class_instances_sbi.v 29.8 KB
Newer Older
1
2
From stdpp Require Import nat_cancel.
From iris.bi Require Import bi tactics.
3
From iris.proofmode Require Import modality_instances classes.
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
Set Default Proof Using "Type".
Import bi.

Section sbi_instances.
Context {PROP : sbi}.
Implicit Types P Q R : PROP.

(* FromAssumption *)
Global Instance from_assumption_later p P Q :
  FromAssumption p P Q  KnownRFromAssumption p P ( Q)%I.
Proof. rewrite /KnownRFromAssumption /FromAssumption=>->. apply later_intro. Qed.
Global Instance from_assumption_laterN n p P Q :
  FromAssumption p P Q  KnownRFromAssumption p P (^n Q)%I.
Proof. rewrite /KnownRFromAssumption /FromAssumption=>->. apply laterN_intro. Qed.
Global Instance from_assumption_except_0 p P Q :
  FromAssumption p P Q  KnownRFromAssumption p P ( Q)%I.
Proof. rewrite /KnownRFromAssumption /FromAssumption=>->. apply except_0_intro. Qed.

Global Instance from_assumption_bupd `{BiBUpd PROP} p P Q :
  FromAssumption p P Q  KnownRFromAssumption p P (|==> Q).
Proof. rewrite /KnownRFromAssumption /FromAssumption=>->. apply bupd_intro. Qed.
Global Instance from_assumption_fupd `{BiBUpdFUpd PROP} E p P Q :
  FromAssumption p P (|==> Q)  KnownRFromAssumption p P (|={E}=> Q)%I.
Proof. rewrite /KnownRFromAssumption /FromAssumption=>->. apply bupd_fupd. Qed.

Global Instance from_assumption_plainly_l_true `{BiPlainly PROP} P Q :
  FromAssumption true P Q  KnownLFromAssumption true ( P) Q.
Proof.
  rewrite /KnownLFromAssumption /FromAssumption /= =><-.
  rewrite intuitionistically_plainly_elim //.
Qed.
Global Instance from_assumption_plainly_l_false `{BiPlainly PROP, BiAffine PROP} P Q :
  FromAssumption true P Q  KnownLFromAssumption false ( P) Q.
Proof.
  rewrite /KnownLFromAssumption /FromAssumption /= =><-.
Ralf Jung's avatar
Ralf Jung committed
39
  rewrite plainly_elim_persistently intuitionistically_into_persistently //.
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
Qed.

(* FromPure *)
Global Instance from_pure_internal_eq af {A : ofeT} (a b : A) :
  @FromPure PROP af (a  b) (a  b).
Proof. by rewrite /FromPure pure_internal_eq affinely_if_elim. Qed.
Global Instance from_pure_later a P φ : FromPure a P φ  FromPure a ( P) φ.
Proof. rewrite /FromPure=> ->. apply later_intro. Qed.
Global Instance from_pure_laterN a n P φ : FromPure a P φ  FromPure a (^n P) φ.
Proof. rewrite /FromPure=> ->. apply laterN_intro. Qed.
Global Instance from_pure_except_0 a P φ : FromPure a P φ  FromPure a ( P) φ.
Proof. rewrite /FromPure=> ->. apply except_0_intro. Qed.

Global Instance from_pure_bupd `{BiBUpd PROP} a P φ :
  FromPure a P φ  FromPure a (|==> P) φ.
Proof. rewrite /FromPure=> <-. apply bupd_intro. Qed.
Global Instance from_pure_fupd `{BiFUpd PROP} a E P φ :
  FromPure a P φ  FromPure a (|={E}=> P) φ.
Proof. rewrite /FromPure. intros <-. apply fupd_intro. Qed.

Global Instance from_pure_plainly `{BiPlainly PROP} P φ :
  FromPure false P φ  FromPure false ( P) φ.
Proof. rewrite /FromPure=> <-. by rewrite plainly_pure. Qed.

(* IntoPure *)
Global Instance into_pure_eq {A : ofeT} (a b : A) :
  Discrete a  @IntoPure PROP (a  b) (a  b).
Proof. intros. by rewrite /IntoPure discrete_eq. Qed.

Global Instance into_pure_plainly `{BiPlainly PROP} P φ :
  IntoPure P φ  IntoPure ( P) φ.
Proof. rewrite /IntoPure=> ->. apply: plainly_elim. Qed.

(* IntoWand *)
Global Instance into_wand_later p q R P Q :
  IntoWand p q R P Q  IntoWand p q ( R) ( P) ( Q).
Proof.
  rewrite /IntoWand /= => HR.
  by rewrite !later_intuitionistically_if_2 -later_wand HR.
Qed.
Global Instance into_wand_later_args p q R P Q :
  IntoWand p q R P Q  IntoWand' p q R ( P) ( Q).
Proof.
  rewrite /IntoWand' /IntoWand /= => HR.
  by rewrite !later_intuitionistically_if_2
             (later_intro (?p R)%I) -later_wand HR.
Qed.
Global Instance into_wand_laterN n p q R P Q :
  IntoWand p q R P Q  IntoWand p q (^n R) (^n P) (^n Q).
Proof.
  rewrite /IntoWand /= => HR.
  by rewrite !laterN_intuitionistically_if_2 -laterN_wand HR.
Qed.
Global Instance into_wand_laterN_args n p q R P Q :
  IntoWand p q R P Q  IntoWand' p q R (^n P) (^n Q).
Proof.
  rewrite /IntoWand' /IntoWand /= => HR.
  by rewrite !laterN_intuitionistically_if_2
             (laterN_intro _ (?p R)%I) -laterN_wand HR.
Qed.

