derived.v 39.8 KB
Newer Older
1
From iris.base_logic Require Export primitive.
2
Set Default Proof Using "Type".
3
Import upred.uPred primitive.uPred.
4
5
6
7
8

Definition uPred_iff {M} (P Q : uPred M) : uPred M := ((P  Q)  (Q  P))%I.
Instance: Params (@uPred_iff) 1.
Infix "↔" := uPred_iff : uPred_scope.

Robbert Krebbers's avatar
Robbert Krebbers committed
9
10
11
12
13
14
15
16
17
18
Definition uPred_laterN {M} (n : nat) (P : uPred M) : uPred M :=
  Nat.iter n uPred_later P.
Instance: Params (@uPred_laterN) 2.
Notation "▷^ n P" := (uPred_laterN n P)
  (at level 20, n at level 9, P at level 20,
   format "▷^ n  P") : uPred_scope.
Notation "▷? p P" := (uPred_laterN (Nat.b2n p) P)
  (at level 20, p at level 9, P at level 20,
   format "▷? p  P") : uPred_scope.

19
20
21
22
23
Definition uPred_always_if {M} (p : bool) (P : uPred M) : uPred M :=
  (if p then  P else P)%I.
Instance: Params (@uPred_always_if) 2.
Arguments uPred_always_if _ !_ _/.
Notation "□? p P" := (uPred_always_if p P)
Robbert Krebbers's avatar
Robbert Krebbers committed
24
  (at level 20, p at level 9, P at level 20, format "□? p  P").
25

26
27
Definition uPred_except_0 {M} (P : uPred M) : uPred M :=  False  P.
Notation "◇ P" := (uPred_except_0 P)
28
  (at level 20, right associativity) : uPred_scope.
29
30
Instance: Params (@uPred_except_0) 1.
Typeclasses Opaque uPred_except_0.
31
32
33
34
35

Class TimelessP {M} (P : uPred M) := timelessP :  P   P.
Arguments timelessP {_} _ {_}.

Class PersistentP {M} (P : uPred M) := persistentP : P   P.
36
Hint Mode PersistentP - ! : typeclass_instances.
37
38
Arguments persistentP {_} _ {_}.

39
Module uPred.
40
41
42
43
44
45
46
47
48
49
Section derived.
Context {M : ucmraT}.
Implicit Types φ : Prop.
Implicit Types P Q : uPred M.
Implicit Types A : Type.
Notation "P ⊢ Q" := (@uPred_entails M P%I Q%I). (* Force implicit argument M *)
Notation "P ⊣⊢ Q" := (equiv (A:=uPred M) P%I Q%I). (* Force implicit argument M *)

(* Derived logical stuff *)
Lemma False_elim P : False  P.
50
Proof. by apply (pure_elim' False). Qed.
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
Lemma True_intro P : P  True.
Proof. by apply pure_intro. Qed.

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  uPred M) a : (P  Ψ a)  P   a, Ψ a.
Proof. intros ->; apply exist_intro. Qed.
Lemma forall_elim' {A} P (Ψ : A  uPred M) : (P   a, Ψ a)   a, P  Ψ a.
Proof. move=> HP a. by rewrite HP forall_elim. Qed.

Hint Resolve pure_intro.
Hint Resolve or_elim or_intro_l' or_intro_r'.
Hint Resolve and_intro and_elim_l' and_elim_r'.
Hint Immediate True_intro False_elim.

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_l P Q : (P  Q)  P  Q.
Proof. apply impl_elim with P; auto. Qed.
Lemma impl_elim_r P Q : P  (P  Q)  Q.
Proof. apply impl_elim with P; auto. Qed.
Lemma impl_elim_l' P Q R : (P  Q  R)  P  Q  R.
Proof. intros; apply impl_elim with Q; auto. Qed.
Lemma impl_elim_r' P Q R : (Q  P  R)  P  Q  R.
Proof. intros; apply impl_elim with P; auto. Qed.
82
Lemma impl_entails P Q : (P  Q)%I  P  Q.
83
Proof. intros HPQ; apply impl_elim with P; rewrite -?HPQ; auto. Qed.
84
85
Lemma entails_impl P Q : (P  Q)  (P  Q)%I.
Proof. intro. apply impl_intro_l. auto. Qed.
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

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  uPred M) :
  ( 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  uPred M) :
  ( 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 (() ==> () ==> ()) (@uPred_and M).
Proof. by intros P P' HP Q Q' HQ; apply and_mono. Qed.
Global Instance and_flip_mono' :
  Proper (flip () ==> flip () ==> flip ()) (@uPred_and M).
Proof. by intros P P' HP Q Q' HQ; apply and_mono. Qed.
Global Instance or_mono' : Proper (() ==> () ==> ()) (@uPred_or M).
Proof. by intros P P' HP Q Q' HQ; apply or_mono. Qed.
Global Instance or_flip_mono' :
  Proper (flip () ==> flip () ==> flip ()) (@uPred_or M).
Proof. by intros P P' HP Q Q' HQ; apply or_mono. Qed.
Global Instance impl_mono' :
  Proper (flip () ==> () ==> ()) (@uPred_impl M).
Proof. by intros P P' HP Q Q' HQ; apply impl_mono. Qed.
128
129
130
Global Instance impl_flip_mono' :
  Proper (() ==> flip () ==> flip ()) (@uPred_impl M).
Proof. by intros P P' HP Q Q' HQ; apply impl_mono. Qed.
131
132
133
Global Instance forall_mono' A :
  Proper (pointwise_relation _ () ==> ()) (@uPred_forall M A).
Proof. intros P1 P2; apply forall_mono. Qed.
134
135
136
Global Instance forall_flip_mono' A :
  Proper (pointwise_relation _ (flip ()) ==> flip ()) (@uPred_forall M A).
Proof. intros P1 P2; apply forall_mono. Qed.
137
Global Instance exist_mono' A :
138
139
140
141
  Proper (pointwise_relation _ (flip ()) ==> flip ()) (@uPred_exist M A).
Proof. intros P1 P2; apply exist_mono. Qed.
Global Instance exist_flip_mono' A :
  Proper (pointwise_relation _ (flip ()) ==> flip ()) (@uPred_exist M A).
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
Proof. intros P1 P2; apply exist_mono. Qed.

