derived_laws.v 102 KB
Newer Older
1
From iris.bi Require Export derived_connectives.
Robbert Krebbers's avatar
Robbert Krebbers committed
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
From iris.algebra Require Import monoid.
From stdpp Require Import hlist.

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.
71
72
Global Instance plainly_proper :
  Proper (() ==> ()) (@bi_plainly PROP) := ne_proper _.
Robbert Krebbers's avatar
Robbert Krebbers committed
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
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.

Jacques-Henri Jourdan's avatar
Typo.    
Jacques-Henri Jourdan committed
209
Lemma exist_impl_forall {A} P (Ψ : A  PROP) :
Robbert Krebbers's avatar
Robbert Krebbers committed
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
  (( 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.


(* 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
497
    apply wand_intro_l. rewrite (forall_elim Hφ) comm. by apply absorbing.
Robbert Krebbers's avatar
Robbert Krebbers committed
498
499
Qed.

500
501
(* Properties of the affinely modality *)
Global Instance affinely_ne : NonExpansive (@bi_affinely PROP).
Robbert Krebbers's avatar
Robbert Krebbers committed
502
Proof. solve_proper. Qed.
503
Global Instance affinely_proper : Proper (() ==> ()) (@bi_affinely PROP).
Robbert Krebbers's avatar
Robbert Krebbers committed
504
Proof. solve_proper. Qed.
505
Global Instance affinely_mono' : Proper (() ==> ()) (@bi_affinely PROP).
Robbert Krebbers's avatar
Robbert Krebbers committed
506
Proof. solve_proper. Qed.
507
508
Global Instance affinely_flip_mono' :
  Proper (flip () ==> flip ()) (@bi_affinely PROP).
Robbert Krebbers's avatar
Robbert Krebbers committed
509
510
Proof. solve_proper. Qed.

511
512
513
514
515
Lemma affinely_elim_emp P : bi_affinely P  emp.
Proof. rewrite /bi_affinely; auto. Qed.
Lemma affinely_elim P : bi_affinely P  P.
Proof. rewrite /bi_affinely; auto. Qed.
Lemma affinely_mono P Q : (P  Q)  bi_affinely P  bi_affinely Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
516
Proof. by intros ->. Qed.
517
518
Lemma affinely_idemp P : bi_affinely (bi_affinely P)  bi_affinely P.
Proof. by rewrite /bi_affinely assoc idemp. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
519

520
521
Lemma affinely_intro' P Q : (bi_affinely P  Q)  bi_affinely P  bi_affinely Q.
Proof. intros <-. by rewrite affinely_idemp. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
522

523
524
525
526
527
528
529
Lemma affinely_False : bi_affinely False  False.
Proof. by rewrite /bi_affinely right_absorb. Qed.
Lemma affinely_emp : bi_affinely emp  emp.
Proof. by rewrite /bi_affinely (idemp bi_and). Qed.
Lemma affinely_or P Q : bi_affinely (P  Q)  bi_affinely P  bi_affinely Q.
Proof. by rewrite /bi_affinely and_or_l. Qed.
Lemma affinely_and P Q : bi_affinely (P  Q)  bi_affinely P  bi_affinely Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
530
Proof.
531
  rewrite /bi_affinely -(comm _ P) (assoc _ (_  _)%I) -!(assoc _ P).
Robbert Krebbers's avatar
Robbert Krebbers committed
532
533
  by rewrite idemp !assoc (comm _ P).
Qed.
534
Lemma affinely_sep_2 P Q : bi_affinely P  bi_affinely Q  bi_affinely (P  Q).
Robbert Krebbers's avatar
Robbert Krebbers committed
535
Proof.
536
  rewrite /bi_affinely. apply and_intro.
537
538
539
  - by rewrite !and_elim_l right_id.
  - by rewrite !and_elim_r.
Qed.
540
Lemma affinely_sep `{BiPositive PROP} P Q :
541
  bi_affinely (P  Q)  bi_affinely P  bi_affinely Q.
542
Proof.
543
  apply (anti_symm _), affinely_sep_2.
544
  by rewrite -{1}affinely_idemp bi_positive !(comm _ (bi_affinely P)%I) bi_positive.
Robbert Krebbers's avatar
Robbert Krebbers committed
545
Qed.
546
547
Lemma affinely_forall {A} (Φ : A  PROP) :
  bi_affinely ( a, Φ a)   a, bi_affinely (Φ a).
Robbert Krebbers's avatar
Robbert Krebbers committed
548
Proof. apply forall_intro=> a. by rewrite (forall_elim a). Qed.
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
Lemma affinely_exist {A} (Φ : A  PROP) :
  bi_affinely ( a, Φ a)   a, bi_affinely (Φ a).
Proof. by rewrite /bi_affinely and_exist_l. Qed.

