primitive.v 27.4 KB
Newer Older
1 2
From iris.base_logic Require Export upred.
From iris.algebra Require Export updates.
3
Set Default Proof Using "Type".
4 5 6 7 8 9 10 11
Local Hint Extern 1 (_  _) => etrans; [eassumption|].
Local Hint Extern 1 (_  _) => etrans; [|eassumption].
Local Hint Extern 10 (_  _) => omega.

(** logical connectives *)
Program Definition uPred_pure_def {M} (φ : Prop) : uPred M :=
  {| uPred_holds n x := φ |}.
Solve Obligations with done.
12 13
Definition uPred_pure_aux : seal (@uPred_pure_def). by eexists. Qed.
Definition uPred_pure {M} := unseal uPred_pure_aux M.
14
Definition uPred_pure_eq :
15
  @uPred_pure = @uPred_pure_def := seal_eq uPred_pure_aux.
16 17 18 19 20 21

Instance uPred_inhabited M : Inhabited (uPred M) := populate (uPred_pure True).

Program Definition uPred_and_def {M} (P Q : uPred M) : uPred M :=
  {| uPred_holds n x := P n x  Q n x |}.
Solve Obligations with naive_solver eauto 2 with uPred_def.
22 23 24
Definition uPred_and_aux : seal (@uPred_and_def). by eexists. Qed.
Definition uPred_and {M} := unseal uPred_and_aux M.
Definition uPred_and_eq: @uPred_and = @uPred_and_def := seal_eq uPred_and_aux.
25 26 27 28

Program Definition uPred_or_def {M} (P Q : uPred M) : uPred M :=
  {| uPred_holds n x := P n x  Q n x |}.
Solve Obligations with naive_solver eauto 2 with uPred_def.
29 30 31
Definition uPred_or_aux : seal (@uPred_or_def). by eexists. Qed.
Definition uPred_or {M} := unseal uPred_or_aux M.
Definition uPred_or_eq: @uPred_or = @uPred_or_def := seal_eq uPred_or_aux.
32 33 34 35 36 37 38 39 40 41

Program Definition uPred_impl_def {M} (P Q : uPred M) : uPred M :=
  {| uPred_holds n x :=  n' x',
       x  x'  n'  n  {n'} x'  P n' x'  Q n' x' |}.
Next Obligation.
  intros M P Q n1 x1 x1' HPQ [x2 Hx1'] n2 x3 [x4 Hx3] ?; simpl in *.
  rewrite Hx3 (dist_le _ _ _ _ Hx1'); auto. intros ??.
  eapply HPQ; auto. exists (x2  x4); by rewrite assoc.
Qed.
Next Obligation. intros M P Q [|n1] [|n2] x; auto with lia. Qed.
42 43
Definition uPred_impl_aux : seal (@uPred_impl_def). by eexists. Qed.
Definition uPred_impl {M} := unseal uPred_impl_aux M.
44
Definition uPred_impl_eq :
45
  @uPred_impl = @uPred_impl_def := seal_eq uPred_impl_aux.
46 47 48 49

Program Definition uPred_forall_def {M A} (Ψ : A  uPred M) : uPred M :=
  {| uPred_holds n x :=  a, Ψ a n x |}.
Solve Obligations with naive_solver eauto 2 with uPred_def.
50 51
Definition uPred_forall_aux : seal (@uPred_forall_def). by eexists. Qed.
Definition uPred_forall {M A} := unseal uPred_forall_aux M A.
52
Definition uPred_forall_eq :
53
  @uPred_forall = @uPred_forall_def := seal_eq uPred_forall_aux.
54 55 56 57

Program Definition uPred_exist_def {M A} (Ψ : A  uPred M) : uPred M :=
  {| uPred_holds n x :=  a, Ψ a n x |}.
Solve Obligations with naive_solver eauto 2 with uPred_def.
58 59 60
Definition uPred_exist_aux : seal (@uPred_exist_def). by eexists. Qed.
Definition uPred_exist {M A} := unseal uPred_exist_aux M A.
Definition uPred_exist_eq: @uPred_exist = @uPred_exist_def := seal_eq uPred_exist_aux.
61

62
Program Definition uPred_internal_eq_def {M} {A : ofeT} (a1 a2 : A) : uPred M :=
63 64
  {| uPred_holds n x := a1 {n} a2 |}.
Solve Obligations with naive_solver eauto 2 using (dist_le (A:=A)).
65 66
Definition uPred_internal_eq_aux : seal (@uPred_internal_eq_def). by eexists. Qed.
Definition uPred_internal_eq {M A} := unseal uPred_internal_eq_aux M A.
67
Definition uPred_internal_eq_eq:
68
  @uPred_internal_eq = @uPred_internal_eq_def := seal_eq uPred_internal_eq_aux.
69 70 71 72 73 74 75 76 77 78

