-
Notifications
You must be signed in to change notification settings - Fork 1
/
Permissions.v
1267 lines (1110 loc) · 35.3 KB
/
Permissions.v
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
Require Import HahnBase.
Require Import List.
Require Import ListSet.
Require Import Permutation.
Require Import PermutationTactic.
Require Import Prelude.
Require Import QArith.
Require Import Qcanon.
Require Import Utf8.
Import ListNotations.
Open Scope Qc_scope.
(** * Permissions *)
(** ** Prerequisites *)
(** Below are some basic properties and results on rational numbers. *)
Lemma Q2Qc_correct :
forall q : Q, Q2Qc q == q.
Proof.
intros. apply Qred_correct.
Qed.
Lemma Qcplus_lt_compat :
forall x y z t : Qc,
x < y -> z < t -> x + z < y + t.
Proof.
ins. unfold Qcplus, Qclt.
repeat rewrite Q2Qc_correct.
apply Qplus_lt_le_compat. easy.
now apply Qlt_le_weak.
Qed.
Lemma Qcplus_le_mono_l :
forall x y z : Qc, x <= y <-> z + x <= z + y.
Proof.
split; intros.
- apply Qcplus_le_compat. apply Qcle_refl. exact H.
- replace x with ((0 - z) + (z + x)) by ring.
replace y with ((0 - z) + (z + y)) by ring.
apply Qcplus_le_compat; intuition. apply Qcle_refl.
Qed.
Lemma Qcplus_le_mono_r :
forall x y z : Qc, x <= y <-> x + z <= y + z.
Proof.
ins. intuition.
rewrite Qcplus_comm with x z.
rewrite Qcplus_comm with y z.
by apply Qcplus_le_mono_l.
apply Qcplus_le_mono_l with z.
rewrite Qcplus_comm with z x.
by rewrite Qcplus_comm with z y.
Qed.
Lemma Qclt_nge :
forall x y : Qc, x < y <-> ~ y <= x.
Proof.
split; auto using Qclt_not_le, Qcnot_le_lt.
Qed.
Lemma Qclt_irrefl :
forall q : Qc, ~ q < q.
Proof.
ins. apply Qcle_not_lt, Qcle_refl.
Qed.
Lemma Qclt_asymm :
forall q1 q2 : Qc,
q1 < q2 -> ~ q2 < q1.
Proof.
intros q1 q2 H1 H2.
assert (q1 < q1). { by apply Qclt_trans with q2. }
by apply Qclt_irrefl with q1.
Qed.
Lemma Qcplus_lt_mono_l :
forall x y z : Qc, x < y <-> z + x < z + y.
Proof.
ins. rewrite !Qclt_nge.
by rewrite <- Qcplus_le_mono_l.
Qed.
Lemma Qcplus_lt_mono_r :
forall x y z : Qc, x < y <-> x + z < y + z.
Proof.
ins. rewrite !Qclt_nge.
by rewrite <- Qcplus_le_mono_r.
Qed.
Lemma Qcplus_lt_compat_le_l :
forall x y z t : Qc,
x <= y -> z < t -> x + z < y + t.
Proof.
intros x y z t H1 H2.
apply Qcle_lt_or_eq in H1.
destruct H1 as [H1 | H1].
- by apply Qcplus_lt_compat.
- clarify. by apply Qcplus_lt_mono_l.
Qed.
Lemma Qcplus_lt_compat_le_r :
forall x y z t : Qc,
x < y -> z <= t -> x + z < y + t.
Proof.
intros x y z t H1 H2.
apply Qcle_lt_or_eq in H2.
destruct H2 as [H2 | H2].
- by apply Qcplus_lt_compat.
- clarify. by apply Qcplus_lt_mono_r.
Qed.
Lemma Qcplus_pos_nonneg :
forall p q : Qc, 0 < p -> 0 <= q -> 0 < p + q.
Proof.
intros p q Hp Hq.
apply Qclt_le_trans with (p + 0).
by rewrite Qcplus_0_r.
by apply Qcplus_le_mono_l.
Qed.
Lemma Qcplus_swap_l :
forall q1 q2 q3 : Qc, q1 + (q2 + q3) = q2 + (q1 + q3).
