class_instances.v 35.9 KB
Newer Older
1
From iris.proofmode Require Export classes.
2
From iris.algebra Require Import gmap.
Ralf Jung's avatar
Ralf Jung committed
3
From stdpp Require Import gmultiset.
4
From iris.base_logic Require Import big_op tactics.
5
Set Default Proof Using "Type".
6 7 8 9 10 11 12
Import uPred.

Section classes.
Context {M : ucmraT}.
Implicit Types P Q R : uPred M.

(* FromAssumption *)
13
Global Instance from_assumption_exact p P : FromAssumption p P P | 0.
14
Proof. destruct p; by rewrite /FromAssumption /= ?always_elim. Qed.
15 16 17
Global Instance from_assumption_False p P : FromAssumption p False P | 1.
Proof. destruct p; rewrite /FromAssumption /= ?always_pure; apply False_elim. Qed.

18 19 20
Global Instance from_assumption_always_r P Q :
  FromAssumption true P Q  FromAssumption true P ( Q).
Proof. rewrite /FromAssumption=><-. by rewrite always_always. Qed.
21 22 23 24

Global Instance from_assumption_always_l p P Q :
  FromAssumption p P Q  FromAssumption p ( P) Q.
Proof. rewrite /FromAssumption=><-. by rewrite always_elim. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
25 26 27 28 29 30
Global Instance from_assumption_later p P Q :
  FromAssumption p P Q  FromAssumption p P ( Q)%I.
Proof. rewrite /FromAssumption=>->. apply later_intro. Qed.
Global Instance from_assumption_laterN n p P Q :
  FromAssumption p P Q  FromAssumption p P (^n Q)%I.
Proof. rewrite /FromAssumption=>->. apply laterN_intro. Qed.
31 32 33
Global Instance from_assumption_except_0 p P Q :
  FromAssumption p P Q  FromAssumption p P ( Q)%I.
Proof. rewrite /FromAssumption=>->. apply except_0_intro. Qed.
34
Global Instance from_assumption_bupd p P Q :
35
  FromAssumption p P Q  FromAssumption p P (|==> Q)%I.
36
Proof. rewrite /FromAssumption=>->. apply bupd_intro. Qed.
37 38 39
Global Instance from_assumption_forall {A} p (Φ : A  uPred M) Q x :
  FromAssumption p (Φ x) Q  FromAssumption p ( x, Φ x) Q.
Proof. rewrite /FromAssumption=> <-. by rewrite forall_elim. Qed.
40 41

(* IntoPure *)
Ralf Jung's avatar
Ralf Jung committed
42
Global Instance into_pure_pure φ : @IntoPure M ⌜φ⌝ φ.
43
Proof. done. Qed.
44
Global Instance into_pure_eq {A : ofeT} (a b : A) :
45 46
  Discrete a  @IntoPure M (a  b) (a  b).
Proof. intros. by rewrite /IntoPure discrete_eq. Qed.
47 48
Global Instance into_pure_cmra_valid `{CMRADiscrete A} (a : A) :
  @IntoPure M ( a) ( a).
49 50
Proof. by rewrite /IntoPure discrete_valid. Qed.

51 52 53
Global Instance into_pure_always P φ : IntoPure P φ  IntoPure ( P) φ.
Proof. rewrite /IntoPure=> ->. by rewrite always_pure. Qed.

Ralf Jung's avatar
Ralf Jung committed
54
Global Instance into_pure_pure_and (φ1 φ2 : Prop) P1 P2 :
Ralf Jung's avatar
Ralf Jung committed
55
  IntoPure P1 φ1  IntoPure P2 φ2  IntoPure (P1  P2) (φ1  φ2).
56
Proof. rewrite /IntoPure pure_and. by intros -> ->. Qed.
Ralf Jung's avatar
Ralf Jung committed
57
Global Instance into_pure_pure_sep (φ1 φ2 : Prop) P1 P2 :
Ralf Jung's avatar
Ralf Jung committed
58
  IntoPure P1 φ1  IntoPure P2 φ2  IntoPure (P1  P2) (φ1  φ2).
59
Proof. rewrite /IntoPure sep_and pure_and. by intros -> ->. Qed.
Ralf Jung's avatar
Ralf Jung committed
60
Global Instance into_pure_pure_or (φ1 φ2 : Prop) P1 P2 :
Ralf Jung's avatar
Ralf Jung committed
61
  IntoPure P1 φ1  IntoPure P2 φ2  IntoPure (P1  P2) (φ1  φ2).
62
Proof. rewrite /IntoPure pure_or. by intros -> ->. Qed.
Ralf Jung's avatar
Ralf Jung committed
63
Global Instance into_pure_pure_impl (φ1 φ2 : Prop) P1 P2 :
Ralf Jung's avatar
Ralf Jung committed
64
  FromPure P1 φ1  IntoPure P2 φ2  IntoPure (P1  P2) (φ1  φ2).
65
Proof. rewrite /FromPure /IntoPure pure_impl. by intros -> ->. Qed.
Ralf Jung's avatar
Ralf Jung committed
66
Global Instance into_pure_pure_wand (φ1 φ2 : Prop) P1 P2 :
Ralf Jung's avatar
Ralf Jung committed
67
  FromPure P1 φ1  IntoPure P2 φ2  IntoPure (P1 - P2) (φ1  φ2).
68 69 70 71
Proof.
  rewrite /FromPure /IntoPure pure_impl always_impl_wand. by intros -> ->.
Qed.