Program Definition uPred_sep_def {M} (P Q : uPred M) : uPred M :=
  {| uPred_holds n x :=  x1 x2, x {n} x1  x2  P n x1  Q n x2 |}.
Next Obligation.
  intros M P Q n x y (x1&x2&Hx&?&?) [z Hy].
  exists x1, (x2  z); split_and?; eauto using uPred_mono, cmra_includedN_l.
  by rewrite Hy Hx assoc.
Qed.
Next Obligation.
  intros M P Q n1 n2 x (x1&x2&Hx&?&?) ?; rewrite {1}(dist_le _ _ _ _ Hx) // =>?.
79
  exists x1, x2; ofe_subst; split_and!;
80 81
    eauto using dist_le, uPred_closed, cmra_validN_op_l, cmra_validN_op_r.
Qed.
82 83 84
Definition uPred_sep_aux : seal (@uPred_sep_def). by eexists. Qed.
Definition uPred_sep {M} := unseal uPred_sep_aux M.
Definition uPred_sep_eq: @uPred_sep = @uPred_sep_def := seal_eq uPred_sep_aux.
85 86 87 88 89 90 91 92 93 94

Program Definition uPred_wand_def {M} (P Q : uPred M) : uPred M :=
  {| uPred_holds n x :=  n' x',
       n'  n  {n'} (x  x')  P n' x'  Q n' (x  x') |}.
Next Obligation.
  intros M P Q n x1 x1' HPQ ? n3 x3 ???; simpl in *.
  apply uPred_mono with (x1  x3);
    eauto using cmra_validN_includedN, cmra_monoN_r, cmra_includedN_le.
Qed.
Next Obligation. naive_solver. Qed.
95 96
Definition uPred_wand_aux : seal (@uPred_wand_def). by eexists. Qed.
Definition uPred_wand {M} := unseal uPred_wand_aux M.
97
Definition uPred_wand_eq :
98
  @uPred_wand = @uPred_wand_def := seal_eq uPred_wand_aux.
99

100 101 102 103 104 105 106 107
Program Definition uPred_plainly_def {M} (P : uPred M) : uPred M :=
  {| uPred_holds n x := P n ε |}.
Solve Obligations with naive_solver eauto using uPred_closed, ucmra_unit_validN.
Definition uPred_plainly_aux : seal (@uPred_plainly_def). by eexists. Qed.
Definition uPred_plainly {M} := unseal uPred_plainly_aux M.
Definition uPred_plainly_eq :
  @uPred_plainly = @uPred_plainly_def := seal_eq uPred_plainly_aux.

108
Program Definition uPred_persistently_def {M} (P : uPred M) : uPred M :=
109 110 111 112 113
  {| uPred_holds n x := P n (core x) |}.
Next Obligation.
  intros M; naive_solver eauto using uPred_mono, @cmra_core_monoN.
Qed.
Next Obligation. naive_solver eauto using uPred_closed, @cmra_core_validN. Qed.
114 115 116 117
Definition uPred_persistently_aux : seal (@uPred_persistently_def). by eexists. Qed.
Definition uPred_persistently {M} := unseal uPred_persistently_aux M.
Definition uPred_persistently_eq :
  @uPred_persistently = @uPred_persistently_def := seal_eq uPred_persistently_aux.
118 119 120 121 122 123 124 125 126

Program Definition uPred_later_def {M} (P : uPred M) : uPred M :=
  {| uPred_holds n x := match n return _ with 0 => True | S n' => P n' x end |}.
Next Obligation.
  intros M P [|n] x1 x2; eauto using uPred_mono, cmra_includedN_S.
Qed.
Next Obligation.
  intros M P [|n1] [|n2] x; eauto using uPred_closed, cmra_validN_S with lia.
Qed.
127 128
Definition uPred_later_aux : seal (@uPred_later_def). by eexists. Qed.
Definition uPred_later {M} := unseal uPred_later_aux M.
129
Definition uPred_later_eq :
130
  @uPred_later = @uPred_later_def := seal_eq uPred_later_aux.
131 132 133 134 135 136 137 138

Program Definition uPred_ownM_def {M : ucmraT} (a : M) : uPred M :=
  {| uPred_holds n x := a {n} x |}.
Next Obligation.
  intros M a n x1 x [a' Hx1] [x2 ->].
  exists (a'  x2). by rewrite (assoc op) Hx1.
Qed.
Next Obligation. naive_solver eauto using cmra_includedN_le. Qed.
139 140
Definition uPred_ownM_aux : seal (@uPred_ownM_def). by eexists. Qed.
Definition uPred_ownM {M} := unseal uPred_ownM_aux M.
141
Definition uPred_ownM_eq :
142
  @uPred_ownM = @uPred_ownM_def := seal_eq uPred_ownM_aux.
143 144 145 146

Program Definition uPred_cmra_valid_def {M} {A : cmraT} (a : A) : uPred M :=
  {| uPred_holds n x := {n} a |}.
Solve Obligations with naive_solver eauto 2 using cmra_validN_le.
147 148
Definition uPred_cmra_valid_aux : seal (@uPred_cmra_valid_def). by eexists. Qed.
Definition uPred_cmra_valid {M A} := unseal uPred_cmra_valid_aux M A.
149
Definition uPred_cmra_valid_eq :
150
  @uPred_cmra_valid = @uPred_cmra_valid_def := seal_eq uPred_cmra_valid_aux.
151 152 153 154 155 156 157 158 159 160 161 162