Proof.
intros q1 q2 q3.
rewrite Qcplus_comm, <- Qcplus_assoc.
by rewrite Qcplus_comm with q1 q3.
Qed.
Lemma Qcplus_swap_r :
forall q1 q2 q3 : Qc, q1 + (q2 + q3) = q3 + (q2 + q1).
Proof.
ins. rewrite Qcplus_comm.
rewrite <- Qcplus_assoc.
apply Qcplus_swap_l.
Qed.
Lemma Qcplus_pos_le_elim :
forall q1 q2 q : Qc, q1 + q2 <= q -> 0 <= q2 -> q1 <= q.
Proof.
intros q1 q2 q H1 H2.
replace q1 with (q1 + 0).
apply Qcle_trans with (q1 + q2); auto.
by rewrite <- Qcplus_le_mono_l.
apply Qcplus_0_r.
Qed.
Lemma Qcplus_pos_lt_elim :
forall q1 q2 q : Qc, q1 + q2 < q -> 0 <= q2 -> q1 < q.
Proof.
intros q1 q2 q H1 H2.
replace q1 with (q1 + 0).
apply Qcle_lt_trans with (q1 + q2); auto.
by rewrite <- Qcplus_le_mono_l.
apply Qcplus_0_r.
Qed.
Lemma Qcplus_le_weaken :
forall q1 q2 q : Qc, 0 < q -> q1 <= q2 -> q1 <= q2 + q.
Proof.
intros q1 q2 q H1 H2.
rewrite Qcplus_comm.
replace q1 with (0 + q1).
apply Qcplus_le_compat; auto.
by apply Qclt_le_weak in H1.
apply Qcplus_0_l.
Qed.
Lemma Qcplus_lt_weaken :
forall q1 q2 q : Qc,
0 < q -> q1 <= q2 -> q1 < q2 + q.
Proof.
intros q1 q2 q H1 H2.
apply Qcle_lt_or_eq in H2.
destruct H2 as [H2 | H2].
(* [q1] is less than [q2] *)
- rewrite Qcplus_comm.
replace q1 with (0 + q1).
apply Qcplus_lt_compat; auto.
apply Qcplus_0_l.
(* [q1] is equal to [q2] *)
- clarify.
apply Qclt_minus_iff.
rewrite <- Qcplus_assoc.
rewrite Qcplus_comm.
rewrite <- Qcplus_assoc.
rewrite Qcplus_comm with (- q2) q2.
by rewrite Qcplus_opp_r, Qcplus_0_r.
Qed.
Lemma Qcle_diff :
forall q1 q2 : Qc, q1 <= q2 -> exists q, q2 = q1 + q.
Proof.
intros q1 q2 H.
apply Qcle_minus_iff in H.
exists (q2 + (- q1)).
rewrite Qcplus_comm.
rewrite <- Qcplus_assoc.
rewrite Qcplus_comm with (-q1) q1.
rewrite Qcplus_opp_r.
by rewrite Qcplus_0_r.
Qed.
Lemma Qclt_diff :
forall q1 q2 : Qc, q1 < q2 -> exists q, q2 = q + q1.
Proof.
intros q1 q2 H.
apply Qclt_minus_iff in H.
exists (q2 + (- q1)).
rewrite <- Qcplus_assoc.
rewrite Qcplus_comm with (-q1) q1.
rewrite Qcplus_opp_r.
by rewrite Qcplus_0_r.
Qed.
Lemma Qclt_mono_pos :
forall q1 q2 : Qc, 0 < q1 -> q2 < q2 + q1.
Proof.
intros q1 q2 H.
replace (q2 < q2 + q1) with (q2 + 0 < q2 + q1).
by apply Qcplus_lt_mono_l.
by rewrite Qcplus_0_r.
Qed.
Lemma Qcle_plus_elim :
forall q1 q2 q3 : Qc, 0 <= q1 -> 0 <= q2 -> q1 + q2 <= q3 -> q1 <= q3.
Proof.
intros q1 q2 q3 H1 H2 H3.
apply Qcle_trans with (q1 + q2); auto.
replace (q1 <= q1 + q2) with (q1 + 0 <= q1 + q2).
by apply Qcplus_le_mono_l.
by rewrite Qcplus_0_r.