Global Instance into_wand_bupd `{BiBUpd PROP} p q R P Q :
  IntoWand false false R P Q  IntoWand p q (|==> R) (|==> P) (|==> Q).
Proof.
  rewrite /IntoWand /= => HR. rewrite !intuitionistically_if_elim HR.
  apply wand_intro_l. by rewrite bupd_sep wand_elim_r.
Qed.
Global Instance into_wand_bupd_persistent `{BiBUpd PROP} p q R P Q :
  IntoWand false q R P Q  IntoWand p q (|==> R) P (|==> Q).
Proof.
  rewrite /IntoWand /= => HR. rewrite intuitionistically_if_elim HR.
  apply wand_intro_l. by rewrite bupd_frame_l wand_elim_r.
Qed.
Global Instance into_wand_bupd_args `{BiBUpd PROP} p q R P Q :
  IntoWand p false R P Q  IntoWand' p q R (|==> P) (|==> Q).
Proof.
  rewrite /IntoWand' /IntoWand /= => ->.
  apply wand_intro_l. by rewrite intuitionistically_if_elim bupd_wand_r.
Qed.

Global Instance into_wand_fupd `{BiFUpd PROP} E p q R P Q :
  IntoWand false false R P Q 
  IntoWand p q (|={E}=> R) (|={E}=> P) (|={E}=> Q).
Proof.
  rewrite /IntoWand /= => HR. rewrite !intuitionistically_if_elim HR.
  apply wand_intro_l. by rewrite fupd_sep wand_elim_r.
Qed.
Global Instance into_wand_fupd_persistent `{BiFUpd PROP} E1 E2 p q R P Q :
  IntoWand false q R P Q  IntoWand p q (|={E1,E2}=> R) P (|={E1,E2}=> Q).
Proof.
  rewrite /IntoWand /= => HR. rewrite intuitionistically_if_elim HR.
  apply wand_intro_l. by rewrite fupd_frame_l wand_elim_r.
Qed.
Global Instance into_wand_fupd_args `{BiFUpd PROP} E1 E2 p q R P Q :
  IntoWand p false R P Q  IntoWand' p q R (|={E1,E2}=> P) (|={E1,E2}=> Q).
Proof.
  rewrite /IntoWand' /IntoWand /= => ->.
  apply wand_intro_l. by rewrite intuitionistically_if_elim fupd_wand_r.
Qed.

Global Instance into_wand_plainly_true `{BiPlainly PROP} q R P Q :
  IntoWand true q R P Q  IntoWand true q ( R) P Q.
Proof. rewrite /IntoWand /= intuitionistically_plainly_elim //. Qed.
Global Instance into_wand_plainly_false `{BiPlainly PROP} q R P Q :
  Absorbing R  IntoWand false q R P Q  IntoWand false q ( R) P Q.
Proof. intros ?. by rewrite /IntoWand plainly_elim. Qed.

(* FromAnd *)
Global Instance from_and_later P Q1 Q2 :
  FromAnd P Q1 Q2  FromAnd ( P) ( Q1) ( Q2).
Proof. rewrite /FromAnd=> <-. by rewrite later_and. Qed.
Global Instance from_and_laterN n P Q1 Q2 :
  FromAnd P Q1 Q2  FromAnd (^n P) (^n Q1) (^n Q2).
Proof. rewrite /FromAnd=> <-. by rewrite laterN_and. Qed.
Global Instance from_and_except_0 P Q1 Q2 :
  FromAnd P Q1 Q2  FromAnd ( P) ( Q1) ( Q2).
Proof. rewrite /FromAnd=><-. by rewrite except_0_and. Qed.

Global Instance from_and_plainly `{BiPlainly PROP} P Q1 Q2 :
  FromAnd P Q1 Q2  FromAnd ( P) ( Q1) ( Q2).
Proof. rewrite /FromAnd=> <-. by rewrite plainly_and. Qed.

(* FromSep *)
Global Instance from_sep_later P Q1 Q2 :
  FromSep P Q1 Q2  FromSep ( P) ( Q1) ( Q2).
Proof. rewrite /FromSep=> <-. by rewrite later_sep. Qed.
Global Instance from_sep_laterN n P Q1 Q2 :
  FromSep P Q1 Q2  FromSep (^n P) (^n Q1) (^n Q2).
Proof. rewrite /FromSep=> <-. by rewrite laterN_sep. Qed.
Global Instance from_sep_except_0 P Q1 Q2 :
  FromSep P Q1 Q2  FromSep ( P) ( Q1) ( Q2).
Proof. rewrite /FromSep=><-. by rewrite except_0_sep. Qed.