Program Definition uPred_bupd_def {M} (Q : uPred M) : uPred M :=
  {| uPred_holds n x :=  k yf,
      k  n  {k} (x  yf)   x', {k} (x'  yf)  Q k x' |}.
Next Obligation.
  intros M Q n x1 x2 HQ [x3 Hx] k yf Hk.
  rewrite (dist_le _ _ _ _ Hx); last lia. intros Hxy.
  destruct (HQ k (x3  yf)) as (x'&?&?); [auto|by rewrite assoc|].
  exists (x'  x3); split; first by rewrite -assoc.
  apply uPred_mono with x'; eauto using cmra_includedN_l.
Qed.
Next Obligation. naive_solver. Qed.
163 164 165
Definition uPred_bupd_aux : seal (@uPred_bupd_def). by eexists. Qed.
Definition uPred_bupd {M} := unseal uPred_bupd_aux M.
Definition uPred_bupd_eq : @uPred_bupd = @uPred_bupd_def := seal_eq uPred_bupd_aux.
166

Ralf Jung's avatar
Ralf Jung committed
167 168 169
(* Latest notation *)
Notation "'⌜' φ '⌝'" := (uPred_pure φ%C%type)
  (at level 1, φ at level 200, format "⌜ φ ⌝") : uPred_scope.
170 171 172 173 174 175 176
Notation "'False'" := (uPred_pure False) : uPred_scope.
Notation "'True'" := (uPred_pure True) : uPred_scope.
Infix "∧" := uPred_and : uPred_scope.
Notation "(∧)" := uPred_and (only parsing) : uPred_scope.
Infix "∨" := uPred_or : uPred_scope.
Notation "(∨)" := uPred_or (only parsing) : uPred_scope.
Infix "→" := uPred_impl : uPred_scope.
177 178 179
Infix "∗" := uPred_sep (at level 80, right associativity) : uPred_scope.
Notation "(∗)" := uPred_sep (only parsing) : uPred_scope.
Notation "P -∗ Q" := (uPred_wand P Q)
180 181
  (at level 99, Q at level 200, right associativity) : uPred_scope.
Notation "∀ x .. y , P" :=
182 183
  (uPred_forall (λ x, .. (uPred_forall (λ y, P)) ..)%I)
  (at level 200, x binder, y binder, right associativity) : uPred_scope.
184
Notation "∃ x .. y , P" :=
185 186
  (uPred_exist (λ x, .. (uPred_exist (λ y, P)) ..)%I)
  (at level 200, x binder, y binder, right associativity) : uPred_scope.
187 188
Notation "■ P" := (uPred_plainly P)
  (at level 20, right associativity) : uPred_scope.
189
Notation "□ P" := (uPred_persistently P)
190 191 192
  (at level 20, right associativity) : uPred_scope.
Notation "▷ P" := (uPred_later P)
  (at level 20, right associativity) : uPred_scope.
193
Infix "≡" := uPred_internal_eq : uPred_scope.
194 195 196
Notation "✓ x" := (uPred_cmra_valid x) (at level 20) : uPred_scope.
Notation "|==> Q" := (uPred_bupd Q)
  (at level 99, Q at level 200, format "|==>  Q") : uPred_scope.
197
Notation "P ==∗ Q" := (P  |==> Q)
198
  (at level 99, Q at level 200, only parsing) : C_scope.
199 200
Notation "P ==∗ Q" := (P - |==> Q)%I
  (at level 99, Q at level 200, format "P  ==∗  Q") : uPred_scope.
201

202
Coercion uPred_valid {M} (P : uPred M) : Prop := True%I  P.
Robbert Krebbers's avatar
Robbert Krebbers committed
203 204
Typeclasses Opaque uPred_valid.

205
Notation "P -∗ Q" := (P  Q)
206
  (at level 99, Q at level 200, right associativity) : C_scope.
207

208
Module uPred.
209
Definition unseal_eqs :=
210
  (uPred_pure_eq, uPred_and_eq, uPred_or_eq, uPred_impl_eq, uPred_forall_eq,
211 212
  uPred_exist_eq, uPred_internal_eq_eq, uPred_sep_eq, uPred_wand_eq,
  uPred_persistently_eq, uPred_plainly_eq, uPred_persistently_eq,
213
  uPred_later_eq, uPred_ownM_eq, uPred_cmra_valid_eq, uPred_bupd_eq).
214
Ltac unseal := rewrite !unseal_eqs /=.
215 216 217 218 219 220 221 222 223 224 225 226

Section primitive.
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 *)
Arguments uPred_holds {_} !_ _ _ /.
Hint Immediate uPred_in_entails.

(** Non-expansiveness and setoid morphisms *)
227
Global Instance pure_proper : Proper (iff ==> ()) (@uPred_pure M) | 0.
228
Proof. intros φ1 φ2 Hφ. by unseal; split=> -[|n] ?; try apply Hφ. Qed.
229 230 231
Global Instance pure_ne n : Proper (iff ==> dist n) (@uPred_pure M) | 1.
Proof. by intros φ1 φ2 ->. Qed.