Qed.
Lemma Qcle_weaken :
forall q1 q2 : Qc, q1 = q2 -> q1 <= q2.
Proof.
intros ?? H. rewrite H.
apply Qcle_refl.
Qed.
Lemma Qcplus_canc_l :
forall q1 q2 q : Qc, q + q1 = q + q2 -> q1 = q2.
Proof.
intros q1 q2 q H.
apply Qcle_antisym.
apply Qcplus_le_mono_l with q.
by apply Qcle_weaken.
apply Qcplus_le_mono_l with q.
by apply Qcle_weaken.
Qed.
Lemma Qcplus_canc_r :
forall q1 q2 q : Qc, q1 + q = q2 + q -> q1 = q2.
Proof.
intros q1 q2 q H.
apply Qcplus_canc_l with q.
rewrite Qcplus_comm with q q1.
by rewrite Qcplus_comm with q q2.
Qed.
Lemma Qcplus_neg_dist :
forall q1 q2 : Qc, -(q1 + q2) = (-q1) + (-q2).
Proof.
intros q1 q2.
apply Qcplus_canc_r with (q1 + q2).
rewrite Qcplus_comm.
rewrite Qcplus_opp_r.
rewrite Qcplus_comm with q1 q2.
rewrite <- Qcplus_assoc.
rewrite Qcplus_comm.
repeat rewrite <- Qcplus_assoc.
rewrite Qcplus_opp_r.
rewrite Qcplus_0_r.
rewrite <- Qcplus_comm.
rewrite Qcplus_opp_r.
reflexivity.
Qed.
Lemma Qcplus_pos_le :
forall q1 q2 : Qc, 0 < q2 -> q1 <= q1 + q2.
Proof.
intros q1 q2 H.
rewrite <- Qcplus_0_r with q1.
rewrite <- Qcplus_assoc.
rewrite <- Qcplus_le_mono_l.
rewrite Qcplus_0_l.
by apply Qclt_le_weak.
Qed.
(** ** Fractional permissions *)
(** Fractional permissions are rational numbers in the range (0..1]. *)
Definition perm_full := 1%Qc.
(** *** Validity *)
(** Any rational number [q] is a _valid fractional permission_,
written [perm_valid q], if [q] is within the range (0..1]. *)
Definition perm_valid (q : Qc) : Prop := 0 < q /\ q <= 1.
Lemma perm_valid_mono :
forall q1 q2, perm_valid q1 -> q2 < q1 + q2.
Proof.
intros q1 q2 H.
unfold perm_valid in H. intuition.
replace q2 with (0 + q2) at 1.
by rewrite <- Qcplus_lt_mono_r.
by rewrite Qcplus_0_l.
Qed.
Lemma perm_valid_pos :
forall q : Qc, perm_valid q -> 0 < q.
Proof.
unfold perm_valid. ins. desf.
Qed.
Hint Resolve perm_valid_pos : core.
Lemma perm_valid_full_neg :
forall q, perm_valid q -> ~ perm_full < q.
Proof.
intros q H1 H2.
unfold perm_valid in H1.
destruct H1 as (H1 & H3).
unfold perm_full in H2.
apply Qcle_lt_or_eq in H3.
destruct H3 as [H3 | H3]; vauto.
by apply Qclt_asymm in H2.
Qed.
Lemma perm_valid_full :
forall q, perm_valid q -> perm_full <= q -> q = perm_full.
Proof.
intros q H1 H2.
apply Qcle_lt_or_eq in H2.
destruct H2 as [H2 | H2]; auto.
by apply perm_valid_full_neg in H2.
Qed.
(** *** Disjointness *)
(** Two permissions [q1] and [q2] are _disjoint_,
written [perm_disj q1 q2], if they are both positive
and their sum does not exceed 1. *)
Definition perm_disj (q1 q2 : Qc) : Prop :=
0 < q1 /\ 0 < q2 /\ q1 + q2 <= 1.
(** Permission disjointness is a symmetric relation. *)
Instance perm_disj_symm :
Symmetric perm_disj.
Proof.
unfold perm_disj. red. intuition.
by rewrite Qcplus_comm.
Qed.