Global Instance and_idem : IdemP () (@uPred_and M).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance or_idem : IdemP () (@uPred_or M).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance and_comm : Comm () (@uPred_and M).
Proof. intros P Q; apply (anti_symm ()); auto. Qed.
Global Instance True_and : LeftId () True%I (@uPred_and M).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance and_True : RightId () True%I (@uPred_and M).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance False_and : LeftAbsorb () False%I (@uPred_and M).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance and_False : RightAbsorb () False%I (@uPred_and M).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance True_or : LeftAbsorb () True%I (@uPred_or M).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance or_True : RightAbsorb () True%I (@uPred_or M).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance False_or : LeftId () False%I (@uPred_or M).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance or_False : RightId () False%I (@uPred_or M).
Proof. intros P; apply (anti_symm ()); auto. Qed.
Global Instance and_assoc : Assoc () (@uPred_and M).
Proof. intros P Q R; apply (anti_symm ()); auto. Qed.
Global Instance or_comm : Comm () (@uPred_or M).
Proof. intros P Q; apply (anti_symm ()); auto. Qed.
Global Instance or_assoc : Assoc () (@uPred_or M).
Proof. intros P Q R; apply (anti_symm ()); auto. Qed.
Global Instance True_impl : LeftId () True%I (@uPred_impl M).
Proof.
  intros P; apply (anti_symm ()).
  - by rewrite -(left_id True%I uPred_and (_  _)%I) impl_elim_r.
  - by apply impl_intro_l; rewrite left_id.
Qed.
178
179
180
181
182
Lemma False_impl P : (False  P)  True.
Proof.
  apply (anti_symm ()); [by auto|].
  apply impl_intro_l. rewrite left_absorb. auto.
Qed.
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

Lemma exists_impl_forall {A} P (Ψ : A  uPred M) :
  (( 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  uPred M) : 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  uPred M) : ( a, Φ a)  P   a, Φ a  P.
Proof.
  rewrite -(comm _ P) and_exist_l. apply exist_proper=>a. by rewrite comm.
Qed.
219
220
221
222
223
224
225
Lemma or_exist {A} (Φ Ψ : A  uPred M) :
  ( 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.
226

227
Lemma pure_elim φ Q R : (Q  ⌜φ⌝)  (φ  Q  R)  Q  R.
228
229
230
231
Proof.
  intros HQ HQR. rewrite -(idemp uPred_and Q) {1}HQ.
  apply impl_elim_l', pure_elim'=> ?. by apply entails_impl, HQR.
Qed.
Ralf Jung's avatar
Ralf Jung committed
232
Lemma pure_mono φ1 φ2 : (φ1  φ2)  ⌜φ1  ⌜φ2.
233
234
235
Proof. intros; apply pure_elim with φ1; eauto. Qed.
Global Instance pure_mono' : Proper (impl ==> ()) (@uPred_pure M).
Proof. intros φ1 φ2; apply pure_mono. Qed.
Ralf Jung's avatar
Ralf Jung committed
236
Lemma pure_iff φ1 φ2 : (φ1  φ2)  ⌜φ1  ⌜φ2.
237
Proof. intros [??]; apply (anti_symm _); auto using pure_mono. Qed.
Ralf Jung's avatar
Ralf Jung committed
238
Lemma pure_intro_l φ Q R : φ  (⌜φ⌝  Q  R)  Q  R.
239
Proof. intros ? <-; auto using pure_intro. Qed.
Ralf Jung's avatar
Ralf Jung committed
240
Lemma pure_intro_r φ Q R : φ  (Q  ⌜φ⌝  R)  Q  R.
241
Proof. intros ? <-; auto. Qed.
Ralf Jung's avatar
Ralf Jung committed
242
Lemma pure_intro_impl φ Q R : φ  (Q  ⌜φ⌝  R)  Q  R.
243
Proof. intros ? ->. eauto using pure_intro_l, impl_elim_r. Qed.
Ralf Jung's avatar
Ralf Jung committed
244
Lemma pure_elim_l φ Q R : (φ  Q  R)  ⌜φ⌝  Q  R.
245
Proof. intros; apply pure_elim with φ; eauto. Qed.
Ralf Jung's avatar
Ralf Jung committed
246
Lemma pure_elim_r φ Q R : (φ  Q  R)  Q  ⌜φ⌝  R.
247
Proof. intros; apply pure_elim with φ; eauto. Qed.
248

Ralf Jung's avatar
Ralf Jung committed
249
Lemma pure_True (φ : Prop) : φ  ⌜φ⌝  True.
250
Proof. intros; apply (anti_symm _); auto. Qed.
Ralf Jung's avatar
Ralf Jung committed
251
Lemma pure_False (φ : Prop) : ¬φ  ⌜φ⌝  False.
252
Proof. intros; apply (anti_symm _); eauto using pure_elim. Qed.
253

Ralf Jung's avatar
Ralf Jung committed
254
Lemma pure_and φ1 φ2 : ⌜φ1  φ2  ⌜φ1  ⌜φ2.
255
256
257
258
259
Proof.
  apply (anti_symm _).
  - eapply pure_elim=> // -[??]; auto.
  - eapply (pure_elim φ1); [auto|]=> ?. eapply (pure_elim φ2); auto.
Qed.
Ralf Jung's avatar
Ralf Jung committed
260
Lemma pure_or φ1 φ2 : ⌜φ1  φ2  ⌜φ1  ⌜φ2.
261
262
263
264
265
Proof.
  apply (anti_symm _).
  - eapply pure_elim=> // -[?|?]; auto.
  - apply or_elim; eapply pure_elim; eauto.
Qed.
Ralf Jung's avatar
Ralf Jung committed
266
Lemma pure_impl φ1 φ2 : ⌜φ1  φ2  (⌜φ1  ⌜φ2).
267
268
269
270
Proof.
  apply (anti_symm _).
  - apply impl_intro_l. rewrite -pure_and. apply pure_mono. naive_solver.
  - rewrite -pure_forall_2. apply forall_intro=> ?.
271
    by rewrite -(left_id True uPred_and (_→_))%I (pure_True φ1) // impl_elim_r.
272
Qed.
Ralf Jung's avatar
Ralf Jung committed
273
Lemma pure_forall {A} (φ : A  Prop) :  x, φ x   x, ⌜φ x.
274
275
276
277
Proof.
  apply (anti_symm _); auto using pure_forall_2.
  apply forall_intro=> x. eauto using pure_mono.
Qed.
Ralf Jung's avatar
Ralf Jung committed
278
Lemma pure_exist {A} (φ : A  Prop) :  x, φ x   x, ⌜φ x.
279
280
281
282
283
284
Proof.
  apply (anti_symm _).
  - eapply pure_elim=> // -[x ?]. rewrite -(exist_intro x); auto.
  - apply exist_elim=> x. eauto using pure_mono.
Qed.

