class_instances.v 15.3 KB
Newer Older
1
From iris.proofmode Require Export classes.
2
From iris.algebra Require Import upred_big_op gmap.
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Import uPred.

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

(* FromAssumption *)
Global Instance from_assumption_exact p P : FromAssumption p P P.
Proof. destruct p; by rewrite /FromAssumption /= ?always_elim. Qed.
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.
Global Instance from_assumption_always_r P Q :
  FromAssumption true P Q  FromAssumption true P ( Q).
Proof. rewrite /FromAssumption=><-. by rewrite always_always. Qed.
18 19 20
Global Instance from_assumption_rvs p P Q :
  FromAssumption p P Q  FromAssumption p P (|=r=> Q)%I.
Proof. rewrite /FromAssumption=>->. apply rvs_intro. Qed.
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

(* IntoPure *)
Global Instance into_pure_pure φ : @IntoPure M ( φ) φ.
Proof. done. Qed.
Global Instance into_pure_eq {A : cofeT} (a b : A) :
  Timeless a  @IntoPure M (a  b) (a  b).
Proof. intros. by rewrite /IntoPure timeless_eq. Qed.
Global Instance into_pure_valid `{CMRADiscrete A} (a : A) : @IntoPure M ( a) ( a).
Proof. by rewrite /IntoPure discrete_valid. Qed.

(* FromPure *)
Global Instance from_pure_pure φ : @FromPure M ( φ) φ.
Proof. intros ?. by apply pure_intro. Qed.
Global Instance from_pure_eq {A : cofeT} (a b : A) : @FromPure M (a  b) (a  b).
Proof. intros ->. apply eq_refl. Qed.
Global Instance from_pure_valid {A : cmraT} (a : A) : @FromPure M ( a) ( a).
Proof. intros ?. by apply valid_intro. Qed.
38 39
Global Instance from_pure_rvs P φ : FromPure P φ  FromPure (|=r=> P) φ.
Proof. intros ??. by rewrite -rvs_intro (from_pure P). Qed.
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

(* IntoPersistentP *)
Global Instance into_persistentP_always_trans P Q :
  IntoPersistentP P Q  IntoPersistentP ( P) Q | 0.
Proof. rewrite /IntoPersistentP=> ->. by rewrite always_always. Qed.
Global Instance into_persistentP_always P : IntoPersistentP ( P) P | 1.
Proof. done. Qed.
Global Instance into_persistentP_persistent P :
  PersistentP P  IntoPersistentP P P | 100.
Proof. done. Qed.

(* IntoLater *)
Global Instance into_later_default P : IntoLater P P | 1000.
Proof. apply later_intro. Qed.
Global Instance into_later_later P : IntoLater ( P) P.
Proof. done. Qed.
Global Instance into_later_and P1 P2 Q1 Q2 :
  IntoLater P1 Q1  IntoLater P2 Q2  IntoLater (P1  P2) (Q1  Q2).
Proof. intros ??; red. by rewrite later_and; apply and_mono. Qed.
Global Instance into_later_or P1 P2 Q1 Q2 :
  IntoLater P1 Q1  IntoLater P2 Q2  IntoLater (P1  P2) (Q1  Q2).
Proof. intros ??; red. by rewrite later_or; apply or_mono. Qed.
Global Instance into_later_sep P1 P2 Q1 Q2 :
  IntoLater P1 Q1  IntoLater P2 Q2  IntoLater (P1  P2) (Q1  Q2).
Proof. intros ??; red. by rewrite later_sep; apply sep_mono. Qed.

Global Instance into_later_big_sepM `{Countable K} {A}
    (Φ Ψ : K  A  uPred M) (m : gmap K A) :
  ( x k, IntoLater (Φ k x) (Ψ k x)) 
  IntoLater ([ map] k  x  m, Φ k x) ([ map] k  x  m, Ψ k x).
Proof.
  rewrite /IntoLater=> ?. rewrite big_sepM_later; by apply big_sepM_mono.
Qed.
Global Instance into_later_big_sepS `{Countable A}
    (Φ Ψ : A  uPred M) (X : gset A) :
  ( x, IntoLater (Φ x) (Ψ x)) 
  IntoLater ([ set] x  X, Φ x) ([ set] x  X, Ψ x).
Proof.
  rewrite /IntoLater=> ?. rewrite big_sepS_later; by apply big_sepS_mono.
Qed.

