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) :