(** Permission disjointness implies validity of its operands. *)
Lemma perm_disj_valid_l :
forall q1 q2, perm_disj q1 q2 -> perm_valid q1.
Proof.
unfold perm_disj, perm_valid. intuition.
apply Qcplus_pos_le_elim in H2. auto.
by apply Qclt_le_weak in H.
Qed.
Lemma perm_disj_valid_r :
forall q1 q2, perm_disj q1 q2 -> perm_valid q2.
Proof.
intros ?? H. symmetry in H.
by apply perm_disj_valid_l in H.
Qed.
Lemma perm_disj_valid :
forall q1 q2, perm_disj q1 q2 -> perm_valid q1 /\ perm_valid q2.
Proof.
intros ?? H. split.
by apply perm_disj_valid_l in H.
by apply perm_disj_valid_r in H.
Qed.
(** Below are several other useful properties of disjointness. *)
Lemma perm_disj_add_l :
forall q1 q2 q3,
perm_disj q1 q2 -> perm_disj (q1 + q2) q3 -> perm_disj q2 q3.
Proof.
unfold perm_disj.
intros q1 q2 q3. intuition.
apply Qcplus_pos_le_elim with (q2 := q1).
by rewrite <- Qcplus_comm, Qcplus_assoc.
by apply Qclt_le_weak.
Qed.
Lemma perm_disj_add_r :
forall q1 q2 q3,
perm_disj q2 q3 -> perm_disj q1 (q2 + q3) -> perm_disj q1 q2.
Proof.
unfold perm_disj.
intros q1 q2 q3. intuition.
apply Qcplus_pos_le_elim with (q2 := q3).
by rewrite <- Qcplus_assoc.
now apply Qclt_le_weak.
Qed.
Lemma perm_disj_assoc_l :
forall q1 q2 q3,
perm_disj q1 q2 -> perm_disj (q1 + q2) q3 -> perm_disj q1 (q2 + q3).
Proof.
unfold perm_disj. intuition.
apply Qcplus_pos_nonneg; auto.
by apply Qclt_le_weak.
by rewrite Qcplus_assoc.
Qed.
Lemma perm_disj_assoc_r :
forall q1 q2 q3,
perm_disj q2 q3 -> perm_disj q1 (q2 + q3) -> perm_disj (q1 + q2) q3.
Proof.
unfold perm_disj. intuition.
apply Qcplus_pos_nonneg; auto.
by apply Qclt_le_weak.
by rewrite <- Qcplus_assoc.
Qed.
Lemma perm_add_valid :
forall q1 q2, perm_disj q1 q2 -> perm_valid (q1 + q2).
Proof.
unfold perm_disj, perm_valid.
intros ?? H. intuition.
apply Qcplus_pos_nonneg. auto.
by apply Qclt_le_weak in H.
Qed.
Lemma perm_lt_weaken :
forall q q1 q2, perm_disj q1 q2 -> q < q1 -> q < q1 + q2.
Proof.
intros q q1 q2 H1 H2.
unfold perm_disj in H1. desf.
apply Qcplus_lt_weaken; auto.
by apply Qclt_le_weak.
Qed.
Lemma perm_le_weaken :
forall q q1 q2, perm_disj q1 q2 -> q <= q1 -> q <= q1 + q2.
Proof.
intros q q1 q2 H1 H2.
unfold perm_disj in H1. desf.
by apply Qcplus_le_weaken.
Qed.
Lemma perm_disj_full_neg_l :
forall q, ~ perm_disj q perm_full.
Proof.
intros q H.
unfold perm_disj in H.
intuition desf.
assert (perm_full < q + perm_full).
unfold perm_full in *.
rewrite Qcplus_comm.
by apply Qclt_mono_pos.
unfold perm_full in *.
by apply Qclt_not_le in H1.
Qed.
Lemma perm_disj_full_neg_r :
forall q, ~ perm_disj perm_full q.
Proof.
intros q H. symmetry in H.
by apply perm_disj_full_neg_l in H.
Qed.
Lemma perm_lt_diff :
forall q1 q2,
perm_valid q1 ->
perm_valid q2 ->
q1 < q2 ->
exists q3, perm_disj q1 q3 /\ q1 + q3 = q2.
