derived.v 76.7 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
From iris.bi Require Export interface.
From iris.algebra Require Import monoid.
From stdpp Require Import hlist.

Definition bi_iff {PROP : bi} (P Q : PROP) : PROP := ((P  Q)  (Q  P))%I.
Arguments bi_iff {_} _%I _%I : simpl never.
Instance: Params (@bi_iff) 1.
Infix "↔" := bi_iff : bi_scope.

Definition bi_wand_iff {PROP : bi} (P Q : PROP) : PROP :=
  ((P - Q)  (Q - P))%I.
Arguments bi_wand_iff {_} _%I _%I : simpl never.
Instance: Params (@bi_wand_iff) 1.
Infix "∗-∗" := bi_wand_iff (at level 95, no associativity) : bi_scope.

16
Class Persistent {PROP : bi} (P : PROP) := persistent : P   P.
Robbert Krebbers's avatar
Robbert Krebbers committed
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
Arguments Persistent {_} _%I : simpl never.
Arguments persistent {_} _%I {_}.
Hint Mode Persistent + ! : typeclass_instances.
Instance: Params (@Persistent) 1.

Definition bi_bare {PROP : bi} (P : PROP) : PROP := (emp  P)%I.
Arguments bi_bare {_} _%I : simpl never.
Instance: Params (@bi_bare) 1.
Typeclasses Opaque bi_bare.
Notation "■ P" := (bi_bare P) (at level 20, right associativity) : bi_scope.
Notation "⬕ P" := (  P)%I (at level 20, right associativity) : bi_scope.

Class Affine {PROP : bi} (Q : PROP) := affine : Q  emp.
Arguments Affine {_} _%I : simpl never.
Arguments affine {_} _%I {_}.
Hint Mode Affine + ! : typeclass_instances.

Class AffineBI (PROP : bi) := absorbing_bi (Q : PROP) : Affine Q.
Existing Instance absorbing_bi | 0.

37
38
39
Class PositiveBI (PROP : bi) :=
  positive_bi (P Q : PROP) :  (P  Q)   P  Q.

Robbert Krebbers's avatar
Robbert Krebbers committed
40
41
42
43
44
45
46
Definition bi_sink {PROP : bi} (P : PROP) : PROP := (True  P)%I.
Arguments bi_sink {_} _%I : simpl never.
Instance: Params (@bi_sink) 1.
Typeclasses Opaque bi_sink.
Notation "▲ P" := (bi_sink P) (at level 20, right associativity) : bi_scope.

Class Absorbing {PROP : bi} (P : PROP) := absorbing :  P  P.
Robbert Krebbers's avatar
Robbert Krebbers committed
47
Arguments Absorbing {_} _%I : simpl never.
Robbert Krebbers's avatar
Robbert Krebbers committed
48
Arguments absorbing {_} _%I.
Robbert Krebbers's avatar
Robbert Krebbers committed
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

Definition bi_persistently_if {PROP : bi} (p : bool) (P : PROP) : PROP :=
  (if p then  P else P)%I.
Arguments bi_persistently_if {_} !_ _%I /.
Instance: Params (@bi_persistently_if) 2.
Typeclasses Opaque bi_persistently_if.
Notation "□? p P" := (bi_persistently_if p P)
  (at level 20, p at level 9, P at level 20,
   right associativity, format "□? p  P") : bi_scope.

Definition bi_bare_if {PROP : bi} (p : bool) (P : PROP) : PROP :=
  (if p then  P else P)%I.
Arguments bi_bare_if {_} !_ _%I /.
Instance: Params (@bi_bare_if) 2.
Typeclasses Opaque bi_bare_if.
Notation "■? p P" := (bi_bare_if p P)
  (at level 20, p at level 9, P at level 20,
   right associativity, format "■? p  P") : bi_scope.
Notation "⬕? p P" := (?p ?p P)%I
  (at level 20, p at level 9, P at level 20,
   right associativity, format "⬕? p  P") : bi_scope.

Fixpoint bi_hexist {PROP : bi} {As} : himpl As PROP  PROP :=
  match As return himpl As PROP  PROP with
  | tnil => id
  | tcons A As => λ Φ,  x, bi_hexist (Φ x)
  end%I.
Fixpoint bi_hforall {PROP : bi} {As} : himpl As PROP  PROP :=
  match As return himpl As PROP  PROP with
  | tnil => id
  | tcons A As => λ Φ,  x, bi_hforall (Φ x)
  end%I.

Definition bi_laterN {PROP : sbi} (n : nat) (P : PROP) : PROP :=
  Nat.iter n bi_later P.
Arguments bi_laterN {_} !_%nat_scope _%I.
Instance: Params (@bi_laterN) 2.
Notation "▷^ n P" := (bi_laterN n P)
  (at level 20, n at level 9, P at level 20, format "▷^ n  P") : bi_scope.
Notation "▷? p P" := (bi_laterN (Nat.b2n p) P)
  (at level 20, p at level 9, P at level 20, format "▷? p  P") : bi_scope.

Definition bi_except_0 {PROP : sbi} (P : PROP) : PROP := ( False  P)%I.
Arguments bi_except_0 {_} _%I : simpl never.
Notation "◇ P" := (bi_except_0 P) (at level 20, right associativity) : bi_scope.
Instance: Params (@bi_except_0) 1.
Typeclasses Opaque bi_except_0.

Class Timeless {PROP : sbi} (P : PROP) := timeless :  P   P.
Arguments Timeless {_} _%I : simpl never.
Arguments timeless {_} _%I {_}.
Hint Mode Timeless + ! : typeclass_instances.
Instance: Params (@Timeless) 1.

Module bi.
Import interface.bi.
Section bi_derived.
Context {PROP : bi}.
Implicit Types φ : Prop.
Implicit Types P Q R : PROP.
Implicit Types Ps : list PROP.
Implicit Types A : Type.

Hint Extern 100 (NonExpansive _) => solve_proper.

(* Force implicit argument PROP *)
Notation "P ⊢ Q" := (@bi_entails PROP P%I Q%I).
Notation "P ⊣⊢ Q" := (equiv (A:=bi_car PROP) P%I Q%I).

(* Derived stuff about the entailment *)
Global Instance entails_anti_sym : AntiSymm () (@bi_entails PROP).
Proof. intros P Q ??. by apply equiv_spec. Qed.
Lemma equiv_entails P Q : (P  Q)  (P  Q).
Proof. apply equiv_spec. Qed.
Lemma equiv_entails_sym P Q : (Q  P)  (P  Q).
Proof. apply equiv_spec. Qed.
Global Instance entails_proper :
  Proper (() ==> () ==> iff) (() : relation PROP).
Proof.
  move => P1 P2 /equiv_spec [HP1 HP2] Q1 Q2 /equiv_spec [HQ1 HQ2]; split=>?.
  - by trans P1; [|trans Q1].
  - by trans P2; [|trans Q2].
Qed.
Lemma entails_equiv_l P Q R : (P  Q)  (Q  R)  (P  R).
Proof. by intros ->. Qed.
Lemma entails_equiv_r P Q R : (P  Q)  (Q  R)  (P  R).
Proof. by intros ? <-. Qed.
 Global Instance bi_valid_proper : Proper (() ==> iff) (@bi_valid PROP).
Proof. solve_proper. Qed.
Global Instance bi_valid_mono : Proper (() ==> impl) (@bi_valid PROP).
Proof. solve_proper. Qed.
Global Instance bi_valid_flip_mono :
  Proper (flip () ==> flip impl) (@bi_valid PROP).
Proof. solve_proper. Qed.