Ralf Jung's avatar
Ralf Jung committed
72
Global Instance into_pure_exist {X : Type} (Φ : X  uPred M) (φ : X  Prop) :
73 74 75 76 77 78
  ( x, @IntoPure M (Φ x) (φ x))  @IntoPure M ( x, Φ x) ( x, φ x).
Proof.
  rewrite /IntoPure=>Hx. apply exist_elim=>x. rewrite Hx.
  apply pure_elim'=>Hφ. apply pure_intro. eauto.
Qed.

Ralf Jung's avatar
Ralf Jung committed
79
Global Instance into_pure_forall {X : Type} (Φ : X  uPred M) (φ : X  Prop) :
80 81 82 83 84
  ( x, @IntoPure M (Φ x) (φ x))  @IntoPure M ( x, Φ x) ( x, φ x).
Proof.
  rewrite /IntoPure=>Hx. rewrite -pure_forall_2. by setoid_rewrite Hx.
Qed.

85
(* FromPure *)
Ralf Jung's avatar
Ralf Jung committed
86
Global Instance from_pure_pure φ : @FromPure M ⌜φ⌝ φ.
87
Proof. done. Qed.
88
Global Instance from_pure_internal_eq {A : ofeT} (a b : A) :
89 90 91 92
  @FromPure M (a  b) (a  b).
Proof.
  rewrite /FromPure. eapply pure_elim; [done|]=> ->. apply internal_eq_refl'.
Qed.
93 94
Global Instance from_pure_cmra_valid {A : cmraT} (a : A) :
  @FromPure M ( a) ( a).
95 96
Proof.
  rewrite /FromPure. eapply pure_elim; [done|]=> ?.
97
  rewrite -cmra_valid_intro //. auto with I.
98
Qed.
99 100 101 102 103 104 105 106 107

Global Instance from_pure_always P φ : FromPure P φ  FromPure ( P) φ.
Proof. rewrite /FromPure=> <-. by rewrite always_pure. Qed.
Global Instance from_pure_later P φ : FromPure P φ  FromPure ( P) φ.
Proof. rewrite /FromPure=> ->. apply later_intro. Qed.
Global Instance from_pure_laterN n P φ : FromPure P φ  FromPure (^n P) φ.
Proof. rewrite /FromPure=> ->. apply laterN_intro. Qed.
Global Instance from_pure_except_0 P φ : FromPure P φ  FromPure ( P) φ.
Proof. rewrite /FromPure=> ->. apply except_0_intro. Qed.
108
Global Instance from_pure_bupd P φ : FromPure P φ  FromPure (|==> P) φ.
109
Proof. rewrite /FromPure=> ->. apply bupd_intro. Qed.
110

Ralf Jung's avatar
Ralf Jung committed
111
Global Instance from_pure_pure_and (φ1 φ2 : Prop) P1 P2 :
112
  FromPure P1 φ1  FromPure P2 φ2  FromPure (P1  P2) (φ1  φ2).
113
Proof. rewrite /FromPure pure_and. by intros -> ->. Qed.
Ralf Jung's avatar
Ralf Jung committed
114
Global Instance from_pure_pure_sep (φ1 φ2 : Prop) P1 P2 :
115
  FromPure P1 φ1  FromPure P2 φ2  FromPure (P1  P2) (φ1  φ2).
116
Proof. rewrite /FromPure pure_and always_and_sep_l. by intros -> ->. Qed.
Ralf Jung's avatar
Ralf Jung committed
117
Global Instance from_pure_pure_or (φ1 φ2 : Prop) P1 P2 :
118
  FromPure P1 φ1  FromPure P2 φ2  FromPure (P1  P2) (φ1  φ2).
119
Proof. rewrite /FromPure pure_or. by intros -> ->. Qed.
Ralf Jung's avatar
Ralf Jung committed
120
Global Instance from_pure_pure_impl (φ1 φ2 : Prop) P1 P2 :
121
  IntoPure P1 φ1  FromPure P2 φ2  FromPure (P1  P2) (φ1  φ2).
122
Proof. rewrite /FromPure /IntoPure pure_impl. by intros -> ->. Qed.
Ralf Jung's avatar
Ralf Jung committed
123
Global Instance from_pure_pure_wand (φ1 φ2 : Prop) P1 P2 :
124
  IntoPure P1 φ1  FromPure P2 φ2  FromPure (P1 - P2) (φ1  φ2).
125 126 127 128
Proof.
  rewrite /FromPure /IntoPure pure_impl always_impl_wand. by intros -> ->.
Qed.

Ralf Jung's avatar
Ralf Jung committed
129
Global Instance from_pure_exist {X : Type} (Φ : X  uPred M) (φ : X  Prop) :
130 131 132 133 134
  ( x, @FromPure M (Φ x) (φ x))  @FromPure M ( x, Φ x) ( x, φ x).
Proof.
  rewrite /FromPure=>Hx. apply pure_elim'=>-[x ?]. rewrite -(exist_intro x).
  rewrite -Hx. apply pure_intro. done.
Qed.
Ralf Jung's avatar
Ralf Jung committed
135
Global Instance from_pure_forall {X : Type} (Φ : X  uPred M) (φ : X  Prop) :
136 137 138 139 140 141
  ( x, @FromPure M (Φ x) (φ x))  @FromPure M ( x, Φ x) ( x, φ x).
Proof.
  rewrite /FromPure=>Hx. apply forall_intro=>x. apply pure_elim'=>Hφ.
  rewrite -Hx. apply pure_intro. done.
Qed.

142
(* IntoPersistentP *)
143 144 145 146
Global Instance into_persistentP_always_trans p P Q :
  IntoPersistentP true P Q  IntoPersistentP p ( P) Q | 0.
Proof. rewrite /IntoPersistentP /==> ->. by rewrite always_if_always. Qed.
Global Instance into_persistentP_always P : IntoPersistentP true P P | 1.
147 148
Proof. done. Qed.
Global Instance into_persistentP_persistent P :
149
  PersistentP P  IntoPersistentP false P P | 100.
150 151 152
Proof. done. Qed.