Global Instance from_sep_bupd `{BiBUpd PROP} P Q1 Q2 :
  FromSep P Q1 Q2  FromSep (|==> P) (|==> Q1) (|==> Q2).
Proof. rewrite /FromSep=><-. apply bupd_sep. Qed.
Global Instance from_sep_fupd `{BiFUpd PROP} E P Q1 Q2 :
  FromSep P Q1 Q2  FromSep (|={E}=> P) (|={E}=> Q1) (|={E}=> Q2).
Proof. rewrite /FromSep =><-. apply fupd_sep. Qed.

Global Instance from_sep_plainly `{BiPlainly PROP} P Q1 Q2 :
  FromSep P Q1 Q2  FromSep ( P) ( Q1) ( Q2).
Proof. rewrite /FromSep=> <-. by rewrite plainly_sep_2. Qed.

(* IntoAnd *)
Global Instance into_and_later p P Q1 Q2 :
  IntoAnd p P Q1 Q2  IntoAnd p ( P) ( Q1) ( Q2).
Proof.
  rewrite /IntoAnd=> HP. apply intuitionistically_if_intro'.
  by rewrite later_intuitionistically_if_2 HP
             intuitionistically_if_elim later_and.
Qed.
Global Instance into_and_laterN n p P Q1 Q2 :
  IntoAnd p P Q1 Q2  IntoAnd p (^n P) (^n Q1) (^n Q2).
Proof.
  rewrite /IntoAnd=> HP. apply intuitionistically_if_intro'.
  by rewrite laterN_intuitionistically_if_2 HP
             intuitionistically_if_elim laterN_and.
Qed.
Global Instance into_and_except_0 p P Q1 Q2 :
  IntoAnd p P Q1 Q2  IntoAnd p ( P) ( Q1) ( Q2).
Proof.
  rewrite /IntoAnd=> HP. apply intuitionistically_if_intro'.
  by rewrite except_0_intuitionistically_if_2 HP
             intuitionistically_if_elim except_0_and.
Qed.

Global Instance into_and_plainly `{BiPlainly PROP} p P Q1 Q2 :
  IntoAnd p P Q1 Q2  IntoAnd p ( P) ( Q1) ( Q2).
Proof.
  rewrite /IntoAnd /=. destruct p; simpl.
  - rewrite -plainly_and -[(  P)%I]intuitionistically_idemp intuitionistically_plainly =>->.
    rewrite [( (_  _))%I]intuitionistically_elim //.
  - intros ->. by rewrite plainly_and.
Qed.

(* IntoSep *)
Global Instance into_sep_later P Q1 Q2 :
  IntoSep P Q1 Q2  IntoSep ( P) ( Q1) ( Q2).
Proof. rewrite /IntoSep=> ->. by rewrite later_sep. Qed.
Global Instance into_sep_laterN n P Q1 Q2 :
  IntoSep P Q1 Q2  IntoSep (^n P) (^n Q1) (^n Q2).
Proof. rewrite /IntoSep=> ->. by rewrite laterN_sep. Qed.
Global Instance into_sep_except_0 P Q1 Q2 :
  IntoSep P Q1 Q2  IntoSep ( P) ( Q1) ( Q2).
Proof. rewrite /IntoSep=> ->. by rewrite except_0_sep. Qed.

(* FIXME: This instance is overly specific, generalize it. *)
Global Instance into_sep_affinely_later `{!Timeless (emp%I : PROP)} P Q1 Q2 :
  IntoSep P Q1 Q2  Affine Q1  Affine Q2 
  IntoSep (<affine>  P) (<affine>  Q1) (<affine>  Q2).
Proof.
  rewrite /IntoSep /= => -> ??.
  rewrite -{1}(affine_affinely Q1) -{1}(affine_affinely Q2) later_sep !later_affinely_1.
  rewrite -except_0_sep /sbi_except_0 affinely_or. apply or_elim, affinely_elim.
  rewrite -(idemp bi_and (<affine>  False)%I) persistent_and_sep_1.
  by rewrite -(False_elim Q1) -(False_elim Q2).
Qed.

Global Instance into_sep_plainly `{BiPlainly PROP, BiPositive PROP} P Q1 Q2 :
  IntoSep P Q1 Q2  IntoSep ( P) ( Q1) ( Q2).
Proof. rewrite /IntoSep /= => ->. by rewrite plainly_sep. Qed.

Global Instance into_sep_plainly_affine `{BiPlainly PROP} P Q1 Q2 :
  IntoSep P Q1 Q2 
  TCOr (Affine Q1) (Absorbing Q2)  TCOr (Absorbing Q1) (Affine Q2) 
  IntoSep ( P) ( Q1) ( Q2).
Proof.
  rewrite /IntoSep /= => -> ??. by rewrite sep_and plainly_and plainly_and_sep_l_1.
Qed.

(* FromOr *)
Global Instance from_or_later P Q1 Q2 :
  FromOr P Q1 Q2  FromOr ( P) ( Q1) ( Q2).
Proof. rewrite /FromOr=><-. by rewrite later_or. Qed.
Global Instance from_or_laterN n P Q1 Q2 :
  FromOr P Q1 Q2  FromOr (^n P) (^n Q1) (^n Q2).
Proof. rewrite /FromOr=><-. by rewrite laterN_or. Qed.
Global Instance from_or_except_0 P Q1 Q2 :
  FromOr P Q1 Q2  FromOr ( P) ( Q1) ( Q2).
Proof. rewrite /FromOr=><-. by rewrite except_0_or. Qed.