Proof.
intros q1 q2 H1 H2 H3.
apply Qclt_minus_iff in H3.
exists (q2 + (- q1)). split; auto.
(* left part of conjunction *)
- red. intuition auto.
rewrite Qcplus_comm.
rewrite <- Qcplus_assoc.
rewrite Qcplus_comm with (-q1) q1.
rewrite Qcplus_opp_r.
rewrite Qcplus_0_r.
unfold perm_valid in H2. desf.
(* right part of conjunction *)
- rewrite Qcplus_comm.
rewrite <- Qcplus_assoc.
rewrite Qcplus_comm with (-q1) q1.
rewrite Qcplus_opp_r.
by rewrite Qcplus_0_r.
Qed.
Lemma perm_disj_lt :
forall q1 q2 q3,
perm_valid q1 ->
perm_disj q2 q3 ->
q1 < q2 ->
perm_disj q1 q3.
Proof.
intros q1 q2 q3 H1 H2 H3.
unfold perm_valid in H1.
destruct H1 as (H1 & H4).
unfold perm_disj in *.
destruct H2 as (H2 & H5 & H6).
intuition.
apply Qcle_trans with (q2 + q3); auto.
apply Qcplus_le_mono_r.
by apply Qclt_le_weak.
Qed.
Lemma perm_disj_le :
forall q1 q2 q3,
perm_valid q1 ->
perm_disj q2 q3 ->
q1 <= q2 ->
perm_disj q1 q3.
Proof.
intros q1 q2 q3 H1 H2 H3.
apply Qcle_lt_or_eq in H3.
destruct H3 as [H3 | H3]; vauto.
by apply perm_disj_lt with q2.
Qed.
(** ** Permission sequences *)
(** This section defines operations on _sequences_ of fractional permissions. *)
(** *** Validity *)
(** A sequence of fractional permissions is _valid_ if all permissions are valid individually. *)
Fixpoint perm_valid_list (xs : list Qc) : Prop :=
match xs with
| nil => True
| q :: xs' => perm_valid q /\ perm_valid_list xs'
end.
Notation "√ qs" := (perm_valid_list qs) (only printing, at level 80).
Lemma perm_valid_list_single :
forall q, perm_valid_list [q] <-> perm_valid q.
Proof.
intuition simpls. desf.
Qed.
Hint Resolve perm_valid_list_single : core.
Lemma perm_valid_list_permutation :
forall xs1 xs2,
Permutation xs1 xs2 -> perm_valid_list xs1 -> perm_valid_list xs2.
Proof.
intros ?? H ?. induction H; simpls; desf; intuition.
Qed.
Add Parametric Morphism : perm_valid_list
with signature @Permutation Qc ==> iff
as perm_valid_list_permut_mor.
Proof.
intros xs ys ?. intuition.
apply perm_valid_list_permutation with xs; auto.
apply perm_valid_list_permutation with ys; auto.
Qed.
(** *** Positivity *)
(** A sequence of fractional permissions is _positive_ if
all rationals in the list are positive. *)
Fixpoint perm_pos_list (xs : list Qc) : Prop :=
match xs with
| nil => True
| q :: xs' => 0 < q /\ perm_pos_list xs'
end.
Lemma perm_pos_list_In :
forall qs q, perm_pos_list qs -> In q qs -> 0 < q.
Proof.
induction qs; ins.
desf. by apply IHqs.
Qed.
Lemma perm_pos_list_permutation :
forall xs ys,
Permutation xs ys -> perm_pos_list xs -> perm_pos_list ys.
Proof.
intros xs ys H1 H2.
induction H1; simpls; intuition.
Qed.
Add Parametric Morphism : perm_pos_list
with signature @Permutation Qc ==> iff
as perm_pos_list_permut_mor.
Proof.
intros xs ys ?. intuition.
apply perm_pos_list_permutation with xs; auto.
apply perm_pos_list_permutation with ys; auto.
Qed.
Ltac perm_pos_list_solve :=
match goal with
| [ _ : (perm_pos_list ?X) |- (perm_pos_list _) ] =>
apply perm_pos_list_permutation with X; auto; list_permutation;
fail "perm_pos_list_solve can not solve this system."
| [ |- _ ] => fail "perm_pos_list_solve can not be applied."
end.