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

Lemma affinely_and_l P Q : bi_affinely P  Q  bi_affinely (P  Q).
Proof. by rewrite /bi_affinely assoc. Qed.
Lemma affinely_and_r P Q : P  bi_affinely Q  bi_affinely (P  Q).
Proof. by rewrite /bi_affinely !assoc (comm _ P). Qed.
Lemma affinely_and_lr P Q : bi_affinely P  Q  P  bi_affinely Q.
Proof. by rewrite affinely_and_l affinely_and_r. Qed.

(* Properties of the absorbingly modality *)
Global Instance absorbingly_ne : NonExpansive (@bi_absorbingly PROP).
Robbert Krebbers's avatar
Robbert Krebbers committed
565
Proof. solve_proper. Qed.
566
Global Instance absorbingly_proper : Proper (() ==> ()) (@bi_absorbingly PROP).
Robbert Krebbers's avatar
Robbert Krebbers committed
567
Proof. solve_proper. Qed.
568
Global Instance absorbingly_mono' : Proper (() ==> ()) (@bi_absorbingly PROP).
Robbert Krebbers's avatar
Robbert Krebbers committed
569
Proof. solve_proper. Qed.
570
571
Global Instance absorbingly_flip_mono' :
  Proper (flip () ==> flip ()) (@bi_absorbingly PROP).
Robbert Krebbers's avatar
Robbert Krebbers committed
572
573
Proof. solve_proper. Qed.

574
Lemma absorbingly_intro P : P  bi_absorbingly P.
575
Proof. by rewrite /bi_absorbingly -True_sep_2. Qed.
576
Lemma absorbingly_mono P Q : (P  Q)  bi_absorbingly P  bi_absorbingly Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
577
Proof. by intros ->. Qed.
578
Lemma absorbingly_idemp P : bi_absorbingly (bi_absorbingly P)  bi_absorbingly P.
Robbert Krebbers's avatar
Robbert Krebbers committed
579
Proof.
580
581
  apply (anti_symm _), absorbingly_intro.
  rewrite /bi_absorbingly assoc. apply sep_mono; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
582
583
Qed.

584
Lemma absorbingly_pure φ : bi_absorbingly  φ    φ .
Robbert Krebbers's avatar
Robbert Krebbers committed
585
Proof.
586
  apply (anti_symm _), absorbingly_intro.
Robbert Krebbers's avatar
Robbert Krebbers committed
587
588
  apply wand_elim_r', pure_elim'=> ?. apply wand_intro_l; auto.
Qed.
589
590
Lemma absorbingly_or P Q :
  bi_absorbingly (P  Q)  bi_absorbingly P  bi_absorbingly Q.
591
Proof. by rewrite /bi_absorbingly sep_or_l. Qed.
592
593
Lemma absorbingly_and P Q :
  bi_absorbingly (P  Q)  bi_absorbingly P  bi_absorbingly Q.
594
Proof. apply and_intro; apply absorbingly_mono; auto. Qed.
595
596
Lemma absorbingly_forall {A} (Φ : A  PROP) :
  bi_absorbingly ( a, Φ a)   a, bi_absorbingly (Φ a).
Robbert Krebbers's avatar
Robbert Krebbers committed
597
Proof. apply forall_intro=> a. by rewrite (forall_elim a). Qed.
598
599
Lemma absorbingly_exist {A} (Φ : A  PROP) :
  bi_absorbingly ( a, Φ a)   a, bi_absorbingly (Φ a).
600
Proof. by rewrite /bi_absorbingly sep_exist_l. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
601