(* FromLater *)
Global Instance from_later_later P : FromLater ( P) P.
Proof. done. Qed.
Global Instance from_later_and P1 P2 Q1 Q2 :
  FromLater P1 Q1  FromLater P2 Q2  FromLater (P1  P2) (Q1  Q2).
Proof. intros ??; red. by rewrite later_and; apply and_mono. Qed.
Global Instance from_later_or P1 P2 Q1 Q2 :
  FromLater P1 Q1  FromLater P2 Q2  FromLater (P1  P2) (Q1  Q2).
Proof. intros ??; red. by rewrite later_or; apply or_mono. Qed.
Global Instance from_later_sep P1 P2 Q1 Q2 :
  FromLater P1 Q1  FromLater P2 Q2  FromLater (P1  P2) (Q1  Q2).
Proof. intros ??; red. by rewrite later_sep; apply sep_mono. Qed.

(* IntoWand *)
95 96 97 98 99 100
Global Instance into_wand_wand P Q Q' :
  FromAssumption false Q Q'  IntoWand (P - Q) P Q'.
Proof. by rewrite /FromAssumption /IntoWand /= => ->. Qed.
Global Instance into_wand_impl P Q Q' :
  FromAssumption false Q Q'  IntoWand (P  Q) P Q'.
Proof. rewrite /FromAssumption /IntoWand /= => ->. by rewrite impl_wand. Qed.
101 102 103 104 105 106
Global Instance into_wand_iff_l P Q : IntoWand (P  Q) P Q.
Proof. by apply and_elim_l', impl_wand. Qed.
Global Instance into_wand_iff_r P Q : IntoWand (P  Q) Q P.
Proof. apply and_elim_r', impl_wand. Qed.
Global Instance into_wand_always R P Q : IntoWand R P Q  IntoWand ( R) P Q.
Proof. rewrite /IntoWand=> ->. apply always_elim. Qed.
107 108 109
Global Instance into_wand_rvs R P Q :
  IntoWand R P Q  IntoWand R (|=r=> P) (|=r=> Q) | 100.
Proof. rewrite /IntoWand=>->. apply wand_intro_l. by rewrite rvs_wand_r. Qed.
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

(* FromAnd *)
Global Instance from_and_and P1 P2 : FromAnd (P1  P2) P1 P2.
Proof. done. Qed.
Global Instance from_and_sep_persistent_l P1 P2 :
  PersistentP P1  FromAnd (P1  P2) P1 P2 | 9.
Proof. intros. by rewrite /FromAnd always_and_sep_l. Qed.
Global Instance from_and_sep_persistent_r P1 P2 :
  PersistentP P2  FromAnd (P1  P2) P1 P2 | 10.
Proof. intros. by rewrite /FromAnd always_and_sep_r. Qed.
Global Instance from_and_always P Q1 Q2 :
  FromAnd P Q1 Q2  FromAnd ( P) ( Q1) ( Q2).
Proof. rewrite /FromAnd=> <-. by rewrite always_and. Qed.
Global Instance from_and_later P Q1 Q2 :
  FromAnd P Q1 Q2  FromAnd ( P) ( Q1) ( Q2).
Proof. rewrite /FromAnd=> <-. by rewrite later_and. Qed.