285
Lemma internal_eq_refl' {A : ofeT} (a : A) P : P  a  a.
286
287
Proof. rewrite (True_intro P). apply internal_eq_refl. Qed.
Hint Resolve internal_eq_refl'.
288
Lemma equiv_internal_eq {A : ofeT} P (a b : A) : a  b  P  a  b.
289
Proof. by intros ->. Qed.
290
Lemma internal_eq_sym {A : ofeT} (a b : A) : a  b  b  a.
291
Proof. apply (internal_eq_rewrite a b (λ b, b  a)%I); auto. solve_proper. Qed.
292
293
294
Lemma internal_eq_rewrite_contractive {A : ofeT} a b (Ψ : A  uPred M) P
  {HΨ : Contractive Ψ} : (P   (a  b))  (P  Ψ a)  P  Ψ b.
Proof.
295
296
  move: HΨ=> /contractiveI HΨ Heq ?.
  apply (internal_eq_rewrite (Ψ a) (Ψ b) id _)=>//=. by rewrite -HΨ.
297
Qed.
298

Ralf Jung's avatar
Ralf Jung committed
299
Lemma pure_impl_forall φ P : (⌜φ⌝  P)  ( _ : φ, P).
300
301
Proof.
  apply (anti_symm _).
302
  - apply forall_intro=> ?. by rewrite pure_True // left_id.
303
304
  - apply impl_intro_l, pure_elim_l=> Hφ. by rewrite (forall_elim Hφ).
Qed.
Ralf Jung's avatar
Ralf Jung committed
305
Lemma pure_alt φ : ⌜φ⌝   _ : φ, True.
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
Proof.
  apply (anti_symm _).
  - eapply pure_elim; eauto=> H. rewrite -(exist_intro H); auto.
  - by apply exist_elim, pure_intro.
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.
  apply and_intro. by rewrite (forall_elim true). by 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.
  apply or_elim. by rewrite -(exist_intro true). by rewrite -(exist_intro false).
Qed.

Global Instance iff_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_iff M).
Proof. unfold uPred_iff; solve_proper. Qed.
Global Instance iff_proper :
  Proper (() ==> () ==> ()) (@uPred_iff M) := ne_proper_2 _.

Lemma iff_refl Q P : Q  P  P.
Proof. rewrite /uPred_iff; apply and_intro; apply impl_intro_l; auto. Qed.
329
Lemma iff_equiv P Q : (P  Q)%I  (P  Q).
330
331
Proof.
  intros HPQ; apply (anti_symm ());
332
    apply impl_entails; rewrite /uPred_valid HPQ /uPred_iff; auto.
333
Qed.
334
Lemma equiv_iff P Q : (P  Q)  (P  Q)%I.
335
Proof. intros ->; apply iff_refl. Qed.
336
Lemma internal_eq_iff P Q : P  Q  P  Q.
337
Proof.
338
339
  apply (internal_eq_rewrite P Q (λ Q, P  Q))%I;
    first solve_proper; auto using iff_refl.
340
341
342
343
Qed.

(* Derived BI Stuff *)
Hint Resolve sep_mono.
344
Lemma sep_mono_l P P' Q : (P  Q)  P  P'  Q  P'.
345
Proof. by intros; apply sep_mono. Qed.
346
Lemma sep_mono_r P P' Q' : (P'  Q')  P  P'  P  Q'.
347
348
349
350
351
352
Proof. by apply sep_mono. Qed.
Global Instance sep_mono' : Proper (() ==> () ==> ()) (@uPred_sep M).
Proof. by intros P P' HP Q Q' HQ; apply sep_mono. Qed.
Global Instance sep_flip_mono' :
  Proper (flip () ==> flip () ==> flip ()) (@uPred_sep M).
Proof. by intros P P' HP Q Q' HQ; apply sep_mono. Qed.
353
Lemma wand_mono P P' Q Q' : (Q  P)  (P'  Q')  (P - P')  Q - Q'.
354
355
356
357
358
Proof.
  intros HP HQ; apply wand_intro_r. rewrite HP -HQ. by apply wand_elim_l'.
Qed.
Global Instance wand_mono' : Proper (flip () ==> () ==> ()) (@uPred_wand M).
Proof. by intros P P' HP Q Q' HQ; apply wand_mono. Qed.
359
360
361
Global Instance wand_flip_mono' :
  Proper (() ==> flip () ==> flip ()) (@uPred_wand M).
Proof. by intros P P' HP Q Q' HQ; apply wand_mono. Qed.
362
363
364
365
366
367
368
369
370
371
372
373