(* IntoLater *)
153
Global Instance into_laterN_later n P Q :
154 155 156
  IntoLaterN n P Q  IntoLaterN' (S n) ( P) Q.
Proof. by rewrite /IntoLaterN' /IntoLaterN =>->. Qed.
Global Instance into_laterN_laterN n P : IntoLaterN' n (^n P) P.
157
Proof. done. Qed.
158
Global Instance into_laterN_laterN_plus n m P Q :
159 160
  IntoLaterN m P Q  IntoLaterN' (n + m) (^n P) Q.
Proof. rewrite /IntoLaterN' /IntoLaterN=>->. by rewrite laterN_plus. Qed.
161

162
Global Instance into_laterN_and_l n P1 P2 Q1 Q2 :
163
  IntoLaterN' n P1 Q1  IntoLaterN n P2 Q2 
Robbert Krebbers's avatar
Robbert Krebbers committed
164
  IntoLaterN' n (P1  P2) (Q1  Q2) | 10.
165
Proof. rewrite /IntoLaterN' /IntoLaterN=> -> ->. by rewrite laterN_and. Qed.
166
Global Instance into_laterN_and_r n P P2 Q2 :
Robbert Krebbers's avatar
Robbert Krebbers committed
167
  IntoLaterN' n P2 Q2  IntoLaterN' n (P  P2) (P  Q2) | 11.
168
Proof.
169
  rewrite /IntoLaterN' /IntoLaterN=> ->. by rewrite laterN_and -(laterN_intro _ P).
Robbert Krebbers's avatar
Robbert Krebbers committed
170
Qed.
171

Robbert Krebbers's avatar
Robbert Krebbers committed
172 173 174 175 176 177 178 179
Global Instance into_later_forall {A} n (Φ Ψ : A  uPred M) :
  ( x, IntoLaterN' n (Φ x) (Ψ x))  IntoLaterN' n ( x, Φ x) ( x, Ψ x).
Proof. rewrite /IntoLaterN' /IntoLaterN laterN_forall=> ?. by apply forall_mono. Qed.
Global Instance into_later_exist {A} n (Φ Ψ : A  uPred M) :
  ( x, IntoLaterN' n (Φ x) (Ψ x)) 
  IntoLaterN' n ( x, Φ x) ( x, Ψ x).
Proof. rewrite /IntoLaterN' /IntoLaterN -laterN_exist_2=> ?. by apply exist_mono. Qed.

180
Global Instance into_laterN_or_l n P1 P2 Q1 Q2 :
181
  IntoLaterN' n P1 Q1  IntoLaterN n P2 Q2 
Robbert Krebbers's avatar
Robbert Krebbers committed
182
  IntoLaterN' n (P1  P2) (Q1  Q2) | 10.
183
Proof. rewrite /IntoLaterN' /IntoLaterN=> -> ->. by rewrite laterN_or. Qed.
184
Global Instance into_laterN_or_r n P P2 Q2 :
185
  IntoLaterN' n P2 Q2 
Robbert Krebbers's avatar
Robbert Krebbers committed
186
  IntoLaterN' n (P  P2) (P  Q2) | 11.
187
Proof.
188
  rewrite /IntoLaterN' /IntoLaterN=> ->. by rewrite laterN_or -(laterN_intro _ P).
189 190 191
Qed.

Global Instance into_laterN_sep_l n P1 P2 Q1 Q2 :
192
  IntoLaterN' n P1 Q1  IntoLaterN n P2 Q2 
Robbert Krebbers's avatar
Robbert Krebbers committed
193 194
  IntoLaterN' n (P1  P2) (Q1  Q2) | 10.
Proof. rewrite /IntoLaterN' /IntoLaterN=> -> ->. by rewrite laterN_sep. Qed.
195
Global Instance into_laterN_sep_r n P P2 Q2 :
196
  IntoLaterN' n P2 Q2 
Robbert Krebbers's avatar
Robbert Krebbers committed
197
  IntoLaterN' n (P  P2) (P  Q2) | 11.
198
Proof.
199
  rewrite /IntoLaterN' /IntoLaterN=> ->. by rewrite laterN_sep -(laterN_intro _ P).
200
Qed.
201 202

Global Instance into_laterN_big_sepL n {A} (Φ Ψ : nat  A  uPred M) (l: list A) :
203 204
  ( x k, IntoLaterN' n (Φ k x) (Ψ k x)) 
  IntoLaterN' n ([ list] k  x  l, Φ k x) ([ list] k  x  l, Ψ k x).
205
Proof.
206
  rewrite /IntoLaterN' /IntoLaterN=> ?.
207
  rewrite big_opL_commute. by apply big_sepL_mono.
208 209
Qed.
Global Instance into_laterN_big_sepM n `{Countable K} {A}
210
    (Φ Ψ : K  A  uPred M) (m : gmap K A) :
211 212
  ( x k, IntoLaterN' n (Φ k x) (Ψ k x)) 
  IntoLaterN' n ([ map] k  x  m, Φ k x) ([ map] k  x  m, Ψ k x).
213
Proof.
214
  rewrite /IntoLaterN' /IntoLaterN=> ?.
215
  rewrite big_opM_commute; by apply big_sepM_mono.
216
Qed.
217
Global Instance into_laterN_big_sepS n `{Countable A}
218
    (Φ Ψ : A  uPred M) (X : gset A) :
219 220
  ( x, IntoLaterN' n (Φ x) (Ψ x)) 
  IntoLaterN' n ([ set] x  X, Φ x) ([ set] x  X, Ψ x).
221
Proof.
222
  rewrite /IntoLaterN' /IntoLaterN=> ?.
223
  rewrite big_opS_commute; by apply big_sepS_mono.
224 225 226
Qed.
Global Instance into_laterN_big_sepMS n `{Countable A}
    (Φ Ψ : A  uPred M) (X : gmultiset A) :
227 228
  ( x, IntoLaterN' n (Φ x) (Ψ x)) 
  IntoLaterN' n ([ mset] x  X, Φ x) ([ mset] x  X, Ψ x).
229
Proof.
230
  rewrite /IntoLaterN' /IntoLaterN=> ?.
231
  rewrite big_opMS_commute; by apply big_sepMS_mono.
232 233 234
Qed.