Global Instance from_or_bupd `{BiBUpd PROP} P Q1 Q2 :
  FromOr P Q1 Q2  FromOr (|==> P) (|==> Q1) (|==> Q2).
Proof.
  rewrite /FromOr=><-.
  apply or_elim; apply bupd_mono; auto using or_intro_l, or_intro_r.
Qed.
Global Instance from_or_fupd `{BiFUpd PROP} E1 E2 P Q1 Q2 :
  FromOr P Q1 Q2  FromOr (|={E1,E2}=> P) (|={E1,E2}=> Q1) (|={E1,E2}=> Q2).
Proof.
  rewrite /FromOr=><-. apply or_elim; apply fupd_mono;
                         [apply bi.or_intro_l|apply bi.or_intro_r].
Qed.

Global Instance from_or_plainly `{BiPlainly PROP} P Q1 Q2 :
  FromOr P Q1 Q2  FromOr ( P) ( Q1) ( Q2).
Proof. rewrite /FromOr=> <-. by rewrite -plainly_or_2. Qed.

(* IntoOr *)
Global Instance into_or_later P Q1 Q2 :
  IntoOr P Q1 Q2  IntoOr ( P) ( Q1) ( Q2).
Proof. rewrite /IntoOr=>->. by rewrite later_or. Qed.
Global Instance into_or_laterN n P Q1 Q2 :
  IntoOr P Q1 Q2  IntoOr (^n P) (^n Q1) (^n Q2).
Proof. rewrite /IntoOr=>->. by rewrite laterN_or. Qed.
Global Instance into_or_except_0 P Q1 Q2 :
  IntoOr P Q1 Q2  IntoOr ( P) ( Q1) ( Q2).
Proof. rewrite /IntoOr=>->. by rewrite except_0_or. Qed.

Global Instance into_or_plainly `{BiPlainly PROP, BiPlainlyExist PROP} P Q1 Q2 :
  IntoOr P Q1 Q2  IntoOr ( P) ( Q1) ( Q2).
Proof. rewrite /IntoOr=>->. by rewrite plainly_or. Qed.

(* FromExist *)
Global Instance from_exist_later {A} P (Φ : A  PROP) :
  FromExist P Φ  FromExist ( P) (λ a,  (Φ a))%I.
Proof.
  rewrite /FromExist=> <-. apply exist_elim=>x. apply later_mono, exist_intro.
Qed.
Global Instance from_exist_laterN {A} n P (Φ : A  PROP) :
  FromExist P Φ  FromExist (^n P) (λ a, ^n (Φ a))%I.
Proof.
  rewrite /FromExist=> <-. apply exist_elim=>x. apply laterN_mono, exist_intro.
Qed.
Global Instance from_exist_except_0 {A} P (Φ : A  PROP) :
  FromExist P Φ  FromExist ( P) (λ a,  (Φ a))%I.
Proof. rewrite /FromExist=> <-. by rewrite except_0_exist_2. Qed.

Global Instance from_exist_bupd `{BiBUpd PROP} {A} P (Φ : A  PROP) :
  FromExist P Φ  FromExist (|==> P) (λ a, |==> Φ a)%I.
Proof.
  rewrite /FromExist=><-. apply exist_elim=> a. by rewrite -(exist_intro a).
Qed.
Global Instance from_exist_fupd `{BiFUpd PROP} {A} E1 E2 P (Φ : A  PROP) :
  FromExist P Φ  FromExist (|={E1,E2}=> P) (λ a, |={E1,E2}=> Φ a)%I.
Proof.
  rewrite /FromExist=><-. apply exist_elim=> a. by rewrite -(exist_intro a).
Qed.

Global Instance from_exist_plainly `{BiPlainly PROP} {A} P (Φ : A  PROP) :
  FromExist P Φ  FromExist ( P) (λ a,  (Φ a))%I.
Proof. rewrite /FromExist=> <-. by rewrite -plainly_exist_2. Qed.

(* IntoExist *)
Global Instance into_exist_later {A} P (Φ : A  PROP) :
  IntoExist P Φ  Inhabited A  IntoExist ( P) (λ a,  (Φ a))%I.
Proof. rewrite /IntoExist=> HP ?. by rewrite HP later_exist. Qed.
Global Instance into_exist_laterN {A} n P (Φ : A  PROP) :
  IntoExist P Φ  Inhabited A  IntoExist (^n P) (λ a, ^n (Φ a))%I.
Proof. rewrite /IntoExist=> HP ?. by rewrite HP laterN_exist. Qed.
Global Instance into_exist_except_0 {A} P (Φ : A  PROP) :
  IntoExist P Φ  Inhabited A  IntoExist ( P) (λ a,  (Φ a))%I.
Proof. rewrite /IntoExist=> HP ?. by rewrite HP except_0_exist. Qed.

Global Instance into_exist_plainly `{BiPlainlyExist PROP} {A} P (Φ : A  PROP) :
  IntoExist P Φ  IntoExist ( P) (λ a,  (Φ a))%I.
Proof. rewrite /IntoExist=> HP. by rewrite HP plainly_exist. Qed.