(* Propers *)
Global Instance pure_proper : Proper (iff ==> ()) (@bi_pure PROP) | 0.
Proof. intros φ1 φ2 Hφ. apply equiv_dist=> n. by apply pure_ne. Qed.
Global Instance and_proper :
  Proper (() ==> () ==> ()) (@bi_and PROP) := ne_proper_2 _.
Global Instance or_proper :
  Proper (() ==> () ==> ()) (@bi_or PROP) := ne_proper_2 _.
Global Instance impl_proper :
  Proper (() ==> () ==> ()) (@bi_impl PROP) := ne_proper_2 _.
Global Instance sep_proper :
  Proper (() ==> () ==> ()) (@bi_sep PROP) := ne_proper_2 _.
Global Instance wand_proper :
  Proper (() ==> () ==> ()) (@bi_wand PROP) := ne_proper_2 _.
Global Instance forall_proper A :
  Proper (pointwise_relation _ () ==> ()) (@bi_forall PROP A).
Proof.
  intros Φ1 Φ2 HΦ. apply equiv_dist=> n.
  apply forall_ne=> x. apply equiv_dist, HΦ.
Qed.
Global Instance exist_proper A :
  Proper (pointwise_relation _ () ==> ()) (@bi_exist PROP A).
Proof.
  intros Φ1 Φ2 HΦ. apply equiv_dist=> n.
  apply exist_ne=> x. apply equiv_dist, HΦ.
Qed.
Global Instance internal_eq_proper (A : ofeT) :
  Proper (() ==> () ==> ()) (@bi_internal_eq PROP A) := ne_proper_2 _.
Global Instance persistently_proper :
  Proper (() ==> ()) (@bi_persistently PROP) := ne_proper _.

(* Derived logical stuff *)
Lemma and_elim_l' P Q R : (P  R)  P  Q  R.
Proof. by rewrite and_elim_l. Qed.
Lemma and_elim_r' P Q R : (Q  R)  P  Q  R.
Proof. by rewrite and_elim_r. Qed.
Lemma or_intro_l' P Q R : (P  Q)  P  Q  R.
Proof. intros ->; apply or_intro_l. Qed.
Lemma or_intro_r' P Q R : (P  R)  P  Q  R.
Proof. intros ->; apply or_intro_r. Qed.
Lemma exist_intro' {A} P (Ψ : A  PROP) a : (P  Ψ a)  P   a, Ψ a.
Proof. intros ->; apply exist_intro. Qed.
Lemma forall_elim' {A} P (Ψ : A  PROP) : (P   a, Ψ a)   a, P  Ψ a.
Proof. move=> HP a. by rewrite HP forall_elim. Qed.

Hint Resolve pure_intro forall_intro.
Hint Resolve or_elim or_intro_l' or_intro_r'.
Hint Resolve and_intro and_elim_l' and_elim_r'.

Lemma impl_intro_l P Q R : (Q  P  R)  P  Q  R.
Proof. intros HR; apply impl_intro_r; rewrite -HR; auto. Qed.
Lemma impl_elim P Q R : (P  Q  R)  (P  Q)  P  R.
Proof. intros. rewrite -(impl_elim_l' P Q R); auto. Qed.
Lemma impl_elim_r' P Q R : (Q  P  R)  P  Q  R.
Proof. intros; apply impl_elim with P; auto. Qed.
Lemma impl_elim_l P Q : (P  Q)  P  Q.
Proof. by apply impl_elim_l'. Qed.
Lemma impl_elim_r P Q : P  (P  Q)  Q.
Proof. by apply impl_elim_r'. Qed.

Lemma False_elim P : False  P.
Proof. by apply (pure_elim' False). Qed.
Lemma True_intro P : P  True.
Proof. by apply pure_intro. Qed.
Hint Immediate False_elim.

Lemma and_mono P P' Q Q' : (P  Q)  (P'  Q')  P  P'  Q  Q'.
Proof. auto. Qed.
Lemma and_mono_l P P' Q : (P  Q)  P  P'  Q  P'.
Proof. by intros; apply and_mono. Qed.
Lemma and_mono_r P P' Q' : (P'  Q')  P  P'  P  Q'.
Proof. by apply and_mono. Qed.

Lemma or_mono P P' Q Q' : (P  Q)  (P'  Q')  P  P'  Q  Q'.
Proof. auto. Qed.
Lemma or_mono_l P P' Q : (P  Q)  P  P'  Q  P'.
Proof. by intros; apply or_mono. Qed.
Lemma or_mono_r P P' Q' : (P'  Q')  P  P'  P  Q'.
Proof. by apply or_mono. Qed.

Lemma impl_mono P P' Q Q' : (Q  P)  (P'  Q')  (P  P')  Q  Q'.
Proof.
  intros HP HQ'; apply impl_intro_l; rewrite -HQ'.
  apply impl_elim with P; eauto.
Qed.
Lemma forall_mono {A} (Φ Ψ : A  PROP) :
  ( a, Φ a  Ψ a)  ( a, Φ a)   a, Ψ a.
Proof.
  intros HP. apply forall_intro=> a; rewrite -(HP a); apply forall_elim.
Qed.
Lemma exist_mono {A} (Φ Ψ : A  PROP) :
  ( a, Φ a  Ψ a)  ( a, Φ a)   a, Ψ a.
Proof. intros HΦ. apply exist_elim=> a; rewrite (HΦ a); apply exist_intro. Qed.

Global Instance and_mono' : Proper (() ==> () ==> ()) (@bi_and PROP).
Proof. by intros P P' HP Q Q' HQ; apply and_mono. Qed.
Global Instance and_flip_mono' :
  Proper (flip () ==> flip () ==> flip ()) (@bi_and PROP).
Proof. by intros P P' HP Q Q' HQ; apply and_mono. Qed.
Global Instance or_mono' : Proper (() ==> () ==> ()) (@bi_or PROP).
Proof. by intros P P' HP Q Q' HQ; apply or_mono. Qed.
Global Instance or_flip_mono' :
  Proper (flip () ==> flip () ==> flip ()) (@bi_or PROP).
Proof. by intros P P' HP Q Q' HQ; apply or_mono. Qed.
Global Instance impl_mono' :
  Proper (flip () ==> () ==> ()) (@bi_impl PROP).
Proof. by intros P P' HP Q Q' HQ; apply impl_mono. Qed.
Global Instance impl_flip_mono' :
  Proper (() ==> flip () ==> flip ()) (@bi_impl PROP).
Proof. by intros P P' HP Q Q' HQ; apply impl_mono. Qed.
Global Instance forall_mono' A :
  Proper (pointwise_relation _ () ==> ()) (@bi_forall PROP A).
Proof. intros P1 P2; apply forall_mono. Qed.
Global Instance forall_flip_mono' A :
  Proper (pointwise_relation _ (flip ()) ==> flip ()) (@bi_forall PROP A).
Proof. intros P1 P2; apply forall_mono. Qed.
Global Instance exist_mono' A :
  Proper (pointwise_relation _ (()) ==> ()) (@bi_exist PROP A).
Proof. intros P1 P2; apply exist_mono. Qed.
Global Instance exist_flip_mono' A :
  Proper (pointwise_relation _ (flip ()) ==> flip ()) (@bi_exist PROP A).
Proof. intros P1 P2; apply exist_mono. Qed.