(* FromLater *)
235
Global Instance from_laterN_later P : FromLaterN 1 ( P) P | 0.
236
Proof. done. Qed.
237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259
Global Instance from_laterN_laterN n P : FromLaterN n (^n P) P | 0.
Proof. done. Qed.

(* The instances below are used when stripping a specific number of laters, or
to balance laters in different branches of ∧, ∨ and ∗. *)
Global Instance from_laterN_0 P : FromLaterN 0 P P | 100. (* fallthrough *)
Proof. done. Qed.
Global Instance from_laterN_later_S n P Q :
  FromLaterN n P Q  FromLaterN (S n) ( P) Q.
Proof. by rewrite /FromLaterN=><-. Qed.
Global Instance from_laterN_later_plus n m P Q :
  FromLaterN m P Q  FromLaterN (n + m) (^n P) Q.
Proof. rewrite /FromLaterN=><-. by rewrite laterN_plus. Qed.

Global Instance from_later_and n P1 P2 Q1 Q2 :
  FromLaterN n P1 Q1  FromLaterN n P2 Q2  FromLaterN n (P1  P2) (Q1  Q2).
Proof. intros ??; red. by rewrite laterN_and; apply and_mono. Qed.
Global Instance from_later_or n P1 P2 Q1 Q2 :
  FromLaterN n P1 Q1  FromLaterN n P2 Q2  FromLaterN n (P1  P2) (Q1  Q2).
Proof. intros ??; red. by rewrite laterN_or; apply or_mono. Qed.
Global Instance from_later_sep n P1 P2 Q1 Q2 :
  FromLaterN n P1 Q1  FromLaterN n P2 Q2  FromLaterN n (P1  P2) (Q1  Q2).
Proof. intros ??; red. by rewrite laterN_sep; apply sep_mono. Qed.
260

261 262 263 264 265 266 267 268 269 270 271 272
Global Instance from_later_always n P Q :
  FromLaterN n P Q  FromLaterN n ( P) ( Q).
Proof. by rewrite /FromLaterN -always_laterN=> ->. Qed.

Global Instance from_later_forall {A} n (Φ Ψ : A  uPred M) :
  ( x, FromLaterN n (Φ x) (Ψ x))  FromLaterN n ( x, Φ x) ( x, Ψ x).
Proof. rewrite /FromLaterN laterN_forall=> ?. by apply forall_mono. Qed.
Global Instance from_later_exist {A} n (Φ Ψ : A  uPred M) :
  Inhabited A  ( x, FromLaterN n (Φ x) (Ψ x)) 
  FromLaterN n ( x, Φ x) ( x, Ψ x).
Proof. intros ?. rewrite /FromLaterN laterN_exist=> ?. by apply exist_mono. Qed.

273
(* IntoWand *)
274 275
Global Instance wand_weaken_assumption p P1 P2 Q :
  FromAssumption p P2 P1  WandWeaken p P1 Q P2 Q | 0.
276
Proof. by rewrite /WandWeaken /FromAssumption /= =>->. Qed.
277 278
Global Instance wand_weaken_later p P Q P' Q' :
  WandWeaken p P Q P' Q'  WandWeaken' p P Q ( P') ( Q').
Robbert Krebbers's avatar
Robbert Krebbers committed
279
Proof.
280 281
  rewrite /WandWeaken' /WandWeaken=> ->.
  by rewrite always_if_later -later_wand -later_intro.
Robbert Krebbers's avatar
Robbert Krebbers committed
282
Qed.
283 284
Global Instance wand_weaken_laterN p n P Q P' Q' :
  WandWeaken p P Q P' Q'  WandWeaken' p P Q (^n P') (^n Q').
Robbert Krebbers's avatar
Robbert Krebbers committed
285
Proof.
286 287
  rewrite /WandWeaken' /WandWeaken=> ->.
  by rewrite always_if_laterN -laterN_wand -laterN_intro.
Robbert Krebbers's avatar
Robbert Krebbers committed
288
Qed.
289 290
Global Instance bupd_weaken_laterN p P Q P' Q' :
  WandWeaken false P Q P' Q'  WandWeaken' p P Q (|==> P') (|==> Q').
Robbert Krebbers's avatar
Robbert Krebbers committed
291 292
Proof.
  rewrite /WandWeaken' /WandWeaken=> ->.
293
  apply wand_intro_l. by rewrite always_if_elim bupd_wand_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
294 295
Qed.

296 297
Global Instance into_wand_wand p P P' Q Q' :
  WandWeaken p P Q P' Q'  IntoWand p (P - Q) P' Q'.
Robbert Krebbers's avatar
Robbert Krebbers committed
298
Proof. done. Qed.
299 300
Global Instance into_wand_impl p P P' Q Q' :
  WandWeaken p P Q P' Q'  IntoWand p (P  Q) P' Q'.
Robbert Krebbers's avatar
Robbert Krebbers committed
301 302
Proof. rewrite /WandWeaken /IntoWand /= => <-. apply impl_wand. Qed.