Global Instance sep_comm : Comm () (@uPred_sep M).
Proof. intros P Q; apply (anti_symm _); auto using sep_comm'. Qed.
Global Instance sep_assoc : Assoc () (@uPred_sep M).
Proof.
  intros P Q R; apply (anti_symm _); auto using sep_assoc'.
  by rewrite !(comm _ P) !(comm _ _ R) sep_assoc'.
Qed.
Global Instance True_sep : LeftId () True%I (@uPred_sep M).
Proof. intros P; apply (anti_symm _); auto using True_sep_1, True_sep_2. Qed.
Global Instance sep_True : RightId () True%I (@uPred_sep M).
Proof. by intros P; rewrite comm left_id. Qed.
374
Lemma sep_elim_l P Q : P  Q  P.
375
Proof. by rewrite (True_intro Q) right_id. Qed.
376
377
378
Lemma sep_elim_r P Q : P  Q  Q.
Proof. by rewrite (comm ())%I; apply sep_elim_l. Qed.
Lemma sep_elim_l' P Q R : (P  R)  P  Q  R.
379
Proof. intros ->; apply sep_elim_l. Qed.
380
Lemma sep_elim_r' P Q R : (Q  R)  P  Q  R.
381
382
Proof. intros ->; apply sep_elim_r. Qed.
Hint Resolve sep_elim_l' sep_elim_r'.
383
Lemma sep_intro_True_l P Q R : P%I  (R  Q)  R  P  Q.
384
Proof. by intros; rewrite -(left_id True%I uPred_sep R); apply sep_mono. Qed.
385
Lemma sep_intro_True_r P Q R : (R  P)  Q%I  R  P  Q.
386
Proof. by intros; rewrite -(right_id True%I uPred_sep R); apply sep_mono. Qed.
387
Lemma sep_elim_True_l P Q R : P  (P  R  Q)  R  Q.
388
Proof. by intros HP; rewrite -HP left_id. Qed.
389
Lemma sep_elim_True_r P Q R : P  (R  P  Q)  R  Q.
390
Proof. by intros HP; rewrite -HP right_id. Qed.
391
Lemma wand_intro_l P Q R : (Q  P  R)  P  Q - R.
392
Proof. rewrite comm; apply wand_intro_r. Qed.
393
Lemma wand_elim_l P Q : (P - Q)  P  Q.
394
Proof. by apply wand_elim_l'. Qed.
395
Lemma wand_elim_r P Q : P  (P - Q)  Q.
396
Proof. rewrite (comm _ P); apply wand_elim_l. Qed.
397
Lemma wand_elim_r' P Q R : (Q  P - R)  P  Q  R.
398
Proof. intros ->; apply wand_elim_r. Qed.
399
Lemma wand_apply P Q R S : (P  Q - R)  (S  P  Q)  S  R.
Ralf Jung's avatar
Ralf Jung committed
400
Proof. intros HR%wand_elim_l' HQ. by rewrite HQ. Qed.
401
Lemma wand_frame_l P Q R : (Q - R)  P  Q - P  R.
402
Proof. apply wand_intro_l. rewrite -assoc. apply sep_mono_r, wand_elim_r. Qed.
403
Lemma wand_frame_r P Q R : (Q - R)  Q  P - R  P.
404
Proof.
405
  apply wand_intro_l. rewrite ![(_  P)%I]comm -assoc.
406
407
  apply sep_mono_r, wand_elim_r.
Qed.
408
Lemma wand_diag P : (P - P)  True.
409
Proof. apply (anti_symm _); auto. apply wand_intro_l; by rewrite right_id. Qed.
410
Lemma wand_True P : (True - P)  P.
411
412
Proof.
  apply (anti_symm _); last by auto using wand_intro_l.
413
  eapply sep_elim_True_l; last by apply wand_elim_r. done.
414
Qed.
415
Lemma wand_entails P Q : (P - Q)%I  P  Q.
416
417
418
Proof.
  intros HPQ. eapply sep_elim_True_r; first exact: HPQ. by rewrite wand_elim_r.
Qed.
419
420
Lemma entails_wand P Q : (P  Q)  (P - Q)%I.
Proof. intro. apply wand_intro_l. auto. Qed.
421
Lemma wand_curry P Q R : (P - Q - R)  (P  Q - R).
422
423
424
425
426
427
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.

428
Lemma sep_and P Q : (P  Q)  (P  Q).
429
Proof. auto. Qed.
430
Lemma impl_wand P Q : (P  Q)  P - Q.
431
Proof. apply wand_intro_r, impl_elim with P; auto. Qed.
Ralf Jung's avatar
Ralf Jung committed
432
Lemma pure_elim_sep_l φ Q R : (φ  Q  R)  ⌜φ⌝  Q  R.
433
Proof. intros; apply pure_elim with φ; eauto. Qed.
Ralf Jung's avatar
Ralf Jung committed
434
Lemma pure_elim_sep_r φ Q R : (φ  Q  R)  Q  ⌜φ⌝  R.
435
436
437
438
439
440
441
Proof. intros; apply pure_elim with φ; eauto. Qed.

Global Instance sep_False : LeftAbsorb () False%I (@uPred_sep M).
Proof. intros P; apply (anti_symm _); auto. Qed.
Global Instance False_sep : RightAbsorb () False%I (@uPred_sep M).
Proof. intros P; apply (anti_symm _); auto. Qed.

442
Lemma entails_equiv_and P Q : (P  Q  P)  (P  Q).
Robbert Krebbers's avatar
Robbert Krebbers committed
443
Proof. split. by intros ->; auto. intros; apply (anti_symm _); auto. Qed.
444
Lemma sep_and_l P Q R : P  (Q  R)  (P  Q)  (P  R).
445
Proof. auto. Qed.
446
Lemma sep_and_r P Q R : (P  Q)  R  (P  R)  (Q  R).
447
Proof. auto. Qed.
448
Lemma sep_or_l P Q R : P  (Q  R)  (P  Q)  (P  R).
449
450
451
452
Proof.
  apply (anti_symm ()); last by eauto 8.
  apply wand_elim_r', or_elim; apply wand_intro_l; auto.
Qed.
453
Lemma sep_or_r P Q R : (P  Q)  R  (P  R)  (Q  R).
454
Proof. by rewrite -!(comm _ R) sep_or_l. Qed.
455
Lemma sep_exist_l {A} P (Ψ : A  uPred M) : P  ( a, Ψ a)   a, P  Ψ a.
456
457
458
459
460
461
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.
462
Lemma sep_exist_r {A} (Φ: A  uPred M) Q: ( a, Φ a)  Q   a, Φ a  Q.
463
Proof. setoid_rewrite (comm _ _ Q); apply sep_exist_l. Qed.
464
Lemma sep_forall_l {A} P (Ψ : A  uPred M) : P  ( a, Ψ a)   a, P  Ψ a.
465
Proof. by apply forall_intro=> a; rewrite forall_elim. Qed.
466
Lemma sep_forall_r {A} (Φ : A  uPred M) Q : ( a, Φ a)  Q   a, Φ a  Q.
467
468
469
470
471
472
473
474
475
476
477
478
479
Proof. by apply forall_intro=> a; rewrite forall_elim. Qed.