(* IntoForall *)
Global Instance into_forall_later {A} P (Φ : A  PROP) :
  IntoForall P Φ  IntoForall ( P) (λ a,  (Φ a))%I.
Proof. rewrite /IntoForall=> HP. by rewrite HP later_forall. Qed.
Global Instance into_forall_except_0 {A} P (Φ : A  PROP) :
  IntoForall P Φ  IntoForall ( P) (λ a,  (Φ a))%I.
Proof. rewrite /IntoForall=> HP. by rewrite HP except_0_forall. Qed.
Global Instance into_forall_impl_pure φ P Q :
  FromPureT false P φ  IntoForall (P  Q) (λ _ : φ, Q).
Proof.
  rewrite /FromPureT /FromPure /IntoForall=> -[φ' [-> <-]].
  by rewrite pure_impl_forall.
Qed.
Global Instance into_forall_wand_pure φ P Q :
  FromPureT true P φ  IntoForall (P - Q) (λ _ : φ, Q).
Proof.
  rewrite /FromPureT /FromPure /IntoForall=> -[φ' [-> <-]] /=.
  apply forall_intro=>? /=.
  by rewrite -(pure_intro True%I) // /bi_affinely right_id emp_wand.
Qed.

Global Instance into_forall_plainly `{BiPlainly PROP} {A} P (Φ : A  PROP) :
  IntoForall P Φ  IntoForall ( P) (λ a,  (Φ a))%I.
Proof. rewrite /IntoForall=> HP. by rewrite HP plainly_forall. Qed.

(* FromForall *)
Global Instance from_forall_later {A} P (Φ : A  PROP) :
  FromForall P Φ  FromForall ( P)%I (λ a,  (Φ a))%I.
Proof. rewrite /FromForall=> <-. by rewrite later_forall. Qed.
Global Instance from_forall_except_0 {A} P (Φ : A  PROP) :
  FromForall P Φ  FromForall ( P)%I (λ a,  (Φ a))%I.
Proof. rewrite /FromForall=> <-. by rewrite except_0_forall. Qed.

Global Instance from_forall_plainly `{BiPlainly PROP} {A} P (Φ : A  PROP) :
  FromForall P Φ  FromForall ( P)%I (λ a,  (Φ a))%I.
Proof. rewrite /FromForall=> <-. by rewrite plainly_forall. Qed.

(* IsExcept0 *)
Global Instance is_except_0_except_0 P : IsExcept0 ( P).
Proof. by rewrite /IsExcept0 except_0_idemp. Qed.
Global Instance is_except_0_later P : IsExcept0 ( P).
Proof. by rewrite /IsExcept0 except_0_later. Qed.
Global Instance is_except_0_embed `{SbiEmbed PROP PROP'} P :
  IsExcept0 P  IsExcept0 P.
Proof. by rewrite /IsExcept0 -embed_except_0=>->. Qed.
Global Instance is_except_0_bupd `{BiBUpd PROP} P : IsExcept0 P  IsExcept0 (|==> P).
Proof.
  rewrite /IsExcept0=> HP.
  by rewrite -{2}HP -(except_0_idemp P) -except_0_bupd -(except_0_intro P).
Qed.
Global Instance is_except_0_fupd `{BiFUpd PROP} E1 E2 P :
  IsExcept0 (|={E1,E2}=> P).
Proof. by rewrite /IsExcept0 except_0_fupd. Qed.

(* FromModal *)
Global Instance from_modal_later P :
  FromModal (modality_laterN 1) (^1 P) ( P) P.
Proof. by rewrite /FromModal. Qed.
Global Instance from_modal_laterN n P :
  FromModal (modality_laterN n) (^n P) (^n P) P.
Proof. by rewrite /FromModal. Qed.
Global Instance from_modal_except_0 P : FromModal modality_id ( P) ( P) P.
Proof. by rewrite /FromModal /= -except_0_intro. Qed.

Global Instance from_modal_bupd `{BiBUpd PROP} P :
  FromModal modality_id (|==> P) (|==> P) P.
Proof. by rewrite /FromModal /= -bupd_intro. Qed.
Global Instance from_modal_fupd E P `{BiFUpd PROP} :
  FromModal modality_id (|={E}=> P) (|={E}=> P) P.
Proof. by rewrite /FromModal /= -fupd_intro. Qed.

Global Instance from_modal_later_embed `{SbiEmbed PROP PROP'} `(sel : A) n P Q :
  FromModal (modality_laterN n) sel P Q 
  FromModal (modality_laterN n) sel P Q.
Proof. rewrite /FromModal /= =><-. by rewrite embed_laterN. Qed.

Global Instance from_modal_plainly `{BiPlainly PROP} P :
  FromModal modality_plainly ( P) ( P) P | 2.
Proof. by rewrite /FromModal. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
419
Global Instance from_modal_plainly_embed `{BiPlainly PROP, BiPlainly PROP',
420
    BiEmbedPlainly PROP PROP', !SbiEmbed PROP PROP'} `(sel : A) P Q :
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
  FromModal modality_plainly sel P Q 
  FromModal modality_plainly sel P Q | 100.
Proof. rewrite /FromModal /= =><-. by rewrite embed_plainly. Qed.