303 304
Global Instance into_wand_iff_l p P P' Q Q' :
  WandWeaken p P Q P' Q'  IntoWand p (P  Q) P' Q'.
Robbert Krebbers's avatar
Robbert Krebbers committed
305
Proof. rewrite /WandWeaken /IntoWand=> <-. apply and_elim_l', impl_wand. Qed.
306 307
Global Instance into_wand_iff_r p P P' Q Q' :
  WandWeaken p Q P Q' P'  IntoWand p (P  Q) Q' P'.
Robbert Krebbers's avatar
Robbert Krebbers committed
308
Proof. rewrite /WandWeaken /IntoWand=> <-. apply and_elim_r', impl_wand. Qed.
309

310 311
Global Instance into_wand_forall {A} p (Φ : A  uPred M) P Q x :
  IntoWand p (Φ x) P Q  IntoWand p ( x, Φ x) P Q.
312
Proof. rewrite /IntoWand=> <-. apply forall_elim. Qed.
313 314
Global Instance into_wand_always p R P Q :
  IntoWand p R P Q  IntoWand p ( R) P Q.
315
Proof. rewrite /IntoWand=> ->. apply always_elim. Qed.
316

317 318 319 320 321 322 323
Global Instance into_wand_later p R P Q :
  IntoWand p R P Q  IntoWand p ( R) ( P) ( Q).
Proof. rewrite /IntoWand=> ->. by rewrite always_if_later -later_wand. Qed.
Global Instance into_wand_laterN p n R P Q :
  IntoWand p R P Q  IntoWand p (^n R) (^n P) (^n Q).
Proof. rewrite /IntoWand=> ->. by rewrite always_if_laterN -laterN_wand. Qed.

324
Global Instance into_wand_bupd R P Q :
325
  IntoWand false R P Q  IntoWand false (|==> R) (|==> P) (|==> Q).
326
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
327
  rewrite /IntoWand=> ->. apply wand_intro_l. by rewrite bupd_sep wand_elim_r.
328
Qed.
329 330 331 332 333
Global Instance into_wand_bupd_persistent R P Q :
  IntoWand true R P Q  IntoWand true (|==> R) P (|==> Q).
Proof.
  rewrite /IntoWand=>->. apply wand_intro_l. by rewrite bupd_frame_l wand_elim_r.
Qed.
334 335

(* FromAnd *)
336 337 338 339
Global Instance from_and_and p P1 P2 : FromAnd p (P1  P2) P1 P2 | 100.
Proof. by apply mk_from_and_and. Qed.

Global Instance from_and_sep P1 P2 : FromAnd false (P1  P2) P1 P2 | 100.
340 341
Proof. done. Qed.
Global Instance from_and_sep_persistent_l P1 P2 :
342
  PersistentP P1  FromAnd true (P1  P2) P1 P2 | 9.
343 344
Proof. intros. by rewrite /FromAnd always_and_sep_l. Qed.
Global Instance from_and_sep_persistent_r P1 P2 :
345
  PersistentP P2  FromAnd true (P1  P2) P1 P2 | 10.
346
Proof. intros. by rewrite /FromAnd always_and_sep_r. Qed.
347 348 349 350 351 352 353 354 355 356 357 358 359 360 361

Global Instance from_and_pure p φ ψ : @FromAnd M p ⌜φ  ψ⌝ ⌜φ⌝ ⌜ψ⌝.
Proof. apply mk_from_and_and. by rewrite pure_and. Qed.
Global Instance from_and_always p P Q1 Q2 :
  FromAnd false P Q1 Q2  FromAnd p ( P) ( Q1) ( Q2).
Proof.
  intros. apply mk_from_and_and.
  by rewrite always_and_sep_l' -always_sep -(from_and _ P).
Qed.
Global Instance from_and_later p P Q1 Q2 :
  FromAnd p P Q1 Q2  FromAnd p ( P) ( Q1) ( Q2).
Proof. rewrite /FromAnd=> <-. destruct p; by rewrite ?later_and ?later_sep. Qed.
Global Instance from_and_laterN p n P Q1 Q2 :
  FromAnd p P Q1 Q2  FromAnd p (^n P) (^n Q1) (^n Q2).
Proof. rewrite /FromAnd=> <-. destruct p; by rewrite ?laterN_and ?laterN_sep. Qed.
362 363 364 365 366
Global Instance from_and_except_0 p P Q1 Q2 :
  FromAnd p P Q1 Q2  FromAnd p ( P) ( Q1) ( Q2).
Proof.
  rewrite /FromAnd=><-. by destruct p; rewrite ?except_0_and ?except_0_sep.
Qed.
367

368
Global Instance from_sep_ownM (a b1 b2 : M) :
369
  IsOp a b1 b2 
370
  FromAnd false (uPred_ownM a) (uPred_ownM b1) (uPred_ownM b2).
371
Proof. intros. by rewrite /FromAnd -ownM_op -is_op. Qed.
372
Global Instance from_sep_ownM_persistent (a b1 b2 : M) :
373
  IsOp a b1 b2  Or (Persistent b1) (Persistent b2) 
374 375 376
  FromAnd true (uPred_ownM a) (uPred_ownM b1) (uPred_ownM b2).
Proof.
  intros ? Hper; apply mk_from_and_persistent; [destruct Hper; apply _|].
377
  by rewrite -ownM_op -is_op.
378
Qed.
379

380
Global Instance from_sep_bupd P Q1 Q2 :
381 382 383
  FromAnd false P Q1 Q2  FromAnd false (|==> P) (|==> Q1) (|==> Q2).
Proof. rewrite /FromAnd=><-. apply bupd_sep. Qed.