(* Always derived *)
Hint Resolve always_mono always_elim.
Global Instance always_mono' : Proper (() ==> ()) (@uPred_always M).
Proof. intros P Q; apply always_mono. Qed.
Global Instance always_flip_mono' :
  Proper (flip () ==> flip ()) (@uPred_always M).
Proof. intros P Q; apply always_mono. Qed.

Lemma always_intro' P Q : ( P  Q)   P   Q.
Proof. intros <-. apply always_idemp. Qed.

Ralf Jung's avatar
Ralf Jung committed
480
Lemma always_pure φ :  ⌜φ⌝  ⌜φ⌝.
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
Proof. apply (anti_symm _); auto using always_pure_2. Qed.
Lemma always_forall {A} (Ψ : A  uPred M) : (  a, Ψ a)  ( a,  Ψ a).
Proof.
  apply (anti_symm _); auto using always_forall_2.
  apply forall_intro=> x. by rewrite (forall_elim x).
Qed.
Lemma always_exist {A} (Ψ : A  uPred M) : (  a, Ψ a)  ( a,  Ψ a).
Proof.
  apply (anti_symm _); auto using always_exist_1.
  apply exist_elim=> x. by rewrite (exist_intro x).
Qed.
Lemma always_and P Q :  (P  Q)   P   Q.
Proof. rewrite !and_alt always_forall. by apply forall_proper=> -[]. Qed.
Lemma always_or P Q :  (P  Q)   P   Q.
Proof. rewrite !or_alt always_exist. by apply exist_proper=> -[]. Qed.
Lemma always_impl P Q :  (P  Q)   P   Q.
Proof.
  apply impl_intro_l; rewrite -always_and.
  apply always_mono, impl_elim with P; auto.
Qed.
501
Lemma always_internal_eq {A:ofeT} (a b : A) :  (a  b)  a  b.
502
503
Proof.
  apply (anti_symm ()); auto using always_elim.
504
  apply (internal_eq_rewrite a b (λ b,  (a  b))%I); auto.
505
  { intros n; solve_proper. }
506
  rewrite -(internal_eq_refl a) always_pure; auto.
507
508
Qed.

509
Lemma always_and_sep P Q :  (P  Q)   (P  Q).
510
Proof. apply (anti_symm ()); auto using always_and_sep_1. Qed.
511
Lemma always_and_sep_l' P Q :  P  Q   P  Q.
512
Proof. apply (anti_symm ()); auto using always_and_sep_l_1. Qed.
513
Lemma always_and_sep_r' P Q : P   Q  P   Q.
514
Proof. by rewrite !(comm _ P) always_and_sep_l'. Qed.
515
Lemma always_sep P Q :  (P  Q)   P   Q.
516
Proof. by rewrite -always_and_sep -always_and_sep_l' always_and. Qed.
517
Lemma always_sep_dup' P :  P   P   P.
518
519
Proof. by rewrite -always_sep -always_and_sep (idemp _). Qed.

520
Lemma always_wand P Q :  (P - Q)   P -  Q.
521
Proof. by apply wand_intro_r; rewrite -always_sep wand_elim_l. Qed.
522
Lemma always_wand_impl P Q :  (P - Q)   (P  Q).
523
524
525
526
527
Proof.
  apply (anti_symm ()); [|by rewrite -impl_wand].
  apply always_intro', impl_intro_r.
  by rewrite always_and_sep_l' always_elim wand_elim_l.
Qed.
528
Lemma always_entails_l' P Q : (P   Q)  P   Q  P.
529
Proof. intros; rewrite -always_and_sep_l'; auto. Qed.
530
Lemma always_entails_r' P Q : (P   Q)  P  P   Q.
531
532
Proof. intros; rewrite -always_and_sep_r'; auto. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
533
534
535
536
Lemma always_laterN n P :  ^n P  ^n  P.
Proof. induction n as [|n IH]; simpl; auto. by rewrite always_later IH. Qed.


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
(* Later derived *)
Lemma later_proper P Q : (P  Q)   P   Q.
Proof. by intros ->. Qed.
Hint Resolve later_mono later_proper.
Global Instance later_mono' : Proper (() ==> ()) (@uPred_later M).
Proof. intros P Q; apply later_mono. Qed.
Global Instance later_flip_mono' :
  Proper (flip () ==> flip ()) (@uPred_later M).
Proof. intros P Q; apply later_mono. Qed.

Lemma later_intro P : P   P.
Proof.
  rewrite -(and_elim_l ( P) P) -(löb ( P  P)).
  apply impl_intro_l. by rewrite {1}(and_elim_r ( P)).
Qed.

Lemma later_True :  True  True.
Proof. apply (anti_symm ()); auto using later_intro. Qed.
Lemma later_forall {A} (Φ : A  uPred M) : (  a, Φ a)  ( a,  Φ a).
Proof.
  apply (anti_symm _); auto using later_forall_2.
  apply forall_intro=> x. by rewrite (forall_elim x).
Qed.
Lemma later_exist `{Inhabited A} (Φ : A  uPred M) :
   ( a, Φ a)  ( a,  Φ a).
Proof.
  apply: anti_symm; [|apply exist_elim; eauto using exist_intro].
  rewrite later_exist_false. apply or_elim; last done.
  rewrite -(exist_intro inhabitant); auto.
Qed.
Lemma later_and P Q :  (P  Q)   P   Q.
Proof. rewrite !and_alt later_forall. by apply forall_proper=> -[]. Qed.
Lemma later_or P Q :  (P  Q)   P   Q.
Proof. rewrite !or_alt later_exist. by apply exist_proper=> -[]. Qed.
Lemma later_impl P Q :  (P  Q)   P   Q.
Proof. apply impl_intro_l; rewrite -later_and; eauto using impl_elim. Qed.
573
Lemma later_wand P Q :  (P - Q)   P -  Q.
574
575
576
577
578
Proof. apply wand_intro_r; rewrite -later_sep; eauto using wand_elim_l. Qed.
Lemma later_iff P Q :  (P  Q)   P   Q.
Proof. by rewrite /uPred_iff later_and !later_impl. Qed.