Lemma perm_pos_list_add_left :
forall q1 q2 qs,
perm_pos_list (q1 :: q2 :: qs) -> perm_pos_list (q1 + q2 :: qs).
Proof.
simpls. intuition.
apply Qcplus_pos_nonneg; auto.
by apply Qclt_le_weak.
Qed.
Lemma perm_pos_list_add :
forall qs1 q1 q2 qs2,
perm_pos_list (qs1 ++ q1 :: q2 :: qs2) ->
perm_pos_list (qs1 ++ q1 + q2 :: qs2).
Proof.
induction qs1; simpls; intuition.
apply Qcplus_pos_nonneg; auto.
by apply Qclt_le_weak.
Qed.
Lemma perm_pos_list_tail :
forall q qs,
perm_pos_list (q :: qs) -> perm_pos_list qs.
Proof.
ins. desf.
Qed.
Lemma perm_pos_list_sub_l :
forall xs ys,
perm_pos_list (xs ++ ys) -> perm_pos_list xs.
Proof.
induction xs; intros ys H; simpls.
intuition. by apply IHxs with ys.
Qed.
Lemma perm_pos_list_sub_r :
forall xs ys,
perm_pos_list (xs ++ ys) -> perm_pos_list ys.
Proof.
intros xs ys H.
rewrite Permutation_app_comm in H.
by apply perm_pos_list_sub_l in H.
Qed.
Lemma perm_pos_list_remove :
forall q xs ys,
perm_pos_list (xs ++ q :: ys) -> perm_pos_list (xs ++ ys).
Proof.
intros ??? H.
rewrite <- Permutation_middle in H.
by apply perm_pos_list_tail in H.
Qed.
Lemma perm_pos_list_remove_list :
forall xs ys zs,
perm_pos_list (xs ++ ys ++ zs) -> perm_pos_list (xs ++ zs).
Proof.
intros xs ys zs H.
rewrite perm_takeit_1 in H.
by apply perm_pos_list_sub_r in H.
Qed.
Lemma perm_valid_list_pos :
forall qs, perm_valid_list qs -> perm_pos_list qs.
Proof.
induction qs; intro H; simpls.
unfold perm_valid in *. intuition.
Qed.
Hint Resolve perm_valid_list_pos : core.
Lemma perm_pos_list_cons :
forall q qs,
0 < q -> perm_pos_list qs -> perm_pos_list (q :: qs).
Proof.
ins.
Qed.
Lemma perm_pos_list_app :
forall xs ys,
perm_pos_list xs -> perm_pos_list ys -> perm_pos_list (xs ++ ys).
Proof.
induction xs; intros ys H1 H2; simpls.
intuition.
Qed.
Lemma perm_pos_list_assoc_l :
forall q1 q2 q3 qs,
perm_pos_list [q2; q3] ->
perm_pos_list (q1 :: q2 + q3 :: qs) ->
perm_pos_list (q1 + q2 :: q3 :: qs).
Proof.
ins. intuition.
apply Qcplus_pos_nonneg; auto.
by apply Qclt_le_weak.
Qed.
Lemma perm_pos_list_assoc_r :
forall q1 q2 q3 qs,
perm_pos_list [q1; q2] ->
perm_pos_list (q1 + q2 :: q3 :: qs) ->
perm_pos_list (q1 :: q2 + q3 :: qs).
Proof.
ins. intuition.
apply Qcplus_pos_nonneg; auto.
by apply Qclt_le_weak.
Qed.
Lemma perm_pos_list_set_remove :
forall xs x,
perm_pos_list xs -> perm_pos_list (set_remove Qc_eq_dec x xs).
Proof.
induction xs; ins; simpls; desf; intuition vauto.
apply perm_pos_list_cons. auto.
by apply IHxs.
Qed.
Lemma perm_pos_list_sublist :
forall xs ys,
sublist Qc_eq_dec xs ys -> perm_pos_list ys -> perm_pos_list xs.
Proof.
induction xs; ins; simpls; desf; intuition vauto.
by apply perm_pos_list_In in H.
apply IHxs in H1; vauto.
by apply perm_pos_list_set_remove.
Qed.