(* FromSep *)
Global Instance from_sep_sep P1 P2 : FromSep (P1  P2) P1 P2 | 100.
Proof. done. Qed.
Global Instance from_sep_always P Q1 Q2 :
  FromSep P Q1 Q2  FromSep ( P) ( Q1) ( Q2).
Proof. rewrite /FromSep=> <-. by rewrite always_sep. Qed.
Global Instance from_sep_later P Q1 Q2 :
  FromSep P Q1 Q2  FromSep ( P) ( Q1) ( Q2).
Proof. rewrite /FromSep=> <-. by rewrite later_sep. Qed.
136 137 138
Global Instance from_sep_rvs P Q1 Q2 :
  FromSep P Q1 Q2  FromSep (|=r=> P) (|=r=> Q1) (|=r=> Q2).
Proof. rewrite /FromSep=><-. apply rvs_sep. Qed.
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

Global Instance from_sep_ownM (a b : M) :
  FromSep (uPred_ownM (a  b)) (uPred_ownM a) (uPred_ownM b) | 99.
Proof. by rewrite /FromSep ownM_op. Qed.
Global Instance from_sep_big_sepM
    `{Countable K} {A} (Φ Ψ1 Ψ2 : K  A  uPred M) m :
  ( k x, FromSep (Φ k x) (Ψ1 k x) (Ψ2 k x)) 
  FromSep ([ map] k  x  m, Φ k x)
    ([ map] k  x  m, Ψ1 k x) ([ map] k  x  m, Ψ2 k x).
Proof.
  rewrite /FromSep=> ?. rewrite -big_sepM_sepM. by apply big_sepM_mono.
Qed.
Global Instance from_sep_big_sepS `{Countable A} (Φ Ψ1 Ψ2 : A  uPred M) X :
  ( x, FromSep (Φ x) (Ψ1 x) (Ψ2 x)) 
  FromSep ([ set] x  X, Φ x) ([ set] x  X, Ψ1 x) ([ set] x  X, Ψ2 x).
Proof.
  rewrite /FromSep=> ?. rewrite -big_sepS_sepS. by apply big_sepS_mono.
Qed.

(* IntoOp *)
Global Instance into_op_op {A : cmraT} (a b : A) : IntoOp (a  b) a b.
Proof. by rewrite /IntoOp. Qed.
Global Instance into_op_persistent {A : cmraT} (a : A) :
  Persistent a  IntoOp a a a.
Proof. intros; apply (persistent_dup a). Qed.
Global Instance into_op_pair {A B : cmraT} (a b1 b2 : A) (a' b1' b2' : B) :
  IntoOp a b1 b2  IntoOp a' b1' b2' 
  IntoOp (a,a') (b1,b1') (b2,b2').
Proof. by constructor. Qed.
Global Instance into_op_Some {A : cmraT} (a : A) b1 b2 :
  IntoOp a b1 b2  IntoOp (Some a) (Some b1) (Some b2).
Proof. by constructor. Qed.

172 173 174 175
(* IntoAnd *)
Global Instance into_and_sep p P Q : IntoAnd p (P  Q) P Q.
Proof. by apply mk_into_and_sep. Qed.
Global Instance into_and_ownM p (a b1 b2 : M) :
176
  IntoOp a b1 b2 
177 178
  IntoAnd p (uPred_ownM a) (uPred_ownM b1) (uPred_ownM b2).
Proof. intros. apply mk_into_and_sep. by rewrite (into_op a) ownM_op. Qed.
179

180
Global Instance into_and_and P Q : IntoAnd true (P  Q) P Q.
181
Proof. done. Qed.
182 183 184 185 186 187 188 189 190 191 192 193
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.

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.