(* IntoInternalEq *)
Global Instance into_internal_eq_internal_eq {A : ofeT} (x y : A) :
  @IntoInternalEq PROP A (x  y) x y.
Proof. by rewrite /IntoInternalEq. Qed.
Global Instance into_internal_eq_affinely {A : ofeT} (x y : A) P :
  IntoInternalEq P x y  IntoInternalEq (<affine> P) x y.
Proof. rewrite /IntoInternalEq=> ->. by rewrite affinely_elim. Qed.
Global Instance into_internal_eq_intuitionistically {A : ofeT} (x y : A) P :
  IntoInternalEq P x y  IntoInternalEq ( P) x y.
Proof. rewrite /IntoInternalEq=> ->. by rewrite intuitionistically_elim. Qed.
Global Instance into_internal_eq_absorbingly {A : ofeT} (x y : A) P :
  IntoInternalEq P x y  IntoInternalEq (<absorb> P) x y.
Proof. rewrite /IntoInternalEq=> ->. by rewrite absorbingly_internal_eq. Qed.
Global Instance into_internal_eq_plainly `{BiPlainly PROP} {A : ofeT} (x y : A) P :
  IntoInternalEq P x y  IntoInternalEq ( P) x y.
Proof. rewrite /IntoInternalEq=> ->. by rewrite plainly_elim. Qed.
Global Instance into_internal_eq_persistently {A : ofeT} (x y : A) P :
  IntoInternalEq P x y  IntoInternalEq (<pers> P) x y.
Proof. rewrite /IntoInternalEq=> ->. by rewrite persistently_elim. Qed.
Global Instance into_internal_eq_embed
       `{SbiEmbed PROP PROP'} {A : ofeT} (x y : A) P :
  IntoInternalEq P x y  IntoInternalEq P x y.
Proof. rewrite /IntoInternalEq=> ->. by rewrite embed_internal_eq. Qed.

(* IntoExcept0 *)
Global Instance into_except_0_except_0 P : IntoExcept0 ( P) P.
Proof. by rewrite /IntoExcept0. Qed.
Global Instance into_except_0_later P : Timeless P  IntoExcept0 ( P) P.
Proof. by rewrite /IntoExcept0. Qed.
Global Instance into_except_0_later_if p P : Timeless P  IntoExcept0 (?p P) P.
Proof. rewrite /IntoExcept0. destruct p; auto using except_0_intro. Qed.

Global Instance into_except_0_affinely P Q :
  IntoExcept0 P Q  IntoExcept0 (<affine> P) (<affine> Q).
Proof. rewrite /IntoExcept0=> ->. by rewrite except_0_affinely_2. Qed.
Global Instance into_except_0_intuitionistically P Q :
  IntoExcept0 P Q  IntoExcept0 ( P) ( Q).
Proof. rewrite /IntoExcept0=> ->. by rewrite except_0_intuitionistically_2. Qed.
Global Instance into_except_0_absorbingly P Q :
  IntoExcept0 P Q  IntoExcept0 (<absorb> P) (<absorb> Q).
Proof. rewrite /IntoExcept0=> ->. by rewrite except_0_absorbingly. Qed.
Global Instance into_except_0_plainly `{BiPlainly PROP, BiPlainlyExist PROP} P Q :
  IntoExcept0 P Q  IntoExcept0 ( P) ( Q).
Proof. rewrite /IntoExcept0=> ->. by rewrite except_0_plainly. Qed.
Global Instance into_except_0_persistently P Q :
  IntoExcept0 P Q  IntoExcept0 (<pers> P) (<pers> Q).
Proof. rewrite /IntoExcept0=> ->. by rewrite except_0_persistently. Qed.
Global Instance into_except_0_embed `{SbiEmbed PROP PROP'} P Q :
  IntoExcept0 P Q  IntoExcept0 P Q.
Proof. rewrite /IntoExcept0=> ->. by rewrite embed_except_0. Qed.

(* ElimModal *)
Global Instance elim_modal_timeless P Q :
  IntoExcept0 P P'  IsExcept0 Q  ElimModal True P P' Q Q.
Proof.
  intros. rewrite /ElimModal (except_0_intro (_ - _)%I).
  by rewrite (into_except_0 P) -except_0_sep wand_elim_r.
Qed.

Global Instance elim_modal_bupd_plain_goal `{BiBUpdPlainly PROP} P Q :
  Plain Q  ElimModal True (|==> P) P Q Q.
Proof. intros. by rewrite /ElimModal bupd_frame_r wand_elim_r bupd_plain. Qed.
Global Instance elim_modal_bupd_plain `{BiBUpdPlainly PROP} P Q :
  Plain P  ElimModal True (|==> P) P Q Q.
Proof. intros. by rewrite /ElimModal bupd_plain wand_elim_r. Qed.
Global Instance elim_modal_bupd_fupd `{BiBUpdFUpd PROP} E1 E2 P Q :
  ElimModal True (|==> P) P (|={E1,E2}=> Q) (|={E1,E2}=> Q) | 10.
Proof.
  by rewrite /ElimModal (bupd_fupd E1) fupd_frame_r wand_elim_r fupd_trans.
Qed.