Global Instance and_idem : IdemP () (@bi_and PROP).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance or_idem : IdemP () (@bi_or PROP).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance and_comm : Comm () (@bi_and PROP).
Proof. intros P Q; apply (anti_symm ()); auto. Qed.
Global Instance True_and : LeftId () True%I (@bi_and PROP).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance and_True : RightId () True%I (@bi_and PROP).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance False_and : LeftAbsorb () False%I (@bi_and PROP).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance and_False : RightAbsorb () False%I (@bi_and PROP).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance True_or : LeftAbsorb () True%I (@bi_or PROP).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance or_True : RightAbsorb () True%I (@bi_or PROP).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance False_or : LeftId () False%I (@bi_or PROP).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance or_False : RightId () False%I (@bi_or PROP).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance and_assoc : Assoc () (@bi_and PROP).
Proof. intros P Q R; apply (anti_symm ()); auto. Qed.
Global Instance or_comm : Comm () (@bi_or PROP).
Proof. intros P Q; apply (anti_symm ()); auto. Qed.
Global Instance or_assoc : Assoc () (@bi_or PROP).
Proof. intros P Q R; apply (anti_symm ()); auto. Qed.
Global Instance True_impl : LeftId () True%I (@bi_impl PROP).
Proof.
  intros P; apply (anti_symm ()).
  - by rewrite -(left_id True%I ()%I (_  _)%I) impl_elim_r.
  - by apply impl_intro_l; rewrite left_id.
Qed.

Lemma False_impl P : (False  P)  True.
Proof.
  apply (anti_symm ()); [by auto|].
  apply impl_intro_l. rewrite left_absorb. auto.
Qed.

Lemma exists_impl_forall {A} P (Ψ : A  PROP) :
  (( x : A, Ψ x)  P)   x : A, Ψ x  P.
Proof.
  apply equiv_spec; split.
  - apply forall_intro=>x. by rewrite -exist_intro.
  - apply impl_intro_r, impl_elim_r', exist_elim=>x.
    apply impl_intro_r. by rewrite (forall_elim x) impl_elim_r.
Qed.

Lemma or_and_l P Q R : P  Q  R  (P  Q)  (P  R).
Proof.
  apply (anti_symm ()); first auto.
  do 2 (apply impl_elim_l', or_elim; apply impl_intro_l); auto.
Qed.
Lemma or_and_r P Q R : P  Q  R  (P  R)  (Q  R).
Proof. by rewrite -!(comm _ R) or_and_l. Qed.
Lemma and_or_l P Q R : P  (Q  R)  P  Q  P  R.
Proof.
  apply (anti_symm ()); last auto.
  apply impl_elim_r', or_elim; apply impl_intro_l; auto.
Qed.
Lemma and_or_r P Q R : (P  Q)  R  P  R  Q  R.
Proof. by rewrite -!(comm _ R) and_or_l. Qed.
Lemma and_exist_l {A} P (Ψ : A  PROP) : P  ( a, Ψ a)   a, P  Ψ a.
Proof.
  apply (anti_symm ()).
  - apply impl_elim_r'. apply exist_elim=>a. apply impl_intro_l.
    by rewrite -(exist_intro a).
  - apply exist_elim=>a. apply and_intro; first by rewrite and_elim_l.
    by rewrite -(exist_intro a) and_elim_r.
Qed.
Lemma and_exist_r {A} P (Φ: A  PROP) : ( a, Φ a)  P   a, Φ a  P.
Proof.
  rewrite -(comm _ P) and_exist_l. apply exist_proper=>a. by rewrite comm.
Qed.
Lemma or_exist {A} (Φ Ψ : A  PROP) :
  ( a, Φ a  Ψ a)  ( a, Φ a)  ( a, Ψ a).
Proof.
  apply (anti_symm ()).
  - apply exist_elim=> a. by rewrite -!(exist_intro a).
  - apply or_elim; apply exist_elim=> a; rewrite -(exist_intro a); auto.
Qed.

Lemma and_alt P Q : P  Q   b : bool, if b then P else Q.
Proof.
   apply (anti_symm _); first apply forall_intro=> -[]; auto.
   by apply and_intro; [rewrite (forall_elim true)|rewrite (forall_elim false)].
Qed.
Lemma or_alt P Q : P  Q   b : bool, if b then P else Q.
Proof.
  apply (anti_symm _); last apply exist_elim=> -[]; auto.
  by apply or_elim; [rewrite -(exist_intro true)|rewrite -(exist_intro false)].
Qed.

Lemma entails_equiv_and P Q : (P  Q  P)  (P  Q).
Proof. split. by intros ->; auto. intros; apply (anti_symm _); auto. Qed.

Global Instance iff_ne : NonExpansive2 (@bi_iff PROP).
Proof. unfold bi_iff; solve_proper. Qed.
Global Instance iff_proper :
  Proper (() ==> () ==> ()) (@bi_iff PROP) := ne_proper_2 _.

Lemma iff_refl Q P : Q  P  P.
Proof. rewrite /bi_iff; apply and_intro; apply impl_intro_l; auto. Qed.

(* Equality stuff *)
Hint Resolve internal_eq_refl.
Lemma equiv_internal_eq {A : ofeT} P (a b : A) : a  b  P  a  b.
Proof. intros ->. auto. Qed.
Lemma internal_eq_rewrite' {A : ofeT} a b (Ψ : A  PROP) P
  {HΨ : NonExpansive Ψ} : (P  a  b)  (P  Ψ a)  P  Ψ b.
Proof.
  intros Heq HΨa. rewrite -(idemp bi_and P) {1}Heq HΨa.
  apply impl_elim_l'. by apply internal_eq_rewrite.
Qed.

Lemma internal_eq_sym {A : ofeT} (a b : A) : a  b  b  a.
Proof. apply (internal_eq_rewrite' a b (λ b, b  a)%I); auto. Qed.
Lemma internal_eq_iff P Q : P  Q  P  Q.
Proof. apply (internal_eq_rewrite' P Q (λ Q, P  Q))%I; auto using iff_refl. Qed.

Lemma f_equiv {A B : ofeT} (f : A  B) `{!NonExpansive f} x y :
  x  y  f x  f y.
Proof. apply (internal_eq_rewrite' x y (λ y, f x  f y)%I); auto. Qed.