602
Lemma absorbingly_sep P Q : bi_absorbingly (P  Q)  bi_absorbingly P  bi_absorbingly Q.
603
Proof. by rewrite -{1}absorbingly_idemp /bi_absorbingly !assoc -!(comm _ P) !assoc. Qed.
604
Lemma absorbingly_True_emp : bi_absorbingly True  bi_absorbingly emp.
605
Proof. by rewrite absorbingly_pure /bi_absorbingly right_id. Qed.
606
Lemma absorbingly_wand P Q : bi_absorbingly (P - Q)  bi_absorbingly P - bi_absorbingly Q.
607
Proof. apply wand_intro_l. by rewrite -absorbingly_sep wand_elim_r. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
608

609
Lemma absorbingly_sep_l P Q : bi_absorbingly P  Q  bi_absorbingly (P  Q).
610
Proof. by rewrite /bi_absorbingly assoc. Qed.
611
Lemma absorbingly_sep_r P Q : P  bi_absorbingly Q  bi_absorbingly (P  Q).
612
Proof. by rewrite /bi_absorbingly !assoc (comm _ P). Qed.
613
Lemma absorbingly_sep_lr P Q : bi_absorbingly P  Q  P  bi_absorbingly Q.
614
Proof. by rewrite absorbingly_sep_l absorbingly_sep_r. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
615

616
Lemma affinely_absorbingly `{!BiPositive PROP} P :
617
  bi_affinely (bi_absorbingly P)  bi_affinely P.
Robbert Krebbers's avatar
Robbert Krebbers committed
618
Proof.
619
620
  apply (anti_symm _), affinely_mono, absorbingly_intro.
  by rewrite /bi_absorbingly affinely_sep affinely_True_emp affinely_emp left_id.
Robbert Krebbers's avatar
Robbert Krebbers committed
621
622
Qed.

623
(* Affine and absorbing propositions *)
Robbert Krebbers's avatar
Robbert Krebbers committed
624
Global Instance Affine_proper : Proper (() ==> iff) (@Affine PROP).
Robbert Krebbers's avatar
Robbert Krebbers committed
625
Proof. solve_proper. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
626
627
Global Instance Absorbing_proper : Proper (() ==> iff) (@Absorbing PROP).
Proof. solve_proper. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
628

629
630
Lemma affine_affinely P `{!Affine P} : bi_affinely P  P.
Proof. rewrite /bi_affinely. apply (anti_symm _); auto. Qed.
631
Lemma absorbing_absorbingly P `{!Absorbing P} : bi_absorbingly P  P.
632
Proof. by apply (anti_symm _), absorbingly_intro. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
633

Robbert Krebbers's avatar
Robbert Krebbers committed
634
635
636
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
637
638
Proof.
  intros. rewrite /Absorbing -{2}(left_id emp%I _ P).
639
  by rewrite -(absorbing emp) absorbingly_sep_l left_id.
Robbert Krebbers's avatar
Robbert Krebbers committed
640
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
641
642

Lemma sep_elim_l P Q `{H : TCOr (Affine Q) (Absorbing P)} : P  Q  P.
Robbert Krebbers's avatar
Robbert Krebbers committed
643
644
645
646
647
Proof.
  destruct H.
  - by rewrite (affine Q) right_id.
  - by rewrite (True_intro Q) comm.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
648
649
650
Lemma sep_elim_r P Q `{H : TCOr (Affine P) (Absorbing Q)} : P  Q  Q.
Proof. by rewrite comm sep_elim_l. Qed.

651
652
Lemma sep_and P Q
    `{HPQ : TCOr (TCAnd (Affine P) (Affine Q)) (TCAnd (Absorbing P) (Absorbing Q))} :
Robbert Krebbers's avatar
Robbert Krebbers committed
653
  P  Q  P  Q.
654
655
656
657
Proof.
  destruct HPQ as [[??]|[??]];
    apply and_intro; apply: sep_elim_l || apply: sep_elim_r.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
658

659
660
Lemma affinely_intro P Q `{!Affine P} : (P  Q)  P  bi_affinely Q.
Proof. intros <-. by rewrite affine_affinely. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
661
662
663
664
665
666
667
668
669
670
671

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
672
Proof. apply (anti_symm _); auto using True_sep_2. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
673
Lemma sep_True P `{!Absorbing P} : P  True  P.
Robbert Krebbers's avatar
Robbert Krebbers committed
674
Proof. by rewrite comm True_sep. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
675