384 385 386 387 388 389 390 391
Global Instance from_and_big_sepL_cons {A} (Φ : nat  A  uPred M) x l :
  FromAnd false ([ list] k  y  x :: l, Φ k y) (Φ 0 x) ([ list] k  y  l, Φ (S k) y).
Proof. by rewrite /FromAnd big_sepL_cons. Qed.
Global Instance from_and_big_sepL_cons_persistent {A} (Φ : nat  A  uPred M) x l :
  PersistentP (Φ 0 x) 
  FromAnd true ([ list] k  y  x :: l, Φ k y) (Φ 0 x) ([ list] k  y  l, Φ (S k) y).
Proof. intros. by rewrite /FromAnd big_opL_cons always_and_sep_l. Qed.

392 393
Global Instance from_and_big_sepL_app {A} (Φ : nat  A  uPred M) l1 l2 :
  FromAnd false ([ list] k  y  l1 ++ l2, Φ k y)
394
    ([ list] k  y  l1, Φ k y) ([ list] k  y  l2, Φ (length l1 + k) y).
395
Proof. by rewrite /FromAnd big_opL_app. Qed.
396 397 398 399 400
Global Instance from_sep_big_sepL_app_persistent {A} (Φ : nat  A  uPred M) l1 l2 :
  ( k y, PersistentP (Φ k y)) 
  FromAnd true ([ list] k  y  l1 ++ l2, Φ k y)
    ([ list] k  y  l1, Φ k y) ([ list] k  y  l2, Φ (length l1 + k) y).
Proof. intros. by rewrite /FromAnd big_opL_app always_and_sep_l. Qed.
401

402
(* FromOp *)
403 404
(* TODO: Worst case there could be a lot of backtracking on these instances,
try to refactor. *)
405
Global Instance is_op_pair {A B : cmraT} (a b1 b2 : A) (a' b1' b2' : B) :
Robbert Krebbers's avatar
Robbert Krebbers committed
406
  IsOp a b1 b2  IsOp a' b1' b2'  IsOp' (a,a') (b1,b1') (b2,b2').
407
Proof. by constructor. Qed.
408 409
Global Instance is_op_pair_persistent_l {A B : cmraT} (a : A) (a' b1' b2' : B) :
  Persistent a  IsOp a' b1' b2'  IsOp' (a,a') (a,b1') (a,b2').
410
Proof. constructor=> //=. by rewrite -persistent_dup. Qed.
411 412
Global Instance is_op_pair_persistent_r {A B : cmraT} (a b1 b2 : A) (a' : B) :
  Persistent a'  IsOp a b1 b2  IsOp' (a,a') (b1,a') (b2,a').
413 414
Proof. constructor=> //=. by rewrite -persistent_dup. Qed.

415 416
Global Instance is_op_Some {A : cmraT} (a : A) b1 b2 :
  IsOp a b1 b2  IsOp' (Some a) (Some b1) (Some b2).
417
Proof. by constructor. Qed.
418 419 420 421
(* This one has a higher precendence than [is_op_op] so we get a [+] instead of
an [⋅]. *)
Global Instance is_op_plus (n1 n2 : nat) : IsOp (n1 + n2) n1 n2.
Proof. done. Qed.
422

423
(* IntoAnd *)
424
Global Instance into_and_sep p P Q : IntoAnd p (P  Q) P Q.
425 426
Proof. by apply mk_into_and_sep. Qed.
Global Instance into_and_ownM p (a b1 b2 : M) :
427 428
  IsOp a b1 b2  IntoAnd p (uPred_ownM a) (uPred_ownM b1) (uPred_ownM b2).
Proof. intros. apply mk_into_and_sep. by rewrite (is_op a) ownM_op. Qed.
429

430
Global Instance into_and_and P Q : IntoAnd true (P  Q) P Q.
431
Proof. done. Qed.
432 433 434 435 436 437 438
Global Instance into_and_and_persistent_l P Q :
  PersistentP P  IntoAnd false (P  Q) P Q.
Proof. intros; by rewrite /IntoAnd /= always_and_sep_l. Qed.
Global Instance into_and_and_persistent_r P Q :
  PersistentP Q  IntoAnd false (P  Q) P Q.
Proof. intros; by rewrite /IntoAnd /= always_and_sep_r. Qed.

439 440 441 442 443 444 445
Global Instance into_and_pure p φ ψ : @IntoAnd M p ⌜φ  ψ⌝ ⌜φ⌝ ⌜ψ⌝.
Proof. apply mk_into_and_sep. by rewrite pure_and always_and_sep_r. Qed.
Global Instance into_and_always p P Q1 Q2 :
  IntoAnd true P Q1 Q2  IntoAnd p ( P) ( Q1) ( Q2).
Proof.
  rewrite /IntoAnd=>->. destruct p; by rewrite ?always_and always_and_sep_r.
Qed.
446 447 448
Global Instance into_and_later p P Q1 Q2 :
  IntoAnd p P Q1 Q2  IntoAnd p ( P) ( Q1) ( Q2).
Proof. rewrite /IntoAnd=>->. destruct p; by rewrite ?later_and ?later_sep. Qed.
449 450 451
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=>->. destruct p; by rewrite ?laterN_and ?laterN_sep. Qed.
452 453 454 455 456
Global Instance into_and_except_0 p P Q1 Q2 :
  IntoAnd p P Q1 Q2  IntoAnd p ( P) ( Q1) ( Q2).
Proof.
  rewrite /IntoAnd=>->. by destruct p; rewrite ?except_0_and ?except_0_sep.
Qed.
457