Lemma prod_equivI {A B : ofeT} (x y : A * B) : x  y  x.1  y.1  x.2  y.2.
Proof.
  apply (anti_symm _).
  - apply and_intro; apply f_equiv; apply _.
  - rewrite {3}(surjective_pairing x) {3}(surjective_pairing y).
    apply (internal_eq_rewrite' (x.1) (y.1) (λ a, (x.1,x.2)  (a,y.2))%I); auto.
    apply (internal_eq_rewrite' (x.2) (y.2) (λ b, (x.1,x.2)  (x.1,b))%I); auto.
Qed.
Lemma sum_equivI {A B : ofeT} (x y : A + B) :
  x  y 
    match x, y with
    | inl a, inl a' => a  a' | inr b, inr b' => b  b' | _, _ => False
    end.
Proof.
  apply (anti_symm _).
  - apply (internal_eq_rewrite' x y (λ y,
             match x, y with
             | inl a, inl a' => a  a' | inr b, inr b' => b  b' | _, _ => False
             end)%I); auto.
    destruct x; auto.
  - destruct x as [a|b], y as [a'|b']; auto; apply f_equiv, _.
Qed.
Lemma option_equivI {A : ofeT} (x y : option A) :
  x  y  match x, y with
           | Some a, Some a' => a  a' | None, None => True | _, _ => False
           end.
Proof.
  apply (anti_symm _).
  - apply (internal_eq_rewrite' x y (λ y,
             match x, y with
             | Some a, Some a' => a  a' | None, None => True | _, _ => False
             end)%I); auto.
    destruct x; auto.
  - destruct x as [a|], y as [a'|]; auto. apply f_equiv, _.
Qed.

Lemma sig_equivI {A : ofeT} (P : A  Prop) (x y : sig P) : `x  `y  x  y.
Proof. apply (anti_symm _). apply sig_eq. apply f_equiv, _. Qed.

Lemma ofe_funC_equivI {A B} (f g : A -c> B) : f  g   x, f x  g x.
Proof.
  apply (anti_symm _); auto using fun_ext.
  apply (internal_eq_rewrite' f g (λ g,  x : A, f x  g x)%I); auto.
  intros n h h' Hh; apply forall_ne=> x; apply internal_eq_ne; auto.
Qed.
Lemma ofe_morC_equivI {A B : ofeT} (f g : A -n> B) : f  g   x, f x  g x.
Proof.
  apply (anti_symm _).
  - apply (internal_eq_rewrite' f g (λ g,  x : A, f x  g x)%I); auto.
  - rewrite -(ofe_funC_equivI (ofe_mor_car _ _ f) (ofe_mor_car _ _ g)).
    set (h1 (f : A -n> B) :=
      exist (λ f : A -c> B, NonExpansive f) f (ofe_mor_ne A B f)).
    set (h2 (f : sigC (λ f : A -c> B, NonExpansive f)) :=
      @CofeMor A B (`f) (proj2_sig f)).
    assert ( f, h2 (h1 f) = f) as Hh by (by intros []).
    assert (NonExpansive h2) by (intros ??? EQ; apply EQ).
    by rewrite -{2}[f]Hh -{2}[g]Hh -f_equiv -sig_equivI.
Qed.

(* BI Stuff *)
Hint Resolve sep_mono.
Lemma sep_mono_l P P' Q : (P  Q)  P  P'  Q  P'.
Proof. by intros; apply sep_mono. Qed.
Lemma sep_mono_r P P' Q' : (P'  Q')  P  P'  P  Q'.
Proof. by apply sep_mono. Qed.
Global Instance sep_mono' : Proper (() ==> () ==> ()) (@bi_sep PROP).
Proof. by intros P P' HP Q Q' HQ; apply sep_mono. Qed.
Global Instance sep_flip_mono' :
  Proper (flip () ==> flip () ==> flip ()) (@bi_sep PROP).
Proof. by intros P P' HP Q Q' HQ; apply sep_mono. Qed.
Lemma wand_mono P P' Q Q' : (Q  P)  (P'  Q')  (P - P')  Q - Q'.
Proof.
  intros HP HQ; apply wand_intro_r. rewrite HP -HQ. by apply wand_elim_l'.
Qed.
Global Instance wand_mono' : Proper (flip () ==> () ==> ()) (@bi_wand PROP).
Proof. by intros P P' HP Q Q' HQ; apply wand_mono. Qed.
Global Instance wand_flip_mono' :
  Proper (() ==> flip () ==> flip ()) (@bi_wand PROP).
Proof. by intros P P' HP Q Q' HQ; apply wand_mono. Qed.

Global Instance sep_comm : Comm () (@bi_sep PROP).
Proof. intros P Q; apply (anti_symm _); auto using sep_comm'. Qed.
Global Instance sep_assoc : Assoc () (@bi_sep PROP).
Proof.
  intros P Q R; apply (anti_symm _); auto using sep_assoc'.
  by rewrite !(comm _ P) !(comm _ _ R) sep_assoc'.
Qed.
Global Instance emp_sep : LeftId () emp%I (@bi_sep PROP).
Proof. intros P; apply (anti_symm _); auto using emp_sep_1, emp_sep_2. Qed.
Global Instance sep_emp : RightId () emp%I (@bi_sep PROP).
Proof. by intros P; rewrite comm left_id. Qed.

Global Instance sep_False : LeftAbsorb () False%I (@bi_sep PROP).
Proof. intros P; apply (anti_symm _); auto using wand_elim_l'. Qed.
Global Instance False_sep : RightAbsorb () False%I (@bi_sep PROP).
Proof. intros P. by rewrite comm left_absorb. Qed.

Lemma True_sep_2 P : P  True  P.
Proof. rewrite -{1}[P](left_id emp%I bi_sep). auto using sep_mono. Qed.
Lemma sep_True_2 P : P  P  True.
Proof. by rewrite comm -True_sep_2. Qed.

Lemma sep_intro_valid_l P Q R : P  (R  Q)  R  P  Q.
Proof. intros ? ->. rewrite -{1}(left_id emp%I _ Q). by apply sep_mono. Qed.
Lemma sep_intro_valid_r P Q R : (R  P)  Q  R  P  Q.
Proof. intros -> ?. rewrite comm. by apply sep_intro_valid_l. Qed.
Lemma sep_elim_valid_l P Q R : P  (P  R  Q)  R  Q.
Proof. intros <- <-. by rewrite left_id. Qed.
Lemma sep_elim_valid_r P Q R : P  (R  P  Q)  R  Q.
Proof. intros <- <-. by rewrite right_id. Qed.

Lemma wand_intro_l P Q R : (Q  P  R)  P  Q - R.
Proof. rewrite comm; apply wand_intro_r. Qed.
Lemma wand_elim_l P Q : (P - Q)  P  Q.
Proof. by apply wand_elim_l'. Qed.
Lemma wand_elim_r P Q : P  (P - Q)  Q.
Proof. rewrite (comm _ P); apply wand_elim_l. Qed.
Lemma wand_elim_r' P Q R : (Q  P - R)  P  Q  R.
Proof. intros ->; apply wand_elim_r. Qed.
Lemma wand_apply P Q R S : (P  Q - R)  (S  P  Q)  S  R.
Proof. intros HR%wand_elim_l' HQ. by rewrite HQ. Qed.
Lemma wand_frame_l P Q R : (Q - R)  P  Q - P  R.
Proof. apply wand_intro_l. rewrite -assoc. apply sep_mono_r, wand_elim_r. Qed.
Lemma wand_frame_r P Q R : (Q - R)  Q  P - R  P.
Proof.
  apply wand_intro_l. rewrite ![(_  P)%I]comm -assoc.
  apply sep_mono_r, wand_elim_r.
Qed.

