':Map A,
MapDisjoint A A (MapDelta A m m') (MapDomRestrTo A A m m').
Proof.
unfold MapDisjoint in |- *. intros m m' a. rewrite in_dom_delta. rewrite in_dom_restrto.
intros. elim (andb_prop _ _ H0). intros. rewrite H1 in H. rewrite H2 in H. discriminate H.
Qed.
Lemma MapDelta_RestrTo_disjoint_2 :
forall m m':Map A,
MapDisjoint A A (MapDelta A m m') (MapDomRestrTo A A m' m).
Proof.
unfold MapDisjoint in |- *. intros m m' a. rewrite in_dom_delta. rewrite in_dom_restrto.
intros. elim (andb_prop _ _ H0). intros. rewrite H1 in H. rewrite H2 in H. discriminate H.
Qed.
Variable D : Set.
Lemma MapSubset_Disjoint :
forall (m:Map A) (m':Map B) (m'':Map C) (m''':Map D),
MapSubset _ _ m m' ->
MapSubset _ _ m'' m''' ->
MapDisjoint _ _ m' m''' -> MapDisjoint _ _ m m''.
Proof.
unfold MapSubset, MapDisjoint in |- *. intros. exact (H1 _ (H _ H2) (H0 _ H3)).
Qed.
Lemma MapSubset_Disjoint_l :
forall (m:Map A) (m':Map B) (m'':Map C),
MapSubset _ _ m m' -> MapDisjoint _ _ m' m'' -> MapDisjoint _ _ m m''.
Proof.
unfold MapSubset, MapDisjoint in |- *. intros. exact (H0 _ (H _ H1) H2).
Qed.
Lemma MapSubset_Disjoint_r :
forall (m:Map A) (m'':Map C) (m''':Map D),
MapSubset _ _ m'' m''' ->
MapDisjoint _ _ m m''' -> MapDisjoint _ _ m m''.
Proof.
unfold MapSubset, MapDisjoint in |- *. intros. exact (H0 _ H1 (H _ H2)).
Qed.
End MapDisjointExtra.
|