458 459 460 461 462 463 464 465
(* We use [IsCons] and [IsApp] to make sure that [frame_big_sepL_cons] and
[frame_big_sepL_app] cannot be applied repeatedly often when having
[ [∗ list] k ↦ x ∈ ?e, Φ k x] with [?e] an evar. *)
Global Instance into_and_big_sepL_cons {A} p (Φ : nat  A  uPred M) l x l' :
  IsCons l x l' 
  IntoAnd p ([ list] k  y  l, Φ k y)
    (Φ 0 x) ([ list] k  y  l', Φ (S k) y).
Proof. rewrite /IsCons=>->. apply mk_into_and_sep. by rewrite big_sepL_cons. Qed.
466 467 468
Global Instance into_and_big_sepL_app {A} p (Φ : nat  A  uPred M) l l1 l2 :
  IsApp l l1 l2 
  IntoAnd p ([ list] k  y  l, Φ k y)
469
    ([ list] k  y  l1, Φ k y) ([ list] k  y  l2, Φ (length l1 + k) y).
470
Proof. rewrite /IsApp=>->. apply mk_into_and_sep. by rewrite big_sepL_app. Qed.
471 472

(* Frame *)
473 474 475 476
Global Instance frame_here p R : Frame p R R True.
Proof. by rewrite /Frame right_id always_if_elim. Qed.
Global Instance frame_here_pure p φ Q : FromPure Q φ  Frame p ⌜φ⌝ Q True.
Proof. rewrite /FromPure /Frame=> ->. by rewrite always_if_elim right_id. Qed.
477

478
Class MakeSep (P Q PQ : uPred M) := make_sep : P  Q  PQ.
479 480 481 482
Global Instance make_sep_true_l P : MakeSep True P P.
Proof. by rewrite /MakeSep left_id. Qed.
Global Instance make_sep_true_r P : MakeSep P True P.
Proof. by rewrite /MakeSep right_id. Qed.
483
Global Instance make_sep_default P Q : MakeSep P Q (P  Q) | 100.
484
Proof. done. Qed.
485 486 487 488 489 490 491 492

Global Instance frame_sep_persistent_l R P1 P2 Q1 Q2 Q' :
  Frame true R P1 Q1  MaybeFrame true R P2 Q2  MakeSep Q1 Q2 Q' 
  Frame true R (P1  P2) Q' | 9.
Proof.
  rewrite /Frame /MaybeFrame /MakeSep /= => <- <- <-.
  rewrite {1}(always_sep_dup ( R)). solve_sep_entails.
Qed.
493
Global Instance frame_sep_l R P1 P2 Q Q' :
494
  Frame false R P1 Q  MakeSep Q P2 Q'  Frame false R (P1  P2) Q' | 9.
495
Proof. rewrite /Frame /MakeSep => <- <-. by rewrite assoc. Qed.
496 497 498
Global Instance frame_sep_r p R P1 P2 Q Q' :
  Frame p R P2 Q  MakeSep P1 Q Q'  Frame p R (P1  P2) Q' | 10.
Proof. rewrite /Frame /MakeSep => <- <-. by rewrite assoc -(comm _ P1) assoc. Qed.
499

500 501 502 503 504
Global Instance frame_big_sepL_cons {A} p (Φ : nat  A  uPred M) R Q l x l' :
  IsCons l x l' 
  Frame p R (Φ 0 x  [ list] k  y  l', Φ (S k) y) Q 
  Frame p R ([ list] k  y  l, Φ k y) Q.
Proof. rewrite /IsCons=>->. by rewrite /Frame big_sepL_cons. Qed.
505 506
Global Instance frame_big_sepL_app {A} p (Φ : nat  A  uPred M) R Q l l1 l2 :
  IsApp l l1 l2 
507
  Frame p R (([ list] k  y  l1, Φ k y) 
508
           [ list] k  y  l2, Φ (length l1 + k) y) Q 
509
  Frame p R ([ list] k  y  l, Φ k y) Q.
510
Proof. rewrite /IsApp=>->. by rewrite /Frame big_opL_app. Qed.
511

512 513 514 515 516
Class MakeAnd (P Q PQ : uPred M) := make_and : P  Q  PQ.
Global Instance make_and_true_l P : MakeAnd True P P.
Proof. by rewrite /MakeAnd left_id. Qed.
Global Instance make_and_true_r P : MakeAnd P True P.
Proof. by rewrite /MakeAnd right_id. Qed.
517
Global Instance make_and_default P Q : MakeAnd P Q (P  Q) | 100.
518
Proof. done. Qed.
519 520
Global Instance frame_and_l p R P1 P2 Q Q' :
  Frame p R P1 Q  MakeAnd Q P2 Q'  Frame p R (P1  P2) Q' | 9.
521
Proof. rewrite /Frame /MakeAnd => <- <-; eauto 10 with I. Qed.
522 523
Global Instance frame_and_r p R P1 P2 Q Q' :
  Frame p R P2 Q  MakeAnd P1 Q Q'  Frame p R (P1  P2) Q' | 10.
524 525 526 527 528 529 530 531 532
Proof. rewrite /Frame /MakeAnd => <- <-; eauto 10 with I. Qed.