Lemma emp_wand P : (emp - P)  P.
Proof.
  apply (anti_symm _).
  - by rewrite -[(emp - P)%I]left_id wand_elim_r.
  - apply wand_intro_l. by rewrite left_id.
Qed.
Lemma False_wand P : (False - P)  True.
Proof.
  apply (anti_symm ()); [by auto|].
  apply wand_intro_l. rewrite left_absorb. auto.
Qed.

Lemma wand_curry P Q R : (P - Q - R)  (P  Q - R).
Proof.
  apply (anti_symm _).
  - apply wand_intro_l. by rewrite (comm _ P) -assoc !wand_elim_r.
  - do 2 apply wand_intro_l. by rewrite assoc (comm _ Q) wand_elim_r.
Qed.

Lemma sep_and_l P Q R : P  (Q  R)  (P  Q)  (P  R).
Proof. auto. Qed.
Lemma sep_and_r P Q R : (P  Q)  R  (P  R)  (Q  R).
Proof. auto. Qed.
Lemma sep_or_l P Q R : P  (Q  R)  (P  Q)  (P  R).
Proof.
  apply (anti_symm ()); last by eauto 8.
  apply wand_elim_r', or_elim; apply wand_intro_l; auto.
Qed.
Lemma sep_or_r P Q R : (P  Q)  R  (P  R)  (Q  R).
Proof. by rewrite -!(comm _ R) sep_or_l. Qed.
Lemma sep_exist_l {A} P (Ψ : A  PROP) : P  ( a, Ψ a)   a, P  Ψ a.
Proof.
  intros; apply (anti_symm ()).
  - apply wand_elim_r', exist_elim=>a. apply wand_intro_l.
    by rewrite -(exist_intro a).
  - apply exist_elim=> a; apply sep_mono; auto using exist_intro.
Qed.
Lemma sep_exist_r {A} (Φ: A  PROP) Q: ( a, Φ a)  Q   a, Φ a  Q.
Proof. setoid_rewrite (comm _ _ Q); apply sep_exist_l. Qed.
Lemma sep_forall_l {A} P (Ψ : A  PROP) : P  ( a, Ψ a)   a, P  Ψ a.
Proof. by apply forall_intro=> a; rewrite forall_elim. Qed.
Lemma sep_forall_r {A} (Φ : A  PROP) Q : ( a, Φ a)  Q   a, Φ a  Q.
Proof. by apply forall_intro=> a; rewrite forall_elim. Qed.

Global Instance wand_iff_ne : NonExpansive2 (@bi_wand_iff PROP).
Proof. solve_proper. Qed.
Global Instance wand_iff_proper :
  Proper (() ==> () ==> ()) (@bi_wand_iff PROP) := ne_proper_2 _.

Lemma wand_iff_refl P : emp  P - P.
Proof. apply and_intro; apply wand_intro_l; by rewrite right_id. Qed.

Lemma wand_entails P Q : (P - Q)%I  P  Q.
Proof. intros. rewrite -[P]left_id. by apply wand_elim_l'. Qed.
Lemma entails_wand P Q : (P  Q)  (P - Q)%I.
Proof. intros ->. apply wand_intro_r. by rewrite left_id. Qed.

Lemma equiv_wand_iff P Q : (P  Q)  (P - Q)%I.
Proof. intros ->; apply wand_iff_refl. Qed.
Lemma wand_iff_equiv P Q : (P - Q)%I  (P  Q).
Proof.
  intros HPQ; apply (anti_symm ());
    apply wand_entails; rewrite /bi_valid HPQ /bi_wand_iff; auto.
Qed.

Lemma entails_impl P Q : (P  Q)  (P  Q)%I.
Proof. intros ->. apply impl_intro_l. auto. Qed.
Lemma impl_entails P Q `{!Affine P} : (P  Q)%I  P  Q.
Proof. intros HPQ. apply impl_elim with P=>//. by rewrite {1}(affine P). Qed.

Lemma equiv_iff P Q : (P  Q)  (P  Q)%I.
Proof. intros ->; apply iff_refl. Qed.
Lemma iff_equiv P Q `{!Affine P, !Affine Q} : (P  Q)%I  (P  Q).
Proof.
  intros HPQ; apply (anti_symm ());
    apply: impl_entails; rewrite /bi_valid HPQ /bi_iff; auto.
Qed.

(* Pure stuff *)
Lemma pure_elim φ Q R : (Q  ⌜φ⌝)  (φ  Q  R)  Q  R.
Proof.
  intros HQ HQR. rewrite -(idemp ()%I Q) {1}HQ.
  apply impl_elim_l', pure_elim'=> ?. apply impl_intro_l.
  rewrite and_elim_l; auto.
Qed.
Lemma pure_mono φ1 φ2 : (φ1  φ2)  ⌜φ1  ⌜φ2.
Proof. auto using pure_elim', pure_intro. Qed.
Global Instance pure_mono' : Proper (impl ==> ()) (@bi_pure PROP).
Proof. intros φ1 φ2; apply pure_mono. Qed.
Global Instance pure_flip_mono : Proper (flip impl ==> flip ()) (@bi_pure PROP).
Proof. intros φ1 φ2; apply pure_mono. Qed.
Lemma pure_iff φ1 φ2 : (φ1  φ2)  ⌜φ1  ⌜φ2.
Proof. intros [??]; apply (anti_symm _); auto using pure_mono. Qed.
Lemma pure_elim_l φ Q R : (φ  Q  R)  ⌜φ⌝  Q  R.
Proof. intros; apply pure_elim with φ; eauto. Qed.
Lemma pure_elim_r φ Q R : (φ  Q  R)  Q  ⌜φ⌝  R.
Proof. intros; apply pure_elim with φ; eauto. Qed.

Lemma pure_True (φ : Prop) : φ  ⌜φ⌝  True.
Proof. intros; apply (anti_symm _); auto. Qed.
Lemma pure_False (φ : Prop) : ¬φ  ⌜φ⌝  False.
Proof. intros; apply (anti_symm _); eauto using pure_mono. Qed.

Lemma pure_and φ1 φ2 : ⌜φ1  φ2  ⌜φ1  ⌜φ2.
Proof.
  apply (anti_symm _).
  - apply and_intro; apply pure_mono; tauto.
  - eapply (pure_elim φ1); [auto|]=> ?. rewrite and_elim_r. auto using pure_mono.
Qed.
Lemma pure_or φ1 φ2 : ⌜φ1  φ2  ⌜φ1  ⌜φ2.
Proof.
  apply (anti_symm _).
  - eapply pure_elim=> // -[?|?]; auto using pure_mono.
  - apply or_elim; eauto using pure_mono.
Qed.
Lemma pure_impl φ1 φ2 : ⌜φ1  φ2  (⌜φ1  ⌜φ2).
Proof.
  apply (anti_symm _).
  - apply impl_intro_l. rewrite -pure_and. apply pure_mono. naive_solver.
  - rewrite -pure_forall_2. apply forall_intro=> ?.
    by rewrite -(left_id True bi_and (_→_))%I (pure_True φ1) // impl_elim_r.
Qed.
Lemma pure_forall {A} (φ : A  Prop) :  x, φ x   x, ⌜φ x.
Proof.
  apply (anti_symm _); auto using pure_forall_2.
  apply forall_intro=> x. eauto using pure_mono.
Qed.
Lemma pure_exist {A} (φ : A  Prop) :  x, φ x   x, ⌜φ x.
Proof.
  apply (anti_symm _).
  - eapply pure_elim=> // -[x ?]. rewrite -(exist_intro x); auto using pure_mono.
  - apply exist_elim=> x. eauto using pure_mono.
Qed.