Global Instance into_and_big_sepM
194
    `{Countable K} {A} (Φ Ψ1 Ψ2 : K  A  uPred M) p m :
195 196
  ( k x, IntoAnd p (Φ k x) (Ψ1 k x) (Ψ2 k x)) 
  IntoAnd p ([ map] k  x  m, Φ k x)
197 198
    ([ map] k  x  m, Ψ1 k x) ([ map] k  x  m, Ψ2 k x).
Proof.
199
  rewrite /IntoAnd=> HΦ. destruct p.
200 201 202 203
  - apply and_intro; apply big_sepM_mono; auto.
    + intros k x ?. by rewrite HΦ and_elim_l.
    + intros k x ?. by rewrite HΦ and_elim_r.
  - rewrite -big_sepM_sepM. apply big_sepM_mono; auto.
204
Qed.
205 206 207
Global Instance into_and_big_sepS `{Countable A} (Φ Ψ1 Ψ2 : A  uPred M) p X :
  ( x, IntoAnd p (Φ x) (Ψ1 x) (Ψ2 x)) 
  IntoAnd p ([ set] x  X, Φ x) ([ set] x  X, Ψ1 x) ([ set] x  X, Ψ2 x).
208
Proof.
209
  rewrite /IntoAnd=> HΦ. destruct p.
210 211 212 213
  - apply and_intro; apply big_sepS_mono; auto.
    + intros x ?. by rewrite HΦ and_elim_l.
    + intros x ?. by rewrite HΦ and_elim_r.
  - rewrite -big_sepS_sepS. apply big_sepS_mono; auto.
214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231
Qed.

(* Frame *)
Global Instance frame_here R : Frame R R True.
Proof. by rewrite /Frame right_id. Qed.

Class MakeSep (P Q PQ : uPred M) := make_sep : P  Q  PQ.
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.
Global Instance make_sep_default P Q : MakeSep P Q (P  Q) | 100.
Proof. done. Qed.
Global Instance frame_sep_l R P1 P2 Q Q' :
  Frame R P1 Q  MakeSep Q P2 Q'  Frame R (P1  P2) Q' | 9.
Proof. rewrite /Frame /MakeSep => <- <-. by rewrite assoc. Qed.
Global Instance frame_sep_r R P1 P2 Q Q' :
  Frame R P2 Q  MakeSep P1 Q Q'  Frame R (P1  P2) Q' | 10.
232
Proof. rewrite /Frame /MakeSep => <- <-. by rewrite assoc (comm _ R) assoc. Qed.
233 234 235 236 237 238

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.
239
Global Instance make_and_default P Q : MakeAnd P Q (P  Q) | 100.
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
Proof. done. Qed.
Global Instance frame_and_l R P1 P2 Q Q' :
  Frame R P1 Q  MakeAnd Q P2 Q'  Frame R (P1  P2) Q' | 9.
Proof. rewrite /Frame /MakeAnd => <- <-; eauto 10 with I. Qed.
Global Instance frame_and_r R P1 P2 Q Q' :
  Frame R P2 Q  MakeAnd P1 Q Q'  Frame R (P1  P2) Q' | 10.
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.
Global Instance frame_or R P1 P2 Q1 Q2 Q :
  Frame R P1 Q1  Frame R P2 Q2  MakeOr Q1 Q2 Q  Frame R (P1  P2) Q.
Proof. rewrite /Frame /MakeOr => <- <- <-. by rewrite -sep_or_l. Qed.

Global Instance frame_wand R P1 P2 Q2 :
  Frame R P2 Q2  Frame R (P1 - P2) (P1 - Q2).
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.

272
Global Instance frame_later R R' P Q Q' :
273
  IntoLater R' R  Frame R P Q  MakeLater Q Q'  Frame R' ( P) Q'.
274
Proof.
275
  rewrite /Frame /MakeLater /IntoLater=>-> <- <-. by rewrite later_sep.
276 277
Qed.

278 279 280 281
Class MakeExceptLast (P Q : uPred M) := make_except_last :  P  Q.
Global Instance make_except_last_True : MakeExceptLast True True.
Proof. by rewrite /MakeExceptLast except_last_True. Qed.
Global Instance make_except_last_default P : MakeExceptLast P ( P) | 100.
282 283
Proof. done. Qed.

284 285
Global Instance frame_except_last R P Q Q' :
  Frame R P Q  MakeExceptLast Q Q'  Frame R ( P) Q'.
286
Proof.
287 288
  rewrite /Frame /MakeExceptLast=><- <-.
  by rewrite except_last_sep -(except_last_intro R).
289 290
Qed.