Global Instance elim_modal_fupd_fupd `{BiFUpd PROP} E1 E2 E3 P Q :
  ElimModal True (|={E1,E2}=> P) P (|={E1,E3}=> Q) (|={E2,E3}=> Q).
Proof. by rewrite /ElimModal fupd_frame_r wand_elim_r fupd_trans. Qed.

Global Instance elim_modal_embed_fupd_goal `{BiEmbedFUpd PROP PROP'}
       φ E1 E2 E3 (P P' : PROP') (Q Q' : PROP) :
  ElimModal φ P P' (|={E1,E3}=> Q)%I (|={E2,E3}=> Q')%I 
  ElimModal φ P P' |={E1,E3}=> Q |={E2,E3}=> Q'.
Proof. by rewrite /ElimModal !embed_fupd. Qed.
Global Instance elim_modal_embed_fupd_hyp `{BiEmbedFUpd PROP PROP'}
       φ E1 E2 (P : PROP) (P' Q Q' : PROP') :
  ElimModal φ (|={E1,E2}=> P)%I P' Q Q' 
  ElimModal φ |={E1,E2}=> P P' Q Q'.
Proof. by rewrite /ElimModal embed_fupd. Qed.

(* AddModal *)
(* High priority to add a ▷ rather than a ◇ when P is timeless. *)
Global Instance add_modal_later_except_0 P Q :
  Timeless P  AddModal ( P) P ( Q) | 0.
Proof.
  intros. rewrite /AddModal (except_0_intro (_ - _)%I) (timeless P).
  by rewrite -except_0_sep wand_elim_r except_0_idemp.
Qed.
Global Instance add_modal_later P Q :
  Timeless P  AddModal ( P) P ( Q) | 0.
Proof.
  intros. rewrite /AddModal (except_0_intro (_ - _)%I) (timeless P).
  by rewrite -except_0_sep wand_elim_r except_0_later.
Qed.
Global Instance add_modal_except_0 P Q : AddModal ( P) P ( Q) | 1.
Proof.
  intros. rewrite /AddModal (except_0_intro (_ - _)%I).
  by rewrite -except_0_sep wand_elim_r except_0_idemp.
Qed.
Global Instance add_modal_except_0_later P Q : AddModal ( P) P ( Q) | 1.
Proof.
  intros. rewrite /AddModal (except_0_intro (_ - _)%I).
  by rewrite -except_0_sep wand_elim_r except_0_later.
Qed.

Global Instance add_modal_bupd `{BiBUpd PROP} P Q : AddModal (|==> P) P (|==> Q).
Proof. by rewrite /AddModal bupd_frame_r wand_elim_r bupd_trans. Qed.
Global Instance add_modal_fupd `{BiFUpd PROP} E1 E2 P Q :
  AddModal (|={E1}=> P) P (|={E1,E2}=> Q).
Proof. by rewrite /AddModal fupd_frame_r wand_elim_r fupd_trans. Qed.

Global Instance add_modal_embed_fupd_goal `{BiEmbedFUpd PROP PROP'}
       E1 E2 (P P' : PROP') (Q : PROP) :
  AddModal P P' (|={E1,E2}=> Q)%I  AddModal P P' |={E1,E2}=> Q.
Proof. by rewrite /AddModal !embed_fupd. Qed.

(* IntoLater *)
Global Instance into_laterN_0 only_head P : IntoLaterN only_head 0 P P.
Proof. by rewrite /IntoLaterN /MaybeIntoLaterN. Qed.
Global Instance into_laterN_later only_head n n' m' P Q lQ :
  NatCancel n 1 n' m' 
  (** If canceling has failed (i.e. [1 = m']), we should make progress deeper
  into [P], as such, we continue with the [IntoLaterN] class, which is required
  to make progress. If canceling has succeeded, we do not need to make further
  progress, but there may still be a left-over (i.e. [n']) to cancel more deeply
  into [P], as such, we continue with [MaybeIntoLaterN]. *)
  TCIf (TCEq 1 m') (IntoLaterN only_head n' P Q) (MaybeIntoLaterN only_head n' P Q) 
  MakeLaterN m' Q lQ 
  IntoLaterN only_head n ( P) lQ | 2.
Proof.
  rewrite /MakeLaterN /IntoLaterN /MaybeIntoLaterN /NatCancel.
  move=> Hn [_ ->|->] <-;
    by rewrite -later_laterN -laterN_plus -Hn Nat.add_comm.
Qed.
Global Instance into_laterN_laterN only_head n m n' m' P Q lQ :
  NatCancel n m n' m' 
  TCIf (TCEq m m') (IntoLaterN only_head n' P Q) (MaybeIntoLaterN only_head n' P Q) 
  MakeLaterN m' Q lQ 
  IntoLaterN only_head n (^m P) lQ | 1.
Proof.
  rewrite /MakeLaterN /IntoLaterN /MaybeIntoLaterN /NatCancel.
  move=> Hn [_ ->|->] <-; by rewrite -!laterN_plus -Hn Nat.add_comm.
Qed.