676
677
Section bi_affine.
  Context `{BiAffine PROP}.
Robbert Krebbers's avatar
Robbert Krebbers committed
678

679
  Global Instance bi_affine_absorbing P : Absorbing P | 0.
680
  Proof. by rewrite /Absorbing /bi_absorbingly (affine True%I) left_id. Qed.
681
  Global Instance bi_affine_positive : BiPositive PROP.
682
  Proof. intros P Q. by rewrite !affine_affinely. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706

  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.
707
End bi_affine.
Robbert Krebbers's avatar
Robbert Krebbers committed
708

709
(* Properties of the persistence modality *)
Robbert Krebbers's avatar
Robbert Krebbers committed
710
711
712
713
714
715
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
716

717
718
Lemma absorbingly_persistently P :
  bi_absorbingly (bi_persistently P)  bi_persistently P.
Robbert Krebbers's avatar
Robbert Krebbers committed
719
Proof.
720
721
  apply (anti_symm _), absorbingly_intro.
  by rewrite /bi_absorbingly comm persistently_absorbing.
Robbert Krebbers's avatar
Robbert Krebbers committed
722
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
723

724
725
Lemma persistently_and_sep_assoc P Q R :
  bi_persistently P  (Q  R)  (bi_persistently P  Q)  R.
Robbert Krebbers's avatar
Robbert Krebbers committed
726
Proof.
727
728
729
  apply (anti_symm ()).
  - rewrite {1}persistently_idemp_2 persistently_and_sep_elim assoc.
    apply sep_mono_l, and_intro.
730
    + by rewrite and_elim_r persistently_absorbing.
731
732
    + by rewrite and_elim_l left_id.
  - apply and_intro.
733
    + by rewrite and_elim_l persistently_absorbing.
734
    + by rewrite and_elim_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
735
Qed.
736
Lemma persistently_and_emp_elim P : emp  bi_persistently P  P.
Robbert Krebbers's avatar
Robbert Krebbers committed
737
Proof. by rewrite comm persistently_and_sep_elim right_id and_elim_r. Qed.
738
Lemma persistently_elim_absorbingly P : bi_persistently P  bi_absorbingly P.
Robbert Krebbers's avatar
Robbert Krebbers committed
739
Proof.
740
  rewrite -(right_id True%I _ (bi_persistently _)%I) -{1}(left_id emp%I _ True%I).
Robbert Krebbers's avatar
Robbert Krebbers committed
741
  by rewrite persistently_and_sep_assoc (comm bi_and) persistently_and_emp_elim comm.
Robbert Krebbers's avatar
Robbert Krebbers committed
742
Qed.
743
Lemma persistently_elim P `{!Absorbing P} : bi_persistently P  P.
744
Proof. by rewrite persistently_elim_absorbingly absorbing_absorbingly. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
745

746
747
Lemma persistently_idemp_1 P :
  bi_persistently (bi_persistently P)  bi_persistently P.
748
Proof. by rewrite persistently_elim_absorbingly absorbingly_persistently. Qed.
749
750
Lemma persistently_idemp P :
  bi_persistently (bi_persistently P)  bi_persistently P.
751
Proof. apply (anti_symm _); auto using persistently_idemp_1, persistently_idemp_2. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
752

753
754
Lemma persistently_intro' P Q :
  (bi_persistently P  Q)  bi_persistently P  bi_persistently Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
755
756
Proof. intros <-. apply persistently_idemp_2. Qed.