232
Global Instance and_ne : NonExpansive2 (@uPred_and M).
233
Proof.
234
  intros n P P' HP Q Q' HQ; unseal; split=> x n' ??.
235 236 237 238
  split; (intros [??]; split; [by apply HP|by apply HQ]).
Qed.
Global Instance and_proper :
  Proper (() ==> () ==> ()) (@uPred_and M) := ne_proper_2 _.
239
Global Instance or_ne : NonExpansive2 (@uPred_or M).
240
Proof.
241
  intros n P P' HP Q Q' HQ; split=> x n' ??.
242 243 244 245
  unseal; split; (intros [?|?]; [left; by apply HP|right; by apply HQ]).
Qed.
Global Instance or_proper :
  Proper (() ==> () ==> ()) (@uPred_or M) := ne_proper_2 _.
246 247
Global Instance impl_ne :
  NonExpansive2 (@uPred_impl M).
248
Proof.
249
  intros n P P' HP Q Q' HQ; split=> x n' ??.
250 251 252 253
  unseal; split; intros HPQ x' n'' ????; apply HQ, HPQ, HP; auto.
Qed.
Global Instance impl_proper :
  Proper (() ==> () ==> ()) (@uPred_impl M) := ne_proper_2 _.
254
Global Instance sep_ne : NonExpansive2 (@uPred_sep M).
255
Proof.
256
  intros n P P' HP Q Q' HQ; split=> n' x ??.
257
  unseal; split; intros (x1&x2&?&?&?); ofe_subst x;
258 259 260 261 262
    exists x1, x2; split_and!; try (apply HP || apply HQ);
    eauto using cmra_validN_op_l, cmra_validN_op_r.
Qed.
Global Instance sep_proper :
  Proper (() ==> () ==> ()) (@uPred_sep M) := ne_proper_2 _.
263 264
Global Instance wand_ne :
  NonExpansive2 (@uPred_wand M).
265
Proof.
266
  intros n P P' HP Q Q' HQ; split=> n' x ??; unseal; split; intros HPQ x' n'' ???;
267 268 269 270
    apply HQ, HPQ, HP; eauto using cmra_validN_op_r.
Qed.
Global Instance wand_proper :
  Proper (() ==> () ==> ()) (@uPred_wand M) := ne_proper_2 _.
271 272
Global Instance internal_eq_ne (A : ofeT) :
  NonExpansive2 (@uPred_internal_eq M A).
273
Proof.
274
  intros n x x' Hx y y' Hy; split=> n' z; unseal; split; intros; simpl in *.
275 276 277
  - by rewrite -(dist_le _ _ _ _ Hx) -?(dist_le _ _ _ _ Hy); auto.
  - by rewrite (dist_le _ _ _ _ Hx) ?(dist_le _ _ _ _ Hy); auto.
Qed.
278
Global Instance internal_eq_proper (A : ofeT) :
279
  Proper (() ==> () ==> ()) (@uPred_internal_eq M A) := ne_proper_2 _.
280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303
Global Instance forall_ne A n :
  Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_forall M A).
Proof.
  by intros Ψ1 Ψ2 HΨ; unseal; split=> n' x; split; intros HP a; apply HΨ.
Qed.
Global Instance forall_proper A :
  Proper (pointwise_relation _ () ==> ()) (@uPred_forall M A).
Proof.
  by intros Ψ1 Ψ2 HΨ; unseal; split=> n' x; split; intros HP a; apply HΨ.
Qed.
Global Instance exist_ne A n :
  Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_exist M A).
Proof.
  intros Ψ1 Ψ2 HΨ.
  unseal; split=> n' x ??; split; intros [a ?]; exists a; by apply HΨ.
Qed.
Global Instance exist_proper A :
  Proper (pointwise_relation _ () ==> ()) (@uPred_exist M A).
Proof.
  intros Ψ1 Ψ2 HΨ.
  unseal; split=> n' x ?; split; intros [a ?]; exists a; by apply HΨ.
Qed.
Global Instance later_contractive : Contractive (@uPred_later M).
Proof.
304 305
  unseal; intros [|n] P Q HPQ; split=> -[|n'] x ?? //=; try omega.
  apply HPQ; eauto using cmra_validN_S.
306 307 308
Qed.
Global Instance later_proper' :
  Proper (() ==> ()) (@uPred_later M) := ne_proper _.
309 310 311 312 313 314 315
Global Instance plainly_ne : NonExpansive (@uPred_plainly M).
Proof.
  intros n P1 P2 HP.
  unseal; split=> n' x; split; apply HP; eauto using @ucmra_unit_validN.
Qed.
Global Instance plainly_proper :
  Proper (() ==> ()) (@uPred_plainly M) := ne_proper _.
316
Global Instance persistently_ne : NonExpansive (@uPred_persistently M).
317
Proof.
318
  intros n P1 P2 HP.
319 320
  unseal; split=> n' x; split; apply HP; eauto using @cmra_core_validN.
Qed.
321 322
Global Instance persistently_proper :
  Proper (() ==> ()) (@uPred_persistently M) := ne_proper _.
323
Global Instance ownM_ne : NonExpansive (@uPred_ownM M).
324
Proof.
325
  intros n a b Ha.
326 327 328
  unseal; split=> n' x ? /=. by rewrite (dist_le _ _ _ _ Ha); last lia.
Qed.
Global Instance ownM_proper: Proper (() ==> ()) (@uPred_ownM M) := ne_proper _.
329 330
Global Instance cmra_valid_ne {A : cmraT} :
  NonExpansive (@uPred_cmra_valid M A).
331
Proof.
332
  intros n a b Ha; unseal; split=> n' x ? /=.
333 334 335 336
  by rewrite (dist_le _ _ _ _ Ha); last lia.
Qed.
Global Instance cmra_valid_proper {A : cmraT} :
  Proper (() ==> ()) (@uPred_cmra_valid M A) := ne_proper _.
337
Global Instance bupd_ne : NonExpansive (@uPred_bupd M).
338
Proof.
339
  intros n P Q HPQ.
340 341 342 343 344
  unseal; split=> n' x; split; intros HP k yf ??;
    destruct (HP k yf) as (x'&?&?); auto;
    exists x'; split; auto; apply HPQ; eauto using cmra_validN_op_l.
Qed.
Global Instance bupd_proper : Proper (() ==> ()) (@uPred_bupd M) := ne_proper _.
345 346 347 348
Global Instance uPred_valid_proper : Proper (() ==> iff) (@uPred_valid M).
Proof. solve_proper. Qed.
Global Instance uPred_valid_mono : Proper (() ==> impl) (@uPred_valid M).
Proof. solve_proper. Qed.
349 350 351
Global Instance uPred_valid_flip_mono :
  Proper (flip () ==> flip impl) (@uPred_valid M).
Proof. solve_proper. Qed.
352 353