(** *** Addition *)
(** The _addition_ of a sequence of fractional permissions is defined as classical summation. *)
Fixpoint perm_add_list (xs : list Qc) : Qc :=
match xs with
| nil => 0
| q :: xs' => q + perm_add_list xs'
end.
Notation "∑ xs" := (perm_add_list xs) (only printing, at level 80).
Lemma perm_add_list_permutation :
forall xs ys,
Permutation xs ys -> perm_add_list xs = perm_add_list ys.
Proof.
intros xs ys H. induction H; simpls.
by rewrite IHPermutation.
by rewrite Qcplus_swap_l.
by rewrite IHPermutation1, IHPermutation2.
Qed.
Add Parametric Morphism : perm_add_list
with signature @Permutation Qc ==> eq
as perm_add_list_permut_mor.
Proof.
ins. by apply perm_add_list_permutation.
Qed.
Lemma perm_add_list_single :
forall q, perm_add_list [q] = q.
Proof.
ins. apply Qcplus_0_r.
Qed.
Hint Rewrite perm_add_list_single : core.
Lemma perm_add_list_nonneg :
forall qs, perm_pos_list qs -> 0 <= perm_add_list qs.
Proof.
intros qs H. induction qs; simpls; desf.
replace 0 with (0 + 0); auto.
apply Qcplus_le_compat.
by apply Qclt_le_weak.
by apply IHqs.
Qed.
Lemma perm_add_list_add_left :
forall q1 q2 qs,
perm_add_list (q1 :: q2 :: qs) = perm_add_list (q1 + q2 :: qs).
Proof.
intros q1 q2 qs. simpls.
apply Qcplus_assoc.
Qed.
Lemma perm_add_list_add :
forall qs1 q1 q2 qs2,
perm_add_list (qs1 ++ q1 :: q2 :: qs2) = perm_add_list (qs1 ++ q1 + q2 :: qs2).
Proof.
induction qs1; intuition simpls.
apply Qcplus_assoc.
by rewrite IHqs1.
Qed.
Lemma perm_add_list_left :
forall x xs,
perm_add_list (x :: xs) = x + perm_add_list xs.
Proof.
simpls.
Qed.
Lemma perm_add_list_set_remove :
forall qs q,
set_In q qs ->
perm_add_list (set_remove Qc_eq_dec q qs) = perm_add_list qs + - q.
Proof.
induction qs; ins; simpls; desf; intuition vauto.
rewrite Qcplus_comm, Qcplus_assoc.
rewrite Qcplus_comm with (- q) q.
by rewrite Qcplus_opp_r, Qcplus_0_l.
rewrite Qcplus_comm, Qcplus_assoc.
rewrite Qcplus_comm with (- a) a.
by rewrite Qcplus_opp_r, Qcplus_0_l.
rewrite perm_add_list_left.
rewrite IHqs; auto.
by rewrite <- Qcplus_assoc.
Qed.
Lemma perm_add_list_sublist :
forall xs ys,
sublist Qc_eq_dec xs ys ->
perm_pos_list ys ->
perm_add_list xs <= perm_add_list ys.
Proof.
induction xs; ins; simpls; desf; intuition vauto.
by apply perm_add_list_nonneg in H0.
apply IHxs in H1.
rewrite perm_add_list_set_remove in H1; auto.
rewrite Qcplus_le_mono_r with (z := - a).
rewrite Qcplus_comm, Qcplus_assoc.
rewrite Qcplus_comm with (- a) a.
by rewrite Qcplus_opp_r, Qcplus_0_l.
by apply perm_pos_list_set_remove.
Qed.
(** *** Disjointness *)
(** A sequence of fractional permissions is _disjoint_ if all the rationals are positive and
the sum of all rationals in the list does not exceed one. *)
Definition perm_disj_list (qs : list Qc) : Prop :=
perm_pos_list qs /\ perm_add_list qs <= 1.
Notation "⫡ xs" := (perm_disj_list xs) (only printing, at level 80).
Lemma perm_disj_list_permutation :
forall xs ys,
Permutation xs ys -> perm_disj_list xs -> perm_disj_list ys.
Proof.
unfold perm_disj_list.
intuition by rewrite <- H.
Qed.