757
Lemma persistently_pure φ : bi_persistently ⌜φ⌝  ⌜φ⌝.
Robbert Krebbers's avatar
Robbert Krebbers committed
758
Proof.
759
760
  apply (anti_symm _).
  { by rewrite persistently_elim_absorbingly absorbingly_pure. }
761
  apply pure_elim'=> Hφ.
762
  trans ( x : False, bi_persistently True : PROP)%I; [by apply forall_intro|].
763
  rewrite persistently_forall_2. auto using persistently_mono, pure_intro.
Robbert Krebbers's avatar
Robbert Krebbers committed
764
Qed.
765
766
Lemma persistently_forall {A} (Ψ : A  PROP) :
  bi_persistently ( a, Ψ a)   a, bi_persistently (Ψ a).
Robbert Krebbers's avatar
Robbert Krebbers committed
767
768
769
770
Proof.
  apply (anti_symm _); auto using persistently_forall_2.
  apply forall_intro=> x. by rewrite (forall_elim x).
Qed.
771
772
Lemma persistently_exist {A} (Ψ : A  PROP) :
  bi_persistently ( a, Ψ a)   a, bi_persistently (Ψ a).
Robbert Krebbers's avatar
Robbert Krebbers committed
773
774
775
776
Proof.
  apply (anti_symm _); auto using persistently_exist_1.
  apply exist_elim=> x. by rewrite (exist_intro x).
Qed.
777
778
Lemma persistently_and P Q :
  bi_persistently (P  Q)  bi_persistently P  bi_persistently Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
779
Proof. rewrite !and_alt persistently_forall. by apply forall_proper=> -[]. Qed.
780
781
Lemma persistently_or P Q :
  bi_persistently (P  Q)  bi_persistently P  bi_persistently Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
782
Proof. rewrite !or_alt persistently_exist. by apply exist_proper=> -[]. Qed.
783
784
Lemma persistently_impl P Q :
  bi_persistently (P  Q)  bi_persistently P  bi_persistently Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
785
786
787
788
789
Proof.
  apply impl_intro_l; rewrite -persistently_and.
  apply persistently_mono, impl_elim with P; auto.
Qed.

790
791
Lemma persistently_sep_dup P :
  bi_persistently P  bi_persistently P  bi_persistently P.
Robbert Krebbers's avatar
Robbert Krebbers committed
792
Proof.
793
794
795
796
797
  apply (anti_symm _).
  - rewrite -{1}(idemp bi_and (bi_persistently _)).
    by rewrite -{2}(left_id emp%I _ (bi_persistently _))
      persistently_and_sep_assoc and_elim_l.
  - by rewrite persistently_absorbing.
Robbert Krebbers's avatar
Robbert Krebbers committed
798
799
Qed.

800
Lemma persistently_and_sep_l_1 P Q : bi_persistently P  Q  bi_persistently P  Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
801
802
803
Proof.
  by rewrite -{1}(left_id emp%I _ Q%I) persistently_and_sep_assoc and_elim_l.
Qed.
804
Lemma persistently_and_sep_r_1 P Q : P  bi_persistently Q  P  bi_persistently Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
805
806
Proof. by rewrite !(comm _ P) persistently_and_sep_l_1. Qed.

807
808
809
Lemma persistently_emp_intro P : P  bi_persistently emp.
Proof. by rewrite -plainly_elim_persistently -plainly_emp_intro. Qed.

810
Lemma persistently_True_emp : bi_persistently True  bi_persistently emp.
Robbert Krebbers's avatar
Robbert Krebbers committed
811
Proof. apply (anti_symm _); auto using persistently_emp_intro. Qed.
812
Lemma persistently_and_sep P Q : bi_persistently (P  Q)  bi_persistently (P  Q).
Robbert Krebbers's avatar
Robbert Krebbers committed
813
Proof.
814
815
816
817
818
  rewrite persistently_and.
  rewrite -{1}persistently_idemp -persistently_and -{1}(left_id emp%I _ Q%I).
  by rewrite persistently_and_sep_assoc (comm bi_and) persistently_and_emp_elim.
Qed.

819
Lemma persistently_affinely P : bi_persistently (bi_affinely P)  bi_persistently P.
820
Proof.
821
  by rewrite /bi_affinely persistently_and -persistently_True_emp
822
             persistently_pure left_id.
Robbert Krebbers's avatar
Robbert Krebbers committed
823
824
Qed.