(** Introduction and elimination rules *)
Ralf Jung's avatar
Ralf Jung committed
354
Lemma pure_intro φ P : φ  P  ⌜φ⌝.
355
Proof. by intros ?; unseal; split. Qed.
356
Lemma pure_elim' φ P : (φ  True  P)  ⌜φ⌝  P.
357
Proof. unseal; intros HP; split=> n x ??. by apply HP. Qed.
Ralf Jung's avatar
Ralf Jung committed
358
Lemma pure_forall_2 {A} (φ : A  Prop) : ( x : A, ⌜φ x)   x : A, φ x.
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
Proof. by unseal. Qed.

Lemma and_elim_l P Q : P  Q  P.
Proof. by unseal; split=> n x ? [??]. Qed.
Lemma and_elim_r P Q : P  Q  Q.
Proof. by unseal; split=> n x ? [??]. Qed.
Lemma and_intro P Q R : (P  Q)  (P  R)  P  Q  R.
Proof. intros HQ HR; unseal; split=> n x ??; by split; [apply HQ|apply HR]. Qed.

Lemma or_intro_l P Q : P  P  Q.
Proof. unseal; split=> n x ??; left; auto. Qed.
Lemma or_intro_r P Q : Q  P  Q.
Proof. unseal; split=> n x ??; right; auto. Qed.
Lemma or_elim P Q R : (P  R)  (Q  R)  P  Q  R.
Proof. intros HP HQ; unseal; split=> n x ? [?|?]. by apply HP. by apply HQ. Qed.

Lemma impl_intro_r P Q R : (P  Q  R)  P  Q  R.
Proof.
  unseal; intros HQ; split=> n x ?? n' x' ????. apply HQ;
    naive_solver eauto using uPred_mono, uPred_closed, cmra_included_includedN.
Qed.
Lemma impl_elim P Q R : (P  Q  R)  (P  Q)  P  R.
Proof. by unseal; intros HP HP'; split=> n x ??; apply HP with n x, HP'. Qed.

Lemma forall_intro {A} P (Ψ : A  uPred M): ( a, P  Ψ a)  P   a, Ψ a.
Proof. unseal; intros HPΨ; split=> n x ?? a; by apply HPΨ. Qed.
Lemma forall_elim {A} {Ψ : A  uPred M} a : ( a, Ψ a)  Ψ a.
Proof. unseal; split=> n x ? HP; apply HP. Qed.

Lemma exist_intro {A} {Ψ : A  uPred M} a : Ψ a   a, Ψ a.
Proof. unseal; split=> n x ??; by exists a. Qed.
Lemma exist_elim {A} (Φ : A  uPred M) Q : ( a, Φ a  Q)  ( a, Φ a)  Q.
Proof. unseal; intros HΦΨ; split=> n x ? [a ?]; by apply HΦΨ with a. Qed.