Lemma pure_impl_forall φ P : (⌜φ⌝  P)  ( _ : φ, P).
Proof.
  apply (anti_symm _).
  - apply forall_intro=> ?. by rewrite pure_True // left_id.
  - apply impl_intro_l, pure_elim_l=> Hφ. by rewrite (forall_elim Hφ).
Qed.
Lemma pure_alt φ : ⌜φ⌝   _ : φ, True.
Proof.
  apply (anti_symm _).
  - eapply pure_elim; eauto=> H. rewrite -(exist_intro H); auto.
  - by apply exist_elim, pure_intro.
Qed.
Lemma pure_wand_forall φ P `{!Absorbing P} : (⌜φ⌝ - P)  ( _ : φ, P).
Proof.
  apply (anti_symm _).
  - apply forall_intro=> Hφ.
    by rewrite -(left_id emp%I _ (_ - _)%I) (pure_intro emp%I φ) // wand_elim_r.
  - apply wand_intro_l, wand_elim_l', pure_elim'=> Hφ.
Robbert Krebbers's avatar
Robbert Krebbers committed
673
    apply wand_intro_l. rewrite (forall_elim Hφ) comm. by apply absorbing.
Robbert Krebbers's avatar
Robbert Krebbers committed
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
Qed.

Lemma pure_internal_eq {A : ofeT} (x y : A) : x  y  x  y.
Proof. apply pure_elim'=> ->. apply internal_eq_refl. Qed.
Lemma discrete_eq {A : ofeT} (a b : A) : Discrete a  a  b  a  b.
Proof.
  intros. apply (anti_symm _); auto using discrete_eq_1, pure_internal_eq.
Qed.

(* Properties of the bare modality *)
Global Instance bare_ne : NonExpansive (@bi_bare PROP).
Proof. solve_proper. Qed.
Global Instance bare_proper : Proper (() ==> ()) (@bi_bare PROP).
Proof. solve_proper. Qed.
Global Instance bare_mono' : Proper (() ==> ()) (@bi_bare PROP).
Proof. solve_proper. Qed.
Global Instance bare_flip_mono' :
  Proper (flip () ==> flip ()) (@bi_bare PROP).
Proof. solve_proper. Qed.

Lemma bare_elim_emp P :  P  emp.
Proof. rewrite /bi_bare; auto. Qed.
Lemma bare_elim P :  P  P.
Proof. rewrite /bi_bare; auto. Qed.
Lemma bare_mono P Q : (P  Q)   P   Q.
Proof. by intros ->. Qed.
Lemma bare_idemp P :   P   P.
Proof. by rewrite /bi_bare assoc idemp. Qed.

Lemma bare_intro' P Q : ( P  Q)   P   Q.
Proof. intros <-. by rewrite bare_idemp. Qed.

Lemma bare_False :  False  False.
Proof. by rewrite /bi_bare right_absorb. Qed.
Lemma bare_emp :  emp  emp.
Proof. by rewrite /bi_bare (idemp bi_and). Qed.
Lemma bare_or P Q :  (P  Q)   P   Q.
Proof. by rewrite /bi_bare and_or_l. Qed.
Lemma bare_and P Q :  (P  Q)   P   Q.
Proof.
  rewrite /bi_bare -(comm _ P) (assoc _ (_  _)%I) -!(assoc _ P).
  by rewrite idemp !assoc (comm _ P).
Qed.
717
Lemma bare_sep_2 P Q :  P   Q   (P  Q).
Robbert Krebbers's avatar
Robbert Krebbers committed
718
Proof.
719
720
721
722
723
724
725
726
  rewrite /bi_bare. apply and_intro.
  - by rewrite !and_elim_l right_id.
  - by rewrite !and_elim_r.
Qed.
Lemma bare_sep `{PositiveBI PROP} P Q :  (P  Q)   P   Q.
Proof.
  apply (anti_symm _), bare_sep_2.
  by rewrite -{1}bare_idemp positive_bi !(comm _ ( P)%I) positive_bi.
Robbert Krebbers's avatar
Robbert Krebbers committed
727
728
729
730
731
732
733
734
735
Qed.
Lemma bare_forall {A} (Φ : A  PROP) :  ( a, Φ a)   a,  Φ a.
Proof. apply forall_intro=> a. by rewrite (forall_elim a). Qed.
Lemma bare_exist {A} (Φ : A  PROP) :  ( a, Φ a)   a,  Φ a.
Proof. by rewrite /bi_bare and_exist_l. Qed.

Lemma bare_True_emp :  True   emp.
Proof. apply (anti_symm _); rewrite /bi_bare; auto. Qed.

736
Lemma bare_and_l P Q :  P  Q   (P  Q).
Robbert Krebbers's avatar
Robbert Krebbers committed
737
Proof. by rewrite /bi_bare assoc. Qed.
738
739
740
741
Lemma bare_and_r P Q : P   Q   (P  Q).
Proof. by rewrite /bi_bare !assoc (comm _ P). Qed.
Lemma bare_and_lr P Q :  P  Q  P   Q.
Proof. by rewrite bare_and_l bare_and_r. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
742

Robbert Krebbers's avatar
Robbert Krebbers committed
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
(* Properties of the sink modality *)
Global Instance sink_ne : NonExpansive (@bi_sink PROP).
Proof. solve_proper. Qed.
Global Instance sink_proper : Proper (() ==> ()) (@bi_sink PROP).
Proof. solve_proper. Qed.
Global Instance sink_mono' : Proper (() ==> ()) (@bi_sink PROP).
Proof. solve_proper. Qed.
Global Instance sink_flip_mono' :
  Proper (flip () ==> flip ()) (@bi_sink PROP).
Proof. solve_proper. Qed.

Lemma sink_intro P : P   P.
Proof. by rewrite /bi_sink -True_sep_2. Qed.
Lemma sink_mono P Q : (P  Q)   P   Q.
Proof. by intros ->. Qed.
Lemma sink_idemp P :   P   P.
Proof.
  apply (anti_symm _), sink_intro.
  rewrite /bi_sink assoc. apply sep_mono; auto.
Qed.

Lemma sink_pure φ :   φ    φ .
Proof.
  apply (anti_symm _), sink_intro.
  apply wand_elim_r', pure_elim'=> ?. apply wand_intro_l; auto.
Qed.
Lemma sink_or P Q :  (P  Q)   P   Q.
Proof. by rewrite /bi_sink sep_or_l. Qed.
Lemma sink_and P Q :  (P  Q)   P   Q.
Proof. apply and_intro; apply sink_mono; auto. Qed.
Lemma sink_forall {A} (Φ : A  PROP) :  ( a, Φ a)   a,  Φ a.
Proof. apply forall_intro=> a. by rewrite (forall_elim a). Qed.
Lemma sink_exist {A} (Φ : A  PROP) :  ( a, Φ a)   a,  Φ a.
Proof. by rewrite /bi_sink sep_exist_l. Qed.