825
826
Lemma and_sep_persistently P Q :
  bi_persistently P  bi_persistently Q  bi_persistently P  bi_persistently Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
827
Proof.
828
829
830
831
  apply (anti_symm _); auto using persistently_and_sep_l_1.
  apply and_intro.
  - by rewrite persistently_absorbing.
  - by rewrite comm persistently_absorbing.
Robbert Krebbers's avatar
Robbert Krebbers committed
832
Qed.
833
834
Lemma persistently_sep_2 P Q :
  bi_persistently P  bi_persistently Q  bi_persistently (P  Q).
Robbert Krebbers's avatar
Robbert Krebbers committed
835
Proof. by rewrite -persistently_and_sep persistently_and -and_sep_persistently. Qed.
836
Lemma persistently_sep `{BiPositive PROP} P Q :
837
  bi_persistently (P  Q)  bi_persistently P  bi_persistently Q.
838
839
Proof.
  apply (anti_symm _); auto using persistently_sep_2.
840
841
842
  rewrite -persistently_affinely affinely_sep -and_sep_persistently. apply and_intro.
  - by rewrite (affinely_elim_emp Q) right_id affinely_elim.
  - by rewrite (affinely_elim_emp P) left_id affinely_elim.
843
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
844

845
846
Lemma persistently_wand P Q :
  bi_persistently (P - Q)  bi_persistently P - bi_persistently Q.
847
Proof. apply wand_intro_r. by rewrite persistently_sep_2 wand_elim_l. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
848

849
850
Lemma persistently_entails_l P Q :
  (P  bi_persistently Q)  P  bi_persistently Q  P.
Robbert Krebbers's avatar
Robbert Krebbers committed
851
Proof. intros; rewrite -persistently_and_sep_l_1; auto. Qed.
852
853
Lemma persistently_entails_r P Q :
  (P  bi_persistently Q)  P  P  bi_persistently Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
854
855
Proof. intros; rewrite -persistently_and_sep_r_1; auto. Qed.

856
857
Lemma persistently_impl_wand_2 P Q :
  bi_persistently (P - Q)  bi_persistently (P  Q).
Robbert Krebbers's avatar
Robbert Krebbers committed
858
859
860
861
862
863
Proof.
  apply persistently_intro', impl_intro_r.
  rewrite -{2}(left_id emp%I _ P%I) persistently_and_sep_assoc.
  by rewrite (comm bi_and) persistently_and_emp_elim wand_elim_l.
Qed.

864
Lemma impl_wand_persistently_2 P Q : (bi_persistently P - Q)  (bi_persistently P  Q).
865
866
Proof. apply impl_intro_l. by rewrite persistently_and_sep_l_1 wand_elim_r. Qed.

867
Section persistently_affinely_bi.
868
  Context `{BiAffine PROP}.
Robbert Krebbers's avatar
Robbert Krebbers committed
869

870
  Lemma persistently_emp : bi_persistently emp  emp.
Robbert Krebbers's avatar
Robbert Krebbers committed
871
872
  Proof. by rewrite -!True_emp persistently_pure. Qed.

873
874
  Lemma persistently_and_sep_l P Q :
    bi_persistently P  Q  bi_persistently P  Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
875
876
877
878
  Proof.
    apply (anti_symm ());
      eauto using persistently_and_sep_l_1, sep_and with typeclass_instances.
  Qed.
879
  Lemma persistently_and_sep_r P Q : P  bi_persistently Q  P  bi_persistently Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
880
881
  Proof. by rewrite !(comm _ P) persistently_and_sep_l. Qed.

882
883
  Lemma persistently_impl_wand P Q :
    bi_persistently (P  Q)  bi_persistently (P - Q).
Robbert Krebbers's avatar
Robbert Krebbers committed
884
885
886
  Proof.
    apply (anti_symm ()); auto using persistently_impl_wand_2.
    apply persistently_intro', wand_intro_l.
887
    by rewrite -persistently_and_sep_r persistently_elim impl_elim_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
888
889
  Qed.

890
  Lemma impl_wand_persistently P Q : (bi_persistently P  Q)  (bi_persistently P - Q).
891
  Proof.
892
    apply (anti_symm ()). by rewrite -impl_wand_1. apply impl_wand_persistently_2.
893
894
  Qed.