393
Lemma internal_eq_refl {A : ofeT} (a : A) : uPred_valid (M:=M) (a  a).
394
Proof. unseal; by split=> n x ??; simpl. Qed.
395
Lemma internal_eq_rewrite {A : ofeT} a b (Ψ : A  uPred M) P
396
  {HΨ : NonExpansive Ψ} : (P  a  b)  (P  Ψ a)  P  Ψ b.
397 398 399 400 401 402 403
Proof.
  unseal; intros Hab Ha; split=> n x ??. apply HΨ with n a; auto.
  - by symmetry; apply Hab with x.
  - by apply Ha.
Qed.

(* BI connectives *)
404
Lemma sep_mono P P' Q Q' : (P  Q)  (P'  Q')  P  P'  Q  Q'.
405 406
Proof.
  intros HQ HQ'; unseal.
407
  split; intros n' x ? (x1&x2&?&?&?); exists x1,x2; ofe_subst x;
408 409
    eauto 7 using cmra_validN_op_l, cmra_validN_op_r, uPred_in_entails.
Qed.
410
Lemma True_sep_1 P : P  True  P.
411 412 413
Proof.
  unseal; split; intros n x ??. exists (core x), x. by rewrite cmra_core_l.
Qed.
414
Lemma True_sep_2 P : True  P  P.
415
Proof.
416
  unseal; split; intros n x ? (x1&x2&?&_&?); ofe_subst;
417 418
    eauto using uPred_mono, cmra_includedN_r.
Qed.
419
Lemma sep_comm' P Q : P  Q  Q  P.
420 421 422
Proof.
  unseal; split; intros n x ? (x1&x2&?&?&?); exists x2, x1; by rewrite (comm op).
Qed.
423
Lemma sep_assoc' P Q R : (P  Q)  R  P  (Q  R).
424 425 426 427 428 429
Proof.
  unseal; split; intros n x ? (x1&x2&Hx&(y1&y2&Hy&?&?)&?).
  exists y1, (y2  x2); split_and?; auto.
  + by rewrite (assoc op) -Hy -Hx.
  + by exists y2, x2.
Qed.
430
Lemma wand_intro_r P Q R : (P  Q  R)  P  Q - R.
431 432 433 434 435
Proof.
  unseal=> HPQR; split=> n x ?? n' x' ???; apply HPQR; auto.
  exists x, x'; split_and?; auto.
  eapply uPred_closed with n; eauto using cmra_validN_op_l.
Qed.
436
Lemma wand_elim_l' P Q R : (P  Q - R)  P  Q  R.
437
Proof.
438
  unseal =>HPQR. split; intros n x ? (?&?&?&?&?). ofe_subst.
439 440 441
  eapply HPQR; eauto using cmra_validN_op_l.
Qed.

442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460
(* The plainness modality *)
Lemma plainly_mono P Q : (P  Q)   P   Q.
Proof. intros HP; unseal; split=> n x ? /=. apply HP, ucmra_unit_validN. Qed.
Lemma plainly_elim' P :  P   P.
Proof. unseal; split; simpl; eauto using uPred_mono, @ucmra_unit_leastN. Qed.
Lemma plainly_idemp P :  P    P.
Proof. unseal; split=> n x ?? //. Qed.

Lemma plainly_forall_2 {A} (Ψ : A  uPred M) : ( a,  Ψ a)  (  a, Ψ a).
Proof. by unseal. Qed.
Lemma plainly_exist_1 {A} (Ψ : A  uPred M) : (  a, Ψ a)  ( a,  Ψ a).
Proof. by unseal. Qed.

Lemma prop_ext P Q :  ((P  Q)  (Q  P))  P  Q.
Proof.
  unseal; split=> n x ? /= HPQ; split=> n' x' ? HP;
    split; eapply HPQ; eauto using @ucmra_unit_least.
Qed.

461
(* Always *)
462
Lemma persistently_mono P Q : (P  Q)   P   Q.
463
Proof. intros HP; unseal; split=> n x ? /=. by apply HP, cmra_core_validN. Qed.
464
Lemma persistently_elim P :  P  P.
465 466 467 468
Proof.
  unseal; split=> n x ? /=.
  eauto using uPred_mono, @cmra_included_core, cmra_included_includedN.
Qed.
469
Lemma persistently_idemp_2 P :  P    P.
470 471
Proof. unseal; split=> n x ?? /=. by rewrite cmra_core_idemp. Qed.

472
Lemma persistently_forall_2 {A} (Ψ : A  uPred M) : ( a,  Ψ a)  (  a, Ψ a).
473
Proof. by unseal. Qed.
474
Lemma persistently_exist_1 {A} (Ψ : A  uPred M) : (  a, Ψ a)  ( a,  Ψ a).
475 476
Proof. by unseal. Qed.

477
Lemma persistently_and_sep_l_1 P Q :  P  Q   P  Q.
478 479 480 481 482
Proof.
  unseal; split=> n x ? [??]; exists (core x), x; simpl in *.
  by rewrite cmra_core_l cmra_core_idemp.
Qed.

483 484 485 486 487 488 489 490 491 492 493 494 495 496
Lemma persistently_impl_plainly P Q : ( P   Q)   ( P  Q).
Proof.
  unseal; split=> /= n x ? HPQ n' x' ????.
  eapply uPred_mono with (core x), cmra_included_includedN; auto.
  apply (HPQ n' x); eauto using cmra_validN_le.
Qed.