Add Parametric Morphism : perm_disj_list
with signature @Permutation Qc ==> iff
as perm_disj_list_permut_mor.
Proof.
intros xs ys ?. intuition.
apply perm_disj_list_permutation with xs; auto.
apply perm_disj_list_permutation with ys; auto.
Qed.
Lemma perm_disj_list_single :
forall q, perm_disj_list [q] <-> perm_valid q.
Proof.
unfold perm_disj_list, perm_valid.
intro q. intuition simpls; desf.
by rewrite <- Qcplus_0_r with q.
by rewrite Qcplus_0_r.
Qed.
Hint Resolve perm_disj_list_single : core.
Lemma perm_disj_list_binary :
forall q1 q2,
perm_disj_list [q1; q2] <-> perm_disj q1 q2.
Proof.
unfold perm_disj_list, perm_disj.
intros q1 q2. intuition simpls; desf.
by rewrite <- Qcplus_0_r with q2.
by rewrite Qcplus_0_r.
Qed.
Lemma perm_disj_list_tail :
forall q qs,
perm_disj_list (q :: qs) -> perm_disj_list qs.
Proof.
unfold perm_disj_list.
intros q qs H. simpls. intuition.
apply Qcplus_pos_le_elim with q.
by rewrite Qcplus_comm.
by apply Qclt_le_weak.
Qed.
Lemma perm_disj_list_sub_r :
forall xs ys,
perm_disj_list (xs ++ ys) -> perm_disj_list ys.
Proof.
induction xs; intros ys H; simpls.
apply IHxs. by apply perm_disj_list_tail in H.
Qed.
Lemma perm_disj_list_sub_l :
forall xs ys,
perm_disj_list (xs ++ ys) -> perm_disj_list xs.
Proof.
intros xs ys H.
rewrite Permutation_app_comm in H.
by apply perm_disj_list_sub_r in H.
Qed.
Lemma perm_disj_list_remove :
forall q xs ys,
perm_disj_list (xs ++ q :: ys) -> perm_disj_list (xs ++ ys).
Proof.
intros q xs ys H.
rewrite perm_takeit_5 in H.
by apply perm_disj_list_tail in H.
Qed.
Lemma perm_disj_list_remove_list :
forall xs ys zs,
perm_disj_list (xs ++ ys ++ zs) -> perm_disj_list (xs ++ zs).
Proof.
intros xs ys zs H.
rewrite perm_takeit_1 in H.
by apply perm_disj_list_sub_r in H.
Qed.
Lemma perm_disj_list_add_left :
forall q1 q2 qs,
perm_disj_list (q1 :: q2 :: qs) -> perm_disj_list (q1 + q2 :: qs).
Proof.
unfold perm_disj_list. intuition.
by apply perm_pos_list_add_left.
by rewrite <- perm_add_list_add_left.
Qed.
Lemma perm_disj_list_add :
forall q1 q2 qs1 qs2,
perm_disj_list (qs1 ++ q1 :: q2 :: qs2) ->
perm_disj_list (qs1 ++ q1 + q2 :: qs2).
Proof.
unfold perm_disj_list. intuition.
by apply perm_pos_list_add.
by rewrite <- perm_add_list_add.
Qed.
Lemma perm_disj_list_valid_head :
forall q qs,
perm_disj_list (q :: qs) -> perm_valid q.
Proof.
intros q qs H.
apply perm_disj_list_sub_l with [q] qs in H.
by apply perm_disj_list_single.
Qed.
Lemma perm_disj_list_valid :
forall qs, perm_disj_list qs -> perm_valid_list qs.
Proof.
induction qs; intro H; simpls. split.
by apply perm_disj_list_valid_head in H.
apply IHqs. by apply perm_disj_list_tail in H.
Qed.
Lemma perm_disj_list_pos :
forall qs,
perm_disj_list qs -> perm_pos_list qs.
Proof.
induction qs; ins; simpls; desf; intuition vauto.
apply perm_disj_list_valid_head in H. auto.
apply IHqs.
by apply perm_disj_list_tail in H.
Qed.
Lemma perm_disj_list_elim_left :
forall q1 q2 qs,
perm_pos_list [q1; q2] ->
perm_disj_list (q1 + q2 :: qs) ->