291 292 293 294 295 296 297
Global Instance frame_exist {A} R (Φ Ψ : A  uPred M) :
  ( a, Frame R (Φ a) (Ψ a))  Frame R ( x, Φ x) ( x, Ψ x).
Proof. rewrite /Frame=> ?. by rewrite sep_exist_l; apply exist_mono. Qed.
Global Instance frame_forall {A} R (Φ Ψ : A  uPred M) :
  ( a, Frame R (Φ a) (Ψ a))  Frame R ( x, Φ x) ( x, Ψ x).
Proof. rewrite /Frame=> ?. by rewrite sep_forall_l; apply forall_mono. Qed.

298 299 300
Global Instance frame_rvs R P Q : Frame R P Q  Frame R (|=r=> P) (|=r=> Q).
Proof. rewrite /Frame=><-. by rewrite rvs_frame_l. Qed.

301 302 303
(* FromOr *)
Global Instance from_or_or P1 P2 : FromOr (P1  P2) P1 P2.
Proof. done. Qed.
304 305 306
Global Instance from_or_rvs P Q1 Q2 :
  FromOr P Q1 Q2  FromOr (|=r=> P) (|=r=> Q1) (|=r=> Q2).
Proof. rewrite /FromOr=><-. apply or_elim; apply rvs_mono; auto with I. Qed.
307 308 309 310 311 312 313 314 315 316 317

(* IntoOr *)
Global Instance into_or_or P Q : IntoOr (P  Q) P Q.
Proof. done. Qed.
Global Instance into_or_later P Q1 Q2 :
  IntoOr P Q1 Q2  IntoOr ( P) ( Q1) ( Q2).
Proof. rewrite /IntoOr=>->. by rewrite later_or. Qed.

(* FromExist *)
Global Instance from_exist_exist {A} (Φ: A  uPred M): FromExist ( a, Φ a) Φ.
Proof. done. Qed.
318 319 320 321 322
Global Instance from_exist_rvs {A} P (Φ : A  uPred M) :
  FromExist P Φ  FromExist (|=r=> P) (λ a, |=r=> Φ a)%I.
Proof.
  rewrite /FromExist=><-. apply exist_elim=> a. by rewrite -(exist_intro a).
Qed.
323 324 325 326 327 328 329 330 331 332

(* IntoExist *)
Global Instance into_exist_exist {A} (Φ : A  uPred M) : IntoExist ( a, Φ a) Φ.
Proof. done. Qed.
Global Instance into_exist_later {A} P (Φ : A  uPred M) :
  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_always {A} P (Φ : A  uPred M) :
  IntoExist P Φ  IntoExist ( P) (λ a,  (Φ a))%I.
Proof. rewrite /IntoExist=> HP. by rewrite HP always_exist. Qed.
333

334 335
(* IntoExceptLast *)
Global Instance into_except_last_except_last P : IntoExceptLast ( P) P.
336
Proof. done. Qed.
337
Global Instance into_except_last_timeless P : TimelessP P  IntoExceptLast ( P) P.
338 339
Proof. done. Qed.

340 341 342 343 344 345
(* IsExceptLast *)
Global Instance is_except_last_except_last P : IsExceptLast ( P).
Proof. by rewrite /IsExceptLast except_last_idemp. Qed.
Global Instance is_except_last_later P : IsExceptLast ( P).
Proof. by rewrite /IsExceptLast except_last_later. Qed.
Global Instance is_except_last_rvs P : IsExceptLast P  IsExceptLast (|=r=> P).
346
Proof.
347 348
  rewrite /IsExceptLast=> HP.
  by rewrite -{2}HP -(except_last_idemp P) -except_last_rvs -(except_last_intro P).
349
Qed.
350 351 352 353 354 355 356 357

(* FromViewShift *)
Global Instance from_vs_rvs P : FromVs (|=r=> P) P.
Proof. done. Qed.

(* ElimViewShift *)
Global Instance elim_vs_rvs_rvs P Q : ElimVs (|=r=> P) P (|=r=> Q) (|=r=> Q).
Proof. by rewrite /ElimVs rvs_frame_r wand_elim_r rvs_trans. Qed.
358
End classes.