Lemma plainly_impl_plainly P Q : ( P   Q)   ( P  Q).
Proof.
  unseal; split=> /= n x ? HPQ n' x' ????.
  eapply uPred_mono with ε, cmra_included_includedN; auto.
  apply (HPQ n' x); eauto using cmra_validN_le.
Qed.

497 498 499 500 501 502 503 504 505 506 507 508 509 510 511
(* Later *)
Lemma later_mono P Q : (P  Q)   P   Q.
Proof.
  unseal=> HP; split=>-[|n] x ??; [done|apply HP; eauto using cmra_validN_S].
Qed.
Lemma löb P : ( P  P)  P.
Proof.
  unseal; split=> n x ? HP; induction n as [|n IH]; [by apply HP|].
  apply HP, IH, uPred_closed with (S n); eauto using cmra_validN_S.
Qed.
Lemma later_forall_2 {A} (Φ : A  uPred M) : ( a,  Φ a)    a, Φ a.
Proof. unseal; by split=> -[|n] x. Qed.
Lemma later_exist_false {A} (Φ : A  uPred M) :
  (  a, Φ a)   False  ( a,  Φ a).
Proof. unseal; split=> -[|[|n]] x /=; eauto. Qed.
512
Lemma later_sep P Q :  (P  Q)   P   Q.
513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528
Proof.
  unseal; split=> n x ?; split.
  - destruct n as [|n]; simpl.
    { by exists x, (core x); rewrite cmra_core_r. }
    intros (x1&x2&Hx&?&?); destruct (cmra_extend n x x1 x2)
      as (y1&y2&Hx'&Hy1&Hy2); eauto using cmra_validN_S; simpl in *.
    exists y1, y2; split; [by rewrite Hx'|by rewrite Hy1 Hy2].
  - destruct n as [|n]; simpl; [done|intros (x1&x2&Hx&?&?)].
    exists x1, x2; eauto using dist_S.
Qed.
Lemma later_false_excluded_middle P :  P   False  ( False  P).
Proof.
  unseal; split=> -[|n] x ? /= HP; [by left|right].
  intros [|n'] x' ????; [|done].
  eauto using uPred_closed, uPred_mono, cmra_included_includedN.
Qed.
529
Lemma persistently_later P :   P    P.
530
Proof. by unseal. Qed.
531 532
Lemma plainly_later P :   P    P.
Proof. by unseal. Qed.
533 534 535

(* Own *)
Lemma ownM_op (a1 a2 : M) :
536
  uPred_ownM (a1  a2)  uPred_ownM a1  uPred_ownM a2.
537 538 539 540 541 542 543 544
Proof.
  unseal; split=> n x ?; split.
  - intros [z ?]; exists a1, (a2  z); split; [by rewrite (assoc op)|].
    split. by exists (core a1); rewrite cmra_core_r. by exists z.
  - intros (y1&y2&Hx&[z1 Hy1]&[z2 Hy2]); exists (z1  z2).
    by rewrite (assoc op _ z1) -(comm op z1) (assoc op z1)
      -(assoc op _ a2) (comm op z1) -Hy1 -Hy2.
Qed.
545
Lemma persistently_ownM_core (a : M) : uPred_ownM a   uPred_ownM (core a).
546 547 548
Proof.
  split=> n x /=; unseal; intros Hx. simpl. by apply cmra_core_monoN.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
549
Lemma ownM_unit : uPred_valid (M:=M) (uPred_ownM ε).
550 551 552 553 554 555 556 557 558 559 560 561
Proof. unseal; split=> n x ??; by  exists x; rewrite left_id. Qed.
Lemma later_ownM a :  uPred_ownM a   b, uPred_ownM b   (a  b).
Proof.
  unseal; split=> -[|n] x /= ? Hax; first by eauto using ucmra_unit_leastN.
  destruct Hax as [y ?].
  destruct (cmra_extend n x a y) as (a'&y'&Hx&?&?); auto using cmra_validN_S.
  exists a'. rewrite Hx. eauto using cmra_includedN_l.
Qed.

(* Valid *)
Lemma ownM_valid (a : M) : uPred_ownM a   a.
Proof.
562
  unseal; split=> n x Hv [a' ?]; ofe_subst; eauto using cmra_validN_op_l.
563
Qed.
564
Lemma cmra_valid_intro {A : cmraT} (a : A) :  a  uPred_valid (M:=M) ( a).
565 566 567
Proof. unseal=> ?; split=> n x ? _ /=; by apply cmra_valid_validN. Qed.
Lemma cmra_valid_elim {A : cmraT} (a : A) : ¬ {0} a   a  False.
Proof. unseal=> Ha; split=> n x ??; apply Ha, cmra_validN_le with n; auto. Qed.
568
Lemma plainly_cmra_valid_1 {A : cmraT} (a : A) :  a    a.
569 570 571 572 573
Proof. by unseal. Qed.
Lemma cmra_valid_weaken {A : cmraT} (a b : A) :  (a  b)   a.
Proof. unseal; split=> n x _; apply cmra_validN_op_l. Qed.