Class MakeOr (P Q PQ : uPred M) := make_or : P  Q  PQ.
Global Instance make_or_true_l P : MakeOr True P True.
Proof. by rewrite /MakeOr left_absorb. Qed.
Global Instance make_or_true_r P : MakeOr P True True.
Proof. by rewrite /MakeOr right_absorb. Qed.
Global Instance make_or_default P Q : MakeOr P Q (P  Q) | 100.
Proof. done. Qed.
533 534 535 536 537 538 539 540 541 542 543

Global Instance frame_or_persistent_l R P1 P2 Q1 Q2 Q :
  Frame true R P1 Q1  MaybeFrame true R P2 Q2  MakeOr Q1 Q2 Q 
  Frame true R (P1  P2) Q | 9.
Proof. rewrite /Frame /MakeOr => <- <- <-. by rewrite -sep_or_l. Qed.
Global Instance frame_or_persistent_r R P1 P2 Q1 Q2 Q :
  MaybeFrame true R P2 Q2  MakeOr P1 Q2 Q 
  Frame true R (P1  P2) Q | 10.
Proof.
  rewrite /Frame /MaybeFrame /MakeOr => <- <-. by rewrite sep_or_l sep_elim_r.
Qed.
544
Global Instance frame_or R P1 P2 Q1 Q2 Q :
545 546
  Frame false R P1 Q1  Frame false R P2 Q2  MakeOr Q1 Q2 Q 
  Frame false R (P1  P2) Q.
547 548
Proof. rewrite /Frame /MakeOr => <- <- <-. by rewrite -sep_or_l. Qed.

549 550
Global Instance frame_wand p R P1 P2 Q2 :
  Frame p R P2 Q2  Frame p R (P1 - P2) (P1 - Q2).
551 552 553 554 555 556 557 558 559 560 561
Proof.
  rewrite /Frame=> ?. apply wand_intro_l.
  by rewrite assoc (comm _ P1) -assoc wand_elim_r.
Qed.

Class MakeLater (P lP : uPred M) := make_later :  P  lP.
Global Instance make_later_true : MakeLater True True.
Proof. by rewrite /MakeLater later_True. Qed.
Global Instance make_later_default P : MakeLater P ( P) | 100.
Proof. done. Qed.

562 563
Global Instance frame_later p R R' P Q Q' :
  IntoLaterN 1 R' R  Frame p R P Q  MakeLater Q Q'  Frame p R' ( P) Q'.
564
Proof.
565 566
  rewrite /Frame /MakeLater /IntoLaterN=>-> <- <-.
  by rewrite always_if_later later_sep.
567 568 569 570 571 572 573 574
Qed.

Class MakeLaterN (n : nat) (P lP : uPred M) := make_laterN : ^n P  lP.
Global Instance make_laterN_true n : MakeLaterN n True True.
Proof. by rewrite /MakeLaterN laterN_True. Qed.
Global Instance make_laterN_default P : MakeLaterN n P (^n P) | 100.
Proof. done. Qed.

575 576 577 578 579 580 581 582 583 584 585 586 587 588 589
Global Instance frame_laterN p n R R' P Q Q' :
  IntoLaterN n R' R  Frame p R P Q  MakeLaterN n Q Q'  Frame p R' (^n P) Q'.
Proof.
  rewrite /Frame /MakeLater /IntoLaterN=>-> <- <-.
  by rewrite always_if_laterN laterN_sep.
Qed.

Class MakeAlways (P Q : uPred M) := make_always :  P  Q.
Global Instance make_always_true : MakeAlways True True.
Proof. by rewrite /MakeAlways always_pure. Qed.
Global Instance make_always_default P : MakeAlways P ( P) | 100.
Proof. done. Qed.

Global Instance frame_always R P Q Q' :
  Frame true R P Q  MakeAlways Q Q'  Frame true R ( P) Q'.
590
Proof.
591 592
  rewrite /Frame /MakeAlways=> <- <-.
  by rewrite always_sep /= always_always.
593 594
Qed.

595 596 597 598
Class MakeExcept0 (P Q : uPred M) := make_except_0 :  P  Q.
Global Instance make_except_0_True : MakeExcept0 True True.
Proof. by rewrite /MakeExcept0 except_0_True. Qed.
Global Instance make_except_0_default P : MakeExcept0 P ( P) | 100.
599 600
Proof. done. Qed.

601 602
Global Instance frame_except_0 p R P Q Q' :
  Frame p R P Q  MakeExcept0 Q Q'  Frame p R ( P) Q'.
603
Proof.
604
  rewrite /Frame /MakeExcept0=><- <-.
605
  by rewrite except_0_sep -(except_0_intro (?p R)).
606 607
Qed.

608 609
Global Instance frame_exist {A} p R (Φ Ψ : A  uPred M) :
  ( a, Frame p R (Φ a) (Ψ a))  Frame p R ( x, Φ x) ( x, Ψ x).
610
Proof. rewrite /Frame=> ?. by rewrite sep_exist_l; apply exist_mono. Qed.
611 612
Global Instance frame_forall {A} p R (Φ Ψ : A  uPred M) :
  ( a, Frame p R (Φ a) (Ψ a))  Frame p R ( x, Φ x) ( x, Ψ x).
613 614
Proof. rewrite /Frame=> ?. by rewrite sep_forall_l; apply forall_mono. Qed.

615
Global Instance frame_bupd p R P Q : Frame p R P Q  Frame p R (|==> P) (|==> Q).
616
Proof. rewrite /Frame=><-. by rewrite bupd_frame_l. Qed.
617

618 619 620
(* FromOr *)
Global Instance from_or_or P1 P2 : FromOr (P1  P2) P1 P2.
Proof. done. Qed.
621
Global Instance from_or_bupd P Q1 Q2 :
622
  FromOr P Q1 Q2  FromOr (|==> P) (|==> Q1) (|==> Q2).
623
Proof. rewrite /FromOr=><-. apply or_elim; apply bupd_mono; auto with I. Qed.
624 625
Global Instance from_or_pure φ ψ : @FromOr M ⌜φ  ψ⌝ ⌜φ⌝ ⌜ψ⌝.
Proof. by rewrite /FromOr pure_or. Qed.
626 627 628
Global Instance from_or_always P Q1 Q2 :
  FromOr P Q1 Q2  FromOr ( P) ( Q1) ( Q2).
Proof. rewrite /FromOr=> <-. by rewrite always_or. Qed.