Robbert Krebbers's avatar
Robbert Krebbers committed
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
(* Iterated later modality *)
Global Instance laterN_ne n m : Proper (dist n ==> dist n) (@uPred_laterN M m).
Proof. induction m; simpl. by intros ???. solve_proper. Qed.
Global Instance laterN_proper m :
  Proper (() ==> ()) (@uPred_laterN M m) := ne_proper _.

Lemma laterN_0 P : ^0 P  P.
Proof. done. Qed.
Lemma later_laterN n P : ^(S n) P   ^n P.
Proof. done. Qed.
Lemma laterN_later n P : ^(S n) P  ^n  P.
Proof. induction n; simpl; auto. Qed.
Lemma laterN_plus n1 n2 P : ^(n1 + n2) P  ^n1 ^n2 P.
Proof. induction n1; simpl; auto. Qed.
Lemma laterN_le n1 n2 P : n1  n2  ^n1 P  ^n2 P.
Proof. induction 1; simpl; by rewrite -?later_intro. Qed.

Lemma laterN_mono n P Q : (P  Q)  ^n P  ^n Q.
Proof. induction n; simpl; auto. Qed.
Global Instance laterN_mono' n : Proper (() ==> ()) (@uPred_laterN M n).
Proof. intros P Q; apply laterN_mono. Qed.
Global Instance laterN_flip_mono' n :
  Proper (flip () ==> flip ()) (@uPred_laterN M n).
Proof. intros P Q; apply laterN_mono. Qed.

Lemma laterN_intro n P : P  ^n P.
Proof. induction n as [|n IH]; simpl; by rewrite -?later_intro. Qed.

Lemma laterN_True n : ^n True  True.
Proof. apply (anti_symm ()); auto using laterN_intro. Qed.
Lemma laterN_forall {A} n (Φ : A  uPred M) : (^n  a, Φ a)  ( a, ^n Φ a).
Proof. induction n as [|n IH]; simpl; rewrite -?later_forall; auto. Qed.
Lemma laterN_exist `{Inhabited A} n (Φ : A  uPred M) :
  (^n  a, Φ a)   a, ^n Φ a.
Proof. induction n as [|n IH]; simpl; rewrite -?later_exist; auto. Qed.
Lemma laterN_and n P Q : ^n (P  Q)  ^n P  ^n Q.
Proof. induction n as [|n IH]; simpl; rewrite -?later_and; auto. Qed.
Lemma laterN_or n P Q : ^n (P  Q)  ^n P  ^n Q.
Proof. induction n as [|n IH]; simpl; rewrite -?later_or; auto. Qed.
Lemma laterN_impl n P Q : ^n (P  Q)  ^n P  ^n Q.
Proof.
  apply impl_intro_l; rewrite -laterN_and; eauto using impl_elim, laterN_mono.
Qed.
Lemma laterN_sep n P Q : ^n (P  Q)  ^n P  ^n Q.
Proof. induction n as [|n IH]; simpl; rewrite -?later_sep; auto. Qed.
Lemma laterN_wand n P Q : ^n (P - Q)  ^n P - ^n Q.
Proof.
  apply wand_intro_r; rewrite -laterN_sep; eauto using wand_elim_l,laterN_mono.
Qed.
Lemma laterN_iff n P Q : ^n (P  Q)  ^n P  ^n Q.
Proof. by rewrite /uPred_iff laterN_and !laterN_impl. Qed.

631
632
633
634
635
636
637
638
639
640
641
642
643
(* Conditional always *)
Global Instance always_if_ne n p : Proper (dist n ==> dist n) (@uPred_always_if M p).
Proof. solve_proper. Qed.
Global Instance always_if_proper p : Proper (() ==> ()) (@uPred_always_if M p).
Proof. solve_proper. Qed.
Global Instance always_if_mono p : Proper (() ==> ()) (@uPred_always_if M p).
Proof. solve_proper. Qed.

Lemma always_if_elim p P : ?p P  P.
Proof. destruct p; simpl; auto using always_elim. Qed.
Lemma always_elim_if p P :  P  ?p P.
Proof. destruct p; simpl; auto using always_elim. Qed.

Ralf Jung's avatar
Ralf Jung committed
644
Lemma always_if_pure p φ : ?p ⌜φ⌝  ⌜φ⌝.
645
646
647
648
649
650
651
Proof. destruct p; simpl; auto using always_pure. Qed.
Lemma always_if_and p P Q : ?p (P  Q)  ?p P  ?p Q.
Proof. destruct p; simpl; auto using always_and. Qed.
Lemma always_if_or p P Q : ?p (P  Q)  ?p P  ?p Q.
Proof. destruct p; simpl; auto using always_or. Qed.
Lemma always_if_exist {A} p (Ψ : A  uPred M) : (?p  a, Ψ a)   a, ?p Ψ a.
Proof. destruct p; simpl; auto using always_exist. Qed.
652
Lemma always_if_sep p P Q : ?p (P  Q)  ?p P  ?p Q.
653
654
655
656
657
658
Proof. destruct p; simpl; auto using always_sep. Qed.
Lemma always_if_later p P : ?p  P   ?p P.
Proof. destruct p; simpl; auto using always_later. Qed.


(* True now *)
659
Global Instance except_0_ne n : Proper (dist n ==> dist n) (@uPred_except_0 M).
660
Proof. solve_proper. Qed.
661
Global Instance except_0_proper : Proper (() ==> ()) (@uPred_except_0 M).
662
Proof. solve_proper. Qed.
663
Global Instance except_0_mono' : Proper (() ==> ()) (@uPred_except_0 M).
664
Proof. solve_proper. Qed.
665
666
Global Instance except_0_flip_mono' :
  Proper (flip () ==> flip ()) (@uPred_except_0 M).
667
668
Proof. solve_proper. Qed.

669
670
671
Lemma except_0_intro P : P   P.
Proof. rewrite /uPred_except_0; auto. Qed.
Lemma except_0_mono P Q : (P  Q)   P   Q.
672
Proof. by intros ->. Qed.
673
674
675
676
677
678
679
680
681
Lemma except_0_idemp P :   P   P.
Proof. rewrite /uPred_except_0; auto. Qed.