Lemma sink_internal_eq {A : ofeT} (x y : A) :  (x  y)  x  y.
Proof.
  apply (anti_symm _), sink_intro.
  apply wand_elim_r', (internal_eq_rewrite' x y (λ y, True - x  y)%I); auto.
  apply wand_intro_l, internal_eq_refl.
Qed.

Lemma sink_sep P Q :  (P  Q)   P   Q.
Proof. by rewrite -{1}sink_idemp /bi_sink !assoc -!(comm _ P) !assoc. Qed.
Lemma sink_True_emp :  True   emp.
Proof. by rewrite sink_pure /bi_sink right_id. Qed.
Lemma sink_wand P Q :  (P - Q)   P -  Q.
Proof. apply wand_intro_l. by rewrite -sink_sep wand_elim_r. Qed.

Lemma sink_sep_l P Q :  P  Q   (P  Q).
Proof. by rewrite /bi_sink assoc. Qed.
Lemma sink_sep_r P Q : P   Q   (P  Q).
Proof. by rewrite /bi_sink !assoc (comm _ P). Qed.
Lemma sink_sep_lr P Q :  P  Q  P   Q.
Proof. by rewrite sink_sep_l sink_sep_r. Qed.

Lemma bare_sink `{!PositiveBI PROP} P :   P   P.
Proof.
  apply (anti_symm _), bare_mono, sink_intro.
  by rewrite /bi_sink bare_sep bare_True_emp bare_emp left_id.
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
805
(* Affine propositions *)
Robbert Krebbers's avatar
Robbert Krebbers committed
806
Global Instance Affine_proper : Proper (() ==> iff) (@Affine PROP).
Robbert Krebbers's avatar
Robbert Krebbers committed
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
Proof. solve_proper. Qed.

Global Instance emp_affine_l : Affine (emp%I : PROP).
Proof. by rewrite /Affine. Qed.
Global Instance and_affine_l P Q : Affine P  Affine (P  Q).
Proof. rewrite /Affine=> ->; auto. Qed.
Global Instance and_affine_r P Q : Affine Q  Affine (P  Q).
Proof. rewrite /Affine=> ->; auto. Qed.
Global Instance or_affine P Q : Affine P  Affine Q  Affine (P  Q).
Proof.  rewrite /Affine=> -> ->; auto. Qed.
Global Instance forall_affine `{Inhabited A} (Φ : A  PROP) :
  ( x, Affine (Φ x))  Affine ( x, Φ x).
Proof. intros. rewrite /Affine (forall_elim inhabitant). apply: affine. Qed.
Global Instance exist_affine {A} (Φ : A  PROP) :
  ( x, Affine (Φ x))  Affine ( x, Φ x).
Proof. rewrite /Affine=> H. apply exist_elim=> a. by rewrite H. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
823

Robbert Krebbers's avatar
Robbert Krebbers committed
824
825
826
827
828
829
Global Instance sep_affine P Q : Affine P  Affine Q  Affine (P  Q).
Proof. rewrite /Affine=>-> ->. by rewrite left_id. Qed.
Global Instance bare_affine P : Affine ( P).
Proof. rewrite /bi_bare. apply _. Qed.

(* Absorbing propositions *)
Robbert Krebbers's avatar
Robbert Krebbers committed
830
831
Global Instance Absorbing_proper : Proper (() ==> iff) (@Absorbing PROP).
Proof. solve_proper. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
832
833

Global Instance pure_absorbing φ : Absorbing (⌜φ⌝%I : PROP).
Robbert Krebbers's avatar
Robbert Krebbers committed
834
Proof. by rewrite /Absorbing sink_pure. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
835
Global Instance and_absorbing P Q : Absorbing P  Absorbing Q  Absorbing (P  Q).
Robbert Krebbers's avatar
Robbert Krebbers committed
836
Proof. intros. by rewrite /Absorbing sink_and !absorbing. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
837
Global Instance or_absorbing P Q : Absorbing P  Absorbing Q  Absorbing (P  Q).
Robbert Krebbers's avatar
Robbert Krebbers committed
838
Proof. intros. by rewrite /Absorbing sink_or !absorbing. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
839
840
Global Instance forall_absorbing {A} (Φ : A  PROP) :
  ( x, Absorbing (Φ x))  Absorbing ( x, Φ x).
Robbert Krebbers's avatar
Robbert Krebbers committed
841
Proof. rewrite /Absorbing=> ?. rewrite sink_forall. auto using forall_mono. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
842
843
Global Instance exist_absorbing {A} (Φ : A  PROP) :
  ( x, Absorbing (Φ x))  Absorbing ( x, Φ x).
Robbert Krebbers's avatar
Robbert Krebbers committed
844
Proof. rewrite /Absorbing=> ?. rewrite sink_exist. auto using exist_mono. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
845

Robbert Krebbers's avatar
Robbert Krebbers committed
846
847
848
849
850
851
852
853
Global Instance internal_eq_absorbing {A : ofeT} (x y : A) :
  Absorbing (x  y : PROP)%I.
Proof. by rewrite /Absorbing sink_internal_eq. Qed.

Global Instance sep_absorbing_l P Q : Absorbing P  Absorbing (P  Q).
Proof. intros. by rewrite /Absorbing -sink_sep_l absorbing. Qed.
Global Instance sep_absorbing_r P Q : Absorbing Q  Absorbing (P  Q).
Proof. intros. by rewrite /Absorbing -sink_sep_r absorbing. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
854
855

Global Instance wand_absorbing P Q : Absorbing Q  Absorbing (P - Q).
Robbert Krebbers's avatar
Robbert Krebbers committed
856
857
858
859
Proof. intros. by rewrite /Absorbing sink_wand !absorbing -sink_intro. Qed.

Global Instance sink_absorbing P : Absorbing ( P).
Proof. rewrite /bi_sink. apply _. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
860
861

(* Properties of affine and absorbing propositions *)
Robbert Krebbers's avatar
Robbert Krebbers committed
862
863
864
865
866
Lemma affine_bare P `{!Affine P} :  P  P.
Proof. rewrite /bi_bare. apply (anti_symm _); auto. Qed.
Lemma absorbing_sink P `{!Absorbing P} :  P  P.
Proof. by apply (anti_symm _), sink_intro. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
867
868
869
Lemma True_affine_all_affine P : Affine (True%I : PROP)  Affine P.
Proof. rewrite /Affine=> <-; auto. Qed.
Lemma emp_absorbing_all_absorbing P : Absorbing (emp%I : PROP)  Absorbing P.
Robbert Krebbers's avatar
Robbert Krebbers committed
870
871
872
873
Proof.
  intros. rewrite /Absorbing -{2}(left_id emp%I _ P).
  by rewrite -(absorbing emp) sink_sep_l left_id.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
874
875

Lemma sep_elim_l P Q `{H : TCOr (Affine Q) (Absorbing P)} : P  Q  P.
Robbert Krebbers's avatar
Robbert Krebbers committed
876
877
878
879
880
Proof.
  destruct H.
  - by rewrite (affine Q) right_id.
  - by rewrite (True_intro Q) comm.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
881
882
883
Lemma sep_elim_r P Q `{H : TCOr (Affine P) (Absorbing Q)} : P  Q  Q.
Proof. by rewrite comm sep_elim_l. Qed.