895
  Lemma wand_alt P Q : (P - Q)   R, R  bi_persistently (P  R  Q).
Robbert Krebbers's avatar
Robbert Krebbers committed
896
897
898
  Proof.
    apply (anti_symm ()).
    - rewrite -(right_id True%I bi_sep (P - Q)%I) -(exist_intro (P - Q)%I).
899
900
      apply sep_mono_r. rewrite -persistently_pure.
      apply persistently_intro', impl_intro_l.
Robbert Krebbers's avatar
Robbert Krebbers committed
901
      by rewrite wand_elim_r persistently_pure right_id.
902
903
    - apply exist_elim=> R. apply wand_intro_l.
      rewrite assoc -persistently_and_sep_r.
904
      by rewrite persistently_elim impl_elim_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
905
  Qed.
906
  Lemma impl_alt P Q : (P  Q)   R, R  bi_persistently (P  R - Q).
Robbert Krebbers's avatar
Robbert Krebbers committed
907
908
909
  Proof.
    apply (anti_symm ()).
    - rewrite -(right_id True%I bi_and (P  Q)%I) -(exist_intro (P  Q)%I).
910
911
      apply and_mono_r. rewrite -persistently_pure.
      apply persistently_intro', wand_intro_l.
Robbert Krebbers's avatar
Robbert Krebbers committed
912
      by rewrite impl_elim_r persistently_pure right_id.
913
914
    - apply exist_elim=> R. apply impl_intro_l.
      by rewrite assoc persistently_and_sep_r persistently_elim wand_elim_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
915
  Qed.
916
End persistently_affinely_bi.
Robbert Krebbers's avatar
Robbert Krebbers committed
917

918
919
920
921
922
923
924
925
(* Properties of the plainness modality *)
Hint Resolve plainly_mono.
Global Instance plainly_mono' : Proper (() ==> ()) (@bi_plainly PROP).
Proof. intros P Q; apply plainly_mono. Qed.
Global Instance plainly_flip_mono' :
  Proper (flip () ==> flip ()) (@bi_plainly PROP).
Proof. intros P Q; apply plainly_mono. Qed.

926
Lemma persistently_plainly P : bi_persistently (bi_plainly P)  bi_plainly P.
927
Proof.
928
929
930
  apply (anti_symm _).
  - by rewrite persistently_elim_absorbingly /bi_absorbingly comm plainly_absorbing.
  - by rewrite {1}plainly_idemp_2 plainly_elim_persistently.
931
Qed.
932
933
934
935
936
937
938
939
Lemma plainly_persistently P : bi_plainly (bi_persistently P)  bi_plainly P.
Proof.
  apply (anti_symm _); first apply plainly_persistently_1.
  by rewrite {1}plainly_idemp_2 (plainly_elim_persistently P).
Qed.

Lemma absorbingly_plainly P : bi_absorbingly (bi_plainly P)  bi_plainly P.
Proof. by rewrite -(persistently_plainly P) absorbingly_persistently. Qed.
940
941
942
943
944

Lemma plainly_and_sep_elim P Q : bi_plainly P  Q - (emp  P)  Q.
Proof. by rewrite plainly_elim_persistently persistently_and_sep_elim. Qed.
Lemma plainly_and_sep_assoc P Q R :
  bi_plainly P  (Q  R)  (bi_plainly P  Q)  R.
945
Proof. by rewrite -(persistently_plainly P) persistently_and_sep_assoc. Qed.
946
947
Lemma plainly_and_emp_elim P : emp  bi_plainly P  P.
Proof. by rewrite plainly_elim_persistently persistently_and_emp_elim. Qed.
948
Lemma plainly_elim_absorbingly P : bi_plainly P  bi_absorbingly P.
949
950
951
952
953
Proof. by rewrite plainly_elim_persistently persistently_elim_absorbingly. Qed.
Lemma plainly_elim P `{!Absorbing P} : bi_plainly P  P.
Proof. by rewrite plainly_elim_persistently persistently_elim. Qed.

Lemma plainly_idemp_1 P : bi_plainly (bi_plainly P)  bi_plainly P.
954
Proof. by rewrite plainly_elim_absorbingly absorbingly_plainly. Qed.
955
956
957
958
959
960
961
962
963
Lemma plainly_idemp P : bi_plainly (bi_plainly P)  bi_plainly P.
Proof. apply (anti_symm _); auto using plainly_idemp_1, plainly_idemp_2. Qed.

Lemma plainly_intro' P Q : (bi_plainly P  Q)  bi_plainly P  bi_plainly Q.
Proof. intros <-. apply plainly_idemp_2. Qed.

Lemma plainly_pure φ : bi_plainly ⌜φ⌝  ⌜φ⌝.
Proof.
  apply (anti_symm _); auto.
964
  - by rewrite plainly_elim_persistently persistently_pure.
965
966
967
968
969
970
971