Lemma except_0_True :  True  True.
Proof. rewrite /uPred_except_0. apply (anti_symm _); auto. Qed.
Lemma except_0_or P Q :  (P  Q)   P   Q.
Proof. rewrite /uPred_except_0. apply (anti_symm _); auto. Qed.
Lemma except_0_and P Q :  (P  Q)   P   Q.
Proof. by rewrite /uPred_except_0 or_and_l. Qed.
682
Lemma except_0_sep P Q :  (P  Q)   P   Q.
683
684
Proof.
  rewrite /uPred_except_0. apply (anti_symm _).
685
686
687
688
  - apply or_elim; last by auto.
    by rewrite -!or_intro_l -always_pure -always_later -always_sep_dup'.
  - rewrite sep_or_r sep_elim_l sep_or_l; auto.
Qed.
689
Lemma except_0_forall {A} (Φ : A  uPred M) :  ( a, Φ a)   a,  Φ a.
690
Proof. apply forall_intro=> a. by rewrite (forall_elim a). Qed.
691
Lemma except_0_exist {A} (Φ : A  uPred M) : ( a,  Φ a)    a, Φ a.
692
Proof. apply exist_elim=> a. by rewrite (exist_intro a). Qed.
693
694
695
696
697
698
Lemma except_0_later P :   P   P.
Proof. by rewrite /uPred_except_0 -later_or False_or. Qed.
Lemma except_0_always P :   P    P.
Proof. by rewrite /uPred_except_0 always_or always_later always_pure. Qed.
Lemma except_0_always_if p P :  ?p P  ?p  P.
Proof. destruct p; simpl; auto using except_0_always. Qed.
699
Lemma except_0_frame_l P Q : P   Q   (P  Q).
700
Proof. by rewrite {1}(except_0_intro P) except_0_sep. Qed.
701
Lemma except_0_frame_r P Q :  P  Q   (P  Q).
702
Proof. by rewrite {1}(except_0_intro Q) except_0_sep. Qed.
703
704
705
706
707
708
709
710
711
712
713
714

(* Own and valid derived *)
Lemma always_ownM (a : M) : Persistent a   uPred_ownM a  uPred_ownM a.
Proof.
  intros; apply (anti_symm _); first by apply:always_elim.
  by rewrite {1}always_ownM_core persistent_core.
Qed.
Lemma ownM_invalid (a : M) : ¬ {0} a  uPred_ownM a  False.
Proof. by intros; rewrite ownM_valid cmra_valid_elim. Qed.
Global Instance ownM_mono : Proper (flip () ==> ()) (@uPred_ownM M).
Proof. intros a b [b' ->]. rewrite ownM_op. eauto. Qed.
Lemma ownM_empty' : uPred_ownM   True.
715
Proof. apply (anti_symm _); first by auto. apply ownM_empty. Qed.
716
717
718
719
720
721
722
723
724
725
726
Lemma always_cmra_valid {A : cmraT} (a : A) :   a   a.
Proof.
  intros; apply (anti_symm _); first by apply:always_elim.
  apply:always_cmra_valid_1.
Qed.

(** * Derived rules *)
Global Instance bupd_mono' : Proper (() ==> ()) (@uPred_bupd M).
Proof. intros P Q; apply bupd_mono. Qed.
Global Instance bupd_flip_mono' : Proper (flip () ==> flip ()) (@uPred_bupd M).
Proof. intros P Q; apply bupd_mono. Qed.
727
Lemma bupd_frame_l R Q : (R  |==> Q) == R  Q.
728
Proof. rewrite !(comm _ R); apply bupd_frame_r. Qed.
729
Lemma bupd_wand_l P Q : (P - Q)  (|==> P) == Q.
730
Proof. by rewrite bupd_frame_l wand_elim_l. Qed.
731
Lemma bupd_wand_r P Q : (|==> P)  (P - Q) == Q.
732
Proof. by rewrite bupd_frame_r wand_elim_r. Qed.
733
Lemma bupd_sep P Q : (|==> P)  (|==> Q) == P  Q.
734
735
736
737
738
739
Proof. by rewrite bupd_frame_r bupd_frame_l bupd_trans. Qed.
Lemma bupd_ownM_update x y : x ~~> y  uPred_ownM x  |==> uPred_ownM y.
Proof.
  intros; rewrite (bupd_ownM_updateP _ (y =)); last by apply cmra_update_updateP.
  by apply bupd_mono, exist_elim=> y'; apply pure_elim_l=> ->.
Qed.
740
Lemma except_0_bupd P :  (|==> P)  (|==>  P).
741
Proof.
742
  rewrite /uPred_except_0. apply or_elim; auto using bupd_mono.
743
744
745
746
  by rewrite -bupd_intro -or_intro_l.
Qed.