Global Instance into_laterN_and_l n P1 P2 Q1 Q2 :
  IntoLaterN false n P1 Q1  MaybeIntoLaterN false n P2 Q2 
  IntoLaterN false n (P1  P2) (Q1  Q2) | 10.
Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> -> ->. by rewrite laterN_and. Qed.
Global Instance into_laterN_and_r n P P2 Q2 :
  IntoLaterN false n P2 Q2  IntoLaterN false n (P  P2) (P  Q2) | 11.
Proof.
  rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_and -(laterN_intro _ P).
Qed.

Global Instance into_laterN_forall {A} n (Φ Ψ : A  PROP) :
  ( x, IntoLaterN false n (Φ x) (Ψ x)) 
  IntoLaterN false n ( x, Φ x) ( x, Ψ x).
Proof. rewrite /IntoLaterN /MaybeIntoLaterN laterN_forall=> ?. by apply forall_mono. Qed.
Global Instance into_laterN_exist {A} n (Φ Ψ : A  PROP) :
  ( x, IntoLaterN false n (Φ x) (Ψ x)) 
  IntoLaterN false n ( x, Φ x) ( x, Ψ x).
Proof. rewrite /IntoLaterN /MaybeIntoLaterN -laterN_exist_2=> ?. by apply exist_mono. Qed.

Global Instance into_laterN_or_l n P1 P2 Q1 Q2 :
  IntoLaterN false n P1 Q1  MaybeIntoLaterN false n P2 Q2 
  IntoLaterN false n (P1  P2) (Q1  Q2) | 10.
Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> -> ->. by rewrite laterN_or. Qed.
Global Instance into_laterN_or_r n P P2 Q2 :
  IntoLaterN false n P2 Q2 
  IntoLaterN false n (P  P2) (P  Q2) | 11.
Proof.
  rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_or -(laterN_intro _ P).
Qed.

Global Instance into_later_affinely n P Q :
  IntoLaterN false n P Q  IntoLaterN false n (<affine> P) (<affine> Q).
Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_affinely_2. Qed.
Global Instance into_later_intuitionistically n P Q :
  IntoLaterN false n P Q  IntoLaterN false n ( P) ( Q).
Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_intuitionistically_2. Qed.
Global Instance into_later_absorbingly n P Q :
  IntoLaterN false n P Q  IntoLaterN false n (<absorb> P) (<absorb> Q).
Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_absorbingly. Qed.
Global Instance into_later_plainly `{BiPlainly PROP} n P Q :
  IntoLaterN false n P Q  IntoLaterN false n ( P) ( Q).
Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_plainly. Qed.
Global Instance into_later_persistently n P Q :
  IntoLaterN false n P Q  IntoLaterN false n (<pers> P) (<pers> Q).
Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_persistently. Qed.
Global Instance into_later_embed`{SbiEmbed PROP PROP'} n P Q :
  IntoLaterN false n P Q  IntoLaterN false n P Q.
Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite embed_laterN. Qed.

Global Instance into_laterN_sep_l n P1 P2 Q1 Q2 :
  IntoLaterN false n P1 Q1  MaybeIntoLaterN false n P2 Q2 
  IntoLaterN false n (P1  P2) (Q1  Q2) | 10.
Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> -> ->. by rewrite laterN_sep. Qed.
Global Instance into_laterN_sep_r n P P2 Q2 :
  IntoLaterN false n P2 Q2 
  IntoLaterN false n (P  P2) (P  Q2) | 11.
Proof.
  rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_sep -(laterN_intro _ P).
Qed.

Global Instance into_laterN_big_sepL n {A} (Φ Ψ : nat  A  PROP) (l: list A) :
  ( x k, IntoLaterN false n (Φ k x) (Ψ k x)) 
  IntoLaterN false n ([ list] k  x  l, Φ k x) ([ list] k  x  l, Ψ k x).
Proof.
  rewrite /IntoLaterN /MaybeIntoLaterN=> ?.
  rewrite big_opL_commute. by apply big_sepL_mono.
Qed.
Global Instance into_laterN_big_sepM n `{Countable K} {A}
    (Φ Ψ : K  A  PROP) (m : gmap K A) :
  ( x k, IntoLaterN false n (Φ k x) (Ψ k x)) 
  IntoLaterN false n ([ map] k  x  m, Φ k x) ([ map] k  x  m, Ψ k x).
Proof.
  rewrite /IntoLaterN /MaybeIntoLaterN=> ?.
  rewrite big_opM_commute. by apply big_sepM_mono.
Qed.
Global Instance into_laterN_big_sepS n `{Countable A}
    (Φ Ψ : A  PROP) (X : gset A) :
  ( x, IntoLaterN false n (Φ x) (Ψ x)) 
  IntoLaterN false n ([ set] x  X, Φ x) ([ set] x  X, Ψ x).
Proof.
  rewrite /IntoLaterN /MaybeIntoLaterN=> ?.
  rewrite big_opS_commute. by apply big_sepS_mono.
Qed.
Global Instance into_laterN_big_sepMS n `{Countable A}
    (Φ Ψ : A  PROP) (X : gmultiset A) :
  ( x, IntoLaterN false n (Φ x) (Ψ x)) 
  IntoLaterN false n ([ mset] x  X, Φ x) ([ mset] x  X, Ψ x).
Proof.
  rewrite /IntoLaterN /MaybeIntoLaterN=> ?.
  rewrite big_opMS_commute. by apply big_sepMS_mono.
Qed.
End sbi_instances.