884
885
Lemma sep_and P Q
    `{HPQ : TCOr (TCAnd (Affine P) (Affine Q)) (TCAnd (Absorbing P) (Absorbing Q))} :
Robbert Krebbers's avatar
Robbert Krebbers committed
886
  P  Q  P  Q.
887
888
889
890
Proof.
  destruct HPQ as [[??]|[??]];
    apply and_intro; apply: sep_elim_l || apply: sep_elim_r.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
891

Robbert Krebbers's avatar
Robbert Krebbers committed
892

Robbert Krebbers's avatar
Robbert Krebbers committed
893
894
895
896
897
898
899
900
901
902
903
904
905
Lemma bare_intro P Q `{!Affine P} : (P  Q)  P   Q.
Proof. intros <-. by rewrite affine_bare. Qed.

Lemma emp_and P `{!Affine P} : emp  P  P.
Proof. apply (anti_symm _); auto. Qed.
Lemma and_emp P `{!Affine P} : P  emp  P.
Proof. apply (anti_symm _); auto. Qed.
Lemma emp_or P `{!Affine P} : emp  P  emp.
Proof. apply (anti_symm _); auto. Qed.
Lemma or_emp P `{!Affine P} : P  emp  emp.
Proof. apply (anti_symm _); auto. Qed.

Lemma True_sep P `{!Absorbing P} : True  P  P.
Robbert Krebbers's avatar
Robbert Krebbers committed
906
Proof. apply (anti_symm _); auto using True_sep_2. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
907
Lemma sep_True P `{!Absorbing P} : P  True  P.
Robbert Krebbers's avatar
Robbert Krebbers committed
908
Proof. by rewrite comm True_sep. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
909
910
911
912

Section affine_bi.
  Context `{AffineBI PROP}.

913
  Global Instance affine_bi_absorbing P : Absorbing P | 0.
Robbert Krebbers's avatar
Robbert Krebbers committed
914
  Proof. by rewrite /Absorbing /bi_sink (affine True%I) left_id. Qed.
915
916
  Global Instance affine_bi_positive : PositiveBI PROP.
  Proof. intros P Q. by rewrite !affine_bare. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
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

  Lemma True_emp : True  emp.
  Proof. apply (anti_symm _); auto using affine. Qed.

  Global Instance emp_and' : LeftId () emp%I (@bi_and PROP).
  Proof. intros P. by rewrite -True_emp left_id. Qed.
  Global Instance and_emp' : RightId () emp%I (@bi_and PROP).
  Proof. intros P. by rewrite -True_emp right_id. Qed.

  Global Instance True_sep' : LeftId () True%I (@bi_sep PROP).
  Proof. intros P. by rewrite True_emp left_id. Qed.
  Global Instance sep_True' : RightId () True%I (@bi_sep PROP).
  Proof. intros P. by rewrite True_emp right_id. Qed.

  Lemma impl_wand_1 P Q : (P  Q)  P - Q.
  Proof. apply wand_intro_l. by rewrite sep_and impl_elim_r. Qed.

  Lemma decide_emp φ `{!Decision φ} (P : PROP) :
    (if decide φ then P else emp)  (⌜φ⌝  P).
  Proof.
    destruct (decide _).
    - by rewrite pure_True // True_impl.
    - by rewrite pure_False // False_impl True_emp.
  Qed.
End affine_bi.

(* Properties of the persistently modality *)
Hint Resolve persistently_mono.
Global Instance persistently_mono' : Proper (() ==> ()) (@bi_persistently PROP).
Proof. intros P Q; apply persistently_mono. Qed.
Global Instance persistently_flip_mono' :
  Proper (flip () ==> flip ()) (@bi_persistently PROP).
Proof. intros P Q; apply persistently_mono. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
950
951
952
953
954

Lemma sink_persistently P :   P   P.
Proof.
  apply (anti_symm _), sink_intro. by rewrite /bi_sink comm persistently_absorbing.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
955
Global Instance persistently_absorbing P : Absorbing ( P).
Robbert Krebbers's avatar
Robbert Krebbers committed
956
Proof. by rewrite /Absorbing sink_persistently. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
957

958
Lemma persistently_and_sep_assoc P Q R :  P  (Q  R)  ( P  Q)  R.
Robbert Krebbers's avatar
Robbert Krebbers committed
959
Proof.
960
961
962
  apply (anti_symm ()).
  - rewrite {1}persistently_idemp_2 persistently_and_sep_elim assoc.
    apply sep_mono_l, and_intro.
Robbert Krebbers's avatar
Robbert Krebbers committed
963
    + by rewrite and_elim_r sep_elim_l.
964
965
966
967
    + by rewrite and_elim_l left_id.
  - apply and_intro.
    + by rewrite and_elim_l sep_elim_l.
    + by rewrite and_elim_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
968
969
970
Qed.
Lemma persistently_and_emp_elim P : emp   P  P.
Proof. by rewrite comm persistently_and_sep_elim right_id and_elim_r. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
971
Lemma persistently_elim_sink P :  P   P.
Robbert Krebbers's avatar
Robbert Krebbers committed
972
973
Proof.
  rewrite -(right_id True%I _ ( _)%I) -{1}(left_id emp%I _ True%I).
Robbert Krebbers's avatar
Robbert Krebbers committed
974
  by rewrite persistently_and_sep_assoc (comm bi_and) persistently_and_emp_elim comm.
Robbert Krebbers's avatar
Robbert Krebbers committed
975
Qed.
976
Lemma persistently_elim P `{!Absorbing P} :  P  P.
Robbert Krebbers's avatar
Robbert Krebbers committed
977
Proof. by rewrite persistently_elim_sink absorbing_sink. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
978
979

Lemma persistently_idemp_1 P :   P   P.
Robbert Krebbers's avatar
Robbert Krebbers committed
980
Proof. by rewrite persistently_elim_sink sink_persistently. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
981
Lemma persistently_idemp P :   P   P.
982
Proof. apply (anti_symm _); auto using persistently_idemp_1, persistently_idemp_2. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
983
984
985
986
987
988

Lemma persistently_intro' P Q : ( P  Q)   P   Q.
Proof. intros <-. apply persistently_idemp_2. Qed.

Lemma persistently_pure φ :  ⌜φ⌝  ⌜φ⌝.
Proof.
989
990
991
992
  apply (anti_symm _); first by rewrite persistently_elim.
  apply pure_elim'=> Hφ.
  trans ( x : False,  True : PROP)%I; [by apply forall_intro|].
  rewrite persistently_forall_2. auto using persistently_mono, pure_intro.
Robbert Krebbers's avatar
Robbert Krebbers committed
993
994
995
996
997
998
999
1000
Qed.
Lemma persistently_forall {A} (Ψ : A  PROP) : (  a, Ψ a)  ( a,  Ψ a).
Proof.
  apply (anti_symm _); auto using persistently_forall_2.
  apply forall_intro=> x. by rewrite (forall_elim x).
Qed.
Lemma persistently_exist {A} (Ψ : A  PROP) : (  a, Ψ a)  ( a,  Ψ a).
Proof.