(* Timeless instances *)
Ralf Jung's avatar
Ralf Jung committed
747
Global Instance pure_timeless φ : TimelessP (⌜φ⌝ : uPred M)%I.
748
749
750
751
752
753
754
Proof.
  rewrite /TimelessP pure_alt later_exist_false. by setoid_rewrite later_True.
Qed.
Global Instance valid_timeless {A : cmraT} `{CMRADiscrete A} (a : A) :
  TimelessP ( a : uPred M)%I.
Proof. rewrite /TimelessP !discrete_valid. apply (timelessP _). Qed.
Global Instance and_timeless P Q: TimelessP P  TimelessP Q  TimelessP (P  Q).
755
Proof. intros; rewrite /TimelessP except_0_and later_and; auto. Qed.
756
Global Instance or_timeless P Q : TimelessP P  TimelessP Q  TimelessP (P  Q).
757
Proof. intros; rewrite /TimelessP except_0_or later_or; auto. Qed.
758
759
760
761
762
Global Instance impl_timeless P Q : TimelessP Q  TimelessP (P  Q).
Proof.
  rewrite /TimelessP=> HQ. rewrite later_false_excluded_middle.
  apply or_mono, impl_intro_l; first done.
  rewrite -{2}(löb Q); apply impl_intro_l.
763
  rewrite HQ /uPred_except_0 !and_or_r. apply or_elim; last auto.
764
765
  by rewrite assoc (comm _ _ P) -assoc !impl_elim_r.
Qed.
766
Global Instance sep_timeless P Q: TimelessP P  TimelessP Q  TimelessP (P  Q).
767
Proof. intros; rewrite /TimelessP except_0_sep later_sep; auto. Qed.
768
Global Instance wand_timeless P Q : TimelessP Q  TimelessP (P - Q).
769
770
771
772
Proof.
  rewrite /TimelessP=> HQ. rewrite later_false_excluded_middle.
  apply or_mono, wand_intro_l; first done.
  rewrite -{2}(löb Q); apply impl_intro_l.
773
  rewrite HQ /uPred_except_0 !and_or_r. apply or_elim; last auto.
774
775
776
777
778
779
780
781
782
  rewrite -(always_pure) -always_later always_and_sep_l'.
  by rewrite assoc (comm _ _ P) -assoc -always_and_sep_l' impl_elim_r wand_elim_r.
Qed.
Global Instance forall_timeless {A} (Ψ : A  uPred M) :
  ( x, TimelessP (Ψ x))  TimelessP ( x, Ψ x).
Proof.
  rewrite /TimelessP=> HQ. rewrite later_false_excluded_middle.
  apply or_mono; first done. apply forall_intro=> x.
  rewrite -(löb (Ψ x)); apply impl_intro_l.
783
  rewrite HQ /uPred_except_0 !and_or_r. apply or_elim; last auto.
784
785
786
787
788
789
  by rewrite impl_elim_r (forall_elim x).
Qed.
Global Instance exist_timeless {A} (Ψ : A  uPred M) :
  ( x, TimelessP (Ψ x))  TimelessP ( x, Ψ x).
Proof.
  rewrite /TimelessP=> ?. rewrite later_exist_false. apply or_elim.
790
  - rewrite /uPred_except_0; auto.
791
792
793
  - apply exist_elim=> x. rewrite -(exist_intro x); auto.
Qed.
Global Instance always_timeless P : TimelessP P  TimelessP ( P).
794
Proof. intros; rewrite /TimelessP except_0_always -always_later; auto. Qed.
795
796
Global Instance always_if_timeless p P : TimelessP P  TimelessP (?p P).
Proof. destruct p; apply _. Qed.
797
Global Instance eq_timeless {A : ofeT} (a b : A) :
798
799
800
801
802
  Timeless a  TimelessP (a  b : uPred M)%I.
Proof. intros. rewrite /TimelessP !timeless_eq. apply (timelessP _). Qed.
Global Instance ownM_timeless (a : M) : Timeless a  TimelessP (uPred_ownM a).
Proof.
  intros ?. rewrite /TimelessP later_ownM. apply exist_elim=> b.
803
  rewrite (timelessP (ab)) (except_0_intro (uPred_ownM b)) -except_0_and.
804
805
  apply except_0_mono. rewrite internal_eq_sym.
  apply (internal_eq_rewrite b a (uPred_ownM)); first apply _; auto.
806
Qed.
807
808
809
Global Instance from_option_timeless {A} P (Ψ : A  uPred M) (mx : option A) :
  ( x, TimelessP (Ψ x))  TimelessP P  TimelessP (from_option Ψ P mx).
Proof. destruct mx; apply _. Qed.
810

811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
(* Derived lemmas for persistence *)
Lemma always_always P `{!PersistentP P} :  P  P.
Proof. apply (anti_symm ()); auto using always_elim. Qed.
Lemma always_if_always p P `{!PersistentP P} : ?p P  P.
Proof. destruct p; simpl; auto using always_always. Qed.
Lemma always_intro P Q `{!PersistentP P} : (P  Q)  P   Q.
Proof. rewrite -(always_always P); apply always_intro'. Qed.
Lemma always_and_sep_l P Q `{!PersistentP P} : P  Q  P  Q.
Proof. by rewrite -(always_always P) always_and_sep_l'. Qed.
Lemma always_and_sep_r P Q `{!PersistentP Q} : P  Q  P  Q.
Proof. by rewrite -(always_always Q) always_and_sep_r'. Qed.
Lemma always_sep_dup P `{!PersistentP P} : P  P  P.
Proof. by rewrite -(always_always P) -always_sep_dup'. Qed.
Lemma always_entails_l P Q `{!PersistentP Q} : (P  Q)  P  Q  P.
Proof. by rewrite -(always_always Q); apply always_entails_l'. Qed.
Lemma always_entails_r P Q `{!PersistentP Q} : (P  Q)  P  P  Q.
Proof. by rewrite -(always_always Q); apply always_entails_r'. Qed.
Lemma always_impl_wand P `{!PersistentP P} Q : (P  Q)  (P - Q).
Proof.
  apply (anti_symm _); auto using impl_wand.
  apply impl_intro_l. by rewrite always_and_sep_l wand_elim_r.
Qed.

834
(* Persistence *)
Ralf Jung's avatar
Ralf Jung committed
835
Global Instance pure_persistent φ : PersistentP (⌜φ⌝ : uPred M)%I.
836
Proof. by rewrite /PersistentP always_pure. Qed.
837
838
839
840
841
842
843
844
845
846
847
Global Instance pure_impl_persistent φ Q :
  PersistentP Q  PersistentP (⌜φ⌝  Q)%I.
Proof.
  rewrite /PersistentP pure_impl_forall always_forall. auto using forall_mono.
Qed.
Global Instance pure_wand_persistent φ Q :
  PersistentP Q  PersistentP (⌜φ⌝ - Q)%I.
Proof.
  rewrite /PersistentP -always_impl_wand pure_impl_forall always_forall.
  auto using forall_mono.
Qed.
848
849
850
851
852
853
854
855
856
Global Instance always_persistent P : PersistentP ( P).
Proof. by intros; apply always_intro'. Qed.
Global Instance and_persistent P Q :
  PersistentP P  PersistentP Q  PersistentP (P  Q).
Proof. by intros; rewrite /PersistentP always_and; apply and_mono. Qed.
Global Instance or_persistent P Q :
  PersistentP P  PersistentP Q  PersistentP (P  Q).
Proof. by intros; rewrite /PersistentP always_or; apply or_mono. Qed.
Global Instance sep_persistent P Q :
857
  PersistentP P  PersistentP Q  PersistentP (P  Q).