(* Basic update modality *)
574
Lemma bupd_intro P : P == P.
575 576 577 578
Proof.
  unseal. split=> n x ? HP k yf ?; exists x; split; first done.
  apply uPred_closed with n; eauto using cmra_validN_op_l.
Qed.
579
Lemma bupd_mono P Q : (P  Q)  (|==> P) == Q.
580 581 582 583 584
Proof.
  unseal. intros HPQ; split=> n x ? HP k yf ??.
  destruct (HP k yf) as (x'&?&?); eauto.
  exists x'; split; eauto using uPred_in_entails, cmra_validN_op_l.
Qed.
585
Lemma bupd_trans P : (|==> |==> P) == P.
586
Proof. unseal; split; naive_solver. Qed.
587
Lemma bupd_frame_r P R : (|==> P)  R == P  R.
588 589 590 591 592 593 594 595 596
Proof.
  unseal; split; intros n x ? (x1&x2&Hx&HP&?) k yf ??.
  destruct (HP k (x2  yf)) as (x'&?&?); eauto.
  { by rewrite assoc -(dist_le _ _ _ _ Hx); last lia. }
  exists (x'  x2); split; first by rewrite -assoc.
  exists x', x2; split_and?; auto.
  apply uPred_closed with n; eauto 3 using cmra_validN_op_l, cmra_validN_op_r.
Qed.
Lemma bupd_ownM_updateP x (Φ : M  Prop) :
Ralf Jung's avatar
Ralf Jung committed
597
  x ~~>: Φ  uPred_ownM x ==  y, ⌜Φ y  uPred_ownM y.
598 599 600 601 602 603 604
Proof.
  unseal=> Hup; split=> n x2 ? [x3 Hx] k yf ??.
  destruct (Hup k (Some (x3  yf))) as (y&?&?); simpl in *.
  { rewrite /= assoc -(dist_le _ _ _ _ Hx); auto. }
  exists (y  x3); split; first by rewrite -assoc.
  exists y; eauto using cmra_includedN_l.
Qed.
605 606 607 608 609 610
Lemma bupd_plainly P : (|==>  P)  P.
Proof.
  unseal; split => n x Hnx /= Hng.
  destruct (Hng n ε) as [? [_ Hng']]; try rewrite right_id; auto.
  eapply uPred_mono; eauto using ucmra_unit_leastN.
Qed.
611 612

(* Products *)
613
Lemma prod_equivI {A B : ofeT} (x y : A * B) : x  y  x.1  y.1  x.2  y.2.
614 615 616 617
Proof. by unseal. Qed.
Lemma prod_validI {A B : cmraT} (x : A * B) :  x   x.1   x.2.
Proof. by unseal. Qed.

618
(* Type-level Later *)
619
Lemma later_equivI {A : ofeT} (x y : A) : Next x  Next y   (x  y).
620 621 622
Proof. by unseal. Qed.

(* Discrete *)
623
Lemma discrete_valid {A : cmraT} `{!CmraDiscrete A} (a : A) :  a  ⌜✓ a.
624
Proof. unseal; split=> n x _. by rewrite /= -cmra_discrete_valid_iff. Qed.
625
Lemma discrete_eq {A : ofeT} (a b : A) : Discrete a  a  b  a  b.
626
Proof.
627
  unseal=> ?. apply (anti_symm ()); split=> n x ?; by apply (discrete_iff n).
628 629 630
Qed.

(* Option *)
631
Lemma option_equivI {A : ofeT} (mx my : option A) :
632 633 634 635 636 637 638 639 640 641
  mx  my  match mx, my with
             | Some x, Some y => x  y | None, None => True | _, _ => False
             end.
Proof.
  unseal. do 2 split. by destruct 1. by destruct mx, my; try constructor.
Qed.
Lemma option_validI {A : cmraT} (mx : option A) :
   mx  match mx with Some x =>  x | None => True end.
Proof. unseal. by destruct mx. Qed.

642 643
(* Contractive functions *)
Lemma contractiveI {A B : ofeT} (f : A  B) :
644
  Contractive f  ( a b,  (a  b)  f a  f b).
645 646
Proof.
  split; unseal; intros Hf.
647 648
  - intros a b; split=> n x _; apply Hf.
  - intros i a b; eapply Hf, ucmra_unit_validN.
649 650
Qed.

651
(* Functions *)
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
652
Lemma ofe_funC_equivI {A B} (f g : A -c> B) : f  g   x, f x  g x.
653
Proof. by unseal. Qed.
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
654
Lemma ofe_morC_equivI {A B : ofeT} (f g : A -n> B) : f  g   x, f x  g x.
655
Proof. by unseal. Qed.
Jacques-Henri Jourdan's avatar
Jacques-Henri Jourdan committed
656 657 658 659 660

(* Sig ofes *)
Lemma sig_equivI {A : ofeT} (P : A  Prop) (x y : sigC P) :
  x  y  proj1_sig x  proj1_sig y.
Proof. by unseal. Qed.
661
End primitive.